Real Estate Modelling and Forecasting Chris Brooks ICMA Centre, University of Reading Sotiris Tsolacos Property and Portfolio Research CAMBRIDGE UNIVERSITY PRESS
Contents list of figures page x List of tables xii List of boxes xiv Preface xv Acknowledgements xix 1 Introduction 1 1.1 Motivation for this book 2 1.2 What is econometrics? 3 1.3 Steps in formulating an econometric model 4 1.4 Model building in real estate 5 1.5 What do we model and forecast in real estate? 6 1.6 Model categorisation for real estate forecasting 8 1.7 Why real estate forecasting? 9 1.8 Econometrics in real estate, finance and economics: similarities and differences 12 1.9 Econometric packages for modelling real estate data 13 1.10 Outline of the remainder of this book 15 Appendix: Econometric software package suppliers 20 2 Mathematical building blocks for real estate analysis 21 2.1 Introduction 21 2.2 Constructing price index numbers " 21 2.3 Real versus nominal series and deflating nominal series 29 2.4 Properties of logarithms and the log transform 32 2.5 Returns 33 2.6 Matrices 34 2.7 The eigenvalues of a matrix 38
vi Contents 3 Statistical tools for real estate analysis 41 3.1 Types of data for quantitative real estate analysis 41 3.2 Descriptive statistics 44 3.3 Probability and characteristics of probability distributions 54 3.4 Hypothesis testing 55 3.5 Pitfalls in the analysis of real estate data 65 4 An overview of regression analysis 72 4.1 Chapter objectives 72 4.2 What is a regression model? 73 4.3 Regression versus correlation 74 4.4 Simple regression 74 4.5 Some further terminology 79 4.6 Linearity and possible forms for the regression function 85 4.7 The assumptions underlying the classical linear regression model 86 4.8 Properties oftheols estimator 87 4.9 Precision and standard errors 88 4.10 Statistical inference and the classical linear regression model 93 Appendix: Mathematical derivations of CLRM results for the bivariate case 104 4A.1 Derivation of the OLS coefficient estimator 104 4A.2 Derivation of the OLS standard error estimators for the intercept and slope 105 5 Further issues in regression analysis 108 5.1 Generalising the simple model to multiple linear regression 108 5.2 The constant term 109 5.3 How are the parameters (the elements of the ft vector) calculated in the generalised case? 5.4 A special type of hypothesis test: the f-ratio 5.5 Goodness of fit statistics 5.6 Tests of non-nested hypotheses 5.7 Data mining and the true size of the test 5.8 Testing multiple hypotheses: the F-test 5.9 Omission of an important variable 5.10 Inclusion of an irrelevant variable Appendix: Mathematical derivations of CLRM results for the multiple regression case 5A.1 Derivation of the OLS coefficient estimator 5A.2 Derivation of the OLS standard error estimator
Contents vii 6 Diagnostic testing 135 6.1 Introduction 135 6.2 Violations of the assumptions of the classical linear regression model 136 6.3 Statistical distributions for diagnostic tests 136 6.4 Assumption 1: E(u t ) = 0 137 6.5 Assumption 2: var(w,) = a 2 < oo 138 6.6 Assumption 3: COV(M,-, UJ) = 0 for i= j 144 6.7 Causes of residual autocorrelation 152 6.8 Assumption 4: the x t are non-stochastic (cov(u,,x t ) 0) 166 6.9 Assumption 5: the disturbances are normally distributed 167 6.10 Multicollinearity 171 6.11 Adopting the wrong functional form 175 6.12 Parameter stability tests 178 6.13 A strategy for constructing econometric models 186 Appendix: Iterative procedures for dealing with autocorrelation 191 7 Applications of regression analysis 194 7.1 Frankfurt office rents: constructing a multiple regression model 194 7.2 Time series regression models from the literature 210 7.3 International office yields: a cross-sectional analysis 214 7.4 A cross-sectional regression model from the literature 222 8 Time series models 225 8.1 Introduction 225 8.2 Some notation and concepts 226 8.3 Moving average processes 230 8.4 Autoregressive processes 231 8.5 The partial autocorrelation function 234 8.6 ARMA processes 235 8.7 Building ARMA models: the Box-Jenkins approach 241 8.8 Exponential smoothing 244 8.9 An ARMA model for cap rates 246 8.10 Seasonality in real estate data 251 8.11 Studies using ARMA models in real estate 257 Appendix: Some derivations of properties of ARMA models 261 8A.1 Deriving the autocorrelation function for an MA process 261 8A.2 Deriving the properties of AR models 263 9 Forecast evaluation 268 9.1 Forecast tests 269
viii Contents 9.2 Application of forecast evaluation criteria to a simple regression model 274 9.3 Forecast accuracy studies in real estate 290 10 Multi-equation structural models 10.1 Simultaneous-equation models 10.2 Simultaneous equations bias 10.3 How can simultaneous-equation models be estimated? 10.4 Can the original coefficients be retrieved from the 7rs? 10.5 A definition of exogeneity 10.6 Estimation procedures for simultaneous equations systems 10.7 Case study: projections in the industrial property market using a simultaneous equations system 316 10.8 A special case: recursive models 322 10.9 Case study: an application of a recursive model to the City of London office market 322 10.10 Example: a recursive system for the Tokyo office market 325 11 Vector autoregressive models 11.1 Introduction 11.2 Advantages of VAR modelling 11.3 Problems with VARs 11.4 Choosing the optimal lag length for a VAR 11.5 Does the VAR include contemporaneous terms? 11.6 AVAR model for real estate investment trusts 11.7 Block significance and causality tests 11.8 VARs with exogenous variables 11.9 Impulse responses and variance decompositions 11.10 A VAR for the interaction between real estate returns and the macroeconomy 11.11 Using VARs for forecasting 12 Cointegration in real estate markets 12.1 Stationarity and unit root testing 12.2 Cointegration 12.3 Equilibrium correction or error correction models 12.4 Testing for cointegration in regression: a residuals-based approach 387 12.5 Methods of parameter estimation in cointegrated systems 388 12.6 Applying the Engle-Granger procedure: the Sydney office market 390
Contents ix 12.7 The Engle and Yoo three-step method 399 12.8 Testing for and estimating cointegrating systems using the Johansen technique 399 12.9 An application of the Johansen technique to securitised real estate 404 12.10 The Johansen approach: a case study 411 13 Real estate forecasting in practice 414 13.1 Reasons to intervene in forecasting and to use judgement 415 13.2 How do we intervene in and adjust model-based forecasts? 418 13.3 Issues with judgemental forecasting 422 13.4 Case study: forecasting in practice in the United Kingdom 424 13.5 Increasing the acceptability of intervention 426 13.6 Integration of econometric and judgemental forecasts 427 13.7 How can we conduct scenario analysis when judgement is applied? 432 13.8 Making the forecast process effective 432 14 The way forward for real estate modelling and forecasting 434 References 441 Index 448