1. Workshop on Residential Property Price Indices
Empirical Issues in the Econometrics of Hedonic House Price Indexes
Esmeralda A. Ramalho
Joaquim J.S. Ramalho
Department of Economics, Universidade de Évora
CEFAGE-UE
Statistics Netherlands, February 2011

2. Introduction The construction of an hedonic index requires:
The specification of an hedonic function that relates the price (or a function of the price) to house characteristics;
The estimation of the parameters for the hedonic function;
The analysis of the suitability of the hedonic function;
The construction of the index, by using the estimates of the price or a function of the price.

3. Introduction Aims of the paper:
Clarify how the choice of the type of price index and the hedonic method determines the specification of the hedonic function
Look at alternative specifications and the major sources of mispecification;
Examine the alternative estimation methods
Propose a simple strategy to assess the validity of the hedonic function.

4. Notation Consider: N observations; Nt observations at period t
pit = price of dwelling i at period t
xit = (k+1)-vector of attributes of dwelling i at t
Hedonic function:
where p* and x* are functions of p and x, u is the error term, and is the (k+1)-vector of parameters to be estimated.

5. Hedonic Price Indexes
Unajusted Jevons and Dutot indexes
Jevons:

6. Hedonic Price Indexes
Unajusted Jevons and Dutot indexes
Dutot:

7. Hedonic Price Indexes Unajusted Jevons and Dutot indexes
Estimators when prices in 0 and s are estimated:

8. Hedonic Price Indexes Link type of index / pit*
Jevons: only if so that
Dutot: only if so that

9. Hedonic Price Indexes Link type of index / pit* Dutot with : since
Jevons with :
Only when the link is taken into account, the adjusted price indexes present the well known Paasche and Laspeyes forms

10. Hedonic Price Indexes Methods Determine essentially:
How often the regression models are estimated
The number of periods included in the regression (pooled data, adjacent periods, one period,…)
The need or not of time dummies
The imputation method includes as particular cases the other three methods considered here (time dummy, re-pricing, characteristics), although only under particular conditions of these methods

11. Hedonic Price Indexes Methods: Decomposition
where (a,b)=(0,s) and (a,b)=(s,0) for Laspeyres and Paasche indexes, respectively.

12. Hedonic Price Indexes Methods: Imputation
In cases where a=s (a=0), the numerator (denominator) simplifies to the mean of , , and only one regression, for 0 (s), is estimated.
Parameter constancy is not required

13. Hedonic Price Indexes Methods: Time Dummy
Let T be the time periods covered by the sample
For property i, define (T-1) dummy variables, Tit, Tit=1 if the property was sold at t and 0 otherwise. Let be the coefficient associated to Tit
The hedonic function is written as
where are attributes, except the period of sale
Requires parameter constancy, but leads to more efficient estimators than separate regressions

14. Hedonic Price Indexes Methods: Time Dummy The price indexes are
does not depend on the reference period
A regression needs to be estimated at each period

15. Hedonic Price Indexes Methods: Time Dummy
Using only data from 0 and s, this method is a particular case of the imputation: combining the separate regressions for this last method in one:
where is a parameter, is a k-vector of parameters, di=1 if data refers to period s and 0 otherwise, and excludes the constant from
Under parameter constancy, , this equals the time dummy regression with
A Chow test for allows the choice between time dummy and imputation (if the parameters are in fact stable, the time dummy is more efficient)

16. Hedonic Price Indexes Methods: Repricing ,
Requires the computation of unadjusted and quality indexes: ,
Is all cases requires only one regression: b can refer to any period or even pooled data (more efficient, but assumes parameter constancy in that period)
Coincides with imputation when (a,b)=(0,s) or (a,b)=(s,0).

17. Hedonic Price Indexes Methods: Characteristics The price indexes are ,
where is a (k+1)-vector of weights which may be averages of attributes across all houses in period a (in this case coincides with imputation)
Requires regressions at each t (unless coincides with imputation)
Parameter constancy is not required

18. Hedonic Price Indexes Methods: summary
Require parameter constancy: time dummy and re-pricing that uses pooled data
Regressions required:
Imputation: regressions at each t (but in a particular case, only one is required at moment 0)
Time dummy and characteristics: regressions at each t
Re-pricing: only one regression

19. Specification of the HF
Although the type of index and the hedonic method chosen impose some restrictions, there is lots of flexibility to define and to define strata for which parameter constancy is reasonable.
Explanatory variables:
Individual attributes: house size, lot size, type, age, .
Location: region/city/postal code
Neighbourhood: quality of public goods, local unemployment rate, …
Environmental: distance to bussiness district, assessibility of radial highway, air polution, …

20. Specification of the HF
How the attributes are included:
levels, logs (only if positive)
with squares (age, size) – influence on the price not constant for all values of the attribute
dummy (region, type of house)
interaction term (dummy*regressor) – joint influence of a qualitative and other attribute
Definition of the sub-sample for which the regression is valid: definition of dwelling strata by region or type of house for each of which the hedonic function is estimated

21. Specification of the HF
Specification failure
Failure in the specification of one of those aspects may cause the inconsistency of the parameters and indexes estimated.
Major mispecification problems:
Omission of relevant regressors (when the omitted variables are correlated with the included covariates)
Pure functional form failure
Parameters instability (across regions, types of houses and time)
Heteroscedasticity (absence of variance constancy across the observations) – merely causes the failure of standard inference procedures and not the inconsistency of the estimators.

22. Specification Analysis
We propose an integrated approach where all specification tests are implemented as tests for the joint significance of a set of artificial regressors
Strategy:
Heteroscedasticity testing – determines if all remaining inference procedures must be made robust to this problem
RESET test – sensitive to a wide variety of misspecifications
Specific tests: Chow test – sensitive to parameter inconstancy (time / space), Machado & Santos Silva test for sampling endogeneity

23. Specification Analysis
Heteroscedasticity
(homoscedasticity)
Breusch & Pagan (1979)
Artificial regression

24. Specification Analysis
RESET
(correct specification)
Ramsey (1969)
Artificial regression

25. Specification Analysis
Chow (1960)
(parameter constancy)
Artificial regression
where di=1 for houses of group s and 0 otherwise.

26. Specification Analysis
Machado & Santos Silva (2006)
Exogenous sampling
Artificial regression
where wi are weights given by the inverse of the probability of observing the house i

27. Concluding Remarks
This paper surveyed the major econometric aspects involved in the construction of hedonic house price indexes
We focused on the clarification of the link between the type of index and hedonic method and the specification of the hedonic functions
A simple strategy for specification analysis was proposed