Model-based small area estimation with no samples within the areas, by benchmarking to marginal cross-sectional and time-series estimates

Figures & data

Table 1. Benchmarked monthly area estimates $w_{c,m}$. ${\mathit{S}}_{\mathit{m}}=\text{monthly}$ CPS State estimates, ${\mathit{A}}_{\mathit{c}}=\text{annual}$ ACS area estimates.

	Months				Year (y)
Areas(c)	1	2	…	12	ACS totals
1	$w_{1,1}$	$w_{1,2}$	…	$w_{1,12}$	$A_1$
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
C	$w_{C,1}$	$w_{C,2}$	…	$w_{C,12}$	$A_C$
CPS totals	$S_1$	$S_2$	…	$S_{12}$

$\sum _{c=1}^C{w}_{c,m}=S_m,m=1,\dots ,12,\sum _{m=1}^{12}{w}_{c,m}=A_c,c=1,\dots ,C$.

Figure 1. Annual ACS & CPS estimates, 2002–2016.

Figure 2. Annual ACS (y) and CPS(x) State shares of annual national totals.

Figure 3. Monthly State CPS CVs (before modelling) and annual ACS CVs, 2005–2016.

Figure 4. Standard errors of change in monthly CPS and annual ACS, 2005–2016.

Figure 5. Autocorrelations of model residuals with approximate 95% confidence intervals.

Table 2. Information about counties (sub-States) in the State of Arizona.

Total number of counties	15
Counties of size 20k ≤ Pop < 65k	6
Counties of size Pop ≥ 65k	9
Number of missing ACS estimates	14

Table 3. Model (6) parameter estimates, Arizona.

Coefficients, Unemployment/Pop16+				Coefficients, Employment/Pop16+
Coefficient	Estimate	t-value	Pr> |t|	Coefficient	Estimate	t-value	Pr> |t|
Intercept	0.02255	8.97	<.0001	Intercept	0.3300	8.74	<.0001
UI/Pop	2.5354	9.92	<.0001	QCEW/Pop	0.4451	4.32	<.0001
*Yr10	9.3 × 10^−3	15.16	<.0001	*Yr10	0.02979	1.96	<.0001
Variances, Unemployment/Pop				Variances, Employment/Pop
Variance	Estimate	z-value	Pr >| z|	Variance	Estimate	z-value	Pr > |z|
Intercept	5.5 × 10^−5	1.60	0.0549	Intercept	0.01105	1.32	0.0929
UI/Pop	0.6135	1.91	0.0282	QCEW/Pop	0.08421	1.27	0.1026

*Yr10 is a dummy variable controlling for the introduction of new ACS estimates in 2010.

Figure 6. Diagnostics based on Conditional Pearson Residuals (CPR).

Figure 7. Annual unemployment rates for Yuma where the ACS estimates and the model-based ACS predictions are adjusted to the annual model-based CPS.

Figure 8. Unemployment rate estimates for Yuma County as obtained by HB, LAUS, Denton LAUS (DLAUS) and benchmarked Denton LAUS (BMK DLAUS). 2005–2016.

Figure 9. Unemployment Rate estimates for Yuma county: Synthetic, Dsynthetic, BMK Dsynthetic and BMK DLAUS. 2005–2016.

Table 4. Average absolute percent change between input series and benchmarked estimates. Arizona, all counties. 2005–2016.

	Unemployment		Employment
Input Series	ACS Controls	Model-based ACS Controls	ACS Controls	Model-based ACS Controls
HB	44.05%	34.60%	8.97%	7.42%
DHB	4.58%	4.57%	0.90%	0.90%
LAUS	26.39%	16.92%	7.44%	5.68%
DLAUS	0.79%	0.76%	0.01%	0.01%
SYNTH	19.06%	0.31%	4.47%	0.08%
DSYNTH	0.05%	0.01%	0.00%	0.00%

Table 5. Model and Bootstrap (BS) Standard Errors (SE), and Coefficients of Variation (CV) for Arizona Unemployment by county, averaged over all monthly values. 10,000 Bootstrap samples. 2005–2016.

		SE BS					CV BS
Column/County	SE Model ${\hat{L}}_{cym}$	${\hat{L}}_{cym}$	${\hat{L}}_{cym}^{\ast }$	${\hat{L}}_{cym}^{\ast \ast }$	Synthetic, Columns (2/3)	CV Model ${\hat{L}}_{cym}$	${\hat{L}}_{cym}$	${\hat{L}}_{cym}^{\ast }$	${\hat{L}}_{cym}^{\ast \ast }$
1	2	3	4	5	6	7	8	9	10
1	217	197	264	264	1.10	0.07	0.06	0.10	0.10
3	419	381	442	443	1.10	0.09	0.08	0.10	0.10
5	451	411	500	499	1.10	0.08	0.07	0.10	0.10
7	182	165	193	194	1.10	0.08	0.08	0.10	0.10
9	120	119	119	120	1.01	0.11	0.11	0.12	0.12
11	32	31	32	32	1.04	0.11	0.10	0.12	0.12
12	76	68	77	78	1.11	0.09	0.08	0.11	0.11
13	12,691	11,919	8,195	8,198	1.06	0.10	0.10	0.08	0.08
15	704	651	815	816	1.08	0.08	0.07	0.10	0.10
17	344	313	423	422	1.10	0.07	0.06	0.10	0.10
19	3,269	3,019	3,477	3,478	1.08	0.08	0.07	0.09	0.09
21	1,152	1,120	1,404	1,404	1.03	0.07	0.07	0.10	0.10
23	146	154	166	166	0.95	0.09	0.09	0.12	0.12
25	753	742	924	923	1.02	0.07	0.07	0.10	0.10
27	649	629	1,255	1,254	1.03	0.04	0.04	0.09	0.09

Estimators: ${\hat{L}}_{cym}={\hat{R}}_{cym}^{b}Po{p}_{cym},{\hat{L}}_{cym}^{\ast }=\text{ratio}$ adjusted ${\hat{L}}_{cym},{\hat{L}}_{cym}^{\ast \ast }=\text{Calibrated}$ ${\hat{L}}_{cym}^{\ast }$

Model Standard Error =$Po{p}_{cym}\sqrt{Var\left({\hat{R}}_{cym}^b\right)}$ (computed for every BS sample); $Var\left({\hat{R}}_{cym}^b\right)=Var\left({\hat{\alpha }}_c^{\ast b}\right)+S{A}^2\left(x_{cym}\right)Var\left({\widehat{\text{β}}}_c^{\ast b}\right)+2SA\left(x_{cym}\right)Cov\left({\hat{\alpha }}_c^{\ast b},{\widehat{\text{β}}}_c^{\ast b}\right);$ ${\hat{\alpha }}_c^{\ast b}=\left({\hat{\alpha }}^b+{\hat{\alpha }}_c^b\right),{\widehat{\text{β}}}_c^{\ast b}=\left({\widehat{\text{β}}}^b+{\widehat{\text{β}}}_c^b\right)$.

Bootstrap (BS) Standard Errors:

BS ${\hat{L}}_{cym}=\sqrt{\left(1/B\right)\sum _{b=1}^B{\left({\hat{L}}_{cym}^b-{\overline{\hat{L}}}_{cym}^b\right)}^2}$, BS ${\hat{L}}_{cym}^{\ast }$= $\sqrt{\left(1/B\right)\sum _{b=1}^B{\left({\hat{L}}_{cym}^{\ast b}-{\overline{\hat{L}}}_{cym}^{\ast b}\right)}^2}$, BS ${\hat{L}}_{cym}^{\ast \ast }=\sqrt{\left(1/B\right)\sum _{b=1}^B{\left({\hat{L}}_{cym}^{\ast \ast b}-{\overline{\hat{L}}}_{cym}^{\ast \ast b}\right)}^2}$. See Appendix 4 for more details.

Figure A1. Predictions of annual ACS ratios (${\hat{R}}_{cy}$) and standard errors as obtained from state-space modelling and Proc Mixed. 2005–2016.