Molina 2015 - Spatial and Spatio-Temporal Fay-Herriot Models

Summary

When neighbouring areas have correlated unexplained effects, the FH area effects can follow a SAR(1) process; with several periods, time effects nested within areas follow an AR(1). Bootstrap MSE is preferred for these richer models. Both reduce CVs relative to direct estimates in the paper’s examples.

Overview

Spatial and spatio-temporal extensions keep the sampling model of Molina 2015 - Fay-Herriot Area-Level Model and EBLUP and change only the covariance of the area effects.

Main Content

Spatial Fay-Herriot (SFH)

Same combined model ${\hat{\delta }}_d^{\text{DIR}}=x_d^{\top }\beta +u_d+e_d$, but $u={\rho }_{1}Wu+ϵ$, $ϵ\sim N(0,{\sigma }_1^2{I}_D)$. ${\rho }_1\in (-1,1)$ is the autoregression parameter; $W$ is a row-standardised proximity matrix (zeros on the diagonal, 1 for neighbours before standardising). Hence $u\sim N(0,{\sigma }_1^2[(I-{\rho }_{1}W{)}^{\top }(I-{\rho }_{1}W){]}^{-1})$. EBLUP: Petrucci & Salvati (2006). Analytic MSE: Singh et al. (2005) under ML/REML; parametric and non-parametric bootstrap MSE: Molina et al. (2009).

Spatio-temporal Fay-Herriot (STFH; Marhuenda et al. 2013)

${\hat{\delta }}_{dt}^{\text{DIR}}=x_{dt}^{\top }\beta +u_d+v_{dt}+e_{dt}$, $e_{dt}\sim N(0,{\psi }_d)$; $u$ follows the SAR(1) of the SFH; for each area, $v_{dt}={\rho }_2{v}_{d,t-1}+{ϵ}_{2dt}$, ${ϵ}_{2dt}\sim N(0,{\sigma }_2^2)$ (AR(1)), independent across areas. Only AR(1) is used since $T$ is typically small; MA processes are not implemented. EBLUP with parametric-bootstrap MSE; REML only; requires all $T$ periods for all $D$ areas, sorted by period within domain; model="ST" (AR(1)) or "S" (independent time effects).

sae calls: eblupSFH, mseSFH, pbmseSFH, npbmseSFH (argument proxmat; use $B>200$ bootstrap replicates, default 100), eblupSTFH, pbmseSTFH.

Examples

Example 2: Tuscany grape production, 274 municipalities (synthetic)

mseSFH with auxiliary variables area and workdays, proximity from grapesprox. EBLUPs are more stable than direct estimates and have smaller CVs for most municipalities, with remarkable gains where the direct CV was largest (Fig 2).

Example 3: spacetime data, $D=11$, $T=3$

Last-period results (selected rows of the printed table):

Area	DIR	EBLUP STFH	CV DIR (%)	CV STFH (%)
2	0.2615	0.2734	10.94	7.65
12	0.1222	0.1374	21.90	14.80
16	0.4121	0.3189	13.58	9.22
17	0.0579	0.0691	25.20	20.31
46	0.1379	0.1435	8.85	8.36

Even with only 11 areas and 3 periods the CV of the EBLUP is smaller in all areas.

Connections

A SAR(1) random effect is the frequentist counterpart of the CAR/BYM spatial priors; in PyMC use pm.CAR or a precision matrix ${\sigma }^{-2}(I-\rho W{)}^{\top }(I-\rho W)$. The AR(1) time effect is a state-space component (Linear-Gaussian State-Space Models).

See Also

Molina Marhuenda 2015 - sae Small Area Estimation - Overview