Real Estate Modelling and Forecasting By Chris Brooks_1 doc

Introduction 11
involved with acquisitions, will rely more on their knowledge of the local-
ity and building to make a buy or sell decision. This has also given rise to
so-called ‘judgemental’ forecasts. Real estate markets have gone through
severe cycles not predicted by bottom-up analysis, however, and thus this
approach to forecasting has been questioned. For many, the winning for-
mula is now not just having good judgement about the future direction
of the market, but also making a careful quantitative analysis explaining
cyclical movements and the impact of broader trends. Therefore, consistent
with evidence from other fields, a view that has increasingly gained popular-
ity is that the optimal approach arises from a combination of judgemental
and quantitative forecasting. Moreover, there is a more generic econometric
and forecasting interest. Do quantitative techniques underperform judge-
mental approaches or is the combination of quantitative and judgemental
forecasts the most successful formula in the real estate market? The book
addresses this issue directly, and the tools presented will give the reader a
framework to assess such quandaries.
Real estate forecasting can also be used for model selection. There are
often competing theories available and it may be the case that there is more
than one theory-consistent model that passes all the diagnostics tests set
by the researcher. The past relative forecasting success of these models will
guide model selection for future forecast production and other uses.
Finally, forecasting is the natural progression in real estate as more data
become available for a larger number of markets. In scholarly activity, the
issue of data availability is highlighted constantly. One would expect that,
with more data and markets, interest in real estate forecasting will continue
to grow. The key objectives of forecasting in real estate are presented in
box 1.1.
Box 1.1 Objectives of forecasting work
(1) Point forecasts. The forecaster is seeking the actual forecast value for rent growth
or capital growth in one, two, three quarters or years, etc.

(2) Direc tion forecasts. The forecaster is interested in the direction of the forecast
and whether the trend is upward or downward (and perhaps an assessment can
be made as to how steep this trend will be).
(3) Turning point forecasts. The aim in this kind of forecast is to identify turning points
or the possibility of a turning point.
(4) Confidence. The modelling and forecasting process is used to attach a confidence
interval to the forecast, how it can vary and with what probability.
(5) Scenario analysis. This is the sensitivity of the forecast to the drivers of the model.
The content of this book is more geared to help the reader to perform tasks one, two
and five.
12 Real Estate Modelling and Forecasting
1.8 Econometrics in real estate, finance and economics:
similarities and differences
The tools that we use when econometrics is applied to real estate are funda-
mentally the same as those in economic and financial applications. The sets
of issues and problems that are likely to be encountered when analysing
data are different, however. To an extent, real estate data are similar to
economic data (e.g. gross domestic product [GDP], employment) in terms of
their frequency, accuracy, seasonality and other properties. On the other
hand, there are some important differences in how the data are generated.
Real estate data can be generated through the valuation process rather than
through surveys or government accounts, as is the case for economic data.
There are some apparent differences with financial data, given their high
frequency. A commonality with financial data, however, is that most real
estate data are not subject to subsequent revisions, or, at least, not to the
extent of economic data.
In economics, a serious problem is often a lack of data to hand for testing
the theory or hypothesis of interest; this is often called a small samples prob-
lem. Such data may be annual and their method of estimation may have
changed at some point in the past. For example, if the methods used to

measure economic quantities changed twenty years ago then only twenty
annual observations at most are usefully available. There is a similar prob-
lem in real estate markets. Here, though, the problem concerns not only
changing methods of calculation but also the point at which the data were
first collected. In the United Kingdom, data can be found back to 1966 or
earlier, but only at the national level. Databases such as the United King-
dom’s Investment Property Databank (IPD) and that of the United States’
National Council of Real Estate Investment Fiduciaries (NCREIF) go back to
the 1970s. In other regions, such as the Asia-Pacific retail markets, however,
data are available only for about ten years. In general, the frequency dif-
fers by country, with monthly data very limited and available only in some
locations.
As in finance, real estate data can come in many shapes and forms. Rents
and prices that are recorded are usually the product of valuations that have
been criticised as being excessively smooth and slow to adjust to changing
market conditions. The problem arises from infrequent trading and trying
to establish values where the size of the market is small. The industry has
recognised this issue, and we see an increasing compilation of transactions
data. We outlined in section 1.5 above that other real estate market data,
such as absorption (a measure of demand), are constructed based on other
market information. These data are subject to measurement error and revi-
sions (e.g. absorption data are subject to stock and vacancy rate revisions
Introduction 13
unless they are observed). In general, measurement error affects most real
estate series; data revisions can be less serious in the real estate context
compared with economics, however.
Financial data are often considered ‘noisy’, which means that it is diffi-
cult to separate underlying trends or patterns from random and uninteresting
features. Noise exists in real estate data as well, despite their smoothness,
and sometimes it is transmitted from the financial markets. We would con-

sider real estate data noisier than economic data. In addition, financial data
are almost always not normally distributed in spite of the fact that most
techniques in econometrics assume that they are. In real estate, normality
is not always established and does differ by the frequency of the data.
The above features need to be considered in the model-building process,
even if they are not directly of interest to the researcher. What should also
be noted is that these issues are acknowledged by real estate researchers,
valuers and investment analysts, so the model-building process is not hap-
pening in a vacuum or with ignorance of these data problems.
1.9 Econometric packages for modelling real estate data
As the title suggests, this section contains descriptions of various computer
packages that may be employed to estimate econometric models. The num-
ber of available packages is large, and, over time, all packages have improved
in the breadth of the techniques they offer, and they have also converged
in terms of what is available in each package. Some readers may already
be familiar with the use of one or more packages, and, if this is the case,
this section may be skipped. For those who do not know how to use any
econometrics software, or have not yet found a package that suits their
requirements – read on.
1.9.1 What packages are available?
Although this list is by no means exhaustive, a set of widely used packages is
given in table 1.1. The programmes can usefully be categorised according to
whether they are fully interactive (menu-driven), command-driven (so that
the user has to write mini-programmes) or somewhere in between. Menu-
driven packages, which are usually based on a standard Microsoft Windows
graphical user interface, are almost certainly the easiest for novices to get
started with, for they require little knowledge of the structure of the pack-
age, and the menus can usually be negotiated simply. EViews is a package
that falls into this category.
On the other hand, some such packages are often the least flexible, since

the menus of available options are fixed by the developers, and hence, if one
14 Real Estate Modelling and Forecasting
Table 1.1 Econometric software packages for modelling
financial data
Package software supplier
EViews QMS Software
Gauss Aptech Systems
LIMDEP Econometric Software
Matlab The MathWorks
RATS Estima
SAS SAS Institute
Shazam Northwest Econometrics
Splus Insightful Corporation
SPSS SPSS
Stata StataCorp
TSP TSP International
Note: Full contact details for all software suppliers can
be found in the appendix at the end of this chapter.
wishes to build something slightly more complex or just different, one is
forced to consider alternatives. EViews has a command-based programming
language as well as a click-and-point interface, however, so it offers flexibility
as well as user-friendliness.
1.9.2 Choosing a package
Choosing an econometric software package is an increasingly difficult task
as the packages become more powerful but at the same time more homoge-
neous. For example, LIMDEP, a package originally developed for the analysis
of a certain class of cross-sectional data, has many useful features for mod-
elling financial time series. Moreover, many packages developed for time
series analysis, such as TSP (‘Time Series Processor’), can also now be used for
cross-sectional or panel data. Of course, this choice may be made for you if

your institution offers or supports only one or two of the above possibilities.
Otherwise, sensible questions to ask yourself are as follows.
●
Is the package suitable for your intended applications – for example, does
the software have the capability for the models that you want to estimate?
Can it handle sufficiently large databases?
●
Is the package user-friendly?
●
Is it fast?
●
How much does it cost?
Introduction 15
●
Is it accurate?
●
Is the package discussed or supported in a standard textbook?
●
Does the package have readable and comprehensive manuals? Is help available
online?
●
Does the package come with free technical support so that you can e-mail
the developers with queries?
A great deal of useful information can be obtained most easily from the
web pages of the software developers. Additionally, many journals (includ-
ing the Journal of Applied Econometrics,theEconomic Journal,theInternational
Journal of Forecasting and the American Statistician) publish software reviews
that seek to evaluate and compare the packages’ usefulness for a given pur-
pose. Three reviews that the first author has been involved with are Brooks
(1997) and Brooks, Burke and Persand (2001, 2003).

1.10 Outline of the remainder of this book
Chapter 2
This chapter aims to illustrate data transformation and computation, which
are key to the construction of real estate series. The chapter also provides
the mathematical foundations that are important for the computation of
statistical tests in the following chapters. It begins by looking at how to
index a single data series and produce a composite index from several series
by different methods. The chapter continues by showing how to convert
nominal data into real terms. The discussion explains why we log data and
reminds the reader of the properties of logs. The calculation of simple and
continuously compounded returns follows, a topic of much relevance in
the construction of real estate series such as capital value (or price) and
total returns. The last section of the chapter is devoted to matrix alge-
bra. Key aspects of matrices are presented for the reader to help his/her
understanding of the econometric concepts employed in the following
chapters.
Chapter 3
This begins with a description of the types of data that may be available for
the econometric analysis of real estate markets and explains the concepts
of time series, cross-sectional and panel data. The discussion extends to the
properties of cardinal, ordinal and nominal numbers. This chapter covers
important statistical properties of data: measures of central tendency, such
as the median and the arithmetic and geometric means; measures of spread,
16 Real Estate Modelling and Forecasting
including range, quartiles, variance, standard deviation, semi-standard
deviation and the coefficient of variation; higher moments – that is, skew-
ness and kurtosis; and normal and skewed distributions. The reader is fur-
ther introduced to the concepts of covariance and correlation and the metric
of a correlation coefficient. This chapter also reviews probability distribu-
tions and hypothesis testing. It familiarises the reader with the t- and normal

distributions and shows how to carry out hypothesis tests using the test of
significance and confidence interval approaches. The chapter finishes by
highlighting the implications of small samples and sampling error, trends
in the data and spurious associations, structural breaks and data that do
not follow the normal distribution. These data characteristics are crucially
important to real estate analysis.
Chapter 4
This chapter introduces the classical linear regression model (CLRM). This
is the first of four chapters we devote to regression models. The material
brought into this chapter is developed and expanded upon in subsequent
chapters. The chapter provides the general form of a single regression model
and discusses the role of the disturbance term. The method of least squares
is discussed in detail and the reader is familiarised with the derivation of
the residual sum of squares, the regression coefficients and their standard
errors. The discussion continues with the assumptions concerning distur-
bance terms in the CLRM and the properties of the least squares estimator.
The chapter provides guidance to conduct tests of significance for variables
in the regression model.
Chapter 5
Chapter 5 develops and extends the material of chapter 4 to multiple regres-
sion analysis. The coefficient estimates in multiple regression are discussed
and derived. This chapter also presents measures of goodness of fit. It intro-
duces the concept of non-nested hypotheses and provides a first view on
model selection. In this chapter, the reader is presented with the F -test
and its relationship to the t-test. With examples, it is illustrated how to
run the F -test and determine the number of restrictions when running this
test. The F -test is subsequently used in this chapter to assess whether a
statistically significant variable is omitted from the regression model or a
non-significant variable is included.
Chapter 6

This focuses on violations of the assumptions of the CLRM. The discussion
provides the causes of these violations and highlights the implications for
Introduction 17
the robustness of the models. It shows the reader how to conduct diagnostic
checks and interpret the results. With detailed examples, the concepts of
heteroscedasticity, residual autocorrelation, non-normality of the residuals,
functional form and multicollinearity are examined in detail. Within the
context of these themes, the role of lagged terms in a regression is studied.
The exposition of diagnostic checks continues with the presentation of
parameter stability tests, and examples are given. The chapter finishes by
critically reviewing two key approaches to model building.
Chapter 7
This chapter is devoted to two examples of regression analysis: a time series
specification and a cross-sectional model. The aim is to illustrate further
practical issues in building a model. The time series model is a rent growth
model. This section begins by considering the data transformations required
to address autocorrelation and trends in the data. Correlation analysis then
informs the specification of a general model, which becomes specific by
applying a number of tests. The diagnostics studied in the previous chapter
are applied to two competing models of rent growth to illustrate compar-
isons. The second example of the chapter has a focus on international yields
and seeks to identify cross-sectional effects on yields. This part of the chapter
shows that the principles that are applied to build and assess a time series
model can extend to a cross-sectional regression model.
Chapter 8
This presents an introduction to pure time series models. The chapter begins
with a presentation of the features of some standard models of stochastic
processes (white noise, moving average (MA), autoregressive (AR) and mixed
ARMA processes). It shows how the appropriate model can be chosen for a
set of actual data with emphasis on selecting the order of the ARMA model.

The most common information criteria are discussed, which can, of course,
be used to select terms in regression analysis as well. Forecasting from ARMA
models is illustrated with a practical application to cap rates. The issue of
seasonality in real estate data is also treated in the context of ARMA model
estimation and forecasting.
Chapter 9
This chapter is wholly devoted to the assessment of forecast accuracy and
educates the reader about the process of and tests for assessing forecasts.
It presents key contemporary approaches adopted for forecast evaluation,
including mean error measures, measures based on the mean squared error
and Theil’s metrics. The material in this chapter goes further to cover the
18 Real Estate Modelling and Forecasting
principles of forecast efficiency and encompassing. It also examines more
complete tools for forecast evaluation, such as the evaluation of rolling
forecasts. Detailed examples are given throughout to help the application
of the suite of tests proposed in this chapter. The chapter also reviews
studies that show how forecast evaluation has been applied in the real estate
field.
Chapter 10
Chapter 10 moves the analysis from regression models to more general
forms of modelling, in which the segments of the real estate market
are simultaneously modelled and estimated. These multivariate, multi-
equation models are motivated by way of explanation of the possible exis-
tence of bidirectional causality in real estate relationships, and the simulta-
neous equations bias that results if this is ignored. The reader is familiarised
with identification testing and the estimation of simultaneous models. The
chapter makes the distinction between recursive and simultaneous mod-
els. Exhaustive examples help the reader to absorb the concept of multi-
equation models. The analysis finally goes a step further to show how fore-
casts are obtained from these models.

Chapter 11
This chapter relaxes the intrinsic restrictions of simultaneous equations
models and focuses on vector autoregressive (VAR) models, which have
become quite popular in the empirical literature. The chapter focuses on
how such models are estimated and how restrictions are imposed and tested.
The interpretation of VARs is explained by way of joint tests of restric-
tions, causality tests, impulse responses and variance decompositions. The
application of Granger causality tests is illustrated within the VAR con-
text. Again, the last part of the chapter is devoted to a detailed example
of obtaining forecasts from VARs for a REIT (real estate investment trust)
series.
Chapter 12
The first section of the chapter discusses the concepts of stationarity, types of
non-stationarity and unit root processes. It presents several procedures for
unit root tests. The concept of and tests for cointegration, and the formula-
tion of error correction models, are then studied within both the univariate
framework of Engle–Granger and the multivariate framework of Johansen.
Practical examples to illustrate these frameworks are given in the context
of an office market and tests for cointegration between international REIT
markets. These frameworks are also used to generate forecasts.
Introduction 19
Chapter 13
Having reviewed frameworks for simple and more complex modelling in
the real estate field and the process of obtaining forecasts from these frame-
works in the previous chapters, the focus now turns to how this knowledge
is applied in practice. The chapter begins with a review on how forecasting
takes place in real estate in practice and highlights that intervention occurs
to bring in judgement. It explains the reasons for such intervention and
how the intervention operates, and brings to the reader’s attention issues
with judgemental forecasting. The reader benefits from the discussion on

how judgement and model-based forecasts can be combined and how the
relative contributions can be assessed. Ways to combine model-based with
judgemental forecasts are critically presented. Finally, tips are given on how
to make both intervention and the forecast process more acceptable to the
end user.
Chapter 14
This summarises the book and concludes. Some recent developments in the
field, which are not covered elsewhere in the book, are also mentioned.
Some tentative suggestions for possible growth areas in the modelling of
real estate series are also given in this short chapter.
Key concepts
The key terms to be able to define and explain from this chapter are
●
real estate econometrics
●
model building
●
occupier market
●
investment market
●
development market
●
take-up
●
net absorption
●
stock
●
physical construction

●
new orders
●
vacancy
●
prime and average rent
●
effective rent
●
income return
●
initial yield
●
equivalent yield
●
capital growth
●
total returns
●
quantitative models
●
qualitative models
●
point forecasts
●
direction forecasts
●
turning point forecasts
●
scenario analysis

●
data smoothness
●
small samples problem
●
econometric software packages
Appendix: Econometric software
package suppliers
Package Contact information
EViews QMS Software, 4521 Campus Drive, Suite 336, Irvine, CA 92612–2621, United States.
Tel: (+1) 949 856 3368; Fax: (+1) 949 856 2044; Web: www.eviews.com.
Gauss Aptech Systems Inc, PO Box 250, Black Diamond, WA 98010, United States.
Tel: (+1) 425 432 7855; Fax: (+1) 425 432 7832; Web: www.aptech.com.
LIMDEP Econometric Software, 15 Gloria Place, Plainview, NY 11803, United States.
Tel: (+1) 516 938 5254; Fax: (+1) 516 938 2441; Web: www.limdep.com.
Matlab The MathWorks Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, United States.
Tel: (+1) 508 647 7000; Fax: (+1) 508 647 7001; Web: www.mathworks.com.
RATS Estima, 1560 Sherman Avenue, Evanson, IL 60201, United States.
Tel: (+1) 847 864 8772; Fax: (+1) 847 864 6221; Web: www.estima.com.
SAS SAS Institute, 100 Campus Drive, Cary, NC 27513–2414, United States.
Tel: (+1) 919 677 8000; Fax: (+1) 919 677 4444; Web: www.sas.com.
Shazam Northwest Econometrics Ltd, 277 Arbutus Reach, Gibsons, BC V0N 1V8, Canada.
Tel: (+1) 604 608 5511; Fax: (+1) 707 317 5364; Web: shazam.econ.ubc.ca.
Splus Insightful Corporation, 1700 Westlake Avenue North, Suite 500, Seattle, WA
98109–3044, United States.
Tel: (+1) 206 283 8802; Fax: (+1) 206 283 8691; Web: www.splus.com.
SPSS SPSS Inc, 233 S. Wacker Drive, 11th Floor, Chicago, IL 60606–6307, United States.
Tel: (+1) 312 651 3000; Fax: (+1) 312 651 3668; Web: www.spss.com.
Stata StataCorp, 4905 Lakeway Drive, College Station, Texas 77845, United States.
Tel: (+1) 800 782 8272; Fax: (+1) 979 696 4601; Web: www.stata.com.

TSP TSP International, PO Box 61015 Station A, Palo Alto, CA 94306, United States.
Tel: (+1) 650 326 1927; Fax: (+1) 650 328 4163; Web: www.tspintl.com.
20
2
Mathematical building blocks for
real estate analysis
Learning outcomes
In this chapter, you will learn how to
●
construct price indices;
●
compare nominal and real series and convert one to the other;
●
use logarithms and work with matrices; and
●
construct simple and continuously compounded returns from
asset prices.
2.1 Introduction
This chapter provides the mathematical foundations for the quantitative
techniques examined in the following chapters. These concepts are, in
the opinions of the authors, fundamental to a solid understanding of the
remainder of the material in this book. They are presented fairly briefly,
however, since it is anticipated that the majority of readers will already
have some exposure to the techniques, but may require some revision.
2.2 Constructing price index numbers
Index numbers are a useful way to present a series so that it is easy to see
how it has changed over time, and they facilitate comparisons of series
with different units of measurement (for example, if one is expressed in
US dollars and another in euros per square metre). They are widely used
in economics, real estate and finance – to display series for GDP, consumer

prices, exchange rates, aggregate stock values, house prices, and so on. They
are helpful in part because the original series may comprise numbers that
are large and therefore not very intuitive. For example, the average UK house
price according to the Halifax was £132,589 in 2004 rising to £165,807 in
21
22 Real Estate Modelling and Forecasting
2006.
1
Does this represent a large increase? It is hard to tell simply by
glancing at the figures.
Index numbers also make comparisons of the rates of change between
series easier to comprehend. To illustrate, suppose that the average house
price in Greater London rose from £224,305 in 2004 to £247,419 in 2006.
Was the increase in prices for London larger than for the country as a whole?
These two questions can easily be answered by constructing an index for
each series. The simplest way to do this is to construct a set of price relatives.
This is usually achieved by establishing a ‘base period’, for which the index
is given a notional value of 100, and then the other values of the index are
defined relative to this and are calculated by the formula
I
t
=
p
t
p
0
× 100 (2.1)
where p
0
is the initial value of the series in the base year, p

t
is the value of
the series in year t and I
t
is the calculated value of the index at time t.The
base figure is usually set to 100 by convention but of course any other value
(e.g. 1 or 1,000 could be chosen). Applying this formula to the two examples
above, both the United Kingdom overall and the Greater London average
house prices would be given a value of 100 in 2004, and the figures for 2006
would be
I
2006,UK
=
p
2006
p
2004
× 100 =
165807
132589
× 100 = 125.1 (2.2)
and
I
2006,London
=
247419
224305
× 100 = 110.3 (2.3)
respectively. Thus the rise in average house prices in Greater London (of
10.3 per cent) over the period failed to keep pace with that of the country as

a whole (of 25.1 per cent). Indices can also be constructed in the same way
for quantities rather than prices, or for the total value of an entity (e.g. the
total market capitalisation of all stocks on an exchange).
An arguably more important use of index numbers is to represent the
changes over time in the values of groups of series together. This would
be termed an aggregate or composite index number – for example, a stock
market index, an index of consumer prices or a real estate market index. In
all three cases, the values of a number of series are combined or weighted
at each point in time and an index formed on the aggregate measure. An
important choice is of the weighting scheme employed to combine the
1
Halifax have produced a number of house price series at the local, regional and national
level dating back to 1983. These are freely available on their website: see
www.hbosplc.com/economy/housingresearch.asp.
Real estate analysis: mathematical building blocks 23
component series, and there are several methods that are commonly used
for this, including:
●
equal weighting of the components;
●
base period weighting by quantity, also known as Laspeyres weighting;
and
●
current period weighting by quantity, also known as Paasche weighting.
Each of these three methods has its own relative advantages and disadvan-
tages; the Laspeyres and Paasche methods are compared in box 2.1. Equal
weighting evidently has simplicity and ease of interpretation on its side; it
may be inappropriate, however, if some components of the series are viewed
as more important than others. For example, if we wanted to compute a UK
national house price index from a set of regional indices, equally weighting

the regions would assign the same importance to Wales and to the south-
east of England, even though the number of property transactions in the
latter area is far higher. Thus an aggregate index computed in this way
could give a misleading picture of the changing value of house prices in the
country as a whole. Similarly, an equally weighted stock index would assign
the same importance in determining the index value to a ‘micro-cap’ stock
as to a vast multinational oil company.
Box 2.1 A comparison of the Laspeyres and Paasche methods
●
The Laspeyres weighting scheme is simpler than the Paasche method and requires
fewer data since the weights need to be calculated only once.
●
Laspeyres indices may also be available earlier in the month or quarter for
precisely this reason.
●
The Laspeyres approach has the disadvantage, however, that the weights are fixed
over time, and it does not take into account changes in market size or sector
importance and technology that affect demand and prices. For example, a
Laspeyres-weighted stock index constr ucted with a base year of 1998 would
assign a high influence, which many researchers would consider inappropriate
nowadays, to IT stocks whose prices fell considerably during the subsequent
bursting of the technology bubble.
●
On the other hand, the Paasche index will allow the weights to change over time,
so it looks to be the superior method, since it uses the appropriate quantity figures
for that period of time.
●
This also means, however, that, under the Paasche approach, the group of entities
being compared is not the same in all time periods.
●

A Paasche index value could rise, therefore, either because the prices are rising or
because the weights on the more expensive items within the data set are rising.
●
These problems can lead to biases in the constructed index series that may be
serious, and they have led to the development of what is known as the Fisher ideal
price index, which is simply the geometric mean of the Laspeyres and Paasche
approaches.
24 Real Estate Modelling and Forecasting
The following example illustrates how an index can be constructed using
the various approaches. The data were obtained from tables 581 and 584 of
the web pages of the Department for Communities and Local Government
2
and comprise annual house prices (shown in table 2.1) and numbers of
property transactions (shown in table 2.2) for the districts of London for the
period 1996 to 2005. The task is to form equally weighed, base-weighted,
current-weighted and Fisher ideal price indices, assuming that the base
year is 2000.
3
Clearly, given the amount of data involved, this task is best
undertaken using a spreadsheet.
The equally weighted index
The easiest way to form an equally weighted index would be to first construct
the average (i.e. unweighted or equally weighted) house price across the
fourteen regions, which is given in table 2.3.
Effectively, the equal weighting method ignores the sales information in
assigning equal importance to all the districts. Then we assign a value of
100 to the 2000 figure for the index (250,770), so that the figures for all
other years are divided by 250,770 and multiplied by 100. Thus the 1996
figure would be (124,719/250,770) × 100 = 49.7, and the 2005 figure would
be (350,549/250,770) ×100 = 139.8.

The base-weighted index
Turning now to the Laspeyres price index, this implies measuring the aver-
age value of a house in each year weighted by the base year quantities
relative to the average price of the same set of houses at the base year. This
translates for the current example into using the base level of house sales
(the 2000 figures) to weight the regional house prices in forming the index.
The relevant formula could be written as
I
t
=
N

i=1
w
i,0
p
i,t
N

i=1
w
i,0
p
i,0
× 100 (2.4)
where w
i,0
is the weight assigned to each district i at the base year (2000),
p
i,0

is the average price in each area at time 0 and p
i,t
is the price in district
i at time t.
So, for this example, we first need to find for 2000 the total number (i.e.
the sum) of sales across all districts, which turns out to be 63,592. Then we
2
See www.communities.gov.uk/index.asp?id=1156110.
3
Note that it does not have to be the case that the first year in the sample (1996 in this
example) must be the base year, although it usually is.
Table 2.1 Mean house prices by district, British pounds
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
Camden 170,030 198,553 225,966 246,130 319,793 340,971 377,347 387,636 414,538 444,165
City of London 136,566 219,722 324,233 290,773 359,332 310,604 272,664 326,592 311,574 326,496
Hackney 75,420 88,592 103,443 126,848 156,571 179,243 203,347 219,545 236,545 251,251
Hammersmith and Fulham 147,342 171,798 195,698 231,982 282,180 302,960 341,841 350,679 381,713 413,412
Haringey 101,971 106,539 124,106 141,050 171,660 193,083 224,232 236,324 259,604 274,531
Islington 123,425 146,684 171,474 206,023 249,636 263,086 290,018 294,163 321,507 332,866
Kensington and Chelsea 296,735 351,567 382,758 434,354 564,571 576,754 617,788 665,634 716,434 754,639
Lambeth 94,191 107,448 128,453 147,891 182,126 208,715 230,255 237,390 253,321 267,386
Lewisham 62,706 72,652 82,747 94,765 119,351 134,003 160,312 183,701 198,567 203,404
Newham 49,815 57,223 65,056 74,345 96,997 114,432 144,907 175,693 191,482 201,672
Southwark 87,057 104,555 123,644 142,527 189,468 208,728 220,433 238,938 252,454 271,614
Tower Hamlets 88,046 110,564 127,976 159,736 189,947 207,944 222,478 234,852 258,747 264,028
Wandsworth 118,767 137,335 155,833 190,309 234,190 252,773 284,367 298,519 335,431 348,870
Westminster 193,993 236,275 295,001 310,335 394,962 410,866 445,010 463,285 507,460 553,355
26 Real Estate Modelling and Forecasting
Table 2.2 Property sales by district
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Camden 3,877 4,340 3,793 4,218 3,642 3,765 3,932 3,121 3,689 3,283
City of London 288 329 440 558 437 379 374 468 307 299
Hackney 2,221 2,968 3,107 3,266 2,840 3,252 3,570 2,711 3,163 2,407
Hammersmith and
Fulham
4,259 4,598 3,834 4,695 3,807 3,790 4,149 3,465 3,761 3,241
Haringey 3,966 4,662 4,248 4,836 4,238 4,658 4,534 3,765 4,233 3,347
Islington 2,516 3,243 3,347 3,935 3,075 3,407 3,365 2,776 2,941 2,900
Kensington and
Chelsea
4,797 5,262 4,576 5,558 4,707 4,195 4,514 3,497 4,043 3,426
Lambeth 4,957 6,128 5,786 6,297 5,966 5,917 6,212 5,209 5,732 5,020
Lewisham 4,357 5,259 5,123 5,842 5,509 5,646 6,122 5,423 5,765 4,679
Newham 3,493 3,894 4,091 4,498 4,920 5,471 5,313 5,103 4,418 3,649
Southwark 3,223 4,523 4,525 5,439 5,191 5,261 4,981 4,441 5,012 4,204
Tower Hamlets 2,537 3,851 4,536 5,631 5,051 4,752 4,557 3,890 5,143 4,237
Wandsworth 7,389 8,647 7,793 9,757 7,693 8,187 8,485 6,935 8,156 7,072
Westminster 5,165 6,885 5,821 7,118 6,516 6,024 6,417 5,014 5,083 4,796
Table 2.3 Average house prices across all districts, British pounds
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
Unweighted 124,719 150,679 179,028 199,791 250,770 264,583 288,214 308,068 331,384 350,549
average
divide the number of sales in each region for 2000 by this total to get the
weights. Note that, for this type of index, the weights are fixed for all time
at the base period values. The weights are given in table 2.4.
The last row checks that the weights do indeed sum to 1 as they should.
Now the formula in (2.4) can be applied as follows. For 2000 (the base period),
the index value is set to 100 as before. For 1996, the calculation would be
I
t

=
(170, 030 × 0.057) +(136, 566 × 0.007) +(75, 420 ×0.045) +···+(193, 993 ×0.102)
(319, 793 × 0.057) +(359, 332 × 0.007) +(156, 571 ×0.045) +···+(394, 962 × 0.102)
× 100
(2.5)
which is 100× (Camden price 1996 × Camden 2000 weight) +···+(West-
minster price 1996 × Westminster weight 2000) / (Camden price 2000 ×
Camden 2000 weight) +···+ (Westminster price 2000 × Westminster
weight 2000).
Real estate analysis: mathematical building blocks 27
Table 2.4 Laspeyres weights in index
Camden 0.057
City of London 0.007
Hackney 0.045
Hammersmith and Fulham 0.060
Haringey 0.067
Islington 0.048
Kensington and Chelsea 0.074
Lambeth 0.094
Lewisham 0.087
Newham 0.077
Southwark 0.082
Tower Hamlets 0.079
Wandsworth 0.121
Westminster 0.102
Sum of weights 1.000
The current-weighted index
The equivalent of equation (2.4) for the Paasche weighted index is
I
t

=
N

i=1
w
i,t
p
i,t
N

i=1
w
i,t
p
i,0
× 100 (2.6)
with all notation as above.
Thus the first step in calculating the current weighted index is to cal-
culate the weights as we did for 2000 above, but now for every year. This
involves calculating the total number of sales across all districts separately
for each year and then dividing the sales for the district by the total sales
for all districts during that year. For example, the 1996 figure for Camden
is 3,877/(3,877 + 288 +···+5,165) = 0.073. The weights for all districts in
each year are given in table 2.5.
Now that we have the weights for each district, equation (2.6) can be
applied to get the index values for each year. For 1996, the calculation
would be
I
t
=

(170, 030 × 0.073) +(136, 566 × 0.005) +(75, 420 ×0.042) +···+(193, 993 × 0.097)
(319, 793 × 0.073) +(359, 332 × 0.005) +(156, 571 ×0.042) +···+(394, 962 ×0.097)
× 100
(2.7)
28 Real Estate Modelling and Forecasting
Table 2.5 Current weights for each year
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
Camden 0.073 0.067 0.062 0.059 0.057 0.058 0.059 0.056 0.060 0.062
City of London 0.005 0.005 0.007 0.008 0.007 0.006 0.006 0.008 0.005 0.006
Hackney 0.042 0.046 0.051 0.046 0.045 0.050 0.054 0.049 0.051 0.046
Hammersmith
and Fulham
0.080 0.071 0.063 0.066 0.060 0.059 0.062 0.062 0.061 0.062
Haringey 0.075 0.072 0.070 0.067 0.067 0.072 0.068 0.067 0.069 0.064
Islington 0.047 0.050 0.055 0.055 0.048 0.053 0.051 0.050 0.048 0.055
Kensington and
Chelsea
0.090 0.081 0.075 0.078 0.074 0.065 0.068 0.063 0.066 0.065
Lambeth 0.093 0.095 0.095 0.088 0.094 0.091 0.093 0.093 0.093 0.096
Lewisham 0.082 0.081 0.084 0.082 0.087 0.087 0.092 0.097 0.094 0.089
Newham 0.066 0.060 0.067 0.063 0.077 0.085 0.080 0.091 0.072 0.069
Southwark 0.061 0.070 0.074 0.076 0.082 0.081 0.075 0.080 0.082 0.080
Tower Hamlets 0.048 0.060 0.074 0.079 0.079 0.073 0.069 0.070 0.084 0.081
Wandsworth 0.139 0.134 0.128 0.136 0.121 0.127 0.128 0.124 0.133 0.135
Westminster 0.097 0.107 0.095 0.099 0.102 0.093 0.096 0.090 0.083 0.091
Sum of weights 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
Table 2.6 Index values calculated using various methods
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
Equally weighted 49.73 60.09 71.39 79.67 100.00 105.51 114.93 122.85 132.15 139.79
Base-weighted 50.79 59.89 69.59 79.48 100.00 107.41 118.77 126.10 137.03 145.04

Current-weighted 51.20 60.02 69.56 79.55 100.00 107.66 119.23 126.95 137.75 145.20
Fischer ideal 51.00 59.96 69.58 79.52 100.00 107.54 119.00 126.53 137.39 145.12
which is 100 × (Camden price 1996 × Camden 1996 weight) +···+(West-
minster price 1996 × Westminster weight 1996) / (Camden price 2000 ×
Camden 1996 weight) +···+ (Westminster price 2000 × Westminster
weight 1996).
The final table of index values calculated using the four methods is given
in table 2.6. As is evident, there is very little difference between the base and
current weighted indices, since the relative number of sales in each district
has remained fairly stable over the sample period. On the other hand, there
Real estate analysis: mathematical building blocks 29
is a slight tendency for the equally weighted index to rise more slowly in
the second half of the period due to the relatively low weightings it gives
to areas where prices were growing fast and where sales were large, such
as Camden and Hammersmith. The Fisher index values were calculated by
multiplying the square roots of the base (Laspeyres) and current weighted
(Paasche) index values together. For example, the 1996 year Fisher value of
51.00 is given by (50.79 × 51.20)
1/2
, and so on.
Finally, it is worth noting that all the indices described above are aggre-
gate price indices – that is, they measure how average prices change over
time when the component prices are weighted by quantities. It is also pos-
sible, however, to construct quantity indices that measure how sales or
transactions quantities vary over time when the prices are used as weights.
Nonetheless, quantity indices are much less common than price indices,
and so they are not discussed further here; interested readers are referred
to Kazmier and Pohl (1984, pp. 468–9) or Watsham and Parramore (1997,
pp. 74–5).
2.3 Real versus nominal series and deflating nominal series

If a newspaper headline suggests that ‘house prices are growing at their
fastest rate for more than a decade. A typical 3-bedroom house is now selling
for £180,000, whereas in 1990 the figure was £120,000’, it is important to
appreciate that this figure is almost certainly in nominal terms. That is, the
article is referring to the actual prices of houses that existed at those points
in time. The general level of prices in most economies around the world has
a general tendency to rise almost all the time, so we need to ensure that we
compare prices on a like-for-like basis. We could think of part of the rise in
house prices being attributable to an increase in demand for housing, and
part simply arising because the prices of all goods and services are rising
together. It would be useful to be able to separate the two effects, and to
be able to answer the question ‘How much have house prices risen when
we remove the effects of general inflation?’, or, equivalently, ‘How much
are houses worth now if we measure their values in 1990 terms?’. We can
do this by deflating the nominal house price series to create a series of real
house prices, which is then said to be in inflation-adjusted terms or at constant
prices.
Deflating a series is very easy indeed to achieve: all that is required (apart
from the series to deflate) is a price deflator series, which is a series measuring
general price levels in the economy. Series such as the consumer price index
(CPI), the producer price index (PPI) or the GDP implicit price deflator are
30 Real Estate Modelling and Forecasting
often used. A more detailed discussion as to the most relevant general price
index to use is beyond the scope of this book, but suffice to say that, if the
researcher is interested only in viewing a broad picture of the real prices
rather than a highly accurate one, the choice of deflator will be of little
importance.
The real price series is obtained by taking the nominal series, dividing it
by the price deflator index and multiplying by 100 (under the assumption
that the deflator has a base value of 100):

real series
t
=
nominal series
t
deflator
t
× 100 (2.8)
It is worth noting that deflation is a relevant process only for series that are
measured in money terms, so it would make no sense to deflate a quantity-
based series such as the number of houses rented or a series expressed as a
proportion or percentage, such as vacancy or the rate of return on a stock.
Example 2.1
Take the series of annual prime office rents in Singapore expressed in local
currency and the consumer price index for Singapore shown in table 2.7. In
this example, we apply equation (2.8) and we conduct further calculations
to help understand the conversion from nominal to real terms.
Column (i) gives the nominal rent series for Singapore offices taken from
the Urban Redevelopment Authority and column (ii) the consumer price
index for Singapore (values for December each year) taken from the Depart-
ment of Statistics. We have rebased this index to take the value 100 in 1991.
Column (iii) contains real rents calculated with equation (2.8). This series of
real rents is also equivalent to rents in constant 1991 prices.
The remaining columns provide additional calculations that give exactly
the same results as column (iii). We report the original Singapore CPI series
(2004 = 100) in column (iv) and we use this series for further calculations.
The results will not change, of course, if we use the rebased CPI (1991 = 100)
series.
For the rent calculation in column (v), we use the formula real rent
t

=
nominal rent
t
/CPI
t
– that is, the value for 2007 is simply 1,274/106.6. This
simple calculation makes the task of reflating the series more straightfor-
ward. Assume we wish to forecast this real index. In order to convert the
forecast real rents to nominal values, we would need to multiply the real
rent by the future CPI. If we wish to convert rents into a particular year’s
prices, we would apply equation (2.8), but instead of 100 we would have the
CPI value that year. Consider that we wish to express nominal rents in 2007
prices (this is our last observation, and converting rents into today’s prices
Real estate analysis: mathematical building blocks 31
Table 2.7 Construction of a real rent index for offices in Singapore
(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)
Rent CPI Rent CPI Rent Rent Rents index
2
nominal
1
1991 = 100 real
1
2004 = 100 real 2007 prices
1
nominal real
1991 877 100.0 877.0 86.0 10.2 1,087.1 100.0 100.0
1992 720 101.7 708.0 87.5 8.2 877.2 82.1 80.7
1993 628 104.3 602.1 89.7 7.0 746.3 71.6 68.7
1994 699 107.3 651.4 92.3 7.6 807.3 79.7 74.3
1995 918 108.3 847.6 93.1 9.9 1,051.1 104.7 96.7

1996 933 110.5 844.3 95.0 9.8 1,046.9 106.4 96.3
1997 878 112.7 779.1 96.9 9.1 965.9 100.1 88.9
1998 727 111.0 655.0 95.5 7.6 811.5 82.9 74.7
1999 660 111.9 589.8 96.2 6.9 731.4 75.3 67.3
2000 743 114.2 650.6 98.2 7.6 806.6 84.7 74.2
2001 685 113.5 603.5 97.6 7.0 748.2 78.1 68.8
2002 600 114.0 526.3 98.0 6.1 652.7 68.4 60.0
2003 537 114.8 467.8 98.7 5.4 580.0 61.2 53.4
2004 556 116.3 478.1 100.0 5.6 592.7 63.4 54.5
2005 626 117.8 531.4 101.3 6.2 658.8 71.4 60.6
2006 816 118.7 687.4 102.1 8.0 852.0 93.0 78.4
2007 1,274 124.0 1,027.4 106.6 12.0 1,274.0 145.3 117.2
Notes:
1
Singapore dollars per square metre per year.
2
Rents index, 1991 = 100.
is appealing). We would base the calculation on a variant of equation (2.8):
real rent
t
=
nominal rent
t
CPI
t
CPI
reference year
(2.9)
Office rents in Singapore in 2007 prices are shown in column (vi). That is,
for 2006, the value of rents in 2007 prices is given by (816/102.1) × 106.6 =

852. The value for 2007 is similar to the nominal value. The last two columns
present a nominal index and a real rent index taking the value of 100 in
1991. This is a way to make some easy visual comparisons. Nominal rents
were 45 per cent higher in 2007 than 1991. The values for the nominal
and real indices in, say, 2005 are calculated as 71.4 = (626/877) × 100 and
60.6 = (6.2/10.2) × 100, respectively. For the real rent index, we use column
(v) for real rents, but an identical result would be obtained if we used
columns (iii) or (vi). A comparison of real and nominal rents is given in
figure 2.1. Interestingly, office rents in real terms in Singapore recovered to
their 1991 level only in 2007.
32 Real Estate Modelling and Forecasting
200
Nominal
Real
150
100
50
0
1993
1991
1995
1997
1999
2001
2003
2005
2007
Figure 2.1
Index of office rents
in Singapore

2.4 Properties of logarithms and the log transform
Logarithms were invented to simplify cumbersome calculations, since expo-
nents can then be added or subtracted, which is easier than multiplying or
dividing the original numbers. While making logarithmic transformations
for computational ease is no longer necessary, they still have important uses
in algebra and in data analysis. For the latter, there are at least three reasons
why log transforms may be useful. First, taking a logarithm can often help
to rescale the data so that their variance is more constant, which overcomes
a common statistical problem. Second, logarithmic transforms can help to
make a positively skewed distribution closer to a normal distribution. Third,
taking logarithms can also be a way to make a non-linear, multiplicative
relationship between variables into a linear, additive one. These issues are
discussed in some detail in chapter 6.
Taking a logarithm is the inverse of taking an exponential. Natural log-
arithms, also known as logs to base e (where e is 2.71828 ), are more
commonly used and more useful mathematically than logs to any other
base. A log to base e is known as a natural or Naperian logarithm, denoted
interchangeably by ln(y)orlog(y).
The properties of logarithms or ‘laws of logs’ describe the way that we can
work with logs or manipulate expressions using them. These are presented
in box 2.2.
Box 2.2 Laws of logs
For variables x and y:
●
ln(xy) = ln(x) + ln(y);
●
ln(x/y) = ln(x) − ln(y);
●
ln(y
c

) = c ln(y);
●
ln(1) = 0; and
●
ln(1/y) = ln(1) − ln(y) =−ln(y).
Real estate analysis: mathematical building blocks 33
2.5 Returns
In many of the problems of interest in real estate, especially when invest-
ment performance is studied, the starting point is a time series of income
and prices (capital values). For a number of statistical reasons, it is prefer-
able not to work directly with the price or index series, so raw price series
are usually converted into series of returns. Additionally, returns have the
added benefit that they are unit-free. So, for example, if an annualised return
is 10 per cent, then investors know that they will get back £110 for a £100
investment, or £1,100 for a £1,000 investment, and so on.
There are essentially two methods used to calculate returns from a series
of prices, and these involve the formation of simple returns or continuously
compounded returns, which are achieved as follows:
Simple returns Continuously compounded returns
R
t
=
p
t
− p
t−1
p
t−1
× 100% (2.10) r
t

= 100% × ln

p
t
p
t−1

(2.11)
where R
t
denotes the simple return at time t, r
t
denotes the continuously
compounded return at time t, p
t
denotes the asset price at time t and ln
denotes the natural logarithm.
If the asset under consideration is a building or portfolio of buildings,
the total return to holding the asset is the sum of the capital gain and
income received during the holding period. Returns in real estate could be
income returns (that is, the change in income between time periods), capital
returns (the change in the price of buildings) or total returns (income plus
the value change). Box 2.3 shows two key reasons for applying the log-
return formulation (also known as log-price relatives, as they are the log of
the ratio of this period’s price to the previous period’s price) to calculate
returns. There is also a disadvantage to using the log-returns, however. The
simple return on a portfolio of assets is a weighted average of the simple
returns on the individual assets:
R
pt

=
N

i=1
w
i
R
it
(2.12)
This does not work for the continuously compounded returns, though, so
that they are not additive across a portfolio. The fundamental reason why
this is the case is that the log of a sum is not the same as the sum of a
log, since the operation of taking a log constitutes a non-linear transformation.
Calculating portfolio returns in this context must be conducted by first esti-
mating the value of the portfolio at each time period and then determining
the returns from the aggregate portfolio values.
34 Real Estate Modelling and Forecasting
Box 2.3 Two advantages of log returns
(1) Log-returns have the nice property that they can be interpreted as continuously
compounded returns – so that the frequency of compounding of the return does not
matter, and thus returns across assets can more easily be compared.
(2) Continuously compounded returns are time-additive. For example, suppose that a
weekly returns series is required and daily log returns have been calculated for five
days, numbered 1 to 5, representing the returns on Monday to Friday. It is valid to
simply add up the five daily returns to obtain the return for the whole week:
Monday return r
1
= ln(p
1
/p

0
) = ln p
1
− ln p
0
Tuesday return r
2
= ln(p
2
/p
1
) = ln p
2
− ln p
1
Wednesday return r
3
= ln(p
3
/p
2
) = ln p
3
− ln p
2
Thursday return r
4
= ln(p
4
/p

3
) = ln p
4
− ln p
3
Friday return r
5
= ln(p
5
/p
4
) = ln p
5
− lnp
4
——————————–
Return over the week ln p
5
− ln p
0
= ln(p
5
/p
0
)
In the limit, as the frequency of the sampling of the data is increased, so
that they are measured over a smaller and smaller time interval, the simple
and continuously compounded returns will be identical.
2.6 Matrices
A matrix is simply a collection or array of numbers. The size of a matrix is given

by its numbers of rows and columns. Matrices are very useful and impor-
tant ways for organising sets of data together, making manipulating and
transforming them much easier than it would be to work with each con-
stituent of the matrix separately. Matrices are widely used in econometrics
for deriving key results and for expressing formulae in a succinct way. Some
useful features of matrices and explanations of how to work with them are
described below.
●
The size of a matrix is quoted as R ×C, which is the number of rows by
the number of columns.
●
Each element in a matrix is referred to using subscripts. For example,
suppose a matrix M has two rows and four columns. The element in the
second row and the third column of this matrix would be denoted m
23
,
so that m
ij
refers to the element in the ith row and the j th column.
●
If a matrix has only one row, it is known as a row vector, which will be of
dimension 1 ×C:
e.g. (2.73.0 −1.50.3)
Real estate analysis: mathematical building blocks 35
●
A matrix having only one column is known as a column vector, which
will be of dimension R ×1:
e.g.
⎛
⎝

1.3
−0.1
0.0
⎞
⎠
●
When the number of rows and columns is equal (i.e. R = C),itissaid
that the matrix is square:
e.g.

0.30.6
−0.10.7

●
A matrix in which all the elements are zero is known as a zero matrix:
e.g.

000
000

●
A symmetric matrix is a special type of square matrix that is symmetric
about the leading diagonal (the diagonal line running through the matrix
from the top left to the bottom right), so that m
ij
= m
ji
∀i, j (where ∀
denotes ‘for all values of’):
e.g.

⎛
⎜
⎜
⎝
1247
2 −369
462−8
79−80
⎞
⎟
⎟
⎠
●
A diagonal matrix is a square matrix that has non-zero terms on the
leading diagonal and zeros everywhere else:
e.g.
⎛
⎜
⎜
⎝
−300 0
010 0
002 0
000−1
⎞
⎟
⎟
⎠
●
A diagonal matrix with one in all places on the leading diagonal and

zero everywhere else is known as the identity matrix, denoted by I .By
definition, an identity matrix must be symmetric (and therefore also
square):
e.g.
⎛
⎜
⎜
⎝
1000
0100
0010
0001
⎞
⎟
⎟
⎠
●
The identity matrix is essentially the matrix equivalent of the number
one. Multiplying any matrix by the identity matrix of the appropriate