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

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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

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