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Consider the following AR(1) model:

\begin{equation} Y_t = \alpha + \phi Y_{t-1} + \varepsilon_t \end{equation}

I want to test the existence of a unit root for $Y_t$, and thus, I intend to implement a Dickey-Fuller test. Could you provide insight into why the standard t-statistic, $\frac{\hat{\phi}-1}{se(\hat{\phi})}$ does not work?

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The problem in the unit root case is that the t-statistic does not follow a t-distribution, not even asymptotically. The issue lies in the distribution of the OLS estimator, $\hat{\phi}$. In the unit root case, the variance of $Y_t$ is not defined. However, for any finite sample size, a finite estimate of the variance for $Y_t$ can be obtained.

To provide some intuition, under($H_0$) true, i.e., in the unit root case, the empirical distribution of the t-statistic differs from the theoretical t-distribution. This discrepancy arises because the variance of $Y_t$ increases as time progresses, causing the empirical distribution of the t-statistic to become increasingly fatter. The t-statistic diverges at a rate of $T^{1/2}$, thereby exceeding any finite critical values with probability 1.

In reality, the distribution is skewed to the left, resulting in critical values smaller than those for the t-distribution. In other words, one requires a table of Dickey-Fuller critical values; otherwise, there is a tendency to reject the null hypothesis too frequently

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You seem to be confused about the issue.

The standard $t$-statistics does work, the test statistics you will get from estimating the AR model is still $t$-statistics.

What does no longer work are the critical values that you usually use with regular $t$-statistics.

The reason why you can no longer use the standard critical values is that under the null hypothesis $\hat{\psi}$ is no longer $t$-distributed (because of unit root), whereas in stationary AR model this issue does not arise.

Hence the critical values for the test have to be drawn from Dickey–Fuller distribution. However, that is not the same as the test statistics (i.e. the number you get) not being t-statics. There are actually tests that do not use $t$-statistics, and that is done when the expected parameter values follow different distribution than $t$-distribution, but that is not really the case for DF test.

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