Hi Folks,
Following up on the comments below (my note from a previous discussion on
the class email list), I wanted to let you all know that I have posted
revised notes for section 5 to the course website. The main thing to note is
that they no longer include the example for calculating the likelihood
confidence interval and comparing the likelihood and Wald based intervals.
Given the re-parametrization in the binomial log likelihood, this code is
incorrect. To get the correct intervals, we need to analytically transform
our estimates back to the original scale.
See you soon!
Best,
Miya
---------- Forwarded message ----------
From: Miya Woolfalk <woolfalk at fas.harvard.edu>
Date: Tue, Mar 10, 2009 at 1:06 AM
Subject: Re: [gov2001-l] EXO 1
To: gov2001-l at
lists.fas.harvard.edu
Some clarification:
(1) Yes, MLEs are invariant to re-parameterization (i.e. going from an
bounded to an unbounded and back to an bounded scale when estimating your
parameters). But, measures of uncertainty are not invariant to
re-parameterization. There are two main ways to get your standard errors
(and confidence intervals) back to the original scale after you have
estimated your model by re-parameterizing your parameters. (1) The delta
method -- this is an analytical method to get the variance for our
parameters back to the original scale. Basically, it involves some math to
convert what we know about the distribution of our estimates on the
unbounded scale -- that is that they are distributed N(theta, Var(theta)) --
to get to what we want to know about the distribution of our estimates on
the bounded scale -- that is that they are distributed N(g(theta),
g(var(theta))). (2) The simpler way is to just simulate our parameters. We
will take repeated draws of our parameters from their distribution on the
unbounded scale (i.e. N(theta, sigma.sq) ) and convert each of these draws
back to the bounded scale. We can then simply calculate our standard errors
and confidence intervals by using this simulated sampling distribution.
(2) In PS 5 question 1.2.b, we recommend you use the "L-BFGS-B" method to
get your estimates on the original bounded scale. However, if you are
interested in the delta method, we encourage you to learn it and use it to
answer this question.
Best,
Miya