Co-breaking, cointegration, and weak exogeneity: Modelling aggregate consumption in Japan

Economic Modelling 27 (2010) 574–584

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Economic Modelling j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c m o d

Co-breaking, cointegration, and weak exogeneity: Modelling aggregate consumption in Japan Takamitsu Kurita Faculty of Economics, Fukuoka University, Bunkei Center Building, 8-19-1 Nanakuma, Johnanku, Fukuoka, 814-0180, Japan

a r t i c l e

i n f o

Article history: Accepted 20 November 2009 Keywords: Co-breaking Cointegration Weak exogeneity General-to-specific approach Aggregate consumption

a b s t r a c t This paper aims to estimate a parsimonious data-congruent model for aggregate real consumption in Japan using quarterly data over the past two decades. Testing co-breaking, cointegration and weak exogeneity plays an important role in pursuing the model reduction. It is demonstrated that co-breaking removes a deterministic shift caused by the collapse of the bubble economy in Japan in the early 1990s. Multivariate cointegration analysis then reveals that inflation plays a critical role in accounting for the long-run behaviour of the aggregate consumption. Further analysis finds that inflation and aggregate income are weakly exogenous with respect to a set of parameters of interest. Finally, a parsimonious data-congruent model for the aggregate consumption is estimated conditional on the set of weakly exogenous variables. © 2009 Elsevier B.V. All rights reserved.

1. Introduction This paper, inspired by Davidson et al. (1978, denoted as DHSY hereafter), aims to estimate a parsimonious data-congruent model for Japan's aggregate real consumption using quarterly data over the past two decades. It is known that Japan enjoyed an asset-price bubble in the late 1980s and then suffered from long stagnation due to the collapse of the bubble economy from the early 1990s to around 2002. See Yoshikawa (2001) for detailed analysis of Japan's prolonged stagnation. The sample period for estimation in this paper, running from 1981 to 2005, covers the era of economic turmoil caused by the creation and burst of the bubble economy. Thus, it is anticipated that econometric modelling will face much difficulty and splitting the sample period in two around the early 1990s may be required for a reliable econometric analysis. General-to-specific (Gets) econometric modelling adopted in this paper, however, shows that such samplesplitting is not necessary in order to arrive at a parsimonious datacongruent model of Japan's aggregate consumption. The introductory section briefly reviews the related literature and then describes the most significant aspects of the present paper. The Gets approach is a progressive research strategy in empirical modelling, in which the true data generating mechanism is unknown to researchers. In the Gets modelling, researchers start with the investigation of general unrestricted models and then proceed to a parsimonious congruent model by using a series of tests for the model reduction. The validity of the approach has been proved in a numerous books and research papers; see Hendry (1995 and 2001), Campos et al. (2005), inter alia. With regard to time series analysis,

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macroeconomic time series data often exhibit non-stationary behaviour and need to be treated as processes integrated of order 1 (denoted as I(1) hereafter) rather than stationary. Cointegration introduced by Granger (1981) therefore plays a critical role in time series econometrics, and a cointegrated vector autoregressive (VAR) model explored by Johansen (1988, 1996) has become a major econometric tool for macroeconomists. See Juselius (2006) for the extensive empirical research using cointegrated VAR models. The cointegrated VAR model is well fitted in the Gets methodology, as its analysis usually commences with the investigation of general unrestricted VAR models. Hendry and Mizon (1993) discuss a model reduction procedure in the framework of the cointegrated VAR model, and present a parsimonious congruent model for UK money demand. See also Hendry and Doornik (1994) and Kurita (2007) for the Gets methodology using the cointegrated VAR model. Modelling consumers' expenditure is one of the important research projects for macroeconomists. In the literature of consumption, DHSY mentioned above, is a seminal paper on the Gets modelling of aggregate consumption. They present parsimonious congruent representations of consumers' expenditure in the UK using equilibrium correction models, which are intimately linked to cointegration (Engle and Granger, 1987). See Hendry and von Ungern-Sternberg (1981), Hendry et al. (1990), Ermini (1999), for the updates of DHSY, and also see Muellbauer and Lattimore (1995) for the importance of DHSY in the applied macroeconomics and consumption literature. The achievements of DHSY and its successors provide motivation for modelling Japan's aggregate consumption using the Gets methodology. Consumption in Japan has been subjected to empirical investigations in various aspects; see Engle et al. (1993), Hamori (1996), Hall et al. (1997), Pagano (2004), Horioka (2006), Kohara and Horioka

T. Kurita / Economic Modelling 27 (2010) 574–584

(2006), and Kubota et al. (2008), inter alia. Aron et al. (2008) is, in particular, a noteworthy piece of work in that a data-congruent representation of Japan's aggregate consumption is achieved in the same framework as DHSY, using annual data covering the period of 1961–2006. The empirical success of Aron et al. (2008) is remarkable, but it would be better, in terms of the model's policy implications, to estimate Japan's aggregate consumption function using higher frequency data such as quarterly data. It seems that modelling recent behaviour of Japan's aggregate consumption using quarterly data is missing in the literature, probably due to the problem of limited availability of quarterly households' asset data. As mentioned above, Japan's economy suffered from stagnation after the burst of the bubble economy. It is of great interest, from the viewpoint of macroeconomic modelling and policy, to answer the question of whether a stable aggregate consumption function can be estimated from Japan's quarterly data covering the pre and post bubble economy, although we face the problem of limited data availability. Furthermore, DHSY adopt a single-equation approach thus all the variables apart from the UK aggregate consumption are assumed to be weakly exogenous for a set of parameters of interest. See Engle et al. (1983) and the Appendix of this paper for weak exogeneity. Aron et al. (2008) also adopt a single-equation approach to modelling Japan's consumption. Weak exogeneity is, however, a system property and thus should be tested using the data in a multivariate VAR framework rather than assumed from the outset of the analysis. In the absence of the guarantee that all the variables apart from consumption are weakly exogenous for parameters of interest, we should adopt a VARsystem approach — instead of a single-equation approach — to estimating an empirical representation for consumption with no loss of information. This paper, applying the Gets modelling methodology to a VAR system for Japanese quarterly data, successfully achieves a parsimonious data-congruent model for Japan's aggregate consumption over the past two decades, a period of economic turmoil, during which the bubble economy occurred and then collapsed. To the best of the author's knowledge, the present paper is the first empirical study that is successful in modelling quarterly aggregate consumption in Japan for the period of recent economic turmoil. It should be noted that the data-congruent model is estimated without splitting the sample period for estimation. Testing co-breaking, cointegration and weak exogeneity plays an important role in achieving the parsimonious representation of the consumption. Co-breaking is, in general, defined as the removal of deterministic shifts by using linear combinations of variables. Hendry (1997) considers the concept of co-breaking, and Clements and Hendry (1999, Ch.9) demonstrate its usefulness in the context of forecasting non-stationary time series subject to deterministic breaks. See Hendry and Massmann (2007) for the recent developments in research on cobreaking. As demonstrated below, the collapse of the bubble economy in Japan in the early 1990s has caused a simultaneous decline in the growth rate of the aggregate consumption and income. This phenomenon may be interpreted as the presence of a co-breaking relationship in the two aggregate variables. Testing for co-breaking is conducted in this paper, with a view to achieving a parsimonious model free from the influence of the deterministic shift caused by the bubble burst. Research on co-breaking in the framework of a cointegrated VAR model does not seem to be matured yet in the empirical literature. This paper can be regarded as a significant contribution to the literature in that it includes the application of a cobreaking analysis to the empirical modelling. Cointegration analysis in this paper reveals that inflation plays an important role in accounting for consumers' expenditure in Japan. Inflation effects on consumption are demonstrated by not only DHSY, but also Hendry and von Ungern-Sternberg (1981), Molana (1990), and Deaton (1992), although the literature on consumption in general appears to pay a little attention to their importance. As shown below

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in this paper, a set of restrictions on the cointegration space finds a long-run linkage between the average propensity to consumption and inflation. According to Hendry and von Ungern-Sternberg (1981), it is possible to interpret that inflation plays the role of a proxy for a liquid asset–income ratio. It is therefore justifiable to estimate an empirical consumption function using inflation data instead of households' asset data, where the availability of the latter data on a quarterly basis is limited in the case of Japan. Thus, the restricted cointegrating relationship can be consistent with the life-cycle hypothesis on consumers' expenditure. The dynamics properties of inflation effects are also of interest, namely, its short-run and long-run effects on aggregate consumption. Using an equilibrium correction model, the paper investigates how inflation effects vary in the short-run and long-run dynamics. Furthermore, this paper shows that inflation and aggregate income are in fact judged to be weakly exogenous for a set of parameters of interest. Hence, a single-equation model for aggregate consumption conditional on inflation and aggregate income, in line with DHSY, is justified in terms of statistical inference without any loss of information. The organization of this paper is as follows. Section 2 reviews the DHSY model and gives a motivation for empirical modelling of Japan's aggregate consumption. Section 3 briefly reviews a likelihood-based analysis of cointegrated VAR models and co-breaking. Section 4 presents an overview of data for total consumption and other related series in Japan, discussing various features of the data. Section 5 aims to achieve an econometric model for Japan's aggregate consumption based on the Gets methodology. This section estimates general unrestricted models allowing for regime shifts and a deterministic break, then proceeding to multivariate cointegration and co-breaking analysis. After the process of the model reduction incorporating tests for weak exogeneity, a parsimonious model for the aggregate consumption is estimated conditional on a set of weakly exogenous variables. The overall summary and conclusion are provided in Section 6. All the empirical analysis and graphics in this paper use Ox (Doornik, 2006) and OxMetrics/PcGive (Doornik and Hendry, 2006). 2. The DHSY model This paper follows DHSY for econometric modelling methodology and model interpretation. The main result of DHSY is reviewed here in the same manner as Banerjee et al. (1993, Ch.2), in order to motivate an empirical investigation of Japanese macroeconomic data. After the model reduction, DHSY achieves the following parsimonious datacongruent model for aggregate real consumption in the UK: Δ 4cˆt = 0:47 Δ4yt − 0:21 Δ1 Δ4yt − 0:10ðct−4 −yt−4 Þ ð0:04Þ

+

ð0:05Þ ð0:02Þ o 0:01 Δ4 Dt − 0:13 πt − 0:28 Δπt ; ð0:09Þ ð0:07Þ ð0:15Þ

ð1Þ

where ct is the natural logarithm of real consumers' expenditure on non-durable goods and services, yt is the natural logarithm of real personal disposable income, πt is annual inflation based on the price deflator for consumption, Dot is a dummy variable for changes in taxation, and figures in parentheses denote standard errors. DHSY use seasonally unadjusted data thus the fourth-order difference is taken with respect to all the variables in the equation. The DHSY model passes a battery of diagnostic tests, thereby being judged to be a satisfactory representation of the data from a statistical viewpoint. The third term in Eq. (1) appears as a combination of two regressors in level, representing average propensity to consume in logarithm and acting as an equilibrium correction mechanism (ecm) in the model. The mechanism corresponds to a cointegrating combination of non-stationary variables, as reviewed in Section 3. Note that πt represents the level of inflation, thus it may be

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incorporated into the equilibrium correction term as well. Also note that no asset-related variable plays any explicit role in the model. Let us assume time-invariant mean such that E(Δ4ct) =gc, E(Δ4yt) = gy, and E(πt) =gπ, where E(⋅) denotes expectation operator. Taking expectations of Eq.(1) above for fixed values of the estimates leads to

called the cointegrating space. Condition (5) implies that there are at least p − r common stochastic trends and cointegration arises when r ≥ 1. The third condition is

gc = 0:47gy −0:13gπ −0:10ðc*−y*Þ;

p×p−r 1 are orthogonal complewhere Γ = Ip −∑k− i= 1 Γi, and α⊥, β⊥ ∈ R ments such that α′α⊥ = 0 and β′β⊥ = 0 with (α, α⊥) and (β, β⊥) being of full rank. The final condition prevents the process from being I(2) or of higher order. If these conditions are satisfied, an I(1) cointegrated VAR model is defined as a sub-model of Eq. (4), for β⁎ = (β′, γ′)′ and X ⁎t− 1 = (X′t − 1,t)′, as follows:

ð2Þ

where c⁎ and y⁎ denote the logs of steady-state consumption and income, that is, c⁎ = log(C ⁎) and y⁎ = log(Y ⁎), respectively. Given the proportional long-run solution gc = gy, Eq. (2) yields C* = expð−5:3gy −1:3gπ ÞY*:

ð3Þ

rankðα′⊥ Γβ⊥ Þ = p−r;



According to Hendry and von Ungern-Sternberg (1981), and Banerjee et al. (1993, Ch.2), this form of solution is in accord with the life-cycle hypothesis by Ando and Modigliani (1963), such that the coefficient of gπ can be interpreted as a negative value of the liquid asset–income ratio. The beauty of the DHSY model is that it is not only a satisfactory representation of the data from a statistical point of view, but is also an econometric model consistent with macroeconomic theory and insight. For this reason, DHSY is regarded as a seminal work in time series econometrics and applied macroeconomics. See Hamilton (1994, Ch.19), and Muellbauer and Lattimore (1995) for a critical role played by DHSY in the literature. Moreover, DHSY provides methodology for modelling aggregate consumption consistent with the life-cycle hypothesis without using any explicit asset variables. For a number of countries and regions it is often difficult to obtain households' asset and wealth data on a quarterly basis. Following DHSY, we are able to model time series data of aggregate consumption in such a manner as consonant with consumption theory.

k−1

ΔXt = αβ*′Xt−1 + ∑ Γi ΔXt−i + μ + ΦDt + εt ; i=1

ð6Þ

This section reviews a likelihood-based analysis of cointegrated VAR models introduced by Johansen (1988) and fully described by Johansen (1996), and also provides a review on co-breaking based on Hendry and Massmann (2007), and Johansen, Mosconi and Nielsen (2000). Let us consider an unrestricted VAR(k) model for p-dimensional time series, X− k + 1, ..., XT, which is given by

which is the basis for the subsequent cointegration analysis and model reduction. Since the cointegrating rank r is usually unknown to investigators, it needs to be determined using the data. A log-likelihood ratio (logLR) test statistic is given by the null hypothesis of r cointegration rank H(r) against the alternative hypothesis H(p). The asymptotic quantiles for the logLR test statistic are provided by Johansen (1996, Ch.15), and see also Nielsen (1997) and Doornik (1998) for the method of gamma approximations to calculate the quantiles. After determining the cointegrating rank, one is able to test various restrictions on α, β and γ in order to pursue the adjustment structure and cointegrating relationships subject to economic interpretation. Next, let us review co-breaking in the framework of a cointegrated VAR model. Hendry and Massmann (2007) is the main reference for the concept of co-breaking. As explained in the Introduction, co-breaking is in general defined as the removal of deterministic shifts by using linear combinations of variables. One needs to adjust the standard model (4) in order to incorporate co-breaking in a cointegrated vector autoregression. Johansen et al. (2000) consider a VAR system allowing deterministic breaks, which proves to be a model useful for the analysis of cobreaking. The number of structural breaks is given by q, and the length of each sub-sample is expressed as Tj − Tj− 1 for j = 1, ..., q and 0 = T0 b T1 b T2 b ⋯ b Tq = T. The VAR model in Johansen et al. (2000) is then formulated conditionally on the first k observations of each sample, XTj − 1 + 1,...,XTj − 1 + k, as follows:

  k−1 X ΔXt = ðΠ; Πl Þ t−1 + ∑ Γi ΔXt−i + μ + ΦDt + εt ; for t = 1; :::; T; t i=1

  k−1 X ΔXt = ðΠ; Πl;j Þ t−1 + ∑ Γi ΔXt−i + μj + ΦDt + εt ; t i=1

3. Cointegrated VAR model and co-breaking

ð4Þ where Dt is a s-dimensional vector of deterministic terms apart from linear trend and intercept, such as seasonal and impulse dummies, and innovations ε1, ..., εT have independent and identical normal N(0, Ω) distributions conditional on the starting values X−k + 1, ..., X0. The parameters Π, Γi, Ω ∈ Rp × p, Πl, μ ∈ Rp and Φ ∈ Rp × s vary freely and Ω is positive definite. For the purpose of performing an I(1) cointegration analysis, three regularity conditions need to be fulfilled. The first condition is that the characteristic roots obey the equation |A(z)| = 0, where k−1

i

AðzÞ = ð1−zÞIp −Πz− ∑ Γi ð1−zÞz ; i=1

ð7Þ

for Tj − 1 + k + 1 ≤ t ≤ Tj. Out of the regularity conditions given above, Eq. (5) needs to be replaced by rankðΠ; Πl;1 ; :::; Πl;q Þ≤r or ðΠ; Πl;1 ; :::; Πl;q Þ = αðβ′; γ′1 ; :::; γ′q Þ:

ð8Þ

The logLR test statistic for the cointegrating rank can then be constructed, with the result that its limiting distribution is different from the conventional one due to the presence of deterministic breaks. Johansen et al. (2000) conduct the analysis of response surfaces in order to provide the asymptotic quantiles of rank test statistics for Eq. (7). For the sake of simplicity of the exposition, the number of breaks is set to be two (q = 2) and the Granger–Johansen representation of Eq. (7) can then be given by

and the roots satisfy |z| N 1 or z = 1. This condition ensures that the process is neither explosive nor seasonally cointegrated. The second condition is given by

Xt = C

rankðΠ; Πl Þ≤r or ðΠ; Πl Þ = αðβ′; γ′Þ;

where C =β⊥(α⊥′Γβ⊥)− 1α⊥′ and υt is a p-dimensional stationary process with zero expectation. The slope parameter τl,j can be expressed as

ð5Þ

where α, β ∈ Rp × r and γ′ ∈ Rr for r b p. The space spanned by α is referred to as the adjustment space, while the space spanned by β is

t

∑

i = Tj + k + 1

εi + υt + τc;j + τl;j t; j = 1; 2;

−1

τl ; j = Cμj + ðCΓ−IÞβðβ′βÞ

γ′j ; j = 1; 2;

ð9Þ

ð10Þ

T. Kurita / Economic Modelling 27 (2010) 574–584

while τc,j depends on the initial values in each sub-sample in such a way that β′τc,j is an identified function of a set of parameters (see Johansen et al., 2000, for details). If the slope parameter in the cointegration space is time-invariant i.e. γ1 =γ2 =γ, one then observes that co-breaking takes place in the cointegrating space: β′τl;1 = β′τl;2 = − γ′:

ð11Þ

That is, the expansion of τl,j in the direction of β′ eliminates Cμj, thereby yielding the time-invariant slope parameter. Thus the cointegrating vectors are interpreted as co-breaking vectors as well in this case. It is possible to consider that Eq. (11) represents the condition for trend co-breaking, by analogy with drift co-breaking discussed by Hendry and Massmann (2007) and described in the Appendix. 4. An overview of Japan's time series data This section presents an overview of quarterly data for total consumption and other related series in Japan, and discusses a possible long-run relationship subject to economic interpretation. See the Appendix A for details of the data. Fig. 1(a) displays time series data of ct, which is the natural logarithm of aggregate real consumption in Japan. The data exhibit upward trending behaviour, appearing to be an integrated process rather than stationary. The trending feature seems to be caused by both stochastic and deterministic trends. Fig. 1(b) shows the firstorder difference of ct, denoted by Δct. Data plots of Δct exhibit meanreverting feature with no systematic trend, so it seems reasonable to treat Δct as a stationary process with non-zero mean. As the firstorder differencing has generated a non-integrated process, the original series ct is considered to be an I(1) process. The dotted vertical line in Fig. 1(a) corresponds to 1991, when the asset-price

277

bubble burst in Japan. It should be noted that ct has a kink around 1991, so the average growth rate of consumption is negatively affected by the collapse of the bubble economy, as shown in Fig. 1(b). In addition, large outliers are detected in Δct around in 1989 and 1997 in Fig. 1(b), corresponding to the introduction of value added tax in 1989 and an increase in the tax rate in 1997, respectively. Fig. 1(a) also shows time series plots of yt, which is the natural logarithm of real GDP in Japan. The scale of ct and yt is matched by their means in the figure. Data of its first-order difference, Δyt, are presented in Fig. 1(b). Analogous to the aggregate consumption, yt appears to be an I(1) process so Δyt is considered to be a nonintegrated process. A large impact of the bubble burst on the real GDP is also visible in Fig. 1(a) and (b) around 1991, in line with that on the total consumption. Time series data of annual (year-on-year) inflation based on the GDP deflator in Japan, denoted by πt, are presented in Fig. 1(c), together with its linear regression line. In contrast to the differenced series discussed above, the data for πt show downward trending behaviour, wandering around the linear regression line. Fig. 1(c) also presents data for the log of the GDP deflator, pt. In line with the behaviour of πt, the data for pt exhibit a smooth hump-shape feature, which seems to be well approximated by quadratic trend. It would therefore be reasonable to treat πt as an I(1) process subject to deterministic linear trend. An economic implication of the presence of linear trend in annual inflation is discussed in Section 5.4. Finally, Fig. 1(d) displays time series data of ct − yt, a difference of the two logged real variables. In contrast to the data in Fig. 1(a) and (c), the series ct − yt exhibits random-walk behaviour with neither deterministic trend nor shift. It seems that the deterministic trend and shift are eliminated by taking the linear combination of ct and yt, although the stochastic trends still remain in ct − yt. The overview of the data thus indicates that co-breaking may have occurred in ct and yt around the early 1990s, as described in the introduction. In

Fig. 1. An overview of macroeconomic time series data in Japan.

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T. Kurita / Economic Modelling 27 (2010) 574–584

addition, Fig. 1(d) also presents plots of trend-adjusted inflation (residuals from the regression of inflation on a deterministic trend and constant), which is denoted πt̅. The scale of ct − yt and πt̅ is matched by their means in the figure. Counter-cyclical behaviour of both series is observed, and it is therefore likely that the common stochastic trends cancel out by taking a linear combination of ct − yt and πt̅. The overview of the data, therefore, leads us to the conjecture that a co-breaking and cointegrating relationship may be embedded in the Japanese quarterly data. The present paper investigates this using a rigorous time series analysis. 5. General-to-specific econometric modelling We are now in a position to pursue Japan's aggregate consumption function using the Gets methodology. The set of macroeconomic data in Japan discussed above are labeled Xt as follows: Xt = ðct ; yt ; πt Þ′:

ð12Þ

The sample period spans from the third quarter of 1981 to the second quarter of 2005 (denoted 1981.3–2005.2 hereafter). As demonstrated below, the lag order of the VAR models is set to be 3, so that the effective sample is 1982.1–2005.2 and the number of observations for estimation is 93. See the Appendix A for details of the data. This section is composed of five sub-sections. Section 5.1 estimates two unrestricted VAR models paying attention to the influences of the bubble burst and some policy changes. Mis-specification tests for the VAR models are examined in order to demonstrate the validity of the models for the subsequent cointegration analysis and model reduction. The choice of cointegrating rank is investigated in Section 5.2 using Johansen's methodology. Section 5.3 then investigates whether co-breaking exists in the cointegration space. Section 5.4 tests a set of hypotheses on the cointegrating space so as to pursue an interpretable long-run economic relationship. Section 5.5 examines if inflation and aggregate income are judged to be weakly exogenous for a set of parameters of interest. In this section, a test for drift co-breaking is also performed to check that the conditional model is not subject to an intercept break. Finally, in Section 5.6, a parsimonious equilibrium correction model for aggregate consumption is estimated conditional on weakly exogenous variables, and an economic interpretation of the estimated model is also presented. 5.1. Unrestricted VAR models This sub-section estimates two unrestricted VAR models to provide a starting point for the subsequent cointegration analysis and model reduction. The first model is a standard one given by Eq. (4), while the second model is a modified version given by Eq. (7) for q = 2, which takes account of a possible structural break in the deterministic terms. The effects of the bubble burst seem to be present in the data, as discussed in Section 4. It would therefore be appropriate to estimate two models with a view to conducting a comparative empirical analysis. The two models are named as follows:

residuals for ct and yt are calculated using structural time series modelling by Koopman, Harvey, Doornik and Shephard (2006). It is known that such auxiliary residuals are useful for detecting structural breaks in the deterministic terms in a univariate setting. Fig. 2 displays t-test plots based on auxiliary residuals of slope equations for both ct and yt after taking account of the impacts of VAT by dummy variables. See Koopman et al. (2006, Ch.5) for details of t-test plots. According to the figure, t-test plots for ct and yt are both greater than 2.0 in absolute value around the early 1990s, indicating the presence of a break in the deterministic trend in line with the conjecture based on the data overview. The choice of the break point is therefore justified in a univariate setting, and it is later examined in a multivariate setting as well in the context of testing for co-breaking. With reference to the lag order of the VAR models, the order of 3 is chosen for both models based on the F-tests for lag length determination. Reduction from lag 3 to 2 was rejected at the 1% level, with the test statistics F(9,187) = 3.03 [0.00] and F(9,175) = 2.73 [0.005] for Model 1 and Model 2, respectively (figures in the square brackets denote p-values). Three impulse dummy variables are added to Model 2 so as to set the residuals from 1991.1 to 1991.3 zero, i.e. the likelihood function for the second sub-sample is conditioned upon the sample data from 1991.1 to 1991.3 (see Johansen et al., 2000, for details of the required procedure). The unrestricted VAR models are purely statistical representations, so the estimated coefficients are not necessarily subject to economic interpretation. After the model reduction it is possible to pursue such interpretation. The unrestricted VAR models should provide a basis for a parsimonious representation of the data, and therefore needs to pass a battery of diagnostic tests such as normality and no autocorrelation in the residuals. It turns out, however, that the estimated models suffer from non-normality and autoregressive conditional heteroscedasticity (ARCH) effects in the residuals. These are due to outliers caused by policy and regime changes occurring in the sample period. These outliers are still present even if the lag length of the models increases. The following blip and impulse dummy variables corresponding to several historical events therefore need to be introduced: Dvat1;t = ½1 ð1989:1Þ; −1 ð1989:2Þ; Dvat2;t = ½1 ð1997:1Þ; −1 ð1997:2Þ; Doil;t = 1 ð1987:2Þ; and 0 otherwise. These variables are included in the model unrestrictedly by following Doornik et al. (1998). The first two blip dummy variables, Dvat1,t and Dvat2,t, capture large outliers found in ct in Fig. 1. These blip dummy variables are included in the model in order to capture two types of effects from value added tax: the first dummy

• Model 1: a model for standard cointegration analysis, Eq. (4), • Model 2: a model for cointegration analysis subject to a deterministic shift, Eq. (7). Regarding Model 2, one needs to specify a location where a structural shift has taken place in its deterministic terms. An overview of the data in Section 4 suggests that the growth rate of the real income and consumption declined in 1991, corresponding to when house prices started falling after a huge decrease in stock prices. Thus it seems appropriate to set the break point to be 1991.1. In order to check if the choice of the break point is statistically justified, auxiliary

Fig. 2. Auxiliary residuals: detecting structural breaks in slope.

T. Kurita / Economic Modelling 27 (2010) 574–584 Table 1 Mis-specification tests for the unrestricted VAR models. Model 1 Single equation tests Autocorr. [Far(5,74)] ARCH [Farch(4,71)] Hetero. [Fhet(20,58)] Normality [χ2nd(2)]

ct 1.60 0.51 0.71 0.78

Vector tests Autocorr. [Far(9,180)] − [Far(45,184)]

0.97 [0.47] 1.22 [0.18]

[0.17] [0.73] [0.80] [0.68]

yt 0.99 0.49 0.49 0.69

[0.43] [0.74] [0.96] [0.71]

Hetero. [Fhet(120,313)] Normality [χ2nd(6)]

πt 2.01 0.18 0.69 1.75

[0.09] [0.95] [0.82] [0.42]

0.75 [0.97] 4.76 [0.58]

279

second panel for each model provides two types of modulus (denoted mod) of the six largest eigenvalues of the companion matrix, unrestricted and restricted with r = 1. These are the reciprocal values of the roots of A(z) discussed in the last section. No eigenvalue over 1.0 suggests that the model does not include any explosive root, and in the restricted case all the eigenvalues apart from the first and second ones appear to be distinct from a unit root. These outcomes suggest that the regularity conditions given in the previous section are satisfied, supporting the validity of I(1) cointegration analysis with r = 1. 5.3. Testing co-breaking in the cointegrating space

Model 2 Single equation tests Autocorr. [Far(5,69)] ARCH [Farch(4,66)] Hetero. [Fhet(22,51)] Normality [χ2nd(2)]

ct 1.27 0.56 1.05 2.03

Vector tests Autocorr. [Far(9,168)] −[Far(45,170)]

0.73 [0.68] 0.97 [0.53]

[0.29] [0.69] [0.42] [0.36]

yt 0.73 0.42 0.49 0.60

[0.61] [0.79] [0.96] [0.74]

Hetero. [Fhet(132,275)] Normality [χ2nd(6)]

πt 1.44 0.38 0.83 4.07

[0.22] [0.82] [0.68] [0.13]

0.77 [0.95] 7.41 [0.28]

Note. Figures in the square brackets are p-values.

corresponds to the introduction of the tax in 1989, while the second dummy to an increase in the tax rate in 1997. The third dummy variable, Doil,t, corresponds to a sharp decrease in inflation in 1987, which appears to be caused by the preceding large reduction of oil prices. Table 1 presents a set of diagnostic tests for the two unrestricted VAR models. PcGive reports most of the test results in the form Fj(k, T −l), which denotes an approximate F-test against the alternative hypothesis j : kth-order serial correlation (Far: see Godfrey, 1978, Nielsen, 2006), kth-order ARCH (Farch: see Engle, 1982), heteroscedasticity (Fhet: see 2 White, 1980). A chi-square test for normality (χnd : see Doornik and Hansen, 1994) is also provided. All the mis-specification test statistics in the table are insignificant at the 5% level, allowing us to conclude that both of the unrestricted VAR models can be subject to the subsequent modelling procedure.

The common cointegrating rank for the two models, together with the satisfactory diagnostic tests in the previous section, indicates that both models seem to have a potential for a data-congruent representation. As discussed in Section 4, there could be a co-breaking relationship between aggregate consumption and income, so that their linear combination may eliminate the deterministic shift caused by the bubble burst. If this conjecture is true, the slope coefficients in the Granger–Johansen representation of Model 2 will be distinct between the two regimes (τl,1 ≠ τl,2), while the slope coefficients in the cointegration space of Model 2 will hold the identical value (γ1 = γ2). Model 2 will then reduce to Model 1 effectively in terms of the treatment of deterministic trend in the cointegration space. The linear trend with no break will then matter for inflation rather than the average propensity to consumption, as demonstrated in Fig. 1(c) and (d). Regarding τl,j for j = 1, 2, the null hypothesis of interest is therefore given by trend:cob:GJ

H0

: τl;1 = τl;2 ;

and it is expected that Htrend.cob.GJ is rejected due to the presence of a 0 trend break, in line with the recursive univariate tests in Section 4.1. Let us define τl = (τl,1, τl,2), γ = (γ1, γ2)′ and μ = (μ1, μ2). Following Johansen et al. (2000) and recalling that τl,j for j = 1, 2 is expressed as Eq. (10), one can find this hypothesis is formulated as trend:cob:GJ

: γ = Gφ and μ = ζM′ + αδM′⊥ ;

5.2. Determination of the Cointegrating rank

H0

This sub-section discusses the choice of cointegrating rank r in the two VAR models. Table 2 presents the logLR test statistics for cointegrating rank, in addition to the modulus of the six largest roots of the companion matrix. The logLR test statistics for the cointegrating rank [−2logQ(H(r)| H(p))] are reported in the first panel for each model. For Model 1, p-values in the square brackets are calculated using the ordinary limiting quantiles, whereas, for Model 2, p-values are obtained using the response surfaces by Johansen et al. (2000). The logLR tests for both models reject the null of r = 0, hence supporting r = 1. The

where G = M = (1, 1)′, while φ, δ ∈ R and ζ ∈ R3 are unknown parameters to be estimated. Since the regularity condition on G and M given in Theorem 4.2 in Johansen et al. (2000) is satisfied, the logLR test for H 0trend.cob.GJ against H(1) is asymptotically χ2distributed. With regard to γj for j = 1, 2 in the cointegrating space, the null hypothesis of interest is given by

trend:cob:CS

Model 1

H0 r=0 69.76[0.00]** 0.99 0.81 1.00 1.00

0.81 0.65

r≤1 21.96[0.14] 0.63 0.63 0.52 0.52

r≤2 5.91[0.48] 0.47 0.40

0.65 0.71

r≤1 32.83[0.13] 0.65 0.47 0.71 0.48

r≤2 10.34[0.50] 0.47 0.47

Model 2 − 2logQ(H(r)|H(p)) mod (unrestricted) mod (r = 1)

r=0 69.55[0.00]** 0.86 0.75 1.00 1.00

Note. ** Denotes significance at the 1% level.

: γ1 = γ2 ;

and it is expected that Htrend.cob.CS is not rejected due to co-breaking in 0 the cointegration space. This hypothesis can also be formulated using G and φ given above as follows:

Table 2 Determination of the cointegration rank.

− 2logQ(H(r)|H(p)) mod (unrestricted) mod (r = 1)

trend:cob:CS

H0

: γ = Gφ:

Johansen et al. (2000) shows that the logLR test for Htrend.cob.CS 0 against H(1) has an asymptotic χ2-distribution. In order to claim the presence of co-breaking, one needs to confirm that Htrend.cob.GJ is 0 rejected whereas Htrend.cob.CS is not rejected. 0 Table 3 provides the results of co-breaking tests, and df represents degree of freedom. The null hypothesis of constancy of τl is rejected at the 5% level, while that of γ is not rejected at the same level. It is therefore justifiable to consider that the deterministic break is removed by co-breaking simultaneously occurring with cointegration.

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T. Kurita / Economic Modelling 27 (2010) 574–584

Table 3 Testing co-breaking in the cointegrating space.

Table 5 Testing weak exogeneity in the cointegrated VAR model.

Test on τl − 2logQ(Htrend.cob.GJ |H(1)) 0 Test on γ − 2logQ(Htrend.cob.CS |H(1)) 0

13.7[0.00]** (df = 3) 0.10[0.75](df = 1)

Note. Figures in the square brackets are p-values.

Model 1 logLR Model 2 logLR

yt

πt

yt and πt

1.49[0.69](df = 3) 2.29[0.68](df = 4)

6.37[0.09](df = 3) 9.37[0.05](df = 4)

8.04[0.09](df = 4) 10.09[0.07](df = 5)

Note. Figures in the square brackets are p-values.

This is an encouraging finding in terms of the pursuit of a parsimonious model and the justification of trend-adjusted inflation. 5.4. Inflation effects and the restricted cointegrating relation The overview of the data in Fig. 1(d) indicates that the average propensity to consume is negatively correlated with trend-adjusted inflation. As a result, their linear combination is conjectured to be stationary rather than an integrated process. If the cointegrating space is normalised for ct, the restriction of interest is then to impose a coefficient −1 on yt, with the intention of checking if the conjectured combination is in fact judged to be a stationary relationship. Fig. 1(d) shows that the trend-adjusted inflation behaves like a mirror image of ct − yt, suggesting that it may be possible to impose a coefficient 1 on πt. The validity of these two restrictions on β* is jointly investigated with regard to Model 1. Concerning Model 2, the restriction of the common slope coefficient investigated in the previous section is also imposed on β*. It is known that a logLR test statistic for restrictions on β* is asymptotically χ2-distributed (see Johansen, 1996, Ch.7 and Johansen et al., 2000). The test results are presented in Table 4. The table reports the restricted estimates for β*′ with the corresponding logLR test statistics. Both of the test statistics are small enough and the restrictions are therefore not rejected at the 5% level. Both Model 1 and Model 2 have yielded the almost identical empirical representation of ecmt. Setting the trend coefficient to be 0.001, one finds ecmt to be

drift:cob

ˆ ′ X = c −y + − πt ; ecmt = β t t t for − π = π + 0:001t: 

t

It is known that a logLR test statistic for restrictions on α has an asymptotic χ2 distribution (see Johansen, 1996, Ch.8). The results of tests for weak exogeneity are presented in Table 5. The set of restrictions on β* as examined in the previous section is preimposed when testing zero-restrictions on α, and the degrees of freedom reflect the total number of restrictions imposed on the parameters. According to the first column of the table, the hypothesis of yt being weakly exogenous for the estimation of the parameters (31) is not rejected at the 5% level with regard to both Model 1 and Model 2. The second column of the table shows that the same hypothesis is not rejected for πt at the 5% level as well, although the pvalues are rather small in comparison with those for yt. The hypothesis of yt and πt being jointly weakly exogenous for the parameters (31) is not rejected for both models, as presented in the third column of the table. The overall finding allows us to model a conditional single-equation model for ct given yt and πt, instead of modelling a joint system, in which all the variables are treated as endogenous variables. The presence of drift co-breaking in the conditional model is also examined in order to demonstrate that Model 2 can reduce to Model 1 effectively. See the Appendix for details of drift co-breaking. A testing procedure for drift co-breaking is to check that the following hypothesis is not rejected using the estimated conditional model for ct:



ð13Þ

t

The deterministic trend may be treated as a local approximation to agents′ long-term expected inflation formed in the sample period, during which steady downward pressure on price growth exists, i.e. the yen's long-term appreciation, increased cheap imports from China and other Asian countries, and a technological progress in the information industry. It is therefore justifiable to use linear trend for accounting for the behaviour of inflation during the sample period of interest. Eq. (13), in line with the DHSY model, represents a long-run economic linkage consisting of average propensity to consume and inflation.

H0

: μ˜ w;1 = μ˜ w;2 ;

where μ˜ w;i for i = 1, 2 denote time-varying intercepts for the conditional model, as given in the Appendix A for drift co-breaking. The logLR test statistic for Hdrift.cob is 1.87[0.17] , in which the figure in 0 the square bracket is a p-value according to χ2(1). The null hypothesis is not rejected at the 5% level, so it can be concluded that the conditional model based on Model 2 is free from an intercept shift, reducing to that based on Model 1 effectively. Weak exogeneity and drift co-breaking permit us to estimate a conditional single-equation model for ct based on Model 1, which is further reduced in the next sub-section. 5.6. A parsimonious model for Japan's aggregate consumption

5.5. Testing weak exogeneity and drift co-breaking This sub-section examines whether any of the variables in the system is judged to be weakly exogenous for parameters of interest. See the Appendix A for details of weak exogeneity. In the context of the formulation given in Section 3, testing weak exogeneity corresponds to testing zero-restrictions on the adjustment space α. Table 4 Restricted cointegrating relationships. Model 1 β̂*′

ct 1 ð−Þ

yt −1 ð−Þ

πt 1 ð−Þ

t 0:0009 ð6:9e−005Þ

Δ cˆ t = − 0:18 ecmt−1 + 0:54 Δyt − 0:36 Δyt−2 + 0:19 Δct−2

logLR 0.52[0.77](df = 2)

ð0:03Þ

ct 1 ð−Þ

ð0:06Þ

ð0:08Þ

ð0:07Þ

− 0:33 Δπt + 0:18 Δπt−1 + 0:17 Δπt−2 ð0:09Þ

ð0:08Þ

ð0:09Þ

+ 0:02 Dvat1;t + 0:02 Dvat2;t − 0:09;

Model 2 β̂*′

The final sub-section presents a parsimonious congruent model for ct conditional on the weakly exogenous variables, yt and πt. The restricted cointegrating relationship ecmt − 1 will act as the equilibrium correction mechanism in the parsimonious model. First, a general autoregressive distributed lag model based on Model 1 is estimated conditional on the weakly exogenous variables. Insignificant regressors are then eliminated from the model step by step, so that the parsimonious representation of aggregate consumption is attained as follows:

yt −1 ð−Þ

πt 1 ð−Þ

t ⋅ 1(t ≤ 1991.1) 0:0008 ð−Þ

t ⋅ 1(t ≥ 1991.2) 0:0008 ð0:0001Þ

Note. Figures in the square brackets are p-values.

ð0:003Þ

logLR 1.79[0.62](df = 3)

ð0:003Þ

ð0:02Þ

ˆ = 0:004; Far ð5; 78Þ = 1:37½0:24; χ2nd ð2Þ = 1:72½0:43; σ Farch ð4; 75Þ = 1:76½0:15; Fhet ð18; 64Þ = 1:25½0:25;

ð14Þ

T. Kurita / Economic Modelling 27 (2010) 574–584

where σ̂ is the standard error of the regression. None of the diagnostic tests is significant at the 5% level, suggesting that the parsimonious model is a satisfactory representation of the data. Fig. 3(a) records the actual and fitted values, and Fig. 3(b) shows the scaled residuals. The residual density function is displayed in Fig. 3(c), and the residual correlogram is presented in Fig. 3(d). Consistent with the above test results, all the graphs indicate no evidence for model mis-specification. Parameter constancy is also required in a data-congruent representation. The recursive residuals are presented in Fig. 4(a). Fig. 4(b) and (c) show recursive one-step and break-point Chow tests (see Chow, 1960), respectively. All the recursive graphics support the parameter constancy of the preferred parsimonious model. Moreover, the model is re-estimated saving the final eight observations for forecasts. Fig. 4(d) presents the sequence of one-step forecasts, and its tracking is satisfactory. The overall evidence allows us to reach the conclusion that the parsimonious model is a data-congruent representation. It should be noted that the representation has been achieved using the data over the past two decades, without splitting the sample period for estimation. In (14), the coefficient of ecmt − 1 is negative and highly significant, indicating the existence of a stable equilibrium correction mechanism in the parsimonious model. The term ecmt − 1 could represent the long-run effects of inflation on average propensity to consume, while its short-run effects could be embodied in the coefficients for the firstorder differenced terms of inflation. The coefficients for inflation in the short-run dynamics add up close to zero (− 0.33 + 0.18 + 0.17 = 0.02). Thus the restriction of the sum being equal to zero deserves consideration. The test statistic for the restriction is 0.02 [0.90], in which the figure in the square bracket is a p-value according

281

to χ2(1). The test result therefore suggests that the total effects of the short-run inflation dynamics reduce to zero. The short-run dynamics in inflation correspond to inflation accelerations, which could influence consumers' behaviour. Nonzero inflation acceleration may imply the presence of rapid price changes. A positive inflation acceleration, for instance, may generate consumers' surprise, having a sharp negative effect on their behaviour. The downward effect, however, could gradually be mitigated as consumers come to distinguish between the short-run and long-run inflation dynamics. With the passage of time, the increase in inflation is incorporated in ecmt − 1 such that it has a negative long-run impact on consumers' expenditure. The short-run effects may then turn to be positive so as to offset the previous decrease in consumption caused by a surprise at the rapid price increase. Finally, the long-run solution of (14) is derived using the same methodology as the DHSY model described in Section 2. Let us assume E(Δct) = gc, E(Δyt) = gy, and E(π t̅ ) = gπ, and define c* = log(C*) and y* = log(Y *), where C * and Y * denote steady-state consumption and income, respectively. Taking expectations of both sides of Eq. (14) for fixed values of the estimates, one finds gc = 0:22gy −0:22gπ −0:22ðc*−y*Þ:

ð15Þ

Under the proportional long-run solution gc = gy, Eq. (15) leads to C* = expð−3:55gy −gπ ÞY*:

ð16Þ

As in the DHSY model, it is possible to consider that the solution Eq. (16) is in line with the life-cycle hypothesis; the coefficient of gπ can be interpreted as a negative value of the liquid asset–income ratio.

Fig. 3. Fitted and actual values, scaled residuals, residual density and correlogram.

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T. Kurita / Economic Modelling 27 (2010) 574–584

Fig. 4. Recursive graphics and the sequence of one-step forecasts.

Thus, (16) could indicate the presence of significant wealth effects on aggregate consumption, as in Aron et al. (2008). Also note that, according to (16), the derived average propensity to consumption C*/Y* is a decreasing function of the growth rate gy, in accord with DHSY and others such as Modigliani (1975). Thus, when income is falling, the propensity to consumption moves upward to prevent the level of consumption from moving downward further. This indicates the presence of a rachet effect in Japan's aggregate consumption (see Cabinet Office, Government of Japan, 2005). This finding also agrees with Horioka (2006), in which it is indicated that Japan's consumption in recent years did not drag the economy but rather prevented it from stagnating further. The presence of such a rachet effect in the Japanese economy seems to have been a very important factor in the prevention of serious economic depression after the bubble burst, although the economy suffered from prolonged slowdown due to such problems as bad loans and stagnated investment. 6. Summary and conclusion The aim of this paper is, inspired by DHSY, estimating a parsimonious data-congruent model for Japan's aggregate consumption for the period of economic turmoil over the past two decades, during which an asset–price bubble occurred and then collapsed in Japan. Testing co-breaking, cointegration and weak exogeneity plays an important role in the attainment of the parsimonious representation. The Gets modelling commences with the estimation of unrestricted dynamic systems, and then proceeds to multivariate cointegration analysis. It is demonstrated that co-breaking in the cointegration space eliminates the deterministic shift caused by the bubble burst in the early 1990s. A set of restrictions on the

cointegrating vector reveals the presence of a long-run linkage between the average propensity to consumption and inflation, leading to the inference that the estimated long-run relation can be consistent with the life-cycle hypothesis on consumers' expenditure. In the cointegrated VAR system, inflation and aggregate income are judged to be weakly exogenous for a set of parameters of interest. Thus the estimation of a conditional model for aggregate consumption given inflation and aggregate income is justified in terms of statistical inference with no loss of information. The drift term in the conditional model is then judged to be time-invariant, supporting the presence of drift co-breaking in the model. Finally, a parsimonious equilibrium correction model for aggregate consumption is estimated conditional on the weakly exogenous variables. The equilibrium correction model has passed a battery of mis-specification and parameter-constancy tests, thereby being judged to be a data-congruent representation. Based on the preferred model, it is discussed that a rachet effect embedded in aggregate consumption seems to have played a critical role in the prevention of serious depression in the Japanese economy. Acknowledgements I am grateful for the helpful comments and suggestions by the editor and anonymous referees. The draft of this paper was presented at Japanese Economic Association Spring Meeting 2008, Sendai, Japan, and at the 6th OxMetrics User conference, London, UK. I would like to thank David F. Hendry, Masato Kagihara, Ryuzo Miyao, Bent Nielsen, and Mitsuo Takase for their constructive comments and suggestions. This research is supported by JSPS KAKENHI (19830111).

T. Kurita / Economic Modelling 27 (2010) 574–584

283

Appendix A

B. Drift co-breaking

A. Weak exogeneity and conditional model

The decomposition of a joint model into conditional and marginal models, that is, (17) and (18) presented in Appendix A, provides a useful setting in considering co-breaking in the drift terms. The timevarying intercept is expressed as μj = (μw,j′, μz,j′)′ for j = 1, 2 in conformity with Wt and Zt, and a p × m matrix Ψw required for the construction of the conditional model is given by

Consider Eq. (6) and let the process be decomposed as Xt = (Wt′, Zt′)′ for Wt ∈ Rm, Zt ∈ Rp − m and m ≥r. The set of parameters and the error terms are also expressed as  α=

         Γw;i εw;t αw μw Φw ; Γi = ;μ = ;Φ= ; εt = ; Γz;i εz;t αz μz Φz

and the normal innovations have the following variance–covariance matrix:  Ω=

Ωww Ωzw

 Ωwz : Ωzz

Ψ′w = ð Im

−ω Þ:

If the intercept in the conditional model is time-invariant i.e. μ˜ w;1 = μ˜ w;2 = μ˜ w , one then regards Ψw as a set of drift co-breaking vectors: Ψ′w μ1 = Ψ′w μ2 = μ˜ w :

Model (6) is then decomposed into a conditional model for Wt and a marginal model for Zt, that is, k−1

˜ D + ε˜ ; ΔWt = ωΔZt + ðαw −ωαz Þβ ′ Xt−1 + ∑ Γ˜ y;i ΔXt−i + μ˜ w + Φ w t w;t 



ð20Þ

Drift co-breaking observed in (20) means that the expansion of μj in the direction of Ψw′ removes the deterministic break, yielding the time-invariant intercept in the conditional model. This phenomenon indicates the presence of stable parameters in the conditional model, even when the marginal model is subject to a shift in the intercept.

i=1

C. Details of the data

k−1

ΔZt = αz β*′ Xt−1 + ∑ Γz;i ΔXt−i + μz + Φz Dt + εz;t ; i=1

(Data)

where

• ct = the log of the real private consumption in Japan (seasonally adjusted), • yt = the log of the real GDP in Japan (seasonally adjusted). • πt = the annual inflation based on the GDP deflator in Japan. • pt = the log of the GDP deflator in Japan.

˜ ω = Ωwz Ω−1 zz ; Γw;i = Γw;i −ωΓz;i ; ˜ = Φ −ωΦ ; ε˜ = ε −ωε ; μ˜ w = μw −ωμz ; Φ w w z w;t w;t z;t

(Source and Note) System of National Accounts, the webpage of Economic and Social Research Institute, (http://www.esri.cao.go.jp/). 93SNA, Kotei Kijyun Houshiki (base year: 1995).

and 

ε˜w;t εz;t

 =N

   0 Ωww:z ; 0 0

0 Ωzz

 ;

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k−1

ΔZt = ∑ Γz;i ΔXt−i + μz + Φz Dt + εz;t ; i=1

ð18Þ

and Zt is then said to be weakly exogenous with respect to a set of parameters of interest, ˜ and Ω αw ; β*; ω; Γ˜w;i ; μ˜ w ; Φ w ww:z :

ð19Þ

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