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13.9 Granger Causality Tests As to binary variables ,Not Granger causality is defined as follows:

13.9 Granger Causality Tests As to binary variables ,Not Granger causality is defined as follows: If the conditional distribution determined by the lags of and is same as the conditional distribution determined by the lags of only: Then does not Granger cause

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13.9 Granger Causality Tests As to binary variables ,Not Granger causality is defined as follows:

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  1. 13.9 Granger Causality Tests As to binary variables ,Not Granger causality is defined as follows: If the conditional distribution determined by the lags of and is same as the conditional distribution determined by the lags of only: Then does not Granger cause Another description of Granger Causality is, If prediction accuracy of doesn’t improve when lags of are added , Then, does not Granger cause .According to the definition above,The equation of Granger Causality Tests is bellow: If necessary, constants,trends,seasonal dummy variables or something can be involved in the formula above.The Null hypothesis to test does not Granger cause is: Obviously,if the regression parameter estimates of lags of in fomula(13-39) are all not significant,the hypothesis above is rejected.In other words, if any regression parameter estimates of lags of is not significant(not equal to zero),then Granger cause .

  2. The tests above can be done with F-statistic: represents restricted residual sum of squares(null hypothesis is correct); represents unrestricted radiual sum of squares.k represents maximum lag of in the model.2k represents Number of parameters to be estimated.T represents sample size. If the original assumption is correct,the F-statistic subject to distribution F(k,T-2k) progressively.Discrimination Rules are as follows: (1) If F≤Fa(k,T-2k) which is calculted with samples,then original assumption is acccepted,that is, does not Granger cause . (2) If F>Fa(k,T-2k) which is calculted with samples,then original assumption is rejected,that is, Granger cause . Cautions: (1)The official name of "Granger Causality " is "not Granger Causality".We call it "Granger Causality " ,just because it's simple; (2)That (does not) Granger cause are always stated that (does not) Granger cause .(Strictly speaking,this statements is not correct); (3)Granger causality here is different from that in philosophy." Granger cause "just means that "information about predicting is involved in "; (4)This type of test is proposed by Granger in 1969 in the first time.Sims proposed the definition in 1972 too.The two definitions are consistent.

  3. Example 13-9: Take shanghai stock index (SH) and The Shenzhen Component Index (SZ) (January 4,1999 to October 5,2001) for example ,we will go for Two-way Granger causality analysis.The figure of SH and SZ are as figure 13-12.The two series are highly relative to each other.(As figure 13-13).So the two series may have one or two-way Granger causality. figure 13-12 :series SH and SZ figure 13-13:scaler map of SH and SZ

  4. First,we test whether one to two lags of SH Granger cause SZ or not.Regressions of unrestricted model and restricted model are as follows: Calculating the value of F-stastic according to fomula (13-40): Because, ,null hypothsis is accepted.SH does not Granger cause SZ.

  5. Second,we test whether one to two lags of SZ Granger cause SH or not.Regressions of unrestricted model and restricted model are as follows: Calculating the value of F-stastic according to fomula (13-40): Because, ,null hypothsis is rejected.SH Granger cause SZ.

  6. 13.9 Granger Causality Tests In EViews,to do Granger causality tests,open the data window of SH and SZ first, click "view", and choose "Granger Causality".Fill "2" in the subsequent dialog window,and click "OK". Cautions: (1)Lag k is randly choosed.In fact,it is a judge issue.Take and for example,if does have significant impact on ,we needn't test it with a longer lag.if does not have significant impact on ,we should test it with a longer lag. Generally,we ought to do Granger causality tests with different lags,and we should not stop,until results of Granger causality tests with different lags are same. (2)When we are testing whether Granger cause or not ,If Granger cause ,and has relation to ,we should add lags of to the right of formula too. (3)We can't do Granger causality tests,if there is no cointegration between the non-stationary variables.

  7. Do Granger causality tests with lags 5,10,15,20 and 25.results are as follows: The results are the same:the Shanghai stock index does not Granger cause the Shenzhen Component Index,while the shenzhen component index Granger cause shanghai stock index.

  8. 13.10 chow breakpoint tests we use n1 and n2 to resprent two sample size,and dedine T=n1+n2.We assume that,the form of multiple regression models are: We estimate the models with sample size T,n1and n2 separately.the corresponding symbols are as follows: Null hypothesis and alternative hypothesis are: The statistic is: In which, is the corresponding residual sum of squares of the regression model with sample T, and are the corresponding residual sum of squares of the regression model with sample n1 and n2. k is the number of parameters to estimate in the regression model.Rules are: (1) If F≤Fa(k,T-2k) which is calculted with samples,then original assumption is acccepted,that is, regression coefficients have no significant changes. (2) If F>Fa(k,T-2k) which is calculted with samples,then original assumption is rejected,that is, regression coefficients have significant changes. In which, a represents significance level.In principle,F-statistic in formula (13-41) and in formula (13-6) are the same. in formula (13-41) is similar to unrestricted residual sum of squares in formula (13-6). is similar to restricted rasidual sum of squares. Degrees of freedom of and are and .

  9. Example 13-10 The scatter map of the area of arable land (land,million hectares) and The output value of agriculture (output,billion yuan) in year 1993 and 1998 in the country's 30 regions(Chongqing,Hongkong ,Macau and Taiwan are not included) is as figure 13-14.In which,dots are the observation points of the logarithm of the agricultural output to the logarithm of the cultivated area in 1998. circles are the observation points of the logarithm of the agricultural output to the logarithm of the cultivated area in 1993.From the figure, in the case of arable land with little change,the regional agricultural output value increased significantly.

  10. Three regression models with data of 1993 and 1998 (each has 30 observations) and merged data (there 60 observations in total) are as below: RSS (rasidual sum of squares) and other statistics in the three models are as below:

  11. F-statistic according to formula (13-41) is: Because ,model structure of the two years in the 30 regions has great change.That is, under the conditions of approximately equal acreage, agricultural output has been greatly improved (Which also contains inflation factors). In Eviews, to do chow breakpoint test, we should click "view" in the output window of formula (13-42), choose "Stability Tests" and "Chow Breakpoint".Fill "31" in the subsequent dialog window. results are as figure 13-16.In which, F=13.59,which is same with the results above.Caution:sample points in the dialog window belong to the back sub-sample.

  12. 13.11 Chow breakpoint test of the stability of regression coefficients After we obtain the estimated regression coefficient value on the basis of sample size T.when we want to test whether the original coefficient value is stationary, if n samples are added,chow breakpoint test can be used as follows: Make regression models of the same type (number of parameters to estimated is k) with sample size T and sample size T+n seprately. Symbols are as table 13-12: Null hypothesis and alterhypothesis: The statistic is defined as below: In which, is the rasidual sum of squares with sample size T+n, is the rasidual sum of squares with sample size T, k is the number of parameters to estimate. n is the number of observations to be added. Rules are: (1) If F≤Fa(n,T-k) which is calculted with samples,then original assumption is acccepted,that is, regression coefficients have no significant changes. (2) If F>Fa(n,T-k) which is calculted with samples,then original assumption is rejected,that is, regression coefficients have significant changes.

  13. Example 13-12: Stability test of the number of China's currency circulation model(1952-1998) This is a comprehensive case with chow test and dummy variables.Data of money flow(Mt) and Logarithm of the money flow(LnMt) (the series is represented by solid line in figure 13-19) from 1952 to 1998 can be seen in table 13-14.From the figure 13-19,year 1978 is a structural mutation point.After the chow breakpoint test, year is a structural mutation point of LnMt too. Now, we want to know whether the regression coefficients have significant changes when data of 1997 and 1998 are added to the model whose samples are from year 1952 to year 1996.

  14. To describe the structural change in 1978, dummy variables can be used. Dummy variable D is defined as below: Do the OLS regression with data of 1952-1996 to the time t. results are as below: That coefficients of D and tD are significant,means it's necessary to add these two variables to the model. Do the OLS regression with data of 1952-1998 to the time t. results are as below: Value of F-statistic can be obtained,when n=2,k=4 in the case:

  15. Results in Eviews are as follows: Operations in Eviews: click "view" in the output of the regression model window,choose "Stability Tests" and "Chow Forecast Tests", and fill a value of year corresponding to n. In this case,it is 1997.

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