Granger causality testing
- 1. 1 557699 Granger-causality testing within the context of the bivariate analysis of stationary macroeconomic time series 1. Introduction Granger proposed the idea of Granger-causality in his 1969 paper to describe the „causal relationships‟ between variables in econometric models.Before this, econometricians and economists understood the idea of „causal relationships‟ as asymmetrical relationships. Causal relations are studied because policy makers need to know the consequencesof the various actions which they will or are considering taking. For example; given a relationship between output and the price level, we need to know whether this relationship is expected to hold if actions controlling output are implemented, when actions controlling the price level are implemented, or when both of these cases occur (Orcutt, 1952). The use of the term causality is identical to that of Weiner who provided the base idea for Granger‟s work in the early fifties. 2. Granger-causality The idea of Granger-causality is that a variable X Granger-causes variable Y if variable Y can be better predicted using the histories of both X and Y than it can be predicted using the history of Y alone. This is shown if the expectation of Y given the history of X is different from the unconditional expectation of Y A second definition for causality has been offered by Granger (1969) which states that if (which means that if the variance of X predicted using the universe of information, U, is less than the variance of X predicted using all information except variable Y) then we can say that Y is causing X, denoted However, he then clarified that using the whole universe of information, U, is unrealistic so it is replaced with all relevant information. However, this change now makes testing more than a statistical procedure as there is a subjective element regarding what information is relevant. Another element to define is that of Feedback. A feedback system occurs if variable X Granger-causes variable Y, and Y Granger-causes X, denoted .All these definitions assume that only stationary series are involved, as non-stationary series stop these definitions being testable. Granger-causality has several components. The first component, temporality, is based on the principle that only past values of X can Granger-cause Y, because the future cannot cause the past or the present. If X occurs after Y, then we know that X cannot cause Y. Similarly though, if X occurred before Y then that does not necessarily imply that X caused Y. The second component of Granger-causality is exogeneity;Sims (1972) stated that for variable X to be exogenous of variable Y, X must fail to Granger-cause Y; this component was confirmed by Engle, Hendry, and Richard (1983). Independence is the third component of Granger-causality because variables X and Y are only independent of each other if both fail
- 2. 2 557699 to Granger-cause the other. The final component of Granger-causality is that of asymmetry; if X 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 series Y, Y precedes X, or if the movements are contemporaneous. The notion of Granger-causality does therefore not imply „true causality‟, but instead identifies whether one variable precedes another. For example; do changes in output occur before changes in money, or does the opposite occur, or do these changes occur at the same time. In his 1972 paper, Sims showed that money Granger-causes output, but output does not Granger-cause money. This result supported existing business cycle models which hypothesize that money plays an important role in output. We can therefore use Granger-causality tests to test for things we might have assumed to occur from elsewhere or which we have taken for granted. The Granger-causality tests being studied in this paper are bivariate, however multivariate tests can be carried out similarly using a Vector Autoregression (VAR), and in fact the Direct Granger Test is a bivariate case of VAR. It is important to remember that when testing for Granger-causality, the models should be fully specified. If the model isn‟t well specified, then spurious relationships maybe found despite the fact that there actually are no relationships between the variables. Another situation 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 time then it will display Granger-causality even though the real causality is obviously different. 3. Granger-causality tests There are three main tests for Granger-causality within the context of the bivariate analysis of stationary 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 be included 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 and Phillips, 1991). Other than the three tests above, there are other tests which can be used; for example a technique developed by Haugh (1976), and later Pierce and Haugh (1977) called „Haugh‟s Residual Cross-correlation test‟ uses a two-step procedure to test for Granger-causality. It uses the Autoregressive Integrated Moving Average (ARIMA)/cross-correlation approach by first estimating the ARIMA models for both series, let and be theARIMA models for series and respectively. Then the estimated residuals are, and , of the ARIMA models are saved and will be , and
- 3. 3 557699 . The second step is to cross correlate these residual series. The significance of the results is computed 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 from zero then these series contain Granger-causality. This method is not used as much as the aforementioned tests because of a number of drawbacks. Firstly, the statistics that this test produces are less powerful than the above tests. This method is also highly sensitive to the selection of lag length. This test‟s biggest flaw is that it can only indicate whether or not Granger-causality exists between the two variables but it cannot explain the direction of the causality. 4. The Direct Granger Test The direct granger test is a very useful tool as it allows econometricians to test for the direction of Granger-causality as well as for its presence. Following the definition for Granger-causality, the direct Granger test regresses each variable on lagged values of itself and the other explanatory variable. Empirically; the direct Granger test has been found to be more powerful than both the Sims and Modified Sims test, outperforming both of these by rejecting 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 are zero then the series fails to Granger-cause So consider the following regression model Where are the deterministics, is the random error term, is the coefficient on the lagged Y values, and is the coefficient on the lagged X values. We start with a one period lag instead of setting because we are not including instantaneous causality in the model(Instantaneous causality is when the changes in Y and X occur at the same time and are correlated).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 values of the other variable, whereas the restricted model will only include lags of the dependent variable. 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 test will be carried out twice, both using Eviews econometrics software, with the first test carried out manually and the second test using the automated option to confirm the findings. Both
- 4. 4 557699 these variables are assumed to be stationary processesand a fourth order lag structure will be used during all tests. First w2 will be the dependent variable to test whether w3 Granger-causes w2. The unrestricted model for this regression will therefore be This 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) 0.012781 Now we have the results for the unrestricted model, this model must be restricted by assuming that the coefficients for the lagged values of w3 are equal to zero. We do this step to compare between the restricted and unrestricted to be able to identify whether w3 does Granger-cause w2 or not. This is achieved by carrying out a Wald test for coefficient restrictions. 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 for this restricted model is
- 5. 5 557699 The 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 hypothesis of no Granger-causality and state that w3 does Granger-cause w2. The F-statistic can be manually calculated using the following formula Where „RRSS‟ is the restricted model‟s residual sum of squares, „URSS‟ is the unrestricted model‟s residual sum of squares, „T‟ is the sample size, „k‟ is the number of lags, and „q‟ is the number of restrictions in place.Inputting the URSS of 1959.181 from the regression run above and then running a restricted regression (results below) to obtain the RRSS of 2031.035. Plugging the rest of the values in provides the correct F-statistic of 4.3919.
- 6. 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) 0.723766 We 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% levels that the null is rejected and so . From these results, we can tell that w3 precedes w2 and that w2 is better predicted when the history of w3 is taken into account than when it is excluded. Since unidirectional Granger-causality has been identified, it is now time to test whether Granger-causality runs in the opposite direction aswell and whether a feedback system is present. The unrestricted model for regressing w3 as the dependent variable on four of its own lags and four lags of w2 is given by the formula Similarly as before, we are testing the null hypothesis of non-causality which is shown if all the coefficients of w2 are jointly equal to zero. The results below show the regression of w3 on four lags of itself and four lags of w2
- 7. 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) 0.154304 Taking these results, a Wald test is then carried out with the same restrictions as above to test the 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 that we fail to reject the null hypothesis even at the 10% level, and will continue to fail until the 24% level. The F-statistic can be manually calculated as before to confirm the finding, shown below
- 8. 8 557699 As such, we can conclude that w2 does not Granger-cause w3 and therefore conclude that there is unidirectional Granger-causality of . This result can be confirmed by using Eviews‟ automated Granger-causality test option, the results 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 0.2323 As you can see, these results are of course identical to the results obtained from undertaking the direct Granger test manually. It can be concluded that w3 does Granger-cause w2 because since the p value of 0.0017 is so low, then we can definitely say that the coefficients of w3 in the model with w2 as the dependent variable are not equal to zero and so they affect the future performance of w2. The converse can be said for the coefficients of w2 when w3 is the dependent variable. As the p value of 0.2323 is higher than the 10% significance level, we can conclude that the coefficients of w2 are all equal to zero and as such offer no other information on predicting the future of w3. The fact that we have identified unidirectional Granger-causality from W3 to W2 would provide policy makers with important knowledge if these were economic variables instead. Using the example of Sims findings (1972), if W3 was the variable explaining the money supply and W2 the variable explaining output, policy makers would be able to use these findings to show that in order to increase output they could increase the monetary supply. However there are some theoretical issues with the direct Granger test. The first issue is that this test assumes the correct specification to be unknown, which is a direct violation of the correct specification assumption of ordinary least squares. The second issue arises because this test relies on overfitting the model to make sure that all the autodependence processes are removed from the data, with in turn reduces the predictive power of the test and causes estimator inefficiency. Overfitting does not affect biasedness though. Because Granger coefficients area not optimal, they should not be used as a source of structural coefficients and instead should only be used to test for significance. The final major issue of the direct Granger test is of its dependence on the right choice of conditioning set and its sensitivity to data included. The conclusion in Sims‟ (1972) paper mentioned at the start regarding money Granger-causing output but output does not Granger-cause money has since been proved incorrect by Sims himself because when interest rates were included in the system then this finding does not hold. Obviously this is a big drawback as being able to reliably predict movements in output and money would need to include other factors such as interest rates.
- 9. 9 557699 6. The Sims Test Sims proposed a new direct test for the existence of unidirectional Granger-causality which hadn‟t been used before in his 1972 paper. At the time, Sims realised that the „direction of causation‟ between two variables is not normally identified and so he created his test to be able 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 its autocovariance (the variance of the variable against a lagged version of itself) depends on time. He achieves this by using only linear predictors and by using the mean squared error of the forecast as his gauge for predictive accuracy. Sims starts by considering two stochastic series X and Y which are both linearly regular so we can write them in the form (1) Where are uncorrelated white noise error terms with unit variance, and a,b,c and d will 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 least squares and prefiltering of the serial correlation. Granger-causality can then be detected because if testing for only, then all future values of X should have coefficients in the regression 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, all variables used in the regression will be measured in natural logarithmic form and prefiltered using the filter Such that each variable will be transformed into . He used this filtering to change the residuals from the regression into white noise. Although he used this prefiltering, he did identify two problems which arose due to their use. The first problem is that if the filter fails to produce white noise residuals, then it is quite unlikely to fail by leaving substantial positive first-order serial correlation. The second problem he identified was that the prefiltering can produce a “perverse effect on approximation error when lag distributions are subject to prior smoothness restrictions” (Sims, 1972, p.545). After this transformation, the following regression is run
- 10. 10 557699 are being used instead of the original variables because they have been transformed. We can then test the null hypothesis of no causality We use a Wald test to compare the restricted and unrestricted models which in turn produces an F-statistic which can be compared to critical values in order to decide whether the null hypothesis is true or false. The Sims test is the weakest of the three main Granger-causality tests and coupled with the flaws regarding spurious regression and its higher costs than the Direct Granger test, it is used least in empirical testing. 7. The Modified Sims Test Geweke, Meese and Dent (1983) suggested a modified version of the Sims test which is based on the ordinary least squares estimation of Where 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 serial correlation by including lagged values of in the regression. To test whether is the test that The equation above is then estimated in both unrestricted and restricted ( forms. The null hypothesis for this test is of no causality from , which is based on comparing the F-statistic to the critical values. The Modified Sims test can be illustrated by applying it to two variables w2 and w3. The test will be applied using Eviews econometrics software with both variables assumed to be stationary processes. First, the null hypothesis of no causality from w2 to w3 will be tested with the following regression model The Eviews output for this regression is
- 11. 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) 0.020612 Next a Wald test for coefficient restrictions is applied to calculate an F-statistic in order to test the null hypothesis. This is undertaken by inputting that the coefficients on the lead values 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. 12 557699 we are unable to reject the null hypothesis of no causality from w2 to w3. This result supports the result from the early direct granger test and it in fact is a stronger non rejection of the null as the modified Sims test would only reject the null at the 28% level, whereas the direct Granger test would reject the null at the 24% level. Now the test will be applied in the opposite direction to discover whether there is unidirectional Granger-causality or whether the movements are contemporaneous. The regression which first must be run will be This 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) 0.007493 Next the Wald test is applied with the restrictions that the coefficients for the lead values of W2 are all equal to zero. This produces the following results
- 13. 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 a p value of 0.0021 shows that we can reject the null hypothesis at the 1% level and can conclude that there is unidirectional Granger-causality from These results support the findings which were generated using the direct Granger test. The modified Sims test shows with great certainty that w3 Granger-causes w2 but w2 does not Granger-cause w3.If w2 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 the findings by Sims (1972) that there is a unidirectional Granger-causal relationship from Money to Income. This finding would prove Monetarists wrong as they hypothesized that money was an exogenous variable in the money-income relationship, however for money to be exogenous it must not Granger-cause income, which Sims proved otherwise. 8. Contrasting features of the Sims and Modified Sims Tests There 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 serial correlation. The Sims test removes it by using the general least squares procedure and prefiltering, whereas the Modified Sims test introduces lagged values of the dependent variable in order to remove serial correlation.Because the modified Sims test uses this lagged dependent variable, it also solves the problem of spurious regression. The Sims test still has a problem with spurious regression and because the filtering it uses can sometimes fail to produce white noise, this can also cause spurious rejection of the null as well as making the Durbin Watson statistic useless. This brings up the second difference which is that the Sims test filters the variables before they can be used, whereas the modified Sims test does not require this step.
- 14. 14 557699 In terms of performance; Guilkey and Salemi (1982) found that when there is unidirectional causation, 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 false null 3.26% and 2.64% more than the Sims and Modified Sims tests respectively, confirming that the modified version is more powerful, even though both are weaker than the direct Granger test. The Sims procedure also had a much higher rate of type 1 errors than the Modified Sims test. They found that when both variables are mutually uncaused by each other, the Modified version still outperformed the Sims test for frequency of correct decisions. 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
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