Mounted results or random results: The Mundlak method

At the moment I’ll focus on Mundlak’s (1978) various to the Hausman check. Not like the latter, the Mundlak method could also be used when the errors are heteroskedastic or have intragroup correlation.

What’s going on?

Say I wish to match a linear panel-data mannequin and have to determine whether or not to make use of a random-effects or fixed-effects estimator. My choice is dependent upon how time-invariant unobservable variables are associated to variables in my mannequin. Listed below are two examples that will yield completely different solutions:

1. A panel dataset of people endowed with innate skill that doesn’t change over time
2. A panel dataset of nations the place the time-invariant unobservables in our mannequin are units of country-specific geographic traits

Within the first case, innate skill can have an effect on observable traits reminiscent of the quantity of education somebody pursues. Within the second case, geographic traits are most likely not correlated with the variables in our mannequin. After all, these are conjectures, and we would like a check to confirm if unobservables are associated to the variables in our mannequin.

First, I’ll let you know compute the check; then, I’ll clarify the speculation and instinct behind it.

What’s going on?

Computing the check

1. Compute the panel-level common of your time-varying covariates.
2. Use a random-effects estimator to regress your covariates and the panel-level means generated in (1) in opposition to your consequence.
3. Take a look at that the panel-level means generated in (1) are collectively zero.

Should you reject that the coefficients are collectively zero, the check suggests that there’s correlation between the time-invariant unobservables and your regressors, specifically, the fixed-effects assumptions are glad. Should you can’t reject the null that the generated regressors are zero, there’s proof of no correlation between the time-invariant unobservable and your regressors; that’s, the random results assumptions are glad.

Under I display the three-step process above utilizing simulated information. The information fulfill the fixed-effects assumptions and have two time-varying covariates and one time-invariant covariate.

STEP 1

. bysort id: egen mean_x2 = imply(x2)

. bysort id: egen mean_x3 = imply(x3)

STEP 2

. quietly xtreg y x1 x2 x3 mean_x2 mean_x3, vce(sturdy) 

. estimates retailer mundlak

STEP 3

. check mean_x2 mean_x3

 ( 1)  mean_x2 = 0
 ( 2)  mean_x3 = 0

           chi2(  2) =    8.94
         Prob > chi2 =    0.0114

We reject the null speculation. This means that time-invariant unobservables are associated to our regressors and that the fixed-effects mannequin is suitable. Be aware that I used a sturdy estimator of the variance-covariance matrix. I couldn’t have achieved this if I had used a Hausman check.

The place all this got here from

A linear panel-data mannequin is given by

[begin{equation*}
y_{it} = x_{it}beta + alpha_i + varepsilon_{it}
end{equation*}]

The index (i) denotes the person and the index (t) time. (y_{it}) is the result of curiosity, (x_{it}) is the set of regressors, (varepsilon_{it}) is the time-varying unobservable, and (alpha_i) is the time-invariant unobservable.

The important thing to the Mundlak method is to find out if (alpha_i) and (x_{it}) are correlated. We all know how to consider this downside from our regression instinct. We are able to consider the imply of (alpha_i) conditional on the time-invariant a part of our regressors in the identical approach that we consider the imply of our consequence conditional on our covariates.

[begin{eqnarray*}
alpha_i &=& bar{x}_itheta + nu_i
Eleft(alpha_i|x_iright) &=& bar{x}_itheta
end{eqnarray*}]

Within the expression above, (bar{x}_i) is the panel-level imply of (x_{it}), and (nu_i) is a time-invariant unobservable that’s uncorrelated to the regressors.

As in regression, if (theta = 0), we all know (alpha_i) and the covariates are uncorrelated. That is what we check. The implied mannequin is given by

[begin{eqnarray*}
y_{it} &=& x_{it}beta + alpha_i + varepsilon_{it}
y_{it} &=& x_{it}beta + bar{x}_itheta + nu_i + varepsilon_{it}
Eleft(y_{it}|x_{it}right) &=& x_{it}beta + bar{x}_itheta
end{eqnarray*}]

The second equality replaces (alpha_i) by (bar{x}_itheta + nu_i). The third equality depends on the truth that the regressors and unobservables are imply impartial. The check is given by

[begin{equation*}
H_{text{o}}: theta = 0
end{equation*}]

Reference

Mundlak, Y. 1978: On the pooling of time sequence and cross part information. Econometrica 46:69-85.