Introduction Categories Continuous Index References

Continuous Time Hedonic MethodsA new way to construct house price indices

Sofie WaltlUniversity of Graz

August 20, 2014

Introduction Categories Continuous Index References

OVERVIEW

1 HEDONIC METHODS TO CONSTRUCT HOUSE PRICE INDICES

2 CATEGORIES OF HEDONIC METHODS

3 CONTINUOUS TIME HEDONIC METHOD

Introduction Categories Continuous Index References

OVERVIEW

1 HEDONIC METHODS TO CONSTRUCT HOUSE PRICE INDICES

2 CATEGORIES OF HEDONIC METHODS

3 CONTINUOUS TIME HEDONIC METHOD

Introduction Categories Continuous Index References

HEDONIC METHODS TO CONSTRUCT HOUSE PRICE INDICES

“The theory of hedonic indexes is built on the proposition that thecharacteristics [of a product] are the variables that the buyer [...]want, and that the characteristics of the product also are costly toproduce.”

Triplett (2006)

Introduction Categories Continuous Index References

HISTORICAL OVERVIEW

Early works by Waugh (1928) about vegetables’ prices, Vial (1932) aboutprices of mix fertilizers, Court (1939) constructing commodity prices andStone (1956) analysing liquor prices.

Griliches (1961) revived hedonic approaches and investigated therelationship of auto mobile prices in the US to the various dimensions of anauto mobile.

Lancaster (1966) and Rosen (1974) constituted the conceptional basis ofhedonic methods.

In 1968 the US Bureau of the Census constructed a hedonic price index fornew single family houses.

Introduction Categories Continuous Index References

OVERVIEW

1 HEDONIC METHODS TO CONSTRUCT HOUSE PRICE INDICES

2 CATEGORIES OF HEDONIC METHODS

3 CONTINUOUS TIME HEDONIC METHOD

Introduction Categories Continuous Index References

CATEGORIES OF HEDONIC METHODS

Following the taxonomy of Hill (2012) there are three categories of hedonicmethods: Time-dummy, imputation and characteristics methods.

Time-dummy method

1.) Run regression explaining (logged) house prices, p, via a vector ofcharacteristics, X , and time dummy variables, D:

log p = Dδ + Xβ + ε, εiid∼ N(0, σ2).

2.) Construct price index through estimated period-specific shadow prices:

P̂t = exp(δ̂t ) or P̂t = exp(δ̂t −

12σ̂2((X̃ t X̃ )−1)tt

)1,

where X̃ = (D,X ) and t ∈ {1, · · · ,T} with T the number of periods.

1This bias-correction results from the properties of a log-normal distribution. For a generaldiscussion for semi-logarithmic equations consult Kennedy (1981).

Introduction Categories Continuous Index References

Imputation methods

Every house is different and the price of a specific dwelling is usually notobserved in every period. This is why standard price formulae can not directlybe applied in a housing context. To overcome this problem, period-wisemodels are estimated and used to predict the price of a specific dwelling forevery period where the true price was not observed.

Using this semi-calculated data, a standard price formulae such as thePaasche, Laspeyres or Fisher formula can be applied.

Introduction Categories Continuous Index References

Characteristics methods

1.) Construct a hypothetical dwelling that is average in its characteristics.

2.) Estimate period-wise hedonic models and predict the price of thehypothetical dwelling in every period.

3.) Calculate bilateral price comparisons by using a standard price formulaand chain them together delivering a price index.

Introduction Categories Continuous Index References

OVERVIEW

1 HEDONIC METHODS TO CONSTRUCT HOUSE PRICE INDICES

2 CATEGORIES OF HEDONIC METHODS

3 CONTINUOUS TIME HEDONIC METHOD

Introduction Categories Continuous Index References

CONTINUOUS TIME HEDONIC METHOD

• Extension of time-dummy method• Why time-dummy?

Single-model approach, no artificial changes in variance structure,standard errors readily available.

• Why continuous?Time is a continuous variable and shall be treated as such.

• Why are periods bad?• discretization of time (discretization error)• Introducing periods leads to averaging over a time interval. Changes in the

index might be averaged out.• It is not clear how to choose an appropriate period length (competing goals:

long enough to guarantee sufficient number of observations vs. shortenough to guarantee precise estimation).

• A priori selection of starting points of periods can have significant influence.

Introduction Categories Continuous Index References

THE BASIC MODEL

1.) Calculate continuous time scale, e.g.,

TIMEi = YEARi +MONTHi − 1 + DAYi−1

30

12.

2.) Use continuous variable instead of time dummies and estimate the timeeffect smoothly2, i.e.,

log p = f (TIME) + Xβ + ε.

3.) Establish the price index3via

P̂t = (exp ◦f̂ )(t) = exp(f̂ (t)), t ∈ [min(TIME),max(TIME)].

2Estimation is based on penalized least squares using thin plate regression splines introducedby Wood (2003) and Wood (2006). Optimal basis dimension according to GCV criterion.

3To gain an unbiased estimator P̂t has to be adapted as for the original time-dummy method.

Introduction Categories Continuous Index References

APPLICATION

Data set

• data set by Australian Property Monitors• transaction prices, dates, house characteristics (including number of

bed- and bathrooms, land area and exact longitudes and latitudes) ofsold houses between 2001 and 2011

• after cleaning: 435, 295 observations• huge problem: missing data

• 201, 571 observations are incomplete, i.e., 46.3%• 46, 747 incomplete observations can be refilled through simple

reconstruction approach

• final sample size: 280, 4714

• incompleteness is a problem of the variables BED and BATH only

4Further approaches to handle missing observations in this context are under way.

Introduction Categories Continuous Index References

Hedonic model

log(p) =β0 + βAREA log(AREA) +6∑

j=2

βBATHj I{j}(BATH) +6∑

j=2

βBEDj I{j}(BED)

+ f1(LONG,LAT) + f2(TIME) + ε,

where

I{j}(VARIABLE) ={

1, VARIABLE = j ,0, VARIABLE 6= j .

Locational effects: two-dimensional surface defined on longitudes andlatitudes (see Hill and Scholz, 2014)

logged area

Introduction Categories Continuous Index References

RESULTS

Introduction Categories Continuous Index References

RESULTS

Average number of observations per period:

Monthly period 3,30010-days period 1,1005-days period 660

Introduction Categories Continuous Index References

RESULTS

• Truncated period: 2004-2011• Normalization with respect to January 1, 2008

Introduction Categories Continuous Index References

Pros and Cons

⊕ Single model approach

⊕ Circumvents problem of choosing appropriate period length and startingpoints

⊕ Peaks and troughs are not averaged out

⊕ Turning points are detected precisely

⊕ Discrete indices approach continuous index

Computationally more costly than time-dummy method

Higher complexity due to smooth estimation

⊗ Basic model does not allow changing shadow prices → interactions of smoothtime function and characteristics as well as regular updates ofgeographical spline

Introduction Categories Continuous Index References

REFERENCES I

Court, A. T. (1939). The Dynamics of Automobile Demand, chapter Hedonic PriceIndexes with Automotive Examples, pages 99–117. The General MotorsCorporation, New York.

Griliches, Z. (1961). The Price Statistics of the Federal Goverment, chapter HedonicPrice Indexes for Automobiles: An Econometric of Quality Change, pages 173–196.National Bureau of Economic Research.

Hill, R. J. (2012). Hedonic price indexes for residential housing: A survey, evaluationand taxonomy. Journal of Economic Surveys, pages 879–914.

Hill, R. J. and Scholz, M. (2014). Incorporating geospatial data in house price indexes:A hedonic imputation approach with splines. Graz Economics Papers 2014-05,University of Graz, Department of Economics.

Kennedy, P. E. (1981). Estimation with correctly interpreted dummy variables insemilogarithmic equations. The American Economic Review, 71:801.

Lancaster, K. J. (1966). A new approach to consumer theory. Journal of PoliticalEconomy, 74(2):132.

Rosen, S. (1974). Hedonic prices and implicit markets: Product differentiation in purecompetition. Journal of Political Economy, 82(1):34.

Stone, R. (1956). Quantity and Price Indexes in National Accounts. Organisation forEuropean Economic Cooperation, Paris.

Introduction Categories Continuous Index References

REFERENCES II

Triplett, J. (2006). Handbook on Hedonic Indexes and Quality Adjustments in PriceIndexes: Special Application to Information Technology Products. Organization forEconomic Co-operation and Development, Paris.

Vial, E. E. (1932). Retail prices of fertilizer materials and mixed fertilizers. CornellUniversity.

Waugh, F. V. (1928). Quality factors influencing vegetable prices. Journal of FarmEconomics, 10(2):185.

Wood, S. N. (2003). Thin-plate regression splines. Journal of the Royal StatisticalSociety (B), 65(1):95–114.

Wood, S. N. (2006). Generalized Additive Models: An introduction with R. Chapmanand Hall / CRC.

Introduction Categories Continuous Index References

RECONSTRUCTION ALGORITHM

• There are many multiply traded dwellings in the data set (that can beidentified through a unique identification number).

• If a dwelling is sold for instance twice and the number of bedroom isavailable at one transaction date but not at the other, the missingobservation can be refilled.

• However, house characteristics might change over time due torenovation! → Refilling is subject to the following constraints:

1.) Constancy constraint: Available numbers of bedrooms (or bathroomsrespectively) are constant.

2.) Time constraint: The time span between two transactions is greater thansix month, i.e.,

TIMEdiff = TIME2 − TIME1 > 0.5 years.

3.) Price growth constraint: The average annual price growth is less than25%, i.e., (

PRICE2

PRICE1

)1/TIMEdiff− 1 < 25%.

Introduction Categories Continuous Index References

RESULTS OF RECONSTRUCTION ALGORITHM

The share of incomplete recordings has been reduced from 46.3% to 35.6%.37, 657 missing records of BED and 47, 293 missing records of BATH werereconstructed.

BED BATH

Constancy constraint 4, 979 5, 188Time constraint 0 0

Price growth constraint 2, 119 2, 081

Table: Number of non-replacements due to restrictions.

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Introduction Categories Continuous Index References

CHANGING SHADOW PRICES

To the basic model

log P = β0+f1(TIME)+f2(LONG, LAT)+

6∑i=2

βBEDi I{i}(BED)+

6∑i=2

βBATHi I{i}(BATH)+βAREA log(AREA)+ε

interaction terms are added,

log P =β0 + f1(TIME) + f2(LONG, LAT) +

6∑i=2

βBEDi I{i}(BED) +

6∑i=2

βBATHi I{i}(BATH) + βAREA log(AREA)

+

6∑i=2

f BEDi (TIME|BED = i)I{i}(BEDt ) +

6∑i=2

f BATHi (TIME|BATH = i)I{i}(BATH)

+

5∑i=2

f AREAi (TIME|AREAcat = i)I{i}(AREAcat ) + ε.

In the interaction term the continuous variable AREA is transformed to adiscrete variable of five categories, AREAcat .

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Introduction Categories Continuous Index References

ESTIMATED LAND AREA EFFECT

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