5. Application to Korean Labor Force survey

Jae-kwang Kim, Seunghwan Park and Seo-young Kim

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We now consider an application of the proposed method to the labor force surveys in Korea. In Korea, two different labor force surveys are used to obtain information about employment. One is the Korean Labor Force (KLF) survey and the other is the Local Area labor force (LALF) survey. The KLF survey has about 7K sample households but LALF has about 200K sample households. Because LALF is a large-scale survey employing a lot of part time interviewers, there is a certain level of measurement errors in the LALF survey. We assume that the KLF has no measurement error, although it has significant sampling errors at the small area level. The KLF sample is a second-phase sample from the LALF sample. Thus, the sampling errors for two survey estimates are correlated. Let ${\overline{X}}_h$  be the (true) unemployment rate for area $h.$  The small area level we considered is called "Gu�. The number of "Gu� in Korea is 229.

We observe ${\overline{x}}_h$  from KLF survey and ${\overline{y}}_{1h}$  from the LALF survey. To construct linking models, we first partition the population into two regions, urban region and rural region, based on the proportion of the households working on agricultural practice. Within each region, we build models separately (same model but allows for different parameter) and estimate the model parameters separately. The structural model is

$${\overline{Y}}_h={\beta }_1{\overline{X}}_h+e_h(5.1)$$

with $e_h\sim \left(\mathrm{0,}{\sigma }_e^2\right).$ Here, we set ${\beta }_0=0$ to guarantee that the GLS estimator of ${\overline{X}}_h$  is nonnegative. The sampling error model remains the same. In this case, ${\beta }_1$  can be estimated by

$${\widehat{\beta }}_1=\frac{{\displaystyle \sum _{h=1}^H}{w}_h\left({\widehat{\beta }}_1\right)\left\{{\overline{x}}_h{\overline{y}}_{1h}-C\left(a_h\mathrm{,}{b}_h\right)\right\}}{{\displaystyle \sum _{h=1}^H}{w}_h\left({\widehat{\beta }}_1\right)\left\{{\overline{x}}_h^2-V\left(a_h\right)\right\}}\mathrm{.}(5.2)$$

The sampling variance of $\left(a_h\mathrm{,}{b}_h\right)$  is computed using the method of reversed two-phase sampling described in the Appendix. The model variance is estimated by the method of moment technique in (3.8) with ${\widehat{\beta }}_0=0.$ The GLS estimator can be computed by (2.9) with ${\tilde{x}}_h={\widehat{\beta }}_1^{-1}{\overline{y}}_{1h}.$

In addition to the two surveys, we can also use the Census information. The GLS model incorporating the three sources of information can be expressed as

$$\left(\begin{array}{l}{\overline{Y}}_{2h}\hfill \\ {\overline{y}}_{1h}\hfill \\ {\overline{x}}_h\hfill \end{array}\right)=\left(\begin{array}{l}{\gamma }_1\hfill \\ {\beta }_1\hfill \\ 1\hfill \end{array}\right){\overline{X}}_h+\left(\begin{array}{c}{\overline{e}}_{2h}\\ b_h+{\overline{e}}_{1h}\\ a_h\end{array}\right)$$

where ${\overline{Y}}_{2h}$  is the census result for area $h.$  Because the Census estimate does not suffer from sampling error, we have only model error $e_{2h}$  which represents the error when we model $E\left({\overline{Y}}_{h2}\right)={\gamma }_1{\overline{X}}_h.$ The model parameters can be obtained using the method in Section 3 with ${\Sigma }_h=\text{diag}\left(\mathrm{0,}V\left(a_h\mathrm{,}{b}_h\right)\right).$ The GLS estimator of ${\overline{X}}_h$  can be obtained easily. The MSE part can be computed by using the fact that

$$V\left({\widehat{\overline{X}}}_h-{\overline{X}}_h\right)={\left(\begin{array}{l}{\gamma }_1\hfill \\ {\beta }_1\hfill \\ 1\hfill \end{array}\right)}^{\prime }{\left\{V\left(\begin{array}{c}{\overline{e}}_{2h}\\ b_h+{\overline{e}}_{1h}\\ a_h\end{array}\right)\right\}}^{-1}\left(\begin{array}{l}{\gamma }_1\hfill \\ {\beta }_1\hfill \\ 1\hfill \end{array}\right):=M_{h1}$$

and applying the jackknife method for bias correction.

Figure 5.1 presents the plot of the unemployment rate of KLF against LALF for urban areas. From Figure 5.1, we can find that there is a linear structural relationship between KLF and LALF. Instead of the usual residual ${\widehat{\overline{e}}}_h$  in the structural error model, ${\widehat{v}}_h$  are used as the residuals in the regression model with measurement errors, where ${\widehat{v}}_h={\overline{y}}_{1h}-{\widehat{\beta }}_1{\overline{x}}_h.$ Figure 5.2 contains a plot of ${\widehat{v}}_h$  against ${\widehat{\overline{X}}}_h$  for urban area. The plot shows that the assumption of equal variance ${\sigma }_e^2$  is slightly violated. The heteroscedastic variance model in Remark 2 was also considered but the results did not change significantly.

Figure 5.1 Plot of unemployment rate for KLF and LALF survey for urban area

Description for Figure 5.1

Figure 5.2 Plot of residuals against estimated values for urban area

Description for Figure 5.2

Table 5.1
Quartile of the MSE performance of the small area estimates for the 229 areas
Table summary
This table displays the results of Quartile of the MSE performance of the small area estimates for the 229 areas. The information is grouped by MSE (appearing as row headers), 1 Q, Median , 3 Q and Mean (appearing as column headers).

MSE	1st Q	Median	3rd Q	Mean
KLF	0.000063	0.000121	0.0002395	0.0002476
LALF	0.0001123	0.000133	0.0001695	0.0001482
GLS 1	0.0000444	0.0000738	0.000121	0.0000893
GLS 2	0.0000405	0.0000543	0.0000721	0.0000575

Table 5.1 presents the performance of the small area estimates in terms of the MSE estimates. We considered four different estimators of ${\overline{X}}_h.$  KLF represents the result derived using only Korea Labor Force survey, LALF represents the result using only Local Area Labor Force survey, GLS 1 represents the result for combining both surveys KLF and LALF, and GLS 2 represents the result for combining KLF, LALF and the Census data. Table 5.1 shows that the GLS 2 method provides the smallest mean squared errors.

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