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Generalized Bayes estimation for a SAR model with linear restrictions binding the coefficients

Anoop Chaturvedi1,a, Sandeep Mishraa

Correspondence to: 1 Department of Statistics, University of Allahabad, Allahabad, 211002, India. E-mail: sandeepstat24@gmail.com
Received October 25, 2020; Revised February 19, 2021; Accepted March 12, 2021.
Abstract

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.

Keywords : spatial autoregressive (SAR) model, generalized Bayes estimator, restricted least squares estimator, Stein rule estimator, minimaxity
1. Introduction

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 et al. (2008), and Lesage and Pace (2009), Baltagi (2011), Lee and Yu (2015) to cite a few. While applying Bayesian procedures for the estimation of parameters, a major drawback is the lack of robustness with respect to underlying prior assumptions. For instance, the Bayes estimator derived under normal prior has infinite Bayes risk when true prior is Cauchy distribution, see Berger (1980). Stein (1973) proposed the generalized Bayes (GB) estimator for the multivariate normal mean under a scale mixture of prior distributions and established its dominance over the James-Stein and positive part James-Stein estimators. These estimators are more robust with respect to underlying prior assumptions and satisfy minimaxity and admissibility properties, see Brown (1971) and Rubin (1977). Berger (1980) obtained confidence region for multivariate normal mean based on GB estimator. Using Brown’s (1971) condition, Maruyama (1998) developed class of admissible minimax GB estimators. Kubokawa (1991, 1994) established the dominates of GB estimator over the James-Stein estimator. Considering the scale mixture of multivariate normal distribution as prior distribution, Maruyama (1999) derived the GB estimator for normal mean vector, established its admissibility and minimaxity and showed that the estimator dominates positive part Stein rule estimator. For estimating the coefficients vector of spatial autoregressive (SAR) model, Pal et al. (2016) proposed a family of shrinkage estimators and investigated its asymptotic properties. Recently, Chaturvedi and Mishra (2019) derived a GB estimator for estimating the parameters of a SAR model and investigated the admissibility and minimaxity properties of the estimator. They applied the results to demographic data on total fertility rate for selected Indian states. However, a major drawback of their work is that they assumed the disturbances variance to be known.

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 et al. (1996) extended the work of Srivastava and Srivastava (1984) to linear model with non-spherical disturbances. Toutenburg and Shalabh (1996) analysed the performance properties of predictors arising from the methods of restricted regression and mixed regression besides least squares under a target function. Toutenburg and Shalabh (2000) considered the family of Stein rule (SR) estimators proposed by Srivastava and Srivastava (1983) and analysed performance properties of this family when the objective is to predict values outside the sample and within the sample.

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 et al. (2001) considered an alternative approach, which utilizes Rao (1973) and used the generalized inverse as a principal tool for estimating coefficients. They proposed a family of shrinkage estimators for the general linear regression model with non-spherical disturbances in the presence of a set of linear restrictions binding the regression coefficients and investigated its asymptotic and finite sample properties.

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 et al. (2016), Tsukamoto (2019) and the references cited their in. For modelling such kind of spatial spill over along with the phenomena of constant return to scale, an SAR model with appropriate set of linear restrictions binding the coefficients provides appropriate alternative.

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 et al. (2001) to derive the GB estimator. The resulting estimator retains the flavour of the GB concept, yet gives rise to an estimator that satisfies the linear constraints. The minimaxity of restricted GB estimator has been established and dominance over the usual restricted least squares estimator and restricted Stein rule estimator have been demonstrated under a quadratic loss structure. For investigating the finite sample behaviour of the class of estimators, a simulation study has been carried out. The findings of the simulation show that the proposed class of restricted GB estimators performs superior to the usual restricted least squares estimator over a wide range of parameters.

2. Generalized Bayes estimator under linear restrictions

### 2.1. The spatial autoregressive model

Let us consider the spatial autoregressive (SAR) model

$y=ρWy+Xβ+u, u~N (0,σ2In),$

where y is an (n × 1) vector of the sample observations on a dependent variable collected at each of n locations, X is a (n × p) matrix of observations on p exogenous variables with rank (X) = p (< n), β is a (p × 1) vector of regression parameters, ρ is the spatial autoregressive parameter, W is known n × n spatial weight matrix which has been standardized to have row sum of unity. Suppose the prior information available in the form of m exact linear restrictions binding the regression

$r=Rβ,$

where r : m × 1 is a vector with known elements, and R : m × p is a matrix of rank m (< p) with known elements. For obtaining the restricted regression estimator for β under the restrictions (2.2) let us transform the model (2.1) as

$y˜=ρWy+X˜μ+u,$

where

$y˜=yN, X˜=XN,N=Ip-R′(RR′)-1R, μ=Nβ, μ˜=R′ (RR′)-1 r.$

We write

$v(ρ)=[y˜-ρWy]′M [y˜-ρWy], M=In-X˜(X˜′X˜)+X˜′.$

When ρ is unknown, we replace it by its estimator

$ρ^=y′ W′My˜y′W′MWy$

in (2.2) to obtain feasible restricted least squares estimator of μ.

### 2.2. Class of generalized Bayes estimators

For obtaining the GB estimator of regression coefficients vector under linear restrictions, we write the model (2.3) as

$y*(ρ)=X˜μ+u,$

where y*(ρ) = ρWy. Then the pdf of y*(ρ) is given by

$p (y*(ρ)∣μ,σ2)=1(2π)n2σn exp {-12σ2(y* (ρ)-X˜μ)′(y*(ρ)-X˜μ)}.$

The prior distribution for β is taken as a g-prior N (μ̃,σ2gXX), with g = (1 − λ) /λ, (0 < λ < 1). Since the prior mean of μ is 0 and prior covariance matrix of μ is

$E (μμ′)=1-λλσ2X˜′X˜.$

We take the prior distribution of μ as normal with mean vector 0 and covariance matrix {(1 − λ)/λ}σ2 . However, rank of is q = pm, implying that the prior distribution of μ is degenerated and concentrated on a lower dimensional Euclidean space q. Hence, the prior pdf of μ is given by

$p (μ∣σ2,λ)∝σ-q (λ1-λ)q2 exp {-λ2σ2 (1-λ)μ′X˜′X˜μ}.$

Further, we assume that

$p(λ)∝λ-a(1-λ)c,$

and σ has the improper prior distribution

$p (σ)∝1σ, 0<σ<∞.$

Then the joint pdf of (y*(ρ), λ) is obtained as

$p (y*(ρ),λ)∝λq2-a(1-λ)c(v(ρ)+λμ^(ρ)′X˜′X˜μ^(ρ))n2.$

The marginal density of y*(ρ) is

$m* (y*(ρ))∝∫01λq2-a(1-λ)c(v(ρ)+λμ^(ρ)′X˜′X˜μ^(ρ))n2dλ∝∫01λq2-a(1-λ)c(v(ρ)+μ^(ρ)′X˜′X˜μ^(ρ))n2[1-(1-λ)μ^(ρ)′X˜′X˜μ^(ρ)(v(ρ)+λμ^(ρ)′X˜′X˜μ^(ρ))]-n2dλ∝∑j=0∞γj(μ^(ρ)′X˜′X˜μ^(ρ))j(v(ρ)+μ^(ρ)′X˜′X˜μ^(ρ))n2+jΓ (q2-a+1) Γ (j+c+1)Γ (q2-a+j+c+2),$

where

$γj=Γ (n2+j)Γ (n2) j!.$

We write δ(ρ) = v(ρ) (1 + w(ρ)) with w(ρ) = μ̂ (ρ)′X̃ μ̂(ρ)/v(ρ). Then the posterior expectation of λ given y*(ρ) is

$E (λ∣y*(ρ))=∑j=0∞γjδ(ρ)-(j+n2)​∫01λq2-a+1(1-λ)j+cdλ∑j=0∞γjδ(ρ)-(j+n2)​∫01λq2-a(1-λ)j+cdλ=F21 (n2,c+1,q2-a+c+3,1δ(ρ)) Γ (q2-a+2) Γ (q2-a+c+2)F21 (n2,c+1,q2-a+c+2,1δ(ρ)) Γ (q2-a+1) Γ (q2-a+c+3)=φac(w(ρ)) (say).$

Here, the confluent hypergeometric function 2F1 (a, b; c; z) is defined as

$F21 (a,b;c;z)=∑j=0∞(a)j(b)j(c)jzjj!.$

Then the GB estimator of μ is

$μ^G(ρ)=[1-φac(w(ρ))] μ^(ρ).$

Hence, the restricted GB estimator of β is given by

$β^RG (ρ)=μ^G(ρ)+μ˜.$

Let us write

$φr(w(ρ))=w(ρ)∫0∞∫01λq2-a+1(1-λ)cσ-(n+1) exp {-v(ρ)2σ2} exp {-λv(ρ)w(ρ)2σ2} dλdσ∫0∞∫01λq2-a(1-λ)cσ-(n+1) exp {-v(ρ)2σ2} exp {-λv(ρ)w(ρ)2σ2} dλdσ=w(ρ)φa,c(w(ρ)).$

Then we can express the GB estimator μ̂(ρ) as

$μ^G(ρ)=[1-v(ρ)μ^(ρ)′X˜′X˜μ^(ρ)φr(w(ρ))] μ^(ρ).$

The restricted GB estimator β̂RG(ρ) is given by

$β^RG(ρ)=[1-v(ρ)μ^(ρ)′X˜′X˜μ^(ρ)φr(w(ρ))] μ^(ρ)+μ˜.$

### 3. Minimaxity conditions

Notice that the rank of matrix (or ) is q = pm. Thus, we can find p × p orthogonal matrix P, such that

$P′X˜′X˜P=[Λq000]=diag (λ1,λ2,…,λq,0,…,0),$

where λ1, λ2, . . . , λq are eigen values of . Define $P=[P1P2]$, where P1 is q × p and P2 is m × p. Then

$P′X˜′X˜P=(P1′X˜′X˜P1P1′X˜′X˜P2P2′X˜′X˜P1P2′X˜′X˜P2).$

Equations (3.1) and (3.2) together imply that $P1′X˜′X˜P2=0, P2′X˜′X˜P2=0$. Now

$P′μ^(ρ)=(P1′μ^(ρ)P2′μ^(ρ))=(Λq-1P1′X˜′y˜(ρ)0)=(η0).$

Then $η~N(P1′β,σ2Λq-1)$. We write

$Z=1σΛq-1η, θ=1σΛq-1P1′β, ω(ρ)=v(ρ)σ2.$

Then Z ~ N(θ, Iq), and ω(ρ) ~ χ2(np) independently of Z. Let us consider the quadratic loss function

$L(β^,β)=1σ2(β^-β)′X′X (β^-β.)$
Theorem 1

Under the loss function (3.4), the restricted GB estimator β̂RG(ρ) has finite risk.

Proof:

We observe that

$(β^RG(ρ)-β)′X′X (β^RG(ρ)-β)=(μ^G (ρ)-μ)′X˜′X˜ (μ^G(ρ)-μ).$

Hence, under the loss function (3.4), the risk of GB estimator β̂RG(ρ) is given by

$R [β^RG(ρ),β]=E [(Z-θ)′(Z-θ)-2ω(ρ)(Z-θ)′Zφr(w(ρ))Z′Z+ω(ρ)2Z′Zφr2(w(ρ))]=q-2E [ω(ρ)(Z-θ)′Zφr(w(ρ))‖Z‖2]+E [ω(ρ)2‖Z‖2φr2(w(ρ))].$

We observe that 0 ≤ φr(w(ρ)) ≤ w, so that

$E [ω(ρ)2‖Z‖2φr2(w(ρ))]≤E [‖Z‖2]=q+‖θ‖2<∞.$

Further by Schwarz’s inequality

$E [ω(ρ)(Z-θ)′Zφr(w(ρ))‖Z‖2]≤[E(Z-θ)′ (Z-θ) E {ω(ρ)2φr2(w(ρ))‖Z‖2}]12≤[qE [‖Z‖2]]12=[q (q+θ′θ)]2<∞,$

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.

### Theorem 2

The restricted GB estimator is minimax whenever

$(3-p-m2)+(p-m)(p-m-2)n+p-m-2≤a≤p-m2+1.$
Proof:

Under the loss function (3.4) the difference between the risks of restricted GB estimator β̂RG(ρ) and the OLS estimator bR(ρ) is given by

$R (β^RG(ρ),β)-R(bR(ρ),β)=E [ω(ρ)2‖Z‖2φr2(w(ρ))-2ω(ρ)(Z-θ)′Zφr (w (ρ))‖Z‖2]=E [ω(ρ)2‖Z‖2φr2(w(ρ))]-2E [∂∂Z′{Zω(ρ)φr (w (ρ))‖Z‖2}]=E [ω(ρ)2‖Z‖2φr2(w(ρ))-2(q-2)ω(ρ)φr(w(ρ))‖Z‖2-4φr′(w(ρ))].$

For notational convenience we write w(ρ) ≡ w and v(ρ) ≡ v. Let us write the density function

$gw(λ)={λq2-a(1-λ)c}{(1+λw)}n2∫01{λq2-a(1-λ)c}{(1+λw)}n2dλ, 0<λ<1.$

Then

$∂∂w [φr(w)w]=-n2 [Egw (λ21+λw)-Egw (λ1+λw) Egw(λ)].$

Since both λ and λ/(1 + λw) are monotone increasing functions of λ, we get Egw [λ2/(1 + λw)] ≥ Egw [λ/(1 + λw)]Egw (λ). This implies that (/∂w)[φr(w)/w] ≤ 0. Again

$φr′(w)= φr(w)w+w∂∂w{φr(w)w}=Egw(λ)-w n2 [Egw (λ21+λw)-Egw (λ1+λw) Egw(λ)].$

Using integration by parts and after few algebraic manipulations, we get

$φr′(w)=c [Egw (λ21-λ)-Egw (λ1-λ) Egw(λ)].$

Since λ/(1 − λ) and λ are monotone increasing functions of λ, we obtain

$Egw (λ21-λ)≥Egw (λ1-λ) Egw(λ).$

Further 0 ≤ λ ≤ 1, implies that Efr (λ)(λ) − Efr (λ)(λ2) ≥ 0. If $ϕ(χr2)$ is a function of Chi-square variate with r degrees of freedom, then $E[χr2ϕ(χr2)]=rE[ϕ(χr+22)]$, and φr(w) and φr(w)/w are monotone in opposite directions. Hence, we obtain

$R (β^RG(ρ),β)-R (bR(ρ),β)≤E [ω2φr(w)-2(q-2)ω] E [φr(w)‖Z‖2]≤(n-q)(n-q+2)E [φr(w)]-2(q-2)E [φr(w)‖Z‖2].$

Since φr(w) is an increasing function of w, an upper bound for φr(w) can be obtained when w is large. For large w, we can approximate

$∫0∞∫01λq2-a+1(1-λ)cσ-(n+1) exp {-v2σ2} exp {-λvw2σ2} dλdσ≈Γ (q2-a+2) Γ (n-q2+a-2) 12 (2vw)(q2-a+2)(2v)(n-q2+a-2),∫0∞∫01λq2-a(1-λ)cσ-(n+1) exp {-v2σ2} exp {-λvw2σ2} dλdσ≈Γ (q2-a+1) Γ (n-q2+a-1) 12 (2vw)(q2-a+1)(2v)(n-q2+a-1).$

So that, φr(w) can be approximated as

$φr(w)≈(q2-a+1)(n-q2+a-2).$

Thus, a sufficient dominance condition is

$0≤(q2-a+1)(n-q2+a-2)≤2 (q-2)n-q+2.$

Now (q/2 − a + 1) ≥ 0 implies that aq/2 + 1. Further

$(q2-a+1)(n-q2+a-2)≤(q-2)n-q2+1,$

implies that

$a≥(3-q2)+q (q-2)n+q-2.$

Hence, we obtain the required sufficient minimaxity condition for the GB estimator. For large n, the minimaxity condition (3.7) reduces to

$3-q2≤a≤q2+1.$
4. Simulation study

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 y are generated by using the SAR model

$y=ρWy+Xβ+u,$

where u ~ N(0, σ2In).

In simulation study we compare the risks of feasible version of restricted feasible generalized Bayes estimator (β̂RG) with the usual restricted feasible least squares (RFLS) estimator bR( ρ̂) = μ̂( ρ̂) + μ̃.

We consider the following restricted feasible generalized Bayes (RFGB) estimator

$β^RG(ρ^)=[1-φa,c (μ^ (ρ^)′X˜′X˜μ^(ρ^)σ^2)] μ^ (ρ^)+μ˜.$

For comparison purpose, we also consider the restricted feasible Stein-rule (RFSR) estimator

$β^S RG(ρ^)=[1-q-2n-q+2v(ρ^)μ^ (ρ^)′X˜′X˜μ^(ρ^)] μ^ (ρ^)+μ˜.$

The matrix X has been generated from multivariate normal distribution MVN[(1, 3, 5, 4, 7, 5, 6, 4, 7, 4)’, diag(0, 1.6, 0.7, 3.2, 1.5, 1, 2.8, 2, 1.4, 2.2)]. In the weight matrix W, the weights assigned to nearest neighbour values are twice the weights assigned to the second nearest neighbour values and other neighbour weights are taken as zero. For ensuring the property that the weight matrix is row stochastic, initially we take wi,i+1 = 2, wi,i+2 = 1 and all other weights as zero. Then, we divide each element of the selected matrix by the corresponding row sum (which is 3 in our case), so that the sum of elements of each row of W is one. The values of ρ are selected in the range (1/Wmax, 1/Wmin), where Wmax and Wmin are the maximum and minimum eigen value of W. The coefficients vector β is selected so that it follows the linear restriction β2 + β3 = 1. The simulation study has been carried out for the value of parameter c = 1, a = 0.5 and the results are depicted in Figures 1–4 (supplementary material) and Tables 14. Figure 1 / Table 1 shows the percentage gain in efficiency of RFGB estimator over RFLS estimator bR( ρ̂) for ββ = 1.745 when p = 5, ββ = 2.1019 when p = 10 and different values of ρ in the range (−0.95, 0.95). Figure 2 / Table 2 shows the percentage gain in efficiency of RFGB estimator over RFSR estimator for ββ = 1.745 when p = 5, ββ = 2.1019 when p = 10 and different values of ρ. Figure 3 / Table 3 depicts the percentage gain in efficiency of RFGB estimator over RFLS estimator for ββ = 6.29945 when p = 5, ββ = 6.810156 when p = 10 and different values of ρ. Figure 4 / Table 4 shows percentage gain in efficiency of RFGB estimator over RFSR estimator for ββ = 6.29945 when p = 5, ββ = 6.810156 when p = 10 and different values of ρ. For each setting of parameters, the experiment is replicated 5,000 times. We have used maximum likelihood estimator of ρ for evaluating RFLS and RFGB estimators.

The empirical mean squared error (EMSE) of any estimator δ̂ of δ is defined as

$EMSE (d^)=15000∑5000j=1(d^(j)-d)′ (d^(j)-d),$

where ( j) is the estimator of d for the j–th replication. Further, for two estimators δ̂ and δ̃ of δ, the percentage gain in efficiency of estimator δ̂ over δ̃ is defined as

$%GE=EMSE (δ˜)-EMSE (δ^)EMSE (δ^)×100.$

The percentage gains in efficiency of RFGB estimator over RFLS estimator and RFSR estimator are tabulated in Tables 14 for different values of n, p, ββ, and ρ.

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 n = 200, p = 5, ρ = 0.05, 0.25.

• From Figure 1 (supplementary material) and Table 1, we observe that for both n = 20 and n = 50, the percentage gain in efficiency of RFGB estimator over RFLS estimator is maximum at ρ = −0.95, and decreases gradually up to ρ ≈ 0.65 (n = 20, p =5 and 10), and ρ ≈ 0.5 (n = 50, p = 5, 10) and then starts increasing. Further, for n = 100 and 200, the percentage gain in efficiency of RFGB estimator over RFLS estimator is maximum at ρ = −0.95, decreases gradually up to ρ ≈ 0.45 (n = 100, p = 5), ρ ≈ 0.35 (n = 100, p = 10), ρ ≈ 0.25 (n = 200, p = 5) and ρ ≈ 0.15 (n = 200, p =10), then starts increasing with increasing ρ.

• It can be seen from Figure 2 (supplementary material) and Table 2 that for both n = 20 and 50 the percentage gain in efficiency of RFGB estimator over RFSR estimator is maximum at ρ = −0.95 and decreases gradually up to ρ ≈ 0.65 (n = 20, p = 5), ρ ≈ 0.55 (n = 20, p = 10), ρ ≈ 0.5 (n = 50, p = 5) and ρ ≈ 0.45 (n = 50, p = 10), then starts increasing with increasing ρ. For n = 100 and 200, the percentage gain in efficiency of RFGB estimator over RFSR estimator is maximum at ρ = −0.95, decreases gradually up to ρ ≈ 0.35 (n = 100, p = 5), ρ ≈ 0.3 (n = 100, p = 10), ρ ≈ 0.25 (n = 200, p = 5) and ρ ≈ 0.15 (n = 200, p = 10), then starts increasing with increasing ρ.

• For fixed n, in all the cases, the gain in efficiency increases as p increases from 5 to 10.

• For fixed n and p, as ββ increases the gain in efficiency usually decreases.

• An interesting observation for sample size n = 50 is that for p = 5, the gain in efficiency of RFGB estimator over both RFLS estimator and RFSR estimator usually lower for ρ > 0 than the corresponding gain in efficiency for ρ < 0 whereas for p = 10, the gain in efficiency for ρ > 0 is more than that corresponding to negative value of ρ.

• 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.

5. Concluding remarks

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.

TABLES

### Table 1

Percentage gain in efficiency of RFGB estimator over RFLS with changing ρ

ρ n

20 50 100 200
p = 5, ββ = 1.745 −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

p = 10
ββ = 2.1019
−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

### Table 2

Percentage gain in efficiency of RFGB estimator over RFSR estimator with changing (ρ)

ρ n

20 50 100 200
p = 5
ββ = 1.745
−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

p = 10
ββ = 2.1019
−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

### Table 3

Percentage gain in efficiency over FLS with changing ρ

ρ n

20 50 100 200
p = 5
ββ= 6.29945
−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

p = 10
ββ= 6.810156
−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

### Table 4

Percentage gain in efficiency over SR estimator with changing ρ

ρ n

20 50 100 200
p = 5
ββ = 6.29945
−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

p = 10
ββ = 6.810156
−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

References
1. Anselin L (1988). Spatial Econometrics: Methods and Models, Boston, USA, Kluwer Academic Publishers.
2. Anselin L, Le Gallo J, and Jayet H (2008). Spatial panel econometrics. The Econometrics of Panel Data, Fundamentals and Recent Developments in Theory and Practice (3rd ed), Berlin, Springer.
3. Baltagi BH (2011). Spatial panels. The Handbook of Empirical Economics and Finance, (pp. 435-454), New York, Chapman and Hall.
4. Berger JO (1980). Statistical Decision Theory: Foundations, Concepts and Methods, Berlin, Springer.
5. Brown LD (1971). Admissible estimators, recurrent diffusions, and insoluble boundary value problems. Annals of Mathematical Statistics, 42, 85-90.
6. Chaturvedi A, Tran VH, and Shukla G (1996). Improved estimation in the restricted regression model with non-spherical disturbances. Journal of Quantitative Economics (12), 115-123.
7. Chaturvedi A, Wan ATK, and Singh SP (2001). Stein-rule restricted regression estimator in a linear regression model with non-spherical disturbances. Communications in Statistics Theory & Methods, 30, 55-68.
8. Chaturvedi A and Mishra S (2019). Generalized Bayes estimation of spatial autoregressive models. Statistics in Transition, 20.
9. Elhorst JP (2003). Specification and estimation of spatial panel data models. International Regional Science Review, 26, 244-268.
10. Elhorst JP (2010). Spatial panel data models. Handbook of Applied Spatial Analysis, (pp. 377-407), Berlin, Springer.
11. Elhorst JP (2014). Spatial Econometrics: from Cross-Sectional Data to Spatial Panels, Berlin, Springer.
12. Glass AJ, Kenjegalieva K, and Sickles RC (2016). A spatial autoregressive stochastic frontier model for panel data with asymmetric efficiency spillovers. Journal of Econometrics, 190, 289-300.
13. Kubokawa T (1991). An approach to improving the James-Stein estimator. Journal of Multivariate Analysis, 36, 121-126.
14. Kubokawa T (1994). A unified approach to improving equivariant estimators. Annals of Statistics, 22, 290-299.
15. Lee LF and Yu J (2015). Spatial panel data models. The Oxford Handbook of Panel Data, (pp. 363-401), New York, Oxford University Press.
16. LeSage JP and Pace RK (2009). Introduction to Spatial Econometrics, New York, Boca Raton.
17. Maruyama Y (1998). A unified and broadened class of admissible minimax estimators of a multivariate normal mean. Journal of Multivariate Analysis, 64, 196-205.
18. Maruyama Y (1999). Improving on the James-Stein estimator. Statistics & Decisions, 17, 137-140.
19. Pal A, Dubey A, and Chaturvedi A (2016). Shrinkage estimation in spatial autoregressive model. Journal of Multivariate Analysis, 143, 362-373.
20. Rao CR (1973). Linear Statistical Inference and Its Applications (2nd ed), New York, Wiley.
21. Rubin H (1977). Robust Bayesian estimation. Proceeding of the Statistical Decision Theory and Related Topics II Indiana, USA. .
22. Srivastava VK and Srivastava AK (1983). Improved estimation of coefficients in regression models with incomplete prior information. Biometrical Journal, 25, 775-782.
23. Srivastava VK and Srivastava AK (1984). Stein-rule estimators in restricted regression models. Estadistica, 36, 89-98.
24. Srivastava AK and Chandra R (1991). Improved Estimation of Restricted Regression Model When Disturbances are not Necessarily Normal, Sankhyā, B, 53, 119-133.
25. Stein C (1973). Estimation of the mean of a multivariate normal distribution, (pp. 345-381), CA, USA.
26. Tsukamoto T (2019). A spatial autoregressive stochastic frontier model for panel data incorporating a model of technical inefficiency. Japan & The World Economy, 50, 66-77.
27. Toutenburg H and Shalabh (1996). Predictive performance of the methods of restricted and mixed regression estimators. Biometrical Journal, 38, 951-959.
28. Toutenburg H and Shalabh (2000). Improved predictions in linear regression models with stochastic linear constraints. Biometrical Journal, 42, 71-86.