GMM with constraints-- any clues

[KDBRG]

I am having some difficulty with the following issue:

We estimated an AIDS model in STATA, and imposed cross-equation constraints-- i.e., homogeneity and symmetry. We estimated the constrained model using REG3. However, with that and other system estimation options in STATA, it's hard to produce corrected standard errors.

So we turned to E-Views and estimated the system using GMM, and specifying all exogenous variables as instruments. And correcting for serial correlation using Newey-West standard errors. When we estimated the unconstrained model, GMM in E-Views produced precisely the same parameter estimates as we obtained from running the unconstrained model in STATA.

When we imposed constraints, however, we get different numbers. Usually not radically different-- most of our elasticity estimates are only slightly different. But in one or two cases, we got larger differences: e.g., -1.44 in STATA, -1.29 in E-Views (the elasticity is a linear combination of various parameter estimates).

I'm puzzled by this. And I'm puzzled also by the fact that the J-statistic is not zero when we estimate the system imposing constraints.

I have two thoughts: (a) the GMM iteration process is thrown off by the constraints, i.e., it searches in the "wrong" region, or (b) and more seriously, the imposition of the constraints means that we need to modify our instrument list to replicate precisely the result we get with STATA.

With respect to (b), imposing the homogeneity constraint in equation 1 that C1+C2+C3+C4=0, where C1, C2, C3 and C4 are coefficients on X1, X2, X3, X4, means that C4=-C2-C3-C1, and that the model is effectively estimating y= C0+C1.(X1-X4)+C2.(X2-X4)+C3.(X3-X4), which might suggest that the instrument list should be X1-X4, X2-X4, X3-X4 instead of X1, X2, X3 and X4.

Seems a bit weird to me-- seems obvious to use all exogenous variables as instruments, but to replicate what I had in STATA with REG3/OLS option, maybe that's what I need to do?

Any thoughts? I figured I'd ask here before embarking on something that could lead to hours or days of frustration.

Kalyan

[startz]

If you haven't already, set the starting values in EViews to the final Stata values and see what happens.

[KDBRG]

Did that, no luck.

[startz]

Hmmm, one possibility is that EViews is finding a better estimate given the constraints than Stata did.

[EViews Glenn]

The fact that the J-stat is a clue that you've got more instruments than is needed to identify the model.

I have no idea what Stata is doing, but my prediction is that your guess about the instruments is correct. Perhaps the best way to think of this is that you could have pre-generated variables for X1-X4, X2-X4, and X3-X4, and run a model with those variables. Because of the coefficient restriction, you really only have the three variables and the natural instruments would have been those three variables. Just as remove the restriction on the coefficients gives you one extra degree dimension for finding a solution, as it were, so too entering the four instruments separately allows you to project into a more general space.

[KDBRG]

Glenn:

Thanks. I believe that's right. I have both symmetry and homogeneity constraints, but I think only the homogeneity constraint is causing the issue.

In other words:

y1=c0+c1.x1+c2.x2+c3.x3+ u1
y2=c4+c5.x1+c6.x2+c7.x3+u2
y3=c8+c9.x1+c10.x2+c11.x3 +u3

The y's are all market shares, so we can avoid estimating the third equation.

Homogeneity in equation 2, for example, means: c5+c6+c7=0, so c7=-(c6+c5). Now, c5=c2, by symmetry, so c7=-(c6+c2). Equation simplifies to:

y2=c4+c2.(x1-x3)+c6.(x2-x3)+u2. And the instruments I should be declaring for this equation are (x1-x3), (x2-x3). The fact that c5=c2 and c2 is also in equation 1 would not seem relevant for the instrument list for equation 2, I think?