Monte Carlo Critical Values for the Augmented Dickey–Fuller Test
Summary
The document outlines how to approximate critical values for an augmented Dickey–Fuller test by simulating the test statistic under its null hypothesis. The test is based on a regression for the change in a series that includes its level, an intercept and trend when specified, lagged changes, and an error term. The null sets the level coefficient to zero, and the resulting coefficient t-statistic is compared with critical values that depend on the sample length, deterministic terms, and lag count.
For the simulation, estimate nuisance parameters from the data, impose the null on the level coefficient, and generate repeated series of the same length using simulated errors. Refit the regression on each series and collect its t-statistic; selected percentiles of this simulated null distribution provide critical values. The example uses Gaussian errors and suggests many repetitions. The description is a simplified procedure, so its results depend on the assumed error distribution, model specification, and parameter estimates.
Key ideas
- The ADF null hypothesis sets the coefficient on the lagged level to zero.
- The test statistic is the estimated level coefficient divided by its estimated standard error.
- Critical values depend on sample length, deterministic terms, and the number of lagged differences.
- Monte Carlo simulation approximates the null distribution by generating and refitting repeated samples.
- Percentiles of the simulated test statistics provide estimated critical values.
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# how to derive critical values for augmented Dickey–Fuller test (ADF) using Monte Carlo method?
# how to derive critical values for augmented Dickey–Fuller test (ADF) using Monte Carlo method?
Can anybody explain in simple terms how the critical value of the ADF test can be derived using Monte Carlo simulation?
## Answer by Kiwiakos (score 4)
https://quant.stackexchange.com/a/26148
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:
- Estimate all parameters by OLS using the data you have, and compute the t-statistic $t$
- Fix all estimated parameters except $\gamma$ which you set to zero (ie parameters under the null)
- 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 $T$
- 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 $t_1$
- Repeat steps (3) and (4) above $M$ times, say 10,000 times, and produce a set $t_1,\cdots,t_M$
- Percentiles of this distribution give the critical valuesShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.