The problem of quality-mix adjustment

Critical to price index measurement is the need to compare, in successive periods, transaction prices of like-with-like representative goods and services. Price index measurement for consumer, producer, and export and import price indexes (CPI, PPI and XMPIs) largely rely on the matched-models method. The detailed specification of one or more representative brand is selected as a high-volume seller in an outlet, for example a single 330 ml. can of regular Coca Cola, and its price recorded. The outlet is then revisited in subsequent months and the price of the self-same item recorded and a geometric average of its price and those of similar such specifications in other outlets form the building blocks of a price index such as the CPI. There may be problems of temporarily missing prices, quality change, say size of can or sold as a bundled part of an offer if bought in bulk, but essentially the price of like is compared with like every month.⁷ RPPIs are much harder to measure.

First, there are no transaction prices every month/quarter on the same property. RPPIs have to be compiled from infrequent transactions on heterogeneous properties. A higher (lower) proportion of more expensive houses sold in one quarter should not manifest itself as a measured price increase (decrease). There is a need in measurement to control for changes in the quality of houses sold, a non-trivial task.

The main methods of quality adjustment are (i) hedonic regressions; (ii) use of repeat sales data only; (iii) mix-adjustment by weighting detailed relatively homogeneous strata; and (iv) the sales price appraisal ratio (SPAR).⁸ The method selected depends on the database used. There needs to be details of salient price-determining characteristics for hedonic regressions, a relatively large sample of transactions for repeat sales, and good quality appraisal information for SPAR. In the US, for example, price comparisons of repeat sales are mainly used, akin to the like-with-like comparisons of the matched models method, Shiller (1991). There may be bias from not taking full account of depreciation and refurbishment between sales and selectivity bias in only using repeat sales and excluding new home purchases and homes purchased only once. However, the use of repeat sales does not require data on quality characteristics and controls for some immeasurable characteristics that are difficult to effectively include in hedonic regressions, such as a desirable or otherwise view from the property.

The problem of source data

Second, the data sources are generally secondary sources that are not tailor-made by the national statistical offices (NSIs), but collected by third parties, including the land registry/notaries, lenders, realtors (estate agents), and builders. The adequacy of these sources to a large extent depends on a country’s institutional and financial arrangements for purchasing a house and varies between countries in terms of timeliness, coverage (type, vintage, and geographical), price (asking, completion, transaction), method of quality-mix adjustment (repeat sales, hedonic regression, SPAR, square meter) and reliability; pros and cons will vary within and between countries. In the short-medium run users may be dependent on series that have grown up to publicize institutions, such as lenders and realtors, as well as to inform users. Metadata from private organizations may be far from satisfactory.

We stress that our concern here is with measuring RPPIs for FSIs and macroeconomic analysis where the transaction price, that includes structures and land, is of interest. However, for the purpose of national accounts and analysis based thereon, such as productivity, there is a need to both separate the price changes of land from structures and undertake adjustments to price changes due to any quality change on the structures, including depreciation. This is far more complex since separate data on land and structures is not available when a transaction of a property takes place. Diewert, de Haan, and Hendriks (2011) and Diewert and Shimizu (2013a) tackle this difficult problem.

Figure 1 shows alternative data sources in its center and coverage, methods for adjusting for quality mix, nature of the price, and reliability in the four quadrants. Land registry data, for example, may have an excellent coverage of transaction prices, but have relatively few quality characteristics for an effective use of hedonic regressions, not be timely, and have a poor reputation. Lender data may have a biased coverage to certain regions, types of loans, exclude cash sales, have “completion” (of loan) price that may differ from transaction price, but have data on characteristics for hedonic quality adjustment. Realtor data may have good coverage, aside from new houses, data on characteristics for hedonic quality adjustment, but use asking prices rather than transaction prices.

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Methodological issues and data sources in RPPI measurement

Citation: IMF Working Papers 2016, 213; 10.5089/9781475552249.001.A001

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The importance of distinguishing between asking and transaction prices will vary between countries as the length of time between asking and transaction varies with the institutional arrangements for buying and selling a house and the economic cycle of a country.

Whether measurement matters

A natural question is whether the differences in source data and methodologies used matters to the overall outcome of the index. Silver (2015) undertook an extensive formal analysis based on the RPPIs and, as explanatory variables, the associated methodological and source data for 157 RPPIs from 2005:Q1 to 2010:Q1 from 24 countries. The resulting panel data had fixed-time and fixed-country effects; the estimated coefficients on the explanatory measurement variables were first held fixed and then relaxed to be time varying. Subsequently, the explanatory variables were interacted with the country dummies.

He found measurement-related variables as explanatory variables for house price inflation had substantial explanatory power, ${\overline{R}}^2$, especially over the period of recession, when it really matters, about 0.45 in mid-2009. He further investigated the impact of measurement on modelling, using an econometric model of house price inflation based on Igan and Loungani (2012). Using the residuals from the regression of house price index on measurement variables as a “measurement-adjusted” house price index he found the measurement-adjusted model to perform better than the unadjusted one. Less formally, he provided some country illustrations.⁹

Figure 2 shows a feast of RPPIs available for the UK including the ONS (UK, hedonic mix-adjusted, completion price); Nationwide and Halifax (both UK, hedonic, own mortgage approvals, mortgage offer price; Halifax weights); the England and Wales (E&W) Land Registry (E&W, repeat sales, all transaction prices); and the ONS Median price index unadjusted for quality mix—given for comparison.¹⁰ Other available RPPIs in the UK are LSL Acadata HPI (Land registry), ¹¹ Right move (realtor) and two RPPIs based on surveys of expert opinion. Measured inflation in 2008Q4 coming into the trough was -8.7 (ONS) -12.3 (Land registry) -16.2 (Halifax) -14.8 (Nationwide): and -4.9 (ONS Median unadjusted (for quality mix change); methodology and data source matter.

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A Feast of UK RPPIs, annual percent rate, quarterly

Citation: IMF Working Papers 2016, 213; 10.5089/9781475552249.001.A001

Source: http://www.bis.org/statistics/pp.htm, http://www.acadata.co.uk/acadHousePrices.php and http://www.ons.gov.uk/ons/rel/hpi/house-price-index/july-2014/stb-juty-2O14.html.

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Figure 3 shows RPPIs available in the US including the: CoreLogic, Federal Housing Finance Agency (FHFA) purchases-only, Case-Shiller, and the FHFA extended-data House Price Index (HPI). CoreLogic, FHFA, and Case-Shiller, the three primary RPPIs in the US, use repeat sales for quality-mix adjustment—the Census Bureau is a (hedonic) new houses only index based on a limited sample. The FHFA extended-data HPI includes, in addition to transaction prices from purchase-money mortgages guaranteed by Fannie Mae and Freddie Mac, transactions records for houses with mortgages endorsed by the Federal Housing Administration (FHA) and county recorder data licensed from CoreLogic, appropriately re-weighted to ensure there is no undue urban over rural bias. This change in source data coverage accounted for the 4.6 percentage point difference in 2008 Q4 between the annual quarterly RPPI respective falls of 6.89 and 11.66 percent for the FHFA “All Purchases” and “Extended-Data” FHFA HPIs. Coverage limited to particular types of mortgages matters.¹²

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US Repeat sales RPPIs, annual percent change, quarterly

Citation: IMF Working Papers 2016, 213; 10.5089/9781475552249.001.A001

Source: http://www.bis.org/statistics/pp.htm, http://us.spindices.com/index-family/real-estate/sp-case-shiller and http://www.fhfa.gov/KeyTopics/Pages/House-Price-Index.aspx.

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Leventis (2008) decomposed into methodological and coverage differences the average difference between the FHFA (then Office of Federal Housing Enterprise Oversight (OFHEO)) and S&P/Case-Shiller HPIs, covering 10 matched metropolitan areas, for the four-quarter price changes over 2006Q3-2007Q3. Among his findings was that of the overall 4.27 percent average difference, FHFA’s use of a more muted down-weighting of larger differences in the lags between repeat sales,¹³ than use in Case-Shiller, accounts for an incremental 1.17 percent of the difference. It is not just that the use of different quality-mix adjustment methods matters, it does also the manner in which a method is applied.

Functional forms of the hedonic regression: a linear form

Consider a linear hedonic functional form. An estimated hedonic regression would have the prices, $p_i^t$ of an individual property i on the LHS and their associated k characteristics, $z_{k,i}^t$ on the RHS as explanatory variables. Such hedonic regressions may be estimated for each defined stratum in a period 0 reference period (index =100.00) and each successive period t (=1,2,..,T). The linear functional form for period t is given by:

where ${\hat{p}}_i^t$ (and $p_i^t$) are the predicted (actual) price of property i in period t; $z_{k,i}^t$ are the values of each k=1,….,K price-determining characteristic for property i in period t; y₀ and y_k (and, below, β₀ and β_k below) are the coefficients from a linear (and log-linear) hedonic equation; ${ϵ}_i^t$ (and $v_i^t$) i.i.d errors; and $h^t{z}_i^t$ a shorthand for a linear hedonic function estimated using period t data and period t characteristics.

Equation (1) has prices explained by a constant, ${\gamma }_0^t$, slope coefficients ${\gamma }_k^t$ for each k price-determining characteristics, $z_{k,i}^t$, of which there are K, and an error term, ${ϵ}_i^t$. It is a linear relationship dictated, in equation (2), by the estimated constant and the slope coefficients, represented as hats “^” over the coefficients; for a single characteristic: ${\hat{p}}_i^t={\hat{\gamma }}_0^t+{\hat{\gamma }}_1^t{z}_{1,i}^t$.

The actual relationship may be non-linear and there will be omitted variable bias in using a linear form to (mis)represent the relationship. To counter this bias one possibility is to introduce some curvature via a squared term, ${\hat{p}}_i^t={\hat{\gamma }}_0^t+{\hat{\gamma }}_1^t{z}_{1,i}^t+{\hat{\gamma }}_2^t{\left(z_{1,i}^t\right)}^2$, and test a null hypothesis as to whether ${\gamma }_2^t=0$, that is, whether the squared term has any explanatory power over and above that due to sampling error, say at a five percent level of significance. Interaction terms between more than one explanatory variable may also be introduced, Maddala and Lahiri (2009).

Functional forms of the hedonic regression: a log(arithmic)-linear form

An alternative functional form is a log(arithmic)-linear—also referred to as a semi-logarithmic—form of the hedonic regression. This form arises from a hedonic relationship between $p_i^t$ and $z_{k,i}^t$ given by:

The log-linear form first allows for curvature in the relationships say between square footage and price, and second, for a multiplicative association between quality characteristics, i.e. that possession of a garage and additional bathroom may be worth more than the sum of the two. The estimation of ordinary least squares regression (OLS) equations requires a linear form; we transform the non-linear functional relationship in equation (3) into a linear form by taking logarithms of both sides of the equation and use OLS:

where the tilde across ${\tilde{h}}^t\left(z_i^t\right)$ designates a log-linear functional form. An OLS regression estimated for the logarithm of prices, ${\hat{p}}_i^t$, on characteristics, $z_{k,i}^t$, is given as:

It is important to note that the log-linear regression output from estimating equation (4), that is ln $p_i^t$ on $z_{k,i}^t$, provides us with the logarithms of the coefficients from the original log-linear formulation in equation (3). Exponents of the estimated coefficients from the output of the software have to be taken if the parameters of the original function, that is equation (3), are to be recovered, that is: $\exp (\ln {\hat{\beta }}_k^t)={\hat{\beta }}_k^t$.¹⁹

Since many explanatory variables are dummy variables taking a value of zero or one—possession or otherwise of a characteristic—and since logarithms cannot be taken of zero values, the log-linear form is more convenient than a double-logarithmic transformation that would require logarithms be taken of the $z_{k,i}^t$ on the RHS. It should be noted that the interpretation of coefficients from a log-linear form differs from that of coefficients from a linear form. For a log-linear form our estimated coefficients are the logarithms of ${\hat{\beta }}_l,{\hat{\beta }}_2$, and ${\hat{\beta }}_3$: a unit change in the say square footage, z_1,i, leads to a ${\hat{\beta }}_1$ percent change in price, while for a dummy explanatory variable, say “possession of a balcony, z_2,i = 1 as opposed to z_2,i = 0 otherwise,” leads to an estimated(exp(β₂)−1)×100 percent change in price, as will be explained in more detail in the next section.

We consider in this paper that hedonic regressions take a generally applicable linear and lo-linear forms given by equations (2) and (5) and that these have been estimated. Outlines of the three main hedonic approaches to deriving constant quality price indexes from these estimated equations, along their relative merits, are given below in sections B, C and E. These approaches are the (i) hedonic time dummy variable, (ii) hedonic characteristics and (iii) hedonic imputation approaches. The approaches are outlined and discussed in the context of bilateral period 0 (reference period =100.00) and current period t price level comparisons where t=1,2,….,T. While our main concern will be with quarter-on-quarter inflation rates, the principles can be readily extended to quarter-on-same quarter in previous year, though see Rambaldi and Rao (2103). The concern of section F is with the periodic updating or chaining of the reference period estimates.

The method

A single hedonic regression equation may be estimated from data across properties over several time periods including the reference period 0 and successive subsequent periods t. Prices of individual properties are regressed on their characteristics, but also on dummy variables for time, taking the values of δ¹ if the house is sold in period 1, and zero otherwise, δ² if the house is sold in period 2 and zero otherwise,…., δ^T if the house is sold in period T and zero otherwise. We exclude in this case a period 0 dummy time variable and interpret the δ^t as the difference between the current period and reference period 0 average prices, having controlled for quality-mix change via the variables in the hedonic regression on their characteristics. The method has been widely applied including Fisher, Geltner, and Webb (1994), Hansen, (2009), and Shimizu et al. (2010).

Consider a linear form of the hedonic regression given by equation (1) but estimated over say two adjacent periods, 0 and 1:

The data for prices and characteristics extend over the two periods 0 and 1, yet only a single parameter, β_k, is estimated for each characteristic’s slope coefficient. The restriction is that the slopes of the regression lines for period 0 and period t are the same: ${\beta }_k={\beta }_k^0={\beta }_k^t$ for each of k=1,….,K characteristics.

For simplicity, consider a single explanatory variable, the square footage of an individual apartment, $z_i^0$ or $z_i^1$ in periods 0 and 1 respectively. Separate regression equations can be estimated for each of period 0 and period 1, but the slope coefficient, the estimated marginal value of an additional square foot, is restricted to be the same in each period, namely β₁:

The estimated coefficients on the intercepts in each period are respectively ${\hat{\beta }}_0^0$ and ${\hat{\beta }}_0^1$. These are estimates of the average price in periods 0 and 1 having controlled for variation in the square footage of the apartments—the “average” is an arithmetic mean for this linear formulation (and a geometric mean for a log-linear formulation).

We can represent equations (7a and b) in a single hedonic regression:

The dummy variable $D_i^1$ in equation (8) is equal to 1 if the data are in period 1, and zero otherwise and its estimated coefficient ${\hat{\delta }}^1=({\hat{\beta }}_0^1-{\hat{\beta }}_0^0)$. This representation of equations 7a and 7b can be seen by inserting $D_i^1=0$ (period 0 data) into the RHS term of equation (8) to give equation (7a) and inserting $D_i^1=1$ (period 0 data) to give equation (7b), assuming

The estimated coefficient on the dummy variable, δ¹, is the basis for an estimate of a constant quality property price index between periods 0 and 1. The estimate is of the difference between the period 0 and period 1 intercepts,²⁰ that is the difference in the average prices of period 1 and period 0 transactions from their regression lines for period 0 and period 1 having controlled for variation in the quality characteristics $\underset{k=1}{\overset{K}{\mathrm{\Sigma }}}{\beta }_k{z}_{k,i}^{0,t}$, as in equation (6), whereby each k characteristic is valued at its associated ${\hat{\beta }}_k$.

A log-linear specification is given by:

The ${\hat{\delta }}^t$ are estimates of the proportionate change in price arising from a change between the reference period t=0—the period not specified as a dummy time variable—and successive periods t=1,…,T having controlled for changes in the quality characteristics via the term $\underset{k=1}{\overset{K}{\mathrm{\Sigma }}}{\beta }_k{z}_{k,i}^{0,t}$.

The constant-quality price index is given for each period t=1,..,T, with respect to period t=0, which equals 100.00, by $100\times \exp \left({\hat{\delta }}^t\right)$. In principle $100\times \exp \left({\hat{\delta }}^t\right)$ requires an adjustment—for it to be a consistent (and almost unbiased) approximation of the proportionate impact of the time dummy. The adjustment is given by: $[\exp \left({\hat{\delta }}_t\right)/\exp \left(V\right)({\hat{\delta }}_t)/2)\left)\right]-1$, where $V\left({\hat{\delta }}_t\right)$ is the variance (standard error squared) of ${\hat{\delta }}_t$ and is generally very small; the estimate of constant-quality price change is given by:²¹

The time dummy method has many positive features. Given data have been collected over time on price and quality characteristics, it is relatively easy to apply simply requiring the inclusion of time dummy variables into the panel (cross-section (property) time series) data set—a data set that requires no matching of properties since $\underset{k=1}{\overset{K}{\mathrm{\Sigma }}}{\beta }_k{z}_{k,i}^{0,t}$ controls for changes in the quality mix over time. The estimates are readily derived from the estimated coefficients of the time-dummy variables, ${\hat{\delta }}^t$.

Features of the method

The method implicitly restricts the coefficients on the quality characteristics to be constant over time: for example, for adjacent period 0 and 1 regressions, ${\beta }_k={\beta }_k^0={\beta }_k^1$, as apparent from equations (6) and (9). This regression line for period 0 is parallel to that of period 1.

The extent of this restriction depends on the length of the time period over which the regression is run.²² If, for example, the regressions are run over quarterly data for a rolling 10-year window, a property price comparison between say 2006Q1 and 2016Q1 with valuations of characteristics held constant may stretch credibility, though this can be alleviated by shorter windows and or adjacent period regressions as outlined below.

The time dummy method is criticized throughout the literature for holding the estimated coefficients constant. However, as will be outlined below in section C and D, a constant quality price index has to hold something constant over time to separate out the price change from the quality-mix change. In what we will term the “direct method,” the quantities of price determining (quality) characteristics are held constant over time, for example for apartment prices, that the average number of bedrooms is held constant at 3.2, the square footage at 1,150, and so forth and re-priced each period.

An advantage of the time dummy approach is that it the estimates are generated for a regression formulation. This facilitates the exploration of how the addition and deletion of explanatory variables, changes in the functional form and estimator have on the resulting price index number estimates. It also allows for confidence intervals²³ to be drawn up around these estimates and, as Hill (2013, section 5) outlines, geo-spatial data ad spatial dependence can be readily integrated into the estimating framework (see also Pace and LeSage, 2004).

The time dummy approach uses the “indirect method” and adjusts (divides) the change in mean prices by changes in the volume of characteristics over time. However, this adjustment requires the estimated coefficients (characteristic prices) to be constant so that only changes in the volume of characteristics are measured. It is difficult to argue that constraining characteristic prices, the marginal value given to an additional bedroom and so forth, is less tenable than constraining average characteristics, the say average number of bedrooms in houses transacted in period 0 compared with 1. There are no grounds for dismissing the time dummy approach on the grounds of constrained coefficients. Indeed, we show in section IV, and in Diewert, Heravi and Silver (2009), an equivalence between the direct and indirect methods.

If used for regular index number production, past values of the index will be revised each period as new data enter the regression. A “problem” with the revision of past values of the index should not be overstated. The three main RPPIs long-established and well-publicized in the United States, the Case-Shiller, FHFA, and CoreLogic indexes, are all repeat-sales indexes whose past values are revised each period without public concern. Second, the estimated coefficients for the quality characteristics are determined using data on price and quantity characteristics over the whole period of the regression. Thus some element of the estimate of property price inflation for the current period compared with the previous period is determined by past, if not quite distant, data. This lends some stability to the property price index, but may also smooth the results and risk some credibility when there is apparent volatility in the prices not mirrored in the index.

The rolling window approaches differ from the time dummy method in the important respect that estimated coefficients are not restricted to be constant over time: they are time varying. A say period t to t+1 rolling adjacent period index is based on data in these two periods of concern, rather than the whole period. Rambaldi and Fletcher (2014) provide an extensive outline, and an empirical study, of the use of a Kalman Filter Smoother (KS)²⁴ as against the rolling adjacent-period window approach. They argue that the Kalman Filter Smoother is preferred on the grounds that it optimally weights past values of the series when estimating the regression rather than just weighting the observations in the current window. The parameter estimates vary over time but are modeled as stochastic processes and can be applied to the time-dummy hedonic indexes (Schwann, 1998 and Francke, 2008) and the hedonic imputation approach (Rambaldi and Rao, 2011 and 2013). There is a trade-off between the extent to which an index is smoothed and volatility dampened, by drawing on more distant data either through a longer rolling window or Kalman Filter Smoother, and its ability to reflect current price changes in the market, albethey subject to more volatility. Smoothing methods are particularly suitable when data are sparse, that is in “thin” markets, as discussed below in section IV.

Chaining, rolling windows and smoothing

We can militate against the criticisms of undue restriction of coefficients, revisability, and stale data by using a chained rolling window—for illustration here, 4 quarters. Consider a fixed base index of the type described by equations (1) and (2) in which each period’s index, say 2015Q4, is derived from the coefficient of the dummy variable on time for the period in question, compared with the (omitted) period t=0, say 2005Q1. The example is thus of the equation (6), or in log-linear form, equation (9), estimated on a quarterly basis over say 10 years. The fixed base estimated index from equations (9) for 2015Q4, where 2005Q1=100.0 is:

The adjacent period index is derived from successive multiplication—chaining—of regression estimates based on successive adjacent periods, i.e. a regression is first run on 2005Q1 and 2005Q2 data with a time dummy that is equal to 1 if the transaction is in 2005Q2 and zero otherwise. The estimated coefficient on this time dummy is an estimate of the change in price between the two periods, controlling for changes in quality.

The chained adjacent period index for 2005Q1 to 2015Q4 is:

The least restrictive formulation, in terms of assumption f constant coefficients, is to use a rolling window of adjacent periods only (Diewert (2005b). However, the method requires an adequate sample size of transactions over the two periods. Given the same number of transactions in each quarter, in this example say 100, the fixed base equations (6) and (9) formulation use 100×10×4 = 4,000 observations over say 10 years while the adjacent period formulation uses 100×2 = 200 each quarter. There may well be degrees of freedom problems in estimating the hedonic regression, especial if there are many locational variables such as dummy variables for each postcode. Further, in using rolling window adjacent period regressions, compilers have to bear in mind two things: (i) it is desirable to compile RPPIs as weighted sums of constant-quality price indexes across strata of different types of houses, locations, and other meaningful and useful factors. Larger samples enable a more detailed stratification; and (ii) sample sizes of transactions for some strata may appear adequate say if the index is developed outside of a recession, but may become inadequate as an economy moves into and during a recession, when measurement really matters.²⁵

A more general formulation is to use a rolling window time dummy regression. For example, for 2005Q1 to 2015Q4, where 2005Q1=100.0, a 4-quarters rolling window has the first regression estimated over the first four quarters, 2005Q1 to 2005Q4, the second regression drops the first observation in this window, 2005Q1, and adds the next quarter, 2006Q1, and so forth. For example, where ${RP}_{RW2005Q1\to Q4}^{2005Q2}$ the index for 2005Q2, with 2005Q1 =100.00, from a rolling window regression based on 2005Q1 to 2005Q4 data, RW 2005Q1 → 2005Q4:

The overlap terms require explanation. Table 1 shows illustrative results for the first four periods of the index simply based on the results for $\exp \left({\hat{\delta }}_t\right)\times 100.0$ from a rolling window regression for 2005Q1 to 2005Q4. The next window regression is estimated from 2005Q2 to 2006Q1 data. This window extends the results into the next quarter, 2006Q1 (2005Q2=100.0). There is a need to similarly extend the 2005Q1=100.0 index. An overlap of the two indexes for 2005Q4 allows us to rescale the 2006Q1 index from the 2005Q2 =100 window to 2005Q1 =100, that is: $\frac{101.9}{101.5}\times 101.0=101.4$.

Table 1.

Illustrative linking of results from rolling window regression

Table 1.

Illustrative linking of results from rolling window regression

Period	Rolling window 2005Q1 to 2005Q4 time dummy 2005Q1=100.0	Rolling window 2005Q2 to 2006Q1 time dummy 2005Q2=100.0	4-quarter rolling window index 2005Q1=100.0
2005Q1	100.0		100.0
2005Q2	101.2	100.0	101.2
2005Q3	101.1	102.3	101.1
2005Q4	100.9	101.5	100.9
2006Q1		101.0	$\frac{101.9}{101.5}\times 101.0=101.4$

View Table

Table 1.

Illustrative linking of results from rolling window regression

Period	Rolling window 2005Q1 to 2005Q4 time dummy 2005Q1=100.0	Rolling window 2005Q2 to 2006Q1 time dummy 2005Q2=100.0	4-quarter rolling window index 2005Q1=100.0
2005Q1	100.0		100.0
2005Q2	101.2	100.0	101.2
2005Q3	101.1	102.3	101.1
2005Q4	100.9	101.5	100.9
2006Q1		101.0	$\frac{101.9}{101.5}\times 101.0=101.4$

There is a trade-off here. The 4-quarters rolling window smoothes and lags the RPPI results to their detriment given the need for a timely indicator. However, with limited sample sizes available, it can provide more reliable results through more detailed stratification and smaller standard errors and thus confidence intervals.

Compilers of the index would gain from experimental RPPIs being estimated at different frequencies of rolling windows, including where possible, adjacent-period regressions and, where appropriate, provide users with studies of/regular data on smoothed as well as adjacent-period results, akin to the spirit of measures of core inflation and consumer price indexes.

Hedonic characteristics indexes: a linear functional form

Consider first two linear hedonic regression, as given by equation (2), and repeated below as equations (16) and (17)—but adopting the simplification that the constants ${\hat{\gamma }}_k^0$ and ${\hat{\gamma }}_k^t$ are included in the summations as k=0 where $z_{0,i}^0=1$ and $z_{0,i}^t=1$ — in their respective reference period 0 and successive periods t=1,….,T:

and for simplicity of exposition, hereafter k=0 designates the constant for which $z_{0,i}^t=1$.

Constant quality hedonic property price indexes can be defined in two immediately apparent ways. Both require a comparison of the price change of a constant basket of characteristics priced from a hedonic regression in period 0 and again in period t, yet in the first definition it is a constant period 0 basket and in the second a constant period t basket.

Consider a constant period 0 basket of characteristics; we take the averages of each k quality characteristic ${\overline{z}}_k^0$ in period 0, and ask what would be the price of a property with these k average characteristics if sold in period t. This predicted price is then compared with a valuation of the self-same average characteristics using the estimated period 0 hedonic regression. We compare estimated prices of constant period 0 average characteristics. A constant period t basket of characteristics ${\overline{z}}_k^{\mathrm{t}}$ is similarly defined.

The Dutot (ratio of arithmetic means) hedonic base (reference) period 0 index (DHB)³⁰ has in the numerator period 0 mean characteristics valued at period t characteristic-prices and in the denominator period 0 mean characteristics valued at period 0 characteristic-prices:

and a Dutot hedonic current period t quality index is defined as:

If, in a perfect market, preferences change and the implicit prices of one characteristic, say an additional bedroom, increase at an above average rate; other things being equal, utility-maximizing buyers would substitute expenditure towards other characteristics, say more overall space. The use of a constant period 0 characteristic basket, ${\overline{z}}_k^0$ would understate price increases—the $\frac{{\hat{\gamma }}_k^0{\overline{z}}_k^0}{\underset{k=0}{\overset{K}{\mathrm{\Sigma }}}{\hat{\gamma }}_k^0{\overline{z}}_k^0}$ expenditure weights in equation (21) do not reflect the substitution away from characteristics with above average price increases—and of a constant period t characteristic basket, ${\overline{z}}_k^t$, overstate it. This is because, as we show in section VC, the constant quality price change of each characteristic from equations (19) and (20) are implicitly weighted by the estimated relative values of the characteristic. For example, using the notation in equation (15) and equations (16) and (19):

For the aforementioned substitution bias relating to characteristics, a geometric mean of equations (19) and (20)—a hedonic Fisher-type price index number—is justifiable on grounds of economic theory, axiomatic properties, and intuition.³¹

The theory of hedonic regressions can be found in Rosen (1974), Triplett (1987), Feenstra (1995)—and for an application, Silver (1999)—Diewert (2003b), and Silver (2004); the theory of Laspeyres and Paasche bounds is in Konūs (1924) and of substitution effects warranting a (superlative) geometric mean of a Laspeyres and Paasche formula, in Diewert (1976, 1978 and 2004).

Note that the denominator in equation (19) is the imputed or predicted price, rather than actual price, in period 0, $h_k^0\left({\overline{z}}_k^0\right)$, and similarly in the numerator of equation (20) we use the imputed or predicted price rather than actual price in period t, $h_k^t\left({\overline{z}}_k^t\right)$. In calculating equation (19) we take the ratio of two imputations: the imputed price of ${\overline{z}}_k^0$ valued at period t characteristic prices in the numerator and at period 0 characteristic prices in the denominator—a dual imputation. For a linear form the average predicted price in period 0 from an Ordinary least squares regression is equal to the average actual price, $h_k^0\left({\overline{z}}_k^0\right)={\overline{p}}_k^0$ and, though equation (19) is hardly complex, it can be calculated with a “single imputation” as the much simpler:

We return to issues of dual versus single imputation later in this section and in section IV.