Introductory Econometrics for Finance
SECOND EDITION
Chris Brooks
The ICMA Centre, University of Reading

Contents
List of figures page xii
List of tables xiv
List of boxes xvi
List of screenshots xvii
Preface to the second edition xix
Acknowledgements xxiv
1 Introduction 1
1.1 What is econometrics? 1
1.2 Is financial econometrics different from ‘economic econometrics’? 2
1.3 Types of data 3
1.4 Returns in financial modelling 7
1.5 Steps involved in formulating an econometric model 9
1.6 Points to consider when reading articles in empirical finance 10
1.7 Econometric packages for modelling financial data 11
1.8 Outline of the remainder of this book 22
1.9 Further reading 25
Appendix: Econometric software package suppliers 26
2 A brief overview of the classical linear regression model 27
2.1 What is a regression model? 27
2.2 Regression versus correlation 28
2.3 Simple regression 28
2.4 Some further terminology 37
2.5 Simple linear regression in EViews -- estimation of an optimal
hedge ratio 40
2.6 The assumptions underlying the classical linear regression model 43
2.7 Properties of the OLS estimator 44
2.8 Precision and standard errors 46
2.9 An introduction to statistical inference 51

Contents
2.10 A special type of hypothesis test: the t-ratio 65
2.11 An example of the use of a simple t-test to test a theory in finance:
can US mutual funds beat the market? 67
2.12 Can UK unit trust managers beat the market? 69
2.13 The overreaction hypothesis and the UK stock market 71
2.14 The exact significance level 74
2.15 Hypothesis testing in EViews -- example 1: hedging revisited 75
2.16 Estimation and hypothesis testing in EViews -- example 2:
the CAPM 77
Appendix: Mathematical derivations of CLRM results 81
3 Further development and analysis of the classical linear
regression model 88
3.1 Generalising the simple model to multiple linear regression 88
3.2 The constant term 89
3.3 How are the parameters (the elements of the β vector) calculated
in the generalised case? 91
3.4 Testing multiple hypotheses: the F-test 93
3.5 Sample EViews output for multiple hypothesis tests 99
3.6 Multiple regression in EViews using an APT-style model 99
3.7 Data mining and the true size of the test 105
3.8 Goodness of fit statistics 106
3.9 Hedonic pricing models 112
3.10 Tests of non-nested hypotheses 115
Appendix 3.1: Mathematical derivations of CLRM results 117
Appendix 3.2: A brief introduction to factor models and principal
components analysis 120
4 Classical linear regression model assumptions and
diagnostic tests 129
4.1 Introduction 129
4.2 Statistical distributions for diagnostic tests 130
4.3 Assumption 1: E(ut ) = 0 131
4.4 Assumption 2: var(ut ) = σ2 < ∞ 132
4.5 Assumption 3: cov(ui , u j ) = 0 for i = j 139
4.6 Assumption 4: the xt are non-stochastic 160
4.7 Assumption 5: the disturbances are normally distributed 161
4.8 Multicollinearity 170
4.9 Adopting the wrong functional form 174
4.10 Omission of an important variable 178
4.11 Inclusion of an irrelevant variable 179

Contents vii
4.12 Parameter stability tests 180
4.13 A strategy for constructing econometric models and a discussion
of model-building philosophies 191
4.14 Determinants of sovereign credit ratings 194
5 Univariate time series modelling and forecasting 206
5.1 Introduction 206
5.2 Some notation and concepts 207
5.3 Moving average processes 211
5.4 Autoregressive processes 215
5.5 The partial autocorrelation function 222
5.6 ARMA processes 223
5.7 Building ARMA models: the Box--Jenkins approach 230
5.8 Constructing ARMA models in EViews 234
5.9 Examples of time series modelling in finance 239
5.10 Exponential smoothing 241
5.11 Forecasting in econometrics 243
5.12 Forecasting using ARMA models in EViews 256
5.13 Estimating exponential smoothing models using EViews 258
6 Multivariate models 265
6.1 Motivations 265
6.2 Simultaneous equations bias 268
6.3 So how can simultaneous equations models be validly estimated? 269
6.4 Can the original coefficients be retrieved from the πs? 269
6.5 Simultaneous equations in finance 272
6.6 A definition of exogeneity 273
6.7 Triangular systems 275
6.8 Estimation procedures for simultaneous equations systems 276
6.9 An application of a simultaneous equations approach to
modelling bid--ask spreads and trading activity 279
6.10 Simultaneous equations modelling using EViews 285
6.11 Vector autoregressive models 290
6.12 Does the VAR include contemporaneous terms? 295
6.13 Block significance and causality tests 297
6.14 VARs with exogenous variables 298
6.15 Impulse responses and variance decompositions 298
6.16 VAR model example: the interaction between property returns and
the macroeconomy 302
6.17 VAR estimation in EViews 308

Contents
7 Modelling long-run relationships in finance 318
7.1 Stationarity and unit root testing 318
7.2 Testing for unit roots in EViews 331
7.3 Cointegration 335
7.4 Equilibrium correction or error correction models 337
7.5 Testing for cointegration in regression: a residuals-based approach 339
7.6 Methods of parameter estimation in cointegrated systems 341
7.7 Lead--lag and long-term relationships between spot and
futures markets 343
7.8 Testing for and estimating cointegrating systems using the
Johansen technique based on VARs 350
7.9 Purchasing power parity 355
7.10 Cointegration between international bond markets 357
7.11 Testing the expectations hypothesis of the term structure of
interest rates 362
7.12 Testing for cointegration and modelling cointegrated systems
using EViews 365
8 Modelling volatility and correlation 379
8.1 Motivations: an excursion into non-linearity land 379
8.2 Models for volatility 383
8.3 Historical volatility 383
8.4 Implied volatility models 384
8.5 Exponentially weighted moving average models 384
8.6 Autoregressive volatility models 385
8.7 Autoregressive conditionally heteroscedastic (ARCH) models 386
8.8 Generalised ARCH (GARCH) models 392
8.9 Estimation of ARCH/GARCH models 394
8.10 Extensions to the basic GARCH model 404
8.11 Asymmetric GARCH models 404
8.12 The GJR model 405
8.13 The EGARCH model 406
8.14 GJR and EGARCH in EViews 406
8.15 Tests for asymmetries in volatility 408
8.16 GARCH-in-mean 409
8.17 Uses of GARCH-type models including volatility forecasting 411
8.18 Testing non-linear restrictions or testing hypotheses about
non-linear models 417
8.19 Volatility forecasting: some examples and results from the
literature 420
8.20 Stochastic volatility models revisited 427

Contents
11 Limited dependent variable models 511
11.1 Introduction and motivation 511
11.2 The linear probability model 512
11.3 The logit model 514
11.4 Using a logit to test the pecking order hypothesis 515
11.5 The probit model 517
11.6 Choosing between the logit and probit models 518
11.7 Estimation of limited dependent variable models 518
11.8 Goodness of fit measures for linear dependent variable models 519
11.9 Multinomial linear dependent variables 521
11.10 The pecking order hypothesis revisited -- the choice between
financing methods 525
11.11 Ordered response linear dependent variables models 527
11.12 Are unsolicited credit ratings biased downwards? An ordered
probit analysis 528
11.13 Censored and truncated dependent variables 533
11.14 Limited dependent variable models in EViews 537
Appendix: The maximum likelihood estimator for logit and
probit models 544
12 Simulation methods 546
12.1 Motivations 546
12.2 Monte Carlo simulations 547
12.3 Variance reduction techniques 549
12.4 Bootstrapping 553
12.5 Random number generation 557
12.6 Disadvantages of the simulation approach to econometric or
financial problem solving 558
12.7 An example of Monte Carlo simulation in econometrics: deriving a
set of critical values for a Dickey--Fuller test 559
12.8 An example of how to simulate the price of a financial option 565
12.9 An example of bootstrapping to calculate capital risk requirements 571
13 Conducting empirical research or doing a project or dissertation
in finance 585
13.1 What is an empirical research project and what is it for? 585
13.2 Selecting the topic 586
13.3 Sponsored or independent research? 590
13.4 The research proposal 590
13.5 Working papers and literature on the internet 591
13.6 Getting the data 591

Contents xi
13.7 Choice of computer software 593
13.8 How might the finished project look? 593
13.9 Presentational issues 597
14 Recent and future developments in the modelling
of financial time series 598
14.1 Summary of the book 598
14.2 What was not covered in the book 598
14.3 Financial econometrics: the future? 602
14.4 The final word 606
Appendix 1 A review of some fundamental mathematical and
statistical concepts 607
A1 Introduction 607
A2 Characteristics of probability distributions 607
A3 Properties of logarithms 608
A4 Differential calculus 609
A5 Matrices 611
A6 The eigenvalues of a matrix 614
Appendix 2 Tables of statistical distributions 616
Appendix 3 Sources of data used in this book 628
References 629
Index 641