Structural Breakpoint Tests in a Markov-Switching ... - Semantic Scholar

Structural Breakpoint Tests in a Markov-Switching Model: An Empirical Application to the EMU Member Countries Cristina Fuentes-Albero September 20, 2006

Abstract This paper aims to estimate a dynamic non-linear model with statedependent mean and volatility for the real GDP growth series of several EMU member countries. We test the existence of a structural breakpoint at the starting point of the monetary union. We reject the null of parameter constancy for the Markov-switching model for only eight countries. Given the small length of the subsample after monetary union available for analysis, we proceed to analyze the empirical power of our test for a grid of alternatives. We conclude that our test is powerful enough if the null and the alternative are sufficiently far apart.

1 Introduction The third and last stage of the European Monetary Union (EMU) started on January 1, 1999, when eleven countries1 fixed their exchange rates and gave up their national monetary policies. Having a supranational monetary authority developing a common monetary policy for twelve countries will have asymmetric effects on them unless they have a common business cycle that is completely synchronized. Some authors, like Krolzig (2001, 2003), Krolzig and Toro (2001), Artis et al. (2004), and Anas et al. (2004), have analyzed the existence, identification, and dating of the so called Euro-zone business cycle. They use Markov switching models to account for the non-linearity of the business cycle and the asymmetries in the dynamics of recessions and expansions. Krolzig proposes a Markov switching vector autoregressive model (MS-VAR(p)) which is a multivariate version of the Markov switching model proposed by Hamilton (1989). Krozlig constructs an MS-VAR for the EMU members 1 Austria (AT), Belgium (BE), Finland (FI), France (FR), Germany (GE), Ireland (IE), Italy (IT), Luxemburg (LU), Netherlands (NL), Portugal (PT), and Spain (ES). Greece (GR) joined the EMU in 2001.

1

such that the parameters depend on a common Markov chain. This common latent variable represents the phases of the common cycle. As Anas et al. (2004) point out, using a MS-VAR does not allow us to evaluate the relationship among phases in the different economies under analysis. Moreover, even if a common business cycle for the EMU members exists, we cannot conclude that all the countries are perfectly synchronized. Hence, a more complete analysis would require a Multiple Markov switching VAR (see Anas et al (2004)). Some authors have proposed alternative generalizations of the univariate Markov-switching model. For example, Diebold and Rudebusch (1996) introduce the switching dynamics using a common latent factor in order to have an econometric model that deals with the two defining characteristics of business cycles (Burns and Mitchell (1946)): commovement and asymmetry2 . In this paper we are interested in focusing on heterogeneity. Seven years after the start of the monetary union, there are some data available that allow us to analyze the cost or benefit derived from joining it. We have observed unsatisfactory post-EMU growth performance for some members (see (8) in the appendix) but not for others. This has raised the debate among policy makers and economists about the heterogeneity of economic performance of countries in the EMU. For a detailed discussion see Monegelli and Vega (2006). In this paper we want to analyze whether the monetary union has translated into a structural change in the growth series. In particular, we are interested in testing the existence of a structural breakpoint affecting the mean and volatility of real GDP growth. We will be analyzing the business cycle of each country3 using univariate data-driven models with a latent variable that follows a two state, first order Markov process. The paper proceeds as follows. In section 2, we provide the results of the BoxJenkins analysis of the data. In section 3, we do a classical analysis of the Markov switching model. In section 4, we test the existence of a structural breakpoint linked to the start of the third stage of the EMU. We provide two versions of the structural stability test for our non-linear model. In section 5, we perform an empirical analysis of the power of the tests. Section 6 concludes.

2 Data and ARIMA analysis We will consider the seasonally adjusted quarterly Gross Domestic Product measured in constant 2000 Purchasing Power Parity (PPP) series provided by the Economic Outlook4 of the OECD. The countries under analysis are eleven European Monetary Union

2 For a discussion on Burns and Mitchell’s method and its statistical equivalent, see Harding and Pagan (1999). 3 We will also report the analysis for the aggregate. 4 The Economic Outlook is based on the information provided by the Quarterly National Accounts. How-

2

(EMU) members: Belgium, Finland, France, Germany5 , Greece, Ireland, Italy, Luxembourg, Netherlands, Portugal, and Spain. Austria is not analyzed since the series starts in 1995 which is insufficient to proceed with our analysis. The data consists of quarterly observations for the period 1970:1-2005:4, which is 144 observations. The EMU aggregate data are extracted from the Quarterly National Accounts. Since this series starts at 1980:1, we only have 104 observations. The presence of unit roots in the data can be tested by using the Augmented DickeyFuller (1981) test. In particular, we cannot reject the existence of unit roots for the series in levels. However, such a hypothesis is rejected for all the growth rate series where the real GDP growth rates are computed as log differences: y˙ t = [ln (yt ) − ln (yt−1 )] ∗ 100 where t = 1970 : 2, ..., 2005 : 4

(1)

We proceed to do a standard Box-Jenkins analysis of the twelve series6 . We have chosen the ARMA specification that minimizes the Schwarz (Bayesian Information) criterion ¡ ¢ BIC = T k/T s2y 1 − R2 =

T k/T

u0 u T

The specifications chosen are: - AR(1) for Italy (IT) - AR(2) for Belgium (BE) and Finland (FI) - AR(3) for France (FR), Ireland (IE), Luxembourg (LU), Netherlands (NL), Portugal (PT), and Spain (SP) - AR(4) for Germany (GE), Greece (GR), and EMU

3 Markov-Switching Models: A Classical Analysis 3.1

The model

Our model is based on Hamilton (1989), but we will be considering that, in addition to the mean, the volatility of the series may also be state dependent. High and low growth ever, the Economic Outlook also provides projections for future periods. We will ignore the projections and consider only the series extracted from the Quarterly National Accounts. 5 Data through 1991:4 apply to west Germany only. Using a Chow test for the linear model we cannot reject the inexistence of a structural break after the effective reunification. Therefore we will consider the high peak at 1992:1 an outlier. Such an outlier has been smoothed away by setting the data point equal to the sample mean. 6 Maximum likelihood estimates using the Kalman filter are available upon request.

3

states of the economy are modeled as switching regimes of a stochastic process generating the growth rate of real output. Hence, we will be considering an autorregressive process of order p with first-order, two-state Markov-switching mean and variance ³ y˙ t − µSt = φ1

´ y˙ t−1 − µSt−1

³ + ... + φk

´ y˙ t−p − µSt−p

+ εt

(2)

¢ ¡ where εt ∼iid N 0, σ 2St . The latent variable is the unobserved regime of the economy so that St = 0 in the low growth regime and St = 1 in the high growth one. Such a latent variable is generated by a stochastic process governed by a first order Markov chain with transition matrix · ¸ p00 p10 Γ= (3) p01 p11 where pij = P {St = j | St−1 = i} and

X

pij = 1

j

Hence, our model is such that within a given regime, the dynamics of the observed data are determined by a conventional dynamic model -the AR(p) . Note that in our model we do not assess the turning points or the size of the states of the economy. Moreover, we do not constrain the mean of the low-growth regime to be negative when estimating the model. Therefore, we consider economic cycles in a broad dimension; we do not restrict our analysis to study classical cycles but growth ones7 . We estimate the model by maximum likelihood using Hamilton’s algorithm to compute the filtered probabilities associated to the two states of the economy and Kim’s algorithm to compute the smoothed probabilities. Since the likelihood functions can have numerous local optima, our estimation results can be sensitive to the initial parameter values. We proceed as follows: φj , j = 1, ..., p are set equal to the estimates obtained using the Kalman filter on the linear model, and pii , µi , and σ 2i are set to the sample moments8 . Constraints were placed on pii to restrict them to belong to [0, 1]. A complete explanation of the estimation procedure including the Matlab codes are available upon request. We follow the procedure specified in chapter 4 of Kim and Nelson (1999). Nevertheless, we will provide a brief explanation of the computation of the filtered and the smoothed regime probabilities. We report in the appendix the estimates obtained for the whole sample, the first subsample (1970:2-1998:4), and the second subsample (1999:1-2005:4). 7 If we focused on classical cycles, we would have very few cycles for the countries under analysis. Therefore, we prefer to focus on growth cycles. 8 Other strategies could be explored like the one proposed by Goodwin (1993).

4

3.1.1 Filtered Probabilities As in Hamilton (1989), the state of the economy is unobservable. However, the probability of a particular state can be computed given the estimated parameters and the data currently available. In particular, we will use the Hamilton filter to compute the filtered probabilities of each state of the economy. The main objective of the filter is to infer the probability distribution of the latent variable given current © information (ψ t ). That ªis, we are interested in computing P {St | ψ t } . Given P St−1 , St−2 , ..., St−p | ψ t−1 , we have that © ª © ª P St , St−1 , St−2 , ..., St−p | ψ t−1 = P {St | St−1 }·P St−1 , St−2 , ..., St−p | ψ t−1 where P {St | St−1 } is the transition probability9 . Once y˙ t is observed, we can use Bayes’ Law to update the probability © ª P {St , St−1 , St−2 , ..., St−p | ψ t } = P St , St−1 , St−2 , ..., St−p | y˙ t , ψ t−1 ¡ ¢ © ª f y˙ t | St , St−1 , St−2 , ..., St−p , ψ t−1 · P St , St−1 , St−2 , ..., St−p | ψ t−1 ¡ ¢ = f y˙ t | ψ t−1 where ¡ ¢ ¸ · ¡ ¢ X X X f y˙ t | St , St−1 , St−2 , ..., St−p , ψ t−1 · © ª f y˙ t | ψ t−1 = ... P St , St−1 , St−2 , ..., St−p | ψ t−1 St St−1

St−p

i.e. the marginal density of y˙ t conditional on the information set ψ t−1 is just a weighted average of conditional densities. Note that ¡ ¢ f y˙ t | St , St−1 , St−2 , ..., St−p , ψ t−1 is easily computed for Gaussian processes. Finally, to get P {St | ψ t } we only need to integrate out {St−1 , St−2 , ..., St−p } X X P {St | ψ t } = ... P {St , St−1 , St−2 , ..., St−p | ψ t } (4) St−1

St−p

We initialize the filter with the steady state probabilities Π. Definition 1 Let Πt be a (2x1) vector µ ¶ π 0t Πt = 3 i02 Πt = 1 π 1t 0

where π 0t = P {St = 0}, π 1t = P {St = 1} and i2 = [1, 1] . Πt is a steady-state probability vector if Πt+1 = Γ Πt (5) 9 As we are dealing with a first-order Markov process, only the previous state matters i.e. the past history is irrelevant.

5

and Πt+1 = Πt

(6)

From (5) and (6) we get ·

(I2 − Γ) Πt ¸ I2 − Γ Πt i2

=

A Πt

=

02 ·

= ·

Hence, the steady state probabilities will be given by · ¸ 02 −1 Πt = (A0 A) A0 1

02 1 02 1

¸ ¸

∀t

(7)

3.1.2 Smoothed Probabilities Smoothed probabilities are inferences about the regime conditional on all the information available in the sample. Therefore, we will compute P {St | ψ T } . Note that whereas the Hamilton filter gives inferences on St conditional only on current information, the smoothed probabilities use the whole sample to do inference. Hence, smoothed probabilities are a better estimate of the latent variable10 . We will use the smoothing algorithm proposed by Kim (1994). The algorithm is initialized with the output of the last iteration of the Hamilton filter P {ST | ψ T } and it iterates backward for t = T −1, ..., 1. So, in each iteration we will compute

= =

P {St−p+1 , ..., St , St+1 | ψ T } P {St−p+2 , ..., St , St+1 | ψ T } · P {St−p+1 , ..., St , St+1 | ψ t } P {St−p+2 , ..., St , St+1 | ψ t } P {St−p+2 , ..., St , St+1 | ψ T } · P {St−p+1 , ..., St | ψ t } · P {St+1 |Pt } P {St−p+2 , ..., St , St+1 | ψ t }

4 Structural breakpoint tests Authors such as Hansen (1992, 1996), Hamilton (1996), and Garcia (1998) have analyzed the performance of tests in Markov-switching models. However, they have focused on testing the linear model against the Markov-switching model. As Coe (2002) points out, such tests present two problems. First of all, there are some parameters that 10

In particular, to asses the current observation to a regime, we will use a decision rule based on the smoothed probabilities. Sˆt = arg max P {St | ψ T } 1,2

6

are not identifiable under the null; in particular, the transition probabilities. Second, under the null, the scores with respect to the nuisance parameters and the parameters associated with the second regime of the economy are zero. Both problems imply that the likelihood ratio test statistic is not distributed as the standard Chi-squared distribution. Hansen (1992,1996) and Garcia (1998) propose different approaches to deal with such problems. Garcia’s (1998) approach presents two advantages with respect to the one proposed by Hansen: it provides a critical value for the likelihood ratio test statistic rather than a bound, and it is less computationally burdensome. Carrasco et al (2004) propose to test parameter constancy against Markov-switching, but in a broader environment since the model under the null need not be linear. As their test statistic depends only on the score, the derivative of the score under the null and on the parameter vector (free of nuisance parameters) under the null, they do not have to estimate under the alternative. This makes the procedure even less computationally intensive. They provide a Monte Carlo study to compare the power of their test statistic and Garcia’s. They conclude their test statistic has a slightly higher power. As long as we think the starting of the EMU can translate into some effect on the growth performance of the member countires, we will perform structural breakpoint tests using the above non-linear model. In particular, we will be doing two different versions of a structural stability test where the model is a univariate Markov-switching model under both the null and the alternative: 1. The null hypothesis is θpre = θpost ¢0 ¡ i.e. there is no structural break. Where θ = p, q, Φ, σ 20 , σ 21 , µ0 , µ1 and ¡ ¢0 Φ = φ1 , ..., φp . The alternative is θpre 6= θpost 2. The null is

˜θpre = ˜θpost ¢0 ¡ where ˜θ = Φ, σ 20 , σ 21 , µ0 , µ1 i.e. here we will proceed assuming the transition matrix Γ is time invariant and more importantly not subject to a structural break. We do this because some countries are only in one regime after 1999:1, which translates into inaccuracy of the estimate for the transition matrix since the data is uninformative about it. We will conduct the estimation by setting the transition probabilities to their estimated value obtained in section 3 by using the whole sample.

We will test the parameter constancy using the Likelihood Ratio (LR) test. The LR test statistic is based on the comparison of the restricted and the unrestricted maximum of the Gaussian log-likelihood function. LR = −2 [ln (LR ) − ln (LU )] ∼ X (m−1)k where k stands for the number of restrictions and m for the number of subsamples under analysis. 7

The restricted model is identical for the two versions we are going to analyze. The restricted maximum of the Gaussian log-likelihood is obtained by performing numerical optimization of the log-likelihood function using the whole sample. The unrestricted log-likelihood value is obtained by adding the maximum of the Gaussian log-likelihood for the first subsample and the second one. First, we will report the LR test statistic when all the parameters are allowed to change. The a stands for rejection of the null hypothesis at 5% and the b for rejection at 10%. BE FI FR GE GR IE LR 44.46a,b 1.37 5.74 54.71a,b 70.70a,b 140.40a,b LR

IT 28.65a,b

LU 60.83a,b

NL 32.48a,b

PT 57.46a,b

SP EM U 15.27 23.05

Second, we consider the second version of the structural breakpoint test. We impose the transition matrix estimated under the null when computing the log-likelihood associated with the alternative. Therefore, the number of restrictions for each series will be the same as before minus 2.

LR

BE 43.19a,b

FI FR GE 14.50 4.01 102.67a,b

LR

IT 41.60a,b

LU 54.03a,b

NL 16.51

GR 52.45a,b

PT 184.41a,b

IE 181.85a,b

SP 25.34a,b

EM U 19.73

In both versions of the test, we cannot reject the null hypothesis of parameter stability for Finland, France, and the aggregate. We reject the null hypothesis for both significance levels for Belgium, Germany, Greece, Ireland, Italy, Luxembourg, and Portugal. The performance of the Netherlands and Spain is different in each version of the test. From the above we can conclude that the starting of the third stage of the European Monetary Union has translated into a structural break for eight of its members. The conclusions derived for those countries do not depend on the specification used to fit the data series. Belgium has improved its performance during recessions with a larger mean growth and a smaller volatility while the expected length11 of recessions remains invariant. Expansions are characterized by a smaller volatility and a larger expected length in the post-EMU period, but the mean growth is smaller. Germany has also improved in terms of mean and volatility during recessions, however the expected duration of such 11

Conditional on being in the low-growth state, the expected duration of the regime is j Σ∞ j=1 j p00 (1 − p00 ) →

1 1 − p00

Similarly, the expected duration of the high-growth regime is given by j Σ∞ j=1 j p11 (1 − p11 ) →

8

1 1 − p11

a regime is larger. German expansions after the EMU are characterized by a smaller volatility, but the mean and the expected duration are also smaller. Greek recessions are shorter, with a larger mean, and a larger volatility. Expansions are characterized by a similar mean, a smaller variance, and a shorter expected length. Irish expected duration of both regimes does not vary with the starting of the EMU. Ireland has improved its mean in both expansion and recession, however while the volatility is larger in the former, it is smaller in the latter. Italy is a special case since all the variables of interest, i.e. expected duration, mean, and variance, are smaller for both regimes in the post-EMU period. For Luxembourg we obtain that the low growth regime is characterized by a smaller volatility, a larger mean, and a shorter duration. The large growth regime has a larger volatility, a smaller mean, and a shorter duration. Finally, Portuguese recessions are shorter, with a smaller variance, and similar mean. However, expansions are also shorter, with a lower mean and a larger volatility. We have observed diverse effects of the monetary union on its members. Moreover, we cannot, from our analysis, conclude there is a clear winner or loser from joining the union. While we observe that almost all countries of such a group are net winners in terms of low growth cycles, we cannot draw such a conclusion when we consider the high growth regime of the economy.

5 Power of the Structural Breakpoint Test: A Monte Carlo Study Rejecting the null hypothesis implies that the data are inconsistent with each parameter point in the null in the sense that the probability of rejecting the null when it is true (probability of type I error) for each point is small, in particular α or less. However, failing to reject the null does not translate into having all points in the alternative being inconsistent with the data, so that the probability of failing to reject the null when it is false (probability of type II error, β) is small. Therefore, we will proceed to analyze the power (1 − β) of the structural breakpoint tests for those series where we cannot reject the null or our conclusion is different for each version of the test. In this section, we will generate artificial data under the alternative hypothesis using 1000 Monte Carlo replications and sample size equal to 143. Since for an AR (p), the parameter vector is θ ∈ R p+6 , the directions of variation from the null are infinite. Hence, we will simulate for a grid of alternatives and compute the restricted and unrestricted maximum of the Gaussian log-likelihood and the finite sample rejection probabilities of the tests under consideration. In particular, we will analyze the empirical power under three alternatives such that the distance between the parameter vector used to simulate the first subsample(θ1 ) and the one used to simulate the second subsample (θ2 ) is equal to 0.5, 0.2, or 0.05.The parameter vector θ1 for each DGP is set to the corresponding one in (9) , (10) , and (11) . Consequently, θ2 is defined such that kθ1 − θ2 k = ∆

f or ∆ = {0.05, 0.2, 0.5}

In each Monte Carlo replication we will construct the LR test statistic to determine 9

if we can reject the null hypothesis of parameter constancy. We will compute the empirical power of both versions of the structural breakpoint test for our non-linear model and two sizes: α = 0.05 and α = 0.10. The empirical rejection probabilities are reported in (22) . We observe the following regularities. First of all, the test for constancy in all the parameters has equal or larger power than the test for the reduced parameter vector. Second, both tests have high power ((1 − β) ≥ 0.80) against alternatives whose distance with respect to the null is large ˜ ∈ [0, 0.5] such that for any ∆ ≥ ∆, ˜ the power enough i.e. there exists a critical value ∆ ˜ (1 − β) → 0.80 as of the test will be (1 − β) ≥ 0.80. In particular, for any ∆ ≥ ∆, ∆ → 0.5. Finally, the power is a non-decreasing function of ∆ i.e. the empirical rejection probabilities decrease as the distance between θ1 and θ2 vanishes. We have also analyzed the power of the tests for those series for which we clearly reject the null. We have performed 100 Monte Carlo replications when simulating under the alternative. Our results confirm the ones previously obtained. In particular, we conclude that the structural breakpoint test on the whole parameter vector has a larger power than the one on a reduced parameter vector. Therefore, we reject the null of parameter constancy for the Netherlands, but we cannot reject this hypothesis for Spain.

6 Conclusion We have observed that giving up the control over the national monetary policy and following a supranational common policy has different effects in the member countries as a consequence of the heterogeneity among them. Whereas some countries seem to not be affected by joining the EMU, others have experienced changes in the nature of their economic growth. However, these changes are also diverse. The only common factor is the better performance during the low growth regime of the economy. Will our conclusions about such a "supranational adventure" change when a larger sample will be available? Once data availability for the post-EMU period is large enough, we will be able to address issues like possible asymmetries on the dating of the structural breakpoint. In particular, the breakpoint could be considered as unknown so that the current procedure would not be available. Hansen (2001) provides an overview on the development of tests for structural stability with unknown break points. More recently, DeJong et al (2006) have developed a more effective procedure in identifying the timing for structural instability. Further analysis of the results provided could address a sensitivity analysis with respect to the initial parameter vector and a test of the linear dynamic model against the non-linear one proposed here. We should also proceed to a broader analysis of the empirical power of the parameter stability test. In particular, we should repeat the analysis using a larger grid on the length of variation from the null. We should also explore the relationship between the empirical power and the sample size. Finally, a theoretical approach to power issues of structural stability tests in Markov-switching 10

models would be desiderable.

7 References [1] Anas, J., M. Billio, L. Ferrara, and M. Lo Duca (2004): ”Business Cycle Analysis with Multivariate Markox Switching Models”. GRETA working paper n.04.02 [2] Andrews, D. (1993) ,"Tests for Parameter Instability and Structural Change with Unknown Change Point", Econometrica, 61, pp. 821-856 [3] Artis, M., H.-M. Krolzig, and J. Toro (2004): "The European Business Cycle", Oxford Economic Papers, 56, pp.1-44 [4] Burns, A. and W. Mitchell (1946): ”Measuring Business Cycles”, National Bureau of Economic Research, New York. [5] Carrasco,M., L. Hu, and W. Ploberger (2004), "Optimal Test for Markov Switching", Mimeo. [6] Coe, P. (2002), "Power issues when testing the Markov switching model with the sup likelihood ratio test using U.S. output", Empirical Economics, 27, pp. 395-401 [7] DeJong, D.N., R. Liesenfeld, and J-F. Richard (2006), "Timing Structural Change: A Conditional Probabilistic Approach", Applied Econometrics, 21, Issue 2, pp. 175-190 [8] Diebold, F. and G. Rudebusch (1996): ”Measuring Business Cycles: A Modern Perspective”. The Review of Economics and Statistics, vol. 78, No.1, pp. 67-77. [9] Garcia, R. (1998), "Asymptotic Null Distribution of the Likelihood Ratio Test in Markov Switching Models", International Economic Review, 39, No.3. pp.763788. [10] Hamilton J.D. (1989): ”A New Approach to the Economic Analysis of Non Stationary Time Series and the Business Cycle”, Econometrica, 57, No.2, pp.357384. [11] Hamilton, J.D. (1996), "Specification testing in Markov-Switching Time Series Models", Journal of Econometrics, 70, pp. 127-157. [12] Hansen, B.E. (1992), "The likelihood ratio test under non-standard conditions:

11

Testing the Markov switching model of GNP", Journal of Applied Econometrics, 7, pp.S61-S82 [13] Hansen, B.E. (1996), "Erratum: The likelihood ratio test under non-standard conditions: Testing the Markov switching model of GNP", Journal of Applied Econometrics, 11, No.2 pp.195-198 [14] Hansen, B.E. (2001), "The New Econometrics of Structural Change: Dating Breaks in U.S. Labor Productivity", Journal of Economic Perspectives, Vol. 15, No.4 pp.117-128 [15] Harding,D. and A. Pagan (1999): ”Dissecting the Cycle”, Melbourne Institue Working Paper No. 13/99. [16] Kim, C.(1994): ”Dynamic linear models with Markov-switching”. Journal of Econometrics, 60, pp. 1-22. [17] Kim, C. and C. Nelson (1999): State-Space Models With Regime Switching. Classical and Gibbs-Sampling Approaches with Applications. The MIT Press. [18] Goodwin, T.H. (1993): "Business-cycle Analysis with a Markov-Switching Model", Journal of Business & Economic Statistics, Vol. 11, No.3, pp.331-339. [19] Krolzig, H.-M. (2001): ”Markov-switching Procedures for Dating the Euro-Zone Business Cycle”. Vierteljahreshefte zur Wirtschaftsforschung, 70 (3), pp. 339-351. [20] Krolzig, H.-M. (2003): ”Constructing Turning Point Chronologies with MarkovSwitching Vector Autorreressive Models: the Europ-Zone Business Cycle”, Eurostat (ed.), Proceedings on Modern Tools for Business Cycle Analysis, Monography in Official Statistics, forthcoming. [21] Krolzig, H.-M. and J. Toro (2001): ”Classical and Modern Business Cycle Measurement: The European Case”. University of Oxford, Discussion Paper in Economics 60. [22] Mongelli, F.P. and J.L. Vega (2006): ”What Effects Is EMU Having On The Euro Area and Its Member Countries? An Overview ”. Working Paper No. 559, European Central Bank.

8 Appendix

12

8.1

Tables

8.1.1 Average growth

BE FI FR GE GR IE IT LU NL PT SP

8.1.2 Sample:

1970 : 2 − 1998 : 4 0.62 0.67 0.61 0.73 0.67 1.13 0.64 0.97 0.67 0.90 0.72

1999 : 1 − 2005 : 4 0.49 0.75 0.57 0.33 0.98 1.71 0.30 1.18 0.47 0.47 0.92

(8)

1970:2-2005:4

Single star (∗ ) stands for significant parameters at 5% and double stars (∗∗ ) for significant parameters at 10%. The standard errors are reported in parenthesis. 13

p00 0.4448∗ (0.2860)

p11 0.9561∗ (0.0313)

0.8424∗ (0.0918)

0.8265∗ (0.0733)

0 (0.0001)

0.9921∗ (0.0080)

GE

0.2269 (0.2369)

0.9910∗ (0.00851)

GR

0.2931∗ (0.0969)

0.7572∗ (0.0898)

IE

0.9950∗ (0.0080)

0.9955∗ (0.0059)

IT

0.5085 (1.5223)

∗

0.9053 (0.27596)

LU

0.9271∗ (0.0459)

0.9832∗ (0.0125)

NL

0.9886∗ (0.0153)

0.9903∗ (0.0123)

PT

0.9526∗ (0.0263)

0.9388∗ (0.0263)

SP

0.8965∗ (0.0571)

0.9425∗ (0.0336)

EMU

0.4637∗ (0.2029)

0.9631∗ (0.0214)

BE

FI

FR

14

µ

µ

Γ ¶ 0.4448 0.0439 0.5552 0.9561 0.8424 0.1735 0.1576 0.8265 µ

µ

µ

µ

µ

µ

µ

µ

µ

µ

0 0.0079 1 0.9921

¶

¶

0.2269 0.0090 0.7731 0.9910 0.2931 0.2438 0.7069 0.7572 0.9950 0.0045 0.0050 0.9955 0.5085 0.0947 0.4915 0.9053 0.9271 0.0168 0.0729 0.9832 0.9886 0.0097 0.0114 0.9903 0.9526 0.0612 0.0474 0.9388 0.8965 0.0575 0.1035 0.9425 0.4637 0.0369 0.5363 0.9631

¶

¶

¶

¶

¶

¶

¶

¶

¶

(9)

µ0 −0.8440∗ (0.3592)

µ1 0.6940∗ (0.1037)

σ 20 0.3036 (0.2821)

σ 21 0.3757∗ (0.0596)

FI

0.3808∗∗ (0.2246)

0.8423∗ (0.2005)

1.5627∗ (0.3517)

0.1521∗ (0.0411)

FR

−1.4757∗ (0.5353)

0.5972∗ (0.0760)

1.1732 (1.6438)

0.2102∗ (0.0254)

GE

−1.7209∗ (0)

0.8547∗ (0.0347)

0 (0)

0.8212∗ (0.1283)

GR

−1.2192∗ (0.2646)

1.3069∗ (0.1257)

0.4859∗ (0.2130)

6.6805∗ (1.1677)

IE

−2.9981∗ (0.3953)

1.2256∗ (0.1971)

27.2731∗ (6.9770)

0.1122∗ (0.0161)

BE

IT

∗

0.7321 (0.7988)

∗

0.6056 (0.2532)

0.5426 (0.1116)

1.2669 (1.4657)

LU

0.2594 (0.3459)

1.0391∗ (0.1236)

8.6516∗ (1.8305)

0.1925∗ (0.0294)

NL

0.5449∗ (0.2315)

0.5766∗ (0.0955)

2.5411∗ (0.4743)

0.3397∗ (0.0816)

PT

−0.8464 (0.6089)

1.3950∗ (0.6612)

0.5381∗ (0.1003)

0.0455∗ (0.00174)

SP

0.1667 (0.2907)

0.8349∗ (0.2405)

0.9811∗ (0.2308)

0.1194∗ (0.0300)

EMU

−0.6718∗ (0.2001)

0.4922∗ (0.0900)

0.0679 (0.0427)

0.1345∗ (0.0210)

15

(10)

φ1 0.3468∗ (0.1360)

φ2 0.0626 (0.1137)

FI

0.2961∗ (0.1080)

0.3508∗ (0.0838)

FR

0.2115∗ (0.0827)

0.1941∗ (0.0845)

0.0804 (0.0785)

GE

−0.0535 (0.0716)

0.0380 (0.0696)

0.2408∗ (0.0707)

0.0396 (0.0814)

GR

−0.2997∗ (0.0796)

−0.3583∗ (0.0744)

−0.1740∗ (0.0718)

0.2659∗ (0.0654)

IE

1.2075∗ (0.0618)

−0.1732∗ (0.0384)

−0.2010∗ (0.0059)

BE

φ3

IT

0.3762∗ (0.0775)

LU

0.9252∗ (0.0877)

−0.1592 (0.1157)

−0.1227∗∗ (0.0724)

NL

−0.0957 (0.0906)

0.0815 (0.0881)

0.1856∗ (0.0981)

PT

1.4413∗ (0.0886)

−0.4729∗ (0.1157)

−0.0354 (0.0756)

SP

0.0733 (0.0854)

0.4422∗ (0.0792)

0.3223∗ (0.0894)

EMU

0.3023∗ (0.1230)

0.0165 (0.1024)

0.0730 (0.0829)

16

φ4

(11)

0.1461∗∗ (0.0819)

8.1.3 First Subsample:

1970:2-1998:1

p00 0.4219∗∗ (0.2257)

p11 0.9342∗ (0.0386)

FI

0 (0)

0.8700∗ (0.0622)

FR

0.0001 (0.0045)

0.9901∗ (0.0100)

GE

0.0981 (0.1392)

0.9745∗ (0.0144)

GR

0.3305∗ (0.1195)

0.5468∗ (0.0706)

IE

0.1836 (0.3842)

0.9895∗ (0.0098)

IT

0.2431 (2.2437)

∗∗

0.8487 (0.4913)

LU

0.9704∗ (0.0582)

0.9922∗ (0.0084)

NL

0.7611∗ (0.0916)

0.5265∗ (0.1152)

PT

0.8936∗ (0.0718)

0.9763∗ (0.0155)

SP

0.9295∗ (0.0451)

0.9538∗ (0.0284)

EMU

0.4934∗ (0.1749)

0.9221∗ (0.0414)

BE

17

µ

Γ ¶ 0.4219 0.0658 0.5781 0.9342 µ

µ

µ

µ

µ

µ

µ

µ

µ

µ

µ

0 1

0.13 0.87

¶

0.0001 0.0009 0.9999 0.9901 0.0981 0.0255 0.9019 0.9745 0.3305 0.4532 0.6695 0.5468 0.1836 0.0105 0.8164 0.9895 0.2431 0.1513 0.7569 0.8487 0.9704 0.0078 0.0296 0.9922 0.7611 0.4735 0.2389 0.5265 0.8936 0.0237 0.1064 0.9763 0.9295 0.0462 0.0705 0.9538 0.4934 0.0779 0.5066 0.9221

¶

¶

¶

¶

¶

¶

¶

¶

¶

¶

(12)

µ0 −0.7833∗ (0.3247)

µ1 0.7679∗ (0.1222)

σ 20 0.2655 (0.2161)

σ 21 0.3859∗ (0.0723)

FI

−0.3868 (0.3533)

0.9371∗ (0.2193)

2.8051∗ (0.2193)

0.4547∗ (0.0936)

FR

−1.5099∗ (0.5185)

0.6185∗ (0.0913)

1.2374 (1.7439)

0.2161∗ (0.0298)

GE

−1.6459∗ (0)

0.6378∗ (0.0347)

0 (0)

0.77∗ (0.0989)

GR

−1.0227∗ (0.2021)

1.4961∗ (0.3303)

1.3466∗ (0.9037)

10.0111∗ (1.7075)

IE

−2.8557∗ (0.3395)

1.3033∗ (0.2010)

0.6052 (0.8583)

0.1033∗ (0.0148)

∗

BE

IT

0.8787 (0.5611)

1.6166 (1.1986)

0.6550 (0.0971)

0.3059 (0.4132)

LU

−2.4949∗ (0.4999)

0.9776∗ (0.1626)

6.1183∗ (2.6507)

0.2141∗ (0.0299)

NL

0.4652∗ (0.0709)

1.0646∗ (0.05)

2.2785∗ (0.4194)

0.0402∗ (0.0189)

PT

−0.7517 (0.5226)

1.2694∗ (0.5266)

1.1930 (1.3692)

0.0601∗ (0.0095)

SP

0.1502 (0.3944)

0.8657∗ (0.3207)

1.0553∗ (0.2540)

0.1544∗ (0.0327)

EMU

−0.4029∗ (0.1053)

0.57∗

0.0459∗∗ (0.0276)

0.1472∗ (0.0311)

(0.0696) 18

(13)

φ1 0.3757∗ (0.1674)

φ2 0.0366 (0.1244)

FI

0.3128∗ (0.0859)

0.3197∗ (0.0845)

FR

0.2578∗ (0.0938)

0.1607 (0.0987)

0.0859 (0.0886)

GE

−0.1775∗ (0.0743)

0.1171∗ (0.0453)

0.2827∗ (0.0687)

0.0537 (0.0609)

GR

−0.1637∗∗ (0.0938)

−0.3127∗ (0.0709)

−0.0016 (0.0487)

0.2699∗ (0.0684)

IE

1.5277∗ (0.1100)

−0.7633∗ (0.1847)

0.0642 (0.1157)

(0.0965)

−0.4074∗ (0.1496)

−0.1352 (0.0961)

NL

−0.3054∗ (0.0516)

−0.1669∗ (0.0537)

−0.0259 (0.0403)

PT

1.6043∗ (0.0820)

−0.6065∗ (0.1244)

−0.0478 (0.0733)

SP

0.0511 (0.1001)

0.4874∗ (0.0808)

0.3033∗ (0.0989)

EMU

−0.0079 (0.1367)

−0.0737 (0.1206)

0.0873 (0.0954)

BE

IT

LU

φ3

φ4

(14)

0.3892∗ (0.0802) 1.26∗

19

0.2402∗ (0.0934)

8.1.4 Second subsample:

1999:1-2005:4

p00 0.6269∗ (0.2191)

p11 0.9282∗ (0.0540)

FI

0 (0)

0.9285∗ (0.0489)

FR

0 (0)

0.8854∗ (0.1302)

GE

0.5997∗ (0.1340)

0.6513∗ (0.1310)

GR

0 (0)

0.1881∗∗ (0.1034)

IE

0.1701 (0.3703)

0.9750∗ (0.0289)

BE

IT

0 (0)

0.3748 (0.3033)

LU

0.7749∗ (0.1377)

0.8853∗ (0.0735)

0 (0.0044)

0.9585∗ (0.0418)

PT

0.3196∗∗ (0.0804)

0.8292∗ (0.1866)

SP

0.0001 (0.0305)

1∗ (0.0012)

EMU

0.9870∗ (0.1842)

1∗ (0)

NL

20

µ

Γ ¶ 0.6269 0.0718 0.3731 0.9282 µ

µ

µ

0.1146 0.8854

¶

0 1

0.4532 0.8119

0 1

0.6252 0.3748

0 1

0.0415 0.9585

µ

0.0001 0 0.9999 1 0.9870 0 0.0130 1

¶ (15)

¶

¶

¶

0.3196 0.1708 0.6804 0.8292 µ

¶

¶

0.7749 0.1147 0.2251 0.8853 µ

µ

0 1

¶

0.1701 0.025 0.8299 0.8765 µ

µ

0.0715 0.9285

0.5997 0.3487 0.4003 0.6513 µ

µ

0 1

¶

¶

¶

µ0 0.0030∗∗ (0.0017)

µ1 0.4950∗ (0.0011)

σ 20 0 (0)

σ 21 0.351∗ (0.0982)

FI

−1.7359∗ (0.4282)

0.84∗ (0.1122)

0.6930 (0.7181)

0.3370∗ (0.0912)

FR

−0.2345 (0.4224)

0.6075∗ (0.1398)

0.2076 (0.1399)

0.1143∗ (0.0388)

GE

−0.4982 (0.6758)

0.4508 (0.6434)

0.0253 (0.0188)

0.05 (0.0376)

GR

−0.0246 (0.1357)

1.2772∗ (0.1673)

1.6892 (1.9226)

0.0526∗ (0.0215)

IE

−2.3312∗ (0.5328)

1.9146∗ (0.4256)

0 (0)

6.4406∗ (1.2657)

IT

∗∗

0.2202 (0.1259)

∗

0.5749 (0.1551)

∗

0.1103 (0.0413)

0.0599 (0.0445)

LU

0.2120 (0.1822)

0.7634∗ (0.1929)

0.0082∗ (0.0041)

4.9146∗ (1.6411)

NL

−0.7440∗∗ (0.2317)

0.3149 (0.2720)

0.5452 (0.7499)

0.1171∗ (0.0322)

PT

−0.7490 (0.7551)

0.3244 (0.7893)

0.0145 (0.0107)

0.2073∗ (0.0656)

SP

−0.0739 (36.0343)

0.8439∗ (0.1205)

0.0261 (17.4543)

0.1211∗ (0.0318)

EMU

−0.1693 (237.3112)

0.3434∗∗ (0.1878)

0.0003 (9.3531)

0.0638∗ (0.017)

BE

21

(16)

φ1 0.2064∗ (0.0023)

φ2 0.0226∗ (0.0033)

FI

0.0331 (0.2014)

−0.0623 (0.1415)

FR

0.2010 (0.2135)

0.1390 (0.2247)

0.1386 (0.1399)

GE

−0.0388 (0.1481)

0.7798∗ (0.1205)

0.1145 (0.1106)

0.0030 (0.1087)

GR

−0.2928∗ (0.1079)

−0.3150∗∗ (0.1627)

−0.0371 (0.1367)

0.3551∗ (0.0845)

IE

−0.1829∗ (0.0518)

0.1108 (0.1146)

−0.1093 (0.2768)

BE

φ3

IT

0.4016∗ (0.1579)

LU

−0.0856∗ (0.0238)

0.3200∗ (0.0210)

0.4719∗ (0.0156)

NL

0.3412∗∗ (0.1747)

0.2010 (0.1725)

0.1966 (0.1357)

PT

−0.3137∗ (0.1)

0.7856∗ (0.1548)

0.4479∗ (0.0935)

SP

0.2144 (0.1556)

−0.1776 (0.1560)

0.4114∗ (0.1558)

EMU

0.6129∗ (0.1899)

0.2496 (0.2102)

−0.3697∗∗ (0.2051)

8.1.5 Fixed Transition Matrix First Subsample: 1970:2-1998:4 22

φ4

(17)

0.1465 (0.1901)

(0.1169)

σ 20 0.3109 (0.2739)

σ 21 0.4039∗ (0.0716)

0.3560 (0.3314)

0.8143∗ (0.2716)

1.4953∗ (0.3037)

0.1508∗ (0.0628)

FR

−1.5181∗ (0.5043)

0.6182∗ (0.0911)

1.241 (1.7574)

0.2163∗ (0.0298)

GE

−1.6436∗ (0)

0.6385∗ (0.1254)

0 (0)

0.7837∗ (0.1279)

GR

−1.3941∗ (0.2839)

1.2842∗ (0.1671)

0.5122∗ (0.2366)

8.4215∗ (1.6309)

IE

−2.9954∗ (0.194)

1.1629∗ (0.2012)

0 (0)

0.0970∗ (0.0176)

IT

0.9489∗ (0.2071)

1.7999∗ (0.5115)

0.6526∗ (0.1236)

0.2203 (0.2707)

LU

−2.4742∗ (0.5050)

0.9742∗ (0.1609)

9.8668 (6.3197)

0.2141∗ (0.0299)

NL

0.5865∗ (0.0982)

0.8629∗ (0.0621)

1.9541∗ (0.2978)

0.1325∗ (0.0522)

PT

−1.1602∗ (0.3269)

1.0189∗ (0.3158)

0.3634∗ (0.0993)

0.0472∗ (0.0075)

SP

0.1554 (0.4072)

0.8719∗ (0.3204)

1.0699∗ (0.2626)

0.1552∗ (0.033)

EMU

−0.4274∗ (0.1254)

0.5544∗ (0.0761)

0.0464 (0.0289)

0.1546∗ (0.0354)

µ0 −0.8681∗ (0.3162)

FI

BE

µ1 0.75∗

23

(18)

φ1 0.3609∗ (0.1494)

φ2 0.0574 (0.1123)

FI

0.3608∗ (0.0962)

0.3738∗ (0.0832)

FR

0.2576∗ (0.0936)

0.1591 (0.0983)

0.0865 (0.0885)

GE

−0.1779∗ (0.0767)

0.1186 (0.087)

0.2855∗ (0.0777)

0.0533 (0.0897)

GR

−0.2916∗ (0.0844)

−0.3558∗ (0.0822)

−0.1862∗ (0.0732)

0.2519∗ (0.0716)

IE

1.5112∗ (0.0899)

−0.7813∗ (0.1351)

0.1113 (0.0084)

BE

φ3

φ4

(19)

∗

IT

0.3695 (0.0833)

LU

1.2568∗ (0.0979)

−0.4040∗ (0.1523)

−0.1379 (0.0976)

NL

−0.3269∗ (0.088)

−0.1714∗∗ (0.0929)

0.0070 (0.0821)

PT

1.5287∗ (0.0779)

−0.6698∗ (0.1064)

0.0634 (0.0638)

SP

0.0595 (0.1015)

0.4848∗ (0.0811)

0.2973∗ (0.1006)

EMU

0.0051 (0.1614)

−0.0661 (0.1177)

0.0822 (0.0988)

24

0.2454∗ (0.0972)

Second Subsample: 1999:1-2005:4

µ0 0.0031∗∗ (0.0017)

µ1 0.495∗ (0.0011)

σ 20 0 (0)

σ 21 0.3478∗ (0.0965)

FI

0.4411∗ (0.1897)

0.7919∗ (0.0734)

2.9519∗∗ (1.6917)

0.0837∗ (0.0293)

FR

−0.4618 (0.9336)

0.5382∗ (0.1381)

0.2652 (0.6934)

0.2163∗ (0.0298)

GE

0.1408 (2.405)

0.3536∗ (0.1635)

0.2901 (1.2531)

0.2240∗ (0..0616)

GR

0.0218 (0.1306)

1.3972∗ (0.0842)

1.0488∗∗ (0.4513)

0.0302∗ (0.0137)

IE

−2.3414∗ (0.6595)

1.9084∗ (0.4623)

0 (0)

6.481∗ (1.5281)

BE

∗

∗

∗

IT

0.32 (0.0769)

0.8057 (0.0134)

0.1527 (0.0451)

0 (0)

LU

0.0957 (0.2048)

0.66∗ (0.2131)

0.0073∗∗ (0.0041)

4.3119∗ (1.312)

NL

−0.6841 (1.827)

0.2072 (1.6721)

0.098∗ (0.0334)

0.287 (0.1661)

PT

−0.9489∗ (0.0611)

0.2238 (0.2649)

0 (0)

0.4512∗ (0.1151)

SP

−0.1637 (0.226)

0.8042∗ (0.1764)

0.0347 (0.0265)

0.0432∗ (0.0129)

EMU

−0.4888∗∗ (0.2783)

0.3429∗∗ (0.1919)

0.0004 (0.006)

0.0614∗ (0.017)

25

(20)

φ1 0.2063∗ (0.0023)

φ2 0.0228∗ (0.0033)

FI

−0.135 (0.088)

0.1853∗ (0.084)

FR

0.0542 (0.1941)

0.2462 (0.1787)

0.114 (0.1804)

GE

0.2111 (0.1887)

0.1069 (0.1937)

0.0238 (0.1836)

−0.0059 (0.1751)

GR

−0.3821∗ (0.1009)

−0.4819∗ (0.1052)

−0.0547 (0.1177)

0.278∗ (0.0694)

IE

−0.1841∗ (0.0629)

0.1113 (0.1481)

−0.1097 (0.2794)

BE

φ3

IT

0.1732∗ (0.0011)

LU

−0.0835∗ (0.0203)

0.3215∗ (0.0197)

0.4764∗ (0.0149)

NL

0.1691 (0.1757)

0.3726∗∗ (0.1915)

0.3684∗ (0.161)

PT

0.0017 (0.0787)

0.5097∗ (0.1594)

−0.0281 (0.1681)

SP

0.2139∗∗ (0.1255)

0.1681 (0.106)

0.3671∗ (0.0974)

EMU

0.6258∗ (0.1956)

0.2424 (0.214)

−0.354∗∗ (0.2027)

8.1.6 Power analysis 26

φ4

(21)

0.1311 (0.1879)

FI FR EMU NL SP

Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2

kθ1 − θ2 k = 0.05 α = 0.05 α = 0.10 0.27 0.33 0.27 0.32

0.41 0.26 0.41 0.24

0.44 0.33 0.46 0.30

kθ1 − θ2 k = 0.2 α = 0.05 α = 0.10 0.72 0.77 0.62 0.67 0.74 0.79 0.61 0.62 0.90 0.90 0.89 0.90 0.62 0.68 0.40 0.50 0.58 0.67 0.48 0.57

kθ1 − θ2 k = 0.5 α = 0.05 α = 0.10 0.99 0.99 0.99 0.99 0.96 0.96 0.86 0.86 0.99 0.99 0.93 0.93 0.95 0.95 0.93 0.95 0.99 0.99 0.99 0.99 (22)

27