Two-step estimation using a probit and GMM
The initial model:
Yit  Yit1  X it   Z it     i   it (1)
P(Wit  1)  Z it   (2)
The problem:
- You need to estimate the probability of a liquidity shock before being able to
regress (1).
- Hence, you build a latent variable to measure the liquidity shock; run a probit
with model (2) to generate an estimated probability of liquidity shocks,
subsequently introduced in (1).
- The main issue when you include the estimated probability of a liquidity shock in
(1), you have a generated regressors problem. Basically, you’re using a proxy to
measure the actual variable (which you don’t observe) and in doing so you’re
measuring it with an error. The implication of generated regressors is biased
standard errors, which impede proper inference making.
The solution:
Step 1: Regress the Probit to get a liquidity shock variable
- Build a latent variable for the probability of liquidity shock and regress it on your
exogenous variable using the probit model (2).
- Collect the fitted values to use them in equation (1).
Step 2: Regress equation (1) using all variables available as instruments together
- In order to regress equation (1), you must solve two problems:
A. A generated regressor problem mentioned above, which requires the use of
instrumental variables.
B. Equation (1) is a dynamic panel, which also requires the use of instrumental
variables.
- Hence, to solve (A) and (B), one can use GMM (Arellano-Bond).
- The crux of using GMM in this case is to use all moment conditions together
from the equations (1) and (2). That means that one should not only use all
the X it and lags as instruments but also all Z it and lags. The exogenous
-
variables from equation (2), Z it , are providing the appropriate instruments for
the estimated liquidity shock variable, whereas the X it are instrumenting for
the dynamic bit equation (1).
So, if the moment conditions for equation (2)
  X it 1 




  

 X 

i1



Y  Yit 1   (Yit 1  Yit 2 )  ( X it  X it 1 )  Z it    Z it 1   0
E
  Z it 1  it




  

 Z 

  i1 

if equation is estimated using first differences to eliminate fixed effects (  i ) for
example.