## Abstract

Some real estate data for sales of detached houses in the Dutch town of “A” is used in order to construct house price indexes using a variety of methods. A main purpose of the paper is to determine whether the different methods generate different empirical results. The data cover 14 quarters of sales, beginning in 2005 and ending in the middle of 2008.

The base of this chapter is Diewert, W.E. 2010. Alternative approaches to measuring house price inflation. Discussion Paper 10-10, Department of Economics, University of British Columbia, Vancouver, Canada.

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## Notes

- 1.
- 2.
Houses which were older than 50 years at the time of sale were deleted from the data set. Two observations which had unusually low selling prices (36,000 and 40,000 Euros) were deleted as were 28 observations which had land areas greater than 1200m\(^{2}\). No other outliers were deleted from the sample.

- 3.
The \(R^{2}\) between the actual and predicted selling prices ranged from 0.83 to 0.89. The fact that it was not necessary to introduce more price determining characteristics for this particular data set can perhaps be explained by the nature of the location of the town of “A” on a flat, featureless plain and the relatively small size of the town; i.e., location was not a big price determining factor since all locations have basically the same access to amenities.

- 4.
The various international manuals on price measurement recommend this unit value approach to the construction of price indexes at the first stage of aggregation; see ILO et al. (2004), IMF et al. (2004, 2009). However, the unit value aggregation is supposed to take place over homogeneous items and this assumption may not be fulfilled in the present context, since there is a fair amount of variability in

*L*,*S*and*A*within each cell. But since there are only a small number of observations in each cell for the data set under consideration, it would be difficult to introduce more cells to improve homogeneity since this would lead to an increased number of empty cells and a lack of matching for the cells. - 5.
However, since there are only 163 or so observations for each quarter and 45 cells to fill, it can be seen that each cell will have only an average of 3 or so observations in each quarter, and some cells were empty for some quarters. This problem will be addressed subsequently.

- 6.
Thus lot size and structure size are positively correlated with a correlation coefficient of 0.6459. Both

*L*and*S*are fairly highly correlated with the selling price variable*P*: the correlation between*P*and*L*is 0.8234 and between*P*and*S*is 0.8100. These high correlations lead to some multicollinearity problems in the hedonic regression models to be considered later. - 7.
A justification for this approach to dealing with a lack of matching in the context of bilateral index number theory can be found in the discussion by Diewert (1980; 498–501) on the related problem of dealing with new and disappearing goods. Other approaches are also possible. For approaches based on imputation methods, see Alterman et al. (1999) and for approaches that are based on maximum matching over all pairs of periods, see Ivancic et al. (2011) and de Haan and van der Grient (2011).

- 8.
The means (and standard deviations) of the 5 series are as follows: \(P_{\text {FCH}}=1.0737\) (0.0375), \(P_{\text {FFB}}=1.0737\) (0.0370), \(P_{\text {Mean}}=1.0785\) (0.0454), \(P_{\text {Median}}=1.0785\) (0.0510), and \(P_{\text {Representative}}=1.0586\) (0.0366). Thus the representative model price index is smoother than the two matched model Fisher indexes but it has a substantial bias relative to the two Fisher indexes: the representative model price index is well below the Fisher indexes for most of the sample period.

- 9.
- 10.
Diewert and von der Lippe (2010) show that with finer and finer stratification schemes, eventually there is a complete lack of matching and index numbers based on highly stratified unit values become meaningless.

- 11.
For additional examples of this rolling year approach, see the chapters on seasonality in ILO et al. (2004), the IMF et al. (2004) and Diewert (1998). In order to theoretically justify the rolling year indexes from the viewpoint of the economic approach to index number theory, some restrictions on preferences are required. The details of these assumptions can be found in Diewert (1999; 56–61). It should be noted that weather and the lack of fixity of Easter can cause “seasons” to vary and a breakdown in the approach; see Diewert et al. (2009). However, with quarterly data, these limitations of the rolling year index are less important.

- 12.
In practice, as we have seen in the previous section, many of the cells are empty in each period.

- 13.
There are 11 rolling year comparisons that can be made with the data for 14 quarters that are available. The number of unmatched or empty cells for rolling years 2, 3, ..., 11 are as follows: 50, 52, 55, 59, 60, 61, 65, 65, 66, 67. The relatively low number of unmatched or empty cells for rolling years 2, 3 and 4 is due to the fact that for rolling year 2, 3/4 of the data are matched, for rolling year 3, 1/2 of the data are matched and for rolling year 4, 1/4 of the data are matched.

- 14.
The stronger is the seasonality, the stronger will be this argument in favour of the accuracy of the rolling year index. The strength of this argument can be seen if all house price sales in a given cell turn out to be strongly seasonal; i.e., the sales for that cell occur in say only one quarter in each year. Quarter to quarter comparisons are obviously impossible in this situation but rolling year indexes will be perfectly well defined.

- 15.
This methodology was developed by Court (1939; 109–111) as his hedonic suggestion number two but there were earlier contributions which were not noticed by the profession until recently.

- 16.
For all the models estimated in this paper, it is assumed that the error terms \(\varepsilon _{n}^{t}\) are independently distributed normal variables with mean 0 and constant variance and maximum likelihood estimation is used in order to estimate the unknown parameters in each regression model. The nonlinear option in Shazam was used for the actual estimation.

- 17.
The 15 parameters \(\alpha ,\tau ^{1},\ldots ,\tau ^{14}\) correspond to variables that are exactly collinear in the regression (3.7) and thus the restriction \(\tau ^{1}=0\) is imposed in order to identify the remaining parameters.

- 18.
This regression is essentially linear in the unknown parameters and hence it is very easy to estimate.

- 19.
It is a net depreciation rate because we have no information on renovation expenditures so \(\delta \) serves as a net depreciation rate; i.e., it is equal to gross wear and tear depreciation of the house less average expenditures on renovations and repairs.

- 20.
If the variation in the independent variables is relatively small, the difference in indexes generated by the various hedonic regression models considered in this section and the following sections is likely to be small since virtually all of the models considered can offer roughly a linear approximation to the “truth”. But when the variation in the independent variables is large (as it is in the present housing context), then the choice of functional form can have a very substantial effect. Thus a priori reasoning should be applied to both the choice of independent variables in the regression as well as to the choice of functional form. For additional discussion on functional form issues, see Diewert (2003a).

- 21.
It is a net depreciation rate because we have no information on renovation expenditures so \(\delta \) serves is equal to average gross wear and tear depreciation of the house less average real expenditures on renovations and repairs.

- 22.
This exposition follows that of Diewert et al. (2010).

- 23.
It is natural to impose some regularity conditions on the characteristics aggregator function

*f*like continuity, monotonicity (if each component of the vector \(\varvec{z}^{1}\) is strictly greater than the corresponding component of \(\varvec{z}^{2}\), then \(f(\varvec{z}^{1})>f(\varvec{z}^{2})\) and \(f(0,0)=0\). - 24.
- 25.
- 26.
The sample average amounts of

*L*and*S*were \(257.6\,\)m\(^{2}\) and \(127.2\,\)m\(^{2}\) respectively and the average age of the detached dwellings sold over the sample period was 1.85 decades. - 27.
This theory dates back to Court (1939; 108) as his hedonic suggestion number one. His suggestion was followed up by Griliches (1971a; 59–60, 1971b; 6) and Triplett and McDonald (1977; 144). More recent contributions to the literature include Diewert (2003b), de Haan (2003, 2009, 2010), Triplett (2004) and Diewert et al. (2009).

- 28.
Due to the fact that the regressions defined by (3.15) have a constant term and are essentially linear in the explanatory variables, the sample residuals in each of the regressions will sum to zero. Hence the sum of the predicted prices will equal the sum of the actual prices for each period. Thus the sum of the actual prices in the denominator of (3.17) will equal the sum of the corresponding predicted prices and similarly, the sum of the actual prices in the numerator of (3.19) will equal the corresponding sum of the predicted prices.

- 29.
This approach was suggested by Diewert (2007) and implemented by Diewert et al. (2010). Thus the model in this section is a supply side model as opposed to the demand side Cobb Douglas model of McMillen (2003) studied earlier. See Rosen (1974) for a discussion of identification issues in hedonic regression models.

- 30.
In order to obtain homoskedastic errors, it would be preferable to assume multiplicative errors in Eq. (3.1) since it is more likely that expensive properties have relatively large absolute errors compared to very inexpensive properties. However, following Koev and Santos Silva (2008), we think that it is preferable to work with the additive specification (3.1) since we are attempting to decompose the aggregate value of housing (in the sample of properties that sold during the period) into additive structures and land components and the additive error specification will facilitate this decomposition.

- 31.
Thorsnes (1997; 101) has a related cost of production model. He assumed that instead of Eq. (3.21), the value of the property under consideration in period

*t*, \(p^{t}\), is equal to the price of housing output in period*t*, \(\rho ^{t}\), times the quantity of housing output*H*(*L*,*K*) where the production function*H*is a CES function. Thus Thorsnes assumed that \(p^{t}=\rho ^{t}H(L,K)=\rho ^{t}[\alpha L^{\sigma }+\beta K^{\sigma }]^{1/\sigma }\) where \(\rho ^{t},\sigma ,\alpha \) and \(\beta \) are parameters ,*L*is the lot size of the property and*K*is the amount of structures capital in constant quality units (the counterpart to our \(S^{*}\)). Our problem with this model is that there is only one independent time parameter \(\rho ^{t}\) whereas our model has two, \(\beta ^{t}\) and \(\gamma ^{t}\) for each*t*, which allow the price of land and structures to vary freely between periods. - 32.
The present model is similar in structure to the hedonic imputation model described in the previous section except that this model is more parsimonious; i.e., there is only one depreciation rate in the present model (as opposed to 14 depreciation rates in the imputation model) and there are no constant terms in the present model. The important factor in both models is that the prices of land and quality adjusted structures are allowed to vary independently across time periods.

- 33.
This approach follows that of Diewert et al. (2010).

- 34.
Comparing Figs. 3.7 and 3.8, it can be seen that in Fig. 3.7, the price index for land is

*above*the overall price index for the most part while the price index for structures is below the overall index but in Fig. 3.8, this pattern*reverses*. This instability is again an indication of a multicollinearity problem. - 35.
Some direct evidence on this assertion will be presented in the following section.

- 36.
This method for imposing monotonicity restrictions was used by Diewert et al. (2010) with the difference that they imposed monotonicity on both structures and land prices, whereas here, we impose monotonicity restrictions on structures prices only.

- 37.
From the Central Bureau of Statistics (2010) online source, Statline, the following series was downloaded for the New Dwelling Output Price Index for the 14 quarters in our sample of house sales in “A” : 98.8, 98.1, 100.3, 102.7, 99.5, 100.5, 100.0, 100.3, 102.2, 103.2, 105.6, 107.9, 110.0, 110.0. This series was normalized to 1 in the first quarter by dividing each entry by 98.8. The resulting series is denoted by \(\mu ^{1}(=1),\mu ^{2},...,\mu ^{14}\).

- 38.
However, the hedonic regression based indexes can be biased as well if important explanatory variables are omitted and if an “incorrect” functional form for the hedonic regression is chosen. But in general, hedonic regression methods are probably preferred over stratification methods.

- 39.
Users may tolerate a few revisions to recent data but typically, users would not like all the numbers to be revised back into the indefinite past as new data become available.

- 40.
- 41.
- 42.
By construction, \(P_{\text {S4}}\) and \(P_{\text {RWS}}\) are both equal to the official CBS construction price index for new dwellings, \(\mu ^{t}/\mu ^{1}\) for \(t=1,...,14\).

- 43.
This approximation would probably be an adequate one if the sample period were a decade or so. Obviously, our sample period of 14 quarters is too short to be a good approximation but the method we are suggesting can be illustrated using this rough approximation. There are also sample selectivity problems with this approximation; i.e., new houses will be over represented using this method.

- 44.
We did not delete the observations for houses that were transacted multiple times over the 14 quarters since the same house transacted during two or more of the quarters is not actually the same house due to depreciation and renovations.

- 45.
- 46.
Fixed base and chained Laspeyres, Paasche and Fisher indexes are also equal under these circumstances.

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Diewert, W.E., Nishimura, K.G., Shimizu, C., Watanabe, T. (2020). A Comparison of Alternative Approaches to Measuring House Price Inflation. In: Property Price Index. Advances in Japanese Business and Economics, vol 11. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55942-9_3

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