The econometric methodology for behavioural equations

This section is devoted to explaining the estimation method for behavioural equations.

In particular, it explains the process of the bounds test for cointegration method adopted in this model.

Most macroeconomic variables, for example GDP, consumption, export, and prices, are non-stationary time series, as they are trended. Terminologically, the non-stationary time-series is said to be integrated. The order of integration is the minimum number of times the series needs to be differenced to make a stationary series. The problem of non-stationary, i.e.

over I(1), which indicates integrated of order one, is that the standard Ordinary Least Square (OLS) regression cannot be applied to estimate the non-stationary data, since the OLS estimation leads to the spurious regression problem: it is well-known that the converge to functionals of Brownian motions, the t-ratios have a non-standard distribution, and the

Durbin-Watson statistic converges in probability to zero. Consequently, it certainly produces incorrect inferences³¹. Since then, testing and estimating the non-stationary time series has been one of the central issues in econometrics. As part of allowing for non-stationary data, the

„cointegration‟ framework has been established and developed. If there may exist some linear combination in the level of two or more series individually integrated, then the series is said to be cointegrated, and the linear relationship in the level variables is interpreted as a long-run equilibrium.

Several cointegration tests have been introduced: typically, Engle and Granger (1987) provides a residual based test, Hansen (1992) and Park‟s (1992) suggest an instability test and an added variables test respectively. In addition, fully efficient estimation methods have been invented: while Fully Modified OLS (Phillips and Hansen 1990), Canonical Cointegrating Regression (Park 1992), and Dynamic OLS (Stock and Watson 1993) have been proposed to estimate a single equation cointegrating relationship, Johansen (1991, 1995) provides a system maximum likelihood approach for multivariate cointegration.

As a single cointegration approach, Pesaran et al. (2001) develop the bounds testing approach using an Autoregressive Distributed Lag (ARDL) framework. The bounds testing method has been widely used for analysing long-run relationships between non-stationary variables, as it has certain advantages over the other approaches. First, the bounds testing is valid regardless of whether the underlying variables are purely I(0), purely I(1), or fractionally integrated with each other. Second, under the ARDL framework, estimators are super-consistent in small sample sizes. Mah (2000) points out that the Engle and Granger (1987) and Johansen (1991, 1995) methods of cointegration are not robust for small sample sizes

31 See Granger and Newbold (1974, 1977), Plosser and Schwert (1978), and Phillips (1986) for the details of the spurious regressions.

(Mah, 2000). Narayan (2005) also argues that bounds testing is far superior to the Johansen‟s multivariate cointegration method. Pesaran and Shin (1999) prove that the OLS estimators of the short-run parameters are consistent in the ARDL framework and estimators of the long-run coefficients are super-consistent in small sample sizes. Third, unlike the Engle and Granger approach, the bounds test does not lead to an endogeneity problem.

The bounds testing procedure is as follows: Let where is a dependent variable and is a vector of regressors. Note that the regressors are selected based on economic theories in order to give meaning to the estimated parameters on the variables in the equation. The first differenced variable is modelled as an Unrestricted Error-Correction Model (UECM) in order to test whether there exists at most one long-run equilibrium relationship between and , which is given by

∑

∑

where represents log-first-difference, and are long-run multipliers, is a intercept coefficient and is time trend, and are short-run dynamic coefficients on the lagged values of and current and lagged values of respectively, and is an error term.

After the UECM equation above by the OLS, the existence of cointegration is tested by exclusion of the lagged level variables and in the equation above. That is, restrict all estimated coefficients of the lagged level variables by the null hypothesis and alternative hypothesis which are defined as:

The Wald test calculates an F-statistic in order to test these hypotheses in the above equation. Pesaran et al. (2001) provides asymptotic critical value bounds which consist of two sets: the lower bound when the regressors are I(0) and the upper bound when the regressors are I(1). If the computed F statistic is higher than the critical values of the upper bound, then the null hypothesis of no cointegration is rejected and it concludes that there exists a long-run equilibrium relationship between the variables without any knowledge of the order of integration. On the other hand, if the F statistic is less than the critical values of the lower bound, then the null hypothesis of no cointegration is accepted. If, however, the F statistic falls between the critical values of the lower and the upper bound, then the relationship cannot be inferred conclusively. The critical value bounds are given in Table C1(v) in Pesaran et al.

(2001) and in the footnote below³².

As discussed earlier, a number of papers employ bounds testing when they have small sample sizes. In our case, fiscal and sector energy demands data are only available since 1990, which is subject to small sample problems when trying to find the long-run equilibrium relationship. Therefore, the bounds testing approach is adopted in order to estimate macroeconomic variables with non-stationary time series.

After rejecting the null hypothesis, a behavioural equation is set up based on a partial adjustment model³³ that includes a lagged dependent variable among the explanatory

32 Table 3.2. The critical values of lower and upper bound

Significance 10% 5% 1%

No. of variables FL FU FL FU FL FU

2 4.19 5.06 4.87 5.85 6.34 7.52

3 3.47 4.45 4.01 5.07 5.17 6.36

4 3.03 4.06 3.47 4.57 4.40 5.72

5 2.75 3.79 3.12 4.25 3.93 5.23

Source: Pesaran et al. (2001)

33 Fundamentally, choosing lag lengths of the model has the following process; the maximum of lags is set equal to 2, as Pesaran and Shin (1999) suggest for annual data. And then, we select lag order of the ARDL that

variables. The estimated parameters provide information on short-run and long-run elasticities, and the speed of adjustment from the short-run to long-run equilibrium. The partial adjustment specification is given by:

∑

where the is a constant. The and are coefficients of the lagged variable and the set of explanatory variables, respectively. Finally, is an error term that is normally distributed with mean zero constant variance. If the equation cannot reject the null hypothesis, the equation is estimated in log first difference form after checking if the variable is stationary by the unit root test.

It is important to note that having established the long-run equilibrium relationship by equation (3.38), the Error Correction Model (ECM) which models deviations from the long-run relationships could be built up in this study, which is able to capture an agent‟s dynamic adjustment behaviour when a new policy is to be imposed. However, there is a trade-off between the ability to explain short-run dynamics and simulation performance. The errors caused from the coefficients on the short-run variables in the ECM equation pass through to all the other equations by the feedback mechanisms in the system model. Consequently, insignificant short-run coefficients in the ECM worsen the simulation results of the system model. For this reason, this study adopts the partial adjustment specification rather than the ECM model in order to obtain stability of the system model by isolating divergences of the simulated values of the differenced variables from their actual values.

minimises Schwarz criterion (SC) or Akaike Information criteria (AIC). Generally, the partial adjustment specification shows a minimum value of Schwarz criterion (SC) and Akaike Information criteria (AIC).

We report the estimation result of each equation in a table that consists of three parts, the first part, (a) describes estimated parameters on variables, standard error, t-value, and p-value for testing significance of coefficient. The second part (b) indicates the Bounds test for cointegration. represents F-statistic of no-cointegration hypothesis, where Y is a dependent variable, and X are a vector of explanatory variables. If the F-statistic, denoted by > the upper bound, FU, the null of no-cointegration can be rejected, IF < the lower bound, FL, the null cannot be rejected, and thus no long-run relationship exists. If FL<

<FU, the inference in inconclusive. Diagnostic tests are described in the last part, (c).

̅ represents adjusted R-squared, ̂ and D.W. indicate standard error of regression and Durbin-Watson test statistic respectively. is F statistic for overall significance of the model, where and are the number of variables and observation respectively. In addition, we implement four major diagnostic tests; the Ramsey Regressions Specification Error Test (RESET) for general misspecification (Ramsey, 1969), the Breusch-Godfrey test for serial correlation in residuals (Breusch, 1978; and Godfrey, 1978), the Breusch-Pagan test for heteroskedasticity in residuals (Breush and Pagan, 1979), and Jarque-Berra test for normality of residuals in order to guarantee robustness of the OLS estimators (Jarque and Berra, 1990).

Notations for the diagnostic tests are summarised in Table 3.3.

Table 3.3. Lists of diagnostic tests

Notation Definition Null hypothesis

F statistic of Ramsey RESET Correct specification

F statistic of the Breusch-Godfrey test No serial correlation in residuals

F statistic of the Breusch-Pagan test No heteroskedasticity in residuals

statistic of Jarque-Berra test Normality of residuals