Real Estate Modelling and Forecasting By Chris Brooks_1 doc

Real estate markets have gone throughsevere cycles not predicted by bottom-up analysis, however, and thus thisapproach to forecasting has been questioned.. Finally, forecasting is the na

Introduction 11

involved with acquisitions, will rely more on their knowledge of the ity and building to make a buy or sell decision This has also given rise toso-called ‘judgemental’ forecasts Real estate markets have gone throughsevere cycles not predicted by bottom-up analysis, however, and thus thisapproach to forecasting has been questioned For many, the winning for-mula is now not just having good judgement about the future direction

local-of the market, but also making a careful quantitative analysis explainingcyclical movements and the impact of broader trends Therefore, consistentwith evidence from other fields, a view that has increasingly gained popular-ity is that the optimal approach arises from a combination of judgementaland quantitative forecasting Moreover, there is a more generic econometricand forecasting interest Do quantitative techniques underperform judge-mental approaches or is the combination of quantitative and judgementalforecasts the most successful formula in the real estate market? The bookaddresses this issue directly, and the tools presented will give the reader aframework 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 morethan one theory-consistent model that passes all the diagnostics tests set

by the researcher The past relative forecasting success of these models willguide 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, theissue 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 inbox 1.1

(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) Direction 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 differencesThe 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 analysingdata are different, however To an extent, real estate data are similar toeconomic data (e.g gross domestic product [GDP], employment) in terms oftheir frequency, accuracy, seasonality and other properties On the otherhand, there are some important differences in how the data are generated.Real estate data can be generated through the valuation process rather thanthrough surveys or government accounts, as is the case for economic data.There are some apparent differences with financial data, given their highfrequency A commonality with financial data, however, is that most realestate data are not subject to subsequent revisions, or, at least, not to theextent 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 tomeasure economic quantities changed twenty years ago then only twentyannual observations at most are usefully available There is a similar prob-lem in real estate markets Here, though, the problem concerns not onlychanging methods of calculation but also the point at which the data werefirst collected In the United Kingdom, data can be found back to 1966 orearlier, 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 tothe 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 somelocations

As in finance, real estate data can come in many shapes and forms Rentsand prices that are recorded are usually the product of valuations that havebeen criticised as being excessively smooth and slow to adjust to changingmarket conditions The problem arises from infrequent trading and trying

to establish values where the size of the market is small The industry hasrecognised this issue, and we see an increasing compilation of transactionsdata We outlined in section 1.5 above that other real estate market data,such as absorption (a measure of demand), are constructed based on othermarket 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 realestate series; data revisions can be less serious in the real estate contextcompared 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 dataare almost always not normally distributed in spite of the fact that mosttechniques 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 computerpackages 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 anyeconometrics software, or have not yet found a package that suits theirrequirements – read on

1.9.1 What packages are available?

Although this list is by no means exhaustive, a set of widely used packages isgiven in table 1.1 The programmes can usefully be categorised according towhether they are fully interactive (menu-driven), command-driven (so thatthe user has to write mini-programmes) or somewhere in between Menu-driven packages, which are usually based on a standard Microsoft Windowsgraphical user interface, are almost certainly the easiest for novices to getstarted with, for they require little knowledge of the structure of the pack-age, and the menus can usually be negotiated simply EViews is a packagethat falls into this category

On the other hand, some such packages are often the least flexible, sincethe 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

LIMDEP Econometric Software

Shazam Northwest EconometricsSplus Insightful Corporation

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 isforced to consider alternatives EViews has a command-based programminglanguage 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 neous For example, LIMDEP, a package originally developed for the analysis

homoge-of a certain class homoge-of cross-sectional data, has many useful features for elling financial time series Moreover, many packages developed for timeseries analysis, such as TSP (‘Time Series Processor’), can also now be used forcross-sectional or panel data Of course, this choice may be made for you ifyour institution offers or supports only one or two of the above possibilities.Otherwise, sensible questions to ask yourself are as follows

mod-● 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 theweb pages of the software developers Additionally, many journals (includ-

ing the Journal of Applied Econometrics, the Economic Journal, the International Journal of Forecasting and the American Statistician) publish software reviews

that seek to evaluate and compare the packages’ usefulness for a given pose Three reviews that the first author has been involved with are Brooks(1997) and Brooks, Burke and Persand (2001, 2003)

pur-1.10 Outline of the remainder of this book

Chapter 2

This chapter aims to illustrate data transformation and computation, whichare key to the construction of real estate series The chapter also providesthe mathematical foundations that are important for the computation ofstatistical tests in the following chapters It begins by looking at how toindex a single data series and produce a composite index from several series

by different methods The chapter continues by showing how to convertnominal data into real terms The discussion explains why we log data andreminds the reader of the properties of logs The calculation of simple andcontinuously compounded returns follows, a topic of much relevance inthe construction of real estate series such as capital value (or price) andtotal 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/herunderstanding of the econometric concepts employed in the followingchapters

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-standarddeviation 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 ofsignificance and confidence interval approaches The chapter finishes byhighlighting the implications of small samples and sampling error, trends

in the data and spurious associations, structural breaks and data that donot follow the normal distribution These data characteristics are cruciallyimportant 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 materialbrought into this chapter is developed and expanded upon in subsequentchapters The chapter provides the general form of a single regression modeland discusses the role of the disturbance term The method of least squares

is discussed in detail and the reader is familiarised with the derivation ofthe residual sum of squares, the regression coefficients and their standarderrors 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 sion analysis The coefficient estimates in multiple regression are discussedand 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

regres-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 anon-significant variable is included

Chapter 6

This focuses on violations of the assumptions of the CLRM The discussionprovides the causes of these violations and highlights the implications for

Introduction 17

the robustness of the models It shows the reader how to conduct diagnosticchecks and interpret the results With detailed examples, the concepts ofheteroscedasticity, residual autocorrelation, non-normality of the residuals,functional form and multicollinearity are examined in detail Within thecontext of these themes, the role of lagged terms in a regression is studied.The exposition of diagnostic checks continues with the presentation ofparameter stability tests, and examples are given The chapter finishes bycritically reviewing two key approaches to model building

Chapter 7

This chapter is devoted to two examples of regression analysis: a time seriesspecification and a cross-sectional model The aim is to illustrate furtherpractical issues in building a model The time series model is a rent growthmodel This section begins by considering the data transformations required

to address autocorrelation and trends in the data Correlation analysis theninforms the specification of a general model, which becomes specific byapplying a number of tests The diagnostics studied in the previous chapterare applied to two competing models of rent growth to illustrate compar-isons The second example of the chapter has a focus on international yieldsand seeks to identify cross-sectional effects on yields This part of the chaptershows that the principles that are applied to build and assess a time seriesmodel can extend to a cross-sectional regression model

Chapter 8

This presents an introduction to pure time series models The chapter beginswith a presentation of the features of some standard models of stochasticprocesses (white noise, moving average (MA), autoregressive (AR) and mixedARMA processes) It shows how the appropriate model can be chosen for aset 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 ARMAmodels is illustrated with a practical application to cap rates The issue ofseasonality in real estate data is also treated in the context of ARMA modelestimation and forecasting

18 Real Estate Modelling and Forecasting

principles of forecast efficiency and encompassing It also examines morecomplete tools for forecast evaluation, such as the evaluation of rollingforecasts Detailed examples are given throughout to help the application

of the suite of tests proposed in this chapter The chapter also reviewsstudies that show how forecast evaluation has been applied in the real estatefield

Chapter 10

Chapter 10 moves the analysis from regression models to more generalforms of modelling, in which the segments of the real estate marketare 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 familiarisedwith identification testing and the estimation of simultaneous models Thechapter 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 equationsmodels and focuses on vector autoregressive (VAR) models, which havebecome quite popular in the empirical literature The chapter focuses onhow 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 Theapplication 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 ofnon-stationarity and unit root processes It presents several procedures forunit root tests The concept of and tests for cointegration, and the formula-tion of error correction models, are then studied within both the univariateframework 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 REITmarkets These frameworks are also used to generate forecasts

Introduction 19

Chapter 13

Having reviewed frameworks for simple and more complex modelling inthe 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 forecastingtakes place in real estate in practice and highlights that intervention occurs

to bring in judgement It explains the reasons for such intervention andhow the intervention operates, and brings to the reader’s attention issueswith judgemental forecasting The reader benefits from the discussion onhow judgement and model-based forecasts can be combined and how therelative contributions can be assessed Ways to combine model-based withjudgemental forecasts are critically presented Finally, tips are given on how

to make both intervention and the forecast process more acceptable to theend user

Chapter 14

This summarises the book and concludes Some recent developments in thefield, which are not covered elsewhere in the book, are also mentioned.Some tentative suggestions for possible growth areas in the modelling ofreal 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

● physical construction ● new orders

● quantitative models ● qualitative models

● point forecasts ● direction forecasts

● turning point forecasts ● scenario analysis

● 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

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 quantitativetechniques examined in the following chapters These concepts are, inthe opinions of the authors, fundamental to a solid understanding of theremainder of the material in this book They are presented fairly briefly,however, since it is anticipated that the majority of readers will alreadyhave 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 seehow it has changed over time, and they facilitate comparisons of serieswith 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, consumerprices, exchange rates, aggregate stock values, house prices, and so on Theyare helpful in part because the original series may comprise numbers thatare large and therefore not very intuitive For example, the average UK houseprice 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 byglancing at the figures

Index numbers also make comparisons of the rates of change betweenseries easier to comprehend To illustrate, suppose that the average houseprice 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 foreach 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 aredefined relative to this and are calculated by the formula

I t = p t

where p0 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 examplesabove, both the United Kingdom overall and the Greater London averagehouse prices would be given a value of 100 in 2004, and the figures for 2006would be

An arguably more important use of index numbers is to represent thechanges over time in the values of groups of series together This would

be termed an aggregate or composite index number – for example, a stockmarket index, an index of consumer prices or a real estate market index Inall 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 Animportant choice is of the weighting scheme employed to combine the

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 usedfor 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 tages; the Laspeyres and Paasche methods are compared in box 2.1 Equalweighting evidently has simplicity and ease of interpretation on its side; itmay be inappropriate, however, if some components of the series are viewed

disadvan-as more important than others For example, if we wanted to compute a UKnational house price index from a set of regional indices, equally weightingthe regions would assign the same importance to Wales and to the south-east of England, even though the number of property transactions in thelatter area is far higher Thus an aggregate index computed in this waycould give a misleading picture of the changing value of house prices in thecountry as a whole Similarly, an equally weighted stock index would assignthe same importance in determining the index value to a ‘micro-cap’ stock

as to a vast multinational oil company

● 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 constructed 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 usingthe various approaches The data were obtained from tables 581 and 584 ofthe web pages of the Department for Communities and Local Government2

and comprise annual house prices (shown in table 2.1) and numbers ofproperty transactions (shown in table 2.2) for the districts of London for theperiod 1996 to 2005 The task is to form equally weighed, base-weighted,current-weighted and Fisher ideal price indices, assuming that the baseyear is 2000.3 Clearly, given the amount of data involved, this task is bestundertaken using a spreadsheet

The equally weighted index

The easiest way to form an equally weighted index would be to first constructthe average (i.e unweighted or equally weighted) house price across thefourteen regions, which is given in table 2.3

Effectively, the equal weighting method ignores the sales information inassigning 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 allother 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 age value of a house in each year weighted by the base year quantitiesrelative to the average price of the same set of houses at the base year Thistranslates 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

where w i,0 is the weight assigned to each district i at the base year (2000),

p i,0is 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

example) must be the base year, although it usually is.

26 Real Estate Modelling and Forecasting

Table 2.2 Property sales by district

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005Camden 3,877 4,340 3,793 4,218 3,642 3,765 3,932 3,121 3,689 3,283

Hackney 2,221 2,968 3,107 3,266 2,840 3,252 3,570 2,711 3,163 2,407Hammersmith and

Fulham

4,259 4,598 3,834 4,695 3,807 3,790 4,149 3,465 3,761 3,241Haringey 3,966 4,662 4,248 4,836 4,238 4,658 4,534 3,765 4,233 3,347Islington 2,516 3,243 3,347 3,935 3,075 3,407 3,365 2,776 2,941 2,900Kensington and

Chelsea

4,797 5,262 4,576 5,558 4,707 4,195 4,514 3,497 4,043 3,426Lambeth 4,957 6,128 5,786 6,297 5,966 5,917 6,212 5,209 5,732 5,020Lewisham 4,357 5,259 5,123 5,842 5,509 5,646 6,122 5,423 5,765 4,679Newham 3,493 3,894 4,091 4,498 4,920 5,471 5,313 5,103 4,418 3,649Southwark 3,223 4,523 4,525 5,439 5,191 5,261 4,981 4,441 5,012 4,204Tower Hamlets 2,537 3,851 4,536 5,631 5,051 4,752 4,557 3,890 5,143 4,237Wandsworth 7,389 8,647 7,793 9,757 7,693 8,187 8,485 6,935 8,156 7,072Westminster 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

Unweighted 124,719 150,679 179,028 199,791 250,770 264,583 288,214 308,068 331,384 350,549average

divide the number of sales in each region for 2000 by this total to get theweights 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 × Westminsterweight 2000)