Small area estimation for unemployment using latent Markov models
Section 3. Time series area level SAE models

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Rao and Yu (1994) propose an area level model involving autocorrelated random effects and sampling errors using both time-series and cross sectional data. It consists of a sampling model

$${\widehat{\theta }}_{it}\mathrm{<mo>=</mo>}{\theta }_{it}+e_{it}\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}m\mathrm{,}t\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}T\mathrm{,}$$

and an area-linking model

$${\theta }_{it}=x_{it}^{\prime }\beta +v_i+u_{it}\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}m\mathrm{,}t\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}T\mathrm{,}$$

where ${\theta }_{it}$ is the true value corresponding to the estimate ${\widehat{\theta }}_{it}$ for the small area mean, $x_{it}$ is a $p-$ dimensional column vector of fixed covariates, and $e_{it}$ are normal sampling errors. Given the true value ${\theta }_{it},$ each vector $e_i\mathrm{<mo>=</mo>}{\left(e_{i1}\mathrm{,}\dots \mathrm{,}{e}_{iT}\right)}^{\prime }$ has multivariate normal distribution with zero mean and with known variance-covariance matrix ${\Psi }_i.$ Moreover, $v_i\sim N\left(\mathrm{0,}{\sigma }_v^2\right)$ is the area effect and $u_{it}=\rho u_{i\text{}\mathrm{,}t-1}+{ϵ}_{it},$ with $\left|\rho \right|\mathrm{<mo><</mo>\; 1}$ and ${ϵ}_{it}\sim N\left(\mathrm{0,}{\sigma }_{ϵ}^2\right)$ is the area-by-time effect. In this model, $e_i,$ $v_i,$ and ${ϵ}_{it}$ are assumed independent of each other. In our application ${\Psi }_i$ is diagonal, with elements ${\psi }_{it},$ for $t\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}T.$

In the previous formulation, the area-linking model is basically a linear model with mixed coefficients. You et al. (2003, YRG) translate this model into an HB framework as follows. Let ${\theta }_i\mathrm{<mo>=</mo>}{\left({\theta }_{i1}\mathrm{,}\dots \mathrm{,}{\theta }_{iT}\right)}^{\prime }$ and ${\widehat{\theta }}_i={\left({\widehat{\theta }}_{i1}\mathrm{,}\dots \mathrm{,}{\widehat{\theta }}_{iT}\right)}^{\prime }\text{},$ then

$$\begin{array}{ll}{\widehat{\theta }}_i|{\theta }_i\hfill & \sim N_T\left({\theta }_i,{\Psi }_i\right)\mathrm{,}\hfill \\ {\theta }_{it}|\beta \mathrm{,}{u}_{it}\mathrm{,}{\sigma }_v^2\hfill & \sim N\left(x_{it}^{\prime }\beta +u_{it}\mathrm{,}{\sigma }_v^2\right)\mathrm{,}(3.1)\hfill \\ u_{it}|u_{i\text{}\mathrm{,}t-1}\mathrm{,}{\sigma }_{ϵ}^2\hfill & \sim N\left(\rho u_{i\text{}\mathrm{,}t-1}\mathrm{,}{\sigma }_{ϵ}^2\right)\mathrm{,}\hfill \end{array}$$

where $\beta ,$ ${\sigma }_v^2,$ and ${\sigma }_{ϵ}^2$ are mutually independent. The model is fully specified once priors are chosen for $\beta ,$ ${\sigma }_v^2,$ and ${\sigma }_{ϵ}^2,$ namely as $f\left(\beta \right)\propto 1,$ ${\sigma }_v^2\sim \text{IG}\left(a_1\mathrm{,}{b}_1\right),$ and ${\sigma }_{ϵ}^2\sim \text{IG}\left(a_2\mathrm{,}{b}_2\right),$ where $a_1,$ $a_2,$ $b_1$ and $b_2$ are known positive hyperparameters and, usually, set to be small and to reflect a vague knowledge about ${\sigma }_v^2$ and ${\sigma }_{ϵ}^2.$

Datta et al. (1999) follow this approach, but introduce a richer structure for the fixed part of the linking model by assuming

$${\theta }_{it}\mathrm{<mo>=</mo>}{x}_{it}^{\prime }{\beta }_i+v_i+u_{it}\mathrm{,}(3.2)$$

where $v_i$ and ${\beta }_i$ are area-specific intercepts and regression coefficients, respectively, and $u_{it}$ is an area-specific error term that follows the random-walk model

$$u_{it}|u_{i\text{}\mathrm{,}t-1},{\sigma }_{ϵ}^2\sim N\left(u_{i\text{}\mathrm{,}t-1}\mathrm{,}{\sigma }_{ϵ}^2\right)\mathrm{.}$$

The column vector of auxiliary variables $x_{it}$ may also include dummy variables for year and/or seasonality adjustments. Note that area-specific regression coefficients considerably increase the estimation complexity and the computational burden. For this reason, the hyperparameters are assumed to be $m$ independent realizations from a common probability distribution specified by $v_i\sim N\left(\mathrm{0,}{\sigma }_v^2\right)$ and ${\beta }_i\sim N\left(\beta \mathrm{,}{W}_{\beta }^{-1}\right),$ which, in turn, depend on appropriate parameters. See Datta et al. (1999) for further details.

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