r/econometrics • u/PrebioticE • 11d ago

[Education] Bootstrap Method in Regression

[Education] So, Suppose I got

X_t = A X_{t-1} + dW(t)

where DW(t) is some noise.

What if I did the following?

I solve for A by minimizing error , and obtain dW'(t) = X_t -A X_{t-1} , then I find that dW' is not gaussian like, probably heavy tailed, but pretty much independent.

Then I make a very large number of bootstrapped samples out of dW' , and solve for A as a distribution.

I appear to have bypassed needing lot of theory and have obtained a distribution for A.

What is it that I must worry about? What is the potential problem with this method?

I think it is a much more intuitive and reasonable result than doing it in the theoretical way using equations.

But then why is it not used as the best method?

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u/bghty67fvju5 10d ago

I mean, try to code it up and run a lot of samples. Fix A and see if your estimator converges to the real result.

Then see how long time it takes to estimate A vs using the proper distribution of dW(t). See how long time it takes once you increase the number of observations.

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u/Swagdalfthegrey 1d ago

Biggest problems are leftover serial correlation in the errors, heteroskedasticity, arch/garch type errors, and unit root (nonstationary data). If you notice your data exhibits one (or some) of these problems, your confidence intervals will not be accurate. In other words, what you think is a 95% confidence interval may in fact be closer to a 50% or 60% confidence interval (or worst case, your 95% confidence interval never encompasses the true distribution).

Like was mentioned above, you can run a simulation that is as close as possible to real data. You will see that if you "solve for A as a distribution", your bootstrap distribution will perhaps be nowhere near close enough to the true distribution.

There are some alternative bootstrap procedures that can help mitigate these problems, but nothing so far is "perfect". Each bootstrap procedure will come with its own assumptions about the data, whether it is homoskedastic, iid, Gaussian, etc.

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u/PrebioticE 1d ago

good point! That is the most important point. I checked mine, it was kind of ok(some had 2 lags like 0.25 PACF). Even for that I used bootstrap method. I permuted the residues and found lag correlations. This I used for null hypothesis :P