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I estimate the following equation using OLS: $y_{t} = a + b*x_{t} + u_t$. I performed ADF tests for both y and x series and found that H0 (the existence of unit root) can not be rejected. I also performed ADF test for residuals $\hat{u}_t$ and got the same result. $R^2 > 0.9$ and t-stat for $\hat{b}$ is sufficiently greater than zero.

I remembered from my university courses that regressing non-stationary variables lead to spurious regression: t-stat is meaningless and no longer follows t-distribution and $\hat{b}$ is not consistent. The only exception is the case of cointegrated variables.

The question is: exactly which assumptions of Gauss-Markov theorem are violated? Why the $\hat{b}$ is not consistent? Why we can’t just adjust $\hat{Var}(\hat{b})$ using standard errors in the Newey-West form to have consistent estimates of standard errors of coefficients in the case of heteroscedasticity and serial correlation?

kissmemiau
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  • Is $x_t$ really stochastic or is it just increasing (e.g., a time trend-type variable)? It makes some difference for the asymptotics... – jbowman Feb 05 '23 at 23:43
  • This is likely a duplicate question (at least the first two out of three questions). Consider searching the website carefully to find a thread that answers it. – Richard Hardy Feb 06 '23 at 08:21
  • $y_t$ looks like slightly deformed part of sinus and $x_t$ has no clear trend and looks like really stochastic. – kissmemiau Feb 06 '23 at 08:23
  • One example for the types of available answers @RichardHardy mentioned: https://stats.stackexchange.com/questions/145864/estimation-of-unit-root-ar1-model-with-ols – Christoph Hanck Feb 06 '23 at 13:11

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