## Thursday, July 30, 2020

### Dirty Secrets of Scientific Peer Review

Is peer review a joke?  Surely not.  Peer review often makes good papers better and makes bad papers go away.  Of course we all wish that "good" and "bad" papers could be so cleanly and cavalierly classified.

Conversely, is peer review something to rely upon as firmly establishing scientific credibility?  Now THAT'S the joke.  Blind acceptance of "peer reviewed" as "trustworthy" is like blind acceptance of myriad other naive administrative check-the-box "solutions" -- a dubious strategy at best.  I fear, however, that significant parts of the public have been fooled into thinking that "passing peer review" equals "trustworthy", and that "not yet having passed peer review" equals "not yet trustworthy". The real test is whether a paper influences the course of thought in the medium and long run. Who decides that? The fiercely competitive market for ideas/researchers is highly, if not perfectly, efficient, and it usually sorts things out correctly.

For young researchers:

Immediately put your new paper in a visible working paper series like SSRN or arXiv, and let the market take over. But of course work hard simultaneously to get your paper published in a top place.  The refereeing process will (hopefully) improve it, and having the imprimatur of Top Journal X will send a valuable signal to the profession. Just don't be too sad, or too worried, if it doesn't work out as hoped with Top Journal X.

## Monday, July 27, 2020

### The Pandemic Recession as a Giant Outlier

When I earlier blogged on Frank Schorfheide and Dongho Song (2020), I was focusing on exact methods for mixed-frequency data in Bayes vs. frequentist forecasting and nowcasting.

Quite apart from that, Schorfheide-Song provides eye-opening discussion of a key issue in "forecasting through" the Pandemic Recession (PR), namely how to treat the PR data in estimation. They find that "... forecasts based on a pre-crisis estimate of the VAR using data up until the end of 2019 appear to be more stable and reasonable than forecasts based on a sequence of recursive estimates that include the most recent observations."

The point is that the PR is in many respects a massive outlier, so that one has to think hard about what to do with it in estimation. That is, as always one wants to fit signal, not noise, and the PR is in certain respects a massive burst of noise, capable of severely distorting parameter estimates and hence forecasts and nowcasts.

Michele Lenza and
Giorgio Primiceri address the same issue in another fine new paper, “How to Estimate a VAR after March 2020”. Their focus differs in lots of interesting ways, but the message is the same: One way or another, we need to heavily downweight the PR data.

The emerging message is a big deal: One should be careful before attempting to re-estimate forecasting and nowcasting models with data spanning the PR. Of course at this point there remain many open questions, but it's great to see the issues raised.

## Wednesday, July 22, 2020

### Online Econometrics Seminars

In my view it makes little sense for individual university departments to plod forward with their econometrics seminars virtually.  (Do we really want to watch twenty seminars from twenty departments each week?)  Sponsorship of a few seminars by a few broad umbrella organizations (societies, associations, ...) makes much more sense. The Society for Financial Econometrics (SoFiE) has done it well, here.  So has the International Association for Applied Econometrics (IAAE), here, as has the Chamberlain Seminar, here.  All those seminars post recordings/slides/papers.

I would like to see a few more.  Certainly there should be something explicitly and exclusively on predictive modeling in its relation to time series and machine learning.  Also, although there are several online "climate economics" seminars, I would like to see something explicitly "climate econometrics", in the style pioneered here.

## Tuesday, July 21, 2020

### Online Economics Seminars

An interesting and useful (and incomplete but growing) list of online economics seminars is here. Many archive their seminar recordings and slides. The downside is overabundance. (And you thought you were bombarded with too many seminar options before the pandemic!) More on that in a future post.

## Thursday, July 16, 2020

### Evaluating Interval Forecasts

Evaluation of point and density forecasts is more-or-less straightforward.  Interval forecasts are another matter entirely.  We raised and tried to resolve (some of) the issues here.  Now, in a new paper, Gneiting make major progress. The shortest interval fails to be elicitable!  Their eventual conclusion (p. 14) precisely matches my own view: "In many ways, interval forecasts can be seen as an intermediate stage in the ongoing, transdiciplinary transition from point forecasts to fully probabilistic or distribution forecasts...Indeed, probabilistic forecasts in the form of predictive distributions are the gold standard, as they allow for full-fledged decision making and well-understood, powerful evaluation methods are available..."

# Scoring Interval Forecasts: Equal-Tailed, Shortest, and Modal Interval

We consider different types of predictive intervals and ask whether they are elicitable, i.e. are unique minimizers of a loss or scoring function in expectation. The equal-tailed interval is elicitable, with a rich class of suitable loss functions, though subject to either translation invariance, or positive homogeneity and differentiability, the Winkler interval score becomes a unique choice. The modal interval also is elicitable, with a sole consistent scoring function, up to equivalence. However, the shortest interval fails to be elicitable relative to practically relevant classes of distributions. These results provide guidance in interval forecast evaluation and support recent choices of performance measures in forecast competitions.
 Comments: 22 pages Subjects: Statistics Theory (math.ST) MSC classes: 62C05, 91B06 Cite as: arXiv:2007.05709 [math.ST] (or arXiv:2007.05709v1 [math.ST] for this version)

## Tuesday, July 14, 2020

### Spurious Factor Analysis

This abstract definitely produced one of those great "ah ha!" moments, at least for me.  So obvious once someone points it out.  Thanks Alexei and Chen.

I'm hungry for more.

With I(1) regression we have:
Q: When will regressions with I(1) variables not produce spurious results?
A: When the variables are not only integrated but also cointegrated.

What is the analog here, with PCA?  That is:
Q: When will PCA with high-dim I(1) variables not produce spurious results?
A: ??? (I'm not yet sure.  Maybe it's addressed in the paper, which I look forward to reading.  Cointegration should again be part of the answer (maybe all of the answer?), as cointegration implies factor structure.)

 By: Onatski, A.; Wang, C. Abstract: This paper draws parallels between the Principal Components Analysis of factorless high-dimensional nonstationary data and the classical spurious regression. We show that a few of the principal components of such data absorb nearly all the data variation. The corresponding scree plot suggests that the data contain a few factors, which is collaborated by the standard panel information criteria. Furthermore, the Dickey-Fuller tests of the unit root hypothesis applied to the estimated “idiosyncratic terms” often reject, creating an impression that a few factors are responsible for most of the non-stationarity in the data. We warn empirical researchers of these peculiar effects and suggest to always compare the analysis in levels with that in differences. Keywords: Spurious regression, principal components, factor models, Karhunen-Loève expansion. Date: 2020–01–13 URL: http://d.repec.org/n?u=RePEc:cam:camdae:2003&r=ecm

## Thursday, July 2, 2020

### Outstanding New Financial Econometrics Book

Financial Econometric Modelling, by Stan Hurn, Vance L. Martin, Peter C. B. Phillips, and Jun Yu, has now been published by Oxford University Press, here.

It's very, very well done.  The depth, breadth, and clarity of coverage are exceptional.  It's completely up to date, very "2020".  See for yourself; the front and back matter appear below (sans formatting).

Financial Econometric Modelling

Stan Hurn, Vance L. Martin, Peter C. B. Phillips, and Jun Yu
November 12, 2019

Preface

Financial econometrics is an exciting young discipline that began to take on
its present form around the turn of the millennium. The subject brings financial
theory and econometric methods together with the power of data
to advance our understanding of the global financial universe upon which
all modern economies depend. Two major developments underscored its
rapid growth and expanding capabilities. First, the massive importance of
well-functioning financial markets to the global economy and to global financial
stability was universally acknowledged following the dot-com bubble of
the late 1990s in the United States and the global financial crisis of 2008 coupled
with its prolonged aftermath. Second, modern methods of econometrics
emerged that proved equal to some of the special challenges presented by financial
data and the ideas of financial theory.

Among the most significant of these challenges are the complex interdependencies
of financial, commodity, and real estate markets, the dynamic and
spatial linkages within financial data, the random wandering behaviour of asset
prices, anomalies such as financial bubbles and market crashes in the data,
the difficulties in modelling rapidly changing volatility in financial returns,
the growth in high dimensional ultra-high frequency data, and the attention
to market microstructure effects that all such data require. While not entirely
unique to financial data, these challenges presented the econometrics profession
with the need to re-fashion methods, develop new tools of inference,
and tackle a wide selection of new empirical goals associated with a growing
number of financial instruments and vast data sets being generated in the
financial world.

This book, like the subject itself, is motivated by all of these challenges. We
seek to provide a broad and gentle introduction to this rapidly developing
subject of financial econometrics where theory, measurement, and data play
equal roles in our development and where empirical applications occupy a
students, honours students who wish to learn about financial econometrics,
and postgraduate students with limited backgrounds in econometrics
who are doing masters courses designed to offer an introduction to finance
and its applications. We hope the book will also prove useful to practitioners
in the industry as an introductory reference source for relevant tools and
approaches to modern empirical work in finance.
Throughout the book special emphasis is placed on the exposition of core
concepts, their illustration using relevant financial data sets, and a hands-on
approach to learning by doing that involves practical implementation. The
guiding principle we have adopted is that only by working through plenty of
applications and exercises can a coherent understanding of the properties of
financial econometric models, interrelationships with the underlying finance
theory, and the role of econometric tools of inference be achieved.

Our philosophy has been to write a book on financial econometrics, not an
econometrics text that illustrates techniques with datasets drawn from finance.
Our goal is centred on the subject of financial econometrics explaining
how evidenced-based research in applied finance is conducted. Econometrics
is viewed as the vehicle that makes the ideas and theories about financial
markets face the reality of observations.
To ensure the book is self contained for a first course in financial econometrics,
some foundational theory and methods of relevant econometric technique
are provided. But the methods covered in this book travel along a customised
path designed to ease the reader’s transition from concepts and methods
to empirical work. The book tracks its way forward from data to modelling
through to estimation, inference, and prediction.
A consistent thematic throughout the book is to motivate each topic with
the presentation of relevant data. The journey begins with data and a simple
grounding in regression and inference. From this foundation, it moves on to
more advanced financial econometric methods that open up empirical applications
with many different types of data from various financial markets. The
path promises to keep readers motivated throughout their journey by means
of many examples and to reinforce their learning by extensive data-based exercises.
Several introductory Appendices are included to assist students with
limited mathematics and no econometric background in understanding more
technical aspects of the discussion particularly in the second half of the book.

Organisation of the Book
Part I – Fundamentals – is designed to form the basis of a semester long first
course in financial econometrics directed at an introductory level. Technical
difficulty is kept to an absolute minimum with an emphasis on the data, financial
concepts, appropriate econometric methodology, and the intuition
that draws these essential components of modelling together. Methodology
is largely confined to descriptive methods and ordinary least squares regression,
a choice that limits the extent of the analysis and promotes heuristic discussion
on some topics which are revisited later in the book for a more complete
and rigorous development.
In Part II – Methods – the level of difficulty steps up slightly in treating the
relevant econometric estimation methods of instrumental variables, generalised
method of moments, and maximum likelihood. These core estimators
are used extensively throughout the second half of the book and knowledge
of them is a key asset in working through the later material. Also included in
Part II are methods that deal with panel data and models with latent factors.
A second course in financial econometrics might usefully begin with these
five chapters, taking Part I as a given foundation.
Part III – Topics – introduces a number of special topics in financial econometiv
rics, covering models of volatility, financial market microstructure, the econometrics
of options, and methods relating to extreme values and copulas. One
of the dominant features of financial time series is their volatility. Financial
theory and empirical experience both demonstrate that there is often much
less to explain in the levels of financial returns than there is to explain in their
variation. Accordingly, three chapters of the book are devoted to modelling
volatility. These chapters treat parametric univariate and multivariate models
of volatility and introduce the more recent nonparametric modelling approach
that is based on market realised volatility measured using high frequency
data.
As in any project of this nature, sacrifices were made to keep the length of the
book manageable. Some topics, for instance, are treated by example and illustration
within a chapter rather than by devoting an entire chapter to their
development. As a result the book is rich in real world examples drawn from
financial markets for stocks, fixed income securities, exchange rates, derivatives,
and real estate. As such, the coverage is intended to be extensive while
not treating every topic in the same depth.
Computation
A fundamental principle guiding the inclusion of material in this book is
whether the methods are available for easy implementation. In consequence,
all results reported in the book may be reproduced using existing software
packages like Stata and EViews.1 This choice is intended to enhance the usefulness
of the material for beginning students. In some cases the programming
languages in these packages need to be used to achieve full implementation
of the illustrations. Of course, for those who actively choose to learn by
programming themselves, the results are also reproducible in any of the common
matrix programming languages, such as R2. The numerical computations
reported in the book are primarily rounded versions of the results generated
using Stata.
The data files are all available for download from the book’s companion website
format (.dta), EViews format (.wf1), comma delimited files (.csv), and as Excel
1Stata is the copyright of StataCorp LP www.stata.com and EViews is the copyright of IHSInc.
www.eviews.com.
2R is a free software environment for statistical computation and graphics which is part of the
GNU Project, see www.r-project.org.

Acknowledgements
Many colleagues, students, and research assistants have read and commented
on parts of the book and in some cases even taught from early versions of the
book. We are particularly grateful to Ahmad Bahir, Kit Baum, Jimmy Chan,
Han Chen, Ye Chen, Xiaohong Chen, Jieyang Chong, Adam Clements, Fulvio
Corsi, Mark Doolan, Ren´ee Fry-McKibbin, Zhuo Huang, Marko Krause,
Bei Luo, Cheng Liu, Andrew Patton, Shuping Shi, Daniel Smith, Chrismin
Tang, Timo Ter¨asvirta, Stephen Thiele, Tomasz Wo´zniak and Lina Xu. A special
thank you goes to Annastiina Silvennoinen and Glen Wade who were relentless
in picking up typographical errors and factual inconsistencies, as well
as suggesting alternative ways of presenting material. All remaining errors
are the responsibility of the authors.
Stan Hurn, Vance L. Martin, Peter C. B. Phillips and Jun Yu
June 2019

Contents

I Fundamentals 1
1 Prices and Returns 3
1.1 What Is Financial Econometrics? . . . . . . . . . . . . . . . . . . 3
1.2 Financial Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Equity Prices and Returns . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Stock Market Indices . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Bond Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Financial Data 29
2.1 A First Look at the Data . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3 Percentiles and Value at Risk . . . . . . . . . . . . . . . . . . . . 45
2.4 The Efficient Market Hypothesis . . . . . . . . . . . . . . . . . . 48
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Linear Regression 57
3.1 The Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . 58
3.2 A Multifactor CAPM . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 Properties of Ordinary Least Squares . . . . . . . . . . . . . . . . 67
3.4 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5 Measuring Portfolio Performance . . . . . . . . . . . . . . . . . . 80
3.6 Minimum Variance Portfolios . . . . . . . . . . . . . . . . . . . . 82
3.7 Event Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4 Stationary Dynamics 95
4.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2 Univariate Time Series Models . . . . . . . . . . . . . . . . . . . 98
4.3 Autocorrelation and Partial Autocorrelations . . . . . . . . . . . 103
4.4 Mean Aversion and Reversion in Returns . . . . . . . . . . . . . 107
4.5 Vector Autoregressive Models . . . . . . . . . . . . . . . . . . . . 109
4.6 Analysing VARs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.7 Diebold-Yilmaz Spillover Index . . . . . . . . . . . . . . . . . . . 126
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5 Nonstationarity 133
5.1 The RandomWalk with Drift . . . . . . . . . . . . . . . . . . . . 134
5.2 Characteristics of Financial Data . . . . . . . . . . . . . . . . . . 137
5.3 Dickey-Fuller Methods and Unit Root Testing . . . . . . . . . . . 140
5.4 Beyond the Simple Unit Root Framework . . . . . . . . . . . . . 147
5.5 Asset Price Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6 Cointegration 161
6.1 The Present Value Model and Cointegration . . . . . . . . . . . . 162
6.2 Vector Error Correction Models . . . . . . . . . . . . . . . . . . . 167
6.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.4 Cointegration Testing . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.5 Parameter Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.6 Cointegration and the Gordon Model . . . . . . . . . . . . . . . 187
6.7 Cointegration and the Yield Curve . . . . . . . . . . . . . . . . . 192
6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7 Forecasting 205
7.1 Types of Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.2 Forecasting Univariate Time Series Models . . . . . . . . . . . . 208
7.3 Forecasting Multivariate Time Series Models . . . . . . . . . . . 212
7.4 Combining Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.5 Forecast Evaluation Statistics . . . . . . . . . . . . . . . . . . . . 220
7.6 Evaluating the Density of Forecast Errors . . . . . . . . . . . . . 224
7.7 Regression Model Forecasts . . . . . . . . . . . . . . . . . . . . . 229
7.8 Predicting the Equity Premium . . . . . . . . . . . . . . . . . . . 230
7.9 Stochastic Simulation of Value at Risk . . . . . . . . . . . . . . . 237
7.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

II Methods 247

8 Instrumental Variables 249
8.1 The Exogeneity Assumption . . . . . . . . . . . . . . . . . . . . . 250
8.2 Estimating the Risk-Return Tradeoff . . . . . . . . . . . . . . . . 251
8.3 The General Instrumental Variables Estimator . . . . . . . . . . 255
8.4 Testing for Endogeneity . . . . . . . . . . . . . . . . . . . . . . . 259
8.5 Weak Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
8.6 Consumption CAPM . . . . . . . . . . . . . . . . . . . . . . . . . 266
8.7 Endogeneity and Corporate Finance . . . . . . . . . . . . . . . . 269
8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

9 Generalised Method of Moments 277
9.1 Single Parameter Models . . . . . . . . . . . . . . . . . . . . . . . 278
9.2 Multiple Parameter Models . . . . . . . . . . . . . . . . . . . . . 281
9.3 Over-Identified Models . . . . . . . . . . . . . . . . . . . . . . . . 288
9.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
9.5 Properties of the GMM Estimator . . . . . . . . . . . . . . . . . . 299
9.6 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
9.7 Consumption CAPM Revisited . . . . . . . . . . . . . . . . . . . 309
9.8 The CKLS Model of Interest Rates . . . . . . . . . . . . . . . . . 312
9.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

10 Maximum Likelihood 323
10.1 Distributions in Finance . . . . . . . . . . . . . . . . . . . . . . . 324
10.2 Estimation by Maximum Likelihood . . . . . . . . . . . . . . . . 331
10.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
10.4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 342
10.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
10.6 Quasi Maximum Likelihood Estimation . . . . . . . . . . . . . . 347
10.7 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
10.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

11 Panel Data Models 365
11.1 Types of Panel Data . . . . . . . . . . . . . . . . . . . . . . . . . . 366
11.2 Reasons for Using Panel Data . . . . . . . . . . . . . . . . . . . . 368
11.3 Two Introductory Panel Models . . . . . . . . . . . . . . . . . . . 372
11.4 Fixed and Random Effects Panel Models . . . . . . . . . . . . . . 377
11.5 Dynamic Panel Models . . . . . . . . . . . . . . . . . . . . . . . . 386
11.6 Nonstationary Panel Models . . . . . . . . . . . . . . . . . . . . . 394
11.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

12 Latent Factor Models 409
12.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
12.2 Principal Components . . . . . . . . . . . . . . . . . . . . . . . . 412
12.3 A Latent Factor CAPM . . . . . . . . . . . . . . . . . . . . . . . . 420
12.4 Dynamic Factor Models: the Kalman Filter . . . . . . . . . . . . 423
12.5 A Parametric Approach to Factors . . . . . . . . . . . . . . . . . 431
12.6 Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . 434
12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

III Topics 445

13 Univariate GARCH Models 447
13.1 Volatility Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . 448
13.2 The GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . 450
13.3 Asymmetric Volatility Effects . . . . . . . . . . . . . . . . . . . . 458
13.4 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
CONTENTS xi
13.5 The Risk-Return Tradeoff . . . . . . . . . . . . . . . . . . . . . . 465
13.6 Heatwaves and Meteor Showers . . . . . . . . . . . . . . . . . . 467
13.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

14 Multivariate GARCH Models 477
14.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
14.2 Early Covariance Estimators . . . . . . . . . . . . . . . . . . . . . 480
14.3 The BEKK Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
14.4 The DCC Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
14.5 Optimal Hedge Ratios . . . . . . . . . . . . . . . . . . . . . . . . 493
14.6 Capital Ratios and Financial Crises . . . . . . . . . . . . . . . . . 495
14.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

15 Realised Variance and Covariance 505
15.1 High Frequency Data . . . . . . . . . . . . . . . . . . . . . . . . . 506
15.2 Realised Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
15.3 Integrated Variance . . . . . . . . . . . . . . . . . . . . . . . . . . 512
15.4 Microstructure Noise . . . . . . . . . . . . . . . . . . . . . . . . . 515
15.5 Bipower Variation and Jumps . . . . . . . . . . . . . . . . . . . . 518
15.6 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
15.7 The Realised GARCH Model . . . . . . . . . . . . . . . . . . . . 525
15.8 Realised Covariance . . . . . . . . . . . . . . . . . . . . . . . . . 527
15.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

16 Microstructure Models 537
16.1 Characteristics of High Frequency Data . . . . . . . . . . . . . . 537
16.2 Limit Order Book . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
16.3 Bid Ask Bounce . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
16.4 Information Content of Trades . . . . . . . . . . . . . . . . . . . . 543
16.5 Modelling Price Movements in Trades . . . . . . . . . . . . . . . 545
16.6 Modelling Durations . . . . . . . . . . . . . . . . . . . . . . . . . 551
16.7 Modelling Volatility in Transactions Time . . . . . . . . . . . . . 555
16.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

17 Options 565
17.1 Option Pricing Basics . . . . . . . . . . . . . . . . . . . . . . . . . 566
17.2 The Black-Scholes Option Price Model . . . . . . . . . . . . . . . 569
17.3 A First Look at Options Data . . . . . . . . . . . . . . . . . . . . 573
17.4 Estimating the Black-Scholes Model . . . . . . . . . . . . . . . . 574
17.5 Testing the Black-Scholes Model . . . . . . . . . . . . . . . . . . 581
17.6 Option Pricing and GARCH Volatility . . . . . . . . . . . . . . . 583
17.7 The Melick-Thomas Option Price Model . . . . . . . . . . . . . . 585
17.8 Nonlinear Option Pricing . . . . . . . . . . . . . . . . . . . . . . 587
17.9 Using Options to Estimate GARCH Models . . . . . . . . . . . . 588
17.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

18 Extreme Values and Copulas 599
18.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
18.2 Evidence of Heavy Tails . . . . . . . . . . . . . . . . . . . . . . . 602
18.3 Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . . . 605
18.4 Modelling Dependence using Copulas . . . . . . . . . . . . . . . 611
18.5 Properties of Copulas . . . . . . . . . . . . . . . . . . . . . . . . . 614
18.6 Estimating Copula Models . . . . . . . . . . . . . . . . . . . . . . 621
18.7 MGARCH Model Using Copulas . . . . . . . . . . . . . . . . . . 624
18.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626

19 Concluding Remarks 633

A Mathematical Preliminaries 635
A.1 Summation Notation . . . . . . . . . . . . . . . . . . . . . . . . . 635
A.2 Expectations Operator . . . . . . . . . . . . . . . . . . . . . . . . 638
A.3 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
A.4 Taylor Series Expansions . . . . . . . . . . . . . . . . . . . . . . . 641
A.5 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
A.6 Transposition of a Matrix . . . . . . . . . . . . . . . . . . . . . . . 650
A.7 Symmetric Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

B Properties of Estimators 661
B.1 Finite Sample Properties . . . . . . . . . . . . . . . . . . . . . . . 661
B.2 Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . . . . 663
C Linear Regression Model in Matrix Notation 669
D Numerical Optimisation 673
E Simulating Copulas 677
Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698
Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

Author index
A¨ıt-Sahalia, Y., 341, 517, 518
Akaike, H., 113
Aldrich, J., 323
Almeida, H., 269, 270, 382
Amenc, N., 477
Andersen, T.G, 517
Anderson, D.R., 218
Anderson, R.C., 269
Anderson, T.W., 390
Andreou, E., 237
Arellano, M., 390
Augustin, N.H., 218
Bae, K.H., 599
Bai, J., 419
Baillie, R.T., 493, 494
Baker, R., 262
Bali, T.G., 255
Baltagi, B.H, 400
Banerjee, A., 203
Barndorff-Nielsen, O.E., 508, 515,
518, 520, 521, 529, 538
Bartlett, M.S., 175, 349
Bates, J., 218, 235
Becker, R., 224
Bekaert, G., 67
Bera, A.K., 79, 229
Bergstrom, A.R., 161
Black, F., 569
Blundell, R., 390
Bollerslev, T., 448, 450, 457, 517,
525, 589
Bond, S., 390
Bonhomme, S., 372
Bouchard, J-P., 544
Bound, J., 262
Bover, O., 390
Brennan, M.J., 315
Brockwell, P.J., 98, 106
Brownlees, C., 495, 498
Buckland, S.T., 218
Burnham, K.P., 218
Bykhovskaya, A., 150
Campbell, J.Y., 131, 154, 237
Carhart, M.M., 65
Chan, K.C., 313
Chao, J.C., 179
Cheng, X., 179
Chiang, M.-H., 400
Choi, I., 399
Christoffersen, P.F., 464
Chu, C.-J., 397
Clemen, R.T., 219
Clements, A.E., 224, 469
Clements, M.P., 219
Cochrane, J.H., 48, 237
Colacito, R., 224
Corbae, D., 199
Corsi, F., 522
Cox, J.C., 315, 329
Cram´er, H., 345
Davidson, R., 260
Davis, R.A., 98, 106
Diba, B.T., 154
Dickey, D.A., 140, 141, 146
Diebold, F.X., 24, 126, 132, 223, 229,
431, 441, 517
Doolan, M.B., 224
Dungey, M., 362, 467
Durbin, J., 129, 298
Eisler, Z., 544
Elliott, G., 149, 217, 219, 224
Engle, R.F., 78, 161, 173, 174, 224,
448–450, 467, 471, 487,
489, 495, 498, 542, 552,
555, 556
Epps, T.W., 528
Evans, G.W., 154
Fakhrutdinova, L., 467
Fama, E.F., 48, 64, 107, 403
AUTHOR INDEX 699
Fan, Y., 612
Ferreira, D., 269, 270, 382
Ferson, W.E., 268, 269
Fisher, R.A., 398
Flannery, M.J., 388, 406
Flemming, J., 224
French, K.R., 64, 107
Fuller, W.A., 140, 141
Galbraith, J.W., 203
Garratt, A., 219
Ghosh, A., 229
Ghysels, E., 255
Gibbons, M., 403
Gibson, M., 589
Glosten, L.R., 458
Goltz, F., 477
Goodhart, C.A.E., 467
Gordon, M.J., 187, 190, 201
Goyal, A., 231, 244
Granger, C.W.J., 115, 161, 174, 203,
218, 235
Grossman, H.I., 154
Gunther, T.A., 229
Haldane, A.G., 442
Hall, A.D., 544
Hall, A.R., 300
Hall, S.G., 442
Hamilton, J.D., 98
Hankins, K.W., 388, 406
Hannan, E.J., 113
Hansen, B.E., 174
Hansen, L.P., 298, 300, 302, 303, 311
Hansen, P.R., 462, 508, 518, 525,
529, 538
Harris, D., 98, 122, 292, 350, 427,
548, 587, 662
Harvey, A.C., 427, 493
Harvey, C.R., 268, 269
Hasbrouck, J., 543, 559
Hausman, J.A., 385, 545
Hautsch, N., 544
Heaton, J., 298
Hendry, D.F., 203, 219
Hodrick, R.J., 54
Hoerova, M., 67
Hoyem, K., 67
Hsiao, C., 390
Hu, W-Y., 67
Huang, R., 544
Huang, Z., 525
Hurn, A.S., 98, 122, 224, 292, 350,
427, 469, 548, 587, 662
Hwang, J., 173
Im, K.S., 397
Ingersoll, J.E., 315, 329
Ito, T., 467
Jacobs, J.P.A.M., 362
Jacod, J., 518
Jaeger, D., 262
Jaganathan, R., 458
Jarque, C.M., 79
Jensen, M.C., 80
Johansen, S., 174, 177, 184
Judson, R.A., 388
Kanniainen, J., 588, 589
Kao, C., 400
Kapetanious, G., 219
Karolyi, G.A., 313, 599
Kasparis, I., 237
Kelly, B., 489
Kim, M.J., 107
Kirby, C., 224
Kiviet, J., 388
Kockelkoren, J., 544
Koop, G., 219
Kwiatkowski, D.P., 151
L¨ utkepohl, H., 113, 122
Labhard, V., 219
Labys, P., 517
Lee, C.C., 150
Lee, J.H., 237
Levin, A., 397
Li, C., 24, 431, 441
Li, D.X., 600
Lim, K-G., 199
Lin, B., 588, 589
700 BIBLIOGRAPHY
Lin, C.F., 397
Lin, W.-L., 467
Liu, L., 517
Lo, A.W., 154, 545
Lo, D.K., 544
Longstaff, F.A., 313
Loretan, M., 173
Lunde, A., 462, 508, 518, 525, 529,
538
MacBeth, J.D., 403
MacKenzie, D., 600
MacKinlay, A.C., 154, 545
MacKinnon, J.G., 144, 181, 260
Malmendier, U., 67
Mann, H.B., 95, 109
Manresa, E., 372
Mariano, R.S., 223
Martin, V.L., 98, 122, 292, 350, 427,
548, 587, 662
Melick, W.R., 586
Merton, R.C., 251, 473, 536
Millar, R.B., 345
Mincer, J., 462
Moon, H.R., 400
Myers, R.J., 493, 494
Mykland, P.A., 517, 518
Nagel, S., 67
Nelson, C.R., 107, 431
Nelson, D.B., 459
Newbold, P., 203
Newey, W.K., 175, 349
Ng, S., 147, 419
Nickell, S., 387, 388
Ostdiek, B., 224
Ouliaris, S., 181, 199
Owen, A.L., 388
P´erignon, C., 45
Patton, A.J., 224, 471, 517, 525, 612
Peng, L., 255
Perron, P., 147, 150
Pesaran, M.H., 397
Phillips, P.C.B., 140, 150–152, 156,
173, 174, 179–181, 203,
237, 264, 372, 400
Poterba, J.M., 108
Pratt, J.W., 323
Prescott, E.C., 54
Price, S., 219
Quaedvlieg, R., 525
Quinn, B.G., 113
Rajan, R.G., 393, 406
Ramanathan, R., 218
Reeb, D.M., 269
Reinhart, C.M., 362
Rogoff, K.S., 362
Roley, V.V., 467
Roll, R., 541, 558
Ross, S.A., 315, 329, 403
Rothenberg, T.J., 149
Roulet, J., 54
Runkle, D, 458
Russell, J.R., 542, 552
Said, S.E., 146
Saikkonen, P., 173
Samuelson, P., 48
Sanders, A.B., 313
Santa-Clara, P., 255
Schmidt, P., 151
Scholes, M., 569
Schwartz, E.S., 315
Schwarz, G., 113
Shanken, J., 403
Sharpe,W.F., 80
Shek, H.H., 525
Shephard, N., 508, 515, 518, 520,
521, 529, 538
Sheppard, K., 224, 517
Shi, S., 156
Shi, Z., 372
Shiller, R.J., 67, 131
Shin, Y., 151, 397
Siegel, A.F., 431
Silvennoinen, A., 488, 493
Sims, C.A., 109
Singleton, K.J., 311
Smith, D.R., 45
Smith, J., 219
Solnik, B., 54
Spears, T., 600
Staiger, D., 262, 264
Stambaugh, R.F., 236
Startz, B., 107
Stigler, S.M., 323
Stock, J.H., 149, 173, 219, 262, 264,
274
Stulz, R.M., 599
Su, L., 372
Summers, L.H., 108
Sun, Y., 173
Tang, L., 477
Taylor, S.J., 435
Ter¨asvirta, T., 488, 493
Thiele, S., 493
Thomas, C.P., 586
Timmerman, A., 217, 224
Toda, H.Y., 180
Treynor, J.L., 80
Trivedi, P.K., 612, 678
Tsay, R.S., 229
Tse, Y.K., 488
Tsui, A.K.C., 488
Vahey, S.P., 219
Vaidyanathan, V., 477
Valkanov, R., 255
Van der Vaart, A.W., 345
Van der Waart, J.W., 301
Vasicek, O., 341
Volkov, V.V., 469
Wald, A., 95, 109
Wallis, K.F., 219
Watson, M.W., 173, 219
Welch, I., 231, 244
Wellner, J.A., 301
West, K.D., 175, 349
White, H., 78, 349
Winkler, R.L., 219
Wooldridge, J.M., 457
Wright, J.H., 274
Wu, S., 399
Wu, Y., 152
Yamamoto, Y., 180
Yang, H., 588, 589
Yaron, A., 298
Yilmaz, K., 126, 132
Yogo, M., 237, 264, 274
Yoo, B.S., 173
Yu, J., 152, 156
Zako¨ıan, J.M., 458
Zarnowitz, V., 462
Zhang, L., 517, 518
Zhou, H., 589
Zimmer, D.M., 612, 678
Zingales, L., 393, 406

Subject index
F test of significance, 74
t test of significance, 73
(R2), 72
Akaike information criterion (AIC),
113
Asset returns
volatility clustering, 448–450
Asymptotic efficiency
GMM, 301
maximum likelihood, 345
ordinary least squares, 69
Asymptotic normality
GMM, 303
maximum likelihood, 346
ordinary least squares, 70
Augmented Dickey-Fuller test,
145–147
GLS detrending, 149
lag length selection, 147
Autocorrelation function, 103
Autocovariance, 103
Autoregressive (AR) model
estimation, 99
specification, 98
Autoregressive conditional duration
model, 551–555
Autoregressive moving average
(ARMA) model
estimation, 102
specification, 101
BFGS algorithm, 675
BHHH algorithm, 674
Bipower variation, 518–522
Black-Scholes option pricing model
currency option, 573
equity option, 573
European call option, 569–572
European put option, 572
testing bias, 581
testing heteroskedasticity, 582
testing smiles and smirks, 582
Bollerslev-Wooldridge standard
errors, 457
Bond yields, 36
term structure, 24
yield curve, 24
yield to maturity, 23
Capital asset pricing model (CAPM),
58–63
consumption (C-CAPM), 309
Coefficient of determination (R2),
72
Cointegration, 163
fully modified estimation,
174–177
estimation, 173–180
Gordon model, 187–192
Johansen reduced rank regression
estimator, 177–180
modelling the yield curve,
192–198
present value model, 163–167
testing, 180–187
testing hypotheses on cointegrating
parameters, 186
Consistency
GMM, 301
instrumental variables, 259
maximum likelihood, 345
ordinary least squares, 68
Consumption capital asset pricing
model (C-CAPM), 266–
269
Copulas
t copula, 616
Clayton copula, 617
estimating copula models,
621–624
Frank copula, 618
SUBJECT INDEX 703
Gaussian copula, 614
Gumbel copula, 618
measuring tail dependence,
618–621
modelling dependence using
copulas, 611–614
properties of copulas, 614–621
Diagnostic tests on disturbances
ARCH, 78
autocorrelation, 76
heteroskedasticity, 77
normality, 79
Dickey-Fuller test, 140–145
Diebold-Mariano test, 223
Diebold-Yilmaz Spillover Index,
125–128
Dividends
discounted future stream of,
34
dividend yield, 34
Durations, 39
Dynamic factor models, 423–429
Efficient market hypothesis, 48–51,
134
return predictability, 48
variance ratio, 50
Endogeneity and corporate finance,
269–272
Equilibrium dynamics, 164
Equity prices, 30–32
effect of dividends, 10
effect of stock splits, 11
quoted prices, 8
Event analysis, 85–87
Extreme value distribution
distribution types, 606–607
Hill estimator, 608
maximum likelihood estimation,
609–610
VaR calculation, 611
Extreme value theory, 605–611
Financial assets
cash, 6
derivatives, 7
equities, 7
Eurodollar deposits, 6
fixed-income securities, 6
Treasury bills, 6
Forecasting
AR(1) model, 208–211
AR(2) model, 211
bivariate VAR(1) model, 213–
214
bivariate VECM(1) model,
214–216
combining forecasts, 216–219
density forecast evaluation,
224–229
ex ante forecasts, 206
ex post forecasts, 207
forecast evaluation, 220–224
predictive regressions, 229–
237
properties, 212
Fully modified estimation OLS,
174–177
Gibbons-Ross-Shanken test, 371,
403
Gordon model, 187–192
Granger causality, 115
Hannan information criterion
(HIC), 113
Hansen-Sargan J test, 306
High frequency data
characteristics, 537–538
cleaning, 507–508
limit order book, 538–540
transactions data, 506–509
Idiosyncratic risk, 59
Impulse response analysis, 116
Instrumental variables estimator
multiple endogenous regressors,
258–259
two-variable regression model,
254
Integrated process, 137
704 BIBLIOGRAPHY
Integrated variance, 512–515
Inter-temporal CAPM, 251–255
Invariance
maximum likelihood, 346
Jensen’s alpha, 80
Johansen reduced rank regression
estimator, 177–180
Johansen tests of cointegration,
182–185
Kalman filter
estimation, 428
factor extraction, 428–429
multivariate, 427
univariate, 424–426
Lag length selection
information criteria, 113–114
Lagrange multiplier (LM) test
test for ARCH, 470
Lagrange multiplier test, 351–352
Leptokurtosis, 38
Likelihood ratio test, 351
Limited dependent variables, 545–
551
linear probability model, 547
ordered probit, 549–551
probit, 546–549
Linear regression model, 58
disturbance term diagnostics,
76–80
explanatory variable diagnostics,
73–75
matrix notation, 669–670
Marginal expected shortfall, 495–
498
Mean absolute error (MAE), 221
Mean absolute percentage error
(MAPE), 221
Mean square error (MSE), 221
Measuring portfolio performance
Jensen’s alpha, 80
Sharpe ratio, 80
Treynor ratio, 80
Melick-Thomas option pricing
model, 585–587
Microstructure noise, 515–518
Minimum variance portfolio, 82–
85
Modelling the yield curve, 192–198
Moving average (MA) model
estimation, 100
specification, 100
Multi-factor CAPM
instrumental variables estimation,
255–258
Multifactor CAPM, 63–67
Multiple regression model, 63
Nelson-Siegel parametric factor
model, 431–434
News impact curve (NIC), 459
Newton-Raphson algorithm, 674
Nonlinear option pricing model,
587–588
Nonstationary process, 137
Optimal hedge ratio, 493–495
Options
Black-Scholes option pricing
model, 569–573
data, 573–574
GARCH volatility, 583–585
Greeks, 578–581
historical volatility, 575–576
implied volatility, 576–578
pricing basics, 566–569
Order of integration, 137
Order statistics, 602–603
Ordinary least squares estimator,
60–61
Panel data
Arellano-Bond estimator, 390
common effects model, 374–
377
fixed effects model, 378–380
Hausman test, 383–385
Nickell bias, 386
no common effects model,
372–374
SUBJECT INDEX 705
panel cointegration, 398–400
panel unit roots, 396–398
random effects model, 380–
382
system GMM estimator, 390
Partial autocorrelation function,
106
Percentiles, 45
230–237
Principal component analysis,
412–420
estimation, 413–417
factor extraction, 417–418
model specification, 412–413
testing, 419–420
Probability integral transform
(PIT), 224
Properties of GMM estimators,
299–305
Properties of instrumental variable
estimators, 259
Properties of instrumental variables
estimators, 254
Properties of maximum likelihood
estimators, 344–347
Properties of ordinary least squares
estimators, 67–71
Quasi maximum likelihood estimator,
347
Random walk with drift model,
134–137
Realised covariance, 527–531
refresh time synchronisation,
529–531
Realised variance
computing, 509–511
forecasting, 522–525
Residual-based tests of cointegration,
180–182
Returns
effect of dividends, 15
excess returns, 16
continuous compounded returns,
13
dollar returns, 11
log returns, 13, 32
mean aversion and reversion,
107
simple returns, 12
467
Root mean square error (RMSE),
221
Safe capital ratio, 495–498
Schwarz information criterion
(SIC), 113
Sharpe ratio, 80
Signature plot, 511
Spurious regression problem, 203
Stationary process
introduced, 96
Stock market index
Deutscher Aktien Index (DAX),
19
Dow Jones Industrial Average
Index (DJIA), 19
Financial Times Stock Exchange
100 Index (FTSE),
19
Hang Seng Index (HSX), 19
Nikkei 225 Index (NKX), 19
price weighted, 19
Standard and Poors Composite
500 Index (S&P 500),
19
value weighted, 19
Strong exogeneity, 187
Summary statistics, 41–45
sample correlation, 44
sample covariance, 44
sample kurtosis, 43
sample mean, 41
sample skewness, 43
sample standard deviation, 42
sample variance, 41
Systematic risk, 59
706 BIBLIOGRAPHY
Term structure of interest rates, 24,
192
Testing for bubbles, 152–157
Testing for endogeneity, 259–261
Transactions data, 39
Treynor ratio, 80
Unit root tests
Augmented Dickey-Fuller
test, 145–147
Dickey-Fuller test, 140–145
GLS detrending, 149–150
KPSS test, 151
Phillips-Peron test, 150
Right-tailed tests, 152–157
Structural breaks, 147
Univariate GARCH model
Bollerslev-Wooldridge standard
errors, 457
estimation, 455–457
forecasting, 460–465
heatwaves and meteor showers,
467–470
normal distribution, 455
t distribution, 456
Value at risk, 45–47, 237–240
Variance decomposition, 122
Vector autoregressive models (VAR),
109–125
Diebold-Yilmaz spillover index,
125
estimation, 111
Granger causality, 115
impulse response analysis,
116
lag length selection, 113–114
specification, 110
transactions time, 543–544
variance decomposition, 122
Vector error correction model
(VECM), 167–173
Relationship with VARs, 171–
173
Volatility
defined, 42
Volatility models
EGARCH, 458
GARCH, 453–460
GARCH-M, 465–467
TARCH, 458
BEKK, 482–486
DCC, 487–493
DECO, 488
exponentially weighted moving
average, 452, 480
historical volatility, 451, 480
IGARCH, 454
in transactions time, 555–558
options data and GARCH,
588–591
realised GARCH, 525–527
stochastic volatility, 434–437
Wald test, 351
Weak exogeneity, 186
Weak instruments, 261–266
White standard errors, 348