166
August 2020

Rolling-Time-Dummy House Price Indexes:
Window Length, Linking and Options for Dealing with
the Covid-19 Shutdown

Robert J. Hill,
University of Graz, Department of Economics

Michael Scholz
University of Graz, Department of Economics

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

Miriam Steurer
University of Graz, Department of Economics

Rolling-Time-Dummy House Price Indexes:
Window Length, Linking and Options for Dealing
with the Covid-19 Shutdown
Robert J. Hill1 , Michael Scholz1 , Chihiro Shimizu2
and Miriam Steurer1

1

Department of Economics, University of Graz, Universitätsstrasse 15/F4, 8010 Graz,

Austria, robert.hill@uni-graz.at, michael.scholz@uni-graz.at, miriam.steurer@
uni-graz.at
2

University of Tokyo, Japan, cshimizu@csis.u-tokyo.ac.jp
17 August 2020

Abstract:
The rolling-time-dummy (RTD) method is used by a number of countries to compute
their official house price indexes (HPIs), since it requires less data and is more flexible
than other hedonic methods. These features also make it well suited for computing
higher frequency HPIs (e.g., monthly or weekly). In this paper we address three key
issues relating to the RTD hedonic method. First, we develop a method for determining
the optimal length of the rolling window. Second, we consider variants on the standard
way of linking the current period with earlier periods, and show how the optimal linking method can be determined. Third, we propose three ways of modifying the RTD
method to make it robust to the distorting effects of the covid-19 shutdown. These
modifications could prove useful for countries using the RTD method in their official
HPIs. (JEL: C43, E31, R31)
Keywords: House price index; Hedonic quality adjustment; Optimal window length; Optimal chain linking; Higher frequency indexes; Covid-19 shutdown
We acknowledge financial support for this project from the Austrian Research Promotion Agency(FFG), grant #10991131. We also wish to thank Niall O’Hanlon for helpful comments.

1

Introduction

The housing market and the broader economy are closely connected. While it is truethat economic booms and recessions can trigger booms and busts in the housing market,
the causation can also run in the opposite direction. The global financial crisis of 20072010 was a case in point. For central banks to effectively maintain financial stability itis therefore important to have reliable and timely house price indexes (HPIs).
To effectively distinguish between genuine price changes and compositional differences, HPIs are typically computed using hedonic methods. The hedonic approachentails estimating shadow prices on the characteristics of properties (such as floor area,
age, and location) so as to ensure that quality is held fixed when measuring pricechanges from one period to the next. For example, Eurostat recommends that countries in Europe should compute their official HPIs using hedonic methods (Eurostat,2016).
A number of hedonic methods for constructing HPIs have been proposed in theliterature (see Hill, 2013). One hedonic method that has been attracting increased
attention in recent years is the rolling-time-dummy (RTD) method. It was first proposedby Shimizu in 1998 as part of a project entitled Construction of Property Price Indexes
using Big Data 1998-2002 at Reitaku University (see Shimizu et al., 2003, 2010).
The RTD method has three desirable properties. First, it is relatively simple tocompute and interpret.
Second, it requires less data than some other hedonic methods, such as the hedonic
imputation or average characteristics methods. This means that it is particularly useful for smaller countries that have less data. In Europe it is used by Croatia, Cyprus,
France, Ireland and Portugal to construct their official HPIs (see Hill et al. 2018). Japan
has recently decided to compute its official residential and commercial property price indexes using the RTD method (see Shimizu and Diewert, 2019). Also, Brunei Darussalam(see https://www.ambd.gov.bn/Site%20Assets%20%20News/RPPI-Technical-Notes.
pdf), Peru and Thailand (see https://www.bot.or.th/App/BTWS_STAT/statistics/
DownloadFile.aspx?file=EC_EI_008_S2_ENG.PDF) are using RTD, and Indonesia is
about to start using it (see Rachman, 2019). RTD’s effectiveness with smaller datasetsmeans that it is also a good candidate for computing higher frequency indexes, such as
1

monthly or weekly.
Third, an index provider using the RTD method can choose the length of the rolling
window. This makes the method very flexible. A longer window increases the robustnessof the index, which can be important when the dataset is small, while a shorter window
increases the current market relevance of the index. Index providers can trade off these
two aspects when choosing the window length.
In this paper we address three key issues relating to the RTD hedonic method.
First, there is no clear consensus regarding window length. For example, France andPortugal use a two-quarter rolling window, Cyprus and Croatia a four-quarter window,
and Ireland a five-quarter window (see Hill et al. 2018). The question is, how can one
determine the optimal window length for any given dataset? We develop an approach
for answering this question and use it to compute the optimal window length for weekly
HPIs in Sydney and Tokyo.
Second, the standard version of the RTD method links the current period to theperiod directly preceding it. It turns out this is just one of many ways that the HPI
can be computed from the estimated hedonic model. We compare a number of different
ways of linking in the current period, and develop an approach for determining which
linking method is optimal. Our approach is then again tested on Sydney and Tokyo
data.
Third, the covid-19 shutdown can generate weak links in the RTD HPI, which canundermine the integrity of the whole time series. We propose three ways of modifying
the RTD method to make it robust to the covid-19 shutdown. We illustrate the problem
using Sydney data. These modifications could prove useful for countries that are alreadyusing the RTD method in their official HPIs.

2

The Rolling Time Dummy (RTD) Method

2.1

The standard version

Consider the standard version of the RTD method with a window length of k + 1
periods, as defined in Shimizu et al. (2010) and O’Hanlon (2011). Supposing that thefirst period in the window is period t, the first step is to estimate a semilog hedonic2

model as follows:
ln ph =CX

βc zhc +

c=1

t+k
X

δs dhs + εhs ,

(1)

s=t+1

where h indexes the housing transactions that fall in the rolling window, ph the transaction price, c indexes the set of available characteristics of the transacted dwellings,
and ε is an identically, independently distributed error term with mean zero. The characteristics of the dwellings are given by the zhc , while dhs is a dummy variables that
equals 1 when s is the period in which the dwelling sold, and zero otherwise.
Estimating this model using ordinary least squares, the change in the price index
from period t + k − 1 to period t + k is then calculated as follows:texp(δ̂t+k
)
Pt+k
=
,
t
Pt+k−1exp(δ̂t+k−1)(2)

where δ̂ denotes the least squares estimate of δ. A superscript t is included on theestimated δ coefficients to indicate that they are obtained from the hedonic model withperiod t as the base (i.e., Pt = 1). As can be seen from (2), the hedonic model withperiod t as the base is only used to compute the change in house prices from periodt + k − 1 to period t + k. The window is then rolled forward one period and the hedonic
model is re-estimated. The change in house prices from period t + k to period t + k + 1
is now computed as follows:
t+1
exp(δ̂t+k+1)
Pt+k+1=,t+1
Pt+kexp(δ̂t+k )(3)

where now the base period in the hedonic model is period t + 1. The price index
over multiple periods is computed by chaining these bilateral comparisons together asfollows:

"
#"
#
"
#
t+1
t−k+1
t−k
)
exp(δ̂t+k+1
exp(δ̂t+1)
exp(δ̂t+2)
Pt+k+1=× ··· ×
.
t−k+1
t+1
Pt
exp(δ̂tt−k )exp(δ̂t+1)
exp(δ̂t+k)(4)

An important feature of the RTD method is that once a price change Pt+k /Pt+k−1 has
been computed, it is never revised. Hence when data for a new period t + k + 1 becomes
available, the price indexes Pt , Pt+1 , . . . , Pt+k are already fixed. The sole objective whenre-estimating the hedonic model to include period t + k + 1 is to compute Pt+k+1 /Pt+k .

3

2.2

Linking variants on the RTD method

Instead of always focusing on the last two estimated δ coefficients in each hedonic model,
an alternative would be to focus on the last and third last coefficients. In this case theprice change from period t + k − 1 to period t + k could be calculated as follows:
Pt+k=
Pt+k−1



Pt+k−2Pt+k−1texp(δ̂t+k)t
exp(δ̂t+k−2
)

,

(5)

where as has been noted above both Pt+k−1 and Pt+k−2 are already fixed by the timethe data for period t + k becomes available. Another alternative is the following:
Pt+k=
Pt+k−1



Pt+k=Pt+k−1



Pt+k−3Pt+k−1



Pt+k−jPt+k−1texp(δ̂t+k)t
exp(δ̂t+k−3
)

,

(6)

,

(7)

and more generally,texp(δ̂t+k)t
exp(δ̂t+k−j)where j ≤ k. In (eq:dvg1), the hedonic model is used to link each new period withtwo periods earlier. In (eq:dvg2), each new period with three periods earlier, whilein (eq:dvg3), each new period with j periods earlier. In other words, given a window
length of k + 1 periods, there are k distinct ways of linking period t + k with the earlier
periods. Each will give a different answer, and one cannot say ex ante that one is better
than another.
Another possibility is to compute the geometric mean of these k sets of results asfollows:"kY
Pt+k−jPt+k=
Pt+k−1 j=1
Pt+k−1texp(δ̂t+k)t
exp(δ̂t+k−j
)

!#1/k
.

(8)

This method uses each single-period link in turn to generate k distinct estimates of
Pt+k /Pt+k−1 , and then takes the geometric mean of these estimates.
A weighted geometric mean could also be computed, with more recent periods being
given more weight. For example, the weights could decline geometrically as follows:"!# (1−λ)λkj−1 


k


1−λ
t
Y
exp(δ̂t+k)
Pt+k
Pt+k−j=,(9)tPt+k−1 j=1 
exp(δ̂t+k−j) Pt+k−1where 0 < λ < 1.
4

There are close similarities here with the literature on constructing monthly or weekly
price indexes for consumer goods using scanner data. Focusing on the case of monthly
indexes, price indexes in this literature are often computed using a 13 month rolling
window. The price index itself in each window is not necessarily computed using ahedonic method. Nevertheless, the same issue arises regarding how the current period
should be linked to earlier periods. In this literature, the standard linking method
used by the RTD method in (2) is sometimes referred to as a movement splice (see forexample de Haan 2015, and Chessa, Verburg and Willenborg 2017). Other possibilities
considered are a window splice which links the new month in by comparing it with thecorresponding month one year earlier (see Krsinich 2014), and a mean splice, which isanalogous to the geometric mean in (8) (see Diewert and Fox 2017). These methods
are all discussed in de Haan, Hendriks and Scholz (2020). In addition, Melser (2018)
proposes a weighted mean splice. Melser’s method is similar in spirit to our weighted
mean described in (9), although the context is rather different. His weighted mean splice
is a solution to a logarithmic weighted least squares problem focused on transitivizing
bilateral price indexes. The weights are derived from the product overlaps between
adjacent periods. No such equivalent weights exist in our context.

2.3

The covid-19 shutdown (and other periods of low transactions)

The covid-19 shutdown has created a situation where there is a lack of data for someperiods. This could create problems when estimating the price index for these periods.
However, it can also cause a level shift or drift in a RTD price index.1 Consider thecase of a monthly index. Suppose the shutdown drastically reduces the number oftransactions in 2020, months 4, 5, and 6. The price index for these months will therefore
contain an unusually high level of noise. For example, suppose the price index for 2020
month 5 contains a large positive random error. This will have a permanent impact on
the RTD price index for all subsequent periods, which will now be subject to upward
drift.
1

We thank Niall O’Hanlon for alerting us to this issue.

5

We consider three ways of mitigating the effect of covid-19 on RTD house price indexes. Our starting point is that the desired window length is known, and that special
action is deemed necessary for any period that has less than N transactions.

Covid 19-Method 1
When computing the price index for period t + k, if any of earlier periods that aresupposed to be in the window have less than N transactions, then these periods aredeleted and replaced by the most recent available earlier period that has at least Ntransactions. If period t + k has less than N transactions, the RTD method is still used
to compute Pt+k /Pt+k−1 (or Pt+k /Pt+k−2 if period t+k−1 has less than N transactions).
But period t + k is then not used to compute the price indexes of later periods.
An example should help clarify the rule. Suppose the window length is set at 3
months, and that 2020, months 4, 5, and 6 have less than N transactions. This meansthat the periods in the rolling window are as follows:
Current period Periods included in the rolling window
2020m3:(2020m1, 2020m2, 2020m3)

2020m4:(2020m2, 2020m3, 2020m4)

2020m5:(2020m2, 2020m3, 2020m5)

2020m6:(2020m2, 2020m3, 2020m6)

2020m7:(2020m2, 2020m3, 2020m7)

2020m8:(2020m3, 2020m7, 2020m8)

2020m9:(2020m7, 2020m8, 2020m9)

The linking structure in this example is graphed in Figure 1. 2020m4, 2020m5,
2020m6 and 2020m7 are all linked into the price index via 2020m3. From then on
normal chronological chaining as described in section 2.1 resumes.
In the scenario described above, the three-month rolling window price indexes arecalculated as follows:
2020m2exp(δ̂2020m4)
P2020m4=,
2020m2
P2020m3exp(δ̂2020m3)

2020m2
P2020m5
exp(δ̂2020m5)=,
2020m2
P2020m3exp(δ̂2020m3)

2020m2
P2020m6
exp(δ̂2020m6)=
2020m2
P2020m3exp(δ̂2020m3)

2020m2
P2020m7exp(δ̂2020m7)=,
2020m2
P2020m3exp(δ̂2020m3 )2020m3
P2020m8exp(δ̂2020m8)=,
2020m3
P2020m7exp(δ̂2020m7 )2020m7
P2020m9exp(δ̂2020m9)=.
2020m7
P2020m8exp(δ̂2020m8 )6Figure 1: The Linking Structure of Covid-19 Method 1 During a Shutdown in
2020m4, 2020m5 and 2020m6

2020m2t

2020m4t

2020m5t

2020m3t

2020m7t

2020m8t

2020m6t

In these equations, the superscript denotes the base month in each hedonic model andthe subscript denotes the month of the estimated δ parameter.
The overall price index with 2020m3 normalized to 1 is then constructed as follows:
1,

P2020m4 P2020m5 P2020m6 P2020m7 P2020m7 P2020m8 P2020m7 P2020m8 P2020m9
,
,
,
,
×
,
×
×
,···
P2020m3 P2020m3 P2020m3 P2020m3 P2020m3 P2020m7 P2020m3 P2020m7 P2020m8
As can be seen the lack of data for 2020m4, 2020m5 and 2020m6 does not contami-nate the longer time series RTD price index after 2020m6. However, the price indexes
for periods 2020m4, 2020m5 and 2020m6 may be unreliable.

Covid 19-Method 2
When a period has less than N transactions, more transactions are drawn fromthe preceding period, until the threshold N is reached. The chronologically closest
transactions from the previous period are used first. If say there are enough transactionsin the last 10 days of the preceding period to reach the N threshold, then only theselast 10 days are added to the current period, when computing the current period priceindex. These 10 days are still also included in the previous period.
As an example, suppose that each of 2020m4, 2020m5 and 2020m6 has more thanN/2 and less than N transactions. The RTD rolling window would now be constructed
as follows:
Current period Periods included in the rolling window
2020m3:(2020m1, 2020m2, 2020m3)
7

2020m4:(2020m2, 2020m3, 2020m4+ )2020m5:(2020m3, 2020m4+ , 2020m5+ )2020m6:(2020m4+ , 2020m5+ , 2020m6+ )2020m7:(2020m5+ , 2020m6+ , 2020m7)

2020m8:(2020m6+ , 2020m7, 2020m8)

2020m9:(2020m7, 2020m8, 2020m9)

The + superscripts above denote that the month is being supplemented with datafrom the previous month. Once each month with less than N transactions has been
supplemented with transactions from the previous month, the RTD price index is computed in the standard way described in section 2.1.

Covid 19-Method 3
Our third alternative approach is to not compute an index for a period with lessthan N transactions. Instead it is merged with the next period. If together these two
periods reach the N transaction threshold, then they are treated as a single period. If
together they still do not reach the N transaction threshold, then again no index iscomputed until the next period becomes available, etc.
As an example, again suppose that each of 2020m4, 2020m5 and 2020m6 has morethan N/2 and less than N transactions. The RTD rolling window would now be constructed as follows:
Current period Periods included in the rolling window
2020m3:(2020m1, 2020m2, 2020m3)

2020m4:NA

2020m4-m5:(2020m2, 2020m3, 2020m4-m5)

2020m6:NA

2020m6-m7:(2020m3, 2020m4-m5, 2020m6-m7)

2020m8:(2020m4-m5, 2020m6-m7, 2020m8)

2020m9:(2020m6-m7, 2020m8, 2020m9)

2020m10:(2020m8, 2020m9, 2020m10)

All three methods ensure that the covid-19 shutdown does not contaminate the8index in later periods. They differ in their treatment of the covid-19 period. Method 1
computes price indexes for each shutdown month using whatever transaction data are
actually available. Method 2 supplements the data for shutdown months with data fromthe previous month (or more if required), while method 3 merges shutdown months andtreats them as a single period if each individually does not contain enough transactions.
Which method is best depends on the needs of users. If it is important that a priceindex is computed for every period (here months) then either method 1 or 2 should be
used. Method 1 will better capture actual price movements during the shutdown unless
it becomes too distorted by noise arising from small sample sizes.
While transaction data are not available yet for 2020m4, 2020m5 and 2020m6, we
can illustrate these covid-19 methods and their impact on the resulting price index using
data for Sydney. The number of transactions in Sydney are much lower in the second
half of December through to late February (corresponding to the Australian summer
holiday). The summer period each year therefore acts like a mini shutdown.

3

Quarterly Benchmarks

3.1

The hedonic imputation method

The hedonic imputation method is an alternative to the RTD method (see Diewert 2011
and Hill 2013). We use the hedonic imputation method here as a reference index forassessing the performance of different versions of the RTD method.
The hedonic imputation approach estimates a separate hedonic model for each period:
ln pt,h = βt · zt,h + εt,h ,(10)where for convenience βt and zt,h now both denote vectors. The hedonic model is thenused to impute prices for individual houses. For example, let p̂t+1,h (zt,h ) denote theimputed price in period t + 1 of a house with the characteristic vector zt,h sold in periodt. This price is imputed by substituting the characteristics zt,h , into the estimated9hedonic model of period t + 1 as follows:2
p̂t+1,h (zt,h ) = expCX!
β̂c,t+1 zc,t,h

.

(11)

c=1

With these imputed prices it is now possible to construct a matched sample, thus
allowing standard price index formulas to be used.
"1/Ht+1 #Ht+1 
Y
p̂
t+1,h
PI
Paasche − Type Imputation : Pt,t+1=
p̂(z
t,h t+1,h )
h=1
(12)
"
1/Ht #
Ht
Y
p̂
(z
)
t+1,h t,hLI
Laspeyres − Type Imputation : Pt,t+1
=
p̂t,h
h=1
(13)
q
TI
PI
LI
Törnqvist − Type Imputation : Pt,t+1 = Pt,t+1
× Pt,t+1(14)
In a comparison between periods t and t + 1, the Laspeyres-type index focuses on the
Ht houses that sold in the earlier period t. Similarly the Paasche-type index focuseson the Ht+1 houses that sold in the later period t + 1. These price indexes give equal
weight to each house sold.3 By taking the geometric mean of Paasche and Laspeyres,the Törnqvist-type index gives equal weight to both periods. The Paasche, Laspeyres
and Törnqvist-type indexes above are of the double imputation variety, meaning thatboth prices in each price relative are imputed. A single imputation approach by contrast
imputes only one price in each pair (since the actual price is always available for oneof the two periods being compared). There has been some discussion in the literature
over the relative merits of the two approaches (see for example de Haan 2004, and Hilland Melser 2008). Empirically we try both approaches. The resulting price indexes are
virtually indistinguishable. Hence to simplify the presentation we focus here only on
double imputation price indexes. The hedonic imputation method is flexible in that itallows the characteristic shadow prices to update each period.
2

See Silver and Heravi (2007), Diewert, Heravi, and Silver (2009), and Rambaldi and Rao (2013)

for a discussion of some of the advantages of the hedonic imputation method.
3This democratic weighting structure is in our opinion more appropriate in a housing context thanweighting each house by its expenditure share. See de Haan (2010) for a discussion on alternative
weighting schemes.

10

In the context of weekly indexes, the hedonic imputation method is unlikely to work
well since the sample sizes in many weeks may be too small to justify estimating a separate hedonic model each week. However, in our context, quarterly hedonic imputation
will provide a useful benchmark for weekly RTD indexes.

3.2

The time-dummy method

We also use the time-dummy index as a reference for assessing the performance of RTD
weekly indexes. The time-dummy method is the limiting case of the RTD method where
the window length is the same as the number of periods in the comparison.
ln ph =CX

βc zhc +c=1

TXδt dht + εht .

(15)

t=2

The price index for period t relative to period 1 is then calculated as follows:Pt= exp(δ̂t ).
P1

3.3

(16)

A performance criterion for weekly indexes derived fromquarterly indexes

We propose a criterion here for determining the optimal window length and linking
method for weekly RTD indexes, by comparing them with reference quarterly hedonic
indexes.4 We consider two reference quarterly hedonic indexes: these are the hedonic
imputation method and the time-dummy method described above. We focus on these
two methods because they are quite different (i.e., one re-estimates the hedonic model
every quarter while the other does not re-estimate at all). Using these quarterly indexes
as benchmarks should avoid biasing the results towards any particular window length.
Empirically we find that the quarterly hedonic imputation and time-dummy methods
approximate each other closely. By contrast for weekly RTD indexes, if we allow thewindow length to vary between 2 and 53 weeks, the range of possible results becomesmuch larger (see section 5).
4

With slight modifications, the same method can be used to determine the optimal window length

for monthly indexes as well.

11

The greater sensitivity of weekly indexes to the choice of hedonic method makes
them a more interesting focus of analysis than quarterly indexes. The choice of window
length really matters for weekly RTD indexes. Furthermore, the greater robustnessof quarterly indexes is a property we can exploit to discriminate between competing
weekly RTD indexes.
The first step of our criterion for assessing the performance of alternative weekly
RTD indexes is to construct a quarterly index from each weekly index. This can bedone in the following way. Let t = 1, . . . , T index the quarters in the data set, andw
is obtained fromv = 1, . . . , V the 13 weeks in a quarter. A quarterly price index Pt,t+1

a weekly price index as follows:w
Pt,t+1
=

1/13
13 
Y
Pt+1,v
Pt,v

v=1

,

(17)

where Pt,v denotes the level of the weekly price index in quarter t, week v. Each element
Pt+1,v /Pt,v in (17) is a price index comparing a particular week with another week one
quarter later. In other words, each of these elements is a price index calculated at aquarterly frequency. A total of 13 such indexes can be computed in each quarter.5 Bytaking the geometric mean of these 13 quarterly frequency price indexes, we obtain an
overall quarterly price index, which can be interpreted as the quarterly equivalent of
the original weekly index.
The index above in (17) is unweighted. It gives equal weight to all weeks irrespectiveof how the observations are distributed across weeks. A weighted variant that weights
according to the number of observations each week is the following:w
Pt,t+1=
13 YPt+1,vv=1

Pt,v

st,v +st+1,v
2

,

(18)

where st,v = St,v /St , with St,v denoting the number of observations in week (t, v), andSt the total number of observations in quarter t. Empirically we find that the weighted
and unweighted results are very similar. Hence in what follows we focus only on theunweighted case represented in (17).
5

The weeks do not overlap exactly with the quarters. So occasionally it is necessary to include 14

weeks in (17).

12Once the quarterly version of the weekly index has been constructed, its performancecan be measured by comparing it with a reference quarterly index. Here we make thecomparison using two alternative metrics proposed by Diewert (2002, 2009).
!
!
#
"
T −1
quart
w
Pt,t+1
Pt,t+1
1 X
+
X1 =
−2 ,
quart
w
T − 1 t=1
Pt,t+1
Pt,t+1"X2 =T −1 X1
T − 1 t=1 w
Pt,t+1quart
Pt,t+1

#2

!

−1

"
+

quart
Pt,t+1w
Pt,t+1

!

#2 

−1
.


The smaller the value of the X metric, the more similar are the two indexes. The
ordinal rankings generated by the two X metrics are almost identical. Hence in ourempirical analysis we only present results for X1 .
Given a reference quarterly index, we can then vary the length of the RTD rolling
window and observe how it affects the X metric. We prefer whichever window length
generates the smallest X metric. An important question then is how robust is the
optimal window length to the choice of reference quarterly index? If it is reasonably
robust, then the selected window length is optimal in the sense that it generates aweekly RTD index that is the most consistent with our reference quarterly indexes.
Similarly, holding the window length fixed at 53 weeks, we can observe how changing
the RTD linking method affects the X metric. Again, we prefer the linking methodwith the smallest X metric.

4

The Data Sets

4.1

The Sydney data set and hedonic model

We use a data set obtained from Australian Property Monitors that consists of prices
and characteristics of houses sold in Sydney (Australia) for the years 2003–2014. For
each house we have the following characteristics: the actual sale price, time of sale,postcode, property type (i.e., detached or semi), number of bedrooms, number of bathrooms, land area, exact address, longitude and latitude. (We exclude all townhouses
from our analysis since the corresponding land area is for the whole strata and not for
the individual townhouse itself.)
13

For a robust analysis it was necessary to remove some outliers. This is because
there is a concentration of data entry errors in the tails of the distribution, caused
for example by the inclusion of erroneous extra zeroes. These extreme observations
can distort the results. Complete data on all our hedonic characteristics are available
for 433 202 observations. To simplify the computations we also merged the number ofbathrooms and number of bedrooms to broader groups (one, two, and three or morebathrooms; one or two, three, four, five or more bedrooms).
Using weekly periods, the hedonic model for Sydney is estimated with a rolling
window ranging between 2 weeks and 53 weeks. The window is then rolled forward one
period and the hedonic model re-estimated. Hence in the case of the 2 week window, a
total of 711 hedonic models are estimated covering the time interval from January 2003
to December 2014.
The hedonic model estimated for Sydney is semilog, and contains the following fivecharacteristics:(i) number of bedrooms,(ii) number of bathrooms,(iii) log of land area,(iv) house type (detached, or semi),(v) postcode.
All these variables with the exception of land area take the form of dummy variables.

4.2

The Tokyo data set and hedonic model

The Tokyo data set consists of 23 wards of the Tokyo metropolitan area (621 square
kilometers), and the analysis period is approximately 30 years between January 1986
and June 2016. The data set covers previously-owned condominiums published in Residential Information Weekly (or Shukan Jyutaku Joho in Japanese) published by RECRUIT, Co. This magazine provides information on the characteristics and asking
prices of listed properties on a weekly basis. Moreover, Shukan Jutaku Joho provides
time-series data on housing prices from the week they were first posted until the week
they were removed as a result of successful transactions. We only use the price in the
final week because this can be safely regarded as sufficiently close to the contract price.
14

The available housing characteristics include floor space and age. The convenienceof public transportation from each housing location is represented by travel time to thecentral business district (CBD), and time to the nearest station. Ward dummies (i.e.,city codes) and a railway dummy to indicate along which railway/subway line a housing
property is located are also available.
The hedonic model for Tokyo is estimated over 242 233 observations. The functional
form is semilog. The explanatory variables used here are:(i) log of floor area,(ii) age,(iii) time to nearest station,(iv) time to Tokyo central station (included as a quadratic),(v) city code,(vi) ward dummy.

5

Results

5.1

The sensitivity of the results to the choice of window
lengthThe spreads of the weekly RTD hedonic price indexes for Sydney and Tokyo as thewindow length is varied between 2 and 53 weeks are shown in Figures 2 and 3. It canbe seen that the weekly indexes are quite sensitive to the choice of window length.

5.2

The sensitivity of the results to the choice of linking method

Holding the window length fixed at 53 weeks, the sensitivity of a weekly RTD method tothe choice of linking method is shown for Sydney and Tokyo in Figures 4 and 5. It canbe seen that the variation in the RTD price indexes from varying the linking method
is smaller than the variation resulting from changing the window length. However, thespread is still significant.

15

Figure 2: The Impact of Varying the Window Length on Weekly RTD House Price

2.2

Indexes for Sydney

1.6
1.4
1.0

1.2

index variation

1.8

2.0

Min/Max
RTD21
RTD24

2003

2005

2007

2009

2011

2013

Note: RTD21 and RTD24 denote the 21 and 24 week RTD price indexes for Sydney. Min and max
denote the lower and upper bounds on all the RTD price indexes with windows ranging between 2 and
53 weeks.

5.3

A quarterly index as a benchmark

The hedonic imputation and time-dummy methods generate very similar quarterly priceindexes. The results are shown in Figures 6 and 7. These results indicate that at aquarterly frequency we have quite a good idea of what the right answer is. Hence these16Figure 3: The Impact of Varying the Window Length on Weekly RTD House Price

3.5

Indexes for Tokyo

2.0
1.0

1.5

index variation

2.5

3.0

Min/Max
RTD19
RTD53

1986 1989 1992 1995 1998 2001 2004 2007 2010 2013 2016

Note: RTD19 and RTD53 denote the 19 and 53 week RTD price indexes for Tokyo. Min and max
denote the lower and upper bounds on all the RTD price indexes with windows ranging between 2 and
53 weeks.

quarterly indexes can be used as a benchmark for discriminating between competing
weekly indexes.

17Figure 4: The Impact of Varying the Linking Method on Weekly RTD House Price
Indexes with a 53 Week Window for Sydney

1.0

1.2

1.4

1.6

1.8

2.0

2.2

index variation

2003

2005

2007

2009

2011

2013

Note: This graph shows the range of RTD price indexes for Sydney resulting from using different
single-period linking methods. With the window length fixed at 53 weeks, there are 52 ways of doing
single-period linking.

5.4

How RTD index performance depends on window length

Here we focus on the standard RTD linking method described in section 2.1. For
this case, the X1 metric for each RTD window length for Sydney with the hedonic
imputation index as the reference quarterly index is shown in Figure 8. The X1 metric18

Figure 5: The Impact of Varying the Linking Method on Weekly RTD House Price

2.0
1.0

1.5

index variation

2.5

3.0

3.5

Indexes with a 53 Week Window for Tokyo

1986 1989 1992 1995 1998 2001 2004 2007 2010 2013 2016

Note: This graph shows the range of RTD price indexes for Tokyo resulting from using different
single-period linking methods. With the window length fixed at 53 weeks, there are 52 ways of doing
single-period linking.

is minimized when the RTD window length is 21 weeks.6 The same answer is obtained
when the time-dummy index is used as the reference quarterly index as shown in the
Appendix in Figure A1.
6

The red curve in Figure 8 and subsequent Figures is a fitted curve obtained by a local-linear

smoother.

19

Figure 6: Quarterly Hedonic Imputation and Time-Dummy House Price Indexes for

2.2

Sydney

1.6
1.0

1.2

1.4

index variation

1.8

2.0

TDH
HDI

2003

2005

2007

2009

2011

2013

Note: TDH and HDI here denote quarterly time-dummy hedonic and hedonic double imputation priceindexes for Sydney. As can be seen, the two indexes closely approximate each other.

For Tokyo, when the hedonic imputation index is used as the reference quarterlyindex, the X1 metric is minimized by an RTD window length of 18 weeks, as shown in
Figure 9. When the time-dummy index is used as the reference quarterly index, theresults for Tokyo are not so clear, as shown in the Appendix in Figure A2.
The optimal window length may also depend on the linking method. To illustrate
this point, we recompute the optimal window length for Sydney, where now the linking is20

Figure 7: Quarterly Hedonic Imputation and Time-Dummy House Price Indexes for

3.5

Tokyo

2.0
1.0

1.5

index variation

2.5

3.0

TDH
HDI

1986 1989 1992 1995 1998 2001 2004 2007 2010 2013 2016

Note: TDH and HDI here denote quarterly time-dummy hedonic and hedonic double imputation priceindexes for Tokyo. As can be seen, the two indexes closely approximate each other.

done using the geometric mean linking method as described in (8). Using the quarterly
hedonic imputation method as the benchmark, the optimal window length is now 19
weeks (see Figure 10). This is quite similar to the optimal window length of 21 weeks
obtained using single-period linking.
In summary, we find that for Sydney the optimal RTD window length is between
19 and 21 weeks depending on the linking method used. For Tokyo, according to the21Figure 8: Performance of Alternative Window Lengths with the Quarterly Hedonic
Imputation Price Index as the Reference: Sydney●

3.5e−05
3.0e−05
2.5e−05

Performance criterion

4.0e−05

4.5e−05

Reference Index: quarterly HDI

●

●

●

●

●

2.0e−05

●
●

● ●

●

●

1.5e−05

●

●
●
●

10

●

●
●

●

●

● ●

● ●
● ●
●

● ●

● ● ● ●

●

● ●

●

● ● ●

●

● ●

●
●

●

●

20

30

40

50

window length
Note: This graph shows which window length generates a weekly RTD price index that most closely
approximates a quarterly hedonic imputation index. For Sydney the optimal window length here is 21
weeks.

quarterly hedonic imputation benchmark, the optimal window length is 18 weeks.

22

Figure 9: Performance of Alternative Window Lengths with the Quarterly Hedonic
Imputation Price Index as the Reference: Tokyo

●

4e−05
3e−05
2e−05

Performance criterion

5e−05

6e−05

Reference Index: quarterly HDI

●

●
●

●
●
●

●
●

●

● ●
●
●
●

10

●

●

● ● ●

●

●

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●

● ●

●

●

● ● ●

●

● ●

●

● ●

● ●

●

●

●

●

●

●

●

●

20

30

40

50

window length
Note: This graph shows which window length generates a weekly RTD price index that most closely
approximates a quarterly hedonic imputation index. For Tokyo the optimal window length here is 18
weeks.

5.5

How RTD index performance depends on the linking method

Now instead we hold the RTD window length fixed at 53 weeks, and compare the impact
on the X1 metric of varying the linking method used by the RTD method. With a 53
week window, there are 52 ways of linking a new period to a single previous period, as

23

Figure 10: Performance of Geometric Mean Linking for Different Window Lengths:
Quarterly Hedonic Imputation Benchmark for Sydney

Reference Index: quarterly HDI

2.0e−05
1.8e−05

Performance criterion

2.2e−05

2.4e−05

●

●
●

●
● ●

●
●
●

● ●

●

●

●

●

●

●

1.6e−05

●
●
●
●
●

●
●

●

1.4e−05

●

● ●
●
●
● ●

10

● ●

●

●

●

●

●

●

●

●

●
●

●

●

20

30

40

50

window length
Note: This graph shows which window length generates a weekly RTD price index that most closely
approximates a quarterly hedonic imputation index when the current period is linked in using thegeometric linking method in (8). For Sydney the optimal window length here is 19 weeks.

described in (6). For Sydney, the X1 metric corresponding to each of these 52 ways oflinking is shown in Figure 11 for the case where the quarterly hedonic imputation index
is used as the reference. Corresponding results with the quarterly time-dummy index
as the reference are shown in A3.
In addition to these 52 single period linking methods, we also consider three average
24linking methods.
(i) Geomean52 is the geometric mean of the 52 single period linking methods as
described in (8).
(ii) Geomean20 is the geometric mean of the 20 chronologically closest single period
linking methods.
(iii) lambda=0.95 is the weighted geometric mean method with λ = 0.95 as describedin (9).
When the quarterly hedonic imputation method is used as the reference index the
optimal linking method is to link week t to week t − 16 (see Figure 11). Linking throughthe period 16 weeks earlier even slightly outperforms the average linking methods (i),(ii) and (iii).
When the quarterly time-dummy method is used as the reference index the optimallink for week t is with week t − 13 (see Figure A3 in the Appendix). In this case, linking
through the period 13 weeks earlier slightly outperforms methods (i) and (iii), but isabout equivalent to taking the geometric mesn of the chronologically most recent 20
single-week links.
The corresponding results for Tokyo are shown in Figures 12 and A4. For the singleweek links, in both Figures 12 (where the quarterly hedonic imputation index is used
as a benchmark) and A4 in the Appendix (where the quarterly time-dummy index isused as the benchmark), the X1 metric is minimized by linking week t with week t − 12
(i.e., linking the current week with 12 weeks earlier).
For Tokyo in Figure 12 the averaging methods (i), (ii) and (iii) perform equivalently
to linking through 12 weeks earlier. In Figure A4, method (i) (i.e., the geometric mean of
the 52 single-period lining methods) outperforms all the single-period linking methods.

5.6

The covid-19 shutdown method illustrated using weekly
Sydney data

Here we focus specifically on covid-19 shutdown method 1, as described in section 2.3.
Setting the minimum number of observations per week to 250, we can see from Figure
13 that every year the last week in December and the first week in January fail to attain25

Figure 11: Performance of Alternative Linking Methods: Quarterly Hedonic Imputation Benchmark for Sydney

1e−04

Reference Index: quarterly HDI
geomean52
geomean20
lambda=0.95

8e−05

●

●
●

●

●

●

●

6e−05

●

●
●
●
●

●●

4e−05

Performance criterion

●

●

●
●●

●

●

●
●
●

●

●

●●
●

●

2e−05

●

●●●●●●●●

0

●●

●
●●

●●

10

●●●●

●
●

20

30

40

50

linking
Note: This graph shows the performance for Sydney (relative to a quarterly hedonic imputation index)
of different single-period linking methods, Geomean52, Geomean20, and the weighted geomean methodin (9) with λ set to 0.95. The best performing method is single-period linking with 16 weeks earlier.

this threshold.
Setting the window length to 7 weeks, the standard RTD method exhibits a slight
upward drift compared with covid-19 method 1, as can be seen in Figure 14. Towards
the end of our sample, the cumulative magnitude of this upward drift is 8.9 percent. It
remains to be seen how big the actual covid-19 shutdown drift will be. Clearly it will
26

Figure 12: Performance of Alternative Linking Methods: Quarterly Hedonic Imputation Benchmark for Tokyo

Reference Index: quarterly HDI

0.00015geomean52
geomean20
lambda=0.95

●

●

●

0.00010

●

●

●

●

●

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●

●●

●
●

●

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●
●
●

●

0.00005

Performance criterion

●

●

●

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●
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0

●

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10

●

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●
●

●
●

●

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●

●
●

●

●

●

●

●

●

20

30

40

50

linking
Note: This graph shows the performance for Tokyo (relative to a quarterly hedonic imputation index)
of different single-period linking methods, Geomean52, Geomean20, and the weighted geomean methodin (9) with λ set to 0.95. The best performing method is single-period linking with 12 weeks earlier.

differ depending on the country or city and frequency of the index. Drift is most likely
to be a problem for smaller countries without much data.

27

Figure 13: Weekly Observations Versus the 250 Observation Threshold: Sydney

600
400
0

200

number of observations

800

1000

threshold=250

2003

2005

2007

2009

2011

2013

Note: This graph shows the weekly number of transactions in Sydney. The low point each year isthe last week in December and the first week in January. When implementing covid-19 Method 1 athreshold of 250 transactions per week is used.

6

Conclusion

We have considered two dimensions over which RTD HPIs can differ. These are thewindow length and the method used for linking the current period to earlier periods. We
have proposed a new criterion for determining the optimal window length and linking
method.

28

Figure 14: Comparing the Standard 7-Week RTD Index with the Covid-19 Method
1 Index

2.2

7 week window

1.0

1.2

1.4

1.6

1.8

2.0

RTD
RTD method 1

2003

2005

2007

2009

2011

2013

Note: RTD here is a standard weekly RTD price index computed with a 7 week window. RTD method 1
is the modified method where weeks with less than 250 transactions are treated separately as explained
in section 2.3 for covid-19 Method 1. Failure to adjust for low transaction weeks seems to cause a slight
upward drift in the RTD index.

Focusing on weekly indexes, we find that for Sydney the optimal window length is
between 19-21 weeks. For Tokyo the optimal window length is about 18 weeks.
We show that it is possible to improve on the standard linking method used by the
RTD method. The linking method that performs best on weekly Sydney data links the29current week with a period between 13-16 weeks earlier. For Tokyo, linking the current
week with the period 12 weeks earlier performs best. Geometric averages of the singleperiod linking methods performs about equally well as the best of the single-period
linking methods.
We have also considered how the RTD method can be adjusted to mitigate thedistorting effects of the covid-19 shutdown on house price indexes. All three of our
proposed modifications of RTD are designed to prevent periods with limited data (suchas the covid-19 shutdown) from undermining the integrity of the overall index. Thesemethods could prove useful for countries in Europe and the rest of the world thatcompute their official HPIs using the RTD method.

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Diewert, W. E. (2009), “Similarity Indexes and Criteria for Spatial Linking,” in Purchasing Power Parities of Currencies: Recent Advances in Methods and Applications, D. S. P. Rao (ed.). Edward Elgar: Cheltenham, UK, . Chapter 8, 183-216.
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Appendix: Results Obtained Using the Quarterly
Time-Dummy Index as the Reference
As a robustness check, here we recompute all the results derived using the quarterly
hedonic imputation method as a benchmark. Now instead we use the quarterly timedummy method as the benchmark. In most cases the results are very similar to thoseobtained using the hedonic imputation method.

32

Figure A1: Performance of Alternative Window Lengths with the Quarterly TimeDummy Price Index as the Reference: Sydney

Reference Index: quarterly TDH

5.0e−05●

4.0e−05●

3.0e−05

Performance criterion

●

●
●
●

●
●
●

●

2.0e−05

●
●

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●

● ●
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10

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20

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● ●

30

●

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40

●
● ● ●

● ●

●

●

50

window length
Note: This graph shows which window length generates a weekly RTD price index that most closely
approximates a quarterly time-dummy hedonic index. For Sydney the optimal window length here is
21 weeks.

33

Figure A2: Performance of Alternative Window Lengths with the Quarterly TimeDummy Price Index as the Reference: Tokyo

●

7e−05

●

6e−05

Performance criterion

8e−05

9e−05

Reference Index: quarterly TDH

●

● ●

●

5e−05

●

●
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20

30

40

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50

window length
Note: This graph shows which window length generates a weekly RTD price index that most closely
approximates a quarterly hedonic imputation index. For Tokyo the optimal window length here is 53
weeks.

34

Figure A3: Performance of Alternative Linking Methods with Standard RTD Linking: Quarterly Time-Dummy Benchmark for Sydney

0.00012Reference Index: quarterly TDH
geomean52
geomean20
lambda=0.95

0.00010●●0.00008

●

●

●

●

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0.00006

Performance criterion

●

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0.00004

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0.00002●●●●●●●●●●●●●

0

●
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10

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20

30

40

50

linking
Note: This graph shows the performance for Sydney (relative to a quarterly tiem-dummy hedonic
index) of different single-period linking methods, Geomean52, Geomean20, and the weighted geomean
method in (9) with λ set to 0.95. The best performing method is single-period linking with 13 weeks
earlier.

35

Figure A4: Performance of Alternative Linking Methods with Standard RTD Linking: Time-Dummy Benchmark for Tokyo

0.00020

Reference Index: quarterly TDH
geomean52
geomean20
lambda=0.95●

0.00015

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●

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0.00010

Performance criterion

●

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linking
Note: This graph shows the performance for Tokyo (relative to a quarterly time-dummy hedonic index)
of different single-period linking methods, Geomean52, Geomean20, and the weighted geomean methodin (9) with λ set to 0.95. The best performing method is Geomean52.

36