```help spreg postestimation                                    also see:  spreg
-------------------------------------------------------------------------------

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,
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)

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