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Estimation Bias in Quality-Adjusted Hedonic Price Indices
Mick Silver
INTRODUCTION
The paper examines least-squares (LS) estimation bias for log-linear hedonic models.
It is a comprehensive treatment doing so for estimates of:
the constant
predicted prices imputation of missing prices
matched models and composite price indices
hedonic indices the expected value of hedonic price ratio and the ratio of expected hedonic prices
weighted and equally-weighted hedonic regressions
pooled (time dummy variable) hedonic price indices
The paper also has extensive, sound empirical support utilising the results from a hedonic study of car prices in the Netherlands over the period 1990-99. The hedonic regressions had high EMBED Equation.3 =0.93+ with sample sizes in 1990: 1,816 and 1999: 2037 models.
This is an important issue.
It is important because the log-linear is useful functional form when used for adjustments to non-comparable replacements based on a hedonic regression estimated in some previous period. It is better, a priori, to assume the (log-linear) proportionate contribution to price has remained constant rather than the (linear) absolute contribution. So tests of functional form should not be the only arbiter of functional form, though such tests in studies by the author generally favour log-linear over linear.
Double-logarithmic function suffers from problems with use of dummy variables.
Also hedonic indices, composite or otherwise, use predicted prices or the coefficient on the time dummy from a log-linear form and this requires interpretation. We need to know when/whether the likely error is serious and what to do if it is.
Diewert (2002) shows that log price hedonic estimates have better axiomatic properties than linear ones.
Things we can do:
1. Preventative: Avoid the problem:
use a non-linear estimator the bias is in least squares (LS).
for hedonic indices use arithmetic (linear) superlative rather than geometric superlative they give similar results - the bias is in the LS log-linear form.
2. Apply treatment: the paper gives expressions for the necessary corrections.
Some points
Interpretation of coefficients on dummy variables from log-linear forms.
Variance estimator
The focus of the paper is on estimation bias for the parameters, and not their standard errors. Garderen and Shah (2002) derive an approximation an unbiased variance estimator for dummy variables from log-linear functions which may be usefully applied. They find the SE (bias and variance), and its MSE (variance), to be smaller for the uncorrected estimates than the corrected ones there is more uncertainty to the corrected values. While we must do the corrections, it is important to keep an eye on SEs.
Coefficient estimator
First note that an interpretation of a dummy variable coefficient from a semi-log form requires transformation (Kennedy, 1981):
EMBED Equation.3 0.0100.01010.0500.05130.0750.07790.1000.10520.1500.16180.2000.22140.2500.28400.2750.3165
But there is an inconsistency in using this result (outlined at the end of this note and in their paper) if we want the expected value of the exponent of the residuals from the log-linear to equal 0, and that of the untransformed model to equal 1. We correct for this inconsistency by using
EMBED Equation.3 . (1)
Not an exact correction, but empirically very good (Garderen and Shah, 2002).
Leverage and predicted prices using LS for log-linear
They show in their equation (10) that
EMBED Equation.3 . (2)
Again we have EMBED Equation.3 downward bias inside the sample, but we also have leverage. Note the nature of the multiplicative relationship. High leverage hr only potentially affects EMBED Equation.3 , it also requires that EMBED Equation.3 is not close to zero.
Here the r can be thought of as an individual model or an average. When we predict we have to recognise that leverage affect the prediction and have to be controlled for. Most econometric software produce leverage values so the adjustments should be easy. It is good to see leverage raised as an issue.
New appearing models and old models being dumped are likely to have high leverages - Heravi and Silver (2002) so the adjustment is a potential issue.
Leverage not just important here. Silver (2002) has shown leverage to be important for the implicit weight given to observations in OLS and WLS estimators. So it is not just an issue for log-linear bias. It applies to linear functional forms. An OLS vector of ( estimates is a weighted average of the individual y elements, the prices of individual models,
EMBED Equation.3 = (XTX)-1XTy (3)
where the matrix X are the explanatory variable and (XTX)-1XT are the weights given to the prices. The EMBED Equation.3 estimate is a weighted average of prices, y. Consider also a WLS estimator where weights are expenditure shares:
EMBED Equation.3 = (XTWX)-1XTWy (4)
It is apparent from (3) and (4) that outliers with unusual values of X will have a stronger influence, or leverage, in determining EMBED Equation.3 , than an observation which is one of a group clustered in a small area. The leverage for observation t is given by:
ht = Xt (XTX)-1XtT where EMBED Equation.3 . (5)
Sometimes we make adjustments to non-comparable models based on previous regressions that exclude the new model. Does this affect the estimated coefficient and its predicted price? Even for linear equations, the difference between the hedonic coefficients with the tth observation omitted and included given by:
EMBED Equation.3 (t) - EMBED Equation.3 (XTX)-1XtT EMBED Equation.3 (6)
where ht and t are relatively large the effect of the tth observation on at least some of EMBED Equation.3 is likely to be substantial. Thus high leverage ht only potentially affects EMBED Equation.3 , it also requires that t are not close to zero. It follows that including the tth observation in the regression affects the fitted value for that observation by:
Xt EMBED Equation.3 = Xt EMBED Equation.3 (t) + EMBED Equation.3 (7)
So leverage is important, not only in the context of this paper, but also for weighting and prediction. The paper by Ball and Fenwick at this conference has some ideas to motivate the sampling to exclude non-comparable replacements with high leverage.
Extent of log-linear bias for hedonic indices
Jan van Dalen and Ben Bode also examine quality-adjusted price indices. We consider the ratio of predicted prices levels for a constant basket (their B) can be Laspeyres or Paasche-type. LS Bias is given by
EMBED Equation.3 (8)
They find that high explanatory power (low EMBED Equation.3 ) is not a sufficient condition to remove the bias.
EMBED Equation.3 (9)
is almost negligible, due to the leverage effects, in their study. But the bias due to
EMBED Equation.3 (10)
is large in spite of the good EMBED Equation.3 (low EMBED Equation.3 ). Why?
If EMBED Equation.3 and EMBED Equation.3 close to zero, then the bias is small;
but if one year is poor and other good, then bias is large; and
if one year is poor and other year poor, then bias is small, but estimates subject to other forms of bias.
So EMBED Equation.3 is not low enough and must vary considerably.
But EMBED Equation.3 was very high???? Triplett (2004: Chapter 4) noted he reconsidered the empirical effects of this adjustment in his previous 1989 study of PCs and found little difference, as did Silver and Heravi (2002) for washing machines.
However, we need homoskedasticity to reduce this form of bias. Functional form must consistently map DGP and specification include relevant variables.
Their empirical work.
An substantial difference in results: for Laspeyres, for example - the index fell to 0.963 while ML fell to 0.509
First, EMBED Equation.3 did change: all the results are for chained indices, smaller for fixed base? Consider their EMBED Equation.3 estimates:
Bias chained linksBias fixed base comparisons EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 2.0281.8440.9121050.9121051.5070.8449310.7706660.9250.7475160.5760850.8560.9660880.5565490.7810.9631940.5360650.690.955520.5122210.5240.9203510.4714230.5981.0376930.4891920.7521.0800420.528348
Fixed base comparisons also need large adjustments.
Consider Berndt and Rappaport (2002) for PCs who come to similar conclusions. They advise a non-linear estimator if it is found that EMBED Equation.3 differs from EMBED Equation.3 . Berndt and Rappaport (2002) first tested for PCs by taking the squared residuals and regression on the time dummies to reject the null of constancy, decisively rejected. But the bias depends on the magnitude of the difference. Their results for PCs found the variances of the residuals falling sharply in, for example, 1997 compared with 1993:
YEAR
1989
1993
1997
2001All(1)
.24188
.20437
.07004
.06929Desktop(2)
.23039
.32602
.14901
.24495Mobile desktop(3)
.14011
.16219
.12121
.07757 Berndt and Rappaport (2002: Table 3)
The annual compound rate of fall for the index was 33.59% (log-linear) compared with 35.38% (non-linear estimator), a difference of 1.79 percentage points. The adjustment in (10) brought it to 34.69 1.1 percentage point increase. For desktop PCs it closed most of the gaps. But they considered a variety of cases and sometimes, but always, the adjustment did little to close the gap. So both (9) and (10) can be important. Use of a parsimonious model generally increased the bias.
Consideration of the results by Berndt and Rappaport (2002) help confirm the potential importance of the bias, but also show that the factors underlying it and its extent will vary according to the product and the regression specification.
So what about the dummy time on a hedonic time dummy index?
EMBED Equation.3 (11)
At first sight this is a worst case scenario because we restrict the coefficients and the residuals to be the same by construction. A visit to (10) would cause concern. But the authors take this in their stride, and I think rightly so. If this restricted model in (11) is correct then the estimate of EMBED Equation.3 required is:
EMBED Equation.3 (12)
which ignores the fact that EMBED Equation.3 and EMBED Equation.3 may differ. Although we can construct EMBED Equation.3 from the residuals of a pooled model, this is not the thinking behind the method. It asks: what if the coefficients and error variances were constrained to be the same? All that could then change would be the intercept? This intercept is then the required estimate of quality-adjusted price change. But what if a log-linear form was used? Then what would be the price change? We only need the adjustment for the dummy variable as above in (12). This answers the counterfactual question.
Now there is a sense in which if the constraints on the error structure were less restrictive, we would have more faith in the index. Indeed the homoskedasticity would lead to less bias in the standard errors and more reliable test results. But these constraints do not lead to bias in the estimated dummy variable coefficient. There is no further bias to the adjustment in (12) above to the answer to the counterfactual question asked and its interpretation. The subject of the paper is not to question the dummy variable method: it is to provide an answer when a log-linear transform is used, and this it does.
SUMMARY
It is a thorough treatment with empirical support. Just about all of what I have said supports their work. I can see in the near future appropriate adjustments being more readily undertaken, followed by the use of non-linear estimators. It is a fortunately tractable problem thanks in part to this paper.
The Inconsistency
Consider
EMBED Equation.3 where EMBED Equation.3 (A.1)
this is the untransformed process, the relationship of interest about which we want to make inferences. To do so we use least squares (LS) estimators of the EMBED Equation.3 in (2), the transformed equation:
EMBED Equation.3 where EMBED Equation.3 ~ EMBED Equation.3 (A.2)
this is what we estimate
and
We would like EMBED Equation.3 and EMBED Equation.3 . But there is an inconsitency
EMBED Equation.3 non-transformed
EMBED Equation.3 transformed
if we want EMBED Equation.3 ,
EMBED Equation.3 then EMBED Equation.3 : it is not zero; it is too low, so the estimated constant is biased downwards, as is the predicted price for a given set of characteristics.
The (LS) bias to a EMBED Equation.3 where EMBED Equation.3 can be estimated using EMBED Equation.3 .
For a semi-log the estimate is downwards biased, but only for the intercept, the LS estimators\of the other coefficients, EMBED Equation.3 , are unbiased.
They note: the bias of EMBED Equation.3 is to the estimated model. The adjustment is undertaken son that EMBED Equation.3 where EMBED Equation.3 is from the semi log. The authors note (page 6) that the bias lies in (1), the estimated semi-log, not in (2), the exponents. The distinction is to use with Kennedys standard advice of using EMBED Equation.3 and EMBED Equation.3 and not EMBED Equation.3 .
References
Berndt, E.R. and Rappaport N.J. (2002) Hedonics for Personal Computers: A Re-examination of Selected Econometric Issues, Draft Paper presented at NBER Summer Institute; see also Berndt, E.R. and Rappaport N.J. (2001). Price and Quality of Desktop and Mobile Personal Computers: A Quarter-Century Historical Overview, American Economic Review, 91, 2, 268-273.
Diewert W.E. (2002). Hedonic Regressions: A Review of Some Unresolved Issues, Mimeo, Department of Economics, University of British Columbia.
Kennedy P.E. (1981) Estimation with Correctly Interpreted Dummy Variables in Semi-Logarithmic Equations, American Economic Review, 71, p. 801.
Silver, M and Heravi, S. (2002), Why the CPI Matched Models Method May Fail Us: Results from an Hedonic and Matched Experiment Using Scanner Data, European Central Bank (ECB) Working Paper No. 144, Frankfurt: ECB.
Silver, M. (2002), The Use of Weights in Hedonic Regressions: the Measurement of Quality-Adjusted Price Changes. Mimeo, Cardiff Business School, Cardiff University.
Triplett, J.E. (2004), Handbook on Hedonic Indexes and Quality Adjustments in Price Indexes. Paris: OECD forthcoming
Van Garderen K.J. and Shah C. (2002) Econometrics Journal, Exact Interpretation of Dummy Variables in Semi-Logarithmic Equations, Econometrics Journal, 5, 149-159.
There would be no such bias in a linear form which is likely to have a more restrictive error form. An advantage of a log-linear functional form against a linear one is that it will reduce herteroskedasticity. It would be a strange ruling to announce further error in the log-linear estimate due to bias in the dummy variable, as against a linear form, when the very use of the log-linear form is to reduce such heteroskedasticity. An advantage of the time dummy variable method is thus that it only requires minimal adjustment as in (12) to the time dummy coefficient estimate compared with hedonic indices.
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