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)