The ardl method in the Energy-Growth Nexus Field; Best Implementation Strategies
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2. The Methodology
For reasons of educative demonstration, we assume two series, the Y t and the X t in this paper but the reader can easily generalize into more variables. Nevertheless, the production function equation in the energy-growth nexus, has more variables. In a bivariate energy-growth nexus model, the Y t stands for economic growth and the X t stands for energy consumption. It is also typical in the energy growth nexus to use logarithms of the variables in order to translate variable coe fficients as elasticities. The series of steps in the ARDL procedure is the investigation of: (i) stationarity, (ii) cointegration, and last but not least (iii) causality. There are other ways to proceed to causality analysis without the first two steps, but this occurs within other methodological frameworks. 2.1. Stationarity After a presentation of the descriptive statistics of the series (mean, median, minimum and maximum values, skewness, kurtosis, as well as the standard deviation, Bera–Jacque normality test and pairwise correlation), the first step in the ARDL analysis, is the unit root analysis. It informs about the degree of integration of each variable. To satisfy the bounds test assumption of the ARDL models, each variable must be I(0) or I(1). Under no circumstances, should it be I(2). De Vita et al. ( 2006
) also noted that the dependent variable should be I(1). However, this is not widely claimed in the current literature. Unit root analysis is performed with a long array of tests such as for example the augmented Dickey Fuller (ADF) and the Kwiatkowski–Phillips–Schmidt–Shin (KPSS), the Phillips–Perron (PP), the Ng–Perron test, the cross-sectional augmented IPS-CIPS ( Pesaran 2007 ), the LS ( Lee and Strazicich 2003 ), and many others. Each one is more compatible with different data characteristics, but this paper will not discuss them for brevity reasons. However, it should be stressed that researchers should apply both the traditional and structural break unit root tests to make sure that the variables are not I(2). Economies 2019, 7, 105 3 of 16
2.2. Cointegration The essence models in the ARDL bounds test framework are the following unrestricted error correction models: ∆LY
t = a 0 + a 1 t + m X i=1 α 2i ∆LY t−i + n X i=0
a 3i ∆LX t−i + a 4 LY t−1 + a 5 LX t−1
+ µ 1t (1) ∆LX
t = β 0 + β 1 t + m X i=1 β 2i ∆LX t−i + n X i=0
β 3i ∆LΥ t−i + β 4 LX t−1 + β 5 L Υ t−1 + µ 2t (2) ∆ is the first difference operator, µ is the error term that must be a white noise or put in other words it represents the residual term which is supposed to be well behaved (serially independent, homoskedastic and normally distributed). All α and β coefficients are non-zero with a 4 and β 4 also being negative (this represents the speed of adjustment). The parameters α 2i and a 3i represent the short-run dynamic coe fficients, while a 4 and a
5 are long-run coe fficients in the energy-growth nexus relationship. The a
0 and
β 0 are drift components, µ 1t and µ 2t are white noise. What type of explanatory variables must be incorporated in the energy-growth relationship is provided in detail by Inglesi-Lotz ( 2018
) in a chapter written specifically on this topic. The interested reader is advised to read that. Generally, one can decide first on the framework one is going to work, namely whether that is a production function approach or a demand function approach or others such as the Kuznets curve hypothesis and then decide on the variables and other components. Other deterministic components are included on a trial and error basis and to corroborate further the stability of an estimated relationship. Overall, we observe in Equations (1) and (2) that each variable is represented as dependent on the past values of itself, the past values of the other variable(s), and the past values of di fferenced values of itself and the past values of di fferenced values of the other variable(s). Models (1) and (2) can be formulated either as intercept or trend ARDL models, or both. Equations (1) and (2) contain both. Halicioglu ( 2007 ) claims that it is possible to end up with two models, one with trend and one without a trend. There is a method described in Bahmani-Oskooee and Goswami ( 2003 ), according to which one ends up with a single long-run relationship through consecutive eliminations of the rest of the relationships. The first stage of the ARDL estimation produces a ( p + 1 ) k number of regressions so that the optimal lag length for each variable is obtained, with p being the maximum number of lags and k is the number of variables in the equation. In our simplistic example, there is only one X t variable. In the framework described in Equations (1) and (2), the ARDL bounds cointegration test is carried out. These equations are estimated with ordinary least squares (OLS). 2.3. More on the ARDL Analysis The ARDL analysis occurs as follows: If the existence of cointegration is confirmed in Equations (1) and (2), then the long-run and the short-run models are estimated and both long and short-run elasticities are derived, namely the ARDL equivalent of the UECM (Unrestricted error correction model). Cointegration, in the ARDL bounds test approach, is examined under the following hypothesis set up: H 0
1 = a 2 = a n = 0 H 1 : a 1 , a
2 , a
n , 0
The setup of the hypotheses reads as follows: there is cointegration if the null hypothesis is rejected. The F-statistics for testing are compared with the critical values developed by Pesaran et al.
( 2001
). Narayan critical values are more appropriate for small samples. Pesaran et al. ( 2001
) provide a table enumerated as CI and entitled: “Asymptotic critical value bounds for the F-statistic. Testing for the existence of a levels relationship” in five versions. These are (i) no intercept and no trend, (ii) restricted intercept and no trend, (iii) unrestricted intercept and no trend, (iv) unrestricted Economies 2019, 7, 105 4 of 16
intercept and restricted trend, (v) unrestricted intercept and unrestricted trend. They also provide a table CII entitled “Asymptotic critical value bounds for the t-statistic. Testing for the existence of a levels relationship” in three versions: (i) No intercept and no trend, (ii) unrestricted intercept and no trend, (iii) unrestricted intercept and unrestricted trend. Next we reproduce a part of these tables (CI-iii and CI-v) in order to explain how the decision for cointegration was made in Bölük and Mert ( 2015
) based on Pesaran tables. Note that Pesaran tables are not valid for I(2) variables ( Ali et al. 2016 ). The interested reader can find these tables in Pesaran et al. ( 2001
). Narayan and Smyth ( 2005
) on the other hand, has estimated critical values for the bounds test for four cases at three significance levels and up to seven independent variables up to eighty observations. The critical values of the four cases are entitled as: (i) Case II: restricted intercept and no trend, (ii) case III: unrestricted intercept and no trend, (iii) case IV: unrestricted intercept and restricted trend, (iv) case V: unrestricted intercept and unrestricted trend. In Narayan tables, k stands for the number of regressors, n is the sample size, I(0): stationary at levels, I(1): stationary at first di fferences. The interested reader can find these tables in Narayan and Smyth ( 2005 ). When no cointegration is confirmed, we can proceed with simple Granger causality (unrestricted VAR). The VAR equation should be specified on stationary data. There are various reasons why cointegration is not confirmed (e.g., no relationship between the examined variables or due to omitted variables). The Toda and Yamamoto ( 1995
) test is a solution for Granger causality testing in this case. After all, even when a long-run relationship does not exist in the data, this does not mean that no short-run relationship exists either. Moreover, it needs to be remembered that the cointegration equation provides the long-run elasticities. Short-run elasticities are presented by the coe fficients of the first di fferenced variables. In cases where more than one coefficient for a particular variable has been estimated for the short-run case, these are added and their joint significance is tested with a Wald test ( Fuinhas and Marques 2012 ). However, if cointegration is the case (which occurs very commonly, when there is a known and established theoretical connection between some variables), then we can proceed with the establishment of the error correction mechanism (ECM). Evidence of cointegration implies that there is a long-run relationship between the variables and their connection is not a short-lived situation, but a more permanent one, which can be recovered every time there is a disturbance. Alternatively to the above described F-test, a Wald test can be applied which is used to test the null hypothesis of no cointegration when there is more than one short-run coe fficient of the same variable ( Tursoy and Faisal 2018 ). 2.4. Diagnostic Tests after Cointegration A model to be trusted, it must be robust. To support robustness of an estimated model, one needs to peruse various diagnostic tests. Typical diagnostic χ 2 tests follow to investigate the goodness of fit, stability, parsimoniality, functional form, and a well-behaved model in general. The Breusch Godfrey serial correlation LM test, the Breusch–Pagan Godfrey Heteroskedasticity test or the White test, and the Jarque–Bera test are some of the tests encountered in these applications. In addition to that, the Ramsey reset test is used for the functional form. Besides the latter, the variance inflation factor (VIF) for multicollinearity might be useful in cases where there is evidence of multicollinearity. The Impulse Response Function (IRF), Shifts, and Dummies The impulse response functions can be of use because they reveal the e ffect of a standard deviation shock on the dependant variable. IRF are formed through the moving average (MA) of the vector autoregressive (VAR) equation 1 . Some energy-growth researchers use them as an indispensable tool 1 A VAR model is a generalization of univariate AR models for multiple time series. Within a VAR framework, all variables are represented by an equation that explains its evolution based on its own lags and the lags of the other variables in the multivariate framework. The number of variables k are measured over a period of time t as a linear evolution of their past values.
Economies 2019, 7, 105 5 of 16
for valuable information in their ARDL models. The impulse response function mainly shows what happens when the model is transferred to the one side of a dummy variable. For example, if the value of 1 represents war time and the value of 0 represents peace time, then if we take the ones or zeros only and separately, we have an impulse response function, one for war time and one for peace time. Thus, they are also a useful tool to test the stability of a model across structural breaks. There are various hypotheses that underlie the models after cointegration is confirmed. After the identification of the long-run relationship in Equations (1) and (2), we can continue with the examination of the short-run and the long-run Granger causality. The Granger causality refers to a situation where the past can be used to predict the future. Thus, if past values of X t significantly contribute to forecasting future values of Y t , the X t is said to Granger cause the Y t . However, evidence of correlation is not necessarily an evidence for causality. 2.5. Combined Cointegration Methods for the Robustness of the ARDL Model In the particular case of a unique order of integration, Bayer and Hanck ( 2013
) have developed a test which borrows elements from a variety of previously developed cointegration tests. The combined test borrows elements from Engle and Granger ( 1987
); Johansen
( 1988
); Boswijk
( 1994
) and Banerjee et al. ( 1998
). The combined cointegration test uses Fisher’s formulae and the p-values of the aforementioned individual tests. Engle and Granger − Johansen = −2 h ln P Engle & Granger
+
P Johansen i Engle and Granger −Johansen − Boswijk − Banerjee et al. = −2 h ln P Engle & Granger
+
P Johansen + ln P Boswijk + ln P Banerjee i The null hypothesis of no cointegration is rejected if the aforementioned Fisher statistic exceeds the critical value as produced by Bayer and Hanck ( 2013
). The above test balances the decisions produced by the independent tests which su ffer from various weaknesses, each one of them. 2.6. Causality after the ARDL Bounds Test and the Importance of the Error Correction Term (ECT) The investigation of causality is the third step in the energy-growth nexus analysis. The lagged error correction term is derived from the cointegration equation. Thus the long-run information that is missed through the di fferencing of the variables for stationarity purposes, is re-introduced in the system of causality equations. This is a necessary step when variables are cointegrated. Cointegration implies that there must be causality of some direction, however, it does not reveal to which direction that causality goes. Therefore, additional causality analysis is required. Thus, before going to the estimation of Equations (3) and (4) below, one needs to run another set of regressions in order to get the residuals which will be inserted to Equations (3) and (4) as the ECT term. There are many strategies to follow in the examination and direction of causality. One such strategy is the VECM approach (vector error correction model), which is a restricted form of unrestricted VAR and is suitable, once the variables are integrated at I(1). According to this model setup, the dependent variable is dependent on its own lagged values, as well as the lagged values of the independent variables, the error correction term, and the residual term. This is shown in the following set of equations. ∆lnY t
a 1 + l X i=1 a 11 ∆LY t−i + m X j=0
a 22 ∆X t− j + n 1 ECT
t−1 + µ 1i (3)
∆lnX t = a 1 + l X i=1 a 21 ∆LX t−i + m X j=0
a 22 ∆Y t− j + n 2 ECT
t−1 + µ 2i (4)
Economies 2019, 7, 105 6 of 16
Residual terms in the above equations, are assumed to distribute normally. The coe fficient of the ECT must be negative to assure system convergence from the short run toward the long run. An ECT equal to x% is interpreted as such that x% of economic growth is corrected by deviations in the short run that lead eventually to the long-run equilibrium path. The significant variables on the right hand side of each equation show short-run causality for the dependent variable. FMOLS and DOLS Estimators for Robustness The FMOLS (fully modified OLS) and the DOLS (dynamic OLS) were developed by Phillips and Hansen
( 1990
) and Stock and Watson ( 1993
). They lead to the generation of asymptotically e fficient
coe fficients, because they take into account the serial autocorrelation and endogeneity. They are applied only in the I(1) case for all variables. The latter makes them less flexible and attractive methods. OLS is biased when variables are cointegrated but nonstationary, while FMOLS is not. DOLS performs better than the FMOLS approach ( Kao and Chiang 2000 ) for several reasons: DOLS is computationally simpler and it reduces bias better than FMOLS. The t-statistic produced from DOLS approximates the standard normal density better than the statistic generated from the OLS or the FMOLS. DOLS estimators are fully parametric and do not require pre-estimation and non parametric correction. Ali et al. ( 2017 ) reports that the most significant benefit of DOLS is that the test considers the mixed order of integration of variables in the cointegration framework. 2.7. Additional Ways to Study Causality Literature reports additional types of causality: (a) The weak causality /short-run causality, (b) the long-run causality, (c) the strong causality (joint causality), (d) the pairwise causality. Each one serves a particular purpose. (a) Weak causality /short-run causality Each variable is caused by its own past only. (b) Long-run causality The error correction term is zero. This is a VAR (vector autoregression) causality leading to Toda and Yamamoto ( 1995 ) method. Granger causality can be checked for existence through a VAR model (note that data are not in di fferences, namely they are in level form): Y t = g 0 + α 1 Υ t−1
+ . . .
+ α ρ Υ t−p
+ b 1 X t−1
+ . . .
+ b p X t−p
+ u t X t = h 0 + c 1 Υ t−1 + . . . + c ρ Υ t−p
+ d 1 X t−1
+ . . .
+ d p X t−p
+ v t H 0 : b 1 = b 2 = . . . = b p = 0 H 1 : X Granger causes Y A similar hypothesis set up can be constructed for the second equation, but this will not be done here for space considerations. Please note the following rationale: If b i
t will lead Y t in the long run. If b i = 0 and d i , 0, then Y t will lead X t in the long run. If b i , 0 and d i , 0, then the feedback relationship is present. If b i
0 and d i = 0, then no cointegration exists. After we have calculated the diagnostics of the model and we have verified that the model is well behaved, then the next step is the bounds test. The existence of a long-run relationship can be further corroborated with the investigation of significance of the individual terms.
Economies 2019, 7, 105 7 of 16
(c) Strong causality: The joint causality investigation process This is altogether the case described in (a) and (b). The joint causality test also known as strong causality test ( Lee and Chang 2008 ) identifies two sources of causation, one the short run and the other the long run, to which the variables re-adjust after a short-run perturbation. This is tested with the short-run coe fficients of the lagged variables and the significance of the lagged error correction term. Granger causality can be investigated in two other known ways: It can be investigated with the F-test to decide about the significance of first di fference stationary variables ( Asafu-Adjaye 2000 ; Masih and Masih 1996 ) or by including the ECT as a source of variation. This is most commonly checked with a t-test. (d) Pairwise Granger causality test This is another solution toward the investigation of causality when cointegration is not confirmed. An additional usage is for the corroboration of VECM results. Menegaki and Tugcu ( 2016 ) have employed this method for the investigation of the energy-sustainable growth nexus in Sub-Saharan African countries for the years 1985–2013. In addition to that, Menegaki and Tugcu ( 2018
) have employed the same method for the investigation of the energy-sustainable growth nexus in Asian countries. Yüklə 254,06 Kb. Dostları ilə paylaş:
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