Producer Price Index Manual: Theory and Practice ... - METAC

7. Treatment of Quality Change
7.129 The first column provides the results from
a linear regression model, the dependent variable
being price. The first variable is processor speed
with a coefficient of 2.731; a unit MHz increase in
processing speed leads to an estimated £2.731 increase
(positive sign) in price. A change from 733
MHz to 933 MHz would be valued at an estimated
200 (2.731) = £546.20. The coefficient is statistically
significant, its difference from zero (no effect)
not being due to sampling errors at a 0.1 percent
level of significance. This estimated coefficient
is based on a multivariate model; the coefficient
measures the effect of a unit change in processing
speed on price having controlled for the effect
of other variables in the equation. The result
of 3.261 in equation (7.22) was based on just one
variable and did not benefit from this. That number
is different from this improved result.
7.130 The brand variables are dummy intercepts
taking values of 1 if, for example, it is a Dell computer
and zero otherwise. While brands are not in
themselves quality characteristics, they may be
proxy variables for other factors such as afterservice
reliability. The inclusion of such brand
dummies also reflects segmented markets as communities
of buyers as discussed in Chapter 21, Appendix
21.1. Similar dummy variables were
formed for other makes and models, including the
Compaq Presario and Compaq Prosignia. The
Compaq Deskpro was, however, omitted to form
the benchmark against which other models are
compared. The coefficient on Dell is an estimate of
the difference between the worth of a Dell and a
Compaq Deskpro, other variables being constant
(that is, £1,330.78 cheaper). Similarly, an Intel
Pentium III commands a premium estimated at
£282.78 over an AMD Athlon.
7.131 The estimate for processor speed was
based on data for Dell and Compaq PCs. If the adjustment
for quality is between two Dell PCs, it
might be argued that data on Compaq PCs should
be ignored. Separate regressions could be estimated
for each make, but this would severely restrict
the sample size. Alternatively an interaction
term or slope dummy can be used for variables that
are believed to have a distinctive brand-interaction
effect. Take Dell × Speed, which takes the value
of speed when the PC is a Dell and zero otherwise.
The coefficient on this variable is 1.714 (see Table
7.4); it is an estimate of the additional (positive
sign) price arising for a Dell PC over and above
that already arising from the standard valuation of
a 1 MHz increase in speed. For Dell PCs, it is
2.731 + 1.714 = £4.445. Therefore, if the replacement
Dell PC is 200 MHz faster than the unavailable
PC, the price adjustment to the unavailable
PC is to add 200 × £4.445 = £889. Interactive
terms for other variables can similarly be defined
and used. The estimation of regression equations is
easily undertaken using econometric or statistical
software, or data analysis functions in spreadsheets.
An understanding of the techniques is given
in many texts, including Kennedy (2003) and
Maddala (1988). In Chapter 21, Appendix 21.1,
econometric concerns particular to the estimation
of hedonic regressions are discussed.
2
7.132 The R is the proportion of variation in
price explained by the estimated equation. More
formally, it is 1 minus the ratio of the variance of
n
t t
the residuals ( p − pˆ ) 2
/ n, of the equation to
∑
i=
1
t t
the variance of prices ∑ ( ) 2
i
−
i
i
n
i
i=
1
p p / n. The bar on
the R 2 denotes that an appropriate adjustment for
degrees of freedom is made to this expression,
which is necessary when comparing equations with
different numbers of explanatory variables. At
2
2
0.934 R is high. However, high R can be misleading
for the purpose of quality adjustment.
First, such values inform us that the explanatory
variables account for much of price variation. This
may be over a relatively large number of varieties
of goods in the period concerned. This is not the
same as implying a high degree of prediction for
an adjustment to a replacement product of a single
brand in a subsequent time period. For their accuracy,
predicted values depend not just on the fit of
the equation, but also on how far the characteristics
of the product whose price is to be predicted
are from the means of the sample. The more unusual
the product, the higher the prediction probability
interval. Second, R 2 informs us as to the
proportion of variation in prices explained by the
estimated equation. It may be that 0.90 is explained,
while 0.10 is not. If the dispersion in
prices is large, this still leaves a large absolute
margin of prices unexplained. Nonetheless, a high
2
R is a necessary condition for the use of hedonic
adjustments.
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