2. Two estimators for the growth rate of the total turnover

Paul Knottnerus

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Consider a population of $N$ enterprises $U=\{1,\mathrm{...},N\},$ and suppose there are no births and deaths in the population. Let $Y_i$ denote the value of the turnover for the $i\text{-th}$ enterprise in a given month (say $t$ ) and $X_i$ the value of the turnover of that enterprise in month $t-12.$ Hence, the variables $y$ and $x$ concern the same variable on two different occasions. Denote their population totals by $Y$ and $X,$ and their population means by $\overline{Y}$ and $\overline{X},$ respectively. That is, $Y={\displaystyle {\sum }_{i\in U}{Y}_i},$ $X={\displaystyle {\sum }_{i\in U}{X}_i},$ $\overline{Y}=Y/N$ and $\overline{X}=X/N.$ Let $s_1,s_2$ and $s_3$ denote three mutually disjoint simple random samples from $U$ without replacement (SRS). Define $s_{12}$ and $s_{23}$ by $s_{12}=s_1\cup s_2$ and $s_{23}=s_2\cup s_3,$ respectively. Denote the size of $s_k$ by $n_k(k=1,2,3,12,23)$ and the corresponding sample means by ${\overline{y}}_k$ and ${\overline{x}}_k.$ Let the variable $x$ be observed in $s_{12}$ on the first occasion and the variable $y$ in $s_{23}$ on the second occasion. Denote the overlap ratios by $\lambda$ $(=n_2/n_{12})$ and $\mu$ $(=n_2/n_{23}).$ The SRS estimators for the population totals $Y$ and $X$ are defined by ${\widehat{Y}}_{SRS}=N{\overline{y}}_{23}$ and ${\widehat{X}}_{SRS}=N{\overline{x}}_{12},$ respectively.

Define the growth rate $g$ of the total turnover between the two occasions by $g=G-1$ with $G=Y/X.$ For estimating $G$ there are two options. One of the standard (STN) options is based on the estimated totals on both occasions, that is

$${\widehat{G}}_{STN}=\frac{{\widehat{Y}}_{SRS}}{{\widehat{X}}_{SRS}}=\frac{{\overline{y}}_{23}}{{\overline{x}}_{12}};(2.1)$$

see Nordberg (2000), Qualité and Tillé (2008) and Knottnerus and Van Delden (2012). Note that the estimator ${\widehat{g}}_{STN}={\widehat{G}}_{STN}-1$ for $g$ has the same variance as ${\widehat{G}}_{STN}.$ For sufficiently large $n$ this variance can be approximated by using a first-order Taylor series expansion of ${\widehat{G}}_{STN}.$ That is,

$$\begin{array}{cc}\mathrm{var}\left({\widehat{G}}_{STN}\right)& \approx \frac{1}{{\overline{X}}^2}\mathrm{var}\left({\overline{y}}_{23}-G{\overline{x}}_{12}\right)\hfill \\ & =\frac{1}{{\overline{X}}^2}\left\{\mathrm{var}\left({\overline{y}}_{23}\right)+G^2\mathrm{var}\left({\overline{x}}_{12}\right)-2G\mathrm{cov}\left({\overline{y}}_{23},{\overline{x}}_{12}\right)\right\}\hfill \\ & =\frac{1}{{\overline{X}}^2}\left\{\left(\frac{1}{n_{23}}-\frac{1}{N}\right)S_y^2+G^2\left(\frac{1}{n_{12}}-\frac{1}{N}\right)S_x^2-2G\left(\frac{\lambda \mu }{n_2}-\frac{1}{N}\right)S_{xy}\right\}\text{,}\text{(2}\text{.2)}\hfill \end{array}$$

where $S_y^2={\displaystyle {\sum }_U{\left(Y_i-\overline{Y}\right)}^2}/\left(N-1\right)$ is the adjusted population variance of the $Y_i$ and $S_x^2$ that of the $X_i$ while $S_{xy}={\displaystyle {\sum }_U\left(X_i-\overline{X}\right)\left(Y_i-\overline{Y}\right)}/\left(N-1\right)$ is their adjusted population covariance. Cochran (1977, page 153) suggests as working rule to use the large-sample result if the sample size exceeds 30 and the coefficients of variation of the numerator and denominator are less than 10%. For (different) derivations of the expression for $\mathrm{cov}\left({\overline{y}}_{23},{\overline{x}}_{12}\right)$ used in (2.2), see Tam (1984) and Knottnerus and Van Delden (2012). The adjusted population (co)variances can be estimated unbiasedly by the sample (co)variances; recall sample (co)variances $s_{yk}^2$ and $s_{yxk}^{}$ from sample $s_k$ $(k=1,2,3,12,23)$ are defined by

$$\begin{array}{cc}{s}_{yk}^2& =\frac{1}{n_k-1}{\displaystyle \sum _{i\in s_k}^{}{\left(Y_i-{\overline{y}}_k\right)}^2}\hfill \\ s_{yxk}^{}& =\frac{1}{n_k-1}{\displaystyle \sum _{i\in s_k}^{}\left(Y_i-{\overline{y}}_k\right)\left(X_i-{\overline{x}}_k\right)}\text{. }\hfill \end{array}$$

An alternative option for estimating $G$ and $g$ is based on enterprises observed on both occasions in overlap $s_2$ (OLP). That is,

$${\widehat{G}}_{OLP}=\frac{{\overline{y}}_2}{{\overline{x}}_2}(2.3)$$

For sufficiently large $n_2,$ the well-known approximation for the variance of this estimator is

$$\begin{array}{cc}\mathrm{var}\left({\widehat{G}}_{OLP}\right)& \approx \frac{1}{{\overline{X}}^2}\mathrm{var}\left({\overline{y}}_2-G{\overline{x}}_2\right)\hfill \\ & =\frac{1}{{\overline{X}}^2}\left(\frac{1}{n_2}-\frac{1}{N}\right)S_{y-Gx}^2\text{,}\text{(2}\text{.4)}\hfill \end{array}$$

where $S_{y-Gx}^2$ stands for $S_y^2+G^2{S}_x^2-2G{S}_{xy};$ see Cochran (1977, page 31). In order to get some more insight into the merits of both ${\widehat{g}}_{STN}$ and ${\widehat{g}}_{OLP},$ consider the following examples.

Example 2.1. The data used in this example are panel observations on the turnover of Dutch supermarkets in February 2011 and 2012 from stratum 3 (size class 3). The stratum size is $N=386.$ Furthermore, $n_1=15,n_2=57$ and $n_3=17.$ For the different samples we have (in thousand euros)

$${\overline{y}}_{23}=97.2,{\overline{x}}_{12}=89.8,s_{y23}^2=3,781,\text{and }{s}_{x12}^2=2,232.$$

The population correlation coefficient ${\rho }_{xy}\left(=S_{xy}/S_x{S}_y\right)$ between the $Y_i$ and the $X_i$ is estimated from overlap $s_2$ by ${\widehat{\rho }}_{xy2}=s_{xy2}/s_{y2}{s}_{x2}=\mathrm{0.876.}$ To avoid negative variance estimates, Knottnerus and Van Delden (2012) propose estimating $S_{xy}$ in (2.2) by ${\widehat{S}}_{xy}={\widehat{\rho }}_{xy2}{s}_{x12}^{}{s}_{y23}^{}=2,545.$ Substituting the above outcomes into (2.1) and (2.2), we obtain ${\widehat{g}}_{STN}=0.082\left(=8.2\%\right)$ and $v\widehat{a}r\left({\widehat{g}}_{STN}\right)=\mathrm{0.00324.}$ Assuming normality and using $u_{0.975}=1.96,$ the 95%-confidence interval is approximately $I_{STN}^{95}\approx \left(-3.0\%,19.4\%\right).$ In contrast, from overlap $s_2$ we get the estimates

$${\overline{y}}_2=102.2,{\overline{x}}_2=97.3\text{ and }{\widehat{g}}_{OLP}=0.050\text{ (}=\text{5}\text{.0\%)}\text{.}$$

Substituting the same estimates as before for $\overline{X}$ and the (co)variances of the $X_i$ and $Y_i$ into (2.4) yields $v\widehat{a}r\left({\widehat{g}}_{OLP}\right)=\mathrm{0.00166.}$ Under the normality assumption this yields a smaller 95%-confidence interval $I_{OLP}^{95}\approx \left(-3.0\text{\%,13}\text{.0\%}\right)\text{. }$

Example 2.2. Among the data of Example 2.1 there were three enterprises with extreme $g\text{-}$ values of -50%, 133% and -91%. It is beyond the scope of this paper to further analyse or correct these outliers. But to illustrate the difference between the estimators ${\widehat{g}}_{STN}$ and ${\widehat{g}}_{OLP}$ once more, we simply omit these enterprises so that $n_2=54$ instead of $n_2=57.$ A first result is that estimate ${\widehat{\rho }}_{xy2}$ increases from 0.876 to 0.970. The latter is fairly high in spite of the fact that the coefficient of variation of the growth rates $g_i=\left(Y_i/X_i-1\right)$ is $c{v}_{g2}=s_{g2}/{\overline{g}}_2=4.1$ which still indicates a rather high volatility among the growth rates in this example. Furthermore, in analogy with the previous example, we get ${\widehat{g}}_{STN}=0.074\left(=7.4\%\right)$ with $\mathrm{var}\left({\widehat{g}}_{STN}\right)=0.00251$ and ${\widehat{g}}_{OLP}=0.039\left(=3.9\%\right)$ with $\mathrm{var}\left({\widehat{g}}_{OLP}\right)=\mathrm{0.00039.}$ The corresponding 95%-confidence intervals in this slightly modified example are approximately $I_{STN}^{95}\approx \left(\text{-2}\text{.4\%,17}\text{.2\%}\right)$ and $I_{OLP}^{95}\approx \left(0.1\text{\%,}7.7\text{\%}\right)\text{. }$ Compared to Example 2.1 the interval $I_{OLP}^{95}$ decreased relatively stronger than $I_{STN}^{95}.$

In addition, Example 2.2 may serve as a warning to be cautious when using sample means as ${\overline{y}}_{23}$ and ${\overline{x}}_{12}$ for estimating growth rates because these estimates may lead to unnecessarily large confidence interval around a suboptimal estimate. In the next section we look more closely at the question of what kind of circumstances may lead to a large interval $I_{STN}^{95}.$

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