5. Aligned composite estimators for growth rates and totals

Paul Knottnerus

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So far we only looked at growth rates because in practice the estimate ${\widehat{X}}_{SRS}$ for the turnover of 12 months ago can be considered more or less as fixed (i.e., can not be changed anymore). When $X$ refers to the total turnover in month $(t-1),$ it is likely that the figures for the preceding month can still be improved and modified. In such a situation the initial estimate ${\widehat{X}}_{SRS}$ might be revised as well.

Before examining a multivariate composite estimator for growth rates and totals, we first look at a multivariate composite estimator for the parameter of absolute change and the corresponding population means or totals; also see Example 4.2. Define the initial vector estimator ${\widehat{\theta }}_0$ by ${\widehat{\theta }}_0={\left({\widehat{\overline{D}}}_{OLP},{\overline{y}}_{23},{\overline{x}}_{12}\right)}^{\prime }.$ Denote the underlying parameter vector to be estimated by $\theta ={\left({\theta }_1,{\theta }_2,{\theta }_3\right)}^{\prime }.$ Let $V_0$ denote the covariance matrix of ${\widehat{\theta }}_0.$ In terms of $\theta$ the problem is now to find an aligned composite (AC) estimator ${\widehat{\theta }}_{AC}$ with elements satisfying the prior restriction ${\theta }_1-{\theta }_2+{\theta }_3=0$ or, equivalently, $\overline{D}-\overline{Y}+\overline{X}=0$ or $D-Y+X=0.$ Although there is one restriction in this situation, we treat in this section the somewhat more general case with $m$ restrictions $\left(1\le m\le 3\right).$ When the prior restrictions are of the linear form $c-R\theta =0$ where $R$ is a $m\times 3$ matrix of rank $m$ $(m\le 3),$ the optimal unbiased composite estimator for $\theta$ is equal to the general restriction (GR) estimator

$$\begin{array}{ccc}{\widehat{\theta }}_{GR}& ={\widehat{\theta }}_0+K\left(c-R{\widehat{\theta }}_0\right)\hfill & \\ K& =V_0{R}^{\prime }{\left(R{V}_0{R}^{\prime }\right)}^{-1}\hfill & \hfill (5.1)\\ V_{GR}& \equiv \mathrm{cov}\left({\widehat{\theta }}_{GR}\right)=\left(I_3-KR\right)V_0,\hfill & \hfill (5.2)\end{array}$$

where $I_3$ stands for the $3\times 3$ identity matrix. The estimator ${\widehat{\theta }}_{GR}$ is optimal in the sense that when ${\widehat{\theta }}_0$ follows a multivariate normal distribution $N\left(\theta ,V_0\right),$ the likelihood of ${\widehat{\theta }}_0$ attains its maximum, under the constraint $c-R\theta =0,$ for ${\theta }_{\max }={\widehat{\theta }}_{GR}.$ Moreover, given the form ${\widehat{\theta }}_K={\widehat{\theta }}_0+K\left(c-R{\widehat{\theta }}_0\right),$ it can be shown that minimizing $\text{tr}\left\{\mathrm{cov}\left({\widehat{\theta }}_K\right)\right\}$ with respect to the $3\times m$ matrix $K$ leads to (5.2). Recall that this means that for any other matrix $K$ the corresponding covariance matrix $\mathrm{cov}\left({\widehat{\theta }}_K\right)$ exceeds $V_{GR}$ by a positive semidefinite matrix; see Magnus and Neudecker (1988, pages 255-256). For further details on the GR estimator, see Knottnerus (2003, pages 328-332). To illustrate how (5.1) and (5.2) can be used for obtaining an aligned composite estimator ${\widehat{\theta }}_{AC},$ consider the following example dealing with the estimation of two population means and their difference.

Example 5.1. We use the same data as in Examples 2.1 and 4.2. The initial vector ${\widehat{\theta }}_0={\left({\widehat{\overline{D}}}_{OLP},{\overline{y}}_{23},{\overline{x}}_{12}\right)}^{\prime }$ is given by ${\left(4.89,97.19,89.84\right)}^{\prime }$ . These estimates do not satisfy the restriction ${\theta }_1-{\theta }_2+{\theta }_3=0;$ note that $R=(1,-1,1)$ and $c=0.$ Most elements of $V_0$ have already been discussed. Similar to Example 4.2, for element $\mathrm{cov}\left({\widehat{\overline{D}}}_{OLP},{\overline{y}}_{23}\right)$ we get

$$\begin{array}{ll}\mathrm{cov}\left({\widehat{\overline{D}}}_{OLP},{\overline{y}}_{23}\right)& =\mathrm{cov}\left({\overline{y}}_2-{\overline{x}}_2,{\overline{y}}_{23}\right)\hfill \\ & =\mathrm{var}\left({\overline{y}}_{23}\right)-\mathrm{cov}\left({\overline{x}}_{23},{\overline{y}}_{23}\right).(5.3)\hfill \end{array}$$

Each term in (5.3) can be estimated from $s_2$ as described before. The other covariances in $V_0$ have a similar form and can be estimated in the same manner. The variance estimates for ${\widehat{\overline{D}}}_{OLP},{\overline{y}}_{23}$ and ${\overline{x}}_{12}$ are 13.12, 38.79 and 22.92, respectively. Next, applying (5.1) and (5.2) with $K$ replaced by $\widehat{K}={\widehat{V}}_0{R}^{\prime }{\left(R{\widehat{V}}_0{R}^{\prime }\right)}^{-1}\text{,}$ we obtain the following aligned composite AC estimates

$${\widehat{\overline{D}}}_{AC}=5.40(12.37),{\widehat{\overline{Y}}}_{AC}=96.28(36.32),\text{and}{\widehat{\overline{X}}}_{AC}=90.88(19.75).$$

Between parentheses the variances are mentioned.

Now three remarks are in order. Firstly, ${\widehat{\overline{D}}}_{COM}$ discussed in the preceding section can also be derived from (5.1) and (5.2) by choosing ${\widehat{\theta }}_0={\left({\widehat{\overline{D}}}_{STN},{\widehat{\overline{D}}}_{OLP}\right)}^{\prime }$ with prior restriction ${\theta }_1-{\theta }_2=0.$ Secondly, by construction, the estimator ${\widehat{\overline{D}}}_{AC}$ is equal to estimator ${\widehat{\overline{D}}}_{COM}$ and, consequently, they have the same variance. Thirdly, were $K$ known, then the AC estimates estimator would be unbiased. But because $K$ is to be replaced by $\widehat{K},$ the AC estimates estimator ${\widehat{\theta }}_{AC}$ is only asymptotically unbiased. The same remark applies to the estimator $\left(I_3-\widehat{K}R\right){\widehat{V}}_0$ of $\mathrm{cov}({\widehat{\theta }}_{AC}).$ Similar to ${\widehat{\theta }}_{COM}$ described in the preceding section, the bias of ${\widehat{\theta }}_{AC}$ is of order $O(1/n_2);$ for the relationship between ${\widehat{\theta }}_{AC}$ and the regression estimator, see Appendix A.3.

In case of $m$ nonlinear restrictions, say $c-R\left(\theta \right)=0,$ a first-order Taylor series approximation around $\theta ={\widehat{\theta }}_0$ yields $c-R\left({\widehat{\theta }}_0\right)-D_R\left({\widehat{\theta }}_0\right)\left(\theta -{\widehat{\theta }}_0\right)=0$ or, equivalently,

$$c\left({\widehat{\theta }}_0\right)-D_R\left({\widehat{\theta }}_0\right)\theta =0,\text{ where }c\left({\widehat{\theta }}_0\right)=c-R\left({\widehat{\theta }}_0\right)+D_R\left({\widehat{\theta }}_0\right){\widehat{\theta }}_0.(5.4)$$

$D_R\left(\theta \right)$ stands for the $m\times 3$ matrix of partial derivatives of $R(\theta )$ $\left(\text{i}\text{.e}\text{., }{D}_R\left(\theta \right)=\partial R\left(\theta \right)/\partial {\theta }^{\prime }\right).$ Subsequently, an iterative procedure can be carried out by repeatedly applying (5.1) and (5.2) to the updated linearized versions of the nonlinear restrictions $c-R\left(\theta \right)=0.$ This yields

$$\begin{array}{cc}\hfill {\widehat{\theta }}_h& ={\widehat{\theta }}_0+K_h{\widehat{e}}_h;\hfill \\ \hfill {\widehat{e}}_h& =c_h-D_h{\widehat{\theta }}_0;\hfill \\ \hfill K_h& =V_0{D^{\prime }}_h{\left(D_h{V}_0{D^{\prime }}_h\right)}^{-1};\hfill \\ \hfill \mathrm{cov}({\widehat{\theta }}_h)& =\left(I_3-K_h{D}_h\right)V_0;\hfill \\ \hfill D_h& =D_R\left({\widehat{\theta }}_{h-1}\right);\hfill \\ \hfill c_h& =c-R\left({\widehat{\theta }}_{h-1}\right)+D_h{\widehat{\theta }}_{h-1}\left(h=1,2,\mathrm{...}\right)\text{.}\hfill \end{array}\}(5.5)$$

For further details, see Appendix A.2 and Knottnerus (2003, pages 351-354). Note that the first equation can be seen as an update of ${\widehat{\theta }}_0$ rather than of ${\widehat{\theta }}_{h-1}.$ This is an important difference with the celebrated Kalman equations; see Kalman (1960). In the present context, the vectors ${\widehat{\theta }}_{h-1}$ are only used in a numerical procedure for finding new (better) Taylor series approximations of the nonlinear restrictions $c-R(\theta )=0$ around $\theta ={\widehat{\theta }}_{h-1}$ $(h=1,2,\mathrm{...})$ until convergence is reached. Furthermore, note that ${\widehat{e}}_h$ can be seen as a $m-$ vector of restriction errors when substituting $\theta ={\widehat{\theta }}_0$ into the linearized restrictions around $\theta ={\widehat{\theta }}_{h-1}.$ To illustrate the use of the Kalman-like equations in (5.5) for deriving aligned composite estimators for growth rates and totals, consider the following example.

Example 5.2. We use the same data as in Example 4.1. The initial vector ${\widehat{\theta }}_0$ is now defined by ${\widehat{\theta }}_0={\left({\widehat{G}}_{OLP},{\overline{y}}_{23},{\overline{x}}_{12}\right)}^{\prime }$ and is given by ${\left(1.050,97.191,89.840\right)}^{\prime }.$ These estimates do not satisfy the (nonlinear) prior restriction ${\theta }_2-{\theta }_1{\theta }_3=0$ $(m=1).$ All elements of $V_0$ and their estimation have already been discussed. For the $(h+1)\text{-th}$ recursion $R\left({\widehat{\theta }}_h\right)$ and the $1\times 3$ matrix $D_{h+1}$ are given by

$$\begin{array}{cc}R\left({\widehat{\theta }}_h\right)& =\left({\widehat{\theta }}_{h2}-{\widehat{\theta }}_{h1}{\widehat{\theta }}_{h3}\right)\hfill \\ D_{h+1}& =\left(\begin{array}{ccc}-{\widehat{\theta }}_{h3}& 1& -{\widehat{\theta }}_{h1}\end{array}\right),\hfill \end{array}$$

respectively; ${\widehat{\theta }}_{hk}$ is the $k\text{-th}$ element of vector ${\widehat{\theta }}_h$ $(1\le k\le 3).$ Recall $V_0$ and ${\widehat{V}}_0$ remain unchanged for all recursions. The first recursion from (5.5) yields

$${\widehat{\theta }}_1={\left(1.0544,95.945,91.000\right)}^{\prime }.$$

The (nonlinear) restriction is almost satisfied, that is, $R\left({\widehat{\theta }}_1\right)=-\mathrm{0.005.}$ The second recursion yields the following aligned composite (AC) estimates

$${\widehat{G}}_{AC}=1.0544(0.00130),{\widehat{\overline{Y}}}_{AC}=95.947(35.55),\text{and}{\widehat{\overline{X}}}_{AC}=90.998(19.85).$$

Between parentheses the variances are mentioned. The (absolute) error of the second restriction further decreased, that is, $R\left({\widehat{\theta }}_2\right)=-0.001$ and we stopped the recursions. Due to the nonlinearity of the restriction, the estimates of ${\widehat{G}}_{AC}$ and its variance are slightly different from those of ${\widehat{G}}_{COM}$ and its variance in Example 4.1.

It is noteworthy that in Example 5.2 ${\widehat{G}}_{AC}$ is not much different of ${\widehat{G}}_{OLP}$ (=1.050). A related method for estimating totals is the so-called matched pair (MP) method; see Smith et al. (2003, page 269-271). The original MP method is purely based on ${\widehat{G}}_{OLP}$ (in our notation) between months $t$ and $t-1$ and used by ONS for estimating the monthly retail sales index. In a simulation study the authors found that the MP method gives a good performance for the short-term growth rates but for terms of more than 15 months the performance was worsening with respect to the bias. The bias could be corrected by benchmarking to growth rates on a regular basis. Another drawback of the MP method seems to be that a formula for the variance of the MP estimator is (still) lacking. In the next section we describe an extension of the AC estimates estimator for incorporating auxiliary information into the AC estimates estimation procedure.

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