Dickey-Fuller test and Monte Carlo method

Asked Jun 12 '17 at 13:31

Active Jun 13 '17 at 08:07

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The Dickey-Fuller test aims to test the presence of unit roots in a process. If I've the following process:

$$y_t = a_1y_{t-1} +\epsilon_t$$

it tests $H_0\colon \ a_1=1$.

If I specify $\theta =a_1-1$ I can rewrite the process as:

$$(a_1-1)y_{t-1} +\epsilon_t = \theta y_{t-1} + \epsilon_t$$

and test $H_0\colon \ \theta=0$.

My question is: why does the distribution of the $t$-statistic have to be computed using Monte Carlo method?

edited Jun 13 '17 at 08:07

Tim

asked Jun 12 '17 at 13:31

zar

Because under the null you would have a stationary process $y_t-y_{t-1}$ regressed on an integrated process $y_{t-1}$. If you look at the $X'X$ matrix and the $X'\epsilon$ matrix of the OLS estimator, you would see funny things happening there due to integratedness of $y_{t-1}$. (Here $X$ is $y_{t-1}$.) – Richard Hardy Jun 12 '17 at 19:41
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Next to @RichardHardy's intuition, I provide some more detail in this answer: https://stats.stackexchange.com/questions/213551/how-is-the-augmented-dickey-fuller-test-adf-table-of-critical-values-calculate/213589#213589 – Christoph Hanck Jun 13 '17 at 07:40
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zar, consider marking your question as duplicate if the linked answer by Christoph Hanck answers your question. – Richard Hardy Jun 13 '17 at 08:03
Possible duplicate of How is the augmented Dickey–Fuller test (ADF) table of critical values calculated? – zar Jun 16 '17 at 08:07
A similar question is also asked here: https://quant.stackexchange.com/questions/26137/how-to-derive-critical-values-for-augmented-dickey-fuller-test-adf-using-monte – zar Jun 16 '17 at 08:07

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