1 557699 Granger-causality testing within the context of the bivariate analysis of stationary macroeconomic time series 1. IntroductionGranger proposed the idea of Granger-causality in his 1969 paper to describe the „causalrelationships‟ between variables in econometric models.Before this, econometricians andeconomists understood the idea of „causal relationships‟ as asymmetrical relationships.Causal relations are studied because policy makers need to know the consequencesof thevarious actions which they will or are considering taking. For example; given a relationshipbetween output and the price level, we need to know whether this relationship is expected tohold if actions controlling output are implemented, when actions controlling the price levelare implemented, or when both of these cases occur (Orcutt, 1952). The use of the termcausality is identical to that of Weiner who provided the base idea for Granger‟s work in theearly fifties. 2. Granger-causalityThe idea of Granger-causality is that a variable X Granger-causes variable Y if variable Ycan be better predicted using the histories of both X and Y than it can be predicted using thehistory of Y alone. This is shown if the expectation of Y given the history of X is differentfrom the unconditional expectation of YA second definition for causality has been offered by Granger (1969) which states thatif (which means that if the variance of X predicted using theuniverse of information, U, is less than the variance of X predicted using all informationexcept variable Y) then we can say that Y is causing X, denoted However, he thenclarified that using the whole universe of information, U, is unrealistic so it is replaced withall relevant information. However, this change now makes testing more than a statisticalprocedure as there is a subjective element regarding what information is relevant. Anotherelement to define is that of Feedback. A feedback system occurs if variable X Granger-causesvariable Y, and Y Granger-causes X, denoted .All these definitions assume that onlystationary series are involved, as non-stationary series stop these definitions being testable.Granger-causality has several components. The first component, temporality, is based on theprinciple that only past values of X can Granger-cause Y, because the future cannot cause thepast or the present. If X occurs after Y, then we know that X cannot cause Y. Similarlythough, if X occurred before Y then that does not necessarily imply that X caused Y. Thesecond component of Granger-causality is exogeneity;Sims (1972) stated that for variable Xto be exogenous of variable Y, X must fail to Granger-cause Y; this component wasconfirmed by Engle, Hendry, and Richard (1983). Independence is the third component ofGranger-causality because variables X and Y are only independent of each other if both fail
2 557699to Granger-cause the other. The final component of Granger-causality is that of asymmetry; ifX Granger-causes Y, then changes in Y have no effect on the future values of X.Granger-causality tests observe two time series to identify whether series X precedes seriesY, Y precedes X, or if the movements are contemporaneous. The notion of Granger-causalitydoes therefore not imply „true causality‟, but instead identifies whether one variable precedesanother. For example; do changes in output occur before changes in money, or does theopposite occur, or do these changes occur at the same time. In his 1972 paper, Sims showedthat money Granger-causes output, but output does not Granger-cause money. This resultsupported existing business cycle models which hypothesize that money plays an importantrole in output. We can therefore use Granger-causality tests to test for things we might haveassumed to occur from elsewhere or which we have taken for granted.The Granger-causality tests being studied in this paper are bivariate, however multivariatetests can be carried out similarly using a Vector Autoregression (VAR), and in fact the DirectGranger Test is a bivariate case of VAR.It is important to remember that when testing for Granger-causality, the models should befully specified. If the model isn‟t well specified, then spurious relationships maybe founddespite the fact that there actually are no relationships between the variables. Anothersituation to be mindful of is that all variables in an economy could be reacting to some un-modelled factor, a war for example, and if the reactions of both X and Y are staggered in timethen it will display Granger-causality even though the real causality is obviously different. 3. Granger-causality testsThere are three main tests for Granger-causality within the context of the bivariate analysis ofstationary time series which this paper will explore: The Direct Granger test, the Sims test,and the Modified Sims test. Each of these three tests will be explained in their own sections.There are other tests for multivariate and non-stationary models however these will not beincluded in the analysis of this paper. There are special problems when testing for Granger-causality in cointegrated relationships, which is why this paper will not cover them (Toda andPhillips, 1991).Other than the three tests above, there are other tests which can be used; for example atechnique developed by Haugh (1976), and later Pierce and Haugh (1977) called „Haugh‟sResidual Cross-correlation test‟ uses a two-step procedure to test for Granger-causality. Ituses the Autoregressive Integrated Moving Average (ARIMA)/cross-correlation approach byfirst estimating the ARIMA models for both series, let and be theARIMAmodels for series and respectively. Then the estimated residuals are, and , of theARIMA models are saved and will be ,and
3 557699 .The second step is to cross correlate these residual series. The significance of the results iscomputed by comparing the cross-correlation estimate at lag k with its standard error(Freeman, 1983). If these cross-correlation values are larger than ±2 standard errors fromzero then these series contain Granger-causality. This method is not used as much as theaforementioned tests because of a number of drawbacks. Firstly, the statistics that this testproduces are less powerful than the above tests. This method is also highly sensitive to theselection of lag length. This test‟s biggest flaw is that it can only indicate whether or notGranger-causality exists between the two variables but it cannot explain the direction of thecausality. 4. The Direct Granger TestThe direct granger test is a very useful tool as it allows econometricians to test for thedirection of Granger-causality as well as for its presence. Following the definition forGranger-causality, the direct Granger test regresses each variable on lagged values of itselfand the other explanatory variable. Empirically; the direct Granger test has been found to bemore powerful than both the Sims and Modified Sims test, outperforming both of these byrejecting a false null 3.26% and 2.64% more respectively.If, in a regression of on lagged values of and , the coefficients of the values arezero then the series fails to Granger-cause So consider the following regression modelWhere are the deterministics, is the random error term, is the coefficient on thelagged Y values, and is the coefficient on the lagged X values. We start with a one periodlag instead of setting because we are not including instantaneous causality in themodel(Instantaneous causality is when the changes in Y and X occur at the same time and arecorrelated).If then X fails to Granger-cause Y.To decide this, an F-test must be carried out to examine the null hypothesis of non-causality, . For the F-test, the unrestricted model will include lagged valuesof the other variable, whereas the restricted model will only include lags of the dependentvariable.The direct Granger test‟s effectiveness is measured by minimizing the mean square error(MSE) of the forecast: , where is the predictor of .The direct Granger test can be illustrated by applying it to two variables w2 and w3. The testwill be carried out twice, both using Eviews econometrics software, with the first test carriedout manually and the second test using the automated option to confirm the findings. Both
4 557699these variables are assumed to be stationary processesand a fourth order lag structure will beused during all tests.First w2 will be the dependent variable to test whether w3 Granger-causes w2. Theunrestricted model for this regression will therefore beThis regression is then carried out in Eviews, by typing “ls w2 c w2(-1 to -4) w3(-1 to -4)”,which produces the following results Dependent Variable: W2 Method: Least Squares Date: 01/27/12 Time: 14:55 Sample (adjusted): 1970M05 2010M12 Included observations: 488 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C -0.191490 0.124689 -1.535735 0.1253 W2(-1) -0.047480 0.045564 -1.042049 0.2979 W2(-2) 0.031968 0.045290 0.705846 0.4806 W2(-3) 0.058368 0.045332 1.287561 0.1985 W2(-4) 0.008925 0.045414 0.196534 0.8443 W3(-1) 0.088020 0.095999 0.916880 0.3597 W3(-2) 0.223407 0.095592 2.337081 0.0198 W3(-3) 0.281564 0.095722 2.941467 0.0034 W3(-4) 0.165180 0.096562 1.710606 0.0878 R-squared 0.039485 Mean dependent var 0.140908 Adjusted R-squared 0.023443 S.D. dependent var 2.046543 S.E. of regression 2.022412 Akaike info criterion 4.264729 Sum squared resid 1959.181 Schwarz criterion 4.342009 Log likelihood -1031.594 Hannan-Quinn criter. 4.295085 F-statistic 2.461352 Durbin-Watson stat 2.003610 Prob(F-statistic) 0.012781Now we have the results for the unrestricted model, this model must be restricted byassuming that the coefficients for the lagged values of w3 are equal to zero. We do this stepto compare between the restricted and unrestricted to be able to identify whether w3 doesGranger-cause w2 or not. This is achieved by carrying out a Wald test for coefficientrestrictions. Eviews considers the coefficients on the 4 lagged values of w3 to be c(6), c(7),c(8) and c(9), so to test this restriction these values must be equated to zero. The equation forthis restricted model is
5 557699The results from this Wald test are Wald Test: Equation: Untitled Test Statistic Value df Probability F-statistic 4.391862 (4, 479) 0.0017 Chi-square 17.56745 4 0.0015 Null Hypothesis: C(6)=C(7)=C(8)=C(9)=0 Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. C(6) 0.088020 0.095999 C(7) 0.223407 0.095592 C(8) 0.281564 0.095722 C(9) 0.165180 0.096562 Restrictions are linear in coefficients.The F-statistic of 4.392 with a p value of 0.0017 means that we can reject the null hypothesisof no Granger-causality and state that w3 does Granger-cause w2. The F-statistic can bemanually calculated using the following formulaWhere „RRSS‟ is the restricted model‟s residual sum of squares, „URSS‟ is the unrestrictedmodel‟s residual sum of squares, „T‟ is the sample size, „k‟ is the number of lags, and „q‟ isthe number of restrictions in place.Inputting the URSS of 1959.181 from the regression runabove and then running a restricted regression (results below) to obtain the RRSS of2031.035. Plugging the rest of the values in provides the correct F-statistic of 4.3919.
6 557699 Dependent Variable: W2 Method: Least Squares Date: 01/27/12 Time: 16:34 Sample (adjusted): 1970M05 2010M12 Included observations: 488 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 0.131640 0.093759 1.404026 0.1610 W2(-1) -0.020913 0.045492 -0.459706 0.6459 W2(-2) 0.040212 0.045585 0.882120 0.3782 W2(-3) 0.048299 0.045724 1.056301 0.2914 W2(-4) -0.006448 0.045741 -0.140975 0.8879 R-squared 0.004258 Mean dependent var 0.140908 Adjusted R-squared -0.003988 S.D. dependent var 2.046543 S.E. of regression 2.050620 Akaike info criterion 4.284354 Sum squared resid 2031.035 Schwarz criterion 4.327288 Log likelihood -1040.382 Hannan-Quinn criter. 4.301219 F-statistic 0.516339 Durbin-Watson stat 1.998142 Prob(F-statistic) 0.723766We can compare this F-statistic to the F-critical value at the 1% level (3.32) and the 5% level(2.37) and can conclude that since the F-statistic is higher than both the 5% and 1% levelsthat the null is rejected and so . From these results, we can tell that w3 precedes w2and that w2 is better predicted when the history of w3 is taken into account than when it isexcluded.Since unidirectional Granger-causality has been identified, it is now time to test whetherGranger-causality runs in the opposite direction aswell and whether a feedback system ispresent.The unrestricted model for regressing w3 as the dependent variable on four of its own lagsand four lags of w2 is given by the formulaSimilarly as before, we are testing the null hypothesis of non-causality which is shown if allthe coefficients of w2 are jointly equal to zero. The results below show the regression of w3on four lags of itself and four lags of w2
7 557699 Dependent Variable: W3 Method: Least Squares Date: 01/27/12 Time: 15:22 Sample (adjusted): 1970M05 2010M12 Included observations: 488 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 0.512293 0.058950 8.690337 0.0000 W3(-1) 0.015801 0.045386 0.348138 0.7279 W3(-2) -0.059553 0.045193 -1.317738 0.1882 W3(-3) -0.029739 0.045255 -0.657140 0.5114 W3(-4) -0.084203 0.045652 -1.844457 0.0657 W2(-1) -0.012713 0.021541 -0.590168 0.5554 W2(-2) -0.036423 0.021412 -1.701064 0.0896 W2(-3) 0.022747 0.021432 1.061356 0.2891 W2(-4) -0.020955 0.021470 -0.975972 0.3296 R-squared 0.024450 Mean dependent var 0.438320 Adjusted R-squared 0.008157 S.D. dependent var 0.960063 S.E. of regression 0.956140 Akaike info criterion 2.766445 Sum squared resid 437.9034 Schwarz criterion 2.843726 Log likelihood -666.0126 Hannan-Quinn criter. 2.796801 F-statistic 1.500638 Durbin-Watson stat 2.021280 Prob(F-statistic) 0.154304Taking these results, a Wald test is then carried out with the same restrictions as above to testthe null hypothesis, with the results of this test in the table below Wald Test: Equation: Untitled Test Statistic Value df Probability F-statistic 1.401791 (4, 479) 0.2323 Chi-square 5.607163 4 0.2305 Null Hypothesis: C(6)=C(7)=C(8)=C(9)=0 Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. C(6) -0.012713 0.021541 C(7) -0.036423 0.021412 C(8) 0.022747 0.021432 C(9) -0.020955 0.021470 Restrictions are linear in coefficients.In contrast to the results above, the F-statistic of 1.402 with a p value of 0.2323 shows thatwe fail to reject the null hypothesis even at the 10% level, and will continue to fail until the24% level. The F-statistic can be manually calculated as before to confirm the finding, shownbelow
8 557699As such, we can conclude that w2 does not Granger-cause w3 and therefore conclude thatthere is unidirectional Granger-causality of .This result can be confirmed by using Eviews‟ automated Granger-causality test option, theresults of which are shown below. Pairwise Granger Causality Tests Date: 01/27/12 Time: 14:44 Sample: 1970M01 2010M12 Lags: 4 Null Hypothesis: Obs F-Statistic Prob. W3 does not Granger Cause W2 488 4.39186 0.0017 W2 does not Granger Cause W3 1.40179 0.2323As you can see, these results are of course identical to the results obtained from undertakingthe direct Granger test manually. It can be concluded that w3 does Granger-cause w2 becausesince the p value of 0.0017 is so low, then we can definitely say that the coefficients of w3 inthe model with w2 as the dependent variable are not equal to zero and so they affect thefuture performance of w2. The converse can be said for the coefficients of w2 when w3 is thedependent variable. As the p value of 0.2323 is higher than the 10% significance level, wecan conclude that the coefficients of w2 are all equal to zero and as such offer no otherinformation on predicting the future of w3. The fact that we have identified unidirectionalGranger-causality from W3 to W2 would provide policy makers with important knowledge ifthese were economic variables instead. Using the example of Sims findings (1972), if W3was the variable explaining the money supply and W2 the variable explaining output, policymakers would be able to use these findings to show that in order to increase output they couldincrease the monetary supply.However there are some theoretical issues with the direct Granger test. The first issue is thatthis test assumes the correct specification to be unknown, which is a direct violation of thecorrect specification assumption of ordinary least squares. The second issue arises becausethis test relies on overfitting the model to make sure that all the autodependence processes areremoved from the data, with in turn reduces the predictive power of the test and causesestimator inefficiency. Overfitting does not affect biasedness though. Because Grangercoefficients area not optimal, they should not be used as a source of structural coefficientsand instead should only be used to test for significance. The final major issue of the directGranger test is of its dependence on the right choice of conditioning set and its sensitivity todata included. The conclusion in Sims‟ (1972) paper mentioned at the start regarding moneyGranger-causing output but output does not Granger-cause money has since been provedincorrect by Sims himself because when interest rates were included in the system then thisfinding does not hold. Obviously this is a big drawback as being able to reliably predictmovements in output and money would need to include other factors such as interest rates.
9 557699 6. The Sims TestSims proposed a new direct test for the existence of unidirectional Granger-causality whichhadn‟t been used before in his 1972 paper. At the time, Sims realised that the „direction ofcausation‟ between two variables is not normally identified and so he created his test to beable to identify the direction of causation.The Sims test starts by assuming that both time series being tested are jointly covariance-stationary. The time series will be covariance-stationary if neither its mean nor itsautocovariance (the variance of the variable against a lagged version of itself) depends ontime. He achieves this by using only linear predictors and by using the mean squared error ofthe forecast as his gauge for predictive accuracy.Sims starts by considering two stochastic series X and Y which are both linearly regular sowe can write them in the form(1)Where are uncorrelated white noise error terms with unit variance, and a,b,c and dwill all vanish for . Expression (1) represents the moving average of the vector .His test is to regress Y on past and future values of X whilst accounting for generalized leastsquares and prefiltering of the serial correlation. Granger-causality can then be detectedbecause if testing for only, then all future values of X should have coefficients in theregression that are not significantly different from zero. Because this test requires accurate F-tests, the assumption of no serial correlation in the residuals must be upheld. As such, allvariables used in the regression will be measured in natural logarithmic form and prefilteredusing the filterSuch that each variable will be transformed into . He used thisfiltering to change the residuals from the regression into white noise. Although he used thisprefiltering, he did identify two problems which arose due to their use. The first problem isthat if the filter fails to produce white noise residuals, then it is quite unlikely to fail byleaving substantial positive first-order serial correlation. The second problem he identifiedwas that the prefiltering can produce a “perverse effect on approximation error when lagdistributions are subject to prior smoothness restrictions” (Sims, 1972, p.545).After this transformation, the following regression is run
10 557699 are being used instead of the original variables because they have been transformed.We can then test the null hypothesis of no causalityWe use a Wald test to compare the restricted and unrestricted models which in turn producesan F-statistic which can be compared to critical values in order to decide whether the nullhypothesis is true or false.The Sims test is the weakest of the three main Granger-causality tests and coupled with theflaws regarding spurious regression and its higher costs than the Direct Granger test, it is usedleast in empirical testing. 7. The Modified Sims TestGeweke, Meese and Dent (1983) suggested a modified version of the Sims test which isbased on the ordinary least squares estimation ofWhere is the coefficient on the leads and lags of , is the coefficient on the lags of ,is the deterministic term and is the stochastic error term. The test deals with serialcorrelation by including lagged values of in the regression.To test whether is the test that The equationabove is then estimated in both unrestricted and restricted (forms. The null hypothesis for this test is of no causality from , which is based oncomparing the F-statistic to the critical values.The Modified Sims test can be illustrated by applying it to two variables w2 and w3. The testwill be applied using Eviews econometrics software with both variables assumed to bestationary processes.First, the null hypothesis of no causality from w2 to w3 will be tested with the followingregression modelThe Eviews output for this regression is
11 557699 Dependent Variable: W2 Method: Least Squares Date: 01/28/12 Time: 14:40 Sample (adjusted): 1970M05 2010M08 Included observations: 484 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C -0.070277 0.171947 -0.408716 0.6829 W2(-1) -0.054761 0.046126 -1.187221 0.2357 W2(-2) 0.026724 0.045756 0.584044 0.5595 W2(-3) 0.055185 0.045527 1.212130 0.2261 W2(-4) 0.014588 0.045570 0.320115 0.7490 W3(-4) 0.164094 0.097654 1.680353 0.0936 W3(-3) 0.278277 0.096503 2.883611 0.0041 W3(-2) 0.218744 0.096467 2.267540 0.0238 W3(-1) 0.085050 0.096788 0.878727 0.3800 W3 -0.056039 0.097155 -0.576799 0.5644 W3(1) -0.030316 0.097338 -0.311448 0.7556 W3(2) -0.131876 0.097096 -1.358203 0.1751 W3(3) 0.121946 0.096853 1.259083 0.2086 W3(4) -0.128530 0.097049 -1.324377 0.1860 R-squared 0.051966 Mean dependent var 0.156198 Adjusted R-squared 0.025743 S.D. dependent var 2.042684 S.E. of regression 2.016220 Akaike info criterion 4.268825 Sum squared resid 1910.618 Schwarz criterion 4.389795 Log likelihood -1019.056 Hannan-Quinn criter. 4.316359 F-statistic 1.981738 Durbin-Watson stat 1.998877 Prob(F-statistic) 0.020612Next a Wald test for coefficient restrictions is applied to calculate an F-statistic in order totest the null hypothesis. This is undertaken by inputting that the coefficients on the leadvalues of w3 are all equal to zero. This generates the following results Wald Test: Equation: Untitled Test Statistic Value df Probability F-statistic 1.281008 (4, 470) 0.2765 Chi-square 5.124033 4 0.2748 Null Hypothesis: C(11)=C(12)=C(13)=C(14)=0 Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. C(11) -0.030316 0.097338 C(12) -0.131876 0.097096 C(13) 0.121946 0.096853 C(14) -0.128530 0.097049 Restrictions are linear in coefficients.The Wald test has returned an F-statistic of 1.281 with a probability of 0.2765, indicating that
12 557699we are unable to reject the null hypothesis of no causality from w2 to w3. This result supportsthe result from the early direct granger test and it in fact is a stronger non rejection of the nullas the modified Sims test would only reject the null at the 28% level, whereas the directGranger test would reject the null at the 24% level.Now the test will be applied in the opposite direction to discover whether there isunidirectional Granger-causality or whether the movements are contemporaneous. Theregression which first must be run will beThis produces the following output Dependent Variable: W3 Method: Least Squares Date: 01/28/12 Time: 15:00 Sample (adjusted): 1970M05 2010M08 Included observations: 484 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 0.502198 0.058863 8.531573 0.0000 W3(-1) -0.012809 0.045784 -0.279780 0.7798 W3(-2) -0.070918 0.045619 -1.554563 0.1207 W3(-3) -0.031645 0.045529 -0.695058 0.4874 W3(-4) -0.085740 0.045561 -1.881887 0.0605 W2(-4) -0.012703 0.021476 -0.591471 0.5545 W2(-3) 0.028604 0.021454 1.333238 0.1831 W2(-2) -0.033810 0.021385 -1.581018 0.1145 W2(-1) -0.012459 0.021573 -0.577530 0.5639 W2 -0.012779 0.021606 -0.591465 0.5545 W2(1) 0.018409 0.021630 0.851063 0.3952 W2(2) 0.050612 0.021426 2.362180 0.0186 W2(3) 0.064199 0.021278 3.017167 0.0027 W2(4) 0.029903 0.021325 1.402243 0.1615 R-squared 0.058349 Mean dependent var 0.434440 Adjusted R-squared 0.032304 S.D. dependent var 0.962381 S.E. of regression 0.946709 Akaike info criterion 2.756849 Sum squared resid 421.2411 Schwarz criterion 2.877818 Log likelihood -653.1574 Hannan-Quinn criter. 2.804383 F-statistic 2.240272 Durbin-Watson stat 2.022328 Prob(F-statistic) 0.007493Next the Wald test is applied with the restrictions that the coefficients for the lead values ofW2 are all equal to zero. This produces the following results
13 557699 Wald Test: Equation: Untitled Test Statistic Value df Probability F-statistic 4.267313 (4, 470) 0.0021 Chi-square 17.06925 4 0.0019 Null Hypothesis: C(11)=C(12)=C(13)=C(14)=0 Null Hypothesis Summary: Normalized Restriction (= 0) Value Std. Err. C(11) 0.018409 0.021630 C(12) 0.050612 0.021426 C(13) 0.064199 0.021278 C(14) 0.029903 0.021325 Restrictions are linear in coefficients.The F-statistic generated by comparing the unrestricted and restricted models of 4.267 with ap value of 0.0021 shows that we can reject the null hypothesis at the 1% level and canconclude that there is unidirectional Granger-causality from These results supportthe findings which were generated using the direct Granger test. The modified Sims testshows with great certainty that w3 Granger-causes w2 but w2 does not Granger-cause w3.Ifw2 was the variable for GDP (for measuring income) and w3 was the variable for money(measured using the money supply and monetary base) then these results would support thefindings by Sims (1972) that there is a unidirectional Granger-causal relationship fromMoney to Income. This finding would prove Monetarists wrong as they hypothesized thatmoney was an exogenous variable in the money-income relationship, however for money tobe exogenous it must not Granger-cause income, which Sims proved otherwise. 8. Contrasting features of the Sims and Modified Sims TestsThere are a number of differences between the standard Sims test and the Modified Sims test.One difference between the Sims test and Modified Sims test is the way they deal with serialcorrelation. The Sims test removes it by using the general least squares procedure andprefiltering, whereas the Modified Sims test introduces lagged values of the dependentvariable in order to remove serial correlation.Because the modified Sims test uses this laggeddependent variable, it also solves the problem of spurious regression. The Sims test still has aproblem with spurious regression and because the filtering it uses can sometimes fail toproduce white noise, this can also cause spurious rejection of the null as well as making theDurbin Watson statistic useless. This brings up the second difference which is that the Simstest filters the variables before they can be used, whereas the modified Sims test does notrequire this step.
14 557699In terms of performance; Guilkey and Salemi (1982) found that when there is unidirectionalcausation, the modified Sims test outperforms the Sims test in its ability to reject a false null.When compared to the Granger test, they found that the Direct Granger test rejected a falsenull 3.26% and 2.64% more than the Sims and Modified Sims tests respectively, confirmingthat the modified version is more powerful, even though both are weaker than the directGranger test. The Sims procedure also had a much higher rate of type 1 errors than theModified Sims test. They found that when both variables are mutually uncaused by eachother, the Modified version still outperformed the Sims test for frequency of correctdecisions. The performance of both tests does improve with increases in sample size;however this improvement is most rapid with the standard Sims test.Word count: 299
15 557699ReferencesEngle, R.F., Hendry, D.F., and Richard, J., 1983. Exogeneity. Econometrica, 51(2), pp.277-304.Freeman, J.R., 1983. Granger Causality and the Times Series Analysis of PoliticalRelationships. American Journal of Political Science, 27(2), pp.327-358.Geweke, J., Meese, R., and Dent, W., 1983. Comparing Alternative Tests of Causality inTemporal Systems. Journal of Econometrics, 21, pp.161-194.Granger, C.W.J., 1969. Investigating Causal Relations by Econometric Models and Cross-spectral Methods. Econometrica, 37(3), pp.424-438.Guilkey, D.K., and Salemi, M.K., 1982. Small Sample Properties of Three Tests for Granger-Causal Ordering in a Bivariate Stochastic System. The Review of Economics and Statistics,64(4), pp.668-680.Haugh, L.D., 1976. Checking the Independence of Two Covariance-Stationary Time Series:A Univariate Residual Cross-Correlation Approach. Journal of the American StatisticalAssociation, 71, pp.265-293.Orcutt, G.H., 1952. Actions, Consequences, and Causal Relations. The Review of Economicsand Statistics, 34(4), pp.305-313.Pierce, D.A., and Haugh, L.D., 1977. Causality in Temporal Systems: Characterisations and aSurvey. Journal of Econometrics, 5(3), pp.265-293.Sims, C.A., 1972. Money, Income, and Causality. The American Economic Review, 62(4),pp.540-552.Toda, H.Y., and Phillips, P.C.B., 1993. Vector Autoregressions and Causality. Econometrica,61(6), pp.1367-1393.