Hedonic Imputation versus Time Dummy Hedonic Indexes
  • 1 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund
  • | 2 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

Contributor Notes

Author’s E-Mail Address: msilver@imf.org

Statistical offices try to match item models when measuring inflation between two periods. However, for product areas with a high turnover of differentiated models, the use of hedonic indexes is more appropriate since they include unmatched new and old models. There are two main competing approaches to hedonic indexes are hedonic imputation (HI) indexes and dummy time hedonic (HD) indexes. This study provides a formal analysis of exactly why the results from the two approaches may differ and discusses the issue of choice between these approaches. An illustrative study for desktop PCs is provided.

Abstract

Statistical offices try to match item models when measuring inflation between two periods. However, for product areas with a high turnover of differentiated models, the use of hedonic indexes is more appropriate since they include unmatched new and old models. There are two main competing approaches to hedonic indexes are hedonic imputation (HI) indexes and dummy time hedonic (HD) indexes. This study provides a formal analysis of exactly why the results from the two approaches may differ and discusses the issue of choice between these approaches. An illustrative study for desktop PCs is provided.

I. Introduction

The purpose of this paper is to compare two main and quite distinct approaches to the measurement of hedonic price indexes: time dummy hedonic indexes and hedonic imputation indexes. Both approaches not only correct price changes for changes in the quality of items purchased, but also allow the indexes to incorporate matched and unmatched models. They provide a means by which price change can be measured in product markets where there is a rapid turnover of differentiated models. However, they can yield quite different results. This paper provides a formal exposition of the factors underlying such differences and the implications for choice of method. We consider both weighted and unweighted hedonic regression models. Unweighted hedonic regression models will be considered in sections 2 and 3 below. These models are of course useful in a sampling context where information on the quantity or value of sales (or purchases) is unavailable. Weighted hedonic regression models are considered in sections 4 and 5 below. The weighting is chosen so that if we are actually in a matched model situation for the two periods being considered, then the resulting hedonic regression measures of price change resemble standard superlative index number formulae.

The standard way price changes are measured by national statistical offices is through the use of a matched models methodology. Using this methodology, the details and prices of a representative selection of items are collected in a base reference period and their matched prices collected in successive periods so that the prices of “like” are compared with “like”. However, if there is a rapid turnover of available models, then the sample of product prices used to measure price changes becomes unrepresentative of the category as a whole. This is as a result of both new unmatched models being introduced (but not included in the sample), and older unmatched models being retired (and thus dropping out of the sample). Hedonic indexes use matched and unmatched models and in doing so put an end to the matched models sample selection bias.2 The need for hedonic indexes can be seen in the context of the need to reduce bias in the measurement of the U.S. consumer price index (CPI), which has been the subject of three major reports: the Stigler (1961) Committee Report, the Boskin (1996) Commission Report and the Schultze and Mackie (2002) Committee on National Statistics Panel Report. Each found the inability to properly remove the effect on price changes of changes in quality to be a major source of bias. Hedonic regressions were considered to be the most promising approach to control for such quality changes, though the Schultze panel cautioned for the need for further research on methodology:

“Hedonic techniques currently offer the most promising approach for explicitly adjusting observed prices to account for changing product quality. But our analysis suggests that there are still substantial unresolved econometric, data, and other measurement issues that need further attention.” C.L. Schultze and C. Mackie (2002; 6).

At first sight the two approaches to hedonic indexes appear quite similar. Both rely on hedonic regression equations to remove the effects on price of quality changes. They can also incorporate a range of weighting systems and can be formulated as a geometric, harmonic or arithmetic aggregator function of quality adjusted prices, and as chained or direct, fixed base comparisons. Yet they can give quite different results, even when using comparable weights, functional forms and the same method of making comparisons over periods. This is due to the fact that they work on different averaging principles. The dummy variable method constrains hedonic regression parameters to be the same over time. A hedonic imputation index conversely allows the quality adjustment parameters to change in each period and undertakes two sets of quality adjustments to prices for each comparison of prices between two periods and then averages over these two comparisons.

There has been some valuable research on the two approaches3 though no formal analysis has been presented, to the authors’ knowledge with one exception4, of the factors governing the differences between the approaches. Berndt and Rappaport (2001) and Pakes (2003) have highlighted the fact that the two approaches can give different results and both of these papers advise the use of hedonic imputation indexes when parameters are unstable, a proposal which will be considered in sections 5 and 7 below.

Section 2 below looks at a simple unweighted two period time dummy variable hedonic regression model. We focus on the estimation of the time dummy estimate of the change in log prices going from period 0 to 1 but we represent this measure of overall log price change as a difference in log price levels for the two periods. In section 3, we take the same unweighted model but run separate hedonic regressions for both periods and use these regression parameters to form two imputed measures of constant quality log price change. These two measures are then averaged to obtain an overall imputed measure of log price change.5 An exact expression for the difference in constant quality log price change between the time dummy and imputation measures is also developed in section 3. It is found that in order for these two overall measures to differ, we require:

  • differences in the two variance covariance matrices pertaining to the model characteristics in each period;

  • differences in average amounts of model characteristics present in each period;6 and

  • differences in estimated hedonic coefficients for the two separate hedonic regressions.

The analysis in sections 2 and 3 is repeated in the weighted context in sections 4 and 5.Section 6 provides an empirical study for desktop PCs and section 7 concludes by discussing the issue of choice between the approaches in light of the theoretical and empirical findings.

Appendix 1 considers two alternative methodologies for constructing measures of overall log price change using the hedonic imputation methodology where two separate hedonic regressions are estimated for the two period under consideration. The first methodological approach is due to Court (1939; 108) where individual prices in each period are quality adjusted using their characteristics vectors and the characteristics prices obtained from one of the two hedonic regressions and then the resulting quality adjusted prices are compared across the two periods. Finally, the resulting two measures of quality adjusted overall log price change are averaged. In the second methodological approach to hedonic imputation indexes, due originally to Griliches (1967), the mean vector of characteristics that pertains to the models observed in period 0 is calculated and then the distance between the two hedonic regressions at this mean characteristics point is calculated, which generates a first measure of overall price change (the Laspeyres measure of log price change). The Paasche measure of overall log price change is calculated using the mean vector of characteristics that pertains to the models observed in period 1 and then the distance between the two hedonic regressions at this mean characteristics point is calculated. Finally the two estimates of overall log price change are averaged. Appendix 1 shows that these two methodological approaches to hedonic imputation indexes lead to exactly the same numerical estimates of overall price change.

It is often thought that a major advantage of the time dummy variable method for obtaining measures of overall log price change is that a standard error for the log price change is obtained. In Appendix 2, a method for obtaining approximate standard errors for the Laspeyres and Paasche hedonic imputation measures of log price change is derived.

II. Unweighted Time Dummy Hedonic Regressions

We begin by considering a simple unweighted two period time dummy variable hedonic regression model. We assume that there are N(t) observations on the prices, pnt, of various models n in period t for t = 0,1. Observation n in period t has a vector of K characteristics associated with it, say [zn1t,zn2t,…,znKt] for t = 0,1 and n = 1,2,…,N(t). The time dummy regression model has the following form:

(1)lnpntynt=αt+Σk=1Kznktγk+εnt;t=0,1;n=1,2,N(t)

where the εnt are independently distributed normal variables with mean 0 and constant variance and α0, α1, γ1, …,γK are parameters to be estimated. The parameters α0 and α1 are measures of the average level of constant quality prices of the items in period 0 and 1 respectively and the γ1, …,γK are quality adjustment factors for the K characteristics; i.e., γk is the contribution to the log price of the product of adding an extra unit of characteristic k. Note that we have parameterized the time dummy hedonic regression model in a slightly different way to the way it is usually done since we do not have an overall constant term in the regression plus a time dummy variable in period 1; instead we have separate constant terms for each period. The overall measure of logarithmic price change going from period 0 to period 1 is α1 − α0.7 Let 1t and 0t be vectors of ones and zeros of dimension N(t), let y0 and y1 be the N(0) and N(1) dimensional vectors of period 0 and 1 logarithms of product prices, let ε0 and ε1 be the N(0) and N(1) dimensional vectors of period 0 and 1 stochastic disturbances and let Z0 and Z1 be matrices of the product characteristics in periods 0 and 1 respectively. Then the model defined by (1) can be written in matrix notation as follows:

(2)y0=10α0+00α1+Z0γ+ε0;
(3)y1=01α0+11α1+Z1γ+ε1.

Let α0*1*1*,…,γK* be the maximum likelihood or least squares estimators for the parameters that appear in (2) and (3). Then letting e0 and e1 be the vectors of least squares residuals for equations (2) and (3) respectively, the following equations will be satisfied by the parameter estimates and the data:

(4)y0=10α0*+00α1*+Z0γ*+e0;
(5)y1=01α0*+11α1*+Z1γ*+e1.

Let y ≡ [y0T,y1T]T and e ≡ [e0T,e1T]T and define ϕ* ≡ [α0*1*1*,…,γK*]T. Now rewrite (4) and (5) as

(6)y=Xϕ*+e.

It is well known that the columns of the X matrix are orthogonal to the vector e of least squares residuals; i.e., we have

(7)XTe=XT[yXϕ*]=02+K.

The first two equations in (7) are equivalent to the following two equations:8

(8)10Ty0=N(0)α0*+10TZ0γ*;
(9)11Ty1=N(1)α1*+11TZ1γ*.

Equations (8) and (9) can be used to solve for the following period 0 and 1 constant quality log price levels:

(10)α0*={10Ty0/N(0)}{10TZ0γ*/N(0)}=10T[y0Z0γ*]/N(0);
(11)α1*={11Ty1/N(1)}{11TZ1γ*/N(1)}=11T[y1Z1γ*]/N(1).

Note that 1tTyt/N(t) is the arithmetic average of the log prices in period t for t = 0,1. Furthermore, note that 10TZ0/N(0) is the arithmetic average of the amounts of each characteristic that are present in the period 0 models and 11TZ1/N(1) is the corresponding arithmetic average amount of each characteristic that is present in the period 1 models. Thus each αt* is equal to the average of the log prices for the models present in period t less a quality adjustment consisting of the inner product of the characteristic prices γ* with the average amount of each characteristic across the models that are present in period t. Alternatively, the second set of equalities in equations (10) and (11) shows that each αt* is equal to the arithmetic average of the quality adjusted log prices, yt − Ztγ*, for the models present in that period. In any case, the (unweighted) hedonic time dummy estimate of the change in log prices going from period 0 to 1, LPHD, is the following difference in the log price levels:9

(12)LPHDα1*α0*.

For later reference, we work out an expression for the estimated characteristic prices, γ*. Recall equation (6), y = Xϕ* + e, which defined the N(0)+N(1) by 2+K matrix X. We rewrite X as follows:

(13)X=[V,Z]

where ZT ≡ [Z0T,Z1T] and V is an N(0)+N(1) by 2 matrix which has the first column equal to [10T,01T]T and second column equal to [00T,11T]T. Now solve the least squares minimization problem that corresponds to (6) in two stages. In the first stage, we condition on γ and minimize with respect to the components of α ≡ [α01]T. The resulting conditional least squares estimator for α is:

(14)α(γ)(VTV)1VT[yZγ].

The second stage minimization problem is the problem of minimizing f(γ) with respect to the components of γ where f is defined as follows:

(15)f(γ)[yZγVα(γ)]T[yZγVα(γ)]=[yZγV(VTV)1VT(yZγ)]T[yZγV(VTV)1VT(yZγ)]using(14)=[MyMZγ]T[MyMZγ]=[yZγ]TMTM[yZγ]=[yZγ]TM[yZγ]since M=MT and M2=M

where the projection matrix M is defined as follows:

(16)MIV(VTV)1VT.

A simple way to solve the problem of minimizing f(γ) with respect to y is to make use of the third equality in (15); i.e., define the projections of y and Z onto M as follows:

(17)y*My;Z*MZ.

Using definitions (17), it can be seen that

(18)f(γ)=[y*Z*γ]T[y*Z*γ].

Thus the solution to the second stage least squares minimization problem is:

(19)γ*(Z*TZ*)1Z*Ty*.

Once γ* has been determined by (19), then we can use (14) or (8) and (9) to determine the least squares estimators for α0* and α1*.

Using the definition of V, it can be shown that the projection matrix M defined by (16) is block diagonal, with the two main diagonal blocks M0 and M1 defined as follows:

(20)M0I01010T/N(0);M1I11111T/N(1)

where I0 and I1 are identity matrices of dimension N(0) and N(1) respectively. Using (20), we can determine more precisely what the vector y* equal to My and the matrix Z* equal to MZ look like. Let y*T = [y0*Ty1*T] and Z*T = [Z0*T,Z1*T]. Then using (20), we have:

(21)yt*=yt1t1tTyt/N(t);t=0,1;
(22)Zt*=Zt1t1tTZt/N(t);t=0,1.

Thus each projected vector yt* is equal to the corresponding period t log price vector yt less a vector of ones times the average of the log prices for period t, Σn=1N(t) ynt/N(t), and each projected matrix Zt* is equal to the corresponding period t characteristics matrix Zt less a column vector of ones times a row vector equal to the average of the characteristics in each model for period t, [Σn=1N(t) zn1t/N(t), Σn=1N(t) zn2t/N(t), …, Σn=1N(t) znKt/N(t)]. Thus yt* and Zt* are simply the corresponding yt and Zt with the period means subtracted from each component.

Using the block diagonal structure of M, it can be verified that we have the following alternative representation for the least squares characteristics prices γ* defined by (19):

(23)γ*(Z0*TZ0*+Z1*TZ1*)1[Z0*Ty0*+Z1*Ty1*].

We now turn our attention to hedonic imputation indexes.

III. Unweighted Hedonic Imputation Indexes

Instead of running one hedonic regression where the same characteristics prices are used to quality adjust prices in each period, we can run two entirely separate hedonic regressions with separate characteristics prices, γ0 in period 0 and γ1 in period 1. Thus using the same notation as in section 2, our models now are:

(24)y0=10β0+Z0γ0+η0;
(25)y1=11β1+Z1γ1+η1

where η0 and η1 are independently distributed normal random variables with means zero and constant variance within each period. Let β0*, γ10*,…,γK0* be the maximum likelihood or least squares estimators for the parameters that appear in (24) and let β1*11*,…,γK1* be the maximum likelihood or least squares estimators for the parameters that appear in (25). Then letting u0 and u1 be the vectors of least squares residuals for equations (24) and (25) respectively, the following equations will be satisfied by the parameter estimates and the data:

(26)y0=10β0*+Z0γ0*+u0;
(27)y1=11β1*+Z1γ1*+u1.

The counterparts to equations (8) and (9) in the present context are:

(28)10Ty0=N(0)β0*+10TZ0γ0*;
(29)11Ty1=N(1)β1*+11TZ1γ1*.

Equations (28) and (29) lead to the following counterparts to equations (8) and (9):

(30)β0*={10Ty0/N(0)}{10TZ0γ0*/N(0)}=10T[y0Z0γ0*]/N(0);
(31)β1*={11Ty1/N(1)}{11TZ1γ1*/N(1)}=11T[y1Z1γ1*]/N(1).

Recall that the hedonic time dummy estimate of the change in log prices going from period 0 to 1, LPHD, was defined by (12) as the difference in the log price levels, α1* − α0*. In the present context, we cannot simply take the difference between β1* and β0* as a measure of constant quality log price change between periods 0 and 1, because the quality adjustment parameters, γ0* and γ1*, are different between the two periods. However, we can use the period 0 parameters, γ0*, to form estimates of quality adjusted log prices for the models present in period 1 and then take the average of the resulting quality adjusted log prices, which we denote by δ1*:

(32)δ1*{11Ty1/N(1)}{11TZ1γ0*/N(1)}=11T[y1Z1γ0*]/N(1).

Note that the above estimate of a period 1 log price level is analogous to β1* defined by (31) except that the period 0 hedonic quality adjustment factors, γ0*, are used in (32) whereas the period 1 hedonic quality adjustment factors, γ1*, were used in (31). Since the period 0 and 1 estimated price levels, β0* and δ1*, use the same quality adjustment factors γ0* in order to form constant quality log prices in each period, we can take the difference δ1* less β0* as a measure of quality adjusted log price change between periods 0 and 1.10 We call this hedonic imputation measure of log price change a Laspeyres type measure of price change and denote it by ϕl* ≡ δ1* − β0*. This measure of overall log price change depends asymmetrically on the characteristics price vector γ0* that was obtained from the period 0 hedonic regression. It can be seen that we can obtain an alternative measure of log price change between the periods using the period 1 hedonic regression characteristics price vector γ1*. Thus use the period 1 characteristics price vector, γ1*, to form estimates of quality adjusted log prices for the models present in period 0 and then take the average of the resulting quality adjusted log prices, which we denote by δ0*:

(33)δ0*{10Ty0/N(0)}{10TZ0γ1*/N(0)}=10T[y0Z0γ1*]/N(0).

Note that the above estimate of a period 0 log price level is analogous to β0* defined by (30) except that the period 1 hedonic quality adjustment factors, γ1*, are used in (33) whereas the period 0 hedonic quality adjustment factors, y0*, were used in (30). Since the period 0 and 1 estimated price levels, δ0* and β1* use the same quality adjustment factors γ1* in order to form constant quality log prices in each period, we can take the difference β1* less δ0* as a second measure of log price change between periods 0 and 1. We call this hedonic imputation measure of log price change a Paasche type measure of price change and denote it by) ϕP* ≡ β1* − δ0*.11

Following Griliches (1971b; 7) and Diewert (2003b; 12), it seems preferable to take a symmetric average of the above two estimates of log price change over the two periods. We choose the arithmetic mean12 as our symmetric average and define the (unweighted) hedonic imputation estimate of the change in log prices going from period 0 to 1, LPHI, as follows:13

(34)LPHI(½)ϕL*+(½)ϕP*=½[δ1*β0*]+½[β1*δ0*]=½{11T[y1Z1γ0*]/N(1)10T[y0Z0γ0*]/N(0)+11T[y1Z1γ1*]/N(1)10T[y0Z0γ1*]/N(0)}using(30)-(33)=11T{y1Z1[½γ0*+½γ1*]}/N(1)10T{y0Z0[½γ0*+½γ1*]}/N(0).

Recall that in the hedonic time dummy method for quality adjusting log prices yt for each period t, we used the characteristics quality adjustments defined by Ztγ*, where γ* was a constant across periods vector of quality adjustment factors. Looking at the right hand side of (34), it can be seen that the hedonic imputation method for quality adjusting log prices in each period is similar but now the period t vector of quality adjustments is Zt[½γ0*+½γ1*] instead of Ztγ*. Thus for the hedonic imputation method of quality adjusting prices, the average of the two separate hedonic regression estimated quality adjustment factors, ½γ0*+½γ1*, replaces the single regression estimated quality adjustment factor vector γ* that was used in the hedonic time dummy method for quality adjustment.

Note also that if the models are the same in the two periods being considered, then N(0) equals N(1) (equals N say) and Z0 equals Z1, so that the two characteristics matrices are identical, then LPhi defined by (34) collapses to 11Ty1/N − 1Ty0/N, which is the arithmetic mean of the period 1 log prices less the arithmetic mean of the period 0 log prices. Thus under these conditions, there is no need to run hedonic regressions; the usual matched model methodology can be used.

Using (12) and (34), we can form the following expression for the difference in the overall log price change using the two methods for quality adjustment:

(35)LPHDLPHI=11TZ1γ*/N(1)+10TZ0γ*/N(0)+11TZ1[½γ0*+½γ1*]/N(1)10TZ0[½γ1*+½γ1*]/N(0)=11TZ1[½γ0*+½γ1*γ*]/N(1)10TZ0[½γ0*+½γ1*γ*]/N(0)={[11TZ1/N(1)][10TZ0/N(0)]}{½γ0*+½γ1*γ*}.

Thus the hedonic time dummy and hedonic imputation measures of log price change will be identical if either of the following two conditions are satisfied:

(36)11TZ1/N(1)=10TZ0/N(0);
(37)γ*=½γ0*+½γ1*.

Condition (36) says that the average amount of each characteristic for the models present in period 1 equals the corresponding average amount of each characteristic for the models present in period 0. Condition (37) says that the time dummy vector of quality adjustment factors, γ*, is equal to the arithmetic average of the two separate hedonic regression estimates for the quality adjustment factors, ½γ0*+½γ1*.

Condition (36) is somewhat unanticipated. It tells us that if the configuration of characteristics present in the models in each period does not change much, then the hedonic time dummy and hedonic imputation estimates of quality adjusted price change will be much the same, even if characteristic valuations change over the two periods.

Condition (37) can be refined. Recall (23) in the previous section, which provided a formula for the hedonic time dummy vector of quality adjustment factors, γ*. The techniques that were used to establish (23) can be used in order to establish the following expressions for the period 0 and 1 least squares estimates γ0* and γ1* that appear in (26) and (27):

(38)γ0*=(Z0*TZ0*)1Z0*Ty0*;γ1*=(Z1*TZ1*)1Z1*Ty1*

where the yt* and Zt* are the demeaned yt and Zt as in the previous section.14 Now premultiply both sides of (23) by the matrix Z0*TZ0* + Z1*TZ1* and we obtain the following equation:15

(39)[Z0*TZ0*+Z1*TZ1*]γ*=Z0*Ty0*+Z1*Ty1*=Z0*TZ0*[Z0*TZ0*]1Z0*Ty0*+Z1*TZ1*[Z1*TZ1*]1Z1*Ty1*=Z0*TZ0*γ0*+Z1*TZ1*γ1*using (38).

Equation (39) tells us that if γ0* equals γ1*, then γ* is necessarily equal to this common vector. We now use equation (39) in order to evaluate the following expression:

(40)2[Z0*TZ0*+Z1*TZ1*][½γ0*+½γ1*γ*]=[Z0*TZ0*+Z1*TZ1*][γ0*+γ1*]2[Z0*TZ0*γ0*+Z1*TZ1*γ1*]using (39)=[Z0*TZ0*][γ1*γ0*][Z1*TZ1*][γ1*γ0*]=[Z1*TZ1*Z0*TZ0*][γ1*γ0*].

Now premultiply both sides of (40) by (1/2)[Z0*TZ0* + Z1*TZ1*]−1 and substitute the resulting expression for ½γ0*+½γ1* − γ* into equation (35) in order to obtain the following expression for the difference between the hedonic dummy estimate of constant quality price change and the corresponding symmetric hedonic imputation estimate:

(41)LPHDLPHI=(1/2){[11TZ1/N(1)][10TZ0/N(0)]}[Z0*TZ0*+Z1*TZ1*]1[Z1*TZ1*Z0*TZ0*][γ1*γ0*].

Using (41), it can be seen that the hedonic time dummy and hedonic imputation measures of log price change will be identical if any of the following three conditions are satisfied:

  • 11TZ1/N(1) equals 10TZ0/N(0) so that the average amount of each characteristic across models in each period stays the same or

  • Z1*TZ1*equals Z0*TZ0*so that the model characteristics variance covariance matrix is the same across periods16 or

  • y1* equals y0* so that separate (unweighted) hedonic regressions in each period give rise to the same characteristics quality adjustment factors.17

In the following sections, we will adapt the above material to cover the case of weighted hedonic regressions.

IV. Weighted Time Dummy Hedonic Regressions

We now consider a weighted two period time dummy variable hedonic regression model. We again assume that there are N(t) observations on the prices, pnt, of various models n in period t for t = 0,1 but we now assume that the quantities purchased for each model n in period t, qnt, are also observable. Model n in period t again has the vector of K characteristics associated with it, [zn1t,zn2t,…,znKt] for t = 0,1 and n = 1,2,…,N(t). The expenditure share of model n in period t is

(42)sntpntqnt/Σi=1N(t)pitqit;t=0,1;n=1,2,N(t).

Let st = [s1t,…,sN(t)t]T denote the period t expenditure share vector for t = 0,1 and let St denote the diagonal period t share matrix that has the elements of the period t expenditure share vector st running down the main diagonal for t = 0,1. The matrix (St)1/2 is the square root matrix of St; i.e., the positive square roots of the elements of the period t expenditure vector st are the diagonal elements of this diagonal matrix for t = 0,1. As in the previous sections, yt is the vector of period t log price changes and Zt is the period t model characteristics matrix for t = 0,1.

With the above notational preliminaries out of the way, the weighted time dummy regression model that is the counterpart to the unweighted model defined earlier by (2) and (3) is defined as follows:18

(43)(S0)1/2y0=(S0)1/2[10α0+00α1+Z0γ]+ε0;
(44)(S1)1/2y1=(S1)1/2[01α0+11α1+Z1γ]+ε1

where the εt vectors have elements εnt that are independently distributed normal variables with zero means and constant variances.

Let α0*1*1*,…,γK* be the maximum likelihood or least squares estimators for the parameters that appear in (43) and (44). Then letting e0 and e1 be the vectors of least squares residuals for equations (43) and (44) respectively, the following equations will be satisfied by the parameter estimates and the data:

(45)(S0)1/2y0=(S0)1/2[10α0*+00α1*+Z0γ*]+e0;
(46)(S1)1/2y1=(S1)1/2[01α0*+11α1*+Z1γ*]+e1

The counterparts to equations (10) and (11) are now:

(47)α0*=s0Ty0s0TZ0γ*=s0T[y0Z0γ*];
(48)α1*=s1Ty1s1TZ1γ*=s1T[y1Z1γ*];

Note that stTyt = Σn=1N(t)snt lnpnt is the period t expenditure share weighted average of the log prices in period t for t = 0,1. Furthermore, note that stTZt is a 1 by K vector whose kth element is equal to the period t expenditure share weighted average Σn=1N(t)sntznkt of the amounts of characteristic k that are present in the period t models for t = 0,1 and k = 1, …,K. Thus each αt* is equal to the expenditure share weighted average of the log prices for the models present in period t less a quality adjustment consisting of the inner product of the characteristic prices γ* with an expenditure share weighted average amount of each characteristic across the models that are present in period t. Alternatively, the second set of equalities in equations (47) and (48) shows that each αt* is equal to the period t expenditure share weighted average of the quality adjusted log prices, yt − Zty*, for the models present in that period. Now use (47) and (48) in order to define the following weighted hedonic time dummy estimate of the change in log prices going from period 0 to 1: LPWHD, is the following difference in the log price levels:

(49)LPWHDα1*α0*=s1T[y1Z1γ*]s0T[y0Z0γ*].

Thus the weighted hedonic time dummy estimate of the change in log prices is equal to a period 1 expenditure share weighted average of the quality adjusted log prices, y1 − Z1γ*, less a period 0 expenditure share weighted average of the quality adjusted log prices, y0 − Z0y*.

For later reference, we can adapt the methodology presented at the end of section 2 in order to work out an explicit expression for the estimated characteristic prices, γ*. Recall the definitions of the diagonal matrices S0 and S1. Define S as a block diagonal matrix that has the blocks S0 and S1 on the main diagonals and let S1/2 be the corresponding square root matrix; i.e., this matrix has the elements of the period 0 expenditure share s0 and the period 1 expenditure share vector s1 running down the main diagonal. Adapting the analysis in section 1, we need to replace the y vector in that section by S1/2y and the Z matrix in that section by S1/2Z, the V matrix by S1/2V and define the counterpart M° to the projection matrix M defined by (16) as follows:

(50)M°IS1/2V(VTSV)1VTS1/2=IS1/2VVTS1/2

where the second equality in (50) follows from the fact that VTSV equals I2, a two by two identity matrix. Now define y° and Z° in terms of M° and the original y vector and Z matrix as follows:

(51)y°M°S1/2y=[IS1/2VVTS1/2]S1/2y using (50)=S1/2[IVVTS]y;
(52)Z°M°S1/2Z=[IS1/2VVTS1/2]S1/2Z using (50)=S1/2[IVVTS]Z.

The new vector of time dummy quality adjustment factors γ* can now be defined as the following counterpart to (19):

(53)γ*(Z°TZ°)1Z°Ty°.

Once γ* has been determined by (53), then we can use (47) and (48) to determine the weighted least squares estimators for α0* and α1*.

It is possible to express the γ* defined by (53) in a more transparent way using our definitions of the matrices Z, V and S. Recall that in section 2, the demeaned log price change vectors yt* defined by (20) and the demeaned characteristics matrices Zt* defined by (21) proved to be useful. In those definitions, we used simple unweighted means. In the present context, we use expenditure share weighted average means as follows:

(54)yt*yt1tstTyt;t=0,1;
(55)Zt*Zt1tstTZt;t=0,1.

Let the nth component of the vector yt* be ynt* and let the nth row of the N(t) by K dimensional matrix Zt* be znt* for t = 0,1 and n = 1,2,…,N(t). Then it can be shown that the vector of characteristics prices γ* defined by (53) can be written in terms of the components of the demeaned log price change vectors yt*, the components of the expenditure share vectors st and the rows of the demeaned characteristics matrices Zt* as follows:

(56)γ*=[Σn=1N(0)sn0zn0*Tzn0*+Σn=1N(1)sn1zn1*Tzn1*]1[Σn=1N(0)sn0zn0*Tzn0*+Σn=1N(1)sn1zn1*Tyn1*].

Note that Σn=1N(t)sntznt*Tznt* can be interpreted as a period t expenditure share weighted sample variance covariance matrix for the model characteristics present in period t and Σn=1N(t)sntznt*Tznt* can be interpreted as a period t expenditure share weighted sample covariance matrix between the period t log prices and the characteristics of the models present in period t.

We now generalize the analysis on unweighted hedonic imputation indexes presented in section 3 to the weighted case.

V. Weighted Hedonic Imputation Indexes

Using the notation explained in the previous section, the two separate weighted hedonic regressions that are counterparts to the separate unweighted regressions (24) and (25) are now (57) and (58) below:

(57)(S0)1/2y0=(S0)1/2[10β0+Z0γ0]+η0;
(58)(S1)1/2y1=(S1)1/2[11β1+Z1γ1]+η1

where η0 and η 1 are independently distributed normal random variables with means zero and constant variance within each period. Let β0*10*,…,γK0* be the maximum likelihood or least squares estimators for the parameters that appear in (57) and let β1*11*, …,γK1* be the maximum likelihood or least squares estimators for the parameters that appear in (58). Then letting u0 and u1 be the vectors of least squares residuals for equations (57) and (58) respectively, the following equations will be satisfied by the parameter estimates and the data:

(59)(S0)1/2y0=(S0)1/2[10β0*+Z0γ0*]+u0;
(60)(S1)1/2y1=(S1)1/2[11β1*+Z1γ1]+u1.

The weighted counterparts to equations (30) and (31) are:

(61)β0*=s0Ty0s0TZ0γ0*=s0T[y0Z0γ0*];
(62)β1*=s1Ty1s1TZ1γ1*=s1T[y0Z1γ1*]

As in section 3, we cannot simply take the difference between β1* and β0* as a measure of constant quality log price change between periods 0 and 1, because the quality adjustment parameters, γ0* and γ1*, are again different between the two periods. As in section 3, we can use the period 0 parameters, γ0*, to form estimates of quality adjusted log prices for the models present in period 1 and then take the period 1 weighted average of the resulting quality adjusted log prices, which we denote by δ1*:

(63)δ1*s1Ty1s1TZ1γ0*=s1T[y1Z1γ0*].

Note that the above estimate of a period 1 log price level is analogous to β1* defined by (62) except that the period 0 hedonic quality adjustment factors, γ0*, are used in (63) whereas the period 1 hedonic quality adjustment factors, γ1*, were used in (62). Since the period 0 and 1 estimated price levels, β0* and δ1*, use the same quality adjustment factors γ0* in order to form constant quality log prices in each period, we can again take the difference δ1* less β0* as a measure of quality adjusted log price change between periods 0 and 1.19 This measure of overall log price change depends asymmetrically on the characteristics price vector γ0* that was obtained from the period 0 hedonic regression. It can be seen that we can obtain an alternative measure of log price change between the periods using the period 1 hedonic regression characteristics price vector γ1*. Thus use the period 1 characteristics price vector, γ1*, to form estimates of quality adjusted log prices for the models present in period 0 and then take the period 0 weighted average of the resulting quality adjusted log prices, which we denote by δ0*:

(64)δ0*s0Ty0s0TZ0γ1*=s0T[y0Z0γ01].

Note that the above estimate of a period 0 log price level is analogous to β0* defined by (61) except that the period 1 hedonic quality adjustment factors, γ1*, are used in (64) whereas the period 0 hedonic quality adjustment factors, γ0*, were used in (61). Since the period 0 and 1 estimated price levels, δ0* and β1* use the same quality adjustment factors γ1* in order to form constant quality log prices in each period, we can take the difference β1* less δ0* as a second measure of log price change between periods 0 and 1.20

Again following Diewert (2003b; 20) and Haan (2003; 14) (2004), it seems preferable to take a symmetric average of the above two estimates of log price change over the two periods. We again choose the arithmetic mean as our symmetric average and define the weighted hedonic imputation estimate of the change in log prices going from period 0 to 1, LPWHI, as follows:21

(65)LPWHI½[δ1*β0*]+½[β1*δ0*].=½{s1T[y1Z1γ0*]s0T[y0Z0γ0*]+s1T[y1Z1γ1*]s0T[y0Z0γ1*]}=s1T{y1Z1[½γ0*+½γ1*]}s0T{y0Z0[½γ0*+½γ1*]}.

Recall that in the weighted hedonic time dummy method for quality adjusting log prices yt for each period t, we used the characteristics quality adjustments defined by Ztγ*, where γ* was a constant across periods vector of quality adjustment factors. Looking at the right hand side of (65), it can be seen that the weighted hedonic imputation method for quality adjusting log prices in each period is similar but now the period t vector of quality adjustments is Zt[½γ0*+½γ1*] instead of Ztγ*.

Using (49) and (65), we can form the following expression for the difference in the overall log price change using the two weighted methods for quality adjustment:

(66)LPWHDLPWHI=s1T[y1Z1γ*]s0T[y0Z0γ*](s1T{y1Z1[½γ0*+½γ1*]}s0T{y0Z0[½γ0*+½γ1*]})={s1TZ1s0TZ0}{½γ0*+½γ1*γ*}.

Thus the hedonic time dummy and hedonic imputation measures of log price change will be identical if either of the following two conditions are satisfied:

(67)s1TZ1=s0TZ0;
(68)γ*=½γ0*+½γ1*.

Condition (67) says that the expenditure share weighted average amount of each characteristic for the models present in period 1 equals the corresponding expenditure share weighted average amount of each characteristic for the models present in period 0. Condition (68) says that the time dummy vector of quality adjustment factors, γ*, is equal to the arithmetic average of the two separate weighted hedonic regression estimates for the quality adjustment factors, ½γ0*+½γ1*.

As in section 3, condition (68) can be strengthened. Recall (56) in the previous section, which provided a formula for the hedonic time dummy vector of quality adjustment factors, γ*. Using the same notation to that used in the previous section, we can establish the following expressions for the period 0 and 1 weighted least squares estimates γ0* and γ1* that appear in (59) and (60):

(69)γ0*=[Σn=1N(0)sn0zn0*Tzn0*]1[Σn=1N(0)sn0zn0*Tzn0*];
(70)γ1*=[Σn=1N(1)sn1zn1*Tzn1*]1[Σn=1N(0)sn1zn1*Tyn1*].

where the yt* and Zt* are the demeaned yt and Zt as in the previous section. Now premultiply both sides of (56) by the matrix [Σn=1N(0)sn0zn0*Tzn0*n=1N(1)sn1zn1*Tzn1*] and we obtain the following equation which is the weighted counterpart to (40):

(71)[Σn=1N(0)sn0zn0*Tzn0*+Σn=1N(1)sn1zn1*Tzn1*]γ*=Σn=1N(0)sn0zn0*Tyn0*+Σn=1N(1)sn1zn1*Tyn1*=[Σn=1N(0)sn0zn0*Tzn0*][Σn=1N(0)sn0zn0*Tzn0*]1[Σn=1N(0)sn0zn0*Tyn0*]+[Σn=1N(1)sn1zn1*Tzn1*][Σn=1N(1)sn1zn1*Tzn1*]1[Σn=1N(1)sn1zn1*Tyn1*]=[Σn=1N(0)sn0zn0*Tzn0*]γ0*+[Σn=1N(1)sn1zn1*Tzn1*]γ1* using (69) and (70).

Equation (71) tells us if γ0* equals γ1*, then γ* is necessarily equal to this common vector. We now use equation (71) in order to evaluate the following expression:

(72)2[Σn=1N(0)sn0zn0*Tzn0*+Σn=1N(1)sn1zn1*Tzn1*][½γ0*+½γ1*γ*]=[Σn=1N(0)sn0zn0*Tzn0*+Σn=1N(1)sn1zn1*Tzn1*][γ0*+γ1*]using(71)2{[Σn=1N(0)sn0zn0*Tzn0*]γ0*+[Σn=1N(1)sn1zn1*Tzn1*]γ1*}=[Σn=1N(0)sn0zn0*Tzn0*][γ1*γ0*][Σn=1N(1)sn1zn1*Tzn1*][γ1*γ0*]=[Σn=1N(1)sn1zn1*Tzn1*Σn=1N(0)sn0zn0*Tzn0*][γ1*γ0*].

Now premultiply both sides of (72) by (1/2)[Σn=1N(0)sn0zn0*Tzn0*n=1N(1)sn1zn1*Tzn1*]−1 and substitute the resulting expression for ½γ0*+½γ1* − γ* into equation (66) in order to obtain the following expression for the difference between the hedonic dummy estimate of constant quality price change and the corresponding symmetric hedonic imputation estimate using weighted hedonic regressions in both cases:

(73)LPWHDLPWHI=[s1TZ1s0TZ0][½γ0*+½γ1*γ*]=½[s1TZ1s0TZ0][Σn=1N(0)sn0zn0*Tzn0*+Σn=1N(1)sn1zn1*Tzn1*]1[Σn=1N(1)sn1zn1*Tzn1*Σn=1N(0)sn0zn0*Tzn0*][γ1*γ0*].

Using (73), it can be seen that the weighted hedonic time dummy and the weighted hedonic imputation measures of log price change will be identical if any of the following three conditions are satisfied:

  • s1TZ1 equals s0TZ0 so that the period expenditure share weighted amount of each characteristic across models in each period stays the same or

  • Σn=1N(1)sn1zn1*Tzn1* equals Σn=1N(0)sn0zn0*Tzn0* so that the expenditure share weighted model characteristics variance covariance matrix is the same in the two periods or

  • γ1* equals γ0* so that separate (weighted) hedonic regressions in each period give rise to the same characteristics quality adjustment factors.

Which weighted method of quality adjustment is “best”? If the weighted average amounts of each characteristic are much the same in the two periods being considered so that s1TZ1 is close to s0TZ0, or if the expenditure share weighted model characteristics variance covariance matrices are similar across periods or if the separate weighted hedonic regression quality adjustment factors do not change much across the two periods, then it will not matter much which method is used, which is the new result that is demonstrated in this paper. If however, s1TZ1 is not close to s0TZ0, the expenditure share weighted model characteristics variance covariance matrices are different across periods and the two separate weighted hedonic regressions (57) and (58) generate very different estimates for the quality adjustment factors, γ0* and γ1*, then the method of quality adjustment could matter. If (57) and (58) are run together as a seemingly unrelated regression model and an F test rejects the equality of γ0 and γ1, then it seems sensible to use the weighted hedonic imputation method which does not depend on having γ0 equal to γ1 as does the hedonic time dummy method. If the F test does not reject the equality of γ0 and γ1 and there are a large number of characteristics in the model, then valuable degrees of freedom will be saved if the weighted time dummy hedonic regression model is used. However, in this case, since γ0 and γ1 are necessarily close, it should not matter much which method is used. Thus it seems that the hedonic imputation methods probably give rise to “better” quality adjustments than dummy variable methods. We will revisit this discussion in section 7 below.

VI. Empirical Illustration: Desktop Personal Computers (PCs)

The empirical study is of the measurement of changes in the quality-adjusted monthly prices of British desktop PCs in 1998. The data are monthly scanner data from the bar-code readers of PC retailers. The data amounted to 7,387 observations (a particular make and model of a PC sold in a given month in an either specialized or non-specialized PC store-type) representing a sales volume of 1.5 million models worth £1.57 billion. Table 1 shows that for the January to February price comparison there were 584 matched models available in both months for the price comparison. However, for the January to December price comparison only 161 matched models were available with 509 unmatched “old” models (available in January, but unmatched in December) and 436 unmatched “new” models (available in December but unavailable in January for matching).22 For product markets where there are a high proportion of unmatched models, Silver and Heravi (2005) demonstrated why matched model indexes suffer from sample selectivity bias and why hedonic indexes should be used instead.

Table 1.

Number of Matched and Unmatched Observations

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The calculation of hedonic indexes requires the estimation of hedonic regression equations. To simplify the illustration we first include only a single explanatory variable in the hedonic (price) regressions, the speed in MHz. The regressions were run separately for each month for the hedonic imputation indexes, and over January and the current month, including a dummy variable for the latter, for the hedonic dummy indexes. The estimated coefficients for speed in the hedonic regressions were statistically significant coefficients with the expected positive signs.23

Table 2 columns (1) and (2) show falls in the unweighted hedonic dummy and hedonic imputation indexes of 73.2 and 76.4 percent respectively, a difference of 3.2 percentage points. Columns (4)-(6) show the constituent elements of equation (41) that make up this difference: column (4) is the change in mean characteristics, [11TZ1/N(1)] − [10TZ0/N(0)]—average speed increased by 127 MHz. over the year; column (5) is the change in the variance-covariance characteristics matrix relative to its sum in the two periods, [Z0*TZ0* + Z1*TZ1*]−1 [Z1*TZ1* − Z0*TZ0*]—with one characteristic in this illustration it is the relative change in the variance of speed, falling in some early months but increasing thereafter; and column (6) is the change in the characteristic parameter estimates, [γ1* − γ0*]—the estimated parameters decreased over time.

Table 2.

Decomposition of Differences Between Unweighted Hedonic Time Dummy and Imputation Indexes for Desktop PCs, 1998

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The decomposition of the difference between the hedonic dummy and imputation indexes given in column (3) is exact as demonstrated in column (7)—the difference is equal to one-half of the product of columns (4), (5) and (6), following equation (41). We stress that the decomposition is based on the product of these constituent parts. If either of these changes is zero, then there will be no difference between the indexes. While it is clear that there were large increases in the mean speed of PCs over the year, column (4), they did not materialize in substantial difference between the formulas, being tempered by the smaller changes in columns (5) and (6). The formulation also provides insights into the factors behind any difference in the results from these methods. For example, in September and December the changes in the estimated parameters were about the same, yet the difference between the hedonic dummy and imputation indexes in column (3) was higher in December than in September, driven by the larger change in mean characteristics in December.

The indexes considered in Table 2 were unweighted and thus unreliable. Conventional index number theory requires that price changes should be weighted by relative expenditures shares and the self same requirement applies to hedonic indexes. In Sections 4 and 5 above weighted hedonic dummy and hedonic imputation indexes were formulated and equation (73) provided a decomposition of the difference between them. The results for PCs for weighted hedonic indexes, their difference, and the constituent elements underlying the difference are given in Table 3. Again the difference depends on the product of three terms: the change in the expenditure share weighted mean of each characteristic, [s1TZ1 − s0TZ0]; the change in the expenditure share weighted characteristics variance-covariance matrix, [Σn=1N(0)sn0zn0*Tzn0* + Σn=1N(1)sn1zn1*Tzn1*]−1n=1N(1)sn1zn1*Tzn1* − Σn=1N(0)sn0zn0*Tzn0*]; and the change in the parameters estimates, [γ1* − γ0*] from the separate (weighted) hedonic regressions in each period. Table 3 shows respective falls for the weighted hedonic dummy and weighted hedonic imputation indexes of 43.1and 50.6 percent (imputation) over the year, compared with falls in the corresponding unweighted indexes of 73.2 and 76.4 percent in Table 2: thus weighting matters. The decomposition in Table 3 follows equation (73) and is exact with (one-half of) the product of columns (4) to (6), in column (7), equaling the difference between the formulas in column (3). The differences are at their highest in September and December at over 7 percentage points, in part due to the relatively high parameter changes. Again comparing September with December, the parameter changes in column (6) are similarly high at −0.0039. But again the change in the mean speed in column (4) at 139.6 MHz. in December is much higher than its September figure of 94.45 MHz.24 This should translate into a very much lower difference between the formulas in September compared with December, but for the weighted results it does not. This is because the much higher December change in characteristics, column (4), is largely offset by the much lower change in the relative variance in column (5).

Table 3.

Decomposition of Differences Between Weighted Hedonic Time Dummy and Imputation Indexes for Desktop PCs, 1998

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The empirical example above was limited to a single price-determining quality characteristic variable for illustrative purposes. The decomposition for more than one characteristic variable is similar in principle to the case of a single explanatory variable, but the constituent items of equation (73) are matrices and it is the product of these matrices that is required to account for the difference between the formulas. The regression estimates are now based on three quality characteristics, speed in MHz., the hard disk capacity (HD), and random access memory (RAM), both in MB. The estimated coefficients for the three variables in the hedonic regressions were statistically significant with the expected positive signs.25 The results of the decomposition of the difference between weighted hedonic dummy and weighted hedonic imputation indexes based on multiple for the January with December comparison is presented below.

The weighted hedonic dummy index in December compared with January, based on the extended specification, fell by 51.5 percent, compared with the fall in the weighted hedonic imputation index of 63.3, a sizable difference of 11.8 percentage points. From Tables 2 and 3 we saw that weighting matters. Here we identify the importance of a fuller specification of the hedonic regression used and its effect on the magnitude of the index change and the spread between the two estimates. The fuller specification has led to an increase in the spread between the two indexes from 7.5 to 11.8 percentage points. Prices are estimated to have fallen further using the extended variable set: a fall for the weighted hedonic dummy hedonic of 51.5 compared with 43.1 and for the hedonic imputation index of 63.3 compared with 50.6 percent. Additional explanatory variables in a quality adjusted hedonic regression based index are probably preferable.26 Yet in spite of this illustration, a fuller specification of the hedonic regression need not necessarily lead to an increase in the difference between the formulas. If any component of an additional variable—its change in relative dispersion, covariance, parameter, or mean value—is negligible, then it will have little effect on the difference. Indeed, the overall impact of additional variables in the product of matrices in equation (73) may take a different sign and reduce the discrepancy. However, in general, product development in high technology products such as PCs takes the form of increased product differentiation (dispersion of characteristics values), improvements in many product dimensions at the same time (which will generally change means and covariances), and decreasing characteristic production costs and marginal utilities, as consumers realign their preferences to the new standards (which will generally lead to changes in parameter estimates). Thus our expectation is that the HI and HD approaches to measuring price change may frequently give different estimates of overall price change in dynamic markets.

The four terms in equation (73) are reproduced below for the December with January comparison, the product of which is equal to the difference between the formulas, i.e.,

½[s1TZ1s0TZ0][n=1N(0)sn0zn0*+n=1N(1)sn1zn1*]1[n=1N(1)sn1zn1*Tzn1*n=1N(0)sn0zn0*Tzn0*][γ1*γ0*]=½[139.593,180.140,14.405][0.000180.000140.000560.0000140.0000240.0000680.0000560,0000680.001136]×[1728.15168.0460.25168.03672.3193.9460.2193.9167.3] [0.007430.001270.00612]=0.1182.

The value of the element in the first row or column of the matrices are for the characteristic variable speed, the second, the capacity of the HD, and third, RAM. The vectors and matrices in turn are the change in the mean value of the characteristics, which were all positive; the sum of the inverted variance-covariance characteristics matrix over the two months; the change in the variance-covariance characteristics matrix over the two months; and the change in the estimated parameters. Thus in the third matrix, for example the diagonal values of: 1,728.1, −3,672.3, and 167.3 are the change in the variances of the characteristics speed, HD and RAM respectively. Note that in the last matrix, the parameter estimates for speed and RAM fall over the period by a similar amount, but the increase in the average value of speed (in the first matrix) is much more than that of the mean RAM size, and thus is a more significant driver of the difference between the formulas. There is also a marked increase in the average HD capacity, yet its parameter difference estimate at 0.00127 is positive, so that a higher marginal value is attached to it in December as compared with January, yet the absolute value of this change, and thus its impact on the difference, is lower than the corresponding change for speed.

VII. Concluding Remarks on the Choice Between Hedonic Imputation Indexes and Hedonic Dummy Hedonic Indexes

Having identified the factors behind the difference between the hedonic dummy and imputation indexes, we turn to consider which if any formula is more appropriate. Given that both approaches make symmetric use of information in the two periods and can be formulated to have the same functional form and weighting system, a plausible stance, when they produce different results, is to take a (geometric) mean of the two. Yet there may be reasons to prefer one against the other.

The main concern with the use of the hedonic time dummy index approach, as given by the respective unweighted and weighted equations (12) and (49), is that by construction, it constrains the parameters on the characteristic variables to be the same. Berndt and Rappaport (2001) found, for example, from 1987 to 1999 for desktop PCs, the null hypothesis of adjacent-year equality of the characteristics parameters to be rejected in all but one case. For mobile PCs the null hypothesis of parameter stability was rejected in eight of the 12 adjacent-year comparisons. Berndt and Rappaport (2001) preferred the use of hedonic imputation indexes if there was evidence of parameter instability. Pakes (2003), using quarterly data for hedonic regressions for desktop PCs over the period 1995 to 1999, rejected just about any hypothesis on the constancy of the coefficients. He also advocated hedonic imputation indexes on the grounds that “.…since hedonic coefficients vary across periods it [the hedonic dummy approach] has no theoretical justification.” Pakes (2003: 1593). The hedonic imputation method is inherently more flexible (in that it can deal with changes in purchasers’ valuations of characteristics over the periods being compared) than the hedonic dummy method and this is a big advantage for the HI method.

At first glance, if the separate period regression coefficients, γ0* and γ1*, are quite different, then there is a sense in which we are likely to have less confidence in an index based on constraining these coefficients to be the same. In this sense the concern over the use of dummy hedonic indexes is warranted. However, consider the respective unweighted and weighted equations (34) and (65) for the hedonic imputation indexes. These two hedonic imputation equations are also based on holding parameter estimates constant over time, but the parameter estimates are held constant at an average of the two separate hedonic regression prices for characteristics, ½γ0*+½γ1*, instead of the average parameter estimate arising from the constrained regression, as in the hedonic dummy approach. It is difficult to criticize the hedonic dummy method for using a constrained (average) value of the parameters for both periods of the comparison when the hedonic imputation index constrains the average of the parameters over the two periods to be the same.

Note also that the difference between the two approaches has been found to depend on three change factors: the change in the mean characteristics, relative variance-covariance characteristics matrix, and parameter estimates. More specifically it was found that the difference depends on the product of such changes. As such, parameter instability by itself need not by itself be a cause for concern. Even if parameters were unstable, the difference between the indexes may be compounded or mitigated by a small change in any of the other components.

Nevertheless, the essence of the HD method is that only one regression is run, with the data in both periods appearing as dependent variables and with the restriction that the characteristics are valued at common “prices” for the two periods. In this interpretation, HD is not as flexible because of these restrictions. Why are these restrictions imposed? Perhaps for three reasons:

  • To conserve degrees of freedom.

  • To give an unambiguous estimate of the amount of price change going from period 0 to 1. Because the regression surfaces are parallel, we can measure the distance between the two surfaces at any characteristics point z and get the same estimate of log price change, which is not the case in the HI methods.

  • To minimize the influence of outliers, particularly in situations where degrees of freedom are small.

In view of the above considerations, the advantages and disadvantages of the two methods can be seen: HI is “better” because it allows for changing characteristics prices over time; i.e., it is more “flexible” but at the cost of:

  • Using up more degrees of freedom and

  • At the cost of leading to a less reproducible estimate of overall price change between the two periods, since we have to condition on one or more “reasonable” z points to measure the distance between the two surfaces.

In practice, the last objection is not very serious; the Laspeyres and Paasche type estimates of price change are well established in index number theory as is the idea that these equally valid estimates of price change should be averaged in order to come up with a single measure of price change.

Thus all things considered, we favor HI methods unless degrees of freedom are very limited.

Triplett (2004) recognized that extensive product differentiation with a high model turnover is an increasing feature of product markets. The motivation for the use of hedonic regression techniques lies in the failure of the matched models method to adequately deal with price measurement in this context. Schultze and Mackie (2002) argued that hedonic indexes were the most promising approach to measuring price changes for such product markets, but advised that further research into such methods was needed: in particular, under what conditions will HD and HI measures of price change be different? This paper has provided answers to this question.

Appendix I. An Alternative Interpretation of the Hedonic Imputation Estimate of Log Price Change

In section 3 above, we derived an estimator for the logarithm of overall price change between the two periods, LPHI defined by (34), which we called the hedonic imputation estimate of the change in log prices going from period 0 to 1. We derived this estimator of price change following the methodology pioneered by Court (1939; 108); i.e., individual prices in each period were quality adjusted using their characteristics vectors and the characteristics prices obtained from one of the two hedonic regressions pertaining to the two periods under consideration and then the resulting quality adjusted prices were compared across the two periods. However, there is an alternative method for estimating price change across two periods when separate hedonic regressions are run for each period. In this second method, we calculate the mean vector of characteristics that pertains to the models observed in period 0 say and then calculate the distance between the two hedonic regressions at this mean characteristics point. This is called a Laspeyres type measure of price change. Then we calculate the mean vector of characteristics that pertains to the models observed in period 1 and calculate the distance between the two hedonic regressions at this second mean characteristics point, leading to a Paasche type measure of price change. Finally these two distances between the hedonic regression surfaces are averaged in order to obtain a final measure of price change between the two periods. This is the methodology originally proposed by Griliches (1967) and Dhrymes (1971; 111-112). In this Appendix, we show that the first and second hedonic imputation methods lead to the same overall estimate of price change.

We first consider the unweighted hedonic imputation model that was described in section 3. Recall the notation used in section 3 above where y0 was the N(0) dimensional vector of log model prices in period 0, y1 was the N(1) dimensional vector of log model prices in period 1 and Zt was the N(t) by K matrix of characteristics by model in period t for t = 0,1. Define the sample average of the log prices in period t, yt*, and the sample average vectors of model characteristics, zt*, as follows:

(A1)y0*10Ty0/N(0);y1*11Ty1/N(1);z0*10TZ0/N(0);z1*11TZ1/N(1).

Using definitions (A1), equations (28) and (29) in section 3 can be rewritten as follows:

(A2)y0*=β0*+z0*T+γ0*;
(A3)y1*=β1*+z1*T+γ1*.

For later reference, it can be seen that equations (A2) and (A3) imply the following expression for the difference in intercepts in the two hedonic regressions:

(A4)β1*β0*=y1*y0*z1*Tγ1*+z0*Tγ0*.

Now we are ready to define some estimators of the distance between the two hedonic regression surfaces. We define the Laspeyres type measure of log price change between periods 0 and 1, LPL, and the Paasche type measure of log price change, LPP, as follows:

(A5)LPLβ1*+z0*Tγ1*[β0*+z0*Tγ0*];
(A6)LPPβ1*+z1*Tγ1*[β0*+z1*Tγ0*].

It can be seen that (A5) and (A6) are both measures of the distance between the two hedonic regression surfaces: the Laspeyres type measure holds the characteristics vector constant at the average of the period 0 levels, z0*, while the Paasche type measure holds the characteristics vector constant at the average of the period 1 levels, z1*. Our final measure of log price change is the arithmetic average of the Laspeyres and Paasche type measures, which we call the Fisher type measure of log price change, LPF:

(A7)LPF(1/2)[LPL+LPP}=(1/2)[β1*+z0*Tγ1*(β0*+z0*Tγ0*)+β1*+z1*Tγ1*(β0*+z1*Tγ0*)]using (A6) and (A7)=β1*β0*+(1/2)[z0*T+z1*T][γ1*γ0*]=y1*y0*z1*Tγ1*+z0*Tγ0*+(1/2)[z0*Tγ1*z0*Tγ0*+z1*Tγ1*z1*Tγ0*]using(A4)=y1*y0*(1/2)[z0*Tγ1*+z0*Tγ0*z1*Tγ1*z1*Tγ0*]=y1*y0*(1/2)[γ1*+γ0*]T[z1*z0*]=LPHI

where LPHI was the hedonic imputation index defined by (34) in section 2. Thus the two hedonic imputation methods for defining an estimate of price change coincide in the unweighted case.27

Now consider the weighted hedonic imputation model that was described in section 5. The equally weighted sample averages of the log prices (y0* and y1*) and of the model characteristics (the vectors z0* and z1*) defined in (A1) above are now replaced by the following expenditure share weighted averages:

(A8)y0*s0Ty0;y1*s1Ty1;z0*s0TZ0;z1*s1TZ1.

Using the new definitions in (A8), it can be seen that equations (61) and (62) in section 5 imply that equations (A2) and (A3) continue to hold so that (A4) also holds using these new definitions. We can again define the Laspeyres and Paasche type measures of log price change by (A5) and (A4) where we use the new hedonic regression estimates for the period 0 weighted regression, β0* and γ0*, and for the period 1 weighted regression, β1* and γ1*, and the period 0 weighted average characteristics vector z0* for the Laspeyres measure LPL and the period 1weighted average characteristics vector z1* for the Paasche measure LPP. Now use LPL and LPP to define the Fisher measure LPF by the first line in (A7) and again we can show that this Fisher measure is equal to the weighted hedonic imputation index LPWHI defined by (65).

Thus we have shown that two rather different looking approaches to hedonic imputation indexes are equivalent.

Appendix II. A Method for Obtaining Approximate Standard Errors for the Hedonic Imputation Laspeyres and Paasche Measures of Log Price Change

We consider the unweighted case first. Recall that the Laspeyres type hedonic imputation measure of log price change was defined as

(A9)ϕL*δ1*β0*

where δ1* and β0* are defined by (32) and (30) respectively. These last two equations can be rewritten as follows:

(A10)N(1)δ1*=11Ty111TZ1γ0*;
(A11)N(0)β0*=10Ty110TZ0γ0*.

Recall also that the period 0 hedonic regression was written as (26) where β0* was the period 0 estimated log price level and γ0* was the period 0 vector of least squares estimates for the characteristics prices. Now use these estimated period 0 regression coefficients to quality adjust the period 1 log prices in the vector y1. After subtracting these quality adjustments from the vector of period 1 log prices y1, we are left with the period 1 vector v1 of quality adjusted prices less the period 0 log price level defined as follows:

(A12)v1y1[11β0*+Z1γ0*]=[y1Z1γ0*]11β0*.

Now run a least squares regression of the period 1 residual vector v1 on a constant term with coefficient ϕ0 The resulting least squares estimator for ϕ0 is:

(A13)ϕ0*11Tv1/N(1)={11T[y1Z1γ0*]/N(1)}N(1)β0*/N(1)using (A12)=δ1*β0*using (A10)=ϕL*using (A9).

Thus the Laspeyres type hedonic imputation measure of log price change ϕL* defined by (A9) is numerically equal to the least squares estimator ϕ0* of the constant term in a regression of period 1 quality adjusted log prices v1 defined by (A12) on a constant and the standard error on this auxiliary regression coefficient can serve as an approximate standard error for the Laspeyres hedonic imputation measure of constant quality log price change over the two periods under consideration.28

The above algebra can be repeated for the Paasche type hedonic imputation measure of log price change, which was defined as

(A14)ϕP*β1*δ0*

where β1* and δ0* are defined by (31) ansd (33) respectively. These last two equations can be rewritten as follows:

(A15)N(1)β1*=11Ty111TZ1γ1*.
(A16)N(0)δ0*=10Ty010TZ0γ1*;

Recall also that the period 0 hedonic regression was written as (27) where β1* was the period β1* estimated log price level and γ1* was the period 1 vector of least squares estimates for the characteristics prices. Now use these estimated period 1 regression coefficients to quality adjust the period 0 log prices in the vector y0. After subtracting these quality adjustments from the vector of period 0 log prices y0, we are left with the period 0 vector v0 of quality adjusted prices less the period 1 log price level defined as follows:

(A17)v0y0[10β1*+Z0γ1*]=[y0Z0γ1*]10β1*.

Now run a least squares regression of the period 0 residual vector v0 on a constant term with coefficient ϕ1. The resulting least squares estimator for ϕ1 is:

(A18)ϕ1*10Tv0/N(0)={10T[y0Z0γ1*]/N(0)}N(0)β1*/N(0)using (A17)=δ0*β1*using (A16)=ϕL*using (A14).

Thus the Paasche type hedonic imputation measure of log price change ϕP* defined by (A14) is numerically equal to minus the least squares estimator ϕ1* of the constant term in a regression of period 0 quality adjusted log prices v0 defined by (A17) on a constant and the standard error on this auxiliary regression coefficient can serve as an approximate standard error for the Paasche hedonic imputation measure of constant quality log price change over the two periods under consideration.

We leave the reader with the task of deriving the counterparts of these results to the case where we have weighted hedonic regressions.

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1

Our thanks to Ernst Berndt (MIT), Jan de Haan (Statistics Netherlands) and Alice Nakamura (University of Alberta) for helpful comments. The authors are at the University of British Columbia, the University of Cardiff, and the International Monetary Fund respectively.

4

The exception is Silver and Heravi (2007a) who considered the case of one characteristic and used a rather different methodological approach based on the bias generated by omitted variables in regression models.

5

An alternative interpretation of this measure of price change is derived in Appendix 1.

6

If the models are exactly the same in the two periods being considered, then this set of differences will be zero and the characteristics variance covariance matrices will also be identical. Hence the hedonic time dummy and hedonic imputation estimates of price change will be identical under these conditions. Thus the two methods will give rise to substantial conflicting estimates only in markets where there are many new models being introduced into the marketplace or many disappearing models (or both).

7

Our method of parameterization is equivalent to the standard method for parameterizing a time dummy hedonic regression model which is to have a common constant term for the two periods and a time dummy variable for the second period regression.

8

Diewert (2003a; 335) (2003b; 39) and Silver and Heravi (2005) used this orthogonality method of proof to provide an interpretation of the hedonic time dummy in terms of quality adjusted prices.

9

This methodology can be traced back to Court (1939; 109-111) as his hedonic suggestion number two. Note also that if the models are the same in the two periods being considered, then N(0) equals N(1) (equals N say) and Z0 equals Z1, so that the two characteristics matrices are identical and thus α1* − α0* = 11T[y1 − Z1γ*]/N(1) − 10T[y0 − Z0γ*]/N(0) = 11Ty1/N − 1Ty0/N, which is the arithmetic mean of the period 1 log prices less the arithmetic mean of the period 0 log prices. Thus under these conditions, there is no need to run a hedonic regression; the usual matched model methodology can be used.

10

This basic idea can be traced 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).

11

In Appendix 2, we develop a simple method for obtaining approximate standard errors for the hedonic imputation Laspeyres and Paasche measures of log price change between the two periods.

12

If we chose to measure price change instead of log price change, then the arithmetic mean estimator of log price

change converts into a geometric mean of the two measures of level price change, exp [δ1* − β0*] and exp [β1* − δ0*]

13

In Appendix 1, we show that the measure of quality adjusted change in log prices defined by (34), which followed the methodology due originally to Court (1939; 108), can also be interpreted as a measure of the distance between the two hedonic regressions. The principles of such measures were discussed in Griliches (1967) and Dhrymes (1971; 111-112) and further developed by Feenstra (1995) and Diewert (2003a; 341-344). Empirical studies using this approach include Berndt, Griliches and Rappaport (1995), Ioannidis and Silver (1999), Berndt and Rappaport (2001; 270), Koskimäki and Vartia (2001) and Silver and Heravi (2003) (2007b).

14

Note that Zt*Tyt* can be interpreted as a vector of covariances between the log prices in period t and the amounts of the characteristics present in the period t models while Zt*Tyt* can be interpreted as a variance covariance matrix for the model characteristics in period t.

15

In the single characteristic case, (40) tells us that γ1* is a weighted average of γ10* and γ11*.

16

This condition for equality is also somewhat unanticipated.

17

Using (38), we can obtain a more “fundamental” condition in terms of variance covariance matrices for the equality of γ0* to γ1*; namely the equality of (Z0*TZ0*)−1Z0*Ty0* to (Z1*TZ1*)−1Z1*Ty1*.

18

Diewert (2003b; 26) explained the logic behind the weighting scheme in the regression model defined by (43) and (44). For additional material on weighting in hedonic regressions, see Diewert (2005; 563) (2006; 13). Basically, the form of weighting that is used in (43) and (44) leads to measures of price change that are comparable (in the case where the characteristics of the models can be defined by dummy variables) to the “best” measures of price change in bilateral index number theory.

19

Haan (2003; 12) defined the exponential of this measure of log price change as the geometric Laspeyres hedonic imputation index of price change, except that for the matched models in both periods, he used actual prices rather than predicted prices.

20

Haan (2003; 13) defined the exponential of this measure of log price change as the geometric Paasche hedonic imputation index of price change, except that for the matched models in both periods, he used actual prices rather than predicted prices. Diewert (2003b; 13-14) considered similar hedonic imputation indexes except that he worked with ordinary Paasche and Laspeyres type indexes rather than geometric Paasche and Laspeyres type indexes.

21

The exponential of this measure of price change is approximately equal to Haan’s (2003; 14) geometric mean of his geometric Paasche and Laspeyres hedonic imputation indexes, which he regarded as an approximation to Törnqvist hedonic imputation index.

22

Bear in mind that some of the indexes estimated in this paper are also weighted by shares of sales values and that the fall off in the coverage of the matched sample by sales is even more dramatic: for the January to December comparison matched models made up only 71% of the January sales value and a mere 12% of the December sales value.

23

The F-statistics for the null hypothesis of coefficients being equal to zero averaged 34.2 for hedonic imputation indexes and 53.4 for hedonic dummy indexes, consistently rejecting the null at a 0.01% level and lower. The explanatory power of the estimated equations were naturally low for this specification with a single explanatory variable, especially since they did not include dummy variables on brand. Details of estimates from a fully specified model are available from the authors.

24

The multiplication is of the change in average speed by the parameter estimate for speed. The result is invariant to the units of measurement used for speed since an accordingly lower coefficient would result if the units of measurement for speed were say doubled.

25

The adjusted-R2 were 0.40 and 0.31 for the hedonic regression equations in January and November respectively. The specification could be extended to include many more variables including dummy variables for brands. The exposition here was simplified to illustrate the decomposition for more than one explanatory variable.

26

There are statistical caveats to this statement, i.e., it may not be useful to include additional explanatory variables if there are insufficient degrees of freedom or if multicollinearity between included and omitted variables is strong.

27

It can also be verified that LPL is equal to β1* − δ0* where δ0* was defined by (33) and LPP is equal to δ0* − β1* where δ1* was defined by (32).

28

This is only an approximate standard error because it is conditional on period 0 estimated parameters, β0* and γ0* which are subject to some sampling uncertainty. If we wanted to use the same methodology to obtain standard errors for the constant quality period 0 log price level, β0, and the constant quality period 1 log price level, δ1, then we would use the period 0 estimated characteristics prices γ0* in order to form period 0 and 1 quality adjusted log price vectors w0 and w1 defined as wt ≡ yt − Ztγ0* for t = 0,1. Now form two auxiliary regressions where w0 is regressed on a constant with coefficient β0 and w1 is regressed on a constant with coefficient δ1. The least squares estimators for β0 and δ1 turn out to be the β0* and δ1* defined by (A10) and (A11). The standard errors for these coefficients in the auxiliary regression can be used as approximate standard errors for the log price levels in the two periods. These standard errors are conditional on the estimated period 0 characteristics prices, γ0*. Of course, the original period 0 hedonic regression can be used in order to obtain an unconditional standard error for the period 0 log price level β0.