Single- and two-stage cross-sectional and time series benchmarking procedures for small area estimation

Abstract

This article is divided into two parts. In the first part, we review and study the properties of single-stage cross-sectional and time series benchmarking procedures that have been proposed in the literature in the context of small area estimation. We compare cross-sectional and time series benchmarking empirically, using data generated from a time series model which complies with the familiar Fay–Herriot model at any given time point. In the second part, we review cross-sectional methods proposed for benchmarking hierarchical small areas and develop a new two-stage benchmarking procedure for hierarchical time series models. The latter procedure is applied to monthly unemployment estimates in Census Divisions and States of the USA.

Appendices

Appendix A: Computation of $\tilde{\Sigma }_{tt}^d =E(\tilde{{\mathbf {e}}}_t^d {\tilde{{\mathbf {e}}}}{'}_t^d )$

The matrix $\tilde{\Sigma }_{tt}^d =\left[ {{\begin{array}{cc} {\Sigma _{tt}^d }&{} {{\mathbf {h}}_{tt}^d }\\ {{{\mathbf {h}}}{'}_{tt}^d } &{} {v_{tt}^d }\\ \end{array} }} \right]$ has as its main block the $S\times S$ diagonal V–C matrix of the State sampling errors, $\Sigma _{tt}^d =E({\mathbf {e}}_t^d {{\mathbf {e}}}{'}_t^d )=\hbox {diag}[\sigma _{ds,tt}^2 ]$, where ${\mathbf {e}}_t^d =(e_{d1,t} ,\ldots ,e_{dS,t})^{\prime }$ (the direct CPS sampling errors are independent between the States). The computation of the other elements of $\tilde{\Sigma }_{tt}^d$; ${\mathbf {h}}_{tt}^d =E({\mathbf {e}}_t^d r_{dt}^\mathrm{bmk} )$ and $v_{tt}^d =\hbox {Var}(r_{dt}^\mathrm{bmk} )$ requires revisiting the first-stage benchmarking.

Consider first the computation of $v_{tt}^d$. Denote $\varvec{l}_{dt}^\mathrm{bmk} =(\hat{\varvec{\alpha }}_{dt}^\mathrm{bmk} -\varvec{\alpha }_{dt} )$, such that $r_{dt}^\mathrm{bmk} ={{\mathbf {z}}}^{\prime }_{dt} \varvec{l}_{dt}^\mathrm{bmk}$ and $v_{tt}^d ={{\mathbf {z}}}^{\prime }_{dt} E(\varvec{l}_{dt}^\mathrm{bmk} \varvec{l}_{dt}^{\mathrm{bmk}^{\prime }}){\mathbf {z}}_{dt}$. Let $\varvec{l}_t^\mathrm{bmk} =(\tilde{\varvec{\alpha }}_t^\mathrm{bmk} -\varvec{\alpha }_t )=(\varvec{l}_{1t}^{\mathrm{bmk}{^\prime }} ,\ldots ,\varvec{l}_{Dt}^{\mathrm{bmk}^{\prime }})^{\prime }$ and $J_d$ be the indicator matrix of zeroes and ones satisfying, $\varvec{l}_{dt}^\mathrm{bmk} =J_d \varvec{l}_t^\mathrm{bmk}$. It follows that $v_{tt}^d ={{\mathbf {z}}}^{\prime }_{dt} J_d P_t^\mathrm{bmk} {J}^{\prime }_d {\mathbf {z}}_{dt}$, where $P_t^\mathrm{bmk} =E(\varvec{l}_t^\mathrm{bmk} \varvec{l}_t^{\mathrm{bmk}^{\prime }})$ is obtained recursively as defined below (15).

Next consider the computation of ${\mathbf {h}}_{tt}^d =E({\mathbf {e}}_t^d r_{dt}^\mathrm{bmk} )$. By Eq. (D.2) in P–T, the error $(\tilde{{\mathbf {\alpha }}}_t^\mathrm{bmk} -{\mathbf {\alpha }}_t )$ when predicting the division state vectors can be written as

$$\begin{aligned} \varvec{l}_t^\mathrm{bmk} =\left( \tilde{\varvec{\alpha }}_t^\mathrm{bmk} -\varvec{\alpha }_t\right) =G_t \tilde{T}\left( \tilde{\varvec{\alpha }}_{t-1}^\mathrm{bmk} -\tilde{\varvec{\alpha }}_{t-1}\right) -G_t \varvec{\eta }_t +K_t \tilde{{\mathbf {e}}}_t, \end{aligned}$$

(35)

with $G_t$ and $K_t$ defined in P–T and $\tilde{{\mathbf {e}}}_t =(e_{1t} ,\ldots ,e_{Dt} ,\sum \nolimits _{d=1}^D {b_{dt} e_{dt} } )^{\prime }$. Let $U_t =G_t \tilde{T}$ such that (35) can be written as $\varvec{l}_t^\mathrm{bmk} =U_t \varvec{l}_{t-1}^\mathrm{bmk} -G_t \varvec{\eta }_t +K_t \tilde{\varvec{e}}_t$. Repeated substitutions in the last equation yields,

$$\begin{aligned} \varvec{l}_t^\mathrm{bmk}&=A_{t,2} \varvec{l}_1^\mathrm{bmk} +B_{t,2} \tilde{{\mathbf {e}}}_2 +\cdots +B_{t,t-1} \tilde{{\mathbf {e}}}_{t-1}\nonumber \\&\quad +K_t \tilde{{\mathbf {e}}}_t -D_{t,2} \varvec{\eta }_2 -\cdots -D_{t,t-1} \varvec{\eta }_{t-1} -G_t \varvec{\eta }_t,\quad t=2,3,\ldots , \end{aligned}$$

(36)

where $A_{t,j} =U_t \times U_{t-1} \times \dots \times U_j ,\,\,j=2,\ldots ,t,\,\,B_{t,j} =A_{t,j+1} K_j ,\,\,D_{t,j} =A_{t,j+1} G_j$. Suppose that the GLS filter is initialized at time $t=1$ by $\tilde{T}\tilde{\varvec{\alpha }}_0^\mathrm{bmk}$, independently of the rest of the series. Then, $C_{1,0}^\mathrm{bmk} =E[(\tilde{T}\tilde{\varvec{\alpha }}_0^\mathrm{bmk} -\varvec{\alpha }_1 ){\tilde{{\mathbf {e}}}}{'}_{1,0} ]=0$ and by Eq. (D.1) in P–T, $\varvec{l}_1^\mathrm{bmk} =[I-P_{1\vert 0}^\mathrm{bmk} {\tilde{Z}}{'}_1 R_{1,0}^{-1} \tilde{Z}_1 ](\tilde{T}\tilde{\varvec{\alpha }}_0^\mathrm{bmk} -\tilde{\varvec{\alpha }}_1 )+P_{1\vert 0}^\mathrm{bmk} {\tilde{Z}}{'}_1 R_{1,0}^{-1} \tilde{{\mathbf {e}}}_1,$ where $R_{1,0}^\mathrm{bmk} =[\tilde{Z}_1 P_{1\vert 0}^\mathrm{bmk} {\tilde{Z}}^{\prime }_1 +\tilde{\Sigma }_{11,0} ]$. Denoting $K_{1\vert 0} =P_{1\vert 0}^\mathrm{bmk} {\tilde{Z}}^{\prime }_1 R_{1,0}^{-1},\,\,\,B_{t,1} =A_{t,2} K_{1\vert 0} ,\,\,\,D_{t,1} =A_{t,2} (I-K_{1\vert 0} \tilde{Z}_1 ),\,\,\,B_{t,t} =K_t ,\,\,\,D_{t,t} =G_t$ and ignoring the term $D_{t,1} \tilde{T}(\tilde{\varvec{\alpha }}_0^\mathrm{bmk} -\varvec{\alpha }_0 )$ which does not enter into any of the computations that follow, (independent of the rest of the series), Eq. (36) can be rewritten as

$$\begin{aligned} \varvec{l}_t^\mathrm{bmk} =\sum \limits _{j=1}^t {B_{t,j} } \tilde{{\mathbf {e}}}_j -\sum \limits _{j=1}^t {D_{t,j} } \varvec{\eta }_{j}. \end{aligned}$$

(37)

The relationship in (37) allows us to express the division prediction error $r_{dt}^\mathrm{bmk} ={{\mathbf {z}}}^{\prime }_{dt} J_d \varvec{l}_t^\mathrm{bmk}$ as a difference between linear functions of the division sampling errors and the state vector errors, which are independent of the sampling errors. We thus have,

$$\begin{aligned} {\mathbf {h}}_{tt}^d =E\left( {\mathbf {e}}_t^d r_{dt}^\mathrm{bmk} \right) =E\left[ {\mathbf {e}}_t^d {l}{'}_t^\mathrm{bmk} {J}{'}_d {\mathbf {z}}_{dt} )\right] =E\left( {\mathbf {e}}_t^d \sum \limits _{j=1}^t {{\tilde{{\mathbf {e}}}}{'}_j {B}^{\prime }_{t,j} } \right) {J}^{\prime }_d {\mathbf {z}}_{dt}. \end{aligned}$$

(38)

Now, ${\mathbf {e}}_t^d =(e_{d1,t},\ldots ,e_{dS,t})^{\prime }$, ${\tilde{{\mathbf {e}}}}{'}_j =(e_{1j} ,\ldots ,e_{Dj} ,\sum \nolimits _{k=1}^D {b_{kj} e_{kj} } )$ and $e_{dj} =\sum \nolimits _{s=1}^S {b_{ds,j} e_{ds,j} }$, implying

$$\begin{aligned} E({\mathbf {e}}_t^d {\tilde{{\mathbf {e}}}}{'}_j )\!=\!\left[ {\mathbf {0}}_{(1)},\ldots ,{\mathbf {0}}_{(d-1)} ,E\left( {\mathbf {e}}_t^d e_{dj} \right) ,{\mathbf {0}}_{(d+1)},\ldots ,{\mathbf {0}}_{(D)} ,b_{dj} E\left( {\mathbf {e}}_t^d e_{dj} \right) \right] \!=\!\tilde{H}_{t,j}^{d},\nonumber \\ \end{aligned}$$

(39)

where ${\mathbf {0}}_{(k)}$ is the null vector of length S in position (column) $k$, and

$$\begin{aligned} E({\mathbf {e}}_t^d e_{dj} )=E\left[ {\mathbf {e}}_t^d \left( \sum \limits _{s=1}^S {b_{ds,j} e_{ds,j} } \right) \right] =\left[ b_{d1,j} \sigma _{d1,tj}^2 ,\ldots ,b_{dS,j} \sigma _{dS,tj}^2\right] ^{\prime }. \end{aligned}$$

(40)

Substituting (40) in (39) and then in (38) gives the expression for the vector ${\mathbf {h}}_{tt}^d$,

$$\begin{aligned} {\mathbf {h}}_{tt}^d =E\left( {\mathbf {e}}_t^d r_{dt}^\mathrm{bmk} \right) =\sum \limits _{j=1}^t \left( \tilde{H}_{t,j}^d {B}{'}_{t,j}\right) {J}{'}_d {\mathbf {z}}_{dt}. \end{aligned}$$

(41)

Appendix B: Computation of $C_{dt}^\mathrm{bmk} =E[(\tilde{T}^d\tilde{\varvec{\alpha }}_{t-1}^{d,\mathrm{bmk}} -\varvec{\alpha }_t^d ){\tilde{{\mathbf {e}}}}{'}_t^d ]$

The computation of the covariance matrix $C_{dt}^\mathrm{bmk}$ is more involved since it requires computing the covariances between the division benchmark error and the State sampling errors. We first express the prediction error $(\tilde{T}^d\tilde{\varvec{\alpha }}_t^{d,\mathrm{bmk}} -\varvec{\alpha }_{t+1}^d)$ corresponding to time $t+1$ as a function of the sampling errors and the state vector errors, similarly to (37). Under the model, $\varvec{\alpha }_{t+1}^d =\tilde{T}^d\varvec{\alpha }_t^d +\varvec{\eta }_{t+1}^d$. Hence,

$$\begin{aligned} {\mathbf {m}}_t^d =\left( \tilde{T}^d \tilde{\varvec{\alpha }}_t^{d,\mathrm{bmk}} -{\varvec{\alpha }}_{t+1}^d \right) =\tilde{T}^d\left( \tilde{\varvec{\alpha }}_t^{d,\mathrm{bmk}} -\varvec{\alpha }_t^d\right) -\varvec{\eta }_{t+1}^d =\tilde{T}^d \varvec{l}_t^{d,\mathrm{bmk}} -\varvec{\eta }_{t+1}^d,\quad \quad \end{aligned}$$

(42)

where $\varvec{l}_t^{d,\mathrm{bmk}} =(\varvec{l}_{t1}^{{d,\mathrm{bmk}}{^\prime }} ,\varvec{l}_{t2}^{{d,\mathrm{bmk}}^{\prime }} ,\ldots ,\varvec{l}_{tS}^{{d,\mathrm{bmk}}^{\prime }})^{\prime }$ is the vector of the benchmark errors for the state vectors in the division. Now, using a similar decomposition to Eq. (D.2) in P–T, we have,

$$\begin{aligned} {\mathbf {m}}_t^d =\tilde{T}^d\left( G_t^d {\mathbf {m}}_{t-1}^d +K_t^d \tilde{{\mathbf {e}}}_t^d\right) -\varvec{\eta }_{t+1}^d =W_t^d {\mathbf {m}}_{t-1}^d +\tilde{T}^d K_t^d \tilde{{\mathbf {e}}}_t^d -\varvec{\eta }_{t+1}^d , \end{aligned}$$

(43)

where $W_t^d =\tilde{T}^dG_t^d$, $\tilde{{\mathbf {e}}}_t^d =(e_{d1,t} ,\ldots ,e_{dS,t} ,r_{d,t}^\mathrm{bmk})^{\prime }=({{\mathbf {e}}}{'}_t^d ,r_{d,t}^\mathrm{bmk} )^{\prime }$ and $r_{d,t}^\mathrm{bmk} ={{\mathbf {z}}}^{\prime }_{dt} J_d \varvec{l}_t^\mathrm{bmk}$ is the division benchmark error at time $t$ with $\varvec{l}_t^\mathrm{bmk}$ defined by (37). The matrices $G_t^d$ and $K_t^d$ are defined similarly to $G_t$ and $K_t$ in Eq. (D.2) of P–T, but referring now to the States in a given division instead of to the divisions. By repeated substitutions in (43) and ignoring the term $\tilde{T}^d(\tilde{\varvec{\alpha } }_0^{d,\mathrm{bmk}} -\varvec{\alpha }_0^d )$ for time $t=0$ which drops out in each of the computations that follow (independent of the rest of the series), we obtain

$$\begin{aligned} {\mathbf {m}}_t^d =\sum \limits _{k=1}^t {B_{t,k}^d } \tilde{{\mathbf {e}}}_k^d -\sum \limits _{k=1}^{t+1} {D_{t,k}^d } \varvec{\eta }_k^d , \end{aligned}$$

(44)

where $D_{t,k}^d =W_t^d \times W_{t-1}^d \times \ldots \times W_k^d ,\,\,k=2,\ldots ,t,\,\,D_{t,t+1}^d =I_{Sq} ,\,\,B_{t,k}^d =D_{t,k+1}^d \tilde{T}^dK_k^d ,\,\,k=1,\ldots ,t-1$, $I_{Sq}$ is the identity matrix of order $Sq$ ($q=\dim (\eta _{ds} ))$ and $B_{t,t}^d =\tilde{T}^dK_t^d$.

At time $t+1$ we need to compute $C_{d,t+1}^\mathrm{bmk} =E[(\tilde{T}^d \tilde{\varvec{\alpha }}_t^{d,\mathrm{bmk}} -\varvec{\alpha }_{t+1}^d ){\tilde{{\mathbf {e}}}}{'}_{t+1}^d ]=E({\mathbf {m}}_t^d \tilde{{{\mathbf {e}}}{'}}_{t+1}^d )$. By (44)

$$\begin{aligned} C_{d,t+1}^\mathrm{bmk}&= \sum \limits _{k=1}^t {B_{t,k}^d } E\left( \tilde{{\mathbf {e}}}_k^d {\tilde{{\mathbf {e}}}}{'}_{t+1}^d\right) -\sum \limits _{k=1}^{t+1} {D_{t,k}^d } E\left( \varvec{\eta }_k^d {\tilde{{\mathbf {e}}}}{'}_{t+1}^d\right) \nonumber \\&= \sum \limits _{k=1}^t {B_{t,k}^d } E\left[ \left( {{\mathbf {e}}}{'}_k^d ,r_{d,k}^\mathrm{bmk}\right) ^{\prime }\times \left( {{\mathbf {e}}}{'}_{t+1}^d ,r_{d,t+1}^\mathrm{bmk} \right) \right] \nonumber \\&-\sum \limits _{k=1}^{t+1} {D_{t,k}^d } E\left[ \varvec{\eta }_k^d \times \left( {{\mathbf {e}}}{'}_{t+1}^d ,r_{d,t+1}^\mathrm{bmk}\right) \right] . \end{aligned}$$

(45)

Next, we evaluate each of the expectations in (45). Define

$$\begin{aligned} \tilde{\Sigma }_{k,t+1}^d =E\left( \tilde{{\mathbf {e}}}_k^d {\tilde{{\mathbf {e}}}}{'}_{t+1}^d \right) =\left[ {{\begin{array}{c@{\quad }c} {\Sigma _{k,t+1}^d }&{} {{\mathbf {h}}_{t+1,k}^d }\\ {{{\mathbf {h}}}{'}_{k,t+1}^d }&{} {v_{k,t+1} }\\ \end{array} }} \right] . \end{aligned}$$

(46)

By (29),

$$\begin{aligned} \Sigma _{k,t+1}^d =E\left( {\mathbf {e}}_k^d {{\mathbf {e}}}{'}_{t+1}^d \right) =\hbox {Diag}\left[ \sigma _{d1,k,t+1}^2,\ldots ,\sigma _{dS,k,t+1}^2 \right] . \end{aligned}$$

(47)

By (37) and (39) and noting that $E({\mathbf {e}}_j^d {\varvec{\eta }}^{\prime }_k )=0$ for all $(j,k)$,

$$\begin{aligned} {\mathbf {h}}_{t+1,k}^d&= E\left( {\mathbf {e}}_k^d r_{d,t+1}^\mathrm{bmk} \right) =E\left[ {\mathbf {e}}_k^d \left( \sum \limits _{j=1}^{t+1} {{\tilde{{\mathbf {e}}}}'_j {B}'_{t+1,j} } \right) \right] {J}'_d {\mathbf {z}}_{d,t+1} =\sum \limits _{j=1}^{t+1} {\tilde{H}_{k,j}^d {B}^{\prime }_{t+1,j} } {J}^{\prime }_d {\mathbf {z}}_{d,t+1},\nonumber \\ \end{aligned}$$

(48)

$$\begin{aligned} {{\mathbf {h}}}{'}_{k,t+1}^d&= E\left( r_{dk}^\mathrm{bmk} {{\mathbf {e}}}{'}_{t+1}^d\right) ={{\mathbf {z}}}'_{dk} J_d \sum \limits _{j=1}^k {B_{k,j} } E\left( \tilde{{\mathbf {e}}}_j {{\mathbf {e}}}{'}_{t+1}^d\right) ={{\mathbf {z}}}{'}_{dk} J_d \sum \limits _{j=1}^k {B_{k,j} \tilde{H}_{j,t+1}^d }. \end{aligned}$$

(49)

Remark 18

${{\mathbf {h}}}{'}_{k,t+1}^d \ne {{\mathbf {h}}}{'}_{t+1,k}^d$.

Recalling that the error vectors $\tilde{{\mathbf {e}}}_j =(e_{1j} ,\ldots ,e_{Dj} ,\sum \nolimits _{d=1}^D {b_{dj} e_{dj} })^{\prime }$ are only functions of the division sampling errors and thus independent of the state error vectors $\{\varvec{\eta }_l\}$, and that under the model $E(\varvec{\eta }_j {\varvec{\eta }}^{\prime }_j )=Q$; $E(\varvec{\eta }_j {\varvec{\eta }}^{\prime }_{\,l} )=0$, $j\ne l$, and defining $\tilde{\Sigma }_{i,j} =E(\tilde{{\mathbf {e}}}_i {\tilde{{\mathbf {e}}}}{'}_j )$,

$$\begin{aligned} v_{k,t+1}&=E\left( r_{dk}^\mathrm{bmk} r_{d,t+1}^\mathrm{bmk}\right) ={{\mathbf {z}}}{'}_{dk} J_d E\left[ \left( \sum \limits _{j=1}^k {B_{k,j}} \tilde{{\mathbf {e}}}_j-\sum \limits _{j=1}^k {D_{k,j} } \varvec{\eta }_j \right) \right. \nonumber \\&\quad \qquad \qquad \qquad \qquad \qquad \left. \times \left( \sum \limits _{l=1}^{t+1} {\tilde{{{\mathbf {e}}}{'}}_l {B}'_{t+1,l} } -\sum \limits _{l=1}^{t+1} {{\varvec{\eta }}{'}_{\,l} {D}'_{t+1,l} }\right) \right] {J}'_d {\mathbf {z}}_{d,t+1}\nonumber \\&={{\mathbf {z}}}^{\prime }_{dk} J_d \left[ \sum \limits _{j=1}^k {\sum \limits _{l=1}^{t+1} {B_{k,j} E\left( \tilde{{\mathbf {e}}}_j \tilde{{{\mathbf {e}}}}{'}_l \right) {B}'_{t+1,l} } } +\sum \limits _{j=1}^k {\sum \limits _{l=1}^{t+1} {D_{k,j} E\left( \varvec{\eta }_j {\varvec{\eta }}^{\prime }_{\,l}\right) {D}^{\prime }_{t+1,l} } } \right] {J}^{\prime }_d {\mathbf {z}}_{d,t+1} \nonumber \\&={{\mathbf {z}}}^{\prime }_{dk} J_d \left[ \sum \limits _{j=1}^k {\sum \limits _{l=1}^{t+1} {B_{k,j} \tilde{\Sigma }_{j,l} {B}^{\prime }_{t+1,l} } } +\sum \limits _{j=1}^k {D_{k,j} Q{D}^{\prime }_{t+1,j} } \right] {J}^{\prime }_d {\mathbf {z}}_{d,t+1}. \end{aligned}$$

(50)

Remark 19

Equation (50) refers to the first step of the benchmarking process.

Finally, by (37) and noting that $E(\varvec{\eta }_k^d {\varvec{\eta } }^{\prime }_j )=0$ for $j\ne k$,

$$\begin{aligned} E\left( \varvec{\eta }_k^d {\tilde{{\mathbf {e}}}}{'}_{t+1}^d\right)&= E\left[ \varvec{\eta }_k^d \left( {{\mathbf {e}}}{'}_{t+1}^d ,r_{d,t+1}^\mathrm{bmk} \right) \right] =\left[ 0_{Sq\times S} ,E\left( {\mathbf {\eta }}_k^d r_{d,t+1}^\mathrm{bmk} \right) \right] \nonumber \\&= \left[ 0_{Sq\times S} ,-E\left( \varvec{\eta }_k^d {\varvec{\eta }}^{\prime }_k \right) {D}^{\prime }_{t+1,k} {J}^{\prime }_d {\mathbf {z}}_{d,t+1} \right] , \end{aligned}$$

(51)

where $0_{Sq\times S}$ is the null matrix of dimension $Sq\times S$. We assume throughout that $\varvec{\alpha }_{dk} = \sum \nolimits _{s=1}^S {b_{ds,k} \varvec{\alpha }_{ds,k}}$, implying $\varvec{\eta }_k^d =J_d \varvec{\eta }_k = \sum \nolimits _{s=1}^S {b_{ds,k} \varvec{\eta }_{ds,k} }$ and hence,

(52)

where is the matrix of dimension $Sq\times Dq$ with $[b_{d1,k} Q_{d1,k},\ldots ,b_{dS,k} Q_{dSk}]^{\prime }$ in columns $(d-1)q+1,\ldots ,dq$ and zeroes elsewhere; $Q_{ds,k} =E(\varvec{\eta }_{ds,k} {\varvec{\eta }}^{\prime }_{ds,k} )$ (Eq. 29).

Substituting (47)–(50) in (46), and then (46) and (52) in (45) completes the computation of the covariance matrix $C_{d,t+1}^\mathrm{bmk}$.

Appendix C: Computation of $P_t^{d,\mathrm{bmk}} =E[(\hat{\varvec{\alpha }}_t^{d,\mathrm{bmk}} -\varvec{\alpha }_t^d )(\hat{\varvec{\alpha }}_t^{d,\mathrm{bmk}} -\varvec{\alpha }_t^d )^{\prime }]$

The computation of the true V–C matrix of the benchmarked state prediction errors follows the same steps as the computation of the V–C matrix $P_t^\mathrm{bmk}$ in the first-stage benchmarking (Appendix in P–T). First, rewrite the benchmarked predictor (33) as,

$$\begin{aligned} \tilde{\varvec{\alpha }}_t^{d,\mathrm{bmk}}&= \left[ I-\left( P_{t\vert t-1}^{\,d,\mathrm{bmk}} \tilde{{Z}'}_t^d -C_{dt,0}^\mathrm{bmk} \right) R_{dt}^{-1} \tilde{Z}_t^d \right] \tilde{T}_d \tilde{\varvec{\alpha }}_{t-1}^{d,\mathrm{bmk}}\nonumber \\&+\left( P_{t\vert t-1}^{\,d,\mathrm{bmk}} \tilde{{Z}'}_t^d -C_{dt,0}^\mathrm{bmk} \right) R_{dt}^{-1} \tilde{{\mathbf {y}}}_t^d =G_t^d \tilde{T}_d \tilde{\varvec{\alpha }}_{t-1}^{d,\mathrm{bmk}} +K_t^d \tilde{{\mathbf {y}}}_t^d. \end{aligned}$$

(53)

Next, substitute $\tilde{{\mathbf {y}}}_t^d =\tilde{Z}_t^d \varvec{\alpha }_t^d +\tilde{{\mathbf {e}}}_t^d$ and decompose $\varvec{\alpha }_t^d =G_t^d \varvec{\alpha }_t^d +K_t^d Z_t^d \varvec{\alpha }_t^d$, implying

$$\begin{aligned} \tilde{\varvec{\alpha }}_t^{d,\mathrm{bmk}} -\varvec{\alpha }_t^d =G_t^d \left( \tilde{T}_d \tilde{\varvec{\alpha } }_{t-1}^{d,\mathrm{bmk}} -\varvec{\alpha }_t^d\right) +K_t^d \tilde{{\mathbf {e}}}_t^{d}. \end{aligned}$$

(54)

The expression for $P_t^{d,\mathrm{bmk}}$ in (34) follows straightforwardly.

Appendix D: Derivation of Eq. (24)

In what follows we drop for convenience the area index $i$ from the notation. Let $\beta _t =T\beta _{t-1} +\eta _t$, where $\eta _t$ is white noise. Repeated substitutions imply

$$\begin{aligned} \beta _t =T^t\beta _0 +\sum \limits _{l=1}^t {T^{t-l}\eta _l }. \end{aligned}$$

(55)

For the model defined by (23b)–(23c), $\beta _t =({\beta }{'}_t^{(1)} ,{\beta }{'}_t^{(2)})^{\prime }$, where $\beta _t^{(1)} =(L_t ,R_t )^{\prime }$ and $\beta _t^{(2)} =(S_{1,t} ,S_{1,t}^*,\ldots ,S_{5,t} ,S_{5,t}^*,S_{6,t} )^{\prime }$. Accordingly, let $\eta _t =({\eta }{'}_t^{(1)} ,{\eta }{'}_t^{(2)} )^{\prime }$, where $\eta _t^{(1)} =(\eta _{_{Lt}} ,\eta _{_{Rt}})^{\prime }$, $\eta _t^{(2)} =(\eta _{_{1t}} ,\eta _{_{1t}}^*,\ldots ,\eta _{_{5t}} ,\eta _{_{5t}}^*,\eta _{_{6t}} )^{\prime }$. Define $h=(1,0,1,0,\ldots 1,0,1)^{\prime }$ such that by (23c) $S_t ={h}^{\prime }\beta _t^{(2)}$. Then, by (23b) the population value at time $t$ is

$$\begin{aligned} Y_t =x_t \beta _1 +L_t +S_t =x_t \beta _1 +(1,0)\beta _t^{(1)} +{h}^{\prime }\beta _t^{(2)}. \end{aligned}$$

(56)

Under the model (23b)–(23c), $T=\left[ {{\begin{array}{c@{\quad }c} {T_{(1)} }&{} 0 \\ 0&{} {T_{(2)} } \\ \end{array} }} \right]$, where $T_{^{(1)}} =\left[ {{\begin{array}{c@{\quad }c} 1&{} 1 \\ 0&{} 1\\ \end{array} }} \right]$ and

$$\begin{aligned} T_{^{(2)}} =\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} {\mathrm{\cos }\frac{\pi }{6}} &{} {\mathrm{sin}\frac{\pi }{6}} &{} 0 &{} 0 &{} 0 &{} 0 \\ {-\mathrm{sin}\frac{\pi }{6}} &{} {\mathrm{cos}\frac{\pi }{6}} &{} &{} &{} &{} 0 \\ 0 &{} &{} 0 &{} &{} &{} 0 \\ 0 &{} &{} &{} {\mathrm{cos}\frac{5\pi }{6}} &{} {\mathrm{sin}\frac{5\pi }{6}} &{} 0 \\ 0 &{} &{} &{} {-\mathrm{sin}\frac{5\pi }{6}} &{} {\mathrm{cos}\frac{5\pi }{6}} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\mathrm{cos}\pi } \\ \end{array} }} \right] . \end{aligned}$$

At time $t, T^t=\left[ {{\begin{array}{c@{\quad }c} {T_{(1)}^t } &{} 0 \\ 0 \ &{} {T_{(2)}^t }\\ \end{array} }} \right]$ where $T_{(1)}^t =\left[ {{\begin{array}{c@{\quad }c} 1 &{} 1 \\ 0 &{} 1 \\ \end{array} }} \right] ^t=\left[ {{\begin{array}{c@{\quad }c} 1 &{} t \\ 0 &{} 1 \\ \end{array} }} \right]$ and $T_{(2)}^t$ is obtained from $T_{(2)}$ by replacing every angle $\frac{j\pi }{6}$ in $T_{(2)}$ by $\frac{j\pi t}{6}$, $j=1,\ldots ,6$.

It follows that $T_{(1)}^t \beta _0^{(1)} =(L_0 +tR_0 ,R_0 )^{\prime }$, and by familiar trigonometric equalities ${h}^{\prime }\beta _t^{(2)} ={h}^{\prime }T_{(2)}^t \beta _0^{(2)} =(\cos \frac{\pi }{6}t,\sin \frac{\pi }{6}t,\ldots ,\cos \pi t)\beta _0^{(2)} =\sum \nolimits _{j=1}^6 {S_{j0} \cos \frac{\pi j}{6}t} +\sum \nolimits _{j=1}^5 {S_{j0}^*\sin \frac{\pi j}{6}t}.$

Similar computations imply,

$$\begin{aligned} T_{(1)}^{t-l} \eta _l^{(1)}&= (\eta _{_{Ll}} +(t-l)\eta _{_{lR}} ,\eta _{_{lR}} )^{\prime };\\{h}^{\prime }T_{(2)}^{t-l} \eta _l^{(2)}&= \sum \limits _{l=1}^6 {\eta _{_{l,j}} \cos \frac{\pi l}{6}(t} -l)+\sum \limits _{l=1}^5 {\eta _{_{l,j}}^*\sin \frac{\pi l}{6}(t-l)}. \end{aligned}$$

By (55) and (56)

$$\begin{aligned} Y_t -x_t \beta _1&= (1,0)\beta _t^{(1)} +{h}'\beta _t^{(2)} =L_0 +tR_0 +R_0 +\sum \limits _{j=1}^6 {S_{j0} \cos \frac{\pi j}{6}t} \nonumber \\&+\sum \limits _{j=1}^5 {S_{j0}^*\sin \frac{\pi j}{6}t}+\sum \limits _{l=1}^t {\eta _{_{Ll}} } +\sum \limits _{l=1}^t {(t-l)\eta _{_{Rl}} } +\sum \limits _{l=1}^t {\eta _{_{Rl}} } \nonumber \\&+\sum \limits _{l=1}^t {\left[ \sum \limits _{j=1}^6 {\eta _{jl} \cos \frac{\pi j}{6}(t} -l)+\sum \limits _{j=1}^5 {\eta _{jl}^*\sin \frac{\pi j}{6}(t-l)}\right] }. \end{aligned}$$

(57)

Equation (24) follows from (57).

Appendix E: Census Divisions and States in the USA

Census divisions	States
New England	Connecticut, Maine, Massachusetts, New Hampshire, Rhode Island, Vermont
Middle Atlantic	New Jersey, New York, Pennsylvania
East North Central	Illinois, Indiana, Michigan, Ohio, Wisconsin
West North Central	Iowa, Kansas, Minnesota, Missouri, Nebraska, North Dakota, South Dakota
South Atlantic	Delaware, District of Columbia, Florida, Georgia, Maryland, North Carolina, South Carolina, Virginia, West Virginia
East South Central	Alabama, Kentucky, Mississippi, Tennessee
West South Central	Arkansas, Louisiana, Oklahoma, Texas
Mountain	Arizona, Colorado, Idaho, Montana, Nevada, New Mexico, Utah, Wyoming
Pacific	Alaska, California, Hawaii, Oregon, Washington