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164
July 2020

Decompositions of Spatially Varying Quantile Distribution
Estimates: The Rise and Fall of Tokyo House Prices

Daniel McMillen
University of Illinois, Department of Economics

Chihiro Shimizu
Nihon University & Center for Spatial Information Science, University of Tokyo

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Decompositions of Spatially Varying Quantile Distribution Estimates:
The Rise and Fall of Tokyo House Prices

Daniel McMillen
Department of Economics
University of Illinois
1407 W. Gregory Dr.
Urbana, IL 61801
mcmillen@illinois.edu
Chihiro Shimizu
Center for Spatial Information Science
University of Tokyo
5-1-5, Kashiwanoha, Kashiwa, Chiba 277-8568
cshimizu@csis.u-tokyo.ac.jp

This Draft: July 2020
(First Draft: April 2017)
Abstract
We extend Machado-Mata’s (2005) approach for decomposing the differences in the distributionof a dependent variable across two samples to account for location when the models are estimatedusing conditional parametric procedures. We find that a substantial portion of the change in the
distribution of condominium prices in Tokyo between the rapid rise in prices in 1986 – 1990 andthe sharp decline in 1991 – 1995 is due to changes in the values of the explanatory variables.
Changes in the locations of sales serve to shift the price distribution to the left because later sales
were more likely to be farther from downtown Tokyo, where prices are lower.
Keywords: Conditionally parametric, quantile regression, decomposition.
JEL codes: C14, C18, R30

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1. Introduction
Although quantile regression is designed to estimate the conditional distribution of adependent variable given the values of the explanatory variables, conditional quantile estimatesalso imply a full marginal distribution when the regressions are estimated at multiple quantiles. In
an influential paper, Machado and Mata (2005) show how quantile regression estimates can beused to estimate counterfactual marginal distributions. Estimating quantile regressions at many
quantiles implies a predicted distribution of the dependent variable at given values of the
explanatory variables. The parametric structure of the conditional quantile model allows thedifferences in the predicted distributions across two samples to be decomposed into portions thatare due to changes in the explanatory variables and changes in the explanatory variables.
Two approaches have been proposed for estimating quantile regression models for spatial
data. A fully parametric approach typically uses the simple spatial autoregressive model, 𝑌𝑌 =𝜌𝜌𝜌𝜌𝜌𝜌 + 𝑋𝑋𝑋𝑋 + 𝑢𝑢 as the base estimating equation, where 𝑊𝑊 is an 𝑛𝑛 × 𝑛𝑛 weight matrix showing theinfluence of each value of 𝑦𝑦𝑗𝑗 on 𝑦𝑦𝑖𝑖 . Quantile regression versions of the model can be estimatedusing the instrumental variable approaches of Chernozhukov and Hansen (2006) or Kim andMuller (2004), which account for the endogeneity of 𝑊𝑊𝑊𝑊. Examples of these approaches include
Kostov (2009), Lei and Zhang (2017), Liao and Wang (2012), Zeitz et al. (2008), and Zhang andLeonard (2014). An alternative is to estimate a “geographically weighted regression” version of
the quantile regression model. This approach, which was proposed by McMillen (2013, 2016), isbased on the conditionally parametric (CPAR) approach of Cleveland, Grosse, and Shyu (1992)
and Cleveland (1994).

The geographic version of the CPAR approach estimates separate

coefficients for various target sets of geographic coordinates. Conditional on location, the model
is a standard quantile regression model, but the coefficients vary smoothly over space. The CPAR

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approach is most useful when a base parametric specific is appropriate for small geographic
regions but the base model does not hold over the entire region covered by the data.
When separate CPAR models are estimated for two samples, the underlying estimating
equations can be written as 𝑦𝑦1 = 𝑥𝑥1′ 𝛽𝛽1 (𝑧𝑧1 ) + 𝑢𝑢1 and 𝑦𝑦2 = 𝑥𝑥2′ 𝛽𝛽2 (𝑧𝑧2 ) + 𝑢𝑢2 where 𝑧𝑧1 and 𝑧𝑧2 are thevectors of geographic coordinates (i.e., longitude and latitude) for samples 1 and 2. Thus,

differences in the distribution of y across the two samples are attributable to three sources –differences in the explanatory variables (x), differences in the coefficients (𝛽𝛽), and differences in

the locations of the observations. In this paper, we show that it is straightforward to extend the
Machado-Mata (2005) approach for decomposing the differences in the distribution of y across
two samples to also account for differences in the location of the observations.
Our data set includes all sales of condominiums in Tokyo from 1986 – 2016. We focus ona particularly interesting time from 1986 – 1995. Prices more than doubled from 1986 to the endof 1990, after which they declined by more than 70% from the beginning of 1991 to the end of
1995. Although the rate of decline moderated afterward, the decline in prices continued throughthe end of 2001, and prices were still not back to their peak levels at the end of 2016. The 1986 –1995 period includes the periods of rapid rise and subsequent rapid decline in prices.
We use the Machado-Mata (2005) approach and our extension to decompose the change inthe distribution of prices between the rapid rise for 1986 – 1990 and the rapid decline for 1991 –1995 into the portions that are due to changes in the estimated quantile coefficients, changes in thevalues of the explanatory variables, and changes in the locations of the sales.

The primary

explanatory variables are the area and age of the unit, the story, and whether the unit has a southern
exposure. In addition, we include controls for location and the quarter of sale. Focusing on the
primary explanatory variables, we find that a large portion of the change in the distributions is5attributable to changes in the values of the variables – primarily the age of the units, which
naturally tends to become higher over time despite a good amount of new construction. Thechange in the locations of sales between 1986 – 1990 and 1991-1995 serves to shift the distributionof sales prices further to the left because sales in the later period were more likely to be in locations
farther from downtown Tokyo, where prices are lower.

2. Spatial Quantile Regression
The standard quantile regression estimating equation can be written as 𝑄𝑄𝑦𝑦 (𝜏𝜏|𝑥𝑥𝑖𝑖 ) = 𝑥𝑥𝑖𝑖′ 𝛽𝛽.

This equation implies that the conditional quantile function for the dependent variable y at quantile𝜏𝜏 is a linear function of 𝑥𝑥𝑖𝑖 . Following Koenker (2005), the standard quantile regression approach

involves finding the values for 𝛽𝛽̂(𝜏𝜏) that minimize ∑𝑖𝑖 𝜌𝜌𝜏𝜏 (𝑦𝑦𝑖𝑖 − 𝑥𝑥𝑖𝑖′ 𝛽𝛽), where 𝜌𝜌𝜏𝜏 (𝑢𝑢) is the piecewise

linear function 𝜌𝜌𝜏𝜏 (𝑢𝑢) = 𝑢𝑢�𝜏𝜏 − 𝐼𝐼(𝑢𝑢 < 0)�. Nonparametric approaches for the quantile regression

model can be implemented by adding a kernel weight function to this expression. At the target, x,
the objective function is ∑𝑖𝑖 𝜌𝜌𝜏𝜏 𝑤𝑤𝑖𝑖 (𝑥𝑥)(𝑦𝑦𝑖𝑖 − (𝑥𝑥𝑖𝑖 − 𝑥𝑥)′ 𝛽𝛽), where 𝑤𝑤𝑖𝑖 (𝑥𝑥) = 𝐾𝐾�(𝑥𝑥𝑖𝑖 − 𝑥𝑥)/ℎ�/ℎ is thekernel weight function and h is the bandwidth (Chaudhuri, 1991; Koenker, 2005, chapter 7; Yu
and Jones, 1998).
The conditionally parametric approach differs from this fully nonparametric locallyweighted quantile regression approach by having different variables in the kernel weight function
and the base regression:𝑛𝑛min � 𝑤𝑤𝑖𝑖 (𝑧𝑧)𝜌𝜌𝜏𝜏 (𝑦𝑦𝑖𝑖 − 𝑥𝑥𝑖𝑖′ 𝛽𝛽)
𝛽𝛽

(1)

𝑖𝑖=1

For spatial models, z typically represents either longitude and latitude or the straight-line distance
between observation i and the target location. A smaller bandwidth leads to more local variation

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in the estimated coefficients, 𝛽𝛽̂(𝜏𝜏, 𝑧𝑧), which vary by both quantile and location. 1 The approach is

discussed in more detail in McMillen (2013, 2015), who used an adaptive decision tree approach(Loader, 1999, section 12.2) to choose the set of target points, after which the results areinterpolated to every point in the data set. The adaptive decision tree approach chooses more
locations in areas with many observations, and smaller bandwidths lead to more target points. In
this application, we choose instead to divide the sample area into a set of square kilometer gridcells, and then interpolate from the cell midpoints to the full set of locations represented in the data
set. The advantage of this approach is that it directly corresponds to the regions covered in themaps we use to summarize the results. 2
After interpolation, there are n coefficient estimates for each value of 𝜏𝜏, one for eachobservation. McMillen (2013, 2015) shows how a procedure based on Machado-Mata (2005) canbe used to estimate the marginal distribution of y implied by the CPAR conditional quantile
estimates. At a given quantile 𝜏𝜏, the quantile regression prediction for observation i is simply𝑥𝑥𝑖𝑖′ 𝛽𝛽̂(𝜏𝜏, 𝑧𝑧𝑖𝑖 ). In matrix form, the full set of n estimates is 𝑋𝑋°𝛽𝛽̂ (𝜏𝜏, 𝑧𝑧)𝑒𝑒𝑘𝑘 , where X and 𝛽𝛽̂(𝜏𝜏, 𝑧𝑧) are both𝑛𝑛 × 𝑘𝑘 matrices, 𝑒𝑒𝑘𝑘 = (1 …

1), and ° represents the Hadamard product. The predictions across

T values of 𝜏𝜏 can then be stored in the 𝑛𝑛 × 𝑇𝑇 matrix 𝑌𝑌� = {𝑋𝑋°𝛽𝛽̂(𝜏𝜏1 , 𝑧𝑧)𝑒𝑒𝑘𝑘… 𝑋𝑋°𝛽𝛽̂(𝜏𝜏 𝑇𝑇 , 𝑧𝑧)𝑒𝑒𝑘𝑘 }. The

implied distribution of 𝑌𝑌� can then be calculated using a standard kernel density estimator for the
implied nT-vector of predictions.

In the empirical section of the paper, we use a tri-cube kernel with a 25% window, which means that the nearest 25%of the observations receive weight when estimating the coefficients at a target location. Weights are based on straight-

1line distances, i.e., 𝑤𝑤𝑖𝑖 (𝑧𝑧) = 𝐼𝐼(𝑑𝑑𝑖𝑖 ≤ 𝑑𝑑.25 ) � 1 − �

𝑑𝑑𝑖𝑖

𝑑𝑑.25

3 3

� � /𝑑𝑑.25 , where 𝑑𝑑𝑖𝑖 is the distance between observation i and thetarget location 𝑑𝑑.25 is the 25th percentile of the distances.
2
The adaptive decision tree approach is much less computation intensive, producing 69 target points for 1986 – 1990
and 71 target points for 1991 – 1990, compared with 845 square-kilometer grid cells.

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Kernel density estimates can also be used to show how the distribution of y responds todiscrete changes in an explanatory variable. For example, consider a two-explanatory variable
model in which 𝑋𝑋 = (1 𝑥𝑥1

𝑥𝑥2 )′ and 𝛽𝛽 = (𝛽𝛽0𝛽𝛽1𝛽𝛽2 )′. Setting 𝑥𝑥1 to an arbitrary value δ,the predicted values at a given quantile are 𝛽𝛽̂0 + 𝛿𝛿𝛽𝛽̂1 + 𝑥𝑥2 °𝛽𝛽̂2 , where 𝛽𝛽̂0 , 𝛽𝛽̂1 , and 𝛽𝛽̂2 are each n-vectors. After grouping these estimates in to the 𝑛𝑛 × 𝑇𝑇 matrix 𝑦𝑦�(𝛿𝛿), a kernel density estimator canbe used to calculate the implied distribution of 𝑦𝑦� when 𝑥𝑥1 = 𝛿𝛿 and 𝑥𝑥2 is set to its actual set ofvalues in the data set. This procedure can be repeated at various values of 𝛿𝛿 to show how the
distribution of predicted values of y changes as 𝑥𝑥1 changes.

3. Counterfactual Decompositions of Distribution Changes
Machado and Mata (2005) present a simple extension of the Oaxaca (1973) approach fordecomposing the difference in two sets of estimates to the portions due to differences in theestimated coefficients and the values of the explanatory variables. For linear regression estimates,
the Oaxaca (1973) decomposition is 𝑦𝑦�1 − 𝑦𝑦�2 = 𝑥𝑥1′ 𝛽𝛽̂1 − 𝑥𝑥2′ 𝛽𝛽̂2 = �𝑥𝑥1′ 𝛽𝛽̂1 − 𝑥𝑥2′ 𝛽𝛽̂1 � + �𝑥𝑥2′ 𝛽𝛽̂1 − 𝑥𝑥2′ 𝛽𝛽̂2 �.

The expression in the first set of parentheses shows the effect of differences in the explanatory
variables on the differences in predicted values, while the terms in the second set of parentheses
represent the effect of differences in the coefficients. These expressions are typically evaluated at
the mean values of the explanatory variables, so the fact that the number of observations willusually differ across the two samples does not affect the calculations.

The order of thedecomposition can be changed so that the change in coefficients precedes the change in variables.
Machado and Mata’s (2005) version of the decomposition shows the effect of differencesin the variables and estimated quantile regression coefficients on differences in the full distribution

8of predicted values of the dependent variables. 3 They propose sampling with replacement fromthe rows of the explanatory variables 𝑋𝑋1 and 𝑋𝑋2 to form the 𝑀𝑀 × 𝑘𝑘 matrices 𝑋𝑋1𝑚𝑚 and 𝑋𝑋2𝑚𝑚 , where
M is the number of draws. They also propose re-estimating the quantile regressions at sets ofrandomly drawn values of 𝜏𝜏. A computationally less burdensome approach is to estimate thequantile regressions at a series of fixed values of 𝜏𝜏, such as 𝜏𝜏 = 0.02, 0.03, … , 0.98, which𝑞𝑞
𝑞𝑞
produces two 𝑘𝑘 × 𝑇𝑇 matrices of coefficients, 𝛽𝛽̂1 and 𝛽𝛽̂2 , where the superscript indicates that thematrices are coefficient vectors estimated at various quantile values. The matrices of predicted
𝑞𝑞

𝑞𝑞

𝑞𝑞

values needed for the decompositions are simply 𝑋𝑋1𝑚𝑚 𝛽𝛽̂1 , 𝑋𝑋2𝑚𝑚 𝛽𝛽̂1 , and 𝑋𝑋2𝑚𝑚 𝛽𝛽̂2 , all of which are ofdimension 𝑀𝑀 × 𝑇𝑇, i.e., M draws from the rows of the X matrices by T quantiles. Each of theseterms can then be treated as MT-vectors, and kernel density estimates can be calculated for each
vector.
This approach extends readily to the case of spatial CPAR quantile regression. Following
Machado and Mata’s (2005) approach, let 𝑋𝑋1𝑚𝑚 and 𝑋𝑋2𝑚𝑚 represent matrices of constructed bymaking M draws with replacement from the rows of 𝑋𝑋1 and 𝑋𝑋2 . Similarly, for each value of 𝜏𝜏,construct the 𝑀𝑀 × 𝑘𝑘 matrices 𝛽𝛽̂1𝑚𝑚 (𝜏𝜏, 𝑧𝑧1 ) and 𝛽𝛽̂2𝑚𝑚 (𝜏𝜏, 𝑧𝑧2 ) by drawing randomly with replacementfrom the rows of 𝛽𝛽̂1 (𝜏𝜏, 𝑧𝑧1 ) and 𝛽𝛽̂2 (𝜏𝜏, 𝑧𝑧2 ). The predicted values needed for the decompositions thenare constructed using equations (2) – (4):𝑌𝑌�1𝑚𝑚 = {𝑋𝑋1𝑚𝑚 °𝛽𝛽̂1𝑚𝑚 (𝜏𝜏1 , 𝑧𝑧1 )𝑒𝑒𝑘𝑘

𝑌𝑌�2𝑚𝑚 = {𝑋𝑋2𝑚𝑚 °𝛽𝛽̂2𝑚𝑚 (𝜏𝜏1 , 𝑧𝑧2 )𝑒𝑒𝑘𝑘

𝑚𝑚
𝑌𝑌�21
= {𝑋𝑋2𝑚𝑚 °𝛽𝛽̂1𝑚𝑚 (𝜏𝜏1 , 𝑧𝑧1 )𝑒𝑒𝑘𝑘…
…
…

𝑋𝑋1𝑚𝑚 °𝛽𝛽̂1𝑚𝑚 (𝜏𝜏 𝑇𝑇 , 𝑧𝑧1 )𝑒𝑒𝑘𝑘 }𝑋𝑋2𝑚𝑚 °𝛽𝛽̂2𝑚𝑚 (𝜏𝜏 𝑇𝑇 , 𝑧𝑧2 )𝑒𝑒𝑘𝑘 }𝑋𝑋2𝑚𝑚 °𝛽𝛽̂1𝑚𝑚 (𝜏𝜏 𝑇𝑇 , 𝑧𝑧1 )𝑒𝑒𝑘𝑘 }

(2)
(3)
(4)

Examples of studies using the Machado and Mata (2005) approach include Carrillo and Yezer (2009) and McMillen(2008).

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Kernel density estimates then show how the distributions vary depending on the values of the
explanatory variables and the estimated coefficients.
The CPAR estimates provide additional information: how much of the difference in thepredictions is due to differences in the locations of the observations across the two samples? Fora given value of 𝜏𝜏, the extended version of the decomposition that accounts for differences in thevalues of z is:(Total)(Variables)

𝑌𝑌�1𝑚𝑚 (𝜏𝜏, 𝑧𝑧1 ) − 𝑌𝑌�2𝑚𝑚 (𝜏𝜏, 𝑧𝑧2 ) = 𝑋𝑋1𝑚𝑚 °𝛽𝛽̂1𝑚𝑚 (𝜏𝜏, 𝑧𝑧1 )𝑒𝑒𝑘𝑘 − 𝑋𝑋2𝑚𝑚 °𝛽𝛽̂2𝑚𝑚 (𝜏𝜏, 𝑧𝑧1 )𝑒𝑒𝑘𝑘 =(Location)(Coefficients)(5a)

𝑋𝑋1𝑚𝑚 °𝛽𝛽̂1𝑚𝑚 (𝜏𝜏, 𝑧𝑧1 )𝑒𝑒𝑘𝑘 − 𝑋𝑋2𝑚𝑚 °𝛽𝛽̂1𝑚𝑚 (𝜏𝜏, 𝑧𝑧1 )𝑒𝑒𝑘𝑘 +(5b)𝑋𝑋2𝑚𝑚 °𝛽𝛽̂1𝑚𝑚 (𝜏𝜏, 𝑧𝑧2 )𝑒𝑒𝑘𝑘 − 𝑋𝑋2𝑚𝑚 °𝛽𝛽̂2𝑚𝑚 (𝜏𝜏, 𝑧𝑧2 )𝑒𝑒𝑘𝑘(5d)

𝑋𝑋2𝑚𝑚 °𝛽𝛽̂1𝑚𝑚 (𝜏𝜏, 𝑧𝑧1 )𝑒𝑒𝑘𝑘 − 𝑋𝑋2𝑚𝑚 °𝛽𝛽̂1𝑚𝑚 (𝜏𝜏, 𝑧𝑧2 )𝑒𝑒𝑘𝑘 +(5c)This decomposition requires one additional set of estimated coefficients, 𝛽𝛽̂1 (𝜏𝜏, 𝑧𝑧2 ) , which

represents the estimated coefficients using the data from sample 1 evaluated at the sample 2
locations. All that is necessary to produce these estimates is to interpolate the estimated sample 1
coefficients to the locations represented in the second sample. 4 The order of the decomposition
can be varied easily.

4. Data and Model Specification
The full data set comprises nearly 235,000 sales of condominiums in Tokyo for 1986 –2016. The dependent variable for our regressions is the natural log of the sale price per square
meter of floor space. Explanatory variables include the log of floor area, the age of the unit, andAn alternative to interpolation – using each location in sample 2 as a target point for estimation using the sample 1
data – may be preferable in situations where locations differ sufficiently to make interpolation unreliable.
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a variable indicating that the unit has a southern view. We also include controls for the quarter ofsale and the census tract. 5 Finally, we include the story on which the unit is located, along withvariables indicating that the unit is on the first or second story. Preliminary data analysis suggested
that the effect of the story on the sale price per square meter is close to being linear beyond thesecond floor. Our base estimating equation is a standard one for hedonic housing models: 𝑦𝑦𝑖𝑖 =𝑥𝑥𝑖𝑖′ 𝛽𝛽 + 𝑑𝑑𝑖𝑖′ 𝛿𝛿 + 𝑠𝑠𝑖𝑖′ 𝛾𝛾 + 𝑢𝑢𝑖𝑖 , where 𝑦𝑦 represents the log of price per square meter; 𝑥𝑥 includes the log offloor area, age, indicators of a southern view, and the story, along with the first and second story
indicator variables; d is a group of variables indicating the quarter of sale; and s is the a group ofvariable indicating the census tract for the building. Sales prices are not adjusted for inflation, but
the rate of inflation was low throughout this period.
Since the number of census tracts is very large (2,184 in 1986 – 1990 and 2,351 in 1991 –1995), we follow a procedure proposed by Canay (2011) and first estimate the location effects, 𝛾𝛾�,by a standard fixed effects regression. We then treat subtract 𝑠𝑠𝑖𝑖′ 𝛾𝛾� from 𝑦𝑦𝑖𝑖 before estimating any
quantile regression. Thus, the dependent variable should be considered 𝑦𝑦𝑖𝑖 − 𝑠𝑠𝑖𝑖′ 𝛾𝛾� for the remainderof the paper.

Figure 1 presents the price index implied by 50% quantile regression estimates using thefull data set. The price index is simply the estimated values of 𝛿𝛿, with a base of zero for 2000:1.

The variation of sales prices over time is striking. The value for the price index is -0.221 in 1986:1,
or approximately 22% lower than the level in 2000:1. Prices rose dramatically over the next five
years, with the index peaking at 0.911 in 1990:4. Prices then declined until the trough of -.037

The data set is drawn from a region of 534 square kilometers. There are 2789 census tracts in the full data set, which
implies an average area of 0.19 square kilometers, or 0.12 square miles. We also experimented with specifications
using building (or “tower”) fixed effects, but the tower indicator is not available before 1990 and approximately 15%of the towers had only one sale over the full time. Approximately half of the buildings for which the tower indicator
is available had only one sale during the 1986-1995 period that is the focus of our analysis.

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was finally reached 2001:4. The rate of decline was particularly high from 1991 – 1995, with theindex declining from 0.911 to 0.176 – a fall of more than 70% in 5 years. The value for the index
is 0.402 for 2015:4, or approximately the same as its value in early 1987.
For the remainder of the paper, we focus on two particularly interesting periods – thestriking rise in prices from 1986 – 1990 and the period of rapid decline from 1991 – 1995.
Summary statistics for these two periods are presented in Table 1. Sales prices average 10.472
million yen for 1986 – 1990, compared with 8.808 million yen for 1991 – 1995. Average floor
areas increased significantly across the two periods, rising from 50.466 square meters for 1986 –1990 to 54.249 square meters in 1991 – 1995. Naturally, the average age also rose over time, from9.721 years in 1986 – 1990 to 12.864 years for 1991 – 1995. Although new units were built during
both periods, the rate of construction slowed considerably as prices declined: 24.22% of the units
sold during 1986 – 1990 were no more than five years old, compared with 10.46% of the units soldin 1991 – 1995. Significantly more units sold had a southern exposure in 1991 – 1995. Stories
range from 1 – 25, with no significance differences across periods.

5. Empirical Results
The base OLS results are shown in the first column of Table 1. The results indicate thatthe price per square meter is lower for larger units. Prices also decline with the age of the unit,
and are lower on the first and second story of a building than on higher floors. Apart from thisinitial discount, prices rise on subsequent floors. A south view is associated with higher sales
prices in the 1991 – 1995 period, but the estimate is statistically insignificant for 1986 – 1990. The
regressions fit the data well, with R2s of 0.899 for 1986 – 1990 and 0.833 in 1991 – 1995.

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The remaining columns of Table 2 present quantile regression results for the 10%, 50%,
and 90% quantiles, along with the difference between the 10% and 90% quantiles. Figure 2 shows
how the coefficients for the log of floor area, age, a southern view, and the continuous story
variable vary by quantile. The negative effect of floor area on the price square meter is higher inmagnitude at higher quantiles, which suggests that the variability of per-meter sales prices is lower
for larger units. In contrast, the negative effect of age is higher at low quantiles, which implies
that the variability of per-meter sales prices is higher in older buildings. Our expectation would
be that the coefficients for story would be higher at lower quantiles because the greater likelihoodof a good view on higher stories should translate into lower variability of sales prices on higher
floors. However, the coefficients for story do not have a clear pattern across quantiles.
Figure 3 displays the coefficients for the 10%, 50%, and 90% quantile regression. Since
the 1986 – 1990 regressions use 1990:4 as the base and the 1991 – 1995 regression have 1991:1
as the base, these plots can be interpreted as 10%, 50%, and 90% quantile price indices, and the
median results are simply a rescaled subset of the results shown in Figure 1. The results are
remarkable for how close they appear to being parallel. However, Figure 4 shows that the spread
between the 90% and 10% quantile widened during the first two years of the price boom: pricesrose much more rapidly at the 90% quantile than at the 10% quantile, after which the rate of pricegrowth for the 10% quantile began to catch up. Overall, these results are much different from theresults found for the United States by Landvoigt, Piazzesi, and Schneider (2015) and McMillen(2016), who find that prices tended to rise more rapidly in low-priced regions of urban areas during
the housing boom of the early 2000s. However, the pattern is consistent with the results of Deng,
McMillen, and Sing (2012), who find that the distribution of residential prices shifted farther tothe right for high-priced homes than for low-priced homes during times when prices in Singapore

13were rising rapidly. The results are also consistent with those found for Sydney, Australia by Waltl(forthcoming).
Previous studies such as Deng, McMillen, and Sing (2012); Landvoigt, and Schneider(2015); McMillen (2016), and Waltl (forthcoming) have found that appreciations can vary
markedly across locations within an urban area. The overall shift in the distribution of the log of
per meter sales prices across the two times is shown in Figure 5. The distribution shifted well tothe left from 1986 – 1990 to 1991 – 1995, with a large increase in the area left of the central
tendency. To allow for spatial variation in the shift in the distribution, we estimate CPAR versionsof the quantile regression models. We use a tri-cube kernel with a 25% window size. The weights
applied to each observation are a declining function of the straight-line distance between eachobservation and the grid cell midpoints. The coefficients are then interpolated to all locations in
the data set. We estimate separate models for each period, for quantiles ranging from 0.04 to 0.96
in increments of 0.02. The estimates are stored in matrices with dimensions 𝑛𝑛 × 𝑘𝑘 × 𝑇𝑇, where n= 32,029 for 1986 – 1990 and n = 62,125 for 1991 – 1995, while k = 26 and T = 47 for both samples.
The spatial variation in the implied appreciation rates for 1986 – 1990 and 1991 – 1995 isshown in Figure 6, along with the difference across periods (1986 – 1990 minus 1991 – 1995).
The estimated appreciation rates for 1986 – 1990 are the coefficients on the 1990:4 variable, as
the base is 1986:1. Similarly, the estimated appreciation rates for 1991 – 1995 are the coefficientsfor the 1995:4 variable, with 1991:1 as the base. Figure 6 shows that there is substantial spatial
variation in the estimated median appreciation rates. The estimated median appreciation rates for1986 – 1990 range from 1.084 to 1.308 (i.e., 108.4% to 130.8%) while the estimated appreciation
rates for 1991 – 1995 range from -0.943 to -0.564 (i.e., a decline ranging from 56.4% to 94.3%).
The difference between the rates varies from 1.666 to 2.226.

For the earlier period, the14appreciation rates are highest for areas near downtown Tokyo and the Tokyo Harbor. These are
the same areas where prices declined most rapidly in 1991 – 1995, and consequently these areas
have the largest spreads between the growth rates across the two periods.
The patterns for the 10% quantile appreciation rates are similar to the median: appreciation
rates are highest near downtown Tokyo in 1986 – 1991 with a range of 1.013 to 1.336, and thesubsequent rates of declines are also highest in these areas, with a range of -0.966 to -0.549. The
range for the absolute differences in 10% quantile appreciation rates over the two periods is 1.094
to 1.593 to 2.207. The patterns are somewhat different for the 90% quantile. While the area near
downtown Tokyo is still the region with the highest rates of appreciation in 1986 – 1990, a large
portion of this region had relatively low depreciation rates in 1991 – 1995. However, the rates ofappreciation were so high in the earlier period that the absolute differences between the rates ofchange are again highest in this region: although the deprecation rates were somewhat modest in
1991 – 1995, they were paired with extremely high rates of appreciation in 1986 – 1990. Acrossthe full region, the range of 90% quantile appreciation rates is 1.094 to 1.318 in 1986 – 1990,-0.966 to -0.525 in 1991 – 1995, and the range of absolute differences is 1.629 – 2.257.
Although the CPAR quantile approach generates a seemingly overwhelming number ofcoefficient estimates, the results are easy to interpret by calculating the implied marginal
distribution of the dependent variable at suitable values of the explanatory variable. We focus on
the four primary explanatory variables – the log of floor area, age, a southern view, and the unit’s
story. 6 We set values for the log of floor area to 3.4, 3.9, and 4.2, which correspond (after
For these counterfactual calculations, we pool the 1986 – 1990 and 1991 – 1995 data sets, which imposes a restrictionthat the coefficients for these variables are constant for a given quantile over time. The advantage of pooling the datais to isolate the effects of changes in the value of the explanatory variable rather than combining the effects of changes
in the variable and changes in the coefficients. The models are again estimated using a 25% window and a tri-cube
kernel for the straight-line distance between each observation and the target points.

6

15

rounding) to 30, 50, and 70 square meters. The other variables are set to their actual values in thedata for these calculations. We then make comparable calculations for other variables: age is set
to 10, 20, and 30 years; a southern view is set to 0 and 1, and the story is set to 1, 10, and 20. Whenthe story is set to 1, the value for the 1st-story variable is also set to 1.
The results for the four primary explanatory variables are shown in Figure 9. Increased
floor area shifts the distribution of log per meter sale price to the left, with lower variability athigher areas. Increasing age from 10 to 20 to 30 years shifts the distribution markedly to the left,
with little discernible effect on the spread of the distribution. A southern view has little effect on
the distribution of sales prices. Moving from the 1st story to 10 and 20 shifts the distribution ofprices well to the right, but again with little effect on the variance of the distribution.
Figure 10 displays the results of a series of similar calculations for the quarter of sale. The
other explanatory variables are set to their actual values in data set, but when the variable for one
quarter is set to 1, the values for all other quarters are set to 0. Figure 10 shows the results for the
first quarters of 1986, 1988, 1991, 1993, and 1995. The distribution of the log of per meter saleprice shifts far to the right from 1986 to 1988 to 1991, with a marked increase in the variance.
From 1991 to 1993, the center of the distribution shifts all the way back to the 1988 point, with areduction in variance that roughly matches the original 1986 level. The distribution shifts further
still to the left in 1995, with an increase in the variance. Between 1993 and 1995, the upper endof the distribution clearly shifts further than the lower end.

6. Decompositions
Figure 5 shows that the distribution of sales prices shifted far to the left between 1986 –1990 and 1991 – 1995 in Tokyo. How much of this change was due to changes in the explanatory

16

variables, changes in the location of the sales, and changes in the coefficients? Before presenting
our decomposition results, note that the descriptive statistics from Table 1 and the graphs of thequantile regression estimates in Figure 2 provide some insight into the patterns that might be
expected. Since all existing buildings are older in the later period, the change in the age variable
will tend to shift the distribution of sales prices to the left. However, Figure 2 suggests that thistendency will be at least somewhat ameliorated by the increase in the magnitude of the age discountfor 1991 – 1995 relative to the earlier period. Second, Table 1 shows that the size of the units
increases significantly over time, and this effect is reinforced by the tendency toward higher
coefficient on the log of floor area in 1991 – 1995.
Our estimated models differ somewhat from our presentation in Section 3 in that many ofthe variables represent the quarter of sale. Since it is not possible to analyze how, for example,
the values of the variable for 1987:1 are different in the later sample, a decomposition of the change
in variable is irrelevant for these variables. If we redefine 𝑑𝑑𝑖𝑖′ 𝛿𝛿(𝑧𝑧) as simply 𝐷𝐷(𝑧𝑧𝑖𝑖 ), the following
two terms are added to the decomposition in equation (5):(Location, Time of Sale)(Time of Sale)

�1𝑚𝑚 (𝜏𝜏, 𝑧𝑧1 )𝑒𝑒𝑘𝑘 − 𝐷𝐷
�1𝑚𝑚 (𝜏𝜏, 𝑧𝑧2 )𝑒𝑒𝑘𝑘 +𝐷𝐷
�1𝑚𝑚 (𝜏𝜏, 𝑧𝑧2 )𝑒𝑒𝑘𝑘 − 𝐷𝐷
�2𝑚𝑚 (𝜏𝜏, 𝑧𝑧2 )𝑒𝑒𝑘𝑘
𝐷𝐷

(6e)
(6f)

Equation (6e) shows the effect of changing the location of sales from the period 1 to period 2
locations on the period 1 time coefficients. For examples, if the locations of the sales in period 2
are concentrated in areas that happened to have relatively lows rates of appreciation in the first�1𝑚𝑚 (𝜏𝜏, 𝑧𝑧2 )𝑒𝑒𝑘𝑘 will be further to the left than the distribution implied byperiod, the distribution of 𝐷𝐷

�1𝑚𝑚 (𝜏𝜏, 𝑧𝑧1 )𝑒𝑒𝑘𝑘 . By holding the sales locations constant, equation (6f) then shows the pure effect of𝐷𝐷
the time coefficients.

17

Figure 11 presents the components of the decomposition for the base explanatory variables,
the log of floor area, age, a southern view, and the three story variables. The intercept is notincluded in this set of variables. The effect of changing the values of these variables to their 1991
– 1995 values is the change from “x1b1” to “x2b1” in Figure 11. The age variable is the dominant
one in this change: since buildings age over time, the direct effect of changing this variable to its
1991 – 1995 values is to shift the distribution of sales prices to the left. Next, the effect of changing
the locations of the sales from the 1986 – 1990 sites to those observed for 1991 – 1995 is the shift
from “x2b1” to “x2b12” in Figure 11. The effect of this change is to move the distribution ofprices still further to the left, except in the very low-priced portion of the distribution. Finally, thechange in the estimated coefficients to their 1991 – 1995 values shifts the distribution back to theright – close to the 1986 – 1995 starting point, but somewhat further to the left. Thus, the change
in coefficients offsets most of the change in the variables and the location of the sales.
Figure 12 adds the effect of the quarter of sale variables to these sale price densities. Thebase is the combination of predicted values from the other explanatory variables for 1991 – 1995(“x2b2”) and the period 1 quarter of sale predictions at their period 1 locations (“D11”). 7
Evaluating these quarter of sale values at the 1991 – 1995 sales locations (“x2b2 + D12) has very
little effect on the distribution of sales prices. Switching the coefficients to their 1991 – 1995
values (“x2b2 + D22”) shifts the distribution of sales prices well to the right, although the fat righttail disappears in the process.

The intercept again is not included in these calculations. Since the base period for the second period is 1991:1,
adding the intercept shifts the distributions in Figure 12 so far to the right that the difference between the otherfunctions is obscured.

7

18

7. Conclusion
Conditionally parametric estimators are a convenient away to allow for spatial variation inmodel coefficients when a global parametric specification is not suitable for the entire area covered
in a data set. The models are estimated by placing more weight on observations close to a set oftarget points. After either estimating the model for every location represented in a data set orinterpolating from the target points to other locations, CPAR models produce a separate set of
coefficients for every point in the data set. Although the large number of coefficients would appear
to make the results difficult to interpret, counterfactual distributions can easily be used to display
the results for discrete changes in the values of an explanatory variables, which readily summarize
the direction of and magnitude of a variable’s effect on the overall distribution of the dependent
variables.
In this paper, we extend Machado and Mata’s (2005) method for decomposing changes inthe distribution of the dependent into the portions explained by differences in the explanatory
variables and coefficients by allowing also for changes in the location of the observations
represented in different samples. The procedure requires only one additional sets of calculations– using the data from one sample to estimate the model at the locations represented in the othersample. The procedure can be applied to either standard CPAR (or “geographically weighted
regression”) estimates or to CPAR quantile models.
We use the CPAR quantile approach to analyze changes in the distribution of per metersales prices for condominiums in Tokyo for 1986 – 1990 and 1991 – 1995, which represent periods
when prices first rose dramatically, followed by a sharp downturn. The decompositions suggest alarge portion of the change in change in distributions across the two periods is due to changes inthe variables. The variable with the most influence on the change in distributions is the age of the19unit, which naturally tends to become larger over time. Although higher ages shift the distributionof sales prices to the left, changes in the floor area have the opposite effect because newly
constructed building tended to be larger than the existing units in Tokyo at the time. Changes in
the coefficients offset much of this leftward shift in the price distribution as the discount associatedwith older units declined from 1986 – 1990 to 1991 – 1995.
The CPAR estimates reveal significant spatial variation in the appreciation rates of sales
prices from 1986 – 1990, and then again for the large rates of depreciation observed for 1991 –1995. Prices rose most rapidly near downtown Tokyo, and subsequently declined most rapidly inthe same areas. However, the patterns were not uniform across time periods: the 90% quantile
price indices reveal much lower depreciation rates near downtown Tokyo for 1991 – 1995 than is
implied by the 10% or 50% quantile. Taking the quarter of sale into account in the pricedistribution decompositions suggests that sales for 1991 – 1995 were located in areas that had
relatively low rates of depreciation in 1986 – 1990, which suggests that standard decomposition
would have attributed some of the effects of location to changes in the coefficients or variables.

20

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22

Table 1: Descriptive Statistics
Variable

Mean
Std. Dev. Minimum Maximum
1986 – 1990 (32,029 obs.)
Price (10,000 yen) per Square Meter 104.721
47.08219.188339.971
Log Price per Square Meter
4.553
0.450
2.954
5.829
Floor Area (square meters)
50.46617.33216.000133.900
Log Building Area
3.854
0.384
2.773
4.897
Age
9.721
5.439
0.416
31.916South View
0.209
0.407
0
1
1st Story
0.098
0.297
0
1
nd
2 Story
0.158
0.364
0
1
Story
4.753
2.951
1
25
1991 – 1995 (62,125 obs.)
Price 10,000 yen) per Square Meter
88.07936.30819.569348.052
Log Price per Square Meter
4.408
0.363
2.974
5.852
Floor Area (square meters)
54.24918.60916.000134.550
Log Building Area
3.928
0.379
2.773
4.902
Age
12.8645.968
0.416
34.416South View
0.383
0.486
0
1
st
1 Story
0.097
0.295
0
1
nd
2 Story
0.157
0.364
0
1
Story
4.766
3.013
1
25

23

Table 2: OLS and Quantile Estimates, 1986 – 1990

Variable

OLS

Quantile 0.1 Quantile 0.5 Quantile 0.9

Difference,
Quantile .9 - .1

1986 – 1990 (32,029 obs.)
Log Floor Area

-0.0431(0.0028)-0.0258(0.0038)-0.0519(0.0022)-0.0584(0.0041)-0.0326(0.0040)Age-0.0223(0.0002)-0.0246(0.0003)-0.0223(0.0002)-0.0218(0.0003)

0.0028(0.0004)

South View

0.0048(0.0025)0.0056(0.0040)0.0088(0.0023)-0.0009(0.0044)-0.0064(0.0037)

1st Story-0.0140(0.0035)-0.0174(0.0057)-0.0168(0.0033)-0.0116(0.0063)0.0058(0.0085)

2nd Story-0.0061(0.0028)-0.0040(0.0046)-0.0062(0.0027)-0.0079(0.0051)-0.0039(0.0070)

Story

0.0062(0.0004)

0.0085(0.0006)

0.0061(0.0004)0.0044(0.0007)-0.0041(0.0008)

1991 – 1995 (62,125 obs.)
Log Floor Area

-0.0440(0.0020)0.0104(0.0026)-0.0472(0.0017)-0.0686(0.0030)-0.0790(0.0048)Age-0.0221(0.0001)-0.0233(0.0002)-0.0221(0.0001)-0.0213(0.0002)0.0020(0.0002)

South View

0.0135(0.0014)0.0132(0.0020)0.0158(0.0013)0.0104(0.0023)-0.0028(0.0026)

1st Story-0.0117(0.0025)-0.0128(0.0038)-0.0129(0.0025)-0.0109(0.0045)0.0020(0.0052)

2nd Story-0.0063(0.0020)-0.0043(0.0031)-0.0072(0.0020)-0.0082(0.0036)-0.0039(0.0040)

Story

0.0057(0.0003)

0.0065(0.0004)

0.0055(0.0003)

0.0049(0.0005)-0.0016(0.0006)

Notes. Significance at the 5% level is indicated by bold face. The regressions also include 20
variables indicating the quarter of sale and census area fixed effects (2,184 in 1986 – 1990 and
2,351 in 1991 – 1995). The R2s for the OLS models are 0.899 in 1986 – 1990 and 0.833 in 2991
– 1995.

24

Figure 1: Quantile Median Price Index, 1986 – 2015

25

Figure 2: Coefficient Estimates by Quantile

26

Figure 3: 10%, 50%, and 90% Quantile Estimates, 1986 – 1995

27

Figure 4: Difference between 90% and 10% Quantile Estimates, 1986 – 1995

28

Figure 5: Kernel Density Estimates for Log Price per Square Meter

29

Figure 6: Spatially Varying Median Appreciation Rates
1986 – 1990

Difference, 1986-90 – 1991-2005

1991-1995

30

Figure 7: Spatially Varying 10% Quantile Appreciation Rates
1986 – 1990

Difference, 1986-90 – 1991-2005

1991-1995

31

Figure 8: Spatially Varying 90% Quantile Appreciation Rates
1986 – 1990

Difference, 1986-90 – 1991-2005

1991-1995

32

Figure 9: Predicted Effects of Discrete Changes in Variables

33

Figure 10: Predicted Effects Changes in Quarter of Sale

34

Figure 11: Decomposition 1986 – 1990 to 1991 – 1995, X

35

Figure 12: Decomposition 1986 – 1990 to 1991 – 1995, Time of Sale