Dear Statalisters
I have the following problem
I'm estimating some parameters a,b,c using the "gmm" command in Stata. I would like to
1) Constrain a,b,c to be non-negative
2) Be able to test a>0 v/s a=0, b>0 v/s b=0 and c>0 v/s c=0
One solution I have observed other researchers in my field (financial economics) use is to re-parametrize the model in order to estimate sqrt(a), sqrt(b), sqrt(c)
This is also the solution I elected to adopt in the beginning.
I would like to get your opinion on the following
1) Does GMM maintain its nice properties (Asymptotic Normality, Consistency, Efficiency) after re-parametrization? My initial instinctive answer was no, since one of the GMM assumptions is that the population moment equation system has a unique solution. Here, if ( sqrt(a),sqrt(b),sqrt(c) ) is a solution, then ( -sqrt(a*),sqrt(b*),sqrt(c*) ) , ( sqrt(a*),-sqrt(b*),sqrt(c*) ) and so on are also solutions. My second thought answer was instead yes: GMM preserves its nice properties. The reason is that, while it is true that from a numerical standpoint we are letting Stata choose ( sqrt(a) sqrt(b)sqrt(c) ), from a theoretical standpoint we are still solving for (a,b,c) because the moment conditions depend on (a,b,c) - not ( sqrt(a) sqrt(b)sqrt(c) ). What is your take on this? Is the re-parametrization described above methodologically sound?
2) Assuming it is, what is the best way to perform the tests described above? My solution would be the following. Because GMM estimates for ( sqrt(a),sqrt(b),sqrt(c) ) are asymptotically normal, by squaring the respective z-stats produced by Stata you should obtain statistics that are asymptotically chi-square(1) distributed. If the chi-square stat for a is above the 1df critical value, you reject a=0. Does this make sense to you?
Thank you very much in advance!
Bruno
I have the following problem
I'm estimating some parameters a,b,c using the "gmm" command in Stata. I would like to
1) Constrain a,b,c to be non-negative
2) Be able to test a>0 v/s a=0, b>0 v/s b=0 and c>0 v/s c=0
One solution I have observed other researchers in my field (financial economics) use is to re-parametrize the model in order to estimate sqrt(a), sqrt(b), sqrt(c)
This is also the solution I elected to adopt in the beginning.
I would like to get your opinion on the following
1) Does GMM maintain its nice properties (Asymptotic Normality, Consistency, Efficiency) after re-parametrization? My initial instinctive answer was no, since one of the GMM assumptions is that the population moment equation system has a unique solution. Here, if ( sqrt(a),sqrt(b),sqrt(c) ) is a solution, then ( -sqrt(a*),sqrt(b*),sqrt(c*) ) , ( sqrt(a*),-sqrt(b*),sqrt(c*) ) and so on are also solutions. My second thought answer was instead yes: GMM preserves its nice properties. The reason is that, while it is true that from a numerical standpoint we are letting Stata choose ( sqrt(a) sqrt(b)sqrt(c) ), from a theoretical standpoint we are still solving for (a,b,c) because the moment conditions depend on (a,b,c) - not ( sqrt(a) sqrt(b)sqrt(c) ). What is your take on this? Is the re-parametrization described above methodologically sound?
2) Assuming it is, what is the best way to perform the tests described above? My solution would be the following. Because GMM estimates for ( sqrt(a),sqrt(b),sqrt(c) ) are asymptotically normal, by squaring the respective z-stats produced by Stata you should obtain statistics that are asymptotically chi-square(1) distributed. If the chi-square stat for a is above the 1df critical value, you reject a=0. Does this make sense to you?
Thank you very much in advance!
Bruno
