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 Netherlands indicate that the SPAR
index is almost as efficient as the GREG index, even for small sub-populations. We have checked this by drawing a
random sample of 50 observations each month from the total number of monthly
sales (15,000 on average). The month-to-month changes of the SPAR index were
only slightly bigger than those of the GREG.
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 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 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
The authors would like to thank the participants at the
Economic Measurement Group Workshop, 1-3 December 2010, University of New South
the participants at an Applied Economics Seminar, 22 November 2011, University of Queensland,
for their helpful comments on preliminary versions of the paper. Comments and
suggestions made by the editor and two anonymous referees also helped to
improve the paper. The assistance of Erna van der Wal, who provided
us with the data, is gratefully acknowledged. The views expressed in this paper
are those of the authors and do not necessarily reflect the views of Statistics
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, and for brevity we delete "OLS�. Using a first-order
expansion, the variance of the index can be approximated by (see e.g., Kendall and Stuart 1976)
where and denote expected values.
The covariance term in (A.1) is equal to 0 since, by
assumption, the samples in periods 0 and 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
Equation (A.2) can be estimated in practice using hence Estimates of the (co)variances are readily available
in most statistical packages from the variance-covariance matrix.
Dividing (A.2) by yields an expression for the relative standard
error or coefficient of variation, of the GREG index:
Of more importance is the relative standard error
of the percentage change of the
index, i.e., This is generally greater than given that and
If both regression lines almost pass through the origin,
hence we have and (A.2) simplifies to
In this particular case the GREG and SPAR indexes
nearly coincide, so (A.4) also holds for the SPAR index (using rather than
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