Logit Misspecification Puzzle
Posted: Fri Jan 13, 2012 10:59 am
Hi All,
I have ran the following model
Y = Ind [ Y* > 0 ],
Y* = a + b X_1 + c X_2 + d X_1 * X_2 +Eps,
where Eps~Logit
i.e. a logit model with the nonlinear index function.
Then by use of the robust standard errors I have tested the hypothesis:
H_0: d = 0
and I got that indeed d is significantly different from 0, which made me very happy!
But then I started to question myself what if my model is misspecified?
I generated new Y** = a + b X_1 + c X_2 (no cross effect and no error term) and a,b,c fixed at their estimates from above
I used the decision rule Y = Ind [ Y** > 0 ], but now allow for 10% of the sample to make a mistake i.e.
for 10% of the observations I changed Y from 0 to 1 and from 1 to 0 (sth. like the Eps above but not exactly).
Then I run the same logit as above on this new data (so it is misspecified because includes the cross effect d X_1 * X_2)
(Y = Ind [ Y** > 0 ], Y** = a + b X_1 + c X_2 + d X_1 * X_2 +Eps, where Eps~Logit)
and surprise, surprise the cross-effect coefficient, d, is significant!
MY QUESTION:
HOW CAN I CHECK THE MODEL FOR SUCH A MISSPECIFICATION? and why at all d is significant in the second test?
Based on this counter example I can only make a conditional statement i.e.
Given that the logit model is correctly specified, then the cross effect X_1 * X_2 is present in the decision problem Y.
OR is something wrong with the counter example?
OR is there any Quasi Maximum Likelihood argument? I've read in Green's book that QML applies only to the logit models with the linear index function i.e. Y* = alpha' * X, so I do not know how to proceed... a way to make an unconditional statement "the cross effect X_1 * X_2 is present in the decision problem Y" would make me very happy again!
Thanks for the hints.
Best,
Black Swan
I have ran the following model
Y = Ind [ Y* > 0 ],
Y* = a + b X_1 + c X_2 + d X_1 * X_2 +Eps,
where Eps~Logit
i.e. a logit model with the nonlinear index function.
Then by use of the robust standard errors I have tested the hypothesis:
H_0: d = 0
and I got that indeed d is significantly different from 0, which made me very happy!
But then I started to question myself what if my model is misspecified?
I generated new Y** = a + b X_1 + c X_2 (no cross effect and no error term) and a,b,c fixed at their estimates from above
I used the decision rule Y = Ind [ Y** > 0 ], but now allow for 10% of the sample to make a mistake i.e.
for 10% of the observations I changed Y from 0 to 1 and from 1 to 0 (sth. like the Eps above but not exactly).
Then I run the same logit as above on this new data (so it is misspecified because includes the cross effect d X_1 * X_2)
(Y = Ind [ Y** > 0 ], Y** = a + b X_1 + c X_2 + d X_1 * X_2 +Eps, where Eps~Logit)
and surprise, surprise the cross-effect coefficient, d, is significant!
MY QUESTION:
HOW CAN I CHECK THE MODEL FOR SUCH A MISSPECIFICATION? and why at all d is significant in the second test?
Based on this counter example I can only make a conditional statement i.e.
Given that the logit model is correctly specified, then the cross effect X_1 * X_2 is present in the decision problem Y.
OR is something wrong with the counter example?
OR is there any Quasi Maximum Likelihood argument? I've read in Green's book that QML applies only to the logit models with the linear index function i.e. Y* = alpha' * X, so I do not know how to proceed... a way to make an unconditional statement "the cross effect X_1 * X_2 is present in the decision problem Y" would make me very happy again!
Thanks for the hints.
Best,
Black Swan