I don't wish to step on anybody's toes here, but this seems to me be a
fascinating discussion. For my research, I was warned off using IVs in my
model specifcations of district-level election outcomes: too messy and too
involved, this professor said. Stick to straight time-series. Needless to
say, I acquiesced.
The trouble is, one of the key variables that has emeraged in my early
model testing is 'campaign intensity', using constituency spending as
surrogates. But since (future) spending can be influenced as much by
(previous) voting outcomes as the other way around, -ivreg- would appear
to be ideal approach of choice to test out the possible endogenity of
spending. Or is it? From what I've seen of it (Wooldridge and elsewhere),
it looks very intimidating!
> The answer to your questions is very nicely and succinctly discussed in
> Wooldridge (2000), Econometric Analysis of Cross Section and Panel Data,
> section 9.5, esp. pp. 236-7.
> The short answer is that you need to go down the route of your Option 1
> and include xsquared as a second endogenous regressor. If you do this,
> you may need additional instruments. One source of additional instruments
> would be squares of some of the other exogenous variables. A quite clever
> idea is suggested by Wooldridge on p. 237. It's similar to your Option 2
> but with an important difference: instead of using xhatsquared as a
> regressor in your second stage equation, use it as an *instrument*, i.e.,
> ivreg2 y q (x xsquared = z xhatsquared)
> In effect this adds a nonlinear function of your exogenous variables to
> your instrument set.
> Your Option 2 is apparently a trap worthy of a special term,
> namely "forbidden regression". In Wooldridge's words, the mistake
> behind "is in thinking that the linear projection of the square is the
> square of the linear projection". See the book for a detailed discussion.
> Quoting "Morris, Stephen" <email@example.com>:
>> Does anyone know of a way to run a 2SLS model in Stata where the
>> endogenous RHS variable would ideally appear in a quadratic form?
>> I am using -ivreg2- to find the effect of an independent variable x
>> on a dependent variable y, where I believe that x and y will be
>> simultaneously determined. I have what I think are a set of
>> non-weak, orthogonal instruments for x, namely z. So, the command I
>> use is:
>> ivreg2 y q (x = z)
>> q is a set of exogenous variables also thought to influence y.
>> I have reason to believe that the true impact of x on y is
>> non-linear, and I would ideally like to estimate a model including x
>> and x squared. Given that x is simultaneously determined with y I am
>> not sure how to proceed.
>> Option 1:
>> One approach would be to run –ivreg2- as normal and instrument
>> both x and x squared. That is, to run:
>> ivreg2 y q (x xsquared = z)
>> This produces a set of results, but the sign and magnitude of the
>> coefficients on x and x squared are counterintuitive. I think this
>> might be because unless my first stage model is able to predict
>> perfectly x and x squared (which it is not) I will not actually be
>> modelling a quadratic form (i.e. the predicted value of x squared
>> from the first stage regressions does not equal the square of the
>> predicted value of x).
>> Option 2:
>> So, the other thing I thought to do was to estimate the first stage
>> equation for x and compute the linear prediction (call this xhat).
>> Then square these predictions (call this xhatsquared) and use these
>> to measure the effects of x squared in my second stage:
>> reg y q xhat xhatsquared
>> The results appear to be more sensible, but I am not sure if the
>> approach is valid.
>> Any thoughts on which option to use, if either, would be greatly
>> appreciated. I am using Stata version 8.2. I have previously
>> searched the FAQ and the Statalist archives, and the question I pose
>> is similar to one posted by Jim Shaw on 18 July, but with respect to
>> non-linear RHS endogenous variables rather than non-linear RHS
>> exogenous variables.
>> Thanks very much.
>> * For searches and help try:
>> * http://www.stata.com/support/faqs/res/findit.html
>> * http://www.stata.com/support/statalist/faq
>> * http://www.ats.ucla.edu/stat/stata/
> Prof. Mark Schaffer
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> School of Management & Languages
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