Partially Linear Model and Consistency?

[Myungun Kim]

Hello.

I am trying to estimate a production function using the semiparametric approach (Olley-Pakes or Levisohn-Petrin).
In this, I control for the error term (observable only to the firm) by using the function of productivity (X) as a proxy for it.

As a result, I end up having a partially linear regression of production function as below


Y = {a0} + {a1}*l + X(m , k) + e

where {a1} is the coefficient for the labour and X(.) is a non-parametric function of m and k.
e is an i.i.d error term not correlated with regressors.

To estimate this, I approximate ‘X(.)’ with a second-order polynomials.

According to Andrews (1991), the estimator on the coefficient on the linear part ({a1} in this case) is consistent.

However, I was wondering whether it is also possible to argue that we can obtain consistent estimators for the coefficients on the linearly approximated lambda function.
Personally, I think there is no problem in arguing so, but my hunch says I’m missing something.

I would be grateful if you could give me any advice in this!

Thank you in advance!

[Phil Bromiley]

You'll increase your chances of a useful answer by following the FAQ on asking questions - provide Stata code in code delimiters, readable Stata output, and sample data using dataex. Also, try to keep your question as short as possible - just what is needed to demonstrate the problem. Also, do not assume we're from your area - if you can explain the issues and not require us to read a paper, that is best. If you must reference a paper, give us the full citation.

If you really want to use a polynomial in the x's, then it is quite easy to specify using factor variable notation. You can certainly get consistent estimates on the parameters on the polynomial, the issue is what those parameters mean in terms of your original model. We could easily have a X function that could not be well approximated by a second-order polynomial - in this case I don't see why any of your parameters would be consistent,

This is not my area, but if you have a mis-specified functional form for any part of the equation, then I don't see why you'd get consistent estimators on any of the parameters (assuming the iv's are not orthogonal).

[Myungun Kim]

Thank you very much for your advice! I'll repost following your advice :D