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2.2 The Granger causality test

In order to assess the existence of a spillover effect from East Asian to Brazilian markets, we employ the statistical test known as the Granger causality test. It measures the significance of past values of variable X in explaining variable Y, taking into account the effect of past values of variable Y itself. Usually causal relations are tested both ways, from X to Y and from Y to X. In the next paragraphs we define the Granger test.

Formally, the test involves estimating two regressions:

where R_1,t is the series RIBOV-ON;
            R_2,t is the series of returns from MSEMF-ASIA;
            a_i, b_i, l_i and d_i are constant coefficients;
                n is the number of lagged terms used; and
            e_1,t and e_1,t are uncorrelated innovations, i.e. S(e_1,t e_1,t)=0.

The series of returns from MSEMF-ASIA is obtained by taking the first difference of the logarithm of the level series.
Causal relations (in the Granger sense) are inferred through the overall statistical significance of coefficients b_i and d_i. Specifically, the null hypotheses tested are:

H0⁽¹⁾: d₁=d₂= ... = d_n=0 à "R1,t does not Granger-cause R2,t."

for the first equation and

H0(2): b₁=b₂= ... = b_n=0 à "R2,t does not Granger-cause R1,t."

for the second one.

The test statistic is the generalized F statistic, which in this case measures whether the inclusion of lagged terms of a variable significantly improves the autoregression of another:

where RSSR is the residual sum of squares in a restricted regression (not including lagged terms of the exogenous variable;
            RSSUR is the residual sum of squares in an unrestricted regression (which includes all lagged terms);
            n is the number of lagged terms used; and
            T is the total number of observations in the time series.

The acceptance or rejection of the two hypotheses defined above opens four possible scenarios. Acceptance of H0⁽¹⁾ and rejection of H0⁽²⁾ is interpreted as causality from R_1,t to R_2,t, while acceptance of H0⁽²⁾ and rejection of H0⁽¹⁾ is interpreted as causality in the reverse direction. If both hypotheses are rejected, it is said that there is feedback between the two variables. Finally, if both are accepted, no Granger causality is said to be detected between the variables.

Although the test is easily understood and implemented, it is also notoriously sensitive to the number of lagged terms included in the regression. In general, the more lags included the better, if the number of observations is sufficiently large. The robustness of causality relationships detected throughout different number of lags is also taken into consideration. Our approach in this study is to run the test including from 1 to 15 lags (roughly three weeks of trading days) and to report the p-values associated with the F statistics obtained in each case. We do this for the same samples A to F already used in Section 1, first for the series of returns and then for the series of squared returns.

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