Alan,
oh, screw the causality -- it is up to you as the researcher to think
as to how your X and Z variables are related, and interpret the
findings. If your question was about how to call those variables so
that economists will understand you -- then I think both Mark and I
gave pretty similar answers: you have a regression with a variable
measured with error (Z is the observed measure of X), and you must
come up with something else that's correlated with X, but not with the
measurement error d.
Note that econometric models usually assume linear relations. The
nonlinearities are far more difficult to deal with. Your main
reference will probably be Carroll and folks book
(http://www.citeulike.org/user/ctacmo/article/1761085). I think a lot
of their motivation comes from similar dose-response kind of stuff.
On Mon, Aug 25, 2008 at 2:22 PM, Feiveson, Alan H. (JSC-SK311)
<[email protected]> wrote:
> Hi Mark, Stas -
>
> Sorry, I think I didn't explain the causative sequence properly. What I
> should have said was that Z affects X through X = h(Z) + d, where d is
> an error term independent of e. For example, Z is a dose and X is a
> (non-observable) effect. That's why I thought that Z would be an
> instrument for X rather than the other way around. Does your
> interpretation still with the above causative sequence?
>
> Thanks
>
> Al
>
>
> -----Original Message-----
> From: [email protected]
> [mailto:[email protected]] On Behalf Of Schaffer,
> Mark E
> Sent: Monday, August 25, 2008 2:07 PM
> To: [email protected]
> Subject: st: RE: instrumental variable nomenclature
>
> Al,
>
>> -----Original Message-----
>> From: [email protected]
>> [mailto:[email protected]] On Behalf Of Feiveson,
>> Alan H. (JSC-SK311)
>> Sent: 25 August 2008 19:26
>> To: [email protected]
>> Subject: st: instrumental variable nomenclature
>>
>> Hi - I am looking for a name/title to describe the following
>> simulatenous-equation model:
>>
>> This starts with a linear regression model Y = X*b + e, but X is not
>> observed. However we know X is correlated with an observable variable
>> Z, with error term independent of e. So at this point, would it be
>> correct to say this is an instrumental variable model with Z as an
>> instrument for X?
>
> Not quite. Say the "true model" is
>
> Y = X*b + e
>
> but you estimate
>
> Y = Z*b + u
>
> Z is an imperfect measure of X. Say that
>
> Z = X*a + v
>
> This is the classic measurement error problem. If you had an instrument
> for X, you could get a consistent estimate of b using linear IV.
>
> HTH.
>
> Cheers,
> Mark
>
>> Furthermore we also observe K = g(X) where g is a step function (for
>> example, K follows an ordered probit model with X as the latent
>> variable). So to get the nomenclature straight, can I say that this is
>
>> a nonlinear simultaneous equation model (one equation for Y given X,
>> and one for K given X), with Z as an "instrumental variable for X"?
>>
>> Of course, how to estimate such a model is another story!
>>
>> Thanks for whatever suggestions (names or estimation
>> approach) you can provide.
>>
>> Al Feiveson
>>
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>
>
> --
> Heriot-Watt University is a Scottish charity registered under charity
> number SC000278.
>
>
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--
Stas Kolenikov, also found at http://stas.kolenikov.name
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