### Archive

Posts Tagged ‘endogeneity’

## Testing model specification and using the program version of gmm

This post was written jointly with Joerg Luedicke, Senior Social Scientist and Statistician, StataCorp.

The command gmm is used to estimate the parameters of a model using the generalized method of moments (GMM). GMM can be used to estimate the parameters of models that have more identification conditions than parameters, overidentified models. The specification of these models can be evaluated using Hansen’s J statistic (Hansen, 1982).

We use gmm to estimate the parameters of a Poisson model with an endogenous regressor. More instruments than regressors are available, so the model is overidentified. We then use estat overid to calculate Hansen’s J statistic and test the validity of the overidentification restrictions.

## Fitting ordered probit models with endogenous covariates with Stata’s gsem command

The new command gsem allows us to fit a wide variety of models; among the many possibilities, we can account for endogeneity on different models. As an example, I will fit an ordinal model with endogenous covariates.

### Parameterizations for an ordinal probit model

The ordinal probit model is used to model ordinal dependent variables. In the usual parameterization, we assume that there is an underlying linear regression, which relates an unobserved continuous variable $$y^*$$ to the covariates $$x$$.

$y^*_{i} = x_{i}\gamma + u_i$

The observed dependent variable $$y$$ relates to $$y^*$$ through a series of cut-points $$-\infty =\kappa_0<\kappa_1<\dots< \kappa_m=+\infty$$ , as follows:

$y_{i} = j {\mbox{ if }} \kappa_{j-1} < y^*_{i} \leq \kappa_j$

Provided that the variance of $$u_i$$ can’t be identified from the observed data, it is assumed to be equal to one. However, we can consider a re-scaled parameterization for the same model; a straightforward way of seeing this, is by noting that, for any positive number $$M$$:

$\kappa_{j-1} < y^*_{i} \leq \kappa_j \iff M\kappa_{j-1} < M y^*_{i} \leq M\kappa_j$

that is,

$\kappa_{j-1} < x_i\gamma + u_i \leq \kappa_j \iff M\kappa_{j-1}< x_i(M\gamma) + Mu_i \leq M\kappa_j$

In other words, if the model is identified, it can be represented by multiplying the unobserved variable $$y$$ by a positive number, and this will mean that the standard error of the residual component, the coefficients, and the cut-points will be multiplied by this number.

Let me show you an example; I will first fit a standard ordinal probit model, both with oprobit and with gsem. Then, I will use gsem to fit an ordinal probit model where the residual term for the underlying linear regression has a standard deviation equal to 2. I will do this by introducing a latent variable $$L$$, with variance 1, and coefficient $$\sqrt 3$$. This will be added to the underlying latent residual, with variance 1; then, the ‘new’ residual term will have variance equal to $$1+((\sqrt 3)^2\times Var(L))= 4$$, so the standard deviation will be 2. We will see that as a result, the coefficients, as well as the cut-points, will be multiplied by 2.

. sysuse auto, clear
(1978 Automobile Data)

. oprobit rep mpg disp , nolog

Ordered probit regression                         Number of obs   =         69
LR chi2(2)      =      14.68
Prob > chi2     =     0.0006
Log likelihood = -86.352646                       Pseudo R2       =     0.0783

------------------------------------------------------------------------------
rep78 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
mpg |   .0497185   .0355452     1.40   0.162    -.0199487    .1193858
displacement |  -.0029884   .0021498    -1.39   0.165     -.007202    .0012252
-------------+----------------------------------------------------------------
/cut1 |  -1.570496   1.146391                      -3.81738    .6763888
/cut2 |  -.7295982   1.122361                     -2.929386     1.47019
/cut3 |   .6580529   1.107838                     -1.513269    2.829375
/cut4 |    1.60884   1.117905                     -.5822132    3.799892
------------------------------------------------------------------------------

. gsem (rep <- mpg disp, oprobit), nolog

Generalized structural equation model             Number of obs   =         69
Log likelihood = -86.352646

--------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
rep78 <-       |
mpg |   .0497185   .0355452     1.40   0.162    -.0199487    .1193858
displacement |  -.0029884   .0021498    -1.39   0.165     -.007202    .0012252
---------------+----------------------------------------------------------------
rep78          |
/cut1 |  -1.570496   1.146391    -1.37   0.171     -3.81738    .6763888
/cut2 |  -.7295982   1.122361    -0.65   0.516    -2.929386     1.47019
/cut3 |   .6580529   1.107838     0.59   0.553    -1.513269    2.829375
/cut4 |    1.60884   1.117905     1.44   0.150    -.5822132    3.799892
--------------------------------------------------------------------------------

. local a = sqrt(3)

. gsem (rep <- mpg disp L@a'), oprobit var(L@1) nolog

Generalized structural equation model             Number of obs   =         69
Log likelihood = -86.353008

( 1)  [rep78]L = 1.732051
( 2)  [var(L)]_cons = 1
--------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
rep78 <-       |
mpg |    .099532     .07113     1.40   0.162    -.0398802    .2389442
displacement |  -.0059739   .0043002    -1.39   0.165    -.0144022    .0024544
L |   1.732051  (constrained)
---------------+----------------------------------------------------------------
rep78          |
/cut1 |  -3.138491   2.293613    -1.37   0.171     -7.63389    1.356907
/cut2 |  -1.456712   2.245565    -0.65   0.517    -5.857938    2.944513
/cut3 |   1.318568    2.21653     0.59   0.552     -3.02575    5.662887
/cut4 |   3.220004   2.236599     1.44   0.150     -1.16365    7.603657
---------------+----------------------------------------------------------------
var(L)|          1  (constrained)
--------------------------------------------------------------------------------`

### Ordinal probit model with endogenous covariates

This model is defined analogously to the model fitted by -ivprobit- for probit models with endogenous covariates; we assume Read more…

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