Estimation of level and change for unemployment using structural time series models
Section 3. Initial estimates

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Let ${\widehat{\overline{Y}}}_{itp}$ denote the initial estimate for $Y_{it}$ based on data from wave $p.$ The initial estimates used as input for the time series small area models are survey regression estimates (Woodruff, 1966; Battese et al., 1988; Särndal et al., 1992)

$${\widehat{\overline{Y}}}_{itp}={\overline{y}}_{itp}+{\widehat{\beta }}_{tp}^{\prime }\left({\overline{X}}_{it}-{\overline{x}}_{itp}\right)\mathrm{,}(3.1)$$

where ${\overline{y}}_{itp},$ ${\overline{x}}_{itp}$ denote sample means, ${\overline{X}}_{it}$ is the vector of population means of the covariates $x,$ and ${\widehat{\beta }}_{tp}$ are estimated regression coefficients. The coefficients are estimated separately for each period and each wave, but they are based on the national samples combining data from all areas. The survey regression estimator is an approximately design-unbiased estimator for the population parameters that, like the GREG estimator, uses auxiliary information to reduce nonresponse bias. See Boonstra and van den Brakel (2016) for more details on the model selected to compute the survey regression estimates. Even though the regression coefficient estimates in (3.1) are not area-specific, the survey regression estimator is a direct domain estimator in the sense that it is primarily based on the data obtained in that particular domain and month, and therefore it has uncacceptably large standard errors due to the small monthly domain sample sizes.

The initial estimates for the different waves give rise to systematic differences in unemployment estimates, generally termed rotation group bias (RGB) (Bailar, 1975). The initial estimates for unemployement for waves 2 to 5 are systematically smaller compared to the first wave. This RGB has many possible causes, including selection, mode and panel effects (van den Brakel and Krieg, 2009). See Boonstra and van den Brakel (2016) for details and graphical illustrations.

The time series models also require variance estimates corresponding to the initial estimates. We use the following cross-sectionally smoothed estimates of the design variances of the survey regression estimates,

$v\left({\widehat{\overline{Y}}}_{itp}\right)\mathrm{<mo>=</mo>}\frac{1}{n_{itp}}\frac{1}{\left(n_{tp}-m_A\right)}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^{m_A}\left(n_{itp}-1\right){\widehat{\sigma }}_{itp}^2}\equiv {\widehat{\sigma }}_{tp}^2/n_{itp},$    with    ${\widehat{\sigma }}_{itp}^2\mathrm{<mo>=</mo>}\frac{1}{\left(n_{itp}-1\right)}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^{n_{itp}}{\widehat{e}}_{ijtp}^2}\mathrm{.}(3.2)$

Here $m_A$ denotes the number of areas, $n_{itp}$ is the number of respondents in area $i,$ period $t$ and wave $p,$ $n_{tp}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^{m_A}{n}_{itp}},$ and ${\widehat{e}}_{ijtp}$ are residuals of the survey regression estimator. The within-area variances ${\widehat{\sigma }}_{itp}^2$ are pooled over the domains to obtain more stable variance approximations. The use of (3.2) can be further motivated as follows. Recall that the sample design is self-weighted. Calculating within-area variances ${\widehat{\sigma }}_{itp}^2$ therefore approximately accounts for the stratification, which is a slightly more detailed regional variable than province. The variance approximation also accounts for calibration and nonresponse correction, since the within-area variances are calculated over the residuals of the survey regression estimator. The variance approximation does not explicitly account for the clustering of persons within households. However, the intra-cluster correlation for unemployment is small. In addition, registered unemployment is used as a covariate in the survey regression estimator. Since this covariate explains a large part of the variation of unemployment, the intra-cluster correlation between the residuals is further reduced.

The panel design induces several non-zero correlations among initial estimates for the same province and different time periods and waves. These correlations are due to partial overlap of the sets of sample units on which the estimates are based. Such correlations exist between estimates for the same province in months $t_1\mathrm{,}{t}_2$ and based on waves $p_1\mathrm{,}{p}_2$ whenever $t_2-t_1\mathrm{<mo>=</mo>\; 3}\left(p_2-p_1\right)\le 12.$ The covariances between ${\widehat{\overline{Y}}}_{i{t}_1{p}_1}$ and ${\widehat{\overline{Y}}}_{i{t}_2{p}_2}$ are estimated as (see e.g., Kish (1965))

$$v\left({\widehat{\overline{Y}}}_{i{t}_1{p}_1}\text{}\mathrm{,}{\widehat{\overline{Y}}}_{i{t}_2{p}_2}\right)=\frac{n_{i{t}_1{p}_1{t}_2{p}_2}}{\sqrt{n_{i{t}_1{p}_1}{n}_{i{t}_2{p}_2}}}{\widehat{\rho }}_{t_1{p}_1{t}_2{p}_2}\sqrt{v\left({\widehat{\overline{Y}}}_{i{t}_1{p}_1}\right)v\left({\widehat{\overline{Y}}}_{i{t}_2{p}_2}\right)}\mathrm{,}(3.3)$$

with

$${\widehat{\rho }}_{t_1{p}_1{t}_2{p}_2}\mathrm{<mo>=</mo>}\frac{1}{\left(n_{t_1{p}_1{t}_2{p}_2}-m_A\right)}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^{m_A}}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^{n_{i{t}_1{p}_1{t}_2{p}_2}}}{\widehat{e}}_{ij{t}_1{p}_1}{\widehat{e}}_{ij{t}_2{p}_2}\mathrm{,}$$

where $n_{i{t}_1{p}_1{t}_2{p}_2}$ is the number of units in the overlap, i.e., the number of observations on the same units in area $i$ between period and wave combinations $\left(t_1\mathrm{,}{p}_1\right)$ and $\left(t_2\mathrm{,}{p}_2\right),$ and $n_{t_1{p}_1{t}_2{p}_2}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^{m_A}{n}_{i{t}_1{p}_1{t}_2{p}_2}}.$ The estimated (auto)correlation coefficient ${\widehat{\rho }}_{t_1{p}_1{t}_2{p}_2}$ is computed as the correlation between the residuals of the linear regression models underlying the survey regression estimators at $\left(t_1\mathrm{,}{p}_1\right)$ and $\left(t_2\mathrm{,}{p}_2\right),$ based on the overlap of both samples over all areas. This way they are pooled over areas in the same way as are the variances ${\widehat{\sigma }}_{tp}^2.$ Together, (3.2) and (3.3) estimate (an approximation of) the design-based covariance matrix for the initial survey regression estimates. See Boonstra and van den Brakel (2016) for more details.

Time series model estimates for monthly provincial unemployment figures will be compared with direct estimates. The procedure for calculating monthly direct estimates is based on the approach that was used before 2010 to calculate official rolling quarterly figures for the labour force. Let ${\widehat{\overline{Y}}}_{it\mathrm{.}}$ denote the monthly direct estimate for provinces, which is calculated as the weighted mean over the five panel survey regression estimates where the weights are based on the variance estimates. To correct for RGB, these direct estimates are multiplied by a ratio, say $f_{it},$ where the numerator is the mean of the survey regression estimates (3.1) for the first wave over the last three years and the denominator is the mean of monthly direct estimates ${\widehat{\overline{Y}}}_{it\mathrm{.}}$ also over the last three years, i.e., ${\tilde{\overline{Y}}}_{it\mathrm{.}}\mathrm{<mo>=</mo>}{f}_{it}{\widehat{\overline{Y}}}_{it\mathrm{.}}.$ See Boonstra and van den Brakel (2016) for details on calculating ${\widehat{\overline{Y}}}_{it\mathrm{.}}$ and ${\tilde{\overline{Y}}}_{it\mathrm{.}}$ including a variance approximation.

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