<p>You seem to be confused about the issue.</p>
<p>The standard <span class="math-container">$t$</span>-statistics does work, the test statistics you will get from estimating the AR model is still <span class="math-container">$t$</span>-statistics.</p>
<p>What does no longer work are the critical values that you usually use with regular <span class="math-container">$t$</span>-statistics.</p>
<p>The reason why you can no longer use the standard critical values is that under the null hypothesis <span class="math-container">$\hat{\psi}$</span> is no longer <span class="math-container">$t$</span>-distributed (because of unit root), whereas in stationary AR model this issue does not arise.</p>
<p>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 <span class="math-container">$t$</span>-statistics, and that is done when the expected parameter values follow different distribution than <span class="math-container">$t$</span>-distribution, but that is not really the case for DF test.</p>

CC BY-SA 4.0
2023-12-20T15:47:29.387Z
2023-12-20T16:04:48.637Z
2023-12-20T16:04:48.637Z
1muflon1
<p>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, <span class="math-container">$\hat{\phi}$</span>. In the unit root case, the variance of <span class="math-container">$Y_t$</span> is not defined. However, for any finite sample size, a finite estimate of the variance for <span class="math-container">$Y_t$</span> can be obtained.</p>
<p>To provide some intuition, under(<span class="math-container">$H_0$</span>) 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 <span class="math-container">$Y_t$</span> increases as time progresses, causing the empirical distribution of the t-statistic to become increasingly fatter.
The t-statistic diverges at a rate of <span class="math-container">$T^{1/2}$</span>, thereby exceeding any finite critical values with probability 1.</p>
<p>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</p>

CC BY-SA 4.0
2023-12-20T16:35:19.460Z
2023-12-20T16:35:19.460Z
Tony
<p>Consider the following AR(1) model:</p>
<p><span class="math-container">\begin{equation}
Y_t = \alpha + \phi Y_{t-1} + \varepsilon_t
\end{equation}</span></p>
<p>I want to test the existence of a unit root for <span class="math-container">$Y_t$</span>, and thus, I intend to implement a Dickey-Fuller test. Could you provide insight into why the standard t-statistic, <span class="math-container">$\frac{\hat{\phi}-1}{se(\hat{\phi})}$</span> does not work?</p>

<econometrics><time-series>
Dickey Fuller Test
Dimitru
# Dickey Fuller Test

(Asked by: username_0 on 2023-12-20)

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?

**Tags:** econometrics, time-series

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## 2 answers

### Answer by username_2 on 2023-12-20. Score: 6 | Accepted answer

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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### Answer by username_1 on 2023-12-20. Score: 3

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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