how to derive critical values for augmented Dickey–Fuller test (ADF) using Monte Carlo method?

Question

Can anybody explain in simple terms how the critical value of the ADF test can be derived using Monte Carlo simulation?

Answer 1

The ADF test assumes the DGP $$ \Delta y_t = \alpha +\beta t +\gamma y_t +\delta_1 \Delta y_{t-1}+\cdots +\delta_k \Delta y_{t-k}+\epsilon_t $$ The parameters are estimated using OLS on a sample of length $T$.

You might impose $\alpha=0$ and/or $\beta=0$, this will give you different null hypotheses to test. But your test is always $\gamma=0$, and the statistic you use to do that is the t-statistic that comes from the regression $t=\hat{\gamma}/\hat{\sigma}_\gamma$.

To perform the test you compare this value to the critical value which depends on the sample size $T$, if the DGP assumes $\alpha$ and/or $\beta$ are zero, and the number or lags $k$. Essentially you want to assess what is the probability that you observed the estimated value $\hat{\gamma}$ due to the randomness of the sample (ie generated by the noise $\epsilon_t$) although the true value that generated the data was $\gamma=0$ (ie the sampling distribution under the null).

In order to produce the sampling distribution using MC you follow the following steps:

1. Estimate all parameters by OLS using the data you have, and compute the t-statistic $t$

2. Fix all estimated parameters except $\gamma$ which you set to zero (ie parameters under the null)

3. Generate Gaussian random numbers $\epsilon_t$, and using the parameters under the null generate random sample paths $y_t$ of the same length as the original data, ie $T$

4. Using this sample re-estimate $\gamma$ and then the t-statustic using OLS, which is a random number drawn from the sampling distribution under the null, say $t_1$

5. Repeat steps (3) and (4) above $M$ times, say 10,000 times, and produce a set $t_1,\cdots,t_M$

6. Percentiles of this distribution give the critical values