Thursday, January 15, 2015

Bootstrapping Standard Errors: Methods and AR(1) processes

The Famous Cameroon/Gelbach paper goes through:

Non parametric bootstraps
1. Pairs Cluster Bootstrap-se and Bootstrap-t
2. Residual and wild bootstrap 
Parametric bootstraps
3. Redidual bootstrap

The first method is just to repeat your regression/correlation X number of times, and plot the distribution of all X estimates to get a nonparametric estimate of the variance. However, what's key here, is that the sampling method mimic whatever issue you're trying to correct for. If it's clustering of standard errors, then you'll want to have a clustered sampler. 

The second and third method (in some sense) simulates "new data" on each run, and then proceeds as in (1). Method 2 reassigns the initial estimated residuals (nonparametric), while method 3 randomly samples new residuals (parametric, because it assumes a distribution on the residuals) from a standard normal. Nonparametric is preferred to parametric, but isn't always possible (like in your case). 

Method (2) is popular for bootstrapping when the # of clusters is small (so you're not just resampling the same data repeatedly, which would give you a crude estimate of the true distribution). 

***Bootstrapping for an AR(1) process*******

Now, in the case of getting se's with an AR(1) process there are two options: non-parametric and parametric. With non-parametric you'll want to mimic the AR(1) process via the sampling method. With parametric you'll mimic the AR(1) process by literally generating data from an AR(1) process (see slide 23/28:http://ocw.mit.edu/courses/sloan-school-of-management/15-450-analytics-of-finance-fall-2010/lecture-notes/MIT15_450F10_lec09.pdf)

I haven't come across time series sampling akin to clustered sampling, so to bootstrap the se's on AR(1) process I don't see any other possibly other than a parametric bootstrap. 

Attached is Stata code for a basic nonparametric X,Y pairwise se-bootstrap. You can find R code here: 
http://www.r-bloggers.com/the-cluster-bootstrap/. My particular code doesn't really help your specific case, but gives an example. 

Some other links I checked out to understand this better:
http://www.stata.com/support/faqs/statistics/bootstrap-with-panel-data/
http://stat.duke.edu/~ab216/report.pdf
http://www.jstor.org/discover/10.2307/1392427?uid=3738224&uid=2&uid=4&sid=21104335031653
http://www.stat.umn.edu/geyer/5102/slides/s8.pdf
http://www.stat.umn.edu/geyer/5601/examp/parm.html
http://magician.ucsd.edu/essentials/WebBookse76.html
http://www.epixanalytics.com/modelassist/CrystalBall/Model_Assist.htm#Analysing_and_using_data/Bootstrap/Parametric_Bootstrap_example.htm
http://www.stat.cmu.edu/~cshalizi/402/lectures/08-bootstrap/lecture-08.pdf
http://www.economics.uci.edu/~dbrownst/bootmi.pdf

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