# Small area estimation for unemployment using latent Markov models

Section 3. Time series area level SAE models

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

$${\widehat{\theta}}_{it}\mathrm{=}{\theta}_{it}+{e}_{it}\mathrm{,}\text{\hspace{1em}}i\mathrm{=1,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}m\mathrm{,}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\mathrm{=1,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}T\mathrm{,}$$

and an area-linking model

$${\theta}_{it}={x}_{it}^{\prime}\beta +{v}_{i}+{u}_{it}\mathrm{,}\text{\hspace{1em}}i\mathrm{=1,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}m\mathrm{,}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\mathrm{=1,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}T\mathrm{,}$$

where ${\theta}_{it}$ is the true value corresponding to the estimate ${\widehat{\theta}}_{it}$ for the small area mean, ${x}_{it}$ is a $p\text{\hspace{0.17em}}-$ dimensional column vector of fixed covariates, and ${e}_{it}$ are normal sampling errors. Given the true value ${\theta}_{it},$ each vector ${e}_{i}\mathrm{=}{\left({e}_{i1}\mathrm{,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}{e}_{iT}\right)}^{\prime}$ has multivariate normal distribution with zero mean and with known variance-covariance matrix ${\Psi}_{i}.$ Moreover, ${v}_{i}\sim N\left(\mathrm{0,}\text{\hspace{0.17em}}{\sigma}_{v}^{2}\right)$ is the area effect and ${u}_{it}=\rho {u}_{i\text{}\mathrm{,}\text{\hspace{0.17em}}t-1}+{\u03f5}_{it},$ with $\left|\text{\hspace{0.17em}}\rho \text{\hspace{0.17em}}\right|\mathrm{<1}$ and ${\u03f5}_{it}\sim N\left(\mathrm{0,}\text{\hspace{0.17em}}{\sigma}_{\u03f5}^{2}\right)$ is the area-by-time effect. In this model, ${e}_{i},$ ${v}_{i},$ and ${\u03f5}_{it}$ are assumed independent of each other. In our application ${\Psi}_{i}$ is diagonal, with elements ${\psi}_{it},$ for $t\mathrm{=1,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}T.$

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 ${\theta}_{i}\mathrm{=}{\left({\theta}_{i1}\mathrm{,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}{\theta}_{iT}\right)}^{\prime}$ and ${\widehat{\theta}}_{i}={\left({\widehat{\theta}}_{i1}\mathrm{,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}{\widehat{\theta}}_{iT}\right)}^{\prime}\text{},$ then

$$\begin{array}{ll}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{1em}}{\widehat{\theta}}_{i}\text{\hspace{0.17em}}|\text{\hspace{0.17em}}{\theta}_{i}\hfill & \sim {N}_{T}\left({\theta}_{i},\text{\hspace{0.17em}}{\Psi}_{i}\right)\mathrm{,}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\theta}_{it}\text{\hspace{0.17em}}|\text{\hspace{0.17em}}\beta \mathrm{,}\text{\hspace{0.17em}}{u}_{it}\mathrm{,}\text{\hspace{0.17em}}{\sigma}_{v}^{2}\hfill & \sim N\left({x}_{it}^{\prime}\beta +{u}_{it}\mathrm{,}\text{\hspace{0.17em}}{\sigma}_{v}^{2}\right)\mathrm{,}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}(3.1)\hfill \\ \text{\hspace{0.17em}}{u}_{it}\text{\hspace{0.17em}}|\text{\hspace{0.17em}}{u}_{i\text{}\mathrm{,}\text{\hspace{0.17em}}t-1}\mathrm{,}\text{\hspace{0.17em}}{\sigma}_{\u03f5}^{2}\hfill & \sim N\left(\rho {u}_{i\text{}\mathrm{,}\text{\hspace{0.17em}}t-1}\mathrm{,}\text{\hspace{0.17em}}{\sigma}_{\u03f5}^{2}\right)\mathrm{,}\hfill \end{array}$$

where $\beta ,$ ${\sigma}_{v}^{2},$ and ${\sigma}_{\u03f5}^{2}$ are mutually independent. The model is fully specified once priors are chosen for $\beta ,$ ${\sigma}_{v}^{2},$ and ${\sigma}_{\u03f5}^{2},$ namely as $f\left(\beta \right)\propto 1,$ ${\sigma}_{v}^{2}\sim \text{IG}\left({a}_{1}\mathrm{,}\text{\hspace{0.17em}}{b}_{1}\right),$ and ${\sigma}_{\u03f5}^{2}\sim \text{IG}\left({a}_{2}\mathrm{,}\text{\hspace{0.17em}}{b}_{2}\right),$ 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 ${\sigma}_{\u03f5}^{2}.$

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

$${\theta}_{it}\mathrm{=}{x}_{it}^{\prime}{\beta}_{i}+{v}_{i}+{u}_{it}\mathrm{,}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}(3.2)$$

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

$${u}_{it}\text{\hspace{0.17em}}|\text{\hspace{0.17em}}{u}_{i\text{}\mathrm{,}\text{\hspace{0.17em}}t-1},\text{\hspace{0.17em}}{\sigma}_{\u03f5}^{2}\sim N\left({u}_{i\text{}\mathrm{,}\text{\hspace{0.17em}}t-1}\mathrm{,}\text{\hspace{0.17em}}{\sigma}_{\u03f5}^{2}\right)\mathrm{.}$$

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 ${v}_{i}\sim N\left(\mathrm{0,}\text{\hspace{0.17em}}{\sigma}_{v}^{2}\right)$ and ${\beta}_{i}\sim N\left(\beta \mathrm{,}\text{\hspace{0.17em}}{W}_{\beta}^{-1}\right),$ which, in turn, depend on appropriate parameters. See Datta et al. (1999) for further details.

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