Data Analysis & Forecasting                                  Faculty of Development Economics                          TIM...
Data Analysis & Forecasting                                      Faculty of Development Economics                         ...
Data Analysis & Forecasting                                    Faculty of Development Economics                           ...
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7. toda yamamoto-granger causality

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7. toda yamamoto-granger causality

  1. 1. Data Analysis & Forecasting Faculty of Development Economics TIME SERIES ANALYSIS TODA-YAMAMOTO VERSION OF GRANGER CAUSALITY (AUGMENTED GRANGER CAUSALITY)According to Toda and Yamamoto (1995), economic series could be either integrated of thedifferent orders or non-cointegrated or both. In these cases, the ECM cannot be applied forGranger causality tests. Hence, they developed an alternative test, irrespective of whether Ytand Xt are I(0), I(1) or I(2), non-cointegrated or cointegrated of an arbitrary order. This iswidely known as the Toda and Yamamoto (1995) augmented Granger causality. Thisprocedure provides the possibility of testing for causality between integrated variablesbased on asymptotic theory.1. THE MODELToda and Yamamoto (1995) augmented Granger causality test method is based on thefollowing equations: h +d k +d Yt = α + ∑ β i Yt −i + ∑ γ j X t − j + u yt (1) i =1 j=1 h+d k +d X t = α + ∑ θ i X t −i + ∑ δ j Yt − j + u xt (2) i =1 j=1where d is the maximal order of integration order of the variables in the system, h and k arethe optimal lag length of Yt and Xt, and are error terms that are assumed to be white noisewith zero mean, constant variance and no autocorrelation. Indeed, all one needs to do is todetermine the maximal order of integration d, which we expect to occur in the model andconstruct a VAR in their levels with a total of (k + d) lags.Important note is the same as other causality models.2. TEST PROCEDURETHE DYNAMIC GRANGER CAUSALITY is performed as follows:Step 1: Testing for the unit root of Yt and Xt, and determining the maximal order ofintegration order (d) (using either DF, ADF, or PP tests)Suppose the test results indicate that Yt and Xt have different integration orders, say I(1)and I(2). Thus, the maximal order of integration is 2.Step 2: Determining the optimal lag length (k) of Yt and Xt a) Automatically determine the optimal lag length of Yt and Xt in their AR models (using AIC or SIC, see Section 8 of my lecture). Optimal lag length of YtPhung Thanh Binh (2010) 1
  2. 2. Data Analysis & Forecasting Faculty of Development Economics h Yt = α + ∑ β i Yt −i + u yt (3) i =1 Restricted model Estimate (4) by OLS, and obtain the RSS of this regression (which is the restricted one) and label it as RSSRY. h +d d Yt = α + ∑ β i Yt −i + ∑ γ j X t − j + u yt (4) i =1 j=1 Optimal lag length of Xt h X t = α + ∑ θ i X t −i + u xt (5) i =1 Restricted model Estimate (6) by OLS, and obtain the RSS of this regression (which is the restricted one) and label it as RSSRX. h + d d X t = α + ∑ θ i X t −i + ∑ δ j Yt − j + u xt (6) i =1 j=1 b) Manually determine the optimal lag length of Xt (k in equation (1)) and Yt (k in equation (2)), (using AIC or SIC, depending on which one you use in step 2a, see Section 8 of my lecture). h +d k +d Yt = α + ∑ β i Yt −i + ∑ γ j X t − j + u yt (7) i =1 j=1 Then estimate (7) by OLS, and obtain the RSS of this regression (which is the unrestricted one) and label it as RSSUY. h + d k + d X t = α + ∑ θ i X t −i + ∑ δ j Yt − j + u xt (8) i =1 j=1 Then estimate (8) by OLS, and obtain the RSS of this regression (which is the unrestricted one) and label it as RSSUX.Step 3: Set the null and alternative hypotheses a) For equation (4) and (7), we set: k H0 : ∑γ j=1 j = 0 or X t does not cause Yt k H1 : ∑γ j=1 j ≠ 0 or X t causes Yt b) For equation (6) and (8), we set: k H0 : ∑δ j=1 j = 0 or Yt does not cause X tPhung Thanh Binh (2010) 2
  3. 3. Data Analysis & Forecasting Faculty of Development Economics k H1 : ∑δ j=1 j ≠ 0 or Yt causes X tStep 4: Calculate the F statistic for the modified Wald test a) For equation (4) and (7), we set: (RSS RY − RSS UY ) / k F= RSS UY /( N − K ) where K is the number of estimated coefficients. b) For equation (6) and (8), we set: (RSS RX − RSS UX ) / k F= RSS UX /( N − K )where K is the number of estimated coefficients. If the computed F value exceeds the critical F value, reject the null hypothesis andconclude that Xt weakly causes Yt, or Yt weakly causes Xt. Questions: How to explain the test results?Phung Thanh Binh (2010) 3

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