Using endogenous explanatory variables in correlated random effect

[Jose Ferreras]

Dear community,

I have a balanced panel data with 4 periods of observations in which I include the following variables:

Dependent variable: col

Explanatory variables: apro, apro_sq(squared term of approp)

Control variables: year, Industry, size, rd_inv , market_size

As I want to control for possible endogeneity in my model (previous values of the dependent variable may influence future values of it) I was recommended to use correlated random. Also, CRE allows to include time-invariant terms in the regression.

When the model includes endogenous explanatory variables, (Papke and Woolbridge, 2008) recommend to use a two step approach:

1. Estimate the reduced form for yit2, in case of “apro” to obtain the residual vit
2. Use the pooled “probit” QMLE to obtain the estimates of the model.

For this procedure, Papke and Woolbridge (2008) introduced an instrumental variable in the model. I have some doubt about how to implement this approach in my case:

1. If I want to use the regions of the firm as an instrumental variable (location), how can I test that this variable satisfies the condition of be strictly exogenous variable?
2. Concerning the control variables, can I include control variables that are time variant? Do the control variables included in the model should also satisfy the condition of being strictly exogenous?
3. In my model I want to test for the possible quadratic effect of the explanatory variable. How can I test it using the two step approach suggested by Papke and Woolbridge (2008)?

Thank you very much in advanced for your help.

[Luca Calianno]

Dear Jose,
I am not completely sure of any of my suggestions, therefore apply them with caution. Any oyher advice/correction from other members may be more useful.

1. You cannot test the the exogeneity condition of instrumental variables, especially when you only have one. At most if you have more than one you can use overidentification tests. You should try to explain why you believe your instrument is exogenous using theory.
2. You can include both time variant and time invariant controls since CRE allows for both. Yes, these variables must be endogenous conditional on other covariates and FE. Read carefully the paper.
3. I would say that you have two possibilities (I am not sure of neither of these approaches):

• Either you run the model as in the paper using apro_sq as all other controls. I don't remember if it was explained in the paper, but, estimate the correct covariance matrix and check what is the distribution of the t-test in this case (I recall there was some discussion in the paper).
• You could try a three step procedure. First you estimate the reduced form for apro, then you estimate the reduced form for apro_sq (the two separately). Store the residuals of both steps and add both residuals in the pooled probit.