Dr. Peter Moser: modelling land prices

Modelling the Spatio-temporalDevelopment of Building LandPrices in the Canton of Zurich:

A Mixed-Effects Approach.

Dr. Peter Moser

Starting Point

• For a variety of reasons (scientific, personal, legal, fiscal etc.) lots ofpeople are interested in actual but also historical „mean prices" paidfor residential building land plots in the municipalities of the Canton ofZurich.

• There is lots of data available: Since 1974 there exists in the Cantonof Zurich a more or less complete data base of all real estatetransactions notified with the authorities, among which there are noless than

• ~ 58 000 Transactions concerning undeveloped land plots forresidential building (unbebautes Wohnbauland)

• There are, however a few problems….

Dr. Peter Moser

1974 1975 1976 1977 1978 1979 1980 1981

1982 1983 1984 1985 1986 1987 1988 1989

1990 1991 1992 1993 1994 1995 1996 1997

1998 1999 2000 2001 2002 2003 2004 2005

Missing values and unmotivated price jumps…

Dr. Peter Moser

….implied in the spatio-temporal sparsity of thedata• For no less than 18% of the 5472 potential

combinations of (171) municipalities Xyears (32; 1974–2005) no transactionstook place at all.

• For a further 20% of these grid pointsmean prices can‘t be published forreasons of data protection (number oftransactions <3)

For a total of 38% of the grid-points nomean price can be published.

• And the n of Transactions is small formuch of the other grid-points, so meansare of limited value (Small-area Problem!)

High temporal variability of communalprices from year to year (dependent onthe actual mix of micro conditions,perhaps also on particular supply-demand constellations) 0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

Number of Transactions per municpality x year

Pro

po

rtio

n<

=x

n:5440 m:0

Dr. Peter Moser

What do we want?

• Estimation of mean prices for all missing year x municipalitygridpoints in a methodically transparent and consistent framework

• Smoothing of the temporal development of mean prices; we want toextract the „signal“ and eliminate the „noise“ created by particularmixes of micro conditions of traded land plots, especially if thenumber of transactions is small.

• Optimal use of all available information about transactions.

• At an acceptable Cost: No back-collection of historical data aboutthe micro-locations of traded land plots – eg. exposition ,distance toinfrastructure like schools, train-stations etc. –: This is out of thequestion.

• Which means: we have to do with covariates relating to themunicipal level or above: a detailed hedonic model of building land-prices, including data on micro-conditions is not feasible - at least forthe more distant past.

Dr. Peter Moser

Some indicators of the market in land plots1974-2005 I

• Number of transactions ...

• ... volume traded (in qm)....

• .... Transaction volume (in SFr).....

• ... Median surface area of landplots traded

1 97 5 1 98 0 1 98 5 1 99 0 1 9 95 2 0 00 2 0 05

1000

3000

5000

Nu

mb

er

of

tran

sacti

on

s

1 97 5 1 98 0 1 98 5 1 99 0 1 9 95 2 0 00 2 0 05

0.5

1.0

1.5

2.0

2.5

are

avo

lum

e(M

io.q

m)

1 97 5 1 98 0 1 98 5 1 99 0 1 9 95 2 0 00 2 0 05

200

400

600

80

0

mo

ne

tary

vo

lum

e(M

io.S

Fr)

1 97 5 1 98 0 1 98 5 1 99 0 1 9 95 2 0 00 2 0 05

200

400

600

80

0

Med

ian

are

ao

fp

lots

(qm

)

Dr. Peter Moser

Price development 1974-2005 II

• ...the cantonal median price persqare metre of land(unweighted)

• and the geometric mean, whichis a quite robust measure forthe central tendency in the caseof left-skewed data, which canbe „normalized“ by taking thelog.

• Caveat: We don‘t want toexplain this generaldevelopment of prices. It isendogenous to our model. Weare only interested in thedivergent developments in themunicipalites of the canton!

1975 1980 1985 1990 1995 2000 2005

20

03

00

40

05

00

60

07

00

Me

dia

np

ric

ep

er

sq

are

me

tre

(SF

r/q

m)

Dr. Peter Moser

ktlogpricejahr

log

(frq

m)

5

6

7

5.0 5.5 6.0

Adliswil Aeugst, Affoltern

5.0 5.5 6.0

BachenbülachBachs, Stadel, Weiach

5.0 5.5 6.0

BassersdorfBauma, Sternenberg

Benken, Dachsen, RheinauElgg, Hagenbuch Embrach Erlenbach Fällanden

5

6

7

Fehraltorf

5

6

7

Feuerthalen Herrliberg HinwilHirzel, Hütten, SchönenbergRorbas Rümlang

Rüti SchwerzenbachSeegräben, Wetzikon Uitikon Uster

5

6

7

Wädenswil

5

6

7

Wald

5.0 5.5 6.0

Wallisellen Winterthur

5.0 5.5 6.0

Zollikon Zumikon

5.0 5.5 6.0

Zürich

Land Prices develop differently in themunicipalities of the Canton

• log(frqm)=β0+β1log(ktpricejahr)

• β0 (Intercepts): differ betweenthe municipalities

general price levels ascompared to the cantonal meandiffer (High on the „Gold-Coast“, low in peripherealregions)

• β1 (Slopes) differ too:

municipal prices „react“differently to the generaldevelopment, elasticities vary(log~log=Δ%~Δ%)

• ~ the relationship is more orless linear

Dr. Peter Moser

Fixed effects Modelling municipality-wise

• A somewhat more schematicview of the group-wiseParameters and theirconfidence intervals

• May be a solution: acollection of fixed effectsmodels for all municipalitesand variables!

• Or we could wrap this up intoa single model with complexinteractions

ste

fbe

z

Adliswil

Aeugst, Affoltern

Bachenbülach

Bachs, Stadel, Weiach

Bassersdorf

Bauma, Sternenberg

Benken, Dachsen, Rheinau

Elgg, Hagenbuch

Embrach

Erlenbach

Fällanden

Fehraltorf

Feuerthalen

Herrliberg

Hinwil

Hirzel, Hütten, Schönenberg

Rorbas

Rümlang

Rüti

Schwerzenbach

Seegräben, Wetzikon

Uitikon

Uster

Wädenswil

Wald

Wallisellen

Winterthur

Zollikon

Zumikon

Zürich

5.0 5.5 6.0 6.5

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(Intercept)

0.6 0.8 1.0 1.2

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ktlogpricejahr2

log(frqm)= β0 + β1(frqmzh)

Dr. Peter Moser

The problem with fixed effects – and a solution

• A fixed-effects-model with interactions becomes complex very rapidlyespecially if the basic model incorporates many variables ( e. g.municipal tax rates etc.) parameters tend to proliferate.

• Municipality-specific parameter estimates are often based on a verysmall number of cases (transactions) and are therefore unreliable:this is especially true for the Slopes, the β1

Solution: Mixed-effects model

• Basic principle: we estimate overall fixed-effect-parameters for theindependent variables. They represent something like mean effectsof a variable.

• The variability of the parameters between the municipalities ismodelled by a distribution – often a normal distribution with thecorresponding fixed effect as a mean and a standard deviationdependent on the variability of the municipality-specific estimates ofthe parameter in question.

Dr. Peter Moser

A simple mixed-effects model• Fixed effects:

• log(frqm)=β0+β1log(frqmzh)+β2steuerfuss

log transaction prices per square metre depend on a cantonal priceindex (exogenous!) and the municipal tax rate.

• Random effects:

• The intercept as well as the slopes of the explanatory variableslog(ktpricejahr) and steuerfuss (tax rate) are allowed to varybetween municipalites; we assume the parameter distribution to benormal.

Transaction prices in the municipalites are allowed to have differentlevels and may also differ in their elasticity towards the general trendin land prices and changes in the municipal tax rates.

• A model specified this way explains about 71% of the variability oflog(frqm) (pseudo-R2; correlation2 between actual transaction pricesand model values). The lion‘s share of the explanatory power of themodel is, unsurprisingly, due to the explanatory power of thecantonal building land price index over time.

Dr. Peter Moser

Implications and advantages of this modellingstrategy

• The normal distribution of the random effects parameters implies thatmunicipal parameter values far from the (fixed effect) mean (in the tailsof the distribution), are rather improbable.

• The random-effects parameters of the explanatory variables (and theintercept) of a particular municipality are (ceteris paribus) relativelyclose to the mean:

If price information (the number of transactions) is sparse.

If the variability of this within-municipality price information is high.

• In difference to fixed-effects model with interactions, the estimates ofthe municipal parameters of the explanatory variables depend not onlyon the information from within this particular spatial unit.

• Municipality-specific parameter values are more or less stronglyinfluenced by the „general situation“, if information within the unit issparse or ambigous. This principle of "borrowing strength" stabilisesestimates and implies a smoothing of the estimates for mean values.

Dr. Peter Moser

1974 1975 1976 1977 1978 1979 1980 1981

1982 1983 1984 1985 1986 1987 1988 1989

1990 1991 1992 1993 1994 1995 1996 1997

1998 1999 2000 2001 2002 2003 2004 2005

The smoothing effect of the model I

Dr. Peter Moser

Smoothing of the communal mean price trajectories

Ro h e M itte lw er te

Tim e

Bo

de

np

reis

inF

r/q

m

1975 1980 1985 1990 1995 2000 2005

05

00

10

00

15

00

M o d e llw e rte

Tim e

Bo

de

np

reis

inF

r/q

m

1 975 1980 1985 1990 1995 2000 2005

05

00

10

00

15

00

•A different view of the temporal aspect:

"raw" mean prices modelled mean prices

Dr. Peter Moser

raw (unmodelled) municpal prices (SFr/ square metre)

mo

de

lva

lue

sm

unic

ipalpri

ce

s(S

Fr/

sq

uare

metr

e)

0

500

1000

1500

0 500 1000 1500 2000

(0,5](1973,1987]

(5,20](1973,1987]

0 500 1000 1500 2000

(20,Inf](1973,1987]

(0,5](1987,1991]

(5,20](1987,1991]

0

500

1000

1500(20,Inf]

(1987,1991]0

500

1000

1500(0,5]

(1991,2005]

0 500 1000 1500 2000

(5,20](1991,2005]

(20,Inf](1991,2005]

The logic of the smoothing

• A small number of trans-actions per year andmunicipality and a largeamount of variabilty:

• Implies a shifting of modelestimates of the meanland price in a munici-pality towards the can-tonal mean (the fixedeffect): In these cases:

• If the raw mean is „toohigh“ model estimates arelower.

• And vice versa

That‘s what "borrowingstrength„ means

Dr. Peter Moser

Some examples ….

• Model estimates

• (Geometric) meanvalues

• Actual transactions

1975 1985 1995 2005

0500

1000

1500

Hombrechtikon

Bodenpre

isin

Fr/

qm

1975 1985 1995 2005

0500

1000

1500

Urdorf

Bodenpre

isin

Fr/

qm

1975 1985 1995 2005

0500

1000

1500

Wald (ZH)

Bodenpre

isin

Fr/

qm

1975 1985 1995 2005

0500

1000

1500

Uitikon

Bodenpre

isin

Fr/

qm

Dr. Peter Moser

….

1975 1980 1985 1990 1995 2000 2005

0500

1000

1500

Stäfa

Bodenpre

isin

Fr/

qm

1975 1980 1985 1990 1995 2000 2005

0500

1000

1500

Meilen

Bodenpre

isin

Fr/

qm

1975 1980 1985 1990 1995 2000 2005

0500

1000

1500

Rüschlikon

Bodenpre

isin

Fr/

qm

1975 1980 1985 1990 1995 2000 2005

0500

1000

1500

Hüttikon

Bodenpre

isin

Fr/

qm

1975 1980 1985 1990 1995 2000

0500

1000

1500

Adlikon

Bodenpre

isin

Fr/

qm

1975 1980 1985 1990 1995 2000 2005

0500

1000

1500

Knonau

Bodenpre

isin

Fr/

qm

Dr. Peter Moser

And some more …

1975 1980 1985 1990 1995

0500

1000

1500

Schleinikon

Bodenpre

isin

Fr/

qm

1975 1980 1985 1990 1995 2000 2005

0500

1000

1500

Truttikon

Bodenpre

isin

Fr/

qm

1975 1980 1985 1990 1995 2000 2005

0500

1000

1500

Hedingen

Bodenpre

isin

Fr/

qm

1975 1980 1985 1990 1995 2000 2005

0500

1000

1500

Elgg

Bodenpre

isin

Fr/

qm

1975 1980 1985 1990 1995 2000 2005

0500

1000

1500

Winterthur

Bodenpre

isin

Fr/

qm

1975 1980 1985 1990 1995 2000 2005

0500

1000

1500

Wallisellen

Bodenpre

isin

Fr/

qm

Dr. Peter Moser

Summary - Perspectives

• Mixed-Effects modelling als a flexible but nevertheless parametricstrategy for this problem, which is at its core a small area estimationproblem

• Estimates are based on a single model, formally implementing somekind of "common sense" – which is what statistics is - or should be -all about in the first place.

• The model actually has an explanatory value, and could be used foranalytical purposes.

• The future:

mixed-effects quantile regression: a L1-norm-strategy would allowus to model medians and quantiles directly, without having totransform the dependent variable – which is always difficult toexplain to lay people.

Endogenization of the general trend in land prices would allow for amuch more powerful model.

Dr. Peter Moser

Thanks for your attention

For further information, do not hesitate to contact me:

Dr. Peter Moser

Forschungsleiter

Statistisches Amt des Kantons Zürich

Bleicherweg 5

8090 Zürich

peter.moser@statistik.ji.zh.ch