Suppose I have a linear regression model as follows: $E[Y_i | X_i] = \beta_0 + \beta_1X_{i1} + \beta_2X_{i2} + \beta_3X_{i3} + \beta_4X_{i4}$, what's the Wald test statistic for testing $H_0: \beta_1 = \beta_2 = \beta_3 = 0$?

  • $\begingroup$ Hint: What is the covariance matrix of $(\hat{\beta}_1,\hat{\beta}_2, \hat{\beta}_3)$ ? $\endgroup$ – air Dec 11 '16 at 2:01
  • $\begingroup$ Is the standard error of the test statistic square root of Var(B1) + Var(B2) + Var(B3) + 2Cov(B1, B2) + 2Cov(B2, B3) + 2Cov(B1, B3)? Also, I am not sure what the numerator of the test statistic should be? $\endgroup$ – Adrian Dec 11 '16 at 2:40
  • $\begingroup$ Yeah, but you can get an explicit expression for it.. The numerator will be a quadratic form of $(\hat{\beta}_0, \hat{\beta}_1, \hat{\beta}_3)$ appropriately scaled by the covariance. $\endgroup$ – air Dec 11 '16 at 4:26
  • $\begingroup$ What about H0: B1 = B2 = 0, would the numerator be B1 - B2 - 0? $\endgroup$ – Adrian Dec 11 '16 at 4:46
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    $\begingroup$ Almost but not quite.. You need to account for the covariance of the estimators as I wrote above. It would be instructive if you updated your post with a definition of a Wald test and your attempts to figure out what the individual terms are in your specific problem. $\endgroup$ – air Dec 11 '16 at 5:00

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