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I use data (adv. labor and wage) through a nonlinear model to recover the unobserved data (use fsolve in Matlab to solve this system of equations) and run a regression of labor (or wage) on these unobserved data and other observed data. The regression equation that I run is the reduced form of the nonlinear model. Is this a problem?

Two eigenfuctions: $\lambda AX = X$ and $\lambda BY = Y$, A and B are given, L and W are the elements in A and B.

I use matlab to solve it for x and y.

I can get the regression equation $L = ax + by + cz + \epsilon$ from these two eigenfunctions.

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Edit1:

Based on the additional information you provided I still believe that my recommendations are valid and that you should analyze your missing data, and attempt to fill it before using a regression to create imputed values.

If you believe the following assumptions about your imputation regression and data analysis to be true, then you are probably in good shape: 1. The specification is correct and based on economic theory in the literature. 2. You correctly identified your missing data as MCAR or MAR. 3. You correctly ran your regression in matlab. (Note: I use SAS and Stata so I'm not able to provide specific advice on how to do your regression.)

Original Answer:

If you are using a regression to fill data gaps in another regression you may want to take a step back. By using estimated values alongside actual values you may run into problems when you try to interpret results. Imputation via regression does not store any way to analyze the precision or accuracy of your imputed values. Your final model may be over identified in that your final regression will not account for the uncertainty in the imputed values.

Data gaps - which it sounds like you are dealing with - are a thorn in the side of econometric analysis. When dealing with data that has significant gaps in it, it is very important to ask why these gaps exist. If it is because of bad data collection on the part of something or someone you can influence, you should call that person or company to find out why there is a gap in their data. Chances are that they either know and have a solution, or had no idea and need to find a solution. Of course sometimes they may not be supporting the data set anymore, or in a rare subset of cases simply don't care that their data is bad.

For those times when you absolutely cannot fill data gaps you have several options. One, which is sometimes advisable if you have significant resources at your disposal (ex. a Fortune 500 company) you can collect the data yourself and make a dataset that works for your specific needs (or pay someone to do it for you).

If you can't fill the gaps, and you can't collect the data yourself, you are stuck with a set of lesser options. The first step is to indent oft the nature of your missing data.

MCAR - Missing Completely At Random If X is your variable of interest, and contains missing data, and the reason that the data is missing is independent of the non-missing data and independent of other data in the dataset, then your missing data is MCAR. MCAR data is pretty harmless, and is common in big datasets. If it comprises less than 5% of your data, you can usually just delete the observations. If it is greater than 5% you will probably want to use imputation (more on this below) to fill the gaps. So if your data were MCAR, your strategy of using regression imputation is actually applicable. However, you should be aware that regression imputation isn't a sure fire strategy. As I mentioned above it can lead to over identification.

MAR - Missing At Random The difference here is that the reason the data are missing is independent with the other data in the variable of interest but may depend on other data in other variables. For MAR data you should use multiple imputation.

NON-IGNORABLE DATA This type of missing data isn't missing at all. For example, say you want to find out if your son is dating a girl at school. So you give him a questionnaire. It asks: "Are you dating a girl at school?" He leaves the question blank. That would tell me: Yes he may be dating a girl at school.

For non ignorable data, the reason they are missing is dependent on the othe data in the variable. In our scenario, my son's non response was actually a form of implied response.

Non ignorable data should not be imputed nor should they be ignored. Every effort should be made to fill them. If you still cannot fill them you should consider switching data sets or, if that is not feasible, at least make detailed note of these missing data in your research.

IMPUTATION TECHNIQUES For more details on imputation techniques, I recommend consulting your statistical package's documentation.

BASIC IMPUTATION METHODS(for MCAR only): Hot decking - This is essentially just swapping in a randomly selected similar value from the non-missing data. I don't recommend this technique. It can lead to false conclusions and is statistically unsound.

Mean substitution - Replace missing values with the mean of the non missing data. This is better than hot decking but not by all that much. It is however an option if you just need an early idea of what your model results might be (ex. for informal paper proposals to colleagues).

Regression imputation - This is basically what you did. It has issues with overidentification but is not all that bad in the scheme of things assuming you are using a good model for these estimates.

Multiple Imputation (best option for MAR) - The most rigorous form of imputation requires that you draw your imputations from many datasets, using a particular procedure such as the one from Donald Rubin's An Overview of Multiple Imputation.

Many statistics packages have imputation methods programmed in already as they are quite technical to set up.

Overall, you have to balance your amount of missing data, type of missing data, time constraints, and need for highly accurate results, and make a decision on what you want to do. As in the rest of economics, this decision is a trade off.

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