The Spatial Autoregressive (SAR) models have drawn considerable attention in recent econometrics literature because of their capability to model the spatial spill overs in a feasible way. While considering the Bayesian analysis of these models, one may face the problem of lack of robustness with respect to underlying prior assumptions. The generalized Bayes estimators provide a viable alternative to incorporate prior belief and are more robust with respect to underlying prior assumptions. The present paper considers the SAR model with a set of linear restrictions binding the regression coefficients and derives restricted generalized Bayes estimator for the coefficients vector. The minimaxity of the restricted generalized Bayes estimator has been established. Using a simulation study, it has been demonstrated that the estimator dominates the restricted least squares as well as restricted Stein rule estimators.
The static spatial econometric models have found wide applications for modelling the spatial spill over, i.e., the dependence of level of response variable on the levels of response variable in the neighbouring regions along with a set of explanatory variables. The classical and Bayesian estimation procedures for different spatial econometric models and their various applications have been extensively discussed in Anselin (1988), Elhorst (2003, 2010, 2014), Anselin
For estimating the coefficients vector of a linear regression model in the presence of linear restrictions binding the coefficients, Srivastava and Srivastava (1983, 1984) and Srivastava and Chandra (1991) considered families of improved restricted estimators obtained by mixing SR with restricted least squares. Chaturvedi
Notice that different families of Stein rule restricted regression estimators discussed above are simply formulated by substituting Stein rule for the OLS and are not Stein rule in the true sense. Chaturvedi
In Cobb-Douglas production function and its different variants such as trans-log production function, the assumption of constant returns to scale lead to a linear restriction on regression parameters associated with logarithm of labour input and logarithm of capital input. It has been observed by several researchers that the neighbouring production activities show interdependence because of several externalities, and the usual assumption of spatial independence in frontier production functions become inappropriate, see Glass
The present paper considers SAR model with exact linear restrictions binding the regression parameters and derives a class of restricted GB estimator for the coefficients vector. Instead of applying GB procedure directly to the restricted regression model, which may not yield an estimator satisfying the restrictions, we followed an alternative approach based on the transformation used by Chaturvedi
Let us consider the spatial autoregressive (SAR) model
where
where
where
We write
When
in (
For obtaining the GB estimator of regression coefficients vector under linear restrictions, we write the model (
where
The prior distribution for
We take the prior distribution of
Further, we assume that
and
Then the joint pdf of (
The marginal density of
where
We write
Here, the confluent hypergeometric function _{2}
Then the GB estimator of
Hence, the restricted GB estimator of
Let us write
Then we can express the GB estimator
The restricted GB estimator
Notice that the rank of matrix
where
Then
Then
We observe that
Hence, under the loss function (
We observe that 0 ≤
Further by Schwarz’s inequality
which proves the required result.
Since the estimators with uniformly smallest risk under a loss function usually do not exist, one of the possible criteria for selecting an estimator is minimaxity, which minimizes the maximum risk. Since the restricted OLS estimator is minimax, any estimator uniformly dominating it is also minimax. The next theorem proves the minimaxity of GB estimator utilizing this logic.
Under the loss function (
For notational convenience we write
Then
Since both
Using integration by parts and after few algebraic manipulations, we get
Since
Further 0 ≤
Since
So that,
Thus, a sufficient dominance condition is
Now (
implies that
Hence, we obtain the required sufficient minimaxity condition for the GB estimator. For large
For studying the finite sample risk performance of restricted GB estimator, we carry out the simulation study using R Software. The observations on response variable
where
In simulation study we compare the risks of feasible version of restricted feasible generalized Bayes estimator (
We consider the following restricted feasible generalized Bayes (RFGB) estimator
For comparison purpose, we also consider the restricted feasible Stein-rule (RFSR) estimator
The matrix
The empirical mean squared error (EMSE) of any estimator
where
The percentage gains in efficiency of RFGB estimator over RFLS estimator and RFSR estimator are tabulated in Tables 1
From the numerical results, we draw the following conclusions:
The RFGB estimator outperforms the restricted RFLS for all the selected parametric settings and restricted RFSR estimators in most of the selected parametric settings. The exceptional parametric values for which RFSR estimator dominates RFGB estimator are
From Figure 1 (supplementary material) and Table 1, we observe that for both
It can be seen from Figure 2 (supplementary material) and Table 2 that for both
For fixed
For fixed
An interesting observation for sample size
For a fixed parametric setting, the gain in efficiency of RFGB estimator over RFLS estimator is more than corresponding gain in efficiency of RFGB estimator over RFSR estimator.
With the objective of achieving robustness with respect to prior distribution and satisfying minimaxity property, a family of RFGB estimators for the regression coefficients vector of a SAR model has been derived in the presence of a set of linear restrictions binding the coefficients vector. The results of the simulation study show that the RFGB estimator outperforms both, the RFLS estimator and RFSR estimator over a wide range of parametric settings. The work on extending the results of the paper for panel data spatial autoregressive models, which also incorporates spatial autoregressive stochastic frontier model for spatio-temporal data, is in progress.
Percentage gain in efficiency of RFGB estimator over RFLS with changing
20 | 50 | 100 | 200 | ||
---|---|---|---|---|---|
−0.95 | 87.652465 | 91.009943 | 92.980846 | 94.559246 | |
−0.75 | 86.816747 | 90.111241 | 92.401142 | 94.213808 | |
−0.55 | 85.892035 | 88.73986 | 91.359453 | 93.488869 | |
−0.35 | 84.820146 | 86.521877 | 88.886109 | 90.992238 | |
0.05 | 81.963294 | 77.318969 | 70.855288 | 60.480241 | |
0.25 | 80.147649 | 70.92225 | 60.433685 | 49.559817 | |
0.45 | 78.307783 | 66.797965 | 59.82711 | 55.822992 | |
0.65 | 76.988437 | 66.766127 | 63.831039 | 62.507009 | |
0.75 | 76.787609 | 67.94647 | 66.155969 | 65.095276 | |
0.95 | 78.395001 | 72.513772 | 71.750403 | 69.452348 | |
−0.95 | 93.550795 | 93.051869 | 94.905196 | 95.744847 | |
−0.75 | 93.349477 | 92.602204 | 94.295924 | 95.352843 | |
−0.55 | 93.060344 | 91.738786 | 93.034165 | 94.243812 | |
−0.35 | 92.688015 | 90.354913 | 90.549374 | 91.213383 | |
0.05 | 91.700071 | 85.674783 | 79.68406 | 68.717889 | |
0.25 | 91.131797 | 82.945035 | 75.356991 | 67.159497 | |
0.45 | 90.614072 | 81.302088 | 76.089054 | 74.992764 | |
0.65 | 90.308848 | 81.728405 | 79.49427 | 80.835451 | |
0.75 | 90.319279 | 82.760914 | 81.376767 | 82.826556 | |
0.95 | 91.106411 | 87.054551 | 85.109257 | 85.691465 |
Percentage gain in efficiency of RFGB estimator over RFSR estimator with changing (
20 | 50 | 100 | 200 | ||
---|---|---|---|---|---|
−0.95 | 82.923094 | 88.84156 | 91.77296 | 93.625716 | |
−0.75 | 81.973603 | 87.814849 | 91.080396 | 93.197345 | |
−0.55 | 80.887227 | 86.163016 | 89.742814 | 92.180977 | |
−0.35 | 79.598078 | 83.438066 | 86.495885 | 88.61416 | |
0.05 | 76.078141 | 72.28575 | 62.832244 | 41.066217 | |
0.25 | 73.881609 | 64.988875 | 51.951933 | 35.561192 | |
0.45 | 71.764217 | 60.832483 | 54.189638 | 50.187771 | |
0.65 | 70.533891 | 61.660405 | 60.108793 | 59.593103 | |
0.75 | 70.548218 | 63.466654 | 63.134815 | 62.817361 | |
0.95 | 72.83734 | 69.540567 | 70.141644 | 68.755346 | |
−0.95 | 63.784184 | 88.609957 | 93.509003 | 95.092562 | |
−0.75 | 63.431254 | 87.973216 | 92.718718 | 94.62062 | |
−0.55 | 62.724952 | 86.680816 | 91.063367 | 93.26248 | |
−0.35 | 61.687076 | 84.614498 | 87.794245 | 89.476292 | |
0.05 | 58.198412 | 78.066565 | 74.062091 | 60.816082 | |
0.25 | 55.953324 | 74.742842 | 69.65128 | 61.821461 | |
0.45 | 54.063984 | 73.209599 | 71.549233 | 72.366209 | |
0.65 | 54.045426 | 74.23149 | 76.008274 | 79.182432 | |
0.75 | 55.197672 | 75.55216 | 78.235483 | 81.385816 | |
0.95 | 59.635264 | 81.133665 | 82.263486 | 84.352788 |
Percentage gain in efficiency over FLS with changing
20 | 50 | 100 | 200 | ||
---|---|---|---|---|---|
−0.95 | 88.250714 | 85.331541 | 88.357915 | 88.237576 | |
−0.75 | 87.280955 | 84.165149 | 86.821451 | 86.933414 | |
−0.55 | 86.086996 | 81.941857 | 83.574269 | 83.424748 | |
−0.35 | 84.660808 | 78.541708 | 77.84365 | 76.181153 | |
0.05 | 81.075901 | 68.320117 | 57.696297 | 45.772802 | |
0.25 | 79.04567 | 62.868848 | 48.676744 | 34.593974 | |
0.45 | 77.193568 | 59.21098 | 46.204104 | 36.247221 | |
0.65 | 76.078811 | 58.552617 | 49.679866 | 44.080881 | |
0.75 | 76.063934 | 59.350188 | 52.759539 | 48.219291 | |
0.95 | 78.286289 | 64.428599 | 61.277897 | 54.571567 | |
−0.95 | 91.035077 | 76.382531 | 81.356926 | 82.434587 | |
−0.75 | 91.054064 | 76.329566 | 79.529575 | 80.673056 | |
−0.55 | 91.015573 | 75.528495 | 76.211548 | 76.29519 | |
−0.35 | 90.835716 | 74.147884 | 71.258038 | 67.899974 | |
0.05 | 90.195831 | 70.663943 | 59.092948 | 41.111684 | |
0.25 | 89.827964 | 69.515726 | 56.170427 | 40.60945 | |
0.45 | 89.541704 | 69.690507 | 57.862524 | 51.794662 | |
0.65 | 89.510794 | 71.929057 | 63.310748 | 63.604441 | |
0.75 | 89.694623 | 74.097215 | 66.934067 | 68.359051 | |
0.95 | 90.946863 | 82.663914 | 75.578299 | 75.556115 |
Percentage gain in efficiency over SR estimator with changing
20 | 50 | 100 | 200 | ||
---|---|---|---|---|---|
−0.95 | 82.263614 | 75.762608 | 81.211744 | 79.709125 | |
−0.75 | 81.273469 | 74.47021 | 78.716413 | 77.191035 | |
−0.55 | 79.950618 | 71.365079 | 72.864911 | 69.425762 | |
−0.35 | 78.326638 | 66.463705 | 61.947065 | 51.094821 | |
0.05 | 74.264449 | 53.03775 | 25.743741 | −31.970086 | |
0.25 | 72.063518 | 47.744477 | 18.176582 | −27.214677 | |
0.45 | 70.215858 | 46.239419 | 25.182331 | 5.164118 | |
0.65 | 69.394017 | 48.748694 | 37.69438 | 29.676562 | |
0.75 | 69.667279 | 51.031933 | 44.066218 | 38.18759 | |
0.95 | 72.708087 | 58.349346 | 57.17885 | 48.729913 | |
−0.95 | 50.916155 | 53.247087 | 71.684003 | 75.747382 | |
−0.75 | 52.027709 | 54.314679 | 68.846558 | 73.036032 | |
−0.55 | 53.116361 | 53.985354 | 63.563583 | 65.938524 | |
−0.35 | 53.667719 | 52.820521 | 55.74647 | 51.497067 | |
0.05 | 53.709882 | 50.586393 | 39.290724 | 6.996654 | |
0.25 | 53.341976 | 51.173714 | 38.519666 | 16.873713 | |
0.45 | 52.968889 | 53.917705 | 44.651203 | 40.432942 | |
0.65 | 53.120161 | 59.268304 | 54.438822 | 58.156965 | |
0.75 | 54.289198 | 63.114882 | 59.832638 | 64.383209 | |
0.95 | 59.266461 | 75.654172 | 71.174588 | 73.133859 |