A Bootstrap test for the dynamic performance of ...

A Bootstrap test for the dynamic performance of Macroeconomic Models — an outline and some experiments¤ Patrick Minforda;b Konstantinos Theodoridisa David Meenagha a Cardi¤ Business School, Cardi¤ University b CEPR

1

Introductory remarks

Let us suppose a model has been estimated appropriately. Say too that its parameter values have not been rejected by the data, in the usual sense that they exceed twice their standard errors. Such a test of a model is rather limited; many rival models can be …tted to data acceptably in this way. Furthermore tests of models against each other through non-nested or encompassing tests are often found to be inconclusive. The reason is easy to see: di¤erent theories can be pushed to explain the same facts by structuring the unexplained errors di¤erently. For this reason economists have tried to test models in other ways. They have in particular considered whether the models could: a) predict within and outside sample, based on lagged data. b) generate e¤ects of shocks (impulse response functions) like those seen in the data c) generate simulated moments like those in the data d) generate VAR and other representations like those estimated on the data. Later in this paper we brie‡y survey the methods that have been used in these four categories. But we may note at once some problems that have been encountered with them.. a) the di¢culty of forecasting tests (which measure success or low standard error in forecasting) is that errors in forecasting are inherent in any model because shocks cannot be forecast. Thus for example in a world of large shocks model errors should be large; a model with a low forecasting error would be misspeci…ed. b) identifying the impulse response functions in the data is problematic because such identifying restrictions can only come from structural models. Thus the models being tested must be used to specify the identifying restrictions, which alters the test since there is now no unambiguous data-given function with which to compare each model. We will argue later for a way of testing the impulse response functions implied by structural models but it di¤ers from such a comparison. c) data moments are apparently theory-free; that is one may calculate them without assuming any structural theory. The main problem faced is the choice of …lter; but this in practice is of great importance. Not …ltering implies that trended data is dominated by its trend; but …ltering by di¤erencing or common …lters like HP or Band-Pass may produce quite di¤erent values. d) VAR and other time-series representations face a similar problem. There are many ways in which one can choose to represent a set of time-series; and for them too a decision must be made on …ltering. We suggest that a way forward is, in line with Popper’s original ideas, to set up as the ’null’ (strictly, the ’working’) hypothesis the structural model (SM) to be tested and derive from this null the relevant restrictions on …lters, on choice of time-series representation, on impulse response functions etc. Thus we should derive from the SM restrictions on the form of the model’s moments and of its time-series as well as on the appropriate …lter. ¤We are grateful for useful comments to James Davidson, Bernard Pearson and Simon Wren-Lewis; also to Mike Wickens for discussions over a long period..

1

A further suggestion is that we should use all available data in our tests. Thus the SM implies what the structural errors are and their values are in general supplied by the actual data; thus testing against the data should allow for the data’s restrictions under the SM on the errors being used in generating the SM’s simulated performance. A further, related, suggestion is that we respect the small sample nature of our tests by using bootstrapping rather than asymptotic distributions which will in general mislead in small samples. Bootstrapping under small sample conditions can be carried out by using the actual data on the errors just mentioned. In this paper we pursue an extended example of such a procedure using the Liverpool Model of the UK and UK data since the 1970s. First, however, we consider the methods that are currently or have recently been in use They divide, following Canova, into four groups: 1) Watson’s R2 measure 2) The Generalised Method of Moments (GMM) and related approaches using the distribution of the actual data 3) The Simulated Method of Moments (SMM) and related approaches using the distribution on the simulated data. 4) Bayesian methods. 1) is not really a test statistic at all. Watson proposes a measure of the ’…t’ of a model against the data in terms of the error process that needs to be added to the model in order for the model’s autocovariance function to match exactly that of the data. The size of this vector error process is used to measure the extent to which the model fails to capture the data. The problem with this measure is that it has no distribution and thus cannot generate a test statistic. 2) Under the GMM method a model is linearised and …tted to the data by the Generalised Method of Moments. The error distributions on the model’s parameters create an implied distribution of the di¤erence of the model’s moments from the data moments. This distribution is  2 : The method relies throughout on asymptotic distributions. It also requires that the model is linear and that the data are made stationary by some …lter. Recently Christiano has pointed out that in small samples the asymptotic distributions are badly misleading. 3) Under SMM the model is again linearised and the data made stationary by some …lter. Those parameters that are regarded as ’free’ , ie available to be varied according to the data, are then moved until the vector of VAR coe¢cients implied by the model (which can be analytically calculated) is as close as possible to that from the data. The di¤erence between these two vectors generates a Â2 statistic whose distribution is based on the simulated behaviour of the model. This statistic is again asymptotic. There are a number of variants of this approach based on other features produced by the model and the data- eg spectral measures and moments. 4) Bayesian methods strictly do not test a model but rather regard the initial model as some sort of approximation to the true model; thus the question is how far this approximation can be improved by using the process of estimation and comparison of the model with the data to modify the prior model. At no point does the researcher consider rejection of the model. Rather one model variant may be compared with another (as ’nested’ hypotheses) and one chosen as giving a greater improvement than the other. Consequently the approach must be considered as quite di¤erent from that of testing being developed here, where the idea is to test quite di¤erent theories- which cannot be nested- against the data. Plainly if economics reached the stage where no such testing was necessary and the only issue remaining was to improve an agreed general model, then testing in our sense would no longer be relevant and instead the Bayesian agenda could naturally replace it. Nevertheless even then a new uncommitted researcher might wish to test the consensus and for that a test would be required in our sense. A fuller account of all these methods is in Appendix A. But in sum, we suggest that the relevant class of existing tests are those in 2) and 3). These tests have points in common with what we propose. In e¤ect what we are suggesting incrementally is the use of bootstrapping to overcome small sample problems; the use of the original nonlinear model rather than linearisation in simulation; the use of implied errors from the data and model structure, rather than assumed error processes; the use of both the data-generated and the model-generated in an overall portmanteau test of the model. With these tools it should be possible to make clear probabilistic statements about models. Our paper is organised as follows. In the …rst section we elaborate on the nature of the test we propose . In the second section we apply our test to a particular model, the Liverpool Model of the UK, as an initial experiment; the reason for choosing this model is simply that we have extensive experience with using it and have FIML estimates of the parameters and error processes.In the last section we draw

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some tentative conclusions from these experiments and discuss further work, some of which is already in hand.

2

Our proposed test

The basic test: comparing the reduced form parameters with the distribution based on the structural model In this paper we propose a method for testing macro models by using the bootstrap; we think of the method as an extension of that used by Real Business Cycle (RBC) modellers. Consider a) a data set: say the UK economy from 1972 to 2004; b) a structural theory of its behaviour stating that it behaves according to certain parameters whose values have been estimated by some means. Notice that in the course of estimation various tests of the model will probably have been performed- notably whether the parameters are statistically signi…cant, that is are su¢ciently large to reject the null hypothesis that they are zero. But let us assume that the model has either passed such tests or is in any case asserted to be a reasonable null hypothesis. Thus we consider models that have already been estimated or calibrated, by whatever means; methods for estimating or calibrating models are not discussed in this paper. Our starting point is the assertions of some modellers to the e¤ect that their model is to be regarded as ‘the truth’, that is as a null hypothesis. We shall use ‘truth’ and ‘true’ to mean that the modellers assert that their model is to be taken as an assertion of how the economy in question works. Such an assertion is made in order for testing to be done of a concrete hypothesis. Thus the ‘truth’ is the same as the economist’s null (strictly, working) hypothesis. Now consider the idea for testing models already in use by RBC modellers. This idea has been to reorganise the facts in the form of correlations and cross-correlations at di¤erent leads and lags. In this form the facts are metamorphosed into ‘stylised facts’ which describe the data in a relevant way but one that is entirely theory-free. We can call this relevant way the ‘time-series behaviour’ of a set of variables, meaning by this the description of the variables in terms of their ‘trends’ and their ‘cycles’, common and individual, that is their correlations (and intercorrelations) over time. Then the RBC proposal has been that the testing of models may be done by comparing the correlations they imply in simulation with the actual ones in the data. This sort of test is quite di¤erent from the testing of parameters (typically against zero) that may be done in estimating a model. The model could pass such a test for every parameter and yet not be a model which implies good dynamic behaviour like that of the economy. Vice versa, a model could fail every such test and yet produce behaviour that resembles the dynamic behaviour of the economy. The method we propose here takes this RBC idea but suggests two changes. First, instead of using the theoretical error distributions postulated by the model to compute the model’s correlations with the facts, we suggest using the actual error distributions implied by the data to compute them, using the bootstrap. Second, instead of using correlations we suggest using a more general time-series representation of the data such as a VAR 1 Let us consider these two proposals in turn. (1) If a model is asserted to be true, then it implies a set of (structural) errors between the theory and the data. Since the model is true, the implied errors are also true. Hence under the null hypothesis the errors driving the model are these implied ones. To test the null we use these errors to see what behaviour in conjunction with the model they imply. Under the bootstrap we can preserve these sample error distributions as the basis for our stochastic draws. It is possible that the modellers only regard certain shocks as relevant in their model. Thus for example in a model with 6 endogenous variables and 6 equations determining them, there will be 6 errors in principle. However, it would be possible to treat some of these as non-stochastic; in e¤ect the modellers are asserting that these errors were the result of some process that would have happened in all states of the world in that sample period. Some RBC modellers for example assert that productivity 1 We also considered using a VECM and carried out a number of experiments with it, reported below. In the end we abandoned this for caes 3 where we restricted the reduced from more tightly in line with the null hypothesis. The reason was that our null hypothesis implies that there should be no cointegrating vectors within our group of 5 endogenous variables: cointegration within the LVP model requires the presence of other exogenous variables not included in the reduced form. Hence in a restricted reduce d form the ’cointegrating vectors’ we ’found’ were spurious statistical artefac ts and were invalid. We could nevertheless interpret them in unrestricted reduced froms, as in cases one and two, as approximate error-correction terms that were usefully correlated with the de pendent vector.

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shocks are the sole driving force of the economy- that tastes are by implication non-stochastic. Under such an assertion only productivity shocks would be treated as stochastic and drawn repeatedly in stochastic simulation. It should also be noted that if a model contains exogenous variables then the modeller must decide whether these too are stochastic or non-stochastic. If stochastic then they too must be modelled as some sort of stochastic process and their errors included in the stochastic simulations. To summarise, a model is an assertion of a set of relationships and of which processes are to be regarded as stochastic, which non-stochastic. Given the model, the data over a sample period then de…ne the actual (true) errors for the stochastic parts as well as the non-stochastic parts. (2) we would like to …nd a general description of the data that emphasises its trend and cyclical features in a reasonably summary way, that is with a set of parameters that can be compared with the implications of the theory we wish to test. However, the potential size of the matrix of correlations and cross-correlations at di¤erent lags becomes very large very quickly; this is not therefore a compact description of the data. However a VAR is a potentially complete but parsimonious representation of the data in time-series form. We may add to this a GARCH process in the error. The VAR/GARCH representation is not the only possibility; we discuss others below. But it is widely used and convenient; our main point is that a more general representation of the facts than the correlation matrices can provide us with a more comprehensive test. The more we can restrict this reduced form representation of the data in conformity with the implications of the structural model, the clearer a test we can perform of the latter as the null hypothesis. Thus in summary the null de…nes the model, the structural ’error’ data and restricts the reduced form in possible ways. We can then estimate the reduced form on the data and compare it with the bootstrapped predictions of the structural model. Our suggested test is thus designed to reject the null hypothesis of the model at some speci…ed con…dence level (say 95%). Under the null hypothesis the model stochastic error processes are de…ned within the sample period. We may then ask whether this model and these processes could have generated the facts as captured by the VAR estimated over the same sample period. Thus under the null hypothesis, the sampling variation would be given by bootstraps of the stochastic errors with the model; these bootstraps would provide us with a large number of pseudosamples on which we can estimate the same VAR to establish the sampling distribution for the VAR parameters. We believe that this test provides a statistical method for rejecting the null of a particular model, in the sense that it answers the question ’could the facts of the economy have been produced by this model?’ An elaborated test: of the equality of two distributions of the reduced form parameters- an initial trial We consider in an initial way a further elaboration of this test in which we make use not merely of the VAR parameter distributions implied by the Structural Model but also that given by the VAR reduced form estimation itself. Under the null hypothesis these two should be the same which provides us with a chi-squared test rather similar in principle to those suggested under the GMM and SMM methods. Perhaps the key di¤erence is that we are using distributions based on the bootstrapped small sample errors for both the SM- and the RF-generated parameters. Testing the model’s impulse responses against those in the reduced form We also test the impulse responses implied by the SM against those in the data. We may note that Christiano, Eichenbaum and Evans:(CEE) have been for some time operating a methodology which is designed to do just this. However we believe it is ‡awed by problems of identi…cation.. CEE set up a calibrated or estimated theoretical structure and generate its impulse response function to particular shocks. This is then compared with the impulse response function obtained from a VAR estimated on the actual data but one that is restricted (or ’identi…ed’) in a certain manner that is only weakly related to the theoretical structure. Thus the idea is to identify the VAR by imposing on it restrictions regarded as universally accepted by economists, non-controversial as it were, so that this VAR could be regarded as an acceptable representation of the facts. The theory can then be tested in respect of its ability to …t selected aspects of this representation that are of interest. For example CEE look at impulse response functions to a monetary policy shock. The theory can then be regarded if it matches this function as being a satisfactory representation for the purpose of monetary policy analysis. However, the identi…cation scheme being used for the VAR.can hardly fail to be controversial and more to the point will in general be inconsistent with the structural model under test. For example CEE identify the VAR/VECM so that a money shock has ’hump-shaped’ response functions on both output and prices; this is a prior assumption emerging from some loose theory they happen to hold. Yet if the particular SM under test is taken as the null it may well not imply such a pattern of response. The SM

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might well be ‘rejected’ (‘accepted’) by such a representation when it would not be if the theory’s own restrictions are applied in identification. An alternative approach, which we use below, is to compare the impulse responses of the model with those of the unrestricted VAR when the model’s restrictions are used to identify the relevant shocks to the VAR. This is done by establishing a mapping from the model shock involved to the model-implied shocks of dyt; these shocks are then input into the VAR and the resulting movement in dyt plotted; the 95% confidence bounds for this are found from the VAR bootstraps.

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Our experiments on the Liverpool model of the UK:

3.1

Our procedure in summary:

1. We begin by subjecting the model to the set of diagnostics used in the RBC literature — in particular the second moments and cross-moments of the main macro series; the univariate properties of the main series; and the cross-moments of the main series with GDP (the overall measure of the ‘cycle’) at various leads and lags. We estimate the ‘facts’ in each case, using a variety of filters and then compare these facts with the LVP-model-generated 95% confidence limits for the same diagnostics. 2. We estimate the VAR with GARCH; again first on the actual data and then on the bootstrapped data, from which we again derive the 95% confidence limits. Our aim from this wide range of diagnostics is to establish whether the model is rejected by the data. We look at three main cases: a) the LVP model, under a monetary targeting regime under floating exchange rates, over the full period from early 1970s to the current time. As we know the monetary regime during this whole period has varied substantially, this is a ‘brave case’; our expectation would be rejection, simply because even if the model was correct, the regime changes would imply substantial specification failures for much of the period. During the 1970s there were no money supply targets and for much of the period incomes policies were used; during the early 1980s there were money supply targets but the money measures targeted switched periodically; from the late 1980s there was ‘DM shadowing’, a sort of fixed exchange rate regime, followed by the ERM proper from 1990-92. Finally there was a switch to inflation targeting from end-1992, which continues to this day. b) the LVP model over the period of 1986 to the current time, under the same regime as above. The same problem of regime variation holds for this period too, as already noted. However, it may be less serious, if an inflation targeting regime is not too different from a money targeting one, which would be the case if velocity was reasonably stable. c) the LVP model over the period of 1979 to the current time, but with the regimes being altered according to our null hypothesis (monetary targeting up to 1986, then exchange rate targeting and finally from 1992 inflation targeting). For this case we impose on the VAR for the full period restrictions obtained from the null hypothesis. Then we check whether the SM can ‘encompass’ the RF. From these three cases we hope to develop some experience of applying these methods of model evaluation.

3.2

CASE A: The LVP model over the whole period

The first thing to be noted about this is that we were only able to get 284 out of 500 bootstrap simulations to solve, using the powerful Powell Hybrid algorithm. What this indicates is that the errors were in some cases very large and when re-sampled these caused the model to crash; when errors were capped at 0.03 (3%) the bootstraps virtually all solved. If we had a large amount of time we could work on the bootstrap simulations that crashed and very likely obtain a solution for each one. However, we know in any case that as a result of this solution failure we must consider the bootstrap simulation variances as a (possibly serious) under-prediction since were we to find the solutions they would be more variable than the ones we have. Anticipating our results below we find 5

in fact that the model’s 95% confidence intervals for the variances for virtually all variables lie above the data variances. Thus the model is rejected decisively on the variance test alone. The VAR/VECM and GARCH parameters are within the 95% confidence limits, as are the serial and cross-correlations and the impulse response functions. Nevertheless we cannot be sure that this result does not simply emerge from the size of the variances within the bootstraps and their ‘blowing-up’ effect on the 95% confidence limits. Plainly the more variable the simulations the larger the 95% bands will be, since these are estimated from the simulation data. Comparing variances and correlations for filtered series As a preliminary diagnostic we follow the RBC method of comparing the variances and correlations generated between data and the model (here its 95% confidence intervals). The actual and simulated data are filtered by the same filter in each case. Variances: UNFILTERED DATA Output Unemployment Inflation Interest Rates Exchange Rates

LOWER BOUND 0.0145 0.2158 0.0005 0.0014 0.0119

UPPER BOUND 0.0829 4.7162 0.0014 0.0050 0.2900

ACTUAL 0.0418 0.2750 0.0026 0.0012 0.0338

STATE IN IN OUT OUT IN

Output Unemployment Inflation Interest Rates Exchange Rates

LOWER BOUND 0.0010 0.0086 0.0003 0.0005 0.0024

UPPER BOUND 0.0031 0.0221 0.0006 0.0010 0.0049

ACTUAL 0.0001 0.0043 0.0002 0.0001 0.0011

STATE OUT OUT OUT OUT OUT

Output Unemployment Inflation Interest Rates Exchange Rates

LOWER BOUND 0.0006 0.0175 0.0002 0.0006 0.0020

UPPER BOUND 0.0022 0.0753 0.0008 0.0024 0.0077

ACTUAL 0.0002 0.0160 0.0005 0.0003 0.0012

STATE OUT OUT IN OUT OUT

Output Unemployment Inflation Interest Rates Exchange Rates

LOWER BOUND 0.0127 0.1324 0.0001 0.0002 0.0056

UPPER BOUND 0.0815 4.6569 0.0005 0.0023 0.2766

ACTUAL 0.0413 0.2428 0.0019 0.0007 0.0296

STATE IN IN OUT IN IN

DIFFERENCE FILTER

BANDPASS FILTER

H-P FILTER

It can be seen in these Tables that only on a few measures — unfiltered and H-P filtered- can the model encompass the actual variances. Allowing for the bias due to omitted bootstraps this rejection is very clear.

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Table 1: Correlations (Christiano-Fitzgerald band-pass filter)

Output Unemployme nt Inflation Interest Rate Exchange Rate

t

Correlation with Own lags -1 -2 -3 -4

1

0.89

0.63

0.33

0.09

1

0.94

0.98

0.53

0.25

0.37

0.13

-0.14

-0.4

-0.63

1

0.91 *

0.64 *

0.34 *

0.05

-0.51 *

-0.55 *

-0.55

-0.51

1

0.90

0.64

0.33

0.64

-0.49

-0.37

-0.22

1 0.55 0.46 0.04 -0.25 -0.31 * Denotes outside 95% confidence band

-0.33

-0.33

+4

Correlation with Output and its lags +3 +2 +1 t -1

-2

-3

-4

-0.78

-0.82

-0.73

-0.65

-0.41 *

-0.24 *

-0.03 *

0.17

0.34

-0.04

0.16

0.33

0.44

0.45 *

0.41

-0.32

-0.29

-0.22

-0.10

0.05

0.22

When one turns to serial and cross-correlations, the model is not rejected, again fairly clearly. It can be seen that the actual values are all inside the 95% confidence limits except most of those involving inflation. The model understates the persistence of inflation. Thus the t-1 lag gives 95% limits of 0.8-0.9 against the data at 0.91; at t-2 it gives 0.31-0.64 against 0.66; and at t-3 it gives – 0.19-0.33 against 0.34. All are just outside the limits. The model also overstates the negative correlation with the business cycle; at t it gives a correlation with output of –0.86 to –0.53 against –0.41; with output at one lag it gives –0.77 to –0.39 against –0.29 and at two lags –0.54 to –0.06 against –0.03. At leads of 3 and 4 it understates the negative correlation; at 4 it gives 0.32 to –0.39 against –0.51 and at 3 it gives 0.67 to –0.52 against – 0.55. In fact it is in the behaviour of inflation that we find regime change explains most. Under regimes of incomes policy, money supply targets and fixed exchange rates persistence is much greater than under inflation targeting. The model as well as the data here are not distinguishing the regimes. Nevertheless the 95% bands are as high as they are because of the excessive variances implied by the model, hence the encompassing noted here may simply be due to the excess variances.

Details of cross-correlations (Christiano-Fitzgerald band-pass filter): Table 2 Correlation of the Output with its (4) lags and leads

yt−1

yt− 2

yt −3

yt− 4

yt Actual Upper Bound Lower Bound State

0.894

0.630

0.330

0.095

0.798

0.320

-0.197

-0.515

0.910 IN

0.678 IN

0.387 IN

0.111 IN

7

Table 3 Correlation of Unemployment with Output and its (4) lags

yt

yt−1

yt− 2

yt −3

yt− 4

ut Actual Lower Bound Upper Bound State

-0.626

-0.775

-0.825

-0.777

-0.655

-0.784

-0.909

-0.910

-0.817

-0.664

-0.477 IN

-0.680 IN

-0.651 IN

-0.424 IN

-0.131 IN

Table 4 Correlation of Inflation with Output and its (4) lags

yt

yt−1

yt− 2

yt −3

yt− 4

pt Actual Lower Bound Upper Bound State

-0.409

-0.242

-0.033

0.174

0.337

-0.864

-0.770

-0.544

-0.280

-0.035

-0.528 OUT

-0.389 OUT

-0.063 OUT

0.286 IN

0.520 IN

Table 5 Correlation of Interest Rates with Output and its (4) lags

yt

yt−1

yt− 2

yt −3

yt− 4

rt Actual Lower Bound Upper Bound State

0.157

0.333

0.435

0.449

0.408

-0.489

-0.420

-0.317

-0.253

-0.230

0.254 IN

0.391 IN

0.419 OUT

0.462 IN

0.482 IN

Table 6 Correlation of Exchange Rates with Output and its (4) lags

yt

yt−1

yt− 2

yt −3

yt− 4

St Actual Lower Bound Upper Bound State

-0.286

-0.219

-0.102

0.054

0.210

-0.315

-0.230

-0.142

-0.107

-0.134

0.411 IN

0.534 IN

0.573 IN

0.566 IN

0.558 IN

Results of the VAR-GARCH analysis Estimating the VAR -GARCH on the data: We begin by describing the data through the prism of the VAR -GARCH. We estimated this over the full period, aiming for maximum parsimony by keeping if possible to one lag vector, eliminating insignificant terms, and in the GARCH using a constant cross-correlation matrix. The results are shown in the Table below (results for the VECM-GARCH are available on request):

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VAR ACTUAL ESTIMATES

* * −0.089 *   yt −1  ety   yt   *   0.981    u   −2.178  0.124 1.020 0.819 * *   ut−1   etu   t    π t  =  0.708  +  −0.036 −0.019 0.806 0.184 −0.034  π t−1  + eπt           * * 0.934 *   rt −1   etr   rt   *   *  St   *   * −0.021 * * 0.908   St −1  etS  hty = a y + 0.849 y ( ety−1 ) + 0.150 hty−1 2

htu = au + 0.657 u ( etu−1 ) + 0.331htu−1 2

htπ = aπ + 0.781π ( etπ−1 ) + 0.203 htπ−1 2

htr = a r + 0.885r ( etr−1 ) + 0.114htr−1 2

htS = a S + 0.140S ( etS−1 ) + 0.074 htS−1 2

Table 9 “SPECIFICATION TESTS ON STANDARIZED RESIDUALS” TEST Q-statistic (16 lags) Adjusted Qstatistic (16 lags) MARCH-LM test

CHI-Square 464.821

DF 395

PVALUES 0.001

495.789

395

0.0002

956.0478

900

0.095

9

Table 10 CONFIDENCE INTERVAL OF THE SIMULATED ESTIMATES1 Coefficients

Lower Bound

Upper Bound

Actual Estimates

State

cu cπ

-6.356

12.471

-1.383

IN

-0.581

1.586

0.691

IN

y y

0.978

1.004

1.001

IN

r y

-0.939

0.557

0.097

IN

y u

-0.119

0.043

-0.035

IN

u u

A

0.838

1.073

1.026

IN

Auπ

-0.019

0.007

-0.016

IN

y π

-0.018

0.066

-0.001

IN

u π

-2.018

4.718

1.117

IN

π π

A

0.477

0.845

0.855

OUT

Aπr

-0.219

3.454

-0.082

IN

S π

-0.040

0.148

0.132

IN

r r

0.897

0.993

0.991

IN

u S

A

-0.023

0.044

-0.036

OUT

ASS

0.909

1.031

1.002

IN

φy

0.001

0.890

0.808

IN

0.001

0.611

0.191

IN

0.001

0.781

0.710

IN

0.001

0.583

0.269

IN

0.001

0.881

0.756

IN

0.001

0.299

0.238

IN

0.001

0.883

0.886

OUT

0.001

0.292

0.113

IN

0.001

0.857

0.288

IN

0.001

0.309

0.023

IN

A

A A

A A

A

A

θy

φu θu φπ θπ φr θr φS θS

1

a=coefficient of loading matrix.

β =coefficient of cointegrated vectors. A=coefficients of the autoregressive

part of the VAR. The subscript denoted the equation, the second subscript denotes the lag, the superscript denotes the variable. ϕ =the ARCH coefficient. θ =the GARCH coefficient.

10

It would seem that the model fits the data rather well apart from generating excessive overall variance. The VAR representation is within the 95% bounds. However, the problem is that because the variances are excessive these bounds may well be exaggerated. So the overall judgement must be one of rejection of the model over this long sample period from the early 1970s to the early 200s. This was to be expected given numerous monetary regime changes over the period.

3.3 2000Q4)

CASE B: The LVP model over the short sample period (1986Q3

Using this later sub-sample 490 out of 500 bootstraps simulations solved. Thus here the size of the errors was reasonable, allowing the model to solve routinely throughout. Variances: Unfiltered Data IN IN IN IN IN

Output Unemployment Inflation Interest Rates Exchange Rates

LOWER BOUND 0.0061 0.0592 0.0002 0.0004 0.0028

UPPER BOUND 0.0206 1.1218 0.0007 0.0024 0.0605

ACTU AL 0.0096 0.1591 0.0003 0.0011 0.0065

Output Unemployment Inflation Interest Rates Exchange Rates

LOWER BOUND 0.0002 0.0016 0.0001 0.0002 0.0008

UPPER BOUND 0.0012 0.0079 0.0003 0.0004 0.0021

ACTU AL 0.0000 0.0017 0.0000 0.0000 0.0008

Output Unemployment Inflation Interest Rates Exchange Rates

LOWER BOUND 0.0001 0.0036 0.0001 0.0002 0.0005

UPPER BOUND 0.0009 0.0379 0.0004 0.0011 0.0034

ACTU AL 0.0001 0.0047 0.0001 0.0001 0.0011

Output Unemployment Inflation Interest Rates Exchange Rates

LOWER BOUND 0.0056 0.0325 0.0000 0.0000 0.0008

UPPER BOUND 0.0199 1.1027 0.0004 0.0011 0.0570

ACTU AL 0.0093 0.1373 0.0002 0.0009 0.0032

Difference Filter

OUT IN OUT OUT IN Bandpass Filter

IN IN IN OUT IN Hodrick-Prescott Filter IN IN IN IN IN

Here it seems that the variances are largely within 95% confidence limits. A few of the correlations are outside the bands; notably those of output with its own lags. Most are within.

11

Table 11: Correlations (Christiano-Fitzgerald band-pass filter) t 1

Correlation with Own lags -1 -2 -3 -4 0.94* 0.78* 0.56* 0.30*

+4

Unemploym ent Inflation

1

0.93

0.75

0.49

0.19

-0.84

-0.89

-0.84

-0.69

1

0.85

0.47

0.02

-0.29

0.30*

-0.10*

0.14*

Interest Rate Exchange Rate

1

0.90

0.64

0.30

0.00

0.21

0.40

1

0.88

0.58

0.20

-0.13

-0.21

-0.22

Output

Correlation with Output and its lags +3 +2 +1 t

-1

-2

-3

-4

-0.46

-0.20

0.04

0.23

0.36

0.37

0.58

0.71

0.75

0.71

0.59

0.52

0.58*

0.56

0.51

0.45

0.40

0.35

-0.22

-0.21

-0.17

-0.09

0.02

0.15

0.27

* Denotes outside 95% confidence band Table 12 Correlation of the Output with its (4) lags and leads

yt−1

yt− 2

yt −3

yt− 4

yt Actual Upper Bound Lower Bound State

0.939

0.779

0.555

0.299

0.770

0.222

-0.312

-0.593

0.929 OUT

0.739 OUT

0.481 OUT

0.222 OUT

Table 13 Correlation of Unemployment with Output and its (4) lags

yt

yt−1

yt− 2

yt −3

yt− 4

ut Actual Lower Bound Upper Bound State

0.932

0.751

0.489

0.188

0.932

0.855

0.487

0.050

-0.299

0.855

0.954 IN

0.825 IN

0.635 IN

0.422 IN

0.954 IN

Table 14 Correlation of Inflation with Output and its (4) lags

yt

yt−1

yt− 2

yt −3

yt− 4

pt Actual Lower Bound Upper Bound State

0.850

0.467

0.024

-0.289

0.850

0.764

0.197

-0.367

-0.636

0.764

0.917 IN

0.699 IN

0.415 IN

0.180 IN

0.917 IN

12

Table 15 Correlation of Interest Rates with Output and its (4) lags

yt

yt−1

yt− 2

yt −3

yt− 4

rt Actual Lower Bound Upper Bound State

0.898

0.636

0.301

0.003

0.898

0.819

0.371

-0.123

-0.482

0.819

0.939 IN

0.774 IN

0.546 IN

0.298 IN

0.939 IN

Table 16 Correlation of Exchange Rates with Output and its (4) lags

yt

yt−1

yt− 2

yt −3

yt− 4

St Actual Lower Bound Upper Bound State

0.884

0.578

0.201

-0.129

0.884

0.807

0.337

-0.172

-0.506

0.807

0.936 IN

0.767 IN

0.531 IN

0.274 IN

0.936 IN

VAR ACTUAL ESTIMATES

 yt   0.390   1.405 −0.036 0.057 −0.158  u   3.663   −1.210 1.851 −1.158 0.998  t    π t  =  1.446  +  −0.210 0.090 0.584 0.228       rt   1.595   −0.122 −0.097 0.114 1.065  St  10.048   −2.229 −0.766 0.817 −2.703  −0.438 0.033 −0.035 0.044  0.930 −0.906 0.389 −0.826  +  0.104 −0.109 −0.130 −0.197  0.102 −0.284  0.108 0.070  1.602 0.591 0.209 1.833

−0.003   yt −1  −0.018   ut−1    −0.007  π t−1    −0.016   rt −1  1.086   St −1 

0.009   yt − 2   ety    0.032   ut −2   etu    −0.001 π t− 2  +  etπ      −0.020   rt − 2   etr  −0.382   St − 2   etS 

Table 19 SPECIFICATION TESTS ON RESIDUALS TEST Q-statistic (16 lags) Adjusted Qstatistic (16 lags) MARCH-LM test

CHI-Square 215.578

DF 350

PVALUES 1.000

259.679 722.0471

350 675

1.000 0.1021

13

Table 20 CONFIDENCE INTERVAL OF THE SIMULATED ESTIMATES Coefficients

Lower Bound

Upper Bound

Actual Estimates

State

cy

-5.470

-2.534

0.390

OUT

-14.162

6.491

3.663

IN

-3.400

-1.369

1.446

OUT

-7.877

-2.600

1.595

OUT

-6.897

-0.718

10.048

OUT

cu cπ cr cS y y ,1 u y ,1

A

0.295

1.188

1.405

OUT

A

-1.387

-0.877

-1.210

IN

π y ,1

-0.316

0.035

-0.210

IN

A

-0.278

0.142

-0.122

IN

A

-1.111

0.212

-2.229

OUT

-0.116

0.254

-0.036

IN

A

r y ,1 S y ,1 y u ,1

A

u u ,1 π u ,1

A

0.603

1.022

1.851

OUT

A

-0.073

0.094

0.090

IN

r u ,1

-0.161

0.046

-0.097

IN

A

-0.603

0.048

-0.766

OUT

A

-0.623

-0.353

0.057

OUT

-1.685

0.812

-1.158

IN

A

S u ,1 y π ,1 u π ,1

A

π π ,1 r π ,1

A

0.103

0.881

0.584

IN

A

-0.237

0.402

0.114

IN

S π ,1

-1.057

-0.480

0.817

OUT

A

-0.776

0.331

-0.158

IN

A

-1.149

-0.698

0.998

OUT

-0.300

-0.133

0.228

OUT

A

y r ,1 u r ,1 π r ,1

A

r r ,1 S r ,1

A

0.148

0.848

1.065

OUT

A

-1.333

0.260

-2.703

OUT

y S ,1

-0.330

-0.252

-0.003

OUT

A

-0.147

0.555

-0.018

IN

A

-0.053

0.064

-0.007

IN

-0.003

0.120

-0.016

OUT

A

u S ,1 π S ,1 r S ,1

A

S S ,1 y y ,2

A

0.427

0.987

1.086

OUT

A

-0.288

0.033

-0.438

OUT

u y ,2

-1.041

0.301

0.930

OUT

π y ,2

-0.302

0.054

0.104

OUT

A A

14

Ayr,2

-0.380

0.032

0.018

IN

S y ,2

-0.823

-0.126

1.602

OUT

y u ,2

-0.325

-0.177

0.033

OUT

A

-0.814

-0.222

-0.906

OUT

A

-0.181

-0.056

-0.109

IN

-0.182

-0.005

0.070

OUT

A

-0.233

-0.004

0.591

OUT

A

-0.541

0.402

-0.035

IN

-1.772

-0.725

0.389

OUT

A

-0.524

-0.510

-0.130

OUT

A

-0.604

-0.210

0.102

OUT

-1.107

-0.137

0.209

OUT

A

-0.537

-0.537

0.044

OUT

A

-1.259

0.954

-0.826

IN

-0.307

0.136

-0.197

IN

A

-0.390

-0.278

-0.284

IN

A

-1.047

-0.488

1.833

OUT

-0.207

0.116

0.009

IN

A

-0.515

-0.177

0.032

OUT

A

A A

u u ,2 π u ,2 r u ,2

A

S u ,2 y π ,2 u π ,2

A

π π ,2 r π ,2 S π ,2

A

y r ,2 u r ,2 π r ,2

A

r r ,2 S r ,2 y S ,2

A

u S ,2 π S ,2

-0.141

-0.050

-0.001

OUT

r S ,2

-0.209

-0.063

-0.020

OUT

S S ,2

-0.296

-0.117

-0.382

OUT

A A

The VAR requires a second lag matrix to fit the data and remove autocorrelation. The two lag matrices also contain a lot of terms where the data reject the model. Thus although now the variances are compatible with the data, the VAR is not.

3.4 CASE C: The LVP Model with three separate regimes from 1979-2003: In this part we take seriously the null hypothesis separation of monetary regimes. We bootstrap each regime separately to obtain pseudo-samples from 1979-2003 that fully embody the different regimes. We then impose these different regimes as far as we can on the reduced form structure. Our null hypothesis states that under each regime the coefficients of the model and regime are constant, implying that for that period we should observe a homoscedastic VAR. Comparing one regime with another the VAR coefficients may change as may the variance/covariance matrix of the VAR errors. We restrict the VAR successively within a Markov-switching framework. We start by presenting some results within a reduced form where Markov switching is of variances only and the switch points are data-determined. What is encouraging in this experiment is that the three states picked out do correspond to a high degree with the a priori regime allocations. Thus state 1 occurs largely from 1979-84, suggesting it is the ‘money targeting’ regime. State 3 occurs mainly in the mid-1990s and then from the late 1990s to 2003, suggesting it is the ‘inflation targeting’ regime. State 2 according to our null hypothesis is the ‘exchange rate targeting’ regime which should

15

occur from 1986-1992. According to the data determination here it occurs in 1986, 1992 and again in 1997; but not in 1987-1991 which the data allocate to money targeting. This last regime also intrudes into the inflation targeting era, in 1998. Of course our regime allocation is far from cast-iron, especially in the period from 1986-1992 when considerable confusion reigned in policy (the prime minister and Chancellor publicly disagreed on whether the UK was targeting the exchange rate or the money supply). If we accept that there is some overlap but some disagreement on regime allocation- and hence the model is strictly rejected on this point-, then we can consider whether the model is otherwise rejected. It does in fact do materially better than the earlier versions, where we did not allow at all for regime change. Thus, with the marginal exception of the variance of inflation in state 2, it picks up the whole variance-covariance matrix in the three switching states. For the VAR parameters it rejects 5 out of 30 at the 95% level; one of these is borderline. Thus taking all the parameters of the Markovswitching model, it rejects 5 (strictly 6) out of 75. This is fairly close to acceptance at the 95% level. Strictly one must go to the 99% level to get non-rejection; at this point 1 parameter out of 75 is outside, which is about 1% of the parameter space. This is a fairly encouraging result, at least in terms of improvement compared with cases A and B. It suggests that allowing for regime change is extremely important if one wishes to explain the behaviour of the UK economy. Plainly the Markov reduced form indicated that there was some ambiguity in the allocation of regimes. Thus these results use a Markov allocation that is inconsistent with the null. We now go on to impose the allocation of regimes by the null onto the reduced form, transforming the latter from a Markov- to a deterministic-switching model, in line with the null. In order to keep as many degrees of freedom as possible, we treat the sample as a whole and in effect introduce dummies for each regime, testing in turn whether there is evidence in the data of shifts across regimes in constants, VAR parameters and the VAR error var-covar matrix. VAR with Markov Switching of Variances between 3 states PERIOD 1979-2003 ESTIMATION RESULTS OF THE ACTUAL DATA

 ∆yt   − 0.003  − 0.238 0.076  ∆u  − 0.0142  2.081 0.148  t    ∆π t  =  0.000  +  0.581 0.067       ∆rt  − 0.0015  0.750 0.0107  ∆et   0.0065  − 0.113 0.213

16

0.066 0.610 − 0.061  ∆yt−1  0.245 − 0.583 − 0.004   ∆u t−1  0.055 − 0.032 − 0.137  ∆π t−1    0.099 − 0.272 − 0.121  ∆rt−1  0.249 0.489 − 0.156   ∆et−1 

SMOOTHED PROBABILITIES STATE 1 1

0.5

0 1975

1980

1985 1990 1995 SMOOTHED PROBABILITIES STATE 2

2000

2005

1980

1985 1990 1995 SMOOTHED PROBABILITIES STATE 2

2000

2005

1980

1985

2000

2005

1

0.5

0 1975 1

0.5

0 1975

1990

1995

Table 12 Diagnostics Statistics

Chi-Square

DF

P-values

1076.20

900

0.000

279.38

150

0.000

234.98

75

0.000

Adj-Q- Statistics lags (4)

240.44

75

0.000

Q- Statistics lags (12)

349.71

275

0.002

Adj-Q- Statistics lags (12)

372.51

275

0.000

Conditional Heteroskedasticity Heteroskedasticity (4)

Q-Statistics

lags

17

ESTIMATION RESULTS OF THE SIMULATED DATA Table 13 Distribution of Parameters Parameters

c ∆y

c ∆u c∆π c ∆r c ∆e

Actual Estimates

Lower Bound

Upper Bound

State

-0.0003

-0.0321

0.0131

IN

-0.0142

-0.0347

0.0351

IN

0.0000

-0.0387

0.0173

IN

-0.0015

-0.0252

0.0116

IN

0.0065

-0.0343

0.0178

IN

∆y ∆y

-0.2377

-0.9734

0.9061

IN

∆y ∆u

A

2.0814

-1.5017

1.5007

OUT

Aπ∆y

0.5809

-1.1047

1.2411

IN

A

∆y ∆r ∆y ∆e

0.7502

-0.8160

0.7364

OUT

-0.1129

-1.1481

0.9939

IN

∆u ∆y

0.0762

-0.2589

0.2895

IN

∆u ∆u

0.1478

-0.4130

0.5709

IN

∆u A∆π

0.0699

-0.2935

0.3651

IN

0.1073

-0.1925

0.2780

IN

A A

A A

∆u ∆r ∆u ∆e

A A

0.2129

-0.4088

0.3353

IN

∆π ∆y

0.0656

-0.6312

0.5890

IN

∆π ∆u

A

0.2451

-1.0365

1.0584

IN

A∆∆ππ

0.0551

-0.6704

0.7620

IN

0.0990

-0.6169

0.3988

IN

A

∆π ∆r ∆π ∆e

A A

0.2492

-0.8630

0.7246

IN

∆r ∆y

0.6101

-0.2411

0.2166

OUT

∆r ∆u

-0.5829

-0.3497

0.4128

OUT

∆r A∆π

-0.0322

-0.3376

0.2661

IN

-0.2724

-0.2999

0.3293

IN

A A

∆r ∆r ∆r ∆e

A A

0.4387

-0.3468

0.3038

OUT

∆e ∆y

-0.0610

-0.6441

0.4252

IN

∆e ∆u

-0.0043

-0.7400

0.8845

IN

∆e A∆π

-0.1366

-0.5907

0.7798

IN

-0.1213

-0.4298

0.5003

IN

-0.1555

-0.6315

0.5220

IN

A A

∆e ∆r ∆e ∆e

A A

(The subscript indicates the equation while the super superscript denotes the lagged variable. g stands for GARCH parameter while a for ARCH one)

18

Table 14 Confidence Range Confidence Range 1 0.995 0.99 0.985 0.98 0.975 0.97 0.965 0.96 0.955 0.95

Number of Parameters Rejected 0 1 1 3 3 4 4 4 4 5 5

Table 20 Covariance Distribution Covariance

Actual Mean

Lower Bound

Upper Bound

Status

σ 1∆y∆y

0.0015

0.0004

0.1648

IN

σ

0.0034

-0.0528

0.0755

IN

σ

1 ∆u∆y 1 ∆π∆y

0.0009

-0.0346

0.0643

IN

σ

1 ∆r∆y

0.0013

-0.0297

0.0251

IN

σ

1 ∆e∆y

0.0007

-0.0133

0.0517

IN

σ

1 ∆u∆ u

0.0123

0.0030

0.7145

IN

σ

1 ∆π∆ u

0.0022

-0.0514

0.0594

IN

σ

1 ∆ r∆ u

0.0028

-0.0650

0.0864

IN

σ

1 ∆ e∆u

0.0029

-0.0892

0.0318

IN

σ

1 ∆ π∆π

0.0010

0.0007

0.2563

IN

σ

1 ∆r ∆ π

0.0008

-0.0570

0.0262

IN

σ

1 ∆e∆ π

0.0004

-0.1038

0.0292

IN

0.0013

0.0006

0.2930

IN

σ 1∆r∆r σ 1∆e∆r

0.0004

-0.0265

0.0399

IN

σ

1 ∆ e∆e

0.0013

0.0005

0.2484

IN

σ

2 ∆y∆y

0.0008

0.0003

0.1238

IN

σ

2 ∆u∆y

0.0000

-0.0299

0.0537

IN

σ

0.0005

-0.0125

0.0297

IN

σ

2 ∆π∆y 2 ∆r∆y

0.0006

-0.0301

0.0271

IN

σ

2 ∆e∆y

0.0008

-0.0169

0.0382

IN

σ

2 ∆ u∆ u

0.0022

0.0017

0.1966

IN

σ

2 ∆π∆u

0.0003

-0.0405

0.0292

IN

σ

2 ∆ r∆u

0.0000

-0.0256

0.0955

IN

σ

2 ∆ e∆ u

-0.0001

-0.0321

0.0224

IN

σ

2 ∆ π∆ π

0.0004

0.0006

0.2037

OUT

19

2 σ ∆r ∆π

0.0004

-0.0462

0.0196

IN

2 σ ∆e ∆π

0.0003

-0.0367

0.0135

IN

0.0005

0.0005

0.1870

IN

0.0007

-0.0308

0.0363

IN

0.0021

0.0006

0.1805

IN

0.0009

0.0003

0.1048

IN

0.0021

-0.0373

0.0258

IN

σ σ σ

2 ∆ r∆r 2 ∆ e∆ r 2 ∆ e∆ e

σ

3 ∆y∆y

σ σ

3 ∆u∆y 3 ∆π∆y

0.0007

-0.0110

0.0300

IN

σ

3 ∆r∆y

0.0007

-0.0109

0.0157

IN

σ

3 ∆e∆y

0.0005

-0.0120

0.0196

IN

σ

3 ∆ u∆ u

0.0063

0.0020

0.1809

IN

3 σ ∆π ∆u

0.0020

-0.0406

0.0169

IN

σ

3 ∆ r∆u

0.0016

-0.0092

0.0477

IN

σ

3 ∆ e∆ u

0.0015

-0.0221

0.0201

IN

σ

3 ∆ π∆ π

0.0007

0.0006

0.1356

IN

σ

3 ∆r∆ π

0.0006

-0.0233

0.0058

IN

σ

3 ∆e∆ π

0.0005

-0.0382

0.0113

IN

σ σ

3 ∆ r∆r 3 ∆ e∆ r 3 ∆ e∆ e

0.0006

0.0005

0.0826

IN

0.0004

-0.0079

0.0195

IN

0.0005

0.0004

0.0852

IN

σ

(The subscript denotes the variables for which the covariance is calculated while the superscript denotes the sate) Table 21 Confidence Range for Covariances Confidence Range 1 0.995 0.99 0.985 0.98 0.975 0.97 0.965 0.96 0.955

Number of Parameters Rejected 0 0 0 0 0 0 1 1 1 1

RESULTS OF IMPOSING 3 REGIMES- AS SET A PRIORI The first regime here is the monetary targeting regime, ending at 1985Q4. This is followed by the Fixed exchange regime until 1992Q3, at which point the inflation targeting regime begins and continues until the end of the sample in 2003. In this specification all the parameters- means, autoregressive, and var-covariance values- are allowed to vary, in effect through a set of dummy shift parameters. Below we show first the actual versus fitted values from the data; and then the parameters compared with their 95% confidence limits from the bootstraps. 20

Output (Mean,Autoregressive & Varince Parameters Vary)Unemployment (Mean,Autoregressive & Varince Parameters Vary) 0.04 0.2 Actual Actual 0.02 Fitted 0.1 Fitted 0

0

-0.02

-0.1

-0.04 -0.2 1970 1980 1990 2000 2010 1970 1980 1990 2000 2010 Inflation (Mean,Autoregressive & Varince Parameters Vary)Interest Rates (Mean,Autoregressive & Varince Parameters Vary) 0.02 0.04 Actual Actual 0.01 0.02 Fitted Fitted 0

0

-0.01

-0.02

-0.02 -0.04 1970 1980 1990 2000 2010 1970 Exchange Rates (Mean,Autoregressive & Varince Parameters Vary) 0.2 Actual 0.1 Fitted

1980

1990

2000

0 -0.1 -0.2 1970

1980

1990

2000

2010

Actual

Lower Bound

Upper Bound

1 ∆y

0.0042

0.0004

µ

1 ∆u

0.0299

µ

1 ∆π

Mean

µ

µ µ

1 ∆r 1 ∆e

Bootstrapped Prob. Values

State

0.0125

0.8465

IN

-0.1239

0.0311

0.0322

IN

-0.0003

-0.0022

0.0015

0.4183

IN

-0.0002

-0.0081

0.0102

0.4332

IN

-0.0008

-0.0455

0.0043

0.0594

IN

2 ∆y

µ

0.0088

0.0074

0.0225

0.9579

IN

µ ∆2u

0.0290

-0.2046

-0.0207

0.0000

OUT

µ

-0.0003

-0.0055

-0.0028

0.0000

OUT

-0.0017

-0.0011

0.0045

0.9901

OUT

2 ∆π

µ µ

2 ∆r 2 ∆e

-0.0076

-0.0456

0.0074

0.2525

IN

3 ∆y

µ

0.0117

0.0094

0.0188

0.8787

IN

µ ∆3u

-0.0082

-0.1367

-0.0161

0.0149

OUT

µ

-0.0003

0.0005

0.0028

0.0000

OUT

-0.0017

-0.0028

0.0020

0.8490

IN

-0.0016

-0.0281

0.0046

0.1436

IN

3 ∆π

µ µ

3 ∆r 3 ∆e

Time from 1979Q1 to 1985Q4

21

2010

Table 10 Autoregressive Parameters Distribution Actual

Lower Bound

Upper Bound

∆y ∆y

0.0128

-0.5577

∆y ∆u

-1.6212

∆y ∆π

Parameters

A A A

∆y ∆r ∆y ∆e

A A

Bootstrapped Prob. Values

State

0.4447

0.3663

IN

-1.6854

1.0895

0.9728

IN

0.1404

-0.3343

0.4907

0.3564

IN

0.0385

-0.6769

0.7618

0.5248

IN

0.0070

-0.8879

1.0028

0.4728

IN

∆u ∆y

-0.0455

-0.0839

0.2260

0.9233

IN

∆u ∆u

0.1606

-0.1373

0.7542

0.8441

IN

∆u ∆π

A

0.0194

-0.1637

0.0577

0.1163

IN

A∆∆ru A∆∆eu

0.0165

-0.2276

0.2384

0.4753

IN

-0.0033

-0.2867

0.2748

0.4629

IN

∆π ∆y

0.1409

-0.5999

0.5317

0.2673

IN

∆π ∆u

-2.9414

-1.4825

1.4928

0.0000

OUT

∆π ∆π

-0.3617

-0.6956

0.1234

0.6609

IN

A A

A A A

∆π ∆r ∆π ∆e

0.6422

-0.7575

0.8289

0.0446

IN

-1.1134

-1.1167

0.7214

0.9753

OUT

∆r ∆y

-0.0536

-0.7020

0.2467

0.2871

IN

∆r ∆u

-0.0684

-0.8594

1.5914

0.8020

IN

∆r ∆π

0.0666

-0.5257

0.1092

0.0347

IN

0.0882

-0.8109

0.6828

0.4505

IN

A A

A A A

∆r ∆r ∆r ∆e

A A

0.8146

-0.8272

0.5971

0.0074

OUT

∆e ∆y

-0.1094

-0.3299

0.1870

0.6485

IN

∆e ∆u

0.0666

-0.6253

0.8872

0.6757

IN

∆e ∆π

0.0257

-0.0461

0.3785

0.8960

IN

-0.0581

-0.1803

0.4167

0.8787

IN

0.2576

-0.4259

0.3434

0.0495

IN

A A A

∆e ∆r ∆e ∆e

A A

22

Table 11 Covariance Distribution Covariance

Actual

σ ∆y∆y

Lower Bound

Upper Bound

Bootstrapped Prob. Values

State

σ ∆u∆y

0.00017

0.00031

0.00122

0.00000

OUT

-0.00012

-0.00285

-0.00053

0.00000

OUT

σ ∆π∆y

-0.00001

-0.00053

0.00000

0.03465

IN

0.00000

-0.00037

0.00039

0.53960

IN

-0.00003

-0.00027

0.00101

0.80941

IN

0.00250

0.00234

0.01119

0.97277

IN

-0.00016

-0.00011

0.00125

0.98020

OUT

-0.00019

-0.00109

0.00117

0.77228

IN

0.00048

-0.00252

0.00140

0.15347

IN

0.00003

0.00018

0.00088

0.00000

OUT

0.00000

-0.00002

0.00085

0.96287

IN

-0.00001

-0.00105

0.00006

0.05941

IN

0.00018

0.00041

0.00322

0.00000

OUT

-0.00014

-0.00124

0.00037

0.42822

IN

0.00156

0.00113

0.00463

0.84406

IN

σ ∆r∆y σ ∆e∆y

σ ∆u∆u σ ∆π∆u σ ∆r∆u σ ∆e∆u σ ∆π∆π σ ∆r∆π σ ∆e∆π σ ∆r∆r σ ∆e∆r σ ∆e∆e

23

Time from 1986Q1 to 1992Q3 Table 12 Autoregressive Parameters Distribution Actual

Lower Bound

Upper Bound

∆y ∆y

0.2194

-0.4603

∆y ∆u

-1.1006

∆y ∆π

A

A∆∆ry A∆∆ey

Parameters

A A

Bootstrapped Prob. Values

State

0.2233

0.0297

IN

-1.0583

1.0384

0.9852

OUT

0.0888

-0.4245

0.3437

0.3119

IN

-0.0236

-1.2564

1.1196

0.5149

IN

0.1560

-0.5401

0.6296

0.3366

IN

∆u ∆y

-0.0275

-0.0839

0.1553

0.7277

IN

∆u ∆u

0.5950

0.1412

0.8215

0.5000

IN

∆u ∆π

0.0086

-0.1607

0.0376

0.0792

IN

-0.0401

-0.3664

0.3115

0.6287

IN

A A A

∆u ∆r ∆u ∆e

A A

-0.0087

-0.1498

0.2690

0.7475

IN

∆π ∆y

0.2565

-0.3542

0.1614

0.0025

OUT

∆π ∆u

-1.3935

-0.4574

0.9537

0.0000

OUT

∆π ∆π

-0.4377

-0.7264

-0.1493

0.5248

IN

A A A

∆π ∆r ∆π ∆e

0.2642

-1.1670

1.1637

0.2995

IN

-0.9034

-0.3492

0.3443

0.0000

OUT

∆r ∆y

A

-0.0715

-0.1031

0.0769

0.9356

IN

A∆∆ur

0.0218

-0.2266

0.2782

0.4604

IN

∆r ∆π

0.0619

-0.1540

0.0900

0.0718

IN

0.0817

-0.4519

0.4811

0.2995

IN

A A

A

∆r ∆r ∆r ∆e

A A

0.6310

-0.1487

0.1007

0.0000

OUT

∆e ∆y

A

-0.0842

-0.3084

0.1680

0.4431

IN

A∆∆ue

-0.0815

-0.6240

0.9357

0.8837

IN

∆e ∆π

0.0272

-0.0163

0.3365

0.9282

IN

0.0095

-0.2134

0.4592

0.7228

IN

0.2444

-0.2999

0.4227

0.1733

IN

A

∆e ∆r ∆e ∆e

A A

24

Table 13 Covariance Distribution Covariance

Actual

σ ∆y∆y

Lower Bound

Upper Bound

Bootstrapped Prob. Values

State

σ ∆u∆y

0.00023

0.00071

0.00200

0.00000

OUT

-0.00012

-0.00468

-0.00137

0.00000

OUT

σ ∆π∆y

-0.00002

-0.00067

0.00019

0.16832

IN

0.00002

-0.00184

0.00121

0.47772

IN

-0.00007

-0.00045

0.00102

0.71535

IN

0.00334

0.00478

0.01606

0.00000

OUT

-0.00016

-0.00047

0.00175

0.90347

IN

-0.00029

-0.00368

0.00472

0.65842

IN

0.00053

-0.00247

0.00196

0.28960

IN

0.00004

0.00087

0.00227

0.00000

OUT

-0.00001

-0.00363

0.00246

0.34158

IN

-0.00001

-0.00148

-0.00015

0.00495

OUT

0.00029

0.00720

0.05440

0.00000

OUT

-0.00016

-0.00258

0.00369

0.74752

IN

0.00288

0.00169

0.00544

0.45792

IN

σ ∆r∆y σ ∆e∆y

σ ∆u∆u σ ∆π∆u σ ∆r∆u σ ∆e∆u σ ∆π∆π σ ∆r∆π σ ∆e∆π σ ∆r∆r σ ∆e∆r σ ∆e∆e

25

Time from 1992Q4 to 2003Q4 Table 14 Autoregressive Parameters Distribution Actual

Lower Bound

Upper Bound

∆y ∆y

0.3733

-0.3820

∆y ∆u

-0.7354

∆y ∆π

A

A∆∆ry A∆∆ey

Parameters

A A

Bootstrapped Prob. Values

State

0.1778

0.0074

OUT

-0.6356

0.7531

0.9852

OUT

0.0561

-0.5523

0.4018

0.2946

IN

-0.0125

-0.5889

0.5776

0.5594

IN

-0.0093

-0.4048

0.6115

0.7079

IN

∆u ∆y

-0.0088

-0.0794

0.1243

0.5644

IN

∆u ∆u

0.6629

0.3482

0.8514

0.4876

IN

∆u ∆π

0.0058

-0.1529

0.0786

0.2302

IN

-0.0399

-0.1886

0.1735

0.7030

IN

A A A

∆u ∆r ∆u ∆e

A A

-0.0689

-0.1449

0.2478

0.9010

IN

∆π ∆y

0.2715

-0.1531

0.0392

0.0000

OUT

∆π ∆u

-1.0064

-0.0661

0.3944

0.0000

OUT

∆π ∆π

-0.4491

-0.7150

-0.2804

0.2995

IN

A A A

∆π ∆r ∆π ∆e

0.2122

-0.1375

0.1683

0.0124

OUT

-1.1928

-0.0594

0.4041

0.0000

OUT

∆r ∆y

A

-0.0099

-0.1433

0.0528

0.4703

IN

A∆∆ur

0.0888

-0.1325

0.2723

0.1312

IN

∆r ∆π

0.0370

-0.2078

0.0899

0.0941

IN

0.1174

-0.2917

0.2477

0.0965

IN

A A

A

∆r ∆r ∆r ∆e

A A

0.3020

-0.1228

0.2146

0.0000

OUT

∆e ∆y

A

-0.0501

-0.2451

0.0995

0.2673

IN

A∆∆ue

-0.1224

-0.3002

0.5749

0.9555

IN

∆e ∆π

0.0128

0.2657

0.8698

0.0000

OUT

0.0212

-0.1720

0.2328

0.5421

IN

0.2147

-0.4580

0.1212

0.0099

OUT

A

∆e ∆r ∆e ∆e

A A

26

Table 15 Covariance Distribution Covariance

Actual

σ ∆y∆y

Lower Bound

Upper Bound

Bootstrapped Prob. Values

State

σ ∆u∆y

0.00016

0.00090

0.00277

0.00000

OUT

-0.00006

-0.00602

-0.00195

0.00000

OUT

σ ∆π∆y

-0.00001

-0.00109

0.00056

0.36386

IN

0.00002

-0.00103

0.00090

0.54950

IN

-0.00004

-0.00054

0.00126

0.69554

IN

0.00225

0.00615

0.01540

0.00000

OUT

-0.00010

-0.00178

0.00141

0.40842

IN

-0.00018

-0.00255

0.00296

0.64356

IN

0.00019

-0.00177

0.00209

0.57921

IN

0.00003

0.00334

0.00758

0.00000

OUT

0.00000

-0.00221

0.00175

0.36386

IN

0.00000

-0.00617

-0.00216

0.00000

OUT

0.00019

0.00466

0.03328

0.00000

OUT

-0.00009

-0.00156

0.00257

0.76733

IN

0.00214

0.00378

0.00906

0.00000

OUT

σ ∆r∆y σ ∆e∆y

σ ∆u∆u σ ∆π∆u σ ∆r∆u σ ∆e∆u σ ∆π∆π σ ∆r∆π σ ∆e∆π σ ∆r∆r σ ∆e∆r σ ∆e∆e

Table 16: The number of parameters rejected by regime and type (rejected/total) Means

Autoregressive

Var-Covar

All

Regime 1

0/5

3/25

5/15

8/45

Regime 2

3/5

5/25

6/15

14/45

Regime 3

2/5

9/25

7/15

18/45

Total

5/15

17/75

18/45

40/135

When we compare the number of parameters rejected at 95% confidence with the total, it is clear that this model, restricted as it is by our a priori allocation of regimes, is a lot less successful than the one where we allowed Markov-switching in the data. In that case we generated the 95% confidence intervals from the model with regimes fixed a priori- of course we have no way of simulating the models with probabilistic regime switching so strictly these confidence limits cannot be computed. What could be being suggested here is that for future work we reconsider the boundaries of the three regimes. For now we continue with our basic null hypothesis given by the 3 regimes’ a priori allocation. We look at two more tests of this model against the data: from the totality of the parameters above using a portmanteau test and from impulse responses.

3.5 The portmanteau chi-squared or LM test: We begin with the portmanteau test for this model. The idea is to compare the multi-variate distribution of the full set of parameters for each regime with its distribution in the data. The test

27

computes first the small-sample distribution of the difference between these two distributions under the null hypothesis (ie that they are the same as that from the model). It then computes the value of the difference between the model distribution from the bootstraps and the VAR distribution in the data. This value should lie below the upper tail of the small-sample distribution. The results we obtain are shown below. The red line shows the actual value and the histograms the small-sample distribution. The results for the portmanteau test echo those from the tests on the individual parameters. The first regime does best, being rejected up to the 98.5% level; the other two are indistinguishably poor, both with rejection up to the 99.9% level. D-Stastic/Monetary Targeting Regime 30

95% Limit D-statistic Distribution

The actual value of D-Statistic coincides with the 98.5% limit of its Small Sample Distribution

20 10 0

0

1

2 3 4 5 D-Stastic/Fixed Exchange Rate Regime

6

7 5

x 10

95% Limit D-statistic Distribution

150 The actual value of D-Statistic coincides with the 99.9% limit of its Small Sample Distribution

100 50 0

0

200

400 600 800 D-Stastic/Inflation Targeting Regime

1000

1200

60

95% Limit D-statistic Distribution

The actual value of D-Statistic 40

coincides with the 99.9% limit of its Small Sample Distribution

20 0

0

20

40

60

80

100

120

140

160

180

200

220

3.6 Impulse responses In what follows we show the impulse responses generated by a supply shock (a rise in the employers’ tax rate on workers, BO) and by a nominal shock (MTEM: this is a fall in the money supply under the Money targeting regime; a fall in foreign prices under Exchange rate targeting; and a temporary rise in interest rates under Inflation targeting regime). The response in the structural model (LVP) is shown in blue; the implied path of the VAR when the VAR shocks are identified via the effect of the structural model (so that in effect their first period effect is the same) is shown in green; and the 95% confidence interval for the VAR effect coming from the model bootstraps is shown by the two red lines. The test is whether the VAR effects lie within the 95% interval. Thus the LVP response it self is not strictly relevant; it is shown for information merely. All the responses shown are for the differences of all the variables, not their levels.

A: Money targeting regime- 1979-85 Here we see that all the VAR effects lie within the 95% interval except for unemployment which is somewhat outside for both the supply and nominal shocks.

28

Output (BO Shock/M.T.R.) 0.02

Lower Upper VAR LVP

0.01 0 -0.01

Unemployment (BO Shock/M.T.R.) 0.1 0.05 0 -0.05

-0.02

0 5 10 15 Inflation (BO Shock/M.T.R.) 0.04

20 Lower Upper VAR LVP

0.02 0

-0.1 0 5 10 15 20 Interest Rates (BO Shock/M.T.R.) 0.04 0.02 0

-0.02

-0.02

-0.04

-0.04 0

0 5 10 15 20 Exchange Rates (BO Shock/M.T.R.) 0.05

0

-0.05

Lower Upper VAR LVP

0

5

10

15

0

0.02 0 -0.02

-0.01

-0.04

0 5 10 15 20 Inflation (MTEM Shock/M.T.R.) 0.01

0.005 0

Lower Upper VAR LVP

0 5 10 15 20 Interest Rates (MTEM Shock/M.T.R.) 0.02 0.01 0

-0.005

-0.01

-0.01

-0.02

0 5 10 15 20 Exchange Rates (MTEM Shock/M.T.R.) 0.02 0.01 0

Lower Upper VAR LVP

-0.01 10

15

20

Unemployment (MTEM Shock/M.T.R.) 0.04

-0.005

5

15

Lower Upper VAR LVP

Lower Upper VAR LVP

0.005

0

10

20

Output (MTEM Shock/M.T.R.) 0.01

-0.02

5

Lower Upper VAR LVP

20

29

0

5

10

15

20

Lower Upper VAR LVP

Lower Upper VAR LVP

B: Exchange rate regime- 1986-92 In this regime all the VAR effects lie within the interval. -3 Output x 10 (BO Shock/F.E.E.R.) 4

Lower Upper VAR LVP

2 0

Unemployment (BO Shock/F.E.E.R.) 0.02 0.01 0

-2 -4

0 5 10 15 20 Inflation (BO Shock/F.E.E.R.) 0.02 0.01 0

-0.01

Lower Upper VAR LVP

0 5 10 15 20 Interest Rates (BO Shock/F.E.E.R.) 0.01

0.005 0

-0.01

-0.005

-0.02

-0.01

0 5 10 15 20 -3 Exchange x 10 Rates (BO Shock/F.E.E.R.) 5

0

-5

0

5

10

15

0

5

10

15

20

Lower Upper VAR LVP

Lower Upper VAR LVP

0.01 0

Unemployment (MTEM Shock/F.E.E.R.) 0.04 0.02 0

-0.01

-0.02

-0.02

-0.04

0 5 10 15 20 Inflation (MTEM Shock/F.E.E.R.) 0.02 0.01 0

Lower Upper VAR LVP

0 5 10 15 20 Interest Rates (MTEM Shock/F.E.E.R.) 0.02 0

-0.02

-0.01 -0.02

0 5 10 15 20 Exchange Rates (MTEM Shock/F.E.E.R.) 0.02 0 -0.02

-0.04

Lower Upper VAR LVP

-0.04

0

5

10

15

Lower Upper VAR LVP

20

Output (MTEM Shock/F.E.E.R.) 0.02

-0.06

Lower Upper VAR LVP

20

30

0

5

10

15

20

Lower Upper VAR LVP

Lower Upper VAR LVP

C: Inflation targeting regime- 1992-2003 In this regime the VAR effects lie outside the 95% interval for several variables under both shocks. Under the supply shock inflation, the exchange rate and unemployment lie outside; under the nominal shock the exchange rate and unemployment lie outside.

D. Comments: What we find here is that, with some exceptions, the model is close to capturing the impulse responses. It is only possible to reject the LVP model for a handful of responses, virtually all confined to the inflation targeting regime. Yet we may notice that none of the impulse responses have a ‘hump shape’; such a shape is the product of particular identifying restrictions at the VAR reduced form stage. Such restrictions do not hold with the LVP model. (In levels, if we integrate these impulses, this would also be true; however, since the level variables follow unit root processes, the intervals tend towards being unbounded.) -3

Output x 10 (BO Shock/I.T.R.) 10

Lower Upper VAR LVP

5

Unemployment (BO Shock/I.T.R.) 0.05

Lower Upper VAR LVP

0

0

-5

0

5

10

15

-0.05

20

0.1 Inflation (BO Shock/I.T.R.) Lower Upper VAR LVP

0.05 0

0

5

10

15

20

Interest Rates (BO Shock/I.T.R.) 0.02

Lower Upper VAR LVP

0.01 0

-0.05 -0.01 -0.1 0

5

10

15

20

-0.02 0

Exchange Rates (BO Shock/I.T.R.) 0.1

Lower Upper VAR LVP

0.05 0 -0.05 -0.1

0

5

10

15

20

31

5

10

15

20

-3 Output x 10 (MTEM Shock/I.T.R.) 2

Lower Upper VAR LVP

0 -2

Unemployment (MTEM Shock/I.T.R.) 0.01 0.005 0 -0.005

-4

0 5 10 15 20 Inflation (MTEM Shock/I.T.R.) 0.04 0.02 0

-0.01

Lower Upper VAR LVP

0 5 10 15 20 -3 Interest x 10Rates (MTEM Shock/I.T.R.) 10 5 0

-0.02 -0.04

0 5 10 15 20 Exchange Rates (MTEM Shock/I.T.R.) 0.02 0

-5

0

5

10

15

20

Lower Upper VAR LVP

-0.02 -0.04

Lower Upper VAR LVP

0

5

10

15

20

3.7 Conclusions In this paper we have looked at three cases; one where we compared the LVP model using the same assumed monetary regime for the whole period (three decades) with the data, one where we did this for a reduced period from the mid-1980s to the early 2000s and one where we looked at the period from 1979 to 2003 and divided it into different monetary regimes. In the first case we found that the model generated large errors which caused large fluctuations in the model bootstrap solutions and prevented the model from solving at all in nearly half the bootstraps; assuming we could get the model to solve in these cases, the variances generated by the model would have been even larger. As it is they were excessive and the data lay outside the 95% confidence limits for these. While otherwise the model generated a VECM-GARCH that lay within the limits, this would likely be because with such large variances the limits were excessively loose. Hence in case one the model is plainly rejected by the data; this should not surprise us because there were several monetary regime changes during it which would have meant the model here was poorly specified for much of the period. In case 2 which covered a smaller period from mid-1980s to early 2000s, the variances were within the 95% limits; however the lags coefficients both of the VECM and the VAR lay outside indicating some failures to pick up the model dynamics. Interestingly this failure was less clearly picked by the correlations usually used as diagnostics in RBC analysis. The failure in case 2 was less easy to forecast; here at least the model chose a monetary regime (money targeting) which did prevail for some of the period; nevertheless for most of it it did not. So again this misspecification could lie at the root of the rejection. In the third case we distinguished between three monetary regimes: floating with monetary targets, exchange rate targeting (implemented by fixing the exchange rate), and floating with inflation targeting. Here we found that the model was rejected for all regimes, but worst for the inflation targeting and fixed rate regimes. Interestingly if we estimated a VAR in which the regimes were switched stochastically in a Markov process we found that the model seemed not to be rejected in so far as we could estimate the relevant 95% confidence intervals; this suggests that altering the regime allocation across periods could be helpful for the model’s success in future tests. In terms of the types of tests we have used, we found that the results of the tests on individual VAR parameters gave similar results to a portmanteau test on the whole VAR parameter distribution. However tests on impulse response functions were less clear; the model mainly passed these tests for the two typical shocks chosen. A comprehensive test on all the shocks to the model might however not have given the same results. Such a comprehensive test, when many shocks are

32

Lower Upper VAR LVP

relevant, is harder to implement and evaluate. Presumably if done appropriately its results should be consistent with those on the VAR parameters; this is a matter left for later work. Our final conclusion is that these methods are practicable on dynamic macro models; we hope to use them in future work on other models.

33

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[45] Kydland, F.,(1988), “On the Econometrics of World Business Cycles”, European Economic Review, 36, 476-482. [46] Kydland, F. and Prescott, E. (1982), “Time To Build and Aggregate Fluctuations”, Econometrica, 50, 1345-1370. [47] Kydland, F. and E. Prescott (1988), “The workweek of capital and its cyclical implications”, Journal of Monetary Economics, 21, 343-360. [48] Labadie, P., 1989, “Stochastic In‡ation and the Equity Premium,” Journal of Monetary Economics 24 : 277-298. [49] Lucas, R., (1990), “Liquidity and interest rates”, Journal of Economic Theory, 50, 237-64. [50] Lutkepohl, H., (1991), Introduction to Multiple Time Series Analysis, Second edition, Springer and Verlag, New York, NY. [51] Lutkepohl, H. and Kratzig, M., (2004). Applied Time Series Econometrics, Cambridge University Press, Cambridge. [52] Ma¤ezzoli, M. (2000) “Human Capital and International Business Cycles”, Review of Economic Dynamics, 3, 137-165. [53] Minford, P. and Webb, B., (2005), “Estimating large rational expectations models by FIML- a new algorithm with bootstrap con…dence limits”, Economic Modelling, 22, 187-205. [54] Merha, R. and Prescott, E. (1985), “The Equity Premium: A Puzzle”, Journal of Monetary Economics, 15, 145-161. [55] Phillips, P. C. B. and Perron, P. (1988), “Testing for a Unit Root in Time Series Regression”, Biometrica, 75, 335-346. [56] Ploberger, W., Kramer, W., and Kontrus, K., (1989), “A New Test for Structural Stability in the Linear Regression Model.” Journal of Econometrics, 40, 307–318. [57] Prescott, E., (1991), “Real Business Cycle Theories: what have we learned?” Federal Research of Minneapolis, Working Paper 486. [58] Rietz, T., (1988), “ The equity premium: A solution”, Journal of Monetary Economics, 22, 17–33. [59] Saikkonen, P. and Lutkepohl, H., (1999), Local Power of Likelihood Ratio Tests for the Cointegrating Rank of a VAR Process, Econometric Theory, 15, 50-78. [60] Schorfheide, F. (2000),“Loss function based evaluation of DSGE models”, Journal of Applied Econometrics, 15, 645-670. [61] Sims, Christopher A. (2003): script,Princeton University.

nProbability Models for Monetary Policy Decisions," Manu-

[62] Soderlind, P. (1994), “Cyclical Properties of a Real Business Cycle Model”, Journal of Applied Econometrics, 9, S113-S122. [63] Smets, F. and R. Wouters (2003), “ An Estimated Stochastic DGE model of the Euro Area”, Journal of the European Economic Association, 1, 1123-1175. [64] Tsionas, E.G., 1994, “Asset Returns in General Equilibrium with Scale-Mixture-of-Normals Endowment Processes,” University of Minnesota unpublished Ph.D. dissertation. [65] Watson, M, (1993), “Measures of Fit for Calibrated Models”, Journal of Political Economy, 101, 1011-1041. [66] White, H. (1982), “Maximum Likelihood Estimation of Misspeci…ed Models”, Econometrica, 50, 1-25.

36

4 4.1

Appendices Appendix A: Other methods in use- a survey

As we noted earlier, following Canova, 4 main methods can be distinguished which we review in turn.. Watson’s R2 procedure Watson’s (1993) working assumption is that the economic model is viewed as an approximation to the stochastic processes generating the actual data and he develops a goodness-of-…t (R2 -type) measure to assess this approximation. The core of these measures is the amount of the error that needs to be added to the data generated by the model so that the autocovariances implied by the model including the error match the autocovariances of the observed data. The economic model generates the evolution of a vector series xt (transformed if necessary to covariance stationary) whose empirical counterpart is denoted by yt . AC Fy is the autocovariance funtion of y t, which is unknown but can be estimated from the data either nonparametrically or parametrically by using a time series representation say a VAR or a VMA process (i.e. y t = £ (L) et where £ (L) is an matrix polynomial in L and et is an vector of i.i.d errors). Di¤erences between the estimator ACF y and ACF y arise solely from sampling error. Watson’s measure of …t is based on the size of the stochastic error,u t; required to reconcile the autocovariance function of x t with that of yt : that is so that the autocovariances of x t + u t are equal to the autocovariances of yt . If the variance of ut is large, then the discrepancy between AC Fy and ACF x is large and vice versa. Loosely speaking, you could think this as the error term in a regression model in which the set of regressors is interpreted as the economic model. The economic model might be viewed as a good approximation to the data if the error term is small (i.e. the R2 of the regression is large) and vice versa. Since y t = x t + u t the error’s autocovariance is de…ned as AF C u = AC Fy + ACF x ¡ AC Fyx ¡ AC Fxy where ACF y is unknown but can be estimated from the data while AC Fx is completely determined by the model. DSGE models may be linearized around a nonstochastic steady state and expressed as a low order VAR and VMA process from which their AC Fx can readily be calculated (e.g. a standard RBC model with a unique technology shock may be expressed as xt = a (L) "t where a (L) is an n ³ 1 matrix polynomial in L and " t is the unique i.i.d technology error). However, ACF yx is unknown because the joint probability function relating xt and y t is unknown. At this point an assumption regarding the AC Fyx is required. The existing literature provides two assumptions regarding this issue, strongly related with the way whereby data is collected or expectations are formed. The researcher could etiher set ACF yx = AC Fx , which implies that xt and u t are are uncorrelated (econometric assumption), or ACF yx = ACF y , which would imply that ut could be interpreted as signal extraction error, with y t an optimal estimate of the unobserved “signal” xt (signal extraction assumption). Obviously, neither of these assumptions seems to …t in the present framework, the error is not the result of imprecise measurement or a forecast/signal extraction error, it represents the approximation or abstraction error in the economic model. Watson’s way of obtaining the autocovariance between yt and x t is to choose that AC Fyx , which minimises the variance of u t . This means that the error process is chosen in order to make the model as close to the data as possible. As Watson (1993) points out this arise from the fact that, independently of any assumption made regarding ACF yx, if the lower bound of the variance of u t is large then the model …ts the data poorly and vice versa. Therefore, the bound on the variance of u t is minimised subject to the constraint that ACF yx is positive semide…nite (a condition that ensures that the variance will be meaningful). For example suppose that yt , x t and u t are scalar and serially uncorrelated random variables. The problem is to choose ¾ yx to minimise ¾ 2u ¢subject to the constraint that ¾ yx is positive semide…nite ¡ 2 min ¾ u = ¾ 2y + ¾ 2x ¡ 2¾ yx s:t: j¾ yx j · ¾ y ¾ x and the solution is ¾ yx = ¾ y ¾ x , which implies that x t and ³ ´ y t are perfectly correlated: xt = ¾¾ xy y t or x t = °yt E (xt yt ) = ¾¾xy E (yt yt ) =) ¾ yx = ¾¾xy ¾ 2y = ¾ y ¾ x . Watson (1993) uses the last equation as a measure of the …t of the model; it shows how to calculate …tted values of x t given the data. Then …tted data is plotted against the actual and this plot o¤ers a way to judge whether or not the former captures growth or cyclical components of the latter. The above analysis is easily extended to multivariate cases (this time the trace of the weighted covariance matrix is minimised) where stochastic singularity is present and to cases where the series are serially correlated (this time the trace of the weighted spectral density matrix is minimised). 37

Canova (2005) notes two shortcomings of Watson’s procedure. First he argues that it is not clear why one should concentrate only on the best possible …t, he suggests using both the best and the worst …t and if the range is narrow and 1 ¡ R2 of the worst outcome small, one could conclude that the model is satisfactory (Canova, Finn and Pagan, 1994). Secondly, the method does not provide information useful in respecifying the model. R 2 could be low for a variety of reasons, the variances of the shocks in the data may be high or the dynamics of the model and of the data are di¤erent or the process for the states has large AR coe¢cients. Obviously, it makes a lot of di¤erence whether it is the …rst or the last of these causes that makes R 2 low. However the basic problem with this approach is that, as Watson himself concedes, the method is not a test of the model since there is no statistical distribution of the resulting R2 under the null hypothesis that the model is correct. The R 2 simply measures the extent of the …t between the model prediction and the actual data; yet this …t has no necessary relationship to the correctness of the model. For example the model could be correct and yet have a poor R2 because the error processes have very high variances. The GMM Method Calibration techniques have been criticised for their assumption that structural parameters are known with certainty. Responding to this criticism Christiano and Eichenbaum (1992) developed a methodology, which evaluates the …t of an RBC / DSGE model, when some uncertainty is assigned They ³ to them. ´ ^ use formal econometric methods to estimate the vector of the structural parameters µ T ; µ , the only

unknown element in their model, and use the sampling variability implied by their estimates to evaluate the …t of the model. It is known from asymptotic theory that under the usual regularity conditions ^µT converges to µ as the sample size tends to becomes very big (T ! 1), however, in small samples these two vectors are unequal because of sampling Another ³p ³ variability. ´ ´ useful asymptotic result shows that ^ this di¤erence has its own distribution T µ T ¡ µ ~N (0; §µ ) ; the authors use this as the basis for a test of the model. Their working hypothesis is that the world is generated by their model (the DGP) so that the moments generated by the model would be identical to the actual moments under an in…nite sample since sampling variability would be eliminated; but would di¤er under a …nite sample because of sampling variability both directly and because it would cause the parameters to be estimated with error. The authors formulate a testing procedure to investigate whether these di¤erences in the moments arise because of sampling variability rather than from model misspeci…cation. Initially the model is linearised around a steady state growth path and then µ is estimated by Hansen’s (1982) Generalised Method of Moments (GMM) 2 . Using the authors’ notation, we let ª 1 denote the ^ 1;T ; and of the assumed stochastic distributions, structural parameters. Thus given the estimate of this, ª the di¤erence of the model’s moments from those of the data, ª2 ;would be distributed as a Wald Jstatistic (called by Canova, 2005, the “economic” Wald statistic).only if the model was speci…ed correctly; typically the second moments are focused on. Given a set of ª1 the model implies particular values for ª2 . Denote these values by f (ª1 ). The test asks whether or not the di¤erences between f (ª1 ) and ª2 are equal to zero (F (ª) = f (ª1 ) ¡ ª2 where ª = [ª1 ; ª2 ]). For example, assume that the structural model is given by yt = axt + u t, x t = pxt¡ 1 + vt ³ ´2 ¡ ¢ 1 (where jpj · 1 and E (x t ut ) = 0); in this case ª1 = a; p; ¾ 2u ; ¾ 2v , ª2 = ¾ y , f (ª1 ) = a2 1¡p ¾ 2v + ¾ 2u µ ³ ¶ ´2 and F (ª) = a2 1¡1 p ¾ 2v + ¾ 2u ¡ ¾ 2y . A second order Taylor expansion of F (ª) about ª0 gives µ ³ ´0 ³ ³ ´´ ³ ´¶ ^ ^ ^ the J-statistic as J = F ª V ar F ª F ª whose main element is the variance-covariance µ ¶ ³ ³ ´´ @F ª ^) @ F (ª (^)0 ^ matrix of the estimated parameters V ar F ª = @ ª §ª . Under the null hypothesis ^ ^ ^ @ª this statistic is distributed as a chi square with degrees of freedom equal to the number of the moments ¡ ¢ tested  2 (dim (ª 2 )) . Thus in the simple example above the stastistic follows a  2 (1) distribution. This distribution is based on asymptotic theory, however.. This implies that this process would work properly if the sample size was big enough. However, big samples are a luxury that, usually, we do not have in macroeconomics. This lack implies the existence of a small sample bias which distorts the inferences obtained from the above procedure . The evidences 3 suggests that estimates of the parameters 2 For a detailed discussion see Hamilton, 1994 chapter 14 and Canova, 2005 chapter 5. 3 Issues related with the sources of this bias, its direction and its importance are explained

5.4.1 he provides a list of studies examined the small sample properties of GMM estimators.

38

by Canova (2005). In section

and of the standard error are biased in small samples; this means that the J-statistic should be used with caution in small samples. The use of GMM methods and the J-statistic require stationary series. This implies that either growth rates (di¤erencing) or some …lter should be used in order to ensure that the above moments exist. However, the use of a …lter is not innocuous, Christiano and De Haan (1995) …nd that HP …ltering induces large and persistent serial correlation in the residuals of the orthogonality conditions and this creates problems in the estimation of the spectral density at frequency zero 4 5 . Clearly, problems are more severe when …ltered data is very persistent as is the case with both the H-P and the Band-Pass …lters. This could be seen as an additional source of bias. Apart from Christiano and Eichenbaum (1992), Burnside, Eichenbaum and Rebelo (1993) and Feve and Langot (1994) have also used this method to evaluate their DSGE models. The SMM method Under the SMM method the model’s simulated moments are brought as close as possible to the data moments by alteration of the free parameters of the model. The proposed test of the model is based on the resulting chi-squared distribution of the di¤erence of these moments. As indicated by footnote …ve of Diebold, Ohanian and Berkowitz (1998), one can assess the sampling error in the model spectrum by simulating repeated realizations from the model. Generally speaking, a wide class of measures of …t can be obtained if a researcher is willing to randomize on the realization of the stochastic process of the model. Gregory and Smith (1991) and (1993) generalized this approach. The sets of simulated data (produced by random realizations of the stochastic process embedded in the model) allow the evaluation of the distance between relevant functions of the model and of actual data using either asymptotic or probabilistic criteria. Canova (2005) argues that measures of …t of this type provide a sense of the ‘economic’ distance between the model and the data, contrary, for example, to what one would obtain with likelihood ratio tests. Gregory and Smith (1991) introduce a metho dology to estimate the probability of falsely rejecting a true economic model (Type I error). The models they use are Mehra and Prescott (1985) model in three di¤erent versions (Rietz,1988; asymmetry in growth transitions; and unusually high risk aversion). They concentrate on the …rst and second moments and wish to …nd the probability with which a model with population moment generates a sample moment µ T > µ. They produce R sets of stochastically simulated data with sample length (N ) equal to that (T ) used to estimate µT (N = T ); they calculate R P the above probability values either by using the empirical distribution R¡1 I (µN r ¡ µ T ) (where I is r=1

an idicator funtion) or by estimating the density non-parametrically with a quadratic kernel. Gregory and Smith (1993) use these same models to discuss issues related to the alternative methods of estimation often used in the RBC literature: the Generalized Metho d of Moments, the Method of Simulated Moments and calibration. The evaluation methods they suggest are the same as in the previous paragraph plus those (J-type statistics) discussed in the previous section when GMM estimation was discussed. Soderlin (1994) addresses the issue of statistical inference that arises with the usual RBC method of comparing model-simulation and data correlations: in other words are the di¤erences really signi…cant? He uses a variety of descriptors (model spectra, coherencies, phase shifts and correlation) to test the null hypothesis that data are realizations of the Kydland and Prescott (1988) model. The log-linear version of the model is expressed as a vector moving average of a three dimensional stochastic process. The latter is simulated 3000 times (random draws from the assumed parametric distribution of the error process) and 90% con…dence intervals are generated at each frequency for all descriptors, to compare with the data-generated estimates. Nason and Cogley (1994) compares ‡uctuations of the US business cycle with those predicted by a class of equilibrium monetary business cycle models (Lucas (1990), Fruest (1992), Christiano and Eichenbaum (1992, a) and Christiano and Eichenbaum (1992, b)). The predictions of the models are generated using the long-run neutrality restrictions implicit in the models ((1) output is independent 4 This matrix is one of the main elements of the parameters’ varianc e covariance matrix, which is estimated by using HACH techniques at zero frequency. The authors conducting a Monte Carlo study found that when the series have been HP …ltered that matrix is downward biased and this a¤ects the coverage probabilities that seem to be very far from the nominal values in this case. For more details see section 5 of Christiano and De Haan (1995). 5 Christiano and De Haan (1996) did a Monte Carlo study for the GMM and for the ”economic” Wald statistic and they note: ”The results are disappointing. The asyptotic theory appears to provide a poor approximation in …nite samples, particularly when the data have been HP …ltered.”

39

of nominal shocks in the long run and (2) money is independent of real shocks in the long run). By imposing these restrictions on sample data, tests of the ability of models to replicate the dynamics of the US business cycle are constructed. The models in log-linear form are expressed as fourth order structural VARs (SVARs) with these long-run restrictions; Blanchard and Quah’s (1989) methodology are used to identify the structural disturbances. When this is done each model is simulated 5000 times and the structural impulse response function (IRF) is produced at each simulation. The average IRF of these simulations is compared (graphically) with the sample’s IRF (the same DGP is used) and they use Quasi – Lagrange multiplier tests to check the signi…cance of the di¤erence between theoretical and sample IRF. Precisely, S(¿ ) = [IRF (¿ ) ¡ IRF A(¿ )]§(¿ )¡1 [IRF (¿ ) ¡ IRF A(¿ )¶] where ¿ refers to horizon, IRF (¿ ) is the mean response of the model (across replications), IRF A (¿ ) L P is the actual response and §(¿ ) = L1 [IRF (l; ¿ ) ¡ IRF A(¿ )][IRF (l; ¿ ) ¡ IRF A (¿ )¶]. This statistic is l=1

asymptotically T £ S (¿ ) ~Â2 (1). Backus, Gregory and Zin (1989) ask whether a monetary extension of Mehra and Prescott (1985) can account for the rejection of the expectations hypothesis with U.S. Treasury bill data. In the usual linear ³ ´ f

f

regression framework rt+1 ¡ rt = a + b rt ¡ rt + ut where rt and rt are one period spot and forward rate respectively, the expectation hypothesis implies that the forward rate is the market expectation of the future spot rate, which implies the coe¢cient restriction a = 0 and b = 1; however if there is a variable risk-premium these restrictions will not hold. The relationship is estimated on U.S. Treasury bill data and various hypotheses are tested (ARCH1, ARCH4, Wald (a = b = 0), Wald (b = 0) and Wald (b = ¡1)). In the arti…cial economy, we know exactly what the risk premium is (something that does not happen in the actual economy) and can determine the extent of its in‡uence by comparing regressions with and without it. Backus et al used pseudo data produced by the Mehra-Prescott model (1000 replications, 200 observations), to estimate these regression equations. Values for the parameters of the structural model are chosen to generate a variable risk premium and autoregressive conditional heteroskedasticity (ARCH) errors. This is an example of using simulation methods to test a structural hypothesis against the data. The problem here remains that we cannot be sure that the errors used were consonant with the data. Canova (1994) starts from the point made by some authors (such as Prescott (1991) and Kydland (1992)) that calibration is an incomplete exercise without sensitivity analysis on the calibrated parameter values. He proposes a methodology that addresses this issue while his evaluation procedure is based on techniques developed by Geweke (1989). He adds one more source of sampling variability to the model by treating the vector of coe¢cients (¯) as random. In his analysis, the density of the parameters is constructed (in a Bayesian way) by smoothing the histograms of prior estimates of available to a simulator (which may include both time series and cross-sectional aspects). The metric used to evaluate the discrepancy of the model is probabilistic: Z Z ³ ´ ¡ ¢ ´ E ¹ (Xt ) =f; h,Ll¸ = ¹ (Xt ) h Xt ; ¯ =^h; = @ (¯; f; =) d¯dXt h

¸l

(Xt is the simulated data produced³from realizations of the exogenous process and the parameters, ´ ^ = is the joint probability of the simulated data and ¹ (Xt ) are the statistics of interest, h Xt ; ¯=h;

parameters using the information set = and @ (¯; f; =) are the weights which depend on the “true” data). He constructs the simulated distribution of the statistics of interest, taking the actual realization of these statistics as the critical values; and he examines (1) in what percentile of the simulated distribution the actual value lies and (2) how much of the simulated distributions is within a k% region centered around the critical value. Extreme values for the percentile of a value smaller than k for the second probability indicates a poor …t in the dimensions examined. He applies his methodology in the equity premium puzzle by using Mehra and Prescott (1985) model and Kydland and Prescott (1982) model to explain the variance of actual US output. Canova (1995) applies the same methodology to three other models: the Brock-Mirman one sector growth model, the Canova and Marrinan (1993) two-country model, and the Cooley and Hansen (1989) optimal taxation model. Ma¤ezzoli (2000) slightly adapts Canova’s methods for use on a two-country, two-sector stochastic endogenous growth model with a human capital externality. While the simulation and sensitivity part remain the same in the present case a whole range for the statistics of interest (correlations for output, investment and labour) is available (one observation for each country or country pair). Therefore the …t of the model is indicated by the degree of the overlap between the actual and the simulated distribution. 40

De Jong, Ingram and Whiteman (1996) argue that in the latter two methods uncertainty is treated asymmetrically. It arises from the prior uncertainty in working with the model and from sampling error in working with the data. Precisely, Canova’s procedure rely on sample means and standard errors to characterize empirical distributions, something that carries the potential to deliver misleading inferences in situations in which these distributions feature skewness and (or) kurtosis. The Bayesian approach De Jong, Ingram and Whiteman (1996) argue that the reason a model may …t the data adequately along some dimensions and yet fail to do so along other dimensions is because the speci…cation of parameter uncertainty is inadequate. They develop a Bayesian approach to the problem. They represent parameter uncertainty via prior distributions and re-compute these distributions using the statistical properties (e.g. correlation patterns) of arti…cial data simulated from the model. In this approach, uncertainty of model and data are treated symmetrically, in that inference about the statistical properties of the actual data requires the speci…cation of a statistical model. They use a VAR (in their empirical example a VAR(6)) because a linearized version of a RBC model implies that the actual data follows a VAR process. Combining a normal likelihood function with a ‡at prior distribution speci…ed over the parameters of the VAR yields a Normal-Wishart posterior distribution over the statistical properties of interest. Under their procedure probability distributions are obtained for both theoretical model P (si ) (where si denotes functions of interest) and data (statistical model) D (si); the proximity and degree of overlap between these distributions are then used to assess the model’s …t. They develop two measures for this "con…dence interval criterion" (C IC). The …rst CIC measure for P (si) is given by the integral of 1 this distribution over the inter a-quantile range of D (si ) normalized by 1¡a (where 1 ¡ a is the integral R b 1 of D (s i) over the same range), CIC ji = 1¡a a Pj (s i). The second measure is analogous to a t statistic for the mean of P (si) in the D (si) distribution, dji =

E( Pj (s i))¡E (Dj ( si)) . (V ar(D j (si)))1=2

They apply these methods to

King, Ploesser and Rebelo’s (1988) RBC model. De Jong, Ingram and Whiteman (2000) also develop a Bayesian method for the estimation of parameters in a DSGE; this method also lends itself to evaluation of the …t of the latter. The standard strategy in Bayesian econometrics is to combine prior and likelihood function to obtain posterior distributions of functions of interest (in the present case the posterior distribution of the structural parameters). In this framework two di¢culties arise (1) the likelihood function for the theoretical model is generally unknown and (2) given an approximation of the likelihood function, posterior analysis cannot be carried out analytically. For the …rst problem they use the Kalman …lter while they conduct posterior analysis by using numerical techniques (importance sampling) developed by Geweke (1989). Having obtained the posterior distribution for the parameters, forecasts produced from the model are comparable with those produced from a four-lag Bayesian vector autoregressive model (with Minnesota priors). This is an out-of-sample forecast exercise and R the measure used to evaluate the performance of the two models is the predictive density (p (y ¤ =y) = p (y ¤ =y; µ) p (µ=y) where y; y ¤ and p (µ=y) are the actual data, the forecasts and the posterior respectively) of which the posterior distribution of the parameters is the main component. Their procedure is applied to Greenwood, Hercowitz and Hu¤man’s (1988) model. Geweke (1999) explains his own approach by considering three alternative econometric interpretations of DSGE models. He de…nes as "Strong econometric interpretation" all those estimated by likelihood functions; as "Weak econometric interpretation" those that are calibrated; and as "Minimal econometric interpretation" suggest to view a DSGE model as representation for the population moments of observable of the data (not their sample counterparts). As it will be obvious below this setup is advantageous because comparisons across models do not require the likelihood function of the data (as in Canova’s procedure). However, since DSGE models have no interpretation for the observables, it is necessary to bridge population and sample statistics (similar to De Jong et all. (1996)). Additionally, he states that assumptions made in the “Weak econometric interpretation” case are not inherently weaker than those that lead to likelihood based econometrics; however in them econometrics has been restricted to an informal comparison of two numbers, the observed and the predictive mean. Yet the sampling distribution of the statistic used for this purpose is often a function of profoundly unrealistic aspects of these models, aspects that lie outside the dimensions of reality the model were intended to mimic. For example, in the equity premium models the sampling distribution of average asset returns over say 100-year period are closely related to the variances of these returns, through the usual arithmetic for standard deviation of a sample mean. In establishing the sampling distribution of these means through repeated simulation of the model, one is taking literally the second moments of returns inherent in the model. There are precisely the dimensions the original model was not intended to capture. He argues 41

that the rescue for DSGE comes from "Minimal econometric interpretation". However, for these DSGE models to be capable of explaining measured aggregate economic behavior, they must be married to atheoretical econometric models with plausible descriptions of measured behavior, linking population moments with observable data. For this reason he introduces a method that generalizes and formalizes the De Jong, Ingram and Whiteman (1996) procedure for joint, multidimensional measures. Precisely, let A and B denote two alternative DSGE model, each describing the same vector of population moments m(the means for riskless rate and equity premium in this example) by means of respective densities p(m=A) and p(m=B). A third atheoretical time series (VAR (1) in the present example) model E is introduced to provide the posterior the data p(m=y; E). The posterior odds R distribution for the m, given R p(A=y;E ) p(A=E) p(m=A)p(m=y; E)dm ratio is given p(B=y;E) = p(B =E) R p(m=B )p(m=y;E) dm where p (m=A) p (m=y; E) dm is the convolution of two densities; the density for m implied by A and the posterior density for m implied by E given y. Loosely speaking, the greater the overlap between these two densities, the greater the Bayes factor R p(y=A;E ) R p(m=A)p(m=y;E) dm in favour of model A. The models used here are Mehra and Prescott = p(y=B ;E) p(m=B)p(m=y; E)dm ¡ ¢ (1985); Rietz (1988); Labadie (1989); and Tsionas (1994). Since the posterior density p m=y 0 ; E is so di¤use, he …nds that there is overlap between this density and each of the model densities. This is true even of the Mehra and Prescott and Labadie models, which received no support under the weak econometric interpretation. The reason is that the weak econometric interpretation takes literally the very small variance in the riskless return and equity premium implied by these two models. Given this small sampling variation the observed averages cannot be accounted for in these models. Given the much greater sampling variation that emerges in the bivariate autoregression used in the method Geweke proposes, the evidence in the historical record is perceived to be much less discriminatory. Schorfheide (2000) uses Bayesian estimation techniques like those introduced by De Jong, Ingram and Whiteman (2000) and resets the evaluation procedure introduced by Geweke (1999) in a more traditional way, where macroeconometric models were evaluated according to their ability to track and forecast economic time series. He starts from the point that DSGE models are misspeci…ed (since the vector of the structural parameters µ is low dimensional) and introduces a reference model M 0 densely parameterized, which imposes weaker restrictions on the data than the former models and achieve an acceptable …t. The model used by Schorfheide is a bivariate (output growth and in‡ation) VAR(4), required to give the actual data a probabilistic representation. In the statistical …t of the various models Mi where i = 0; 1; 2 ¼ i;0 p(Y T =Mi) (DSGE and VAR models) is given by the posterior model probabilities ¼ i;T = P (after the 2 i=0

¼i;0p( YT =Mi)

Rderivation of the posterior distribution for the structural parameters p (µ i =Y T ; M i) since p (Y T =M i ) = p (Y T =µi ; Mi) p (µi =Mi )) which are also the mixture weights for DSGE and reference models. In the next step for each model given the p (µ i= YT ; Mi ) one can calculate the posterior distribution for the population characteristics Á, p (Ái =Y T ; Mi); in this case this vector consists of impulse response dynamics and the correlation between output growth and in‡ation. p (Ái= YT ; Mi ) used for the construction of the 2 P overall posterior of Á, p (Á=Y T ) = ¼ i;T p (Ái=Y T ; M i ). The next step, the innovation of this paper, ³ i=0´ ^ that penalizes the deviations from the actual Á and predicted is to introduce a loss function L Á; Á ´ R ³ ^ population characteristics (where the optimal predictor is ^Á = arg min L Á; ~Á p (Á =Y T ; M i ) dÁ). Á i i ~ Á ³ ´ R ³ ´ ^ i= YT = L Á; Á ^ p (Á= YT ) dÁ is calculated and the DSGE model that Finally the expected loss R Á attains the smallest posterior prediction loss wins the comparison. This procedure is applied to cash-inadvance and portfolio adjustment-cost models. All the methods studied in this category so far assume that the sampling variability of both actual and simulated arise only from the variability of the parameters. Canova and De Nicolo (1995) allow for variability both in the parameters and the exogenous processes; they study the equity premium puzzle in a number of industrialized countries (G7). Their time-series study shows that the moments for the equity premium and risk free rate vary across country and across time sub-periods. The structural model used to explain this behaviour is a standard consumption CAPM model. Two maintained hypotheses are considered. First, countries can di¤er in preferences and technological parameters, but not in institutional setups or market arrangements. Second, countries’ …nancial markets are autarkic. Following Aiyagary (1993) the model has a closed form solution for the risk-free rate and the equity premium; given an additional regression equation where consumption is regressed on a constant, Canova and De Nicolo use these three equations to derive the empirical distribution of the actual vector of statistics in which they are interested (the mean, the standard deviation and the autoregressive parameter) for both the 42

equity premium and the risk free rate; they derive the distribution of the structural estimates by using regular bootstrapping techniques. To construct the simulated distribution of the statistics of interest they draw with replacement from the empirical distribution of the structural parameters created above and simulate the model. The size of the overlap between the two distributions (for each element of the vector of statistics) provides a measure of …t for the model. Canova (2005) remarks that Geweke’s (1999) method provided the statistical foundation for this approach. Stubbornly, the traditional RBC literature wants to explain the world with one maybe two or at maximum three shocks. Smets and Wouters’s (2003) work desires to indicate that the current generation model with sticky prices and wages is su¢ciently to capture the stochastics and the dynamics in the data, as long as a su¢cient number of structural shocks is considered. They augment Christiano, Eichenbaum and Evans’s (2005) model with a substantial number of structural shocks (not measurements errors) and compare the empirical performance of the DSGE model with those of standard and Bayesian VARs on the basis of the marginal likelihood and Bayes factors. Initially, they estimate by using Bayesian methods the DSGE model and compare their estimates for crucial parameters (such as parameters characterising the degree of priceRand wage stickiness) with those given by other studies. Next by using the marginal likelihood M A = µ p (µ=A) p (Y T =µ; A) of a model say A, which re‡ects the prediction performance TQ +m R of the model (^ p TT +m = p (µ=Y ; A) p (y t= Y T ; µ; A) dµ as p^T0 = M T ), they use the Bayes factor T +1 µ t=T + 1

(Bij =

Mi Mj )

and posterior odds ratio (P Oi =

PpiMi j pj Mj

) to compare the empirical performance between

the DSGE and VARs/BVARs with one, two and three lags. Subsequently, they compare the empirical cross covariances with those obtained from the model. The empirical cross covariances are based on a VAR(3) estimated on sample of 115 observation. In order to be consistent, the model-based cross covariavces are also calculating by estimating a VAR(3) on 10.000 random sample of 115 observations generated from DSGE model. They provide a graphical judgment by plotting the median and the 5% and 95% interval for the covariance sample of the SDGE with the empirical ones. Their work ends with a impulse response and variance decomposition study where the qualitatively analysis of the responses and the estimates for the variance decomposition are compared with those existed in the literature. After Sims (2003) who provides some very important guides how Bayesian model comparison methods should be applied a sequence of papers [Negro and Schorfheide (2004), Negro, Schorfheide, Smets and Wouters (2004) and Negro and Schorfeide (2005)] introduces a methodology {built on ideas initially expressed by Ingram and Whiteman (1994) and DeJong, Ingram and Whitman (1996)} that evaluates the …t of a DSGE model through the cross equation restrictions imposed by the structural model on unrestricted VAR. Precisely, the model is linearized around a non-stochastic steady state (st = As t¡1 +!t ; y t = ¦s t) and following Ingram and Whitman (1994) it is expressed (by using the generalised inverse) ³ ´¡1 ¶ ¶ and µ is the vector o fthe structural initially as VAR(1) (yt = F (µ) yt¡ 1 + ´ ; F (µ) = ¦A ¦¦ ¦ t

parameters) and sebsequently as a …nite VAR(p) (y t = B1¤ (µ) yt¡1 +:::+B ¤p (µ) yt¡ p +»t or Y = B¤ X +¥) which is treated as the structural model. The following assumptions regarding the misspeci…cation of the DSGE model (since µ is low dimensional) are made. There is a vector µ and matrices B ¢ and §¢ u such that the actual data are generated for the vector autoregressive model Y = BX + ª where ¤ B = B ¤ (µ) + B¢ and § u = § ¤u (µ) + § ¢ u and there does not exist a µ 2 £ such that B = B (µ) and § u = § ¤u (µ). Since the procedure is developed in Bayesian framework there are beliefs (priors) regarding the misspeci…cations (B ¢ ; § ¢ u ) implied by the structural model. In every paper there is an update in the way whereby the priors are formulated. Initially, Ingram and Whitman (1994) assume normal priors for µ with a diagonal covariance matrix and they derive the prior for F (µ) by using …rst order Taylor expansion around the mean of . Negro and Schorfheide (2004) specify a hierarchical prior of the form p (B; § u ; µ) = p (B; § u =µ) p (µ), where p (B; §u =µ) is generated by augmenting the actual data with T ¤ = ¸T arti…cial (the role of ¸ is going to be analysed below) observations generated from the DSGE model. Actually, rather than generating random observations y ¤1 ; :::; yT¤ ¤ from Y = B ¤ X + ª and augmenting the actual data Y , that is, pre-multiplying the likelihood function by © £ ¤ª ¤ ¤ ¤ ¤ 0 p~ (©; §u =µ) _ j§u j¡T =2 exp ¡ 12 tr § ¡1 , they replace the arti…cial sample u (Y ¡ BX ) (Y ¡ BX ) moments (E(Y ¤ Y ¶¤ ) = ¡¤yy ; E (Y ¤ X ¶¤ ) = ¡¤yx and E(X ¤ X ¶¤ ) = ¡¤xx ) by population analogous. p (©; § u=µ) _

¡( ¸T +n+1)=2

c¡1 (µ) j§u j ½ · µ ¶¸¾ ¡¤yy (µ) ¡ B¡¤xy (µ) 1 ¡1 exp ¡ tr ¸T § u ¶ + B¡¤xx (µ) B ¡¡¤yx (µ) B 2 43

which is proper provided that ¸T = k + n where n is the length of yt , k = np and the proportionality factor c (µ) ensures that the density integrates to one (see appendix of Negro and Schorfheide (2002) for more details. (Additionally, as this is analysed in the same paper if ¸ > 0 then there is a map from the posterior estimates of the VAR parameters to the space traced out by B ¤ (µ) and § ¤u (µ) and its mode can be interpreted as minimum-distance estimator of µ). While Negro et al. (2005) construct a prior that has the property that its density is proportional to the expected likelihood ratio of B evaluated at its misspeci…ed restricted value B¤ (µ) versus the true value B = B ¤ (µ) + B¢ .The log-likelihood · ¸ h ³ ´i ¤ ¢ ¤ L(B +B ;§ u;µ=Y ;X ) ¶ ¢¶ ¡ 2B ¢ Xª taking expectation over X and ratio is ln = ¡ 12 tr §¤u B ¢ X XB L(B ¤; §¤ ;µ=Y ;X) u

Y using the distribution induced by the data generating process yields (minus) Kullback-Leibler · · the ¸¸ L(B ¤+ B¢ ;§ ¤ u;µ=Y ;X ) V AR distance between the data generating process and the DSGE model E B ¤;§ ¤u ln = L( B¤ ;§¤ u ;µ=Y ;X ) h ³ ´i ¶ ¡ 12 tr § ¤u B ¢ ¸T ¡xx B¢ . They choose a prior that is proportional to Kullback-Leibler distance n h ³ ´io ¡ ¢ ¤ ¢ ¶ p B =§u ; µ _ exp 12 tr ¸T §¤u B ¢ ¡ xx B¢ . For convenience this prior is transformed into a prior ¡ ¡ £ ¡1 ¤¢¢ 1 for B B=§ ¤u ; µ~N B ¤ (µ) ; ¸T § u - ¡ xx (µ) . The hyperparameter ¸, which determines the length of the hypothetical sample as a multiple of the actual sample size T , scales the variance of the distribution that generates B ¢ and B. If ¸ is close to zero, the prior variance of the discrepancy B¢ is large. Large values of ¸, on the other hand correspond to small model misspeci…cation. Formally, the posterior µ distribution of ¸ provides an overall assessment ¶ of the DSGE mo del restrictions. Posterior R mass ^¸ = arg max p (Y =µ; §; B) p ¸ (µ; §; B) d (µ; §; B) concentrated on large values of ¸ provides ¸

evidence in support of the DSGE model restrictions. Since the marginal data density of ¸ loosely speaking indicates the one step ahead pseudo out of sample ^ indicates at which extend these cross equation restriction could forecast performance; its optimal value ¸ ³ ³ ´´ be relaxed in order the forecasting performance to be improved DSGE ¡ V AR ^¸ . Next the impulse ³ ´ ^ . responses produced by the DSGE model are compared with those produced by the DSGE ¡ V AR ¸ ³ ´ ^ ( ). The authors introduce an identi…cation scheme so the responses produced by the DSGE ¡ V AR ¸ to correspond to those produced by the DSGE model (which are identi…ed). This idea is based on the way used to identify the structural residuals à t = §C -»t where §C is the Cholesky decomposition of § ¶ = I. Since the DSGE model itself is identi…ed and - is an orthonormal matrix with the property -in the sense that for each value of µ there is a unique matrix A0 (µ), obtained from the state space representation of the DSGE model, that determines the contemporaneous e¤ect of »t on y t. Using a QR factorization ³ ´ of A0 (µ) the initial response of y t to the structural shocks can be uniquely decomposed into @@ y»t ³ ´t by @@»yt t

= A0 (µ) = §¤C (µ) - ¤ (µ) while the initial impact of »t on y t in the VAR is given ³ ´ = § C - to identify the DSGE ¡ V AR ^¸ , they maintain the triangularization of its

DSGE

V AR

covariance matrix§C and replace the rotation matrix - with -¤ (µ). Loosely speaking, the rotation matrix is such that in absence of misspeci…cations the DSGE’s and the DSGE–VAR’s impulse responses to all shocks would coincide.

44

4.2 Appendix B: Notes on the VECM/VAR-GARCH estimation: Quarterly data between of 1970 to 2003 is used here. It consists of log of output, log of unemployment, inflation, nominal interest rates and exchange rates. The null hypothesis2 of stochastic trend cannot be rejected for the above series by using Phillips-Perron (1986) unit root test with Andrews and Monahan’s (1992) quadratic spectral kernel HAC3 estimator, while the same hypothesis is rejected for their first difference. A candidate multivariate time series model will accepted here if and only the estimated residuals are i.i.d. To reach this target two tests for normality, three tests for serial correlation, two test for heteroscedasticity, three selection criteria, one test for the cointegration rank and one test for parameter constancy have been implemented45. Under the existence of cointegration it is well known that the asymptotic distribution of these tests may constitute a very poor approximation in the small sample6.To overcome this problem a small sample study is contacted here for a number of lags p ∈ 1 7 ⊂ Ν , for a number of

[ ]

[ ]

cointegration relationships r ∈ 1 4 ⊂ Ν and for all “Cases” 7 of VECM as they described by Saikkonen and Lutkepohl (1999) (SL). Parametric bootstrapping 8 procedure is used were 49999 pseudo samples have been generated under the null hypothesis aiming to create the bootstrapped probability values. The result of this process is a VECM with six lags p = 6 , two steady state

(

equilibriums

)

(r = 2 ) and a time trend into the ECM term (“Case 4” using SL terminology):   ∆y t = c + α  β ′ y t −1 − τ (t − 1)  + ∑ Γi ∆y t− i + u t (5 ×2 ) ( 2×5 ) ( 2×1)  i=1 6

However, the estimated residuals are still heteroscedastic a fact that indicates model’s miss specification. The first though was to allow the parameter to switch stochastically (by following a first order Markov Chain), however, Hansen and Johansen’s (1999) parameters’ constancy test does not reject the null9.Under this fact the second though is to specify a process that could capture the that empirical volatilities tend to cluster (this is supported by the column 3 and 4 of table A6). For this purposes Engle and Kroner (1995) BEKK form of GARCH(1,1) has been adopted

H t = CC ′ + A1′u t −1ut′−1 A1 + G1′ H t −1G1 and the choice is based on two reasons. Firstly that under very weak condition10 (either C or G have full rank) the covariance matrix is positive definite and secondly because this specification allows for direct dependence of the conditional variance of one variable on the history of other variables within the system. The information matrix is block diagonal, which implies that efficient estimates of the VECM model can be calculated independently of the parameters of the GARCH process and vice versa. Engle and Kroner (1995) suggested and iterative process (adopted here) whereby the estimates will solve the first order conditions for the full information maximum likelihood estimation.

4.3 Appendix C: The Liverpool Model LPV consists of a set of exogenous variable processes (all univariate time-series) and the structural model of endogenous variables. (Recently the model was re-estimated by FIML (Minford and Webb, 2004); the version used here is one already in use prior to this re-estimation. However, it turns out that the parameters all lie comfortably within the FIML confidence limits.

2

The estimation is done by including initially both trend and constant and if they any of them (or both) was insignificant it was excluded and the test was applied again 3 Heteroscedasticity Autocorrelation Consistent Covariance 4 A properly should passes all the above mentioned models. 5 In the appendix is explained the reason why so many test have been used. 6 There is a bias of over rejecting the null and the null here is no serial correlation, no heteroscedasticity and etc. 7 There are five different “Cases” of the functional form of the VECM each of them tries to capture characteristic of data. We study all of for a range (1 to 7) of lags and a range (1 to 4) coinegration relation and we adopt that specification whose estimated residuals are i.i.d. 8 As it is described by Jacobson and etc (2001) 9 See appendix II 10 They could easily be constructed numerically. 45

Table: Comparison of LVP Parameter and FIML Parameters

LVP FIML LVP FIML LVP FIML LVP FIML LVP FIML LVP FIML

A1 1.528 1.22 A11 -0.290 -0.303 A21 0.640 0.654 A31 0.103 0.095 A41 0.170 -0.324 A51 0.011

A2 -0.003 -0.003 A12 0.189 0.196 A22 -0.215 -0.221 A32 0.193 0.160 A42 25.262 24.475 A52 0.750

A3 -2.150 -2.132 A13 0.870 0.843 A23 0.056 0.050 A33 0.000 0.000 A43 0.102 0.132 A53 -0.750 0.000

A4 0.792 0.760 A14 0.150 0.149 A24 0.153 0.141 A34 0.000 0.000 A44 -0.337 -0.420 A54 0.300

A5 0.010 0.011 A15 0.000 0.000 A25 0.870 0.870 A35 0.931 0.722 A45 0.013 0.004 A55 -1.000

A6 0.804 0.783 A16 -0.002 0.000 A26 0.000 0.000 A36 0.271 0.310 A46 0.666 0.780 A56 -1.000

A7 0.470 0.490 A17 -0.349 -0.345 A27 0.529 0.521 A37 1.000 1.000 A47 11.503 11.201

A8 0.210 0.210 A18 0.839 -0.313 A28 -1.205 -1.223 A38 0.012 A48 -0.016

A9 -0.018 -0.021 A19 -0.016 -0.021 A29 -0.388 -0.350 A39 -0.125 -0.094 A49 -.011

A10 -0.244 -0.420 A20 -0.004 -0.003 A30 0.429 0.420 A40 0.320 0.320 A50 0.017

LVP can be thought of in terms of the interesting trade-off frontier suggested by the Bank of England between rigorously-constrained theoretical models and entirely data-generated models — see diagram. Theory Coherence

Data Coherence LVP lies somewhere along from the theory end. It is an IS/LM Phillips Curve model that can be derive with some approximation from a tightly-specified representative agent model in the manner of McCallum and Nelson. It was set up as a compromise model for forecasting and policy analysis some 25 years ago and has been fairly successfully used for both. In that sense perhaps it is not that surprising that at the 95% level it cannot be rejected. What we would like to know is how it fares against other models. Here we envisage a battle mediated by the confidence band, as described in appendix 3. Essentially we ask for each model: how low is the confidence band at which it is just accepted? The model with the lowest band ‘wins’. However we should caution that we must compare apples with apples — that is theory-based with theory-based, compromise with compromise, etc. The absolute data winner should always be the more data-generated model. But that doesn’t help us to choose. The confidence limits for the structural parameters come from the pseudo-samples generated by drawing the vector of endogenous variable errors repeatedly for each period of the sample with its associated exogenous variable actual/projection. The model is re-estimated for each pseudo-sample. Notice that across every pseudo-sample the exogenous variables are the same; in estimation these exogenous variables are held fixed and the structural parameters are estimated conditional on these variables The parameters of the VECM are a product of the structural model, the exogenous variable processes, and the errors both in the endogenous and the exogenous variables. As equation (2) above shows the VECM parameters are a combination of the parameters of the exogenous processes and of the structural model. Thus the pseudo-data on the endogenous variables implied by the Model must 46

be generated by drawing errors for both endogenous and exogenous variables and computing their effect within the Model. Thus the pseudo-samples will be stochastic simulations of the full Model including its exogenous processes over the sample period. The VECM parameters’ variability can be assessed by computing how they would differ across these pseudo-samples which summarise the potential variability coming from the errors. For this purpose we used the Liverpool Model as re-estimated by FIML in Minford and Webb (2004).We created two sets of data; one extending over the sample period of FIML estimation, quarterly1974-1993 and a second extending quarterly from 1986-2002. These sample periods capture the two main periods of recent UK economic history: the first was that of floating exchange rates but poor monetary management when inflation first became very high and subsequently was brought down to low single figures. The second is that during which monetary policy first began to operate at low rates of inflation, soon moving to inflation targeting. It is likely that, with the substantial changes in exogenous forcing processes between these two periods, the VECM would have changed behaviour in line with these regime changes. Thus we generate the pseudo-samples for both periods separately. For both periods we extracted the implied errors for both structural model equations and the exogenous processes.. Any lack of whiteness in the errors was treated by estimating further univariate processes for them (which were now included in the Model). The remaining vector of white errors across all variables was drawn repeatedly with replacement and re-inputted into the model to generate 284 pseudo-samples each of 136 quarters in length, see Minford and Webb (2003) for full details). The VECM is now re-estimated for each of these pseudo-samples to yield the 95% confidence limits from sampling variation. At this stage we have had the problem that only 284 out of our 500 bootstrapped scenarios of 136 periods converged and are therefore used here. We ascribe this to solution algorithm weakness; we could apply effort to each simulation to arrange solution but clearly this would be very time-intensive. Meanwhile we should note that using this restricted set of bootstraps biases the variances of the model simulations downwards and also the size of the 95% confidence bands.

4.4 APPENDIX D: Choosing between two models If one is only interested in one model, (e.g. it alone passes all the relevant theoretical tests of acceptability), then one sets the size of test at some high level of confidence, such as 95%. This means that the chances of wrongly rejecting this (‘good’) model are small. One implicitly does not care that some other model, were it to be true, would be wrongly rejected in a high percentage of outcomes; such ‘low power’ against such an alternative is not interesting because the alternative is not seriously in contention — that is, a priori one is not entertaining the possibility that it might be true. Suppose however the situation is that there are two models each of which passes the above theoretical hurdles and so must be treated as equally valid and therefore possibly true. Then one is interested in the power of the test as well as the size; that is, one wants the chances of each model to be rejected to be equal for in this case the chances of being wrong in one’s choice of model are equalised across the two models. (The power=size of the test for each model). This situation is illustrated below for the case where there is just one parameter for the data — say a single autoregressive parameter, ρ .

47

The reason for this situation is that one would wish to treat each model in the same way; that is, one would want its chance of being wrongly rejected (i.e. rejected when it is true) and wrongly accepted (i.e. the other model being rejected when it is true) to be the same as for the other model. This is uniquely the case when the test’s size is set equal for each model. One can think of choosing a critical ρ * such that the two tails of the two distributions have an equal percentage coverage. It is also clear that an equivalent way of carrying out such a test with such a critical value would be to accept the model for which the actually estimated ? cuts off the highest percentage of the distribution. This will occur for model 1 if ρ lies to the left of ρ * and for model 2 if ρ lies to the right of ρ *.

48

Now suppose that the data is encapsulated by a large number of parameters (e.g. of a VECM). Plainly in this case selecting a critical vector of these parameters will be difficult to calculate. The reason is that for each parameter p the p * will occur at in principle different levels of confidence. Thus one will be unable to find a unique level of confidence for all parameters such that it is equalised across the two models. This is illustrated below. What in principle one would like to find is the vector of the p s that encloses an equal percentage of the joint distributions. This is however numerically difficult to evaluate. Instead however we can use the equivalent principle that the model for which the actual vector of the p s cuts off the highest percentage of the joint distribution is the model to choose. If there are more than two models, then they can be evaluated by the above principle unambiguously pairwise. Illustrated below is the situation with 3 models and 1 data parameter. Here there are 3 critical pairwise comparisons. With πˆ given as shown, the models are ranked as follows: 2,3,1. Again the ranking principle is according to the percentage of each distribution cut off by πˆ .In this case it cuts off a large percentage (about half) of model 2’s distribution, about 5% of model 3’s and none of model 1’s.

49

Some conclusions on model choice We have seen that if we have a universe of models that are admitted to that universe on some principle and exclusively populate it, then we can test between these models comprehensively and obtain an unambiguous ranking. The last point we should clarify concerns the nature of the universe in the context of macro models. Here we can distinguish between different types of model — or ‘generations’. Examples would be 1) IS/LM/PP (Phillips Curve) models which should be compared only with other models at the same level of aggregation — i.e. supply and demand curves. 2) Micro-founded models (closed economy), such as RBC and RBC plus various sorts of nominal rigidities. 3) Micro-founded open economy models. In each case we should only compare like with like. For example were we to compare an IS/LM/PP model at level 1) with a micro-founded model at level 2), the comparison would be between different versions of the same model: an IS/LM/PP model is derived from a micro-founded model and yet is less restricted by theory because its parameters are various combinations of ‘deep structural’ parameters whereas the formers parameters are just the structural parameters.

50

4.5 APPENDIX E: Data Generation Process (DGP) Approximation Under revision Stationarity Test The followed methodology aiming to approximate the true DGP is described in the present section. It should be empathised from the beginning that none restriction raised by the economic theory will be imposed in this study. The analysis proceeds by following these steps dictated by the econometric theory. The first of these steps is to investigate whether or not the above mentioned series contain a stochastic trend. The test used for this purpose is the Phillips Perron (1986) (PP) Test 11 that uses Heteroscedasticity Autocorrelation Consistent (HAC) Covariance Matrix12 (introduced by White (1982)). The data used here, from 1970 until 2003, covers a broad width of the British economic evolution, that would allow more general forms of dependency, such as heteroscedasticity (either conditional or unconditional) and not only serial correlation a typical characteristic of the economic time series. Obviously there is the necessity for a test that incorporates this information during the estimation and provides a valid inference for the stochastic nature of the series studied here. The test is applied by including time trend and intercept if any of them is statistically insignificant then it is excluded and the test is applied again. The HAC estimator used is Andrews and Monahan13 (AM) (1992)

)

(

T −1  T   j  Sˆ = γˆ j, T + γˆ j,T ′  (1) γˆ0,T + ∑ k   T −k  j =1  q + 1  

Where

γˆ j ,T is the autocovariance function and k ( z ) is the quadratic spectral kernel

k ( z) =

3

( 6π z 5 )

2

 sin ( 6π z 5 )  − cos ( 6π z 5)  (2)   6π z 5 

used by another estimator introduced by Andrews 1991., An issue that arises by the use of these kinds of covariance estimators is the choice of bandwidth at which should be extended until the dependency to be captured by the estimator. The latter estimator the AM HAC estimator provides some degrees of freedom regarding this issue, it selects the bandwidth automatically. For instance the bandwidth is chosen, say q = 5 , then by using the above formulas we can see that

T Sˆ = T −k

(

)

(

)

(

)

γˆ + 0.944 γˆ + γˆ ′ + 0.790 γˆ + γˆ ′ + 0.574 γˆ + γˆ ′ + ... (3) 1,T 1,T 2 ,T 2,T 3,T 3,T  0,T 

Even though (1) makes use of all computed autocovariances, there is still a bandwidth parameter q to be chosen for constructing kernel. Table A1 and A2 illustrate the result of this procedure and provides the conclusion that the null hypothesis cannot be rejected for the levels of these series but it is rejected for their first differences. In other words our series a integrated of order one

( I (1) )

something that was fully

anticipated. Specification of Time Series Model Since the above results indicate that our series are non stationary the analysis to proceed requires one more time series property named cointegration. The existence of cointegration will be studied into a multivariate time series framework. The authors choose as the starting point of this procedure, which will eventually, gives us the multivariate time series model that will be used for our simulation exercise, a Vector Error Correction Model (VECM) instead of a Vector Autoregressive Model (VAR). Certainly, this choice is vulnerable to Sim’s (1982) criticism; however, our choice is

11

For details see Hamilton chapter XXX For details see Hamilton chapter XXX 13 For details see Hamilton chapter XXX 12

51

based on two reasons as they arise by Jacobson, etc (2001). They also justified their choice of VECM instead of an unrestricted VAR by adducing two very important econometric arguments. One is that the specifications tests applied here are valid for stationary process only. While the second argument is based on the fact that their (tests) asymptotic distributions may constitute a very poor approximation to the small sample distribution required for reliable inference. This is a problem that can be resolved by a small sample study which requires the application of bootstrapping techniques which in turn require stationary series. Since no prior economic theory is used, we do not have any idea regarding this multivariate model (except the fact that it is going to be based on VECM). Given the uncertainty for this specification process (the determination of the DGP) and our willingness to reduce this at the minimum level, we would accept as satisfactory description of the above mentioned series only that model for which the estimated residuals, obtained by fitting this to the data, convey some very desirable properties such no serial correlation and no heteroscedasticity (both conditional and unconditional). Normality, here is not an issue sine the candidate models will be estimated by using Maximum Likelihood (ML) techniques 14. There is no doubt that most of test statistics used here for the determination of the best fitting time series model have extensively been studied by theoretical econometricians who derived their asymptotic distributions. However, they all agree to the fact that there may exist a lot of bias (rejecting the null) in the small samples under the existence of the cointegration. The present analysis to overcome this problem contacts a small sample study for every test applied in this section15. The process described below is applied to five different functional forms of the VECM process. We use Saikkonen and Lutkepohl (1999) terminology for all these “5 Cases” as they are listed to the above mentioned paper. Any of these five cases aims to capture specific characteristics of the data. Since no prior economic theory is used and for consistency of this study all these “5 Cases” should be investigated. However, it was shown above (unit root test) that a number of these series contain independently of the stochastic trend, a linear trend as well. Immediately, one would expect that these functional forms of the VECM model that do not allow for linear trend in the data (“Case 1”16 and “Case 2” 17) would not fit to the data satisfactory. On the other hand “Case 5” 18 can generate quadratic determinist trends, something that does not characterise macroeconomic time series (Johansen, 1995). The last “Case 3”19 and “Case 4”20 that allow for linear trend in the data and do not pose quadratic deterministic trend would be expected to provide a better description of the data. This turns to be the case, “Case 1”, “Case 2” and “Case 5” cannot provide us “clean” estimated residuals, probably because of the above mentioned characteristics. “Case 3” seems to provide a very good explanation of the data while “Case 4” does even better.

( )

(

)

Our vector yt consists of five series 5 ×1 , log of output, log of unemployment, inflation, nominal interest rates and nominal exchange rates. All the sets of multivariate specification tests, which will be described shortly (for a full discussion see the appendix XXX) below, are applied for a number of lags p ∈ [17] ⊂ Ν and for a number of cointegration relationships r ∈ [17] ⊂ Ν (since

yt is ( 5 ×1) then the maximum number of cointegration relationships is 4). The idea of this testing

procedure and small sample study is exactly the same as this described by Jacobson and etc (2001). However, the present analysis has extended the number of specifications test applied here and, additionally, includes two more tests aiming to investigate the stability of the proposed model.

14

All the estimation & simulations routines taken place in this work have been done by using Matlab 6.5 and be downloaded (including the data) by http://www.cf.ac.uk/carbs/econ/XXX. We have not tested; however, we believe that these routines would work with earlier versions of Matlab. 15 The study uses a considerable amount of statistics, which are applied for a number of alternative VECM and for a great number of samples. The completion of this exercise requires a significant amount of time. 16 17 18 19 20

∆yt = a β ' yt −1 + zt for more details see Saikkonen and Lutkepohl (1999)

∆yt = a ( β ' yt−1 + δ ) + zt for more details see Saikkonen and Lutkepohl (1999) ∆yt =v0 + v1t + aβ ' yt −1 + zt for more details see Saikkonen and Lutkepohl (1999) ∆yt = v + a β ' yt−1 + z t for more details see Saikkonen and Lutkepohl (1999)

∆yt = v + a ( β ' yt −1 + τ ( t− 1) ) + zt for more details see Saikkonen and Lutkepohl (1999) 52

Before saying anything for the tests applied here it should be explained the way whereby the small sample study is contacted. It was mentioned above that the existence of cointegration has a consequence the null hypothesis to be over rejected (and the null here is no serial correlation, no heteroscedasticity (conditional and unconditional) no normality and no instability).Therefore, the authors next to Probability Values (p values) implied by the asymptotic theory provide the p values implied by this study. A number (49999) of pseudo samples where the null hypothesis has been imposed are generated by using parametric bootstrap. The parameters of the VECM model for p ∈ [17] ⊂ Ν and r ∈ [17] ⊂ Ν have been estimated by the data and the residuals are replaced 49999

 j  ˆ  where Σˆ is the estimated by a sequence of random vector  ut  obtained by N ~  0 , Σ 5×1) ( 5×5 )  (   (5×1)  j=1 49999

 j covariance matrix. This sequence  ut  and the parametric format for all “Cases” and for all  (5×1)  j=1 p ∈ [17] ⊂ Ν for all r ∈ [17] ⊂ Ν are used to contract the sequence of the pseudo samples 49999

 j  ∆yt   ( 5×1)  j=1

and these sets are used to approximate the distribution of the following statistics.

Briefly, two tests are applied to investigate normality; three tests are applied for serial correlation, one for heteroscedasticity, one for conditional heteroscedasticity, three selection criteria, one test to investigate the number of the cointegration relationships and two tests studying the structural stability of the model21. The application of all these tests has as unique aim the robustness of this study. The asymptotic behaviour of every statistic into the same category is special. For example, for two of the three tests applied for serial correlation the null hypothesis cannot be rejected for a VECM model with four lags and two cointegration relationships while the null is rejected for the third. We believe that a proper specified model should pass all these tests for all categories. The fact that actual data is observed quarterly is used to determine the number of lags for the application of serial correlation tests. All the tests employed for this purpose are obtained by Bruggemann, Lutkepohl and Saikkonen’s (2004) work. The first test studied by them is BreuschGodfrey Lagrange Multipliers (LM) test

(Q = Tcˆ′ ( I

hK

) ( (

* d QBG  → χ 2 hK 2

(Proposition 1) to have an asymptotic distribution

))

ˆ −1 Σγγ I ⊗ Ω ˆ −1 cˆ which is shown ⊗Ω hK

)

22

. Proposition 1 requires

the cointegration rank to be known (something that happens here since we impose it), however, the authors show that this is not necessary. Two asymptotically equivalent tests, the likelihood ratio (LR)

(Q

LR

( ( ( ))

( ( ) ))) and Wald Test (Q

ˆ − log det Ω ˆe = T log det Ω

LR

( (

) ))

ˆ −1Ω ˆ −K = T tr Ω e

23

are

utilized for the same purpose. Table 4 presents the results for these statistics and shows that the null hypothesis of non serial correlation for a model with four lags and either three or four cointegrated vector cannot be rejected. Based only on the parsimonious principle this would be a candidate specification however, as it is indicated by the Maximum Eigenvalue Statistic — Table A5 — the number of the steady state equilibrium (or cointegration relationships) for a model with 4 lags and time trend including both in data and error correction term (ECM) cannot be more than one, which contradicts with the above specification The next step is to investigate whether or not the estimated residuals for all these model specifications are homoscedastic. The first test used for this purpose, which is a multivariate version of White’s (1982) test for heteroscedasticity, is described by Doornik (1996). A vector (in vec24

21

A short description of all these tests could be found into the appendix XXX. A more detailed discussion could be founded into Lutkepohl and Kratzig 2004 and to the references mentioned for each test. 22 h is the number of the maximum lag aiming to capture the serial correlation in the data in the present case h=4 and K is the dimension of the vector yt equals to 5 23

The operator (tr) denotes the trace of the matrix and

ˆ e is the covariance matrix for the unrestricted model Ω

ˆ for the restricted one. while Ω 24 vec is used for column vectorozation operator. 53

format) of squares and cross product of the estimated residuals is regressed on a matrix which is block diagonal and each block consists of different function of regressors 25 (see appendix XXX). Authors numerical advises for the application of this statistic facilitate its implementation a lot. The second test (Lutkepohl and Kratzig, 2004) utilised in this paragraph is a chi square

(χ ) 2

( hK ( K + 1) 4) degrees of freedom (see the 2

2

asymptotically distributed ARCH-LM test with

appendix XXX). Table A6 suggests that the null hypothesis that the errors are homoscedastic is rejected for any specification. This leads to the conclusion that a linear VECM cannot explain the actual observed data efficiently. It seems that higher order dynamics existed in the data cannot be captured by the proposed linear multivariate model. However, it was shown (by Table A4 and Table A5) that the following model

  ∆yt = c + a  β ′ yt −1 −τ ( t − 1) + ( 5× 2)  ( 2×5) 

6

∑ Γ ∆y i =1

i

t− i

+ ut

does not suffer from serial correlation, shortly it will be obvious how this characteristic is utilised. A possible reason for the failure of the above model could the fact that the parameters are not allowed to change regarding the evolution of the state of the economy. An alternative specification that would incorporate this element is known as Markov Switching Vector Error Correction (MSVEC) 26

  6 ∆yt = cst + ast  β s′t yt−1 − τ ss ( t −1)  + ∑ Γ st ,i ∆yt− i + ut   i=1 (5×2)  ( 2×5)  the parameters and the covariance matrix state is assumed to follow a Markov Process

(u ~ N (0, Σ )) here are state dependent and the t

(s

t

st

= pst −1 + vt with p < 1) . Clearly, it is necessary

the hypothesis that parameters of the linear VECM are constant to be tested. Fortunately, tests introduced by Hansen and Johansen (1999) based on recursive estimation of the eigenvalues answer to the above question. The first statistic that checks of parameters’ compares the ith eigenvalue obtained from the full sample to the one estimated from the first t observation only. The above authors show that this statistic has a limiting distribution that depends on a Brownian bridge and is tabulated by Ploberger, Kramer and Kontrus (1989). This test requires a number of observations until the first recursively estimated eigenvalues to be obtained. In the present case there are thirty seven parameters in each equation plus the fact that six observations have been used as initial values (since we have six lags). The following figure presents these estimates — starting from 1980 quarter 3 — and their asymptotic value. Clearly, early in eighties the estimates of the second eigenvalue show some instability (the dashed live exceeds the solid line).

X = ( β ′ yt −1 )′ ∆yt −1′ ....∆yt −p′  is the matrix of the regressors and the function of them considered here   are X and X.X where . denotes element -vector- by element multiplication. 25

26

For more details see Krolzig 1997 54

Hansen & Johansen Tau Statistic for the First Eigenvalue 1.5

1

0.5 Estimated Asymptotic 0 1980

1985

1990

1995

2000

2005

Hansen & Johansen Tau Statistic for the Second Eigenvalue 2

1.5

1

0.5 Estimated Asymptotic 0 1980

1985

1990

1995

2000

2005

Since the error assumptions (independent, Gaussian with mean zero and constant covariance matrix) made by the authors are strongly violated (see above discussion) a small sample investigation is contacted to this statistic as well. The simulated probability values show that the null hypothesis cannot be rejected Hanses & Johansen Tau statistic for the First Eigenvalue 3 2.5 2 1.5 1 0.5 0 1980

Estimated Bootstraped 1985

1990

1995

2000

2005

Hanses & Johansen Tau statistic for the Second Eigenvalue 2

1.5

1

0.5 Estimated Bootstraped 0 1980

1985

1990

1995

2000

2005

which implies that the model given by equation (XXX) cannot be supported by the data as well. The last try to address heteroscedasticity in the present model is to fit in the errors process another process that has the ability to capture the fact that empirical volatilities tend to cluster (columns three and four of Table A6 support this idea). For this purpose a Multivariate Generalised

55

Autoregressive Conditional Heteroscedastic (MGARCH (p,q)) process is fitted into the errors

ut = H t1 / 2ε t , ( ε t is an i.i.d vector).GARCH model is strongly related with high frequency data and financial studies, however, in macroeconomics it has been argued 27 that price level uncertainty may affect the behaviour of other series. In the literature varies functional forms of the MGARCH (r,q) process has been proposed. The present analysis used BEKK 28 q

p

i =1

i =1

′ ′ ′ Ht = CC′ + ∑ Au i t − iut − iAi + ∑ Gi H t −i Gi — introduced by Engel and Kroner (1995) — for two reasons. Firstly, under very weak conditions29 (at least one of the matrixes C or Gi has full rank) H t is positive definite and secondly, the BEKK representation allows for direct dependence pf the conditional variance of one variable on the history of other variables within the system. In multivariate systems the number of the parameter that should be estimated increases dramatically as p and q increases

 n ( n + 1)  + pn 2 + qn 2  . In the present analysis a GARCH (1,  2  

1)30 is adopted with the hope that this will be a sufficient to explain the actual observed data satisfactory. The entire (time series) model estimated here, after the acceptance of the GARCH process, is summarised y the following equations:

  ∆yt = c + a  β ′ yt −1 −τ ( t − 1) + ( 5× 2)  ( 2×5) 

6

∑ Γ ∆y i =1

i

t− i

+ ut

Ht = CC′ + A1′ut −1 ut′−1 A1 + G1′H t−1G1 Engle and Kroner (1995) mentions that the information is block diagonal between the parameters of the VECM part and the parameters of the covariance equations. This means that efficient estimates of the VECM model can be calculated independently of the parameters of GARCH process and vice versa. They suggest an iterative procedure (adopted here) whereby the estimates will solve the first order conditions for full information maximum likelihood estimation. The estimated residuals of the above model are no more heteroskedastic (Table A7) and in addition the hypothesis of normality cannot be rejected (Table A8). Table A9 shows the eigenvalues

(

) (

)

of matrix A1 ⊗ A1 + G1 ⊗ G1 which all of them have modules less than one, implying that the above process is covariance stationary. Therefore the above specification seems to explain the actual observed data in a satisfactory way. This model will be the benchmark model used for our testing procedure described in the next section.

27

See Fridman (1977)

28

After the work of Baba, Engel, Kraft and Kroner

29

That could be very easily established numerically

30

This implies that sixty parameters should be estimated.

56

4.6 Appendix F: The data Log of Output 12.2 12 11.8 11.6 11.4 11.2 1965

1970

1975

1980

1985

1990

1995

2000

2005

1995

2000

2005

Log of Unemployment 8.5 8 7.5 7 6.5 6 1965

1970

1975

1980

1985

1990

Inflation 0.25 0.2 0.15 0.1 0.05 0 1965

1970

1975

1980

1985

1990

1995

2000

2005

1995

2000

2005

Nominal Interest Rates 0.2 0.15 0.1 0.05 0 1965

1970

1975

1980

1985

57

1990

Nominal Exchange Rates 5.2 5 4.8 4.6 4.4 4.2 1965

1970

1975

1980

1985

1990

1995

2000

2005

Table A1 Phillips-Perron Stationarity Test by Using AM HAC estimator (Levels) Series Trend and Intercept None Intercept Output 0.4264 Unemployment 0.9265 Inflation 0.0794 Interest Rates 0.2498 Exchange Rates 0.1498 Table A2 Phillips-Perron Stationarity Test by Using AM HAC estimator (1st Differences) Series Trend and Intercept None Intercept Output 0.0000 Unemployment 0.0000 Inflation 0.0000 Interest Rates 0.0000 Exchange Rates 0.0000

58

4.7 Appendix G: Liverpool Model Listing Behavioural Equations log(EGt )=log(EG* t ) +A 39 log(Yt /Y* t )

(1)

XVOLt =A 40 Y* t {A 27 {log(WTt )}+A 28 *log(Yt ) +A 47 + A 29 {E* t +0.6{RXRt -E*t }} +A 30 {XVOLt-1 / {A 40 Y* t-1 }}}

(2)

XVALt =XVALt-4 +(XVOLt -XVOLt-4 )+ A 31 {.32Y* t (RXRt -RXRt-4 -E*+E* t-4 )} +A 32 XVALres t-1

(3)

log(M0t )=A 44 +A 13 log(M0t-1 ) +A 14 {log(Yt ) +log(1.-TAXt-1 )} +A 15 UBt-2 +A 16 TREN t +A 17 NRSt +A 18 {VATt }

(4)

log(Ut )=A 42 +A 3 log(Yt ) +A 4 {log(RW t ) +log(1.0 +BOt ) + log(1.0 +VATt )}-0.095(UNRt )-0.04{{TRENt -108.}/4.}D87t +A 5 TREN t +A 6 *log(Ut-1 ) +A 36 Urest-1

(5)

log(Gt )=A 45 +A 19 RLt +A 20 {log(Gt-1 )-log(FINt-1 )} +A 21 {log(Gt-1 )-log(Gt-2 )} +log(Gt-1 )

(6)

log(Ct )=A 46 +A 22 RLt +A 23 log(Wt ) +A 24 Et Qt+1 +A 25 log(Ct-1 )

(7)

log(RW t )=A 43 +A 7 UNRt +A 8 {log(UBt )+log(1.0 +LO t } +A 9 log(Ut ) +A 37 log(RW t-1 ) +0.095{UNRt }{-A 10 } +{0.04/4.0}{TREN t -108.}D87t {-A 10 } +A 10 log(RW t-2 ) +A 11 ETA t +A 12 ETA t-1

(8)

RXRt =A 41 +A 1 {log(RW t ) +log(1.0 +BOt )} +A 53 {log(Pt )-log(Pt-4 )} +{1.0+A 1 }log(1. +VATt ) +A 2 TREN t +A 35 RXRres t-1 -0.064{UNRt } -{0.0525}{{TRENt -108.}/4.}D87t

(9)

59

Identities and Calibrated Relationships RSt ={RXRt -Et RXRt+1 } +RSUSt

(10)

NRSt = Et INFLt+1 +RSt

(11)

RLt ={RXRt - Et RXRt+20}/5.0 +RLUSt

(12)



 / 5.0  

(13)

Yt =GINVt +Ct +EGt +XVOLt -AFCt

(14)

INFLt =log(M ONt )-log(MONt-4 )-log(M0t ) +log(M0t-4 )

(15)

log(Pt )=log(Pt-4 ) +INFLt

(16)

W t =FINt +Gt

(17)

BDEF t =EGt -{2.0*TAXt }*Yt +TAX0*Y0

(18)

AFCt =Yt {0.6588318(AFCt-1 /Yt-1 ) +0.1966416(AFCt-3 /Yt-3 ) +0.1454006(AFCt-4 /Yt-4 )}

(19)

PSBRt =BDEFt +RDIt

(20)

RDIt =-0.5{NRLt-1 /4.0}FINt-1 {{{Pt /Pt-1 }0.66 }-1.0} +PSBRt {0.32{NRSt /4.0} +0.5{NRLt /4.0}} +0.32{NRSt /4.}FINt-1 -0.32{NRSt-1 /4.}FINt-1 +RDIt-1

(21)

GINVt =Gt -Gt-1 +A 38 *Gt-1

(22)

5

∑ INFL 

NRLt =RLt +Et 

i =1

t+ i

FINt =EGt -Yt {TAXt } +XVALt +A 54 FINt-1 +{1.-A 54 }{FINt-1 {{Pt-1 /Pt }0.66}} {1.0-0.155{{NRLt /NRLt-1 }-1.0}} +RDIt

(23)

Equilibrium Variable (-star) The -star variables YSTAR, USTAR, ESTAR and WSTAR are the equilibrium values of Y, U, RXR and RW respectively, found by solving equations 2,5,8 and 9 under the conditions that XVOL=0 and exogenous variables maintain their current values; EGSTAR is the value of EG that would produce a constant debt/GDP ratio with Y=YSTAR. Coefficient values in order A1-56: 1.528 -0.290 0.640 0.103 0.170 0.011

-0.003 0.189 -0.215 0.193 25.26 0.750

-2.150 0.870 0.056 0.000 0.102 -0.750

0.792 0.150 0.153 0.000 -0.337 0.300

0.010 0.000 0.870 0.931 0.013 -1.000

0.804 -0.002 0.000 0.271 0.666 -1.000

0.470 -0.349 0.529 1.000 11.50

60

0.210 0.839 -1.205 0.012 -0.016

-0.018 -0.016 -0.388 -0.125 -0.011

-0.224 -0.004 0.429 0.320 0.017

(Exogenous variables — e = error term) RSUSt =c+0.899 RSUSt-1 +e t EUNRSt =c+0.997 EUNRSt-1 +e t ∆ BOt =c+e t ∆ VATt =c -0.286 ∆ VATt-1 +et ∆ UNRt =c+0.869 ∆ UNRt-1 +et ∆ UBt =c+e t ∆ LOt =c+e t ∆ TAXt =c-0.365 ∆ TAXt-1 +et ∆ logMONt =PEQ t +MTEM t +et Model Notation Endogenous Variable Y XVOL C GINV EG AFC DY G FIN W NRL NRS RL RS P INFL XVAL MON M0 RXRN RXR AE RW Y** E** W** U** U Q PSBR RDI YAFC EG* UR* BDEF

GDP at factor cost Net exports of goods and services Non-Durable Consumption Private Sector Gross Investment Expenditure Government Expenditure Adjustment to Factor Cost Percentage growth rate of Y (year-on-year) Stock of Physical Goods Financial Assets Wealth Nominal 5 year interest rate Nominal deposits with local authorities (3 month) Real 5 year interest rate Real deposits with local authorities (3 month) Consumer Price Level Percentage growth rate of P (year-on-year) Current account deflated by P Nominal Money Stock (M0) MON divided by P Trade-weighted exchange rate Real exchange rate Average Earnings Real wages (AE/P) Equilibrium output Equilibrium real exchange rate Equilibrium real wage Equilibrium unemployment Unemployment Output deviation from trend (Y/Y**) Public Sector Borrowing Requirement Real Debt Interest GDP at market prices Equilibrium government expenditure Equilibrium unemployment rate Budget deficit

Exogenous Variable TREND TAX WT RLUS PEQ RSUS MTEM BO LO UB UNR POP RPI

Time trend Tax Rate Volume of world trade Foreign real long interest rate Growth of money supply Foreign real short-term interest rate Temporary growth of money supply Employers national insurance contributions Average amount lost in taxes and national insurance Unemployment benefit Trade Unionisation rate Working population Growth of retail prices (year-on-year) 61