In Section 3 of Spurious Regressions in Econometrics (1974), by C.W.J. Granger and P. Newbold, the authors present a standard linear regression model and F-Test.

Then they discard the assumption that the dependent variable is characterized by a white noise process. This results in the inability of the null hypothesis of the F-Test to be true. In their words:

"However, to the extent that the $Y_t's$ do not constitute a white noise process, the null hypothesis (8) cannot be true, and tests of it are inappropriate."

(8) $H_0:\beta_1=\beta_2=...=\beta_{k-1}=0$

Why does changing the white noise assumption disable the null hypothesis of the F-Test to be true?



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