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# Re: st: Using ivregress when the endogenous variable is used in an interaction term in the main regression

 From Austin Nichols To statalist@hsphsun2.harvard.edu Subject Re: st: Using ivregress when the endogenous variable is used in an interaction term in the main regression Date Wed, 21 Dec 2011 11:57:19 -0500

```Tirthankar Chakravarty <tirthankar.chakravarty@gmail.com>:
No conditional independence assumed, though of course an independence
assumption lets you form all kinds of transformations of Z to use as
excluded instruments.

We need Z, Z*X1, and X1 uncorrelated with e, but Z and e were already
assumed uncorrelated and X1 is exogenous by assumption as well, in the
original post.  It does not seem too much of a stretch to assume Z*X1
uncorrelated with e as well (which implies X2hat*X1 uncorrelated with
e), but if we use all 3 as instruments we will see evidence of any
violations of assumptions in the overid test (assuming no weak
instruments problem).

On Wed, Dec 21, 2011 at 11:44 AM, Tirthankar Chakravarty
<tirthankar.chakravarty@gmail.com> wrote:
> Austin,
>
> I agree re: well-cited papers.
>
> Note that the efficiency you mention comes at a cost. As I pointed out
> in my previous Statalist reply:
> http://www.stata.com/statalist/archive/2011-08/msg01496.html
> the instrumenting strategy you suggest requires the instruments to be
> conditionally independent rather than just uncorrelated with the
> structural errors.
>
> T
>
> On Wed, Dec 21, 2011 at 7:57 AM, Austin Nichols <austinnichols@gmail.com> wrote:
>> Nick Kohn <coffeemug.nick@gmail.com>:
>> Or better, instrument for X1*X2 using Z, Z*X1, and X1.
>> For maximal efficiency given your assumptions you may prefer
>> to instrument for X1*X2 using Z*X1, or even
>> to instrument for X1*X2 using X2hat*X1,
>> but you should build in an overid test whenever feasible.
>>
>> Just because a well-cited paper does something wrong does not mean you
>> have to, though.
>>
>> Including the main effects of X1 and X2 makes for harder interpretation, but
>> will make you a lot more confident of your answers once you have worked out the
>> interpretation.
>>
>> On Wed, Dec 21, 2011 at 9:20 AM, Tirthankar Chakravarty
>> <tirthankar.chakravarty@gmail.com> wrote:
>>> In that case, none of this is necessary. Just instrument for X1*X2
>>> using Z. All standard results apply.
>>>
>>> T
>>>
>>> On Wed, Dec 21, 2011 at 6:03 AM, Nick Kohn <coffeemug.nick@gmail.com> wrote:
>>>> Hmmm I see what you mean, but I'm following the methodology of a well
>>>> cited paper that does the same thing.
>>>>
>>>> I'll be sure to discuss this limitation, but in terms of using this
>>>> model, would the 3 steps in my last message be correct?
>>>>
>>>> On Wed, Dec 21, 2011 at 2:56 PM, Tirthankar Chakravarty
>>>> <tirthankar.chakravarty@gmail.com> wrote:
>>>>> I wanted to indirectly confirm that you did have the main effect in
>>>>> the regression because even though I don't know the nature of your
>>>>> study, a hard-to-defend methodological position arises when you
>>>>> include interaction terms without including the main effect. You might
>>>>> want to take that on the authority of someone who (literally) wrote
>>>>> the book on the subject:
>>>>>
>>>>> http://www.stata.com/statalist/archive/2011-03/msg00188.html
>>>>>
>>>>> and reconsider your decision to not include the main effect.
>>>>>
>>>>> T
>>>>>
>>>>> On Wed, Dec 21, 2011 at 5:46 AM, Nick Kohn <coffeemug.nick@gmail.com> wrote:
>>>>>> My model doesn't have X2 as a separate term, so in terms of the model
>>>>>> you had it looks like:
>>>>>>  Y = b*X1*X2 + controls
>>>>>> So the only place the endogenous variable comes up is the interaction term
>>>>>>
>>>>>> At the risk of being repetitive, would these be the correct steps (so
>>>>>> essentially only step 3 changes from what you said):
>>>>>> 1) regress X2 on all instruments, exogenous variables and controls
>>>>>> 2) Form interactions of X2hat with the exogenous variable X1, that is, X2hat*X1
>>>>>> 3) ivregress instrumenting for X2*X1 using X2hat*X1.
>>>>>>
>>>>>> On Wed, Dec 21, 2011 at 1:44 PM, Tirthankar Chakravarty
>>>>>> <tirthankar.chakravarty@gmail.com> wrote:
>>>>>>> Not quite; here is the recommended procedure (I am assuming that you
>>>>>>> have the main effect of the endogenous variable in there as in Y =
>>>>>>> a*X2 + b*X1*X2 + controls):
>>>>>>>
>>>>>>> 1) -regress- X2 on _all_ instruments (included exogenous controls and
>>>>>>> excluded instruments) and get predictions X2hat.
>>>>>>>
>>>>>>> 2) Form interactions of X2hat with the exogenous variable X1, that is, X2hat*X1.
>>>>>>>
>>>>>>> 3) -ivregress- instrumenting for X2 and X2*X1 using X2hat and X2hat*X1.
>>>>>>>
>>>>>>> Note that there is distinction between two calls to -regress- and
>>>>>>> using -ivregress- for 3).
>>>>>>>
>>>>>>> T
>>>>>>>
>>>>>>> On Wed, Dec 21, 2011 at 3:43 AM, Nick Kohn <coffeemug.nick@gmail.com> wrote:
>>>>>>>>
>>>>>>>> My simplified model is (X2 is endogenous):
>>>>>>>> Y = b*X1*X2 + controls
>>>>>>>>
>>>>>>>> In regards to the third option you suggest, would I do the following?
>>>>>>>>
>>>>>>>>  1) First stage regression to get X2hat using the instrument Z
>>>>>>>>  2) Run the first stage again but use X1*X2hat as the instrument for
>>>>>>>> X1*X2 (so Z is no longer used)
>>>>>>>>  3) Run the second stage using (X1*X2)hat (so the whole product is
>>>>>>>> fitted from step 2))
>>>>>>>>
>>>>>>>> On Wed, Dec 21, 2011 at 12:24 PM, Tirthankar Chakravarty
>>>>>>>> <tirthankar.chakravarty@gmail.com> wrote:
>>>>>>>>> You can see my previous reply to a similar question here:
>>>>>>>>> http://www.stata.com/statalist/archive/2011-08/msg01496.html
>>>>>>>>>
>>>>>>>>> T
>>>>>>>>>
>>>>>>>>> On Wed, Dec 21, 2011 at 2:24 AM, Nick Kohn <coffeemug.nick@gmail.com> wrote:
>>>>>>>>>> Hi,
>>>>>>>>>>
>>>>>>>>>> I have a specification in which the endogenous variable is interacted
>>>>>>>>>> with an exogenous variable. Since I cannot multiply the variables
>>>>>>>>>> directly in the regression, I create a new variable. In ivregress it
>>>>>>>>>> makes no sense to use the entire interaction term as the endogenous
>>>>>>>>>> variable.
>>>>>>>>>>
>>>>>>>>>> I can do the first stage manually (and then use the fitted value in
>>>>>>>>>> the main regression), however, from what I remember the standard
>>>>>>>>>> errors will be wrong when doing it manually.
>>>>>>>>>>
>>>>>>>>>> Is there a way to overcome this?
>>>>>>>>>>
>>>>>>>>>> Thanks

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