SPURIOUS REGRESSIONS 1 In a famous Monte Carlo experiment, Granger and Newbold fitted the model Y t =  1 +  2 X t + u t where Y t and X t were independently-generated random walks.

2 Obviously, a regression of one random walk on another ought not to yield significant results except as a matter of Type I error. SPURIOUS REGRESSIONS

============================================================= Dependent Variable: Y Method: Least Squares Sample: Included observations: 100 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C X ============================================================= R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criteri Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================= 3 Regressing the upper random walk in the figure on the lower one, the output shown above is obtained. SPURIOUS REGRESSIONS

============================================================= Dependent Variable: Y Method: Least Squares Sample: Included observations: 100 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C X ============================================================= R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criteri Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================= 4 The true slope coefficient is 0, because Y was generated independently of X. And yet the estimate of the slope coefficient appears to be significantly different from 0 at the 1% level and above. SPURIOUS REGRESSIONS

============================================================= Dependent Variable: Y Method: Least Squares Sample: Included observations: 100 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C X ============================================================= R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criteri Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================= 5 Using a 5 percent significance test, you would expect to encounter Type I error 5 percent of the time. However, performing the experiment with 100 pairs of random walks, Granger and Newbold found that the null hypothesis of a 0 slope coefficient was rejected 77 times. SPURIOUS REGRESSIONS

============================================================= Dependent Variable: Y Method: Least Squares Sample: Included observations: 100 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C X ============================================================= R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criteri Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================= 6 Using a 1 percent test made little difference. At that level the null hypothesis should be rejected 1 in 100 times, but in Granger and Newbold's experiment it was rejected 70 times. SPURIOUS REGRESSIONS

7 The reason for this is that the disturbance term cannot satisfy the regression model conditions if the null hypothesis H 0 :  2 = 0 is true. SPURIOUS REGRESSIONS

8 If H 0 is true (and we know it to be true), u t is a random walk. In fact, since  1 = 0, it is the same as Y t. As a consequence, the standard errors and the t statistics in the regression of Y t on X t are invalidated. SPURIOUS REGRESSIONS

9 This would also be the case if Y t were a stationary autoregressive process. The number of apparent Type I errors is most dramatic when  is large, which is why Granger and Engel chose  = 1 for their experiment. SPURIOUS REGRESSIONS

============================================================= Dependent Variable: Y Method: Least Squares Sample: Included observations: 100 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C X ============================================================= R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criteri Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================= 10 Of course, if the disturbance term is highly autocorrelated or a random walk, the Durbin– Watson statistic should provide a warning, as in this case. SPURIOUS REGRESSIONS

============================================================= Dependent Variable: Y Method: Least Squares Sample: Included observations: 100 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C X ============================================================= R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criteri Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================= 11 The main point of Granger and Newbold's experiment was to show that it was easy to obtain apparently significant results, even if the model were nonsense, if evidence of autocorrelation were ignored. SPURIOUS REGRESSIONS

============================================================= Dependent Variable: Y Method: Least Squares Sample: Included observations: 100 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C X ============================================================= R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criteri Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================= 12 However it also helped stimulate research into the theory of nonstationary processes and regression models using them. SPURIOUS REGRESSIONS

Copyright Christopher Dougherty 2000–2006. This slideshow may be freely copied for personal use