# 6 Conclusion

Jan de Haan and Rens Hendriks

The simple GREG method outlined in this paper, which is
based on OLS regressions of selling prices on appraisals, substantially reduces
the volatility of a house price index as compared to the ratio of sample means.
The SPAR index can be viewed as an estimator of the OLS GREG index (which
itself is an estimator, of course) where the base period population mean of
appraisals is replaced by the sample means in the base period and the
comparison period. Our empirical results for the

Due to compositional change of the properties sold, the GREG (and SPAR) time series exhibit strong short-term volatility. An increase in a particular month is typically followed by a decrease in the next month. Put differently, the month-to-month changes do not tell us much about the true price change of the housing stock which, except under unusual circumstances, should behave smoothly. An improved outlier detection method might help reduce the index volatility, but the effect will probably be limited. Applying a smoothing procedure would seem to be an option. However, that will typically lead to revisions of previously published price index numbers, and the lack of revisions is one of the strenghts of the GREG and SPAR approaches. Another option would be to reduce the frequency of observation, for example to quarters, but that may be undesirable as well.

From a purely statistical point of view, in our two-variable model the variability of ${R}^{2}$ seems to be responsible for a large part of the volatility of the slope coefficient and therefore of the volatility of the price index series. Future research could focus on the relation between compositional changes in terms of the property characteristics and changes in ${R}^{2}.$ As many housing characteristics are unavailable, we cannot investigate this issue with our data. Fortunately, Statistics Netherlands has access to a data set from the largest Dutch association of real estate agents that might be useful for this purpose. This data set covers around 70% of all housing sales in the Netherlands during 1999-2008, includes many property characteristics and has been enriched with appraisal data. In the past we already used the data set to compare the SPAR index with various types of hedonic indexes.

## Acknowledgements

The authors would like to thank the participants at the
Economic Measurement Group Workshop, 1-

## Appendix

### Approximate Standard Errors of the GREG Index

The GREG index defined by equation (3.10) in the main
text is a ratio of two estimators, ${\widehat{\overline{p}}}_{\text{GREG}}^{t}$ and ${\widehat{\overline{p}}}_{\text{GREG}}^{0};$ for brevity we delete "OLS�. Using a first-order
*e.g.*, Kendall and Stuart 1976)

$$\mathrm{var}({\widehat{P}}_{\text{GREG}}^{0t})\cong {\left[\frac{E({\widehat{\overline{p}}}_{\text{GREG}}^{t})}{E({\widehat{\overline{p}}}_{\text{GREG}}^{0})}\right]}^{2}\left[\frac{\mathrm{var}({\widehat{\overline{p}}}_{\text{GREG}}^{t})}{{\{E({\widehat{\overline{p}}}_{\text{GREG}}^{t})\}}^{2}}+\frac{\mathrm{var}({\widehat{\overline{p}}}_{\text{GREG}}^{0})}{{\{E({\widehat{\overline{p}}}_{\text{GREG}}^{0})\}}^{2}}+\frac{\mathrm{cov}({\widehat{\overline{p}}}_{\text{GREG}}^{t},{\widehat{\overline{p}}}_{\text{GREG}}^{0})}{E({\widehat{\overline{p}}}_{\text{GREG}}^{t})E({\widehat{\overline{p}}}_{\text{GREG}}^{0})}\right],\text{}\text{}\text{}\left(A.1\right)$$

where $E({\widehat{\overline{p}}}_{\text{GREG}}^{t})$ and $E({\widehat{\overline{p}}}_{\text{GREG}}^{0})$ denote expected values.

The covariance term in (A.1) is equal to 0 since, by assumption, the samples in periods 0 and $t$ are independently drawn. Replacing the expected values in (A.1) by the estimators and subsequently taking the square root leads to the following expression for the standard error of ${\widehat{P}}_{\text{GREG}}^{0t}:$

$$se({\widehat{P}}_{\text{GREG}}^{0t})\cong {\widehat{P}}_{\text{GREG}}^{0t}{\left[\frac{\mathrm{var}({\widehat{\overline{p}}}_{\text{GREG}}^{t})}{{({\widehat{\overline{p}}}_{\text{GREG}}^{t})}^{2}}+\frac{\mathrm{var}({\widehat{\overline{p}}}_{\text{GREG}}^{0})}{{({\widehat{\overline{p}}}_{\text{GREG}}^{0})}^{2}}\right]}^{1/2}.\text{}\text{}\text{}\left(A.2\right)$$

Equation (A.2) can be estimated in practice using ${\widehat{\overline{p}}}_{\text{GREG}}^{s}={\widehat{\alpha}}^{s}+{\widehat{\beta}}^{s}{\overline{a}}^{0}(s=0,t),$ hence $\mathrm{var}({\widehat{\overline{p}}}_{\text{GREG}}^{s})=\mathrm{var}({\widehat{\alpha}}^{s})+{({\overline{a}}^{0})}^{2}\mathrm{var}({\widehat{\beta}}^{s})+2{\overline{a}}^{0}\mathrm{cov}({\widehat{\alpha}}^{s},{\widehat{\beta}}^{s}).$ Estimates of the (co)variances are readily available in most statistical packages from the variance-covariance matrix.

Dividing (A.2) by ${\widehat{P}}_{\text{GREG}}^{0t}$ yields an expression for the relative standard error or coefficient of variation, $CV({\widehat{P}}_{\text{GREG}}^{0t})=se({\widehat{P}}_{\text{GREG}}^{0t})/{\widehat{P}}_{\text{GREG}}^{0t},$ of the GREG index:

$$CV({\widehat{P}}_{\text{GREG}}^{0t})\cong {\left[\frac{\mathrm{var}({\widehat{\overline{p}}}_{\text{GREG}}^{t})}{{({\widehat{\overline{p}}}_{\text{GREG}}^{t})}^{2}}+\frac{\mathrm{var}({\widehat{\overline{p}}}_{\text{GREG}}^{0})}{{({\widehat{\overline{p}}}_{\text{GREG}}^{0})}^{2}}\right]}^{1/2}={\left[{\{CV({\widehat{\overline{p}}}_{\text{GREG}}^{t})\}}^{2}+{\{CV({\widehat{\overline{p}}}_{\text{GREG}}^{0})\}}^{2}\right]}^{1/2}.\text{}\text{}\text{}\left(A.3\right)$$

Of more importance is the relative standard error
of the *percentage change* of the
index, *i.e.*, $CV({\widehat{P}}_{\text{GREG}}^{0t}-1)=se({\widehat{P}}_{\text{GREG}}^{0t}-1)/({\widehat{P}}_{\text{GREG}}^{0t}-1).$ This is generally greater than $CV({\widehat{P}}_{\text{GREG}}^{0t}),$ given that $se({\widehat{P}}_{\text{GREG}}^{0t}-1)=se({\widehat{P}}_{\text{GREG}}^{0t})$ and ${\widehat{P}}_{\text{GREG}}^{0t}-1<{\widehat{P}}_{\text{GREG}}^{0t}.$

If both regression lines almost pass through the origin, hence ${\widehat{\alpha}}^{s}\cong 0(s=0,t),$ we have ${\widehat{P}}_{\text{GREG}}^{0t}\cong {\widehat{\beta}}^{t}/{\widehat{\beta}}^{0}$ and (A.2) simplifies to

$$se({\widehat{P}}_{\text{GREG}}^{0t})=se({\widehat{P}}_{\text{GREG}}^{0t}-1)\cong {\widehat{P}}_{\text{GREG}}^{0t}{\left[\frac{\mathrm{var}({\widehat{\beta}}^{t})}{{\left({\widehat{\beta}}^{t}\right)}^{2}}+\frac{\mathrm{var}({\widehat{\beta}}^{0})}{{\left({\widehat{\beta}}^{0}\right)}^{2}}\right]}^{1/2}.\text{}\text{}\text{}\left(A.4\right)$$

In this particular case the GREG and SPAR indexes nearly coincide, so (A.4) also holds for the SPAR index (using ${\widehat{P}}_{\text{SPAR}}^{0t}$ rather than ${\widehat{P}}_{\text{GREG}}^{0t}).$

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