Why is the dickey fuller test different from a simple t-test

Question

I am trying to understand why should there be different distribution for t-statistic, in case of AR model, Dickey-Fuller test

For e.g. Say, the model is $Y_t = \beta_lY_{t-1} + \varepsilon_{t}$.

Why should I not use Simple linear regression model like $y_i = \beta_0 + \beta_1x_i+\epsilon_i$, where $x_i = Y_{t-1} $ and $y_i = Y_t$, and get the coefficient estimate as $$\hat\beta_1=\frac{\sum_ix_iy_i-n\bar x\bar y}{n\bar x^2-\sum_ix_i^2}$$ and its standard error estimate as $$s_{\hat\beta_1}=\sqrt{\frac{\sum_i\hat\epsilon_i^2}{(n-2)\sum_i(x_i-\bar x)^2}}.$$

Once we get the coefficient estimate and its standard error estimate, why can not we say the t-stat ($\frac{{\hat\beta_1}}{s_{\hat\beta_1}}$) follows a t-distribution, just like how we do in the case of simple linear regression. Are we violating any particular assumption in doing so?

Answer 1

You are right that the test statistic is just a standard t-statistic.

It, however, follows a different null distribution, i.e., using critical values from the t or normal distribution would lead to tests that would not reject in $\alpha$% of the cases when the null is true.

See Estimation of unit-root AR(1) model with OLS for an assumption that is violated and How is the augmented Dickey–Fuller test (ADF) table of critical values calculated? for some information on the asymptotic null distribution.

From the first link, we note that $$ T^{-1}\sum_{t=1}^Tx_{t-1}\epsilon_{t}\Rightarrow\sigma^2/2\{W(1)^2-1\}. $$ In particular, $W(1)^2-1$ is a demeaned $\chi^2_1$ random variable (as the Wiener process has $W(s)\sim N(0,s)$), which has probability 0.682 of being less than zero, leading to the skew in the distribution of the DF statistic.