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A question that frequently arises in time series analysis is whether or not one economic variable can help forecast another economic variable. For instance, it has been well documented that nearly all of the postwar economic recessions have been preceded by large increases in the price of petroleum. Does this imply that oil shocks cause recessions?
One way to address this question was proposed by Granger (1969) and popularized by Sims (1972). Testing causality, in the Granger sense, involves using F-tests to test whether lagged information on a variable Y provides any statistically significant information about a variable X in the presence of lagged X. If not, then \"Y does not Granger-cause X.\"
There are many ways in which to implement a test of Granger causality. On particularly simple approach uses the autoregressive specification of a bivariate vector autoregression. Assume a particular autoregressive lag length p, and estimate the following unrestricted equation by ordinary least squares (OLS):
Producing the desired test statistics requires some preliminary data manipulation. Two Citibase data sets are read from the SASHELP library: CITIQTR, from which the variables DATE and GDPQ are kept, and CITIMON, from which DATE and EEGP are kept.
A problem arises from the fact that GDPQ is quarterly gross domestic product, measured in billions of 1987 dollars from 1980:1 to 1991:4, while EEGP is an index, with base year 1987, of monthly retail gas prices from January 1980 to December 1991. In order to use these two series in this analysis, it is necessary for the observations to be of the same frequency. The EXPAND procedure can be used to transform the monthly gas prices observations into quarterly observations and merge them with the GDP data to create a combined data set with quarterly observations. The OBSERVED= option enables you to control the observation characteristics of the input time series and of the output series. In this example, the average of three monthly observations are used to create each quarterly observation. Then, the data are lagged two periods using the LAG function.
proc expand data=gp out=temp from=month to=qtr; convert eegp / observed=average; id date; run; data combined; merge gdp temp; by date; run; data causal; set work.combined; gdpq_1 = lag(gdpq); gdpq_2 = lag2(gdpq); eegp_1 = lag(eegp); eegp_2 = lag2(eegp); run;
ods select Iml._LIT1010 Iml.TEST1_P_VAL1 Iml.TEST2_P_VAL2; ods html body='exgran01.htm'; * compute test; proc iml; start main; use out1; read all into e1 vare1; close out1; use out2; read all into e0 vare0; close out2; p = 2; /* # of lags */ T = nrow(e1); /* # of observations */ sse1 = ssq(e1); sse0 = ssq(e0); * F test; test1 = ((sse0 - sse1)/p)/(sse1/(T - 2*p - 1)); p_val1 = 1 - probf(test1,p,T - 2*p - 1); * asymtotically equivalent test; test2 = (T * (sse0 - sse1))/sse1; p_val2 = 1 - probchi(test2,p); print \"IML Result\",, test1 p_val1,, test2 p_val2; finish; run; quit; ods html close;IML Resulttest1p_val13.86234940.0286651test2p_val28.62291960.013414 Figure 2: Bivariate Granger Causality Test Results
As shown in Figure 2, with p (the number of lags included in the regressions) set equal to two, both test statistics are significant at the 5% level. Thus, it would seem that past values of petroleum prices help to predict GDP. 076b4e4f54