help spreg postestimation                                    also see:  spreg  
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Title

    spreg postestimation -- Postestimation tools for spreg


Description

    The following postestimation commands are available after spreg:

    command          description
    -------------------------------------------------------------------------
INCLUDE help post_estat
INCLUDE help post_estimates
INCLUDE help post_lincom
INCLUDE help post_lrtest
INCLUDE help post_nlcom
        predict      predicted values
INCLUDE help post_predictnl
INCLUDE help post_test
INCLUDE help post_testnl
    -------------------------------------------------------------------------


Syntax for predict

        predict [type] newvar [if] [in] [, statistic]

    statistic                     Description
    -------------------------------------------------------------------------
    Main
      rform                       reduced-form predicted values; the default
      limited                     predictions based on a limited information
                                    set
      naive                       predictions based on the observed values of
                                    y
      xb                          linear prediction
      rftransform(real matrix T)  user-provided (I-lambda*W)^(-1)
    -------------------------------------------------------------------------


Options for predict

        +------+
    ----+ Main +-------------------------------------------------------------

    rform predicted values calculated from the reduced-form equation, y =
        (I-lambda*W)^(-1)*X*b.

    limited predicted values based on the limited information set.  This
        option is available only for a model with
        homoskedastically-distributed errors.

    naive predicted values based on the observed values of y, lambda*W*y +
        X*b.

    xb calculates the linear prediction X*b.

    See Remarks below for a detailed explanation of the predict options.

    rftransform() tells predict use the user-specified inverse of
        (I-lambda*W).  The matrix T should reside in Mata memory.  This
        option is available only with the reduced-form predictor.


Remarks

    The methods implemented in predict after spreg are documented in Drukker,
    Prucha, and Raciborski (2011) which can be downloaded from 
    http://econweb.umd.edu/~prucha/Papers/WP_spreg_2011.pdf.


    Recall the spatial-autoregressive spatial-error (SARAR) model

        y = lambda*W*y + X*b + u

        u = rho*M*u + e

    This model specifies a system of n simultaneous equations for the
    dependent variable y.

    The predictor based on the reduced-form equation is obtained by solving
    the model for the endogenous variable y which gives (I-lambda*W)^(-1)*X*b
    for the SAR and SARAR models and X*b for the SARE model.

    The limited information set predictor is described in Kalejian and Prucha
    (2007).  Let

        U = (I-rho*M)^(-1) * (I-rho*M')^(-1)
        
        Y = (I-lambda*W)^(-1) * (I-lambda*W')^(-1)
        
        E(w_i*y) = w_i * (I-lambda*W)^(-1) * X*b
        
        cov(u_i,w_i*y) = sigma^2 * w_i*Y*w_i'
        
        var(w_i*y) = sigma^2 * u_i*(I-lambda*W')^(-1)*w_i'

    where w_i and u_i denote the ith row of W and U, respectively.  The
    limited information set predictor for observation i is given by


                               cov(u_i,w_i*y)
        lambda*w_i*y + x_i*b + -------------- * [w_i*y - E(w_i*y)]
                                 var(w_i*y)


    where x_i denotes the ith row of X.  Because the formula involves the
    sigma^2 term, this predictor is available only for a model with
    homoskedastically-distributed errors.

    The reduced-form predictor is based on the information set {X,W}.  The
    limited information set predictor includes additionally the linear
    combination W*y, thus it is more efficient than the reduced-form
    predictor.  Both predictors are unbiased predictors conditional on their
    information set.

    The naive predictor is obtained by treating the values of y on the
    right-hand side as given, which results in the formula lambda*W*y + X*b
    for the SAR and SARAR models, and X*b for the SARE model.  Note that this
    predictor is a special case of the limited information set predictor with
    cov(u_i,w_i*y) = 0, but this this is true only when lambda = rho = 0.

    The naive predictor ignores the feedback that the neighboring
    observations may have on the value of y in a given observation.  The
    reduced-form and limited information set predictors factor this feedback
    into the computations through the (I-lambda*W)^(-1)*X*b term.  If you are
    interested in how a change to a covariate in an observation affects the
    entire system, you should use the reduced-form or the limited information
    set predictor.


Examples

    Setup
        . use pollute
        . spmat use cobj using pollute.spmat
        . spreg ml pollution factories area, id(id) dlmat(cobj) elmat(cobj)

    Obtain predicted values based on the reduced-form equation
        . predict y0

    Increase factories in observation 50 by 1 and obtain a new set of
    predicted values
        . replace factories = factories+1 in 50
        . predict y1

    Compare the two sets of predicted values
        . gen deltay = abs(y1-y0)
        . count if deltay!=0

    Note that a change in one observation resulted in a total of 25 changes.


References

    Drukker, D. M., I. R. Prucha, and R. Raciborski. 2011.
        Maximum-likelihood and generalized spatial two-stage least-squares
        estimators for a spatial-autoregressive model with
        spatial-autoregressive disturbances.  Working paper, University of
        Maryland, Department of Economics, 
        http://econweb.umd.edu/~prucha/Papers/WP_spreg_2011.pdf.


    Kelejian H. H., and I. R. Prucha. 2007.  The relative efficiencies of
        various predictors in spatial econometric models containing spatial
        lags. Regional Science and Urban Economics 37, 363-374.


Also see

    Online:  spreg, spivreg (if installed)