Time-varying parameter error correction models: the

Applied Economics, 2001, 33, 771 ± 782

Time-varying parameter error correction models: the demand for money in Venezuela, 1983.I± 1994.IV J U L I AÂ N R A M A J O Department of Applied Economics, Faculty of Economics, University of Extremadura, E-06071 Badajoz, Spain. E-mail: ramajo@ unex.es

The possibility of using time-varying parameter models in the context of error correction models is studied empirically. As an application, a money demand relationship (M1) for Venezuela is estimated from 1983 to 1994 within a cointegrated VAR framework. First, the stochastic properties of the series are analysed, studying each corresponding order of integration. Second, the existence of a long-run stable relation between the variables involved has been investigated, and then the cointegration relation and the short-run adjustment mechanism estimated. As both relations are identi® ed in the context of constant parameters a stability analysis is performed. Finally, the technique of Kalman ® ltering is used to estimate a model that permits the short-run parameters to vary, while the parameters of the long-run relation are kept constant.

I. INTRODUCTION In recent years, testing for unit roots and vector autoregressive (VAR) methodologies have become standard tools in modelling time series. Normally, the strategy followed is to apply some type of unit root test (Dickey and Fuller, 1979; Phillips and Perron, 1988; Osborn et al., 1988; Hylleberg et al., 1990) to the set of series under consideration, and then to use the Johansen (1988, 1991) maximum likelihood approach to localize the possible stationary linear combinations of these series. Once the stable linear combination (obviously, there can exist more than one) has been tested and estimated, the `® nal’ step in the econometric modelling is to estimate the process of dynamic adjustment to that long-run equilibrium relationship in the form of an vector error correction model (VECM). When no evidence is found for stability (cointegration) between a set of variables for which one expected a priori a long-run equilibrium relationship, it has been common

practice among researchers to adopt an approach of dummy or proxy variables to explain the possible structural changes that gave rise to this instability.1 Similarly, even assuming the existence of a stable long-run combination, one may ® nd signs of instability in the short-run adjustment mechanism. This latter phenomenon is far more understandabl e than the former, since it may be justi® ed by the complex (and unstable) economic and ® nancial environment in which the activity is being carried out, and which would lead to temporary deviations from the equilibrium path marked out by the cointegration equation. These discretely occurring instabilities have, in most cases, again been modelled by means of dummy or proxy variables. Granger (1986) suggested the possibility of using timevarying parameter (TVP) models to estimate the cointegration relations, which implies a `continuously’ variable longrun structure. Hall (1993) investigates the eŒects that the presence of structural changes has on standard econo-

1

Nonlinear modelling can also be introduced in estimating a logistic smooth transition (LSTR) function (see Gennari, 1999) to account for instability due to ® nancial innovation in the money market in the mentioned case. LSTR functions represent a class of nonlinearities used to model transitions from one regime to another which are not discrete but continuous (Granger and TeraÈsvirta, 1993). Applied Economics ISSN 0003± 6846 print/ISSN 1466± 4283 online # 2001 Taylor & Francis Ltd http://www.tandf.co.uk/journals

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772 metric practice and proposes the use of the Kalman ® lter estimation technique to model the data generation process in an unstable context. As with the methodology proposed by Muscatelli and Hurn (1995), Hall allows the short-run parameters to vary to accommodate the possible discretely occurring instabilities, but maintains a constant structure in the long-run relationship (this does not mean that such a function cannot incorporate information, by way of dummy or proxy variables, on some structural change that has caused it to vary during some period of the sample). In this way, the applicability of the cointegration relation is maintained for purposes of simulation or design of medium-term policies, while allowing an adaptive dynamic structure for the design of short-term strategies. The goal of the present work is to apply this approach (constant-paramete r long-run structure and time-varying short-run dynamic) to the speci® c case of the M1 demand function in Venezuela. The economies of Latin America in general, and of Venezuela in particular, have been characterized by a great degree of instability in recent years. If to this we add the strong ® nancial innovation process and the opening-up of capital markets observed over the past several years, it is easy to understand that there may appear problems of stability in the money demand function associated with the underlying economic `volatility’ . If this instability is not taken into account in ® nancial programming, it is very di cult to comply with intermediate objectives (e.g., in the form of bands for M1 growth) set out by the Venezuelan Central Bank, so that problems will arise in keeping strict control in the short-term over monetary aggregates. The paper is organized as follows. Section II presents a brief historical perspective and discusses some recent studies on the demand for money in Venezuela. Section III is dedicated to specify the money demand function, and to introduce the formulation of the ECM which will be used in the empirical part of the work. Section IV studies the stochastic properties of the time series to be used, investigating in particular the presence of unit roots in both the regular part and the seasonal part. Section V is devoted to the estimation of the Venezuelan money demand function. For the long-run relationship, the estimation is performed using Johansen’s (1988) maximum likelihood methodology, while to construct the dynamic short-run model we use Hendry’ s (1986) general-to-speci® c procedure to develop a parsimonious, single-equation conditional model with an error correction mechanism (ECM),2 (Johansen, 1991; Stock and Watson, 1988). In Section VI, the stability of the dynamic speci® cation found in the previous section is analysed. In order to evaluate the constancy of such relationship, recursive estima2

J. Ramajo tion techniques are used (Brown et al., 1975), in order to implement forecast error tests, CUSUM and CUSUMSQ tests, and nonconstanc y Hansen (1992)-type tests (which took into account the bias to parameter constancy statistics which is induced by searching for a structural break). In Section VII, the results obtained on estimating the shortrun dynamic speci® cation allowing its coe cients to vary with time are presented. Finally, the main conclusions that can be drawn from the work are described.

I I . H I S T O R I C A L BA C K G R O U N D Some special episodes in Venezuela during 1989± 94 In March of 1989 the Venezuelan government developed a stabilization macroeconomic programme in conjunction with several importants ® nancial market reforms. These reforms ¯ oated the exchange rate, removed interest rate ceiling and capital account transactions restrictions, and permitted the access to the foreign exchange market. Finally, new types of private ® nancial instruments (such as savings certi® cates and time deposits) and Central Bank bonds (zero-coupon) were also introduced. One of the eŒects of these measures was a sharp drop in real money balances (that never returned to its previous level) in the beginning of 1989 (Figures 1, 2 or 3), which support evidence for the hypothesis that ® nancial innovation caused the decline in real balances. This decline was accompanied by a temporary high in¯ ation (which reached 36.8% in the second quarter of 1989, Figure 1), a sharp increase in the nominal interest rates (Figure 2) and an initial fall in the level of gross domestic product (Figure 3).

Fig. 1. Real M1 and inXation in Venezuela

The ECM generalizes the traditional partial adjustment model, allowing for diŒerent speeds of reaction to the diŒerent determinants of money demand, yet through the error correction term ensures that the long-run relationship holds in steady state.

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Error correction models

From the political perspective, the ® rst attempt of `coup d’ e tat’ at the beginning of 1992 led to a period of social and economic instability. This volatile environment increased with the ® nancial crisis that Venezuela suŒered at the end of 1993 (Figure 4). Previous empirical work

Fig. 2. Real M1 and interest rate in Venezuela

Fig. 3. Real M1 and GDP in Venezuela

Fig. 4. Volume of real transactions and their change

A large number of empirical studies of the demand for money for both developed and less developed countries have appeared in recent years. In the case of Venezuelan economy there are not so many empirical investigations, and in most cases the estimated models use either samples that are nonoverlapping with those used here or traditional speci® cations (as forward looking or partial adjustment functions) now questioned in view of the modern cointegration approach. For this reason the results will be compared with the quarterly models estimated by PaÂez (1995), SaÂnchez (1995) and Copelman (1996). PaÂez (1995) used Engle and Granger’s (1987) approach for testing cointegration and for estimating the cointegration vector for quarterly data from 1983 to 1992. In her static regression for the Engle± Granger procedure she did not ® nd evidence of cointegration between the (log) real money balances (M1 de® nition), the logarithm of total real income (GDP) and the alternative interest rate (three-month nominal yields at Commercial Banks). However, once a dummy variable (D89 which is 0 prior to year 1989 and 1 thereafter) is introduced to capture the eŒects of ® nancial innovation (only on the intercept of the money demand function) the Augmented Dickey± Fuller statistic is signi® cant at 95% critical level, indicating that this equation provides a stable long-run demand function. Following the same univariate approach, Copelman (1996) estimates a single long-run equation model from 1978.I to 1991.IV with the log of real non-oil GDP and in¯ ation rate as determinants of the log of real M1 (de¯ ated by the Consumer Price Index). In this case, the evidence fails to reject the null hypothesis of no cointegration. However, as in PaÂez (1995) , when Copelman allows ® nancial innovation to aŒect the intercept as well as the slope of the demand function through the introduction of two dummy variables (D89 same as described before and D89I which is 0 prior to year 1989 and equal to in¯ ation thereafter), the modi® ed money demand equation becomes stable. Lastly, within the multivariate framework, another recent contribution is that of SaÂnchez (1995) who estimates a VAR with quarterly data over the period 1980.I± 1994.II and, using Johansen’s cointegration procedure on a system of four variables (log of real M1, log of real GDP, 90-days nominal yields at Commercial Banks and in¯ ation), identi® es the money demand relationship as one of the three vectors of the cointegration space. He eliminates two cointegration vectors using economic arguments and reduces

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J. Ramajo

Table 1. Long-run results obtained by Pa ez (1995), SaÂnchez (1995) and Copelman (1996) Variable

y

PaÂez (1995) SaÂnchez (1995) Copelman (1996)

0.466 0.834 3.32

r

º ± ¡1.006 ¡0.97

¡0.012 ¡0.396 ±

Table 2. Total short-run eVects estimated by Pa ez (1995), SaÂnchez (1995) and Copelman (1996) Variable PaÂez (1995) SaÂnchez (1995) Copelman (1996)

y 0.392 0.259 º 1 (not signi® cant)

the system to a simultaneous equation model with only one long-run relationship. On the long-run, the elasticities of real money balances with respect to income and the semi-elasticities of interest rates and in¯ ation of the three studies above mentioned are listed in Table 1. With respect to the short run demand, each one of the above-mentioned three empirical works ends up with the estimation of a single equation dynamic speci® cation for real balances. Table 2 shows the estimated eŒects obtained for the error correction model used in each work, after all the nonsigni® cant variables have been eliminated or some interactive dummy variables are included to take into account ® nancial innovation in the short run.

III. THE LONG-RUN M1 DEMAND FUNC TION AND THE ERROR C O R R E C TI O N M E C H A N I S M The choice of money demand function is made on the basis of the transaction theoretical framework, where the demand for money is due to its property as a medium of exchange. Therefore, the concept of money refers to a narrow range of instruments which ful® l, in the strict sense, the property of being means of payment. The opportunity cost of having liquidity is the interest rate of short-term instruments, and the scale variable is income as a measure of the volume of transactions (MauleoÂn, 1989). Hence, if we denote the real transactions demand by md , the logarithm of the level of internal activity by y, and by r an interest rate (in levels) which represents the gain/loss from holding money instead of an alternative asset, the basic money 3

r ¡0.011 ¡0.434 ±

º ± ¡0.745 ¡0.66

Zt¡1 (error correction term) ¡0.738 ¡0.127 ¡0.68

demand function will be represented at each instant of time by the relation mdt ˆ m…yt ; rt †. Friedman (1956) proposed the inclusion of the expected in¯ ation, ºe , in the above basic speci® cation, arguing that this variable can be regarded as the nominal yield of real goods in so far as their real value remains constant. Hence, it is an opportunity cost of holding idle balances, and it will be negatively correlated with real money balances. Thus, the augmented money demand function, which will serve as a potential long-run equilibrium relationship, is mdt ˆ mt …yt ; rt ; ºt † ˆ ­ 0 ‡ ­ 1 yt ‡ ­ 2 rt ‡ ­ 3 ºt

…1†

where the expected in¯ ation rate has been replaced by the real observed in¯ ation rate (in levels).3 The equation described for md guides the long-run behaviour between the real balances and the variables which cause their variations (naturally, this claim implies that income, interest rate, and in¯ ation are weakly exogenous in the conditional demand model). In the short term, the observed deviations from the equilibrium values will be explained by the relation ¢…m ¡ p†t ˆ k ¡ ¬‰…m ¡ p† ¡ m d Št¡1 ‡ ¯0 …L†¢…m ¡ p†t¡1 ‡ ¯1 …L†¢yt ‡ ¯2 …L†¢rt ‡ ¯3 …L†¢ºt ‡ "t

…2†

where ¢ is the ® rst diŒerence operator, ¯i …L† are polynomials in the lag operator L, and the function md is given by Equation 1; m ¡ p is the measure of the level of real balances, de® ned as the real value (in logarithms) of the money in circulation (M1). Equation 2 can also be written in the form

As will be seen in Section VI, the level of prices in the Venezuelan economy is an I (2) process. One could therefore set up the hypothesis of a process of expectative formation of the type ºet ˆ ¿0 ‡ ¿1 ¢¿t¡1 (Alogoskou® s and Smith, 1995), so that the model would include the lagged in¯ ation rate, ºt¡1 , instead of the current rate.

775

Error correction models ¢…m ¡ p†t ˆ k0 ¡ ¬…m ¡ p†t¡1 ‡ ¬1 yt¡1

Table 3. Dickey and Fuller (ADF) unit roots test

‡ ¬2 rt¡1 ‡ ¬3 ºt¡1 ‡ ¯0 …L†¢…m ¡ p†t¡1 ‡ ¯1 …L†¢yt ‡ ¯2 …L†¢rt ‡ ¯3 …L†¢ºt ‡ "t

…3†

where ¬i ˆ ¬­ i . The two representations of the ECM have been distinguished (Equations 2 and 3) since they will serve as the basis for relaxing the hypothesis of constant coe cients. Thus, if it is assumed that the long-run structure remains constant (­ it ˆ ­ i i ˆ 0; 1; 2; 3; 8t), but the short-run parameters are allowed to vary at each instant of time, then Equation 2 is the more suitable. However, if the cointegration regression coe cients are also allowed to vary in time, then Equation 3 is the more appropriate, since it maintains linearity in the parameters.4

V I . I N T E G R A T I O N O R D E R O F T H E M O N EY DEMAND FUNCTION VARIABLES The data used in the present work are nonseasonally adjusted homogeneous quarterly series obtained from publications of the Venezuelan Central Bank. The variables that will be used in the money demand function are: the logarithm of the real circulating supply (expressed in millions of Bolivars in 1984 prices) de® ned as m ¡ p ˆ log…M1=P), with P being the consumer price index (1984 ˆ 100); the logarithm of the real total GDP (also expressed in millions of Bolivars in 1984 prices), y ˆ log…Y †; the interest rate r de® ned as the 90-days nominal returns at the Commercial Banks; and the interquarterly in¯ ation rate (º ˆ ¢P=P¡1 ). The period used as the basis for the analysis runs from the ® rst quarter of 1983 to the fourth quarter of 1994. Order of regular integration For the analysis of the integrability of the economic series that are the object of the study, one of the most usual tests, the Augmented Dickey± Fuller test (ADF) shall be employed. This test (Fuller, 1976; Dickey and Fuller, 1979, 1981; Dickey and Said, 1984) for the order of integration of an individual series xt is based on the following regression equations: p

¢xt ˆ ¬0 ‡ ¬1 xt¡1 ‡

jˆ1

®j ¢xt¡j ‡ "t p

¢xt ˆ ¬*0 ‡ ¬*1 xt¡1 ‡ ¬*2 t ‡ 4

jˆ1

®*j ¢xt¡j ‡ "*t

Variable

t¬1

…m ¡ p† y r ºr D…m ¡ p† ¢y ¢r ¢º

¡0.97 ¡1.66 ¡2.10 ¡3.47* ¡3.03* ¡2.80* ¡2.58* ¡4.76*

F©1

t¬ ¤ 1

0.74 1.77 2.20 6.10* 4.62* 3.92* 3.44 11.32*

¡2.69 ¡3.71* ¡2.75 ¡4.15* ¡2.96* ¡2.76 ¡2.43 ¡4.70*

F©3

p

2.63 5.04* 2.81 5.78* 3.01 2.58 2.42 7.36*

3.64 7.06* 4.21 8.61* 4.48 3.87 3.50 11.04*

5 4 2 0 4 4 2 2

d 41 4 1 or 0 41 0 0 0 0 0

Notes: t¬1 and t¬¤ 1 represent, respectively, the t-ratios of the parameters ¬1 and ¬*1 . F©1 is the F-test of the null H0 ˆ f¬0 ˆ ¬1 ˆ 0g. F©2 is the F-test of the null H0 ˆ f¬0¤ ˆ ¬1¤ ˆ ¬2¤ ˆ 0g. F©3 is the F-test of the null H0 ˆ f¬1¤ ˆ ¬2¤ ˆ 0g. p represents the longest lag of ¢xt included in the regression. d is the integration order obtained from the ADF test, using the critical values of McKinnon (1991). (*) denotes the signi® cance of the test at 10% level. r: This parameter is not signi® cant at the 1% level.

where it is assumed that the errors are Gaussian `white noise’ perturbations, since the value of p has been chosen to ensure this property. Table 3 lists the results of the above regressions for the variables of the money demand function, including the t statistics of individual signi® cance, and some F statistics to test multiple restrictions on the parameters of the auxiliary regressions (Dickey and Fuller, 1981). In the light of these results, it seems clear that there exists a unit root in the interest rate (r) and real balance series (m ¡ p). They also indicate that real income (y) is a ® rst-order integrated process if a deterministic trend is not incorporated into the stochastic process that describes it; otherwise the series does not seem to contain any unit root. Lastly, the in¯ ation series (º) does not seem to contain unit roots.5 On analysing the graph corresponding to º (Figure 1), however, one observed that a sharp change occurred in the ® rst two quarters of 1989. The presence of these atypical observations led to a reconsideration of the usual Dickey± Fuller test since, as shown by Franses and Haldrup (1994), the presence of additive outliers can produce spurious stationarity, with the too frequent rejection of the presence of unit roots. To take into account the eŒect of deterministic structural changes, these authors suggest including dummy variables in the auxiliary ADF regression, showing that such inclusion does not aŒect the limiting distribution of the statistics. The regression that is suggested is the following in the case that there exist only additive outliers (the modi® cation for the case in which a deterministic trend exists is immediate) p‡1

…4†

F©2

¢xt ˆ ¬0 ‡ ¬1 x t¡1 ‡

m

iˆ0 jˆ1

AO

j !AO ij Dt¡i ‡

p jˆ1

®j ¢xt¡j ‡ "t …5†

As noted by Hall (1994), this generalization may lead to a model having a di cult interpretation in economic terms, and even to variation of the sign of the long-term relationship’s parameters. 5 Although the results corresponding to the price level series (P) are not listed in Table 1, the ADF test does not reject the hypothesis of the existence of unit roots.

776

J. Ramajo Table 4. Osborn et al. (OCSB) seasonal unit roots test Variable

t­ 1

…m ¡ p† y r º

0.30 0.11 ¡1.27 ¡2.11

t­ 2 ¡4.39*** ¡4.83*** ¡4.44*** ¡6.11***

F©4

p

13.64*** 15.49*** 17.02*** 34.61***

4 0 2 0

Notes: t­ 1 and t­ 2 represent, respectively, the t-ratios of the parameters ­ 1 and ­ 2 . F©4 is the F -test of the null H0 ˆ f­ 1 ˆ ­ 2 ˆ 0g. p represents the longest lag of ¢¢4 xt included in the regression. The critical values reported in Osborn (1990) were used. (***) denotes the signi® cance of the test at the 1% level.

Fig. 5. 1-Period forecast test

where the impulse variables (DAO; j ) take the value 1 at the instant when the outlier appears and zero for the rest of the observations. On performing the auxiliary regressions by introducing the dummy variables D89.1 and D89.2, ADF statistics of ¡1:24 and ¡2:18 were obtained for the equations with and without the trend term, respectively, so that the null hypothesis of the existence of a unit root cannot be rejected. Order of seasonal integration The analysis of the previous section was centred on the search for unit modulus roots at zero frequency in the spectrum of the series. On working with quarterly series, however, there exists the possibility that unit roots appear at higher frequencies of the spectrum. Osborn et al. (1988, 1990) propose a test of the I (1,1) hypothesis as against the alternative(s) I (1,0) and I (0,1). This test is based on the following auxiliary regression: 4

¢¢4 xt ˆ

iˆ1

¬j Qi;t ‡ ­ 1 ¢4 xt¡1 ‡ ­ 2 ¢xt¡4

Table 4 lists the results obtained on applying the test proposed by Osborn et al. (1988) (henceforth OCSB) to the variables speci® ed for the money demand function. In all cases, the I (1,1) hypothesis is rejected; also, in light of the t­ 2 statistics, the existence of seasonal unit roots is rejected. The t­ 1 statistics con® rm the results established in the previous section, i.e., that all the series are I(1) (I (1,0) in the OCSB terminology).

V . E S T I M A T I O N O F T H E V E N E Z U E LA N MONEY DEMAND FUNCTIONS Having studied the stochastic properties of the series involved in the money demand function, in this section the possible existence of cointegration relations between such variables are analysed and formulate the corresponding ECM. Initially the explanatory variables y, r, and º are assumed weakly exogenous, so that their contemporaneous values can appear in these correction mechanisms. The methodology employed for the search and for the estimation of the cointegration relation(s) is the one developed by Johansen (1988, 1991). Let Y represent the vector …m ¡ p0 ; y0 ; r0 ; º0 †0 . We shall assume that this vector has an autoregressive representation (VAR) of the type Yt ˆ m ‡ U Qt ‡ P

1 Yt¡1

‡P

2 Yt¡2

‡ ¢¢¢‡ P

k Yt¡k

p

‡

jˆ1

¿j ¢¢4 xt¡j ‡ "t

…6†

where Qi are dummy seasonal variables and ¢ 4 is the seasonal diŒerence operator. The above equation allows the null hypothesis to be tested by a simple one-tailed t statistic: a nonsigni® cant t statistic for the estimate of ­ 1 would be indicative of the presence of a nonseasonal root (with which one would have that xt ¹ I (1,0)), while a nonsigni® cant t statistic for ­ 2 would indicate the presence of `some’ seasonal root (i.e., xt ¹ I (0,1)). Lastly, if the F statistic of the hypothesis H0 ˆ f­ 1 ˆ ­ 2 ˆ 0g was not signi® cant, this would be an indication that xt ¹ I (1,1).

‡e

t

…7†

where Q is a vector that contains s ¡ 1 seasonal dummy variables (where s is the periodicity of the data) and the vectors e t are assumed to be independent and normally distributed, i.e., e ¹ N…0; K †. This VAR(k) model has a representation in the form of a cointegrated VAR (VECM) of the type (see Johansen (1991) for details): D Y t ˆ m ‡ U Qt ‡ C ‡C

k¡1 D

1D

Yt¡1 ‡ C

2D

Yt¡k‡1 ‡ P Yt¡k ‡ e

Yt¡2 ‡ ¢ ¢ ¢ t

…8†

The cointegration hypothesis is formulated by way of a test on the rank of the long-run impact matrix, P . For the VECM representation to be `balanced’, all the components

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Error correction models of the term P Yt¡k must be I(0), which implies that the rank of the matrix P is less than the number of variables included in the vector Y.6 This rank (which we shall denote by rk) is called the cointegration order. If the matrix P is decomposed in the form P ˆ a b 0 , then (if rk < N) the components of the vector b 0 Yt will be stationary and will de® ne the long-run relationships in the variables contained in Y; the vector a will de® ne the weights with which each cointegration vector will enter in the VECM. Johansen (1988, 1991) developed a maximum likelihood (ML) procedure to test the rank of the matrix P and to estimate the matrices a and b . Subsequently, Johansen and Juselius (1990) modi® ed that procedure, applying multivariate techniques of canonical partial correlations, to test linear constraints on the coe cients of the cointegration relations. Table 5 lists the results obtained on applying Johansen and Juselius’ s modi® ed procedure to the speci® c case under analysis. The values of the statistics show that the null hypothesis frk ˆ 0g of the nonexistence of long-run stable relationships between the components of the vector Y must be rejected in favour of the alternative hypothesis rk ˆ 1, and that this latter can not be rejected in favour of rk ˆ 2. The conclusion is that there exists a single cointegration relation for the Venezuelan data set. The estimates of this equilibrium relationship are given by the ® rst column of the matrix b , so that the empirical counterpart of Equation 1 is transformed into …m ¡ p†t ˆ 1:004 ‡ 1:078yt ¡ 2:651rt ¡ 2:483ºt …0:181†

…0:136†

…0:361†

Rank…¦)

¶i

Trace ˆ ¡T

rk ˆ 0 rk 4 1 rk 4 2 rk 4 3

0.5825 0.2380 0.1851 0.0200

57.544* * 20.858 9.443 0.848

rk‡1

log…1 ¡ ¶i †

¶max ˆ ¡T log…1 ¡ ¶rk‡1 † 36.687*** 11.415 8.594 0.848

Notes: The de® nition of statistics used are in Johansen (1988, 1991). The longest lag included in the VECM mechanism was k ˆ 4. The critical values reported in Johansen and Juselius (1990, Table A3) were used. (***) denotes the signi® cance of the test at the 1% level and (**) the signi® cance at the 5% level.

ables Y* ˆ …m ¡ p ¡ y0 ; r0 ; º0 †0 , obtaining the following restricted cointegration equation …m ¡ p ¡ y†t ˆ ¡0:062 ¡ 2:636rt ¡ 2:907ºt …0:085†

…0:372†

…10†

The next step after having found the long-run M1 demand equation was to construct a dynamic model for the short-run M1 demand. One began by estimating a quite general dynamic ECM, starting from lag polynomials of fourth order in the explanatory variables and in the lagged dependent variable. Following the `general-tospeci® c’ methodology (Hendry, 1986), several tests of signi® cance were realized, arriving ® nally at the following model (see Table 6):7 ¢…m ¡ p†t ˆ 0:009 ‡ 0:433¢yt ¡ 0:825¢rt …0:220†

…9†

where the numbers in parentheses represent the estimates of the standard asymptotic errors. In light of the estimated cointegratio n relation, the question arises of the possibility of unit income elasticity (this result would agree with the obtained by HoŒman and Tieslau (1995) with US and G-7 data). The methodology developed by Johansen and Juselius (1990) to test general hypotheses of the type b ˆ H} was applied to test this constraint. Under the null hypothesis, H 0 ˆ fb ˆ H}g, Johansen and Juselius’ s statistic asymptoticall y follows a À2 …rk £ g† distribution, g being the number of constraints under H0 . The value of this statistic in this speci® c case was 2.71, which on comparison with the critical value of a À2 …1† distribution (at the 5% level, the critical value is 3.84) indicates the nonrejection of the unit income elasticity constraint. Given this result, that constraint was imposed and Johansen’ s procedure again applied to the set of vari6

Table 5. Johansen cointegration rank test

…0:201†

¡ 0:904¢ºt ¡ 0:363Zt¡1 ‡ "t …0:204†

…11†

…0:081†

Table 6. Estimation of the error correction mechanism ( Wnal model) Variable 0

k ¢yt ¢rt ¢pt Zt¡1

Coe cient

Std. error

T-statistic

0.008934 0.432886 ¡0.825280 ¡0.903893 ¡0.362834

0.01133 0 0.21965 7 0.20090 7 0.20384 5 0.08148 5

0.788512 1.970733 ¡4.107770 ¡4.434206 ¡4.452771

Regression statistics R2 0.569070 Adjusted R2 0.527029 ¼ of regression 0.070271 Log likelihood 59.5238 8 Durbin± Watson 2.053889 Jarque± Bera 1.234620

RESET (1) F-Chow (89.I± 94.IV) Breusch± Godfrey (4) ARCH (4) White Ljung± Box (12)

1.381725 2.316055** * 6.438483 6.698123 12.00494 17.49300

Notes: The equation was estimated by Ordinary Least Squares (OLS). (***) denotes the signi® cance of the test at the 1% level.

If the rank of this matrix were 0 or N (the number of components of Y), it would be an indication that the variables included in the vector Y are not cointegrated and the standard VAR methodology could be applied (using ® rst diŒerences or levels, respectively). 7 The F test of the null hypothesis of zero coe cients in the lagged variables (except Z) was 2.09, which is below the 5% critical value of a F(16,21) distribution.

778

J. Ramajo

where Z represents the restricted cointegration vector estimated previously, and the numbers in parentheses are the estimated standard errors. All the above regression diagnostics were found to be satisfactory except Chow’ s predictive test, which indicated the possibility of structural change (at the 1% level) in the period 1989± 1994.

V I . S T A B I L I T Y A N A LY S I S The previous section closed with a brief commentary on the stability of the estimated relationship for the real M1 demand in Venezuela. In this section, this hypothesis will be explicitly analysed. The ® rst method to be applied is based on estimating the short-run dynamic model by Recursive Least Squares, which starts from a small data subset (of size K, the number of regressors in the model), and enlarges this subset iteratively by adding the next observation of the time series to the last subset used. After completion of this recursive estimation of the model, various elements have been used to judge the degree to which the hypothesis of parameter constancy has not been ful® lled (Brown et al., 1975; Harvey, 1981; Beggs, 1988): the 1-period forward prediction errors (1-step Chow test) or N-period forward prediction errors (Break-point Chow test), the CUSUM and CUSUMSQ tests, and, ® nally, the estimated recursive coe cients themselves. These statistics are represented in Figures 5± 9. They lead one to draw the following conclusions. Both the 1-period and N-period forward prediction error tests show systematically biased forecasts, above all from 1989 onwards. Also, the CUSUMSQ test rejects the null stability hypothesis at the 5% signi® cance level. The CUSUM test, however, does not reject the parameter stability hypothesis. Lastly, the trajectories of the recursive

Fig. 7. Cumulative sum (CUSUM) of recursive residuals

Fig. 8. CUSUM of squares test

coe cients again con® rm the rupture of the constant parameter hypothesis. To complement the above stability analysis, the two constancy statistics described in Hansen (1992) were applied, denoted V and J for variance and joint parameter constancy respectively; the result was V ˆ 0.80 and J ˆ 1.31, indicating the signi® cance of the V test at 1% level.

V I I . TI M E - V A R Y I N G C O EF F I C I EN T S : T H E K A L M A N F I L T ER TE C H N I Q U E

Fig. 6. N-Period forecast test

In this section, the constant coe cient hypothesis is relaxed, since the results of the previous section have shown this hypothesis has to be rejected in the present application. From among the diŒerent time-varying par-

779

Error correction models

Fig. 9. Recursive coeYcient estimates

ameter (TVP) models, a model was chosen that can be classi® ed as `moderately adaptive’ as against models of `no adaptivity’ (® xed-parameter models) or `high adaptivity’ (TVP models with large coe cient evolution) (Stock and Watson, 1996). Using the notation of Harvey (1993), adapted to the nomenclature used in this investigation, the general TVP model that serves as the basis for the speci® cation used in the work is expressed by the following equations: ¢…m ¡ p†t ˆ H0t c c ¹t ¹ N…0; ¼2 †

t

ˆ Tc

t

‡ ¹t

t¡1

²t ¹ N…0; Q†

‡g

…12†

t

®0 ¹ N…a0 ;

0†

where Ht contains all the explanatory variables of Equation 11. The ® rst of these equations (known as the measurement equation) is similar to that of the classical regression model

except in that the parameter vector c (in systems theory, this vector is called the state variable) is allowed to vary with time according to the second expression (known as the transition equation), which is a multivariate AR(1) model for the state vector. The last equation describes the properties of the errors of measurement and transition equations (which are, furthermore, mutually and serially uncorrelated), as well as the a priori distribution of the initial state vector (initial conditions). In the present application, it is assumed that T ˆ I, where I is the identity matrix of order (K £ K). Therefore, c t follows a multivariate random walk, and, since it is not stationary, evolves in time to accommodate all the structural changes that have taken place during the sampling period. Also, a diagonal matrix Q was speci® ed, the elements of which were estimated (using the maximum likelihood method) together with the rest of the parameters of the model.

780

J. Ramajo

Fig. 10. Slope state vectors estimated using the Kalman Wlter algorithm

The Kalman ® lter algorithm provides a posteriori recursive estimates, at , of the vector c t by means of the expression for the expectation of this vector constrained by the information available up to the period t, X t , and the hyperparameter vector x 0 ˆ …a0 ; 0 †. This conditional mean provides an optimal estimator of c t , in the sense that it minimizes the mean square error (Granger and Newbold, 1986; Harvey, 1989). The most correct way, theoretically, to implement the Kalman ® lter algorithm consists in estimating the vector x 0 by maximum likelihood, and then using the updating equations of the algorithm to estimate b t . However, some studies (e.g., Hackl and Westlund, 1996) have shown that the results of such a procedure present a high degree of structural variability. Hence, they suggest specifying an a priori reasonable vector x 0 , instead of simultaneously estimating c t and x 0 (and the elements of matrix Q). In our case, a diŒuse prior initial state vector was chosen, with a0 ˆ b (where b is the OLS estimate of Section V), and 0 ˆ j I where j ˆ 100. Figure 10 shows the plots of the state vectors for income, interest rate and in¯ ation variables, estimated according to the afore mentioned procedure. It can be seen that the estimated income elasticity has the most unstable coe cients during the sampling period, and that the period of greatest instability begins in 1989. The most stable trajectories correspond to the short-term interest and in¯ ation semielasticities, the former with an increasing trend from 1989 onwards, the latter decreasing from 1987. With respect to the estimated feedback parameters it can be observed (Figure 11) that there is a downward trend and a variability increasing with time. This fact con® rms not only that the speed of adjustment of money demand to its determinants have increased when there is ® nancial innovation and other reforms (Copelman, 1996) but also that there exists most variability in the velocity at the agents are taking into account the long run equilibrium error when adjusting their demand in the short run. The estimate of the ® nal state vector, aT , allows the equation:

Fig. 11. Speed of convergence to equilibrium

¢…m ¡ p†t ˆ 0:046 ‡ 0:475¢yt ¡ 0:573¢rt …0:235†

…0:243†

¡ 0:938¢ºt ¡ 0:461Zt¡1 ‡ "t …0:254†

…13†

…0:163†

to be written which contains all the information necessary to make forecasts of future observations of the change in real M1 demand in Venezuela.

VIII. CONCLUSIONS In this work, the usefulness of a relaxation of the hypothesis of constant coe cients in regression models has been shown. In particular, and in the cointegration framework, whose use has become standardized in recent years, a `stable’ long-run relationship can be allowed (after testing for its existence), while the ECM parameters are left to vary over time. This approach has allowed study of the shortrun eŒects of ® nancial innovations and other reforms in the

Error correction models ® nancial and capital markets on the demand for money in Venezuela. In the application it has been shown that ® nancial innovation has basically aŒected to the income elasticity and the speed of convergence to equilibrium. Comparing this investigation with the works of PaÂez (1995), SaÂnchez (1995) and Copelman (1996), the shortrun eŒects8 of ® nancial innovation have now been modelled in a continuous way rather than in a discrete one. According to this approach, it is possible to have in each instant revised income and price elasticities what is extremely important for economies as the Venezuelan one, subject to frequent macroeconomic shocks.

A C K N O W LE D G E M E N TS I wish to thank the Instituto de Investigaciones EconoÂmicas y Sociales of the Universidad CatoÂlica Andro s Bello (UCAB) of Venezuela for the technical and human support I received during my stay in this institution with the fellowship INTERCAMPUS E/AL 950456. In particular, I would like to express my gratitude to Matõ  as Riutort for his invaluable help, as well as for providing the data used in the present work. Also, I thank Eduard Berenguer, at the University of Barcelona, the editor and the anonymous referees for their helpful comments on the paper.

R E F E R E N C ES Alogoskou® s, G. and Smith, R. (1995) On error correction models: speci® cation, interpretation, estimation, in Surveys in Econometrics (Eds) L. Oxley, D. George, C. Roberts and S. Sayer, Blackwell, Oxford UK and Cambridge USA, pp. 139± 70. Beggs, J. (1988) Diagnostic testing in applied econometrics, The Economic Record, 25, 81± 101. Brown, R., Durbin, J. and Evans, J. (1975) Techniques for testing the constancy of regression relationships over time (with discussion), Journal of the Royal Statistical Society, Series B, 37 (2), 149± 63. Copelman, M. (1996) Financial innovation and the speed of adjustment of money demand: evidence from Bolivia, Israel and Venezuela, Board of Governors of the Federal Reserve System, International Finance Discussion Papers, No. 567. Dickey, D. and Fuller, W. (1979) Distribution of the estimators for autoregressive time series with a unit root, Journal of the American Statistical Association, 84, 427± 31. Dickey, D. and Fuller, W. (1981) Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica, 50, 1057± 72. Dickey, D. and Said, S. (1984) Testing for unit roots in autoregressive-moving average models of unknown order, Biometrika, 71, 153± 74.

8

781 Franses, P. and Haldrup, N. (1994) The eŒects of additive outliers on tests of unit roots cointegration, Journal of Business and Economic Statistics, 12, 471± 8. Friedman, M. (1956) The quantity theory of money: a restatement, in Studies in the Quantity Theory of Money, (Ed.) M. Friedman, University of Chicago Press, Chicago, 424± 38. Fuller, W. (1976) Introduction to Statistical Time Series, Wiley, New York. Gennari, E. (1999) Estimating money demand in Italy: 1970± 1994, Working Paper in Economics No. 99/7, European University Institute. Granger, C. (1986) Developments in the study of cointegrated economic variables, Oxford Bulletin of Economic and Statistics, 48, 213± 28. Granger, C. and Newbold, P. (1986) Forecasting Economic Time Series, Academic Press, San Diego, California. Granger, C. and TeraÈsvirta, T. (1993) Modelling Nonlinear Economic Relationships, Oxford University Press, New York. Hackl, P. and Westlund, A. (1996) Demand for international telecommunication. Time-varying price elasticity, Journal of Econometrics, 70, 243± 60. Hall, S. (1993) Modelling structural change using the Kalman ® lter, Economics of Planning, 26, 1± 13. Hall, S. (1994) Modelling economies in transition, Working Paper of the Centre for Economic Forecasting, London Business School. Hansen, B. (1992) Testing for parameter instability in linear models, Journal of Policy Modeling, 14, 517± 33. Harvey, A. (1989) Forecasting, Structural Time Series Models, and the Kalman Filter, Cambridge University Press, Cambridge. Harvey, A. (1993) Time Series Models, Second Edn, Harvester Wheatsheaf, Great Britain, London). Hendry, D. (1986) Empirical modelling in dynamic econometrics: the new-construction sector, Applied Mathematics and Computation, 20, 201± 36. HoŒman, R. and Tieslau, H. (1995) The stability of long-run money demand in ® ve industrial countries, Journal of Monetary Economics, 35, 317± 39. Hyllerberg, S., Engle, R., Granger, C. and Yoo, B. (1990) Seasonal integration and cointegration, Journal of Econometrics, 44, 215± 38. Johansen, S. (1988) Statistical analysis of cointegration vectors, Journal of Economic Dynamic s and Control, 12, 231± 54. Johansen, S. (1991) Estimation and hypothesis testing of cointegrating vectors in gaussian vector autoregressive models, Econometrica, 59, 1551± 80. Johansen, S. and Juselius, K. (1990) Maximum likelihood estimation and inference on cointegration with applications to the demand for money, Oxford Bulletin of Economics and Statistics, 52(2), 169± 210. Maule on, I. (1989) Oferta y Demanda de Dinero: Teorõ  a y Evidencia Empõ  rica, Alianza Editorial, Madrid. McKinnon, J. (1991) Critical values for cointegration tests in long-run economic relationships, in Readings in Cointegration, (Eds): R. Engle and C. Granger, Oxford University Press, New York, pp. 267± 76. Muscatelli, V. and Hurn, S. (1995) Econometric modelling using cointegrated time series, in Surveys in Econometrics, (Eds), L. Oxley, D. George, C. Roberts and S. Sayer, Blackwell, Oxford UK and Cambridge USA, pp. 171± 214.

It appears unnecessary in this application, in a diŒerent way as occurred in the works above mentioned, to model the instability in the long-run where a `stable’ relationship has been found.

782 Osborn, D., Chui, A., Smith, J. and Birchenhall, C. (1988) Seasonality and the order of integration for consumption, Oxford Bulletin of Economics and Statistics, 50, 361± 77. Osborn, D. (1990) A survey of seasonality in UK macroeconomic variables, International Journal of Forecasting, 6, 327± 36. PaÂez, K. (1995) Demanda de dinero: un enfoque de cointegracioÂn. Caso Venezuela 1983:I± 1992:IV, Temas de Coyuntur a (Instituto de Investigaciones EconoÂmicas y Sociales, UCAB, Caracas-Venezuela), 31, 93± 113.

J. Ramajo Phillips, P. and Perron, P. (1988) Testing for unit roots in time series regressions, Biometrika, 75, 335± 46. SaÂnchez, G. (1995) Un modelo de demanda de dinero para Venezuela: 1982± 1994, Revista del Banco Central de Venezuela, No. 9, 31± 59. Stock, J. and Watson, M. (1988) Testing for common trends, Journal of the American Statistical Association, 83, 1097± 107. Stock, J. and Watson, M. (1996) Evidence on structural instability in macroeconomic time series relations, Journal of Business and Economic Statistics, 14(1), 11± 30.