Table Of ContentCheng-Few Lee · Hong-Yi Chen · John Lee
Financial
Econometrics,
Mathematics and
Statistics
Theory, Method and Application
Financial Econometrics, Mathematics
and Statistics
Cheng-Few Lee Hong-Yi Chen
(cid:129) (cid:129)
John Lee
Financial Econometrics,
Mathematics
and Statistics
Theory, Method and Application
123
Cheng-Few Lee Hong-Yi Chen
Department ofFinance andEconomics Department ofFinance
RutgersBusiness School National Chengchi University
RutgersUniversity Taipei, Taiwan
Piscataway, NJ,USA
JohnLee
Centerfor PBBEF Research
Morris Plains,NJ, USA
ISBN978-1-4939-9427-4 ISBN978-1-4939-9429-8 (eBook)
https://doi.org/10.1007/978-1-4939-9429-8
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Preface
Wedrawuponouryearsofteaching,research,andpracticeonthesubjectsof
financial econometrics, mathematics and statistics for this textbook. Overall,
our goal is to provide an advanced level book that reviews, discusses, and
integrates financial econometrics, mathematics and statistics.
We focus on five principles to frame our presentation of this book:
(1) To discuss the basic methodology of financial econometrics, mathe-
matics and statistics,
(2) To show how econometric methodologies can be used in finance and
accounting-related research, which includes single equation, multiple
regression, simultaneous regression, panel data analysis, time-series
analysis, spectral analysis, nonparametric analysis, semiparametric
analysis, GMM analysis, and other methods,
(3) To show how financial mathematics such as Itô’s calculus is important
to derive the intertemporal capital asset pricing model and option
pricing model,
(4) To demonstrate how statistics distribution, such as normal distribution,
stabledistribution,andlognormaldistribution,hasbeenusedinresearch
related to portfolio theory and risk management,
(5) To show how binomial distribution, lognormal distribution, noncentral
chi-squaredistribution,Poissondistribution,andothershavebeenused
in studies related to option and futures.
In order to comprehend this book, the reader needs two semesters of
econometrics, two semesters of mathematical statistics, and one semester of
multivariate statistics.
We divide this book into four parts: Regression and Financial Econo-
metrics; Time-Series Analysis; Statistical Distributions, Option Pricing
Model, and Risk Management; and Statistics, Itô’s Calculus, and Option
Pricing Model.
PART I: Regression and Financial Econometrics
Therearesevenchaptersinthispart.InChap.2,wediscusstheassumptions
of the multiple regression model, estimated parameters of the multiple
regression model, the standard error of the residual estimate, and the
coefficient of determination. We also investigate tests on sets and individual
v
vi Preface
regressioncoefficientsandtheconfidenceintervalforthemeanresponseand
prediction interval for the individual response.
In Chap. 3, we discuss various topics associated with the regression
analysis, including multicollinearity, heteroscedasticity, autocorrelation,
modelspecificationandspecificationbiasoftheregressionmodel,nonlinear
regression models, lagged dependent variables in the regression model,
dummy variables in the regression model, and regression model with inter-
action variables. We also apply the regression approach to investigate the
effect of alternative business strategies and apply the logistic regression
model to credit risk analysis.
In Chap. 4, we extend single-equation models to simultaneous equation
models. Specifically, we discuss simultaneous equation system, two-stage
least squares method, and three-stage least squares method. In Chap. 5, we
discuss an econometric approach to financial analysis, planning, and fore-
casting. The issue of simultaneity and the dynamics of corporate-budgeting
decisions will be explored by using finance theory. We also investigate the
interrelationships among the programming, the simultaneous equations, and
the econometrics approaches.
Chapter 6 addresses one of the important issues related to panel data
analysis. We introduce the dummy variable technique and the error com-
ponentmodelforanalyzingpooleddata.Weinvestigatethepossibleimpacts
of firm effect and time effect on choosing the optimal functional form of a
financial research study.In this chapter, we also discuss thecriteria ofusing
fixed effects or random effects approach.
Chapter 7 discusses how errors-in-variables estimation methods are used
infinanceresearch.Weshowhowerrors-in-variablesproblemscanaffectthe
estimators of the linear regression model, as well as discuss the effects they
haveontheempiricalresearchcostofcapital,assetpricing,capitalstructure,
and investment decision. Chapter 8 provides three alternative
errors-in-variables estimation models in testing the capital asset pricing
model. Specifically, we present three alternative correction methods for the
errors-in-variables problem. In Chap. 9, we discuss the issue of spurious
regression and data mining in both conditional asset pricing models and
simple predictive regression. We also discuss the impact of spurious
regression and data mining on conditional asset pricing.
PART II: Time-Series Analysis and Its Applications
The purpose of Chap. 10 is to describe the components of time-series
analyses and to discuss alternative methods of economic and business fore-
casting in terms of time-series data. Specifically, we discuss a classical
description of three time-series components, the moving-average and sea-
sonally adjusted time series, linear and log-linear time trend regressions,
exponential smoothing and forecasting, autoregressive forecasting model,
ARIMA model, and composite forecasting.
InChap.11,weattempttoachievetwogoals.First,wepresentalternative
theories for deriving optimal hedge ratios. We discuss various estimation
methodsandtherelationshipamong lengths ofhedginghorizon,maturity of
Preface vii
futurescontract,datafrequency,andhedgingeffectiveness.Second,weshow
how SAS program can be used to estimate hedge ratio in terms of ARCH
method, GARCH method, EGARCH method, GJR-GARCH method, and
TGARCH method.
PART III: Statistical Distributions, Option Pricing Model and Risk
Management
Statistical distributions such as binomial distribution, multinomial distribu-
tion,normaldistribution,lognormaldistribution,Poissondistribution,central
chi-square distribution, noncentral chi-square distribution, copula distribu-
tion, nonparametric distribution, and other distributions are important in
financeresearch.InChap.12,wewilldiscusshowbinomialandmultinomial
distribution canbe usedtoderivetheoptionpricing model. InChap.13,we
show how to use two alternative binomial option pricing model approaches
to derive Black–Scholes option pricing model.
In Chap. 14, we will discuss how normal and lognormal distribution can
be used to derive the option pricing model. In Chap. 15, we will show how
copuladistributioncanbeusedtodocreditriskanalysis.InChap.16,wewill
show how multivariate analyses such as factor analysis and discriminant
analysis can be used to do financial rating analysis. In Part IV, we will
continue to discuss how statistics distribution can be used to derive option
pricingmodel.Inaddition,wewillalsoshowhowItô’scalculuscanbeused
to derive option pricing model.
PART IV: Statistics, Itô’s Calculus, and Option Pricing Model
In Chap. 17, we will show how characteristic function and noncentral
chi-square can be used to analyze stochastic volatility option pricing model.
In Chap. 18, we will discuss alternative methods to estimate implied vari-
ance. In Chap. 19, we will show the numerical valuation of Asian options
with higher moments in the underlying distribution. Both European and
Americanoptionswillbediscussedinthischapter.InChap.20,wewillfirst
reviewItôLemmaandstochasticdifferentialequation,andthenwewillshow
howthismathematicaltechniquecanbeusedtoderiveoptionpricingmodel.
InChap.21,wewilldiscusstherelationshipbetweenbinomialoptionpricing
model and Black–Scholes option pricing model. In addition, we also show
how to use stochastic calculus to derive Black–Scholes model in detail. In
Chap. 22, we will show how to use noncentral chi-square distribution to
derive constant elasticity of variance option pricing model. In Chap. 23, we
will discuss option pricing and hedging performance under stochastic
volatilityandstochasticinterestrates.Finally,inChap.24,wewillshowhow
nonparametric distribution can be used to derive option bounds. Some
empirical studies or option bounds are also provided.
ThistextbookcanbeusedforthequantitativefinanceprogramandPh.D.
programs ineconomics, statistics,andfinance. It demonstrates how toapply
different econometrics and statistical methods in finance research. In
viii Preface
addition, applications of Itô’s calculus in deriving option pricing model are
also discussed in some detail.
Itiswellknownthatfinancialeconometrics,mathematicsandstatisticsare
three of the most important tools to solve theoretical and practical issues of
quantitative finance. This textbook uses real-world data to show how these
three quantitative tools can be used to solve quantitative finance issues in
assetpricing,optionpricingmodels,riskmanagement,andotherquantitative
finance issues. This textbook uses a traditional approach by combining
researchpapers from journals, handbooks, andtextbooks tostreamline these
topics. We take advantage of using our own research papers, edited hand-
book, and textbook to formulate a meaningful, unique, comprehensive
textbook.WeheavilydrawuponHandbookofQuantitativeFinanceandRisk
Management (Springer 2009), Statistics for Business and Financial Eco-
nomics, 3rd ed. (Springer 2013), and Essentials of Excel, Excel VBA, SAS
and Minitab for Statistical and Financial Analyses (Springer 2016), as well
as Review of Quantitative Finance and Accounting and other journals with
which we have published relevant papers.
Note that this textbook is intended to be used in its entirety instead of
chapterbychapter.Readers may findtheaforementioned Springervolumes’
useful references during the learning process. For example, there areseveral
chapterswhereinwedrawfromHandbookofQuantitativeFinanceandRisk
Management—here, the reader can refer to the handbook. Similarly, Chaps.
2 and 3 are expanded versions of Statistics for Business and Financial
Economics, 3rd ed.
There are undoubtedly some errors in the finished product such as con-
ceptual, grammatical, or methodological. We would like to invite readers to
send their suggestions, comments, criticisms, and corrections to the author,
Professor Cheng F. Lee at the Department of Finance and Economics, Rut-
gersUniversityat100RockafellerRoad,Room5188,Piscataway,NJ08854.
Alternatively,readerscansendthisinformationbyemailtoeitherChengFew
Lee(cfl[email protected])orHong-YiChen([email protected]).
Piscataway, USA Cheng-Few Lee
Taipei, Taiwan Hong-Yi Chen
Morris Plains, USA John Lee
May 2019
Contents
1 Introduction to Financial Econometrics, Mathematics,
and Statistics . .... ..... .... .... .... .... .... ..... .... 1
1.1 Introduction. ..... .... .... .... .... .... ..... .... 2
1.2 Regression and Financial Econometrics .... ..... .... 2
1.2.1 Single-Equation Regression Methods.... .... 2
1.2.2 Simultaneous Equation Models ... ..... .... 4
1.2.3 Panel Data Analysis.... .... .... ..... .... 4
1.2.4 Alternative Methods to Deal
with Measurement Error .... .... ..... .... 4
1.2.5 Time-Series Analysis... .... .... ..... .... 5
1.3 Financial Statistics. .... .... .... .... .... ..... .... 5
1.3.1 Statistical Distributions . .... .... ..... .... 5
1.3.2 Principle Components and Factor Analysis ... 6
1.3.3 Nonparametric and
Semiparametric Analyses.... .... ..... .... 6
1.3.4 Cluster Analysis... .... .... .... ..... .... 6
1.4 Applications of Financial Econometrics, Mathematics
and Statistics..... .... .... .... .... .... ..... .... 6
1.4.1 Asset Pricing . .... .... .... .... ..... .... 6
1.4.2 Corporate Finance . .... .... .... ..... .... 6
1.4.3 Financial Institution.... .... .... ..... .... 7
1.4.4 Investment and Portfolio Management... .... 7
1.4.5 Option Pricing Model .. .... .... ..... .... 7
1.4.6 Futures and Hedging ... .... .... ..... .... 7
1.4.7 Mutual Fund . .... .... .... .... ..... .... 7
1.4.8 Credit Risk Modeling... .... .... ..... .... 7
1.4.9 Other Applications. .... .... .... ..... .... 7
1.5 Overall Discussion of This Book . .... .... ..... .... 8
1.5.1 Regression and Financial Econometrics.. .... 8
1.5.2 Time-Series Analysis and Its Application .... 9
1.5.3 Statistical Distributions and Option Pricing
Model .. .... .... .... .... .... ..... .... 9
1.5.4 Statistics, Itô’s Calculus and Option Pricing
Model .. .... .... .... .... .... ..... .... 10
ix
x Contents
1.6 Conclusion . ..... .... .... .... .... .... ..... .... 10
Appendix: Keywords for Chaps. 2–24 ... .... .... ..... .... 11
Bibliography .. .... ..... .... .... .... .... .... ..... .... 12
Part I Regression and Financial Econometrics
2 Multiple Linear Regression... .... .... .... .... ..... .... 19
2.1 Introduction. ..... .... .... .... .... .... ..... .... 20
2.2 The Model and Its Assumptions.. .... .... ..... .... 20
2.3 Estimating Multiple Regression Parameters.. ..... .... 23
2.4 The Residual Standard Error and the Coefficient
of Determination.. .... .... .... .... .... ..... .... 24
2.5 Tests on Sets and Individual Regression Coefficients ... 26
2.6 Confidence Interval for the Mean Response
and Prediction Interval for the Individual Response .... 30
2.7 Business and Economic Applications .. .... ..... .... 33
2.8 Using Computer Programs to Do Multiple Regression
Analyses... ..... .... .... .... .... .... ..... .... 39
2.8.1 SAS Program for Multiple Regression
Analysis. .... .... .... .... .... ..... .... 39
2.9 Conclusion . ..... .... .... .... .... .... ..... .... 47
Appendix 1: Derivation of the Sampling Variance
of the Least Squares Slope Estimations... .... .... ..... .... 48
Appendix 2: Cross-sectional Relationship Among Price Per
Share, Dividend Per Share, and Return Earning Per Share........ 49
Bibliography .. .... ..... .... .... .... .... .... ..... .... 52
3 Other Topics in Applied Regression Analysis .... ..... .... 55
3.1 Introduction. ..... .... .... .... .... .... ..... .... 56
3.2 Multicollinearity .. .... .... .... .... .... ..... .... 57
3.3 Heteroscedasticity. .... .... .... .... .... ..... .... 59
3.4 Autocorrelation... .... .... .... .... .... ..... .... 64
3.5 Model Specification and Specification Bias.. ..... .... 70
3.6 Nonlinear Models. .... .... .... .... .... ..... .... 74
3.7 Lagged Dependent Variables. .... .... .... ..... .... 79
3.8 Dummy Variables. .... .... .... .... .... ..... .... 89
3.9 Regression with Interaction Variables.. .... ..... .... 92
3.10 Regression Approach to Investigating the Effect
of Alternative Business Strategies. .... .... ..... .... 96
3.11 Logistic Regression and Credit Risk Analysis:
Ohlson’s and Shumway’s Methods for Estimating
Default Probability .... .... .... .... .... ..... .... 96
3.12 Conclusion . ..... .... .... .... .... .... ..... .... 100
Appendix 1: Dynamic Ratio Analysis.... .... .... ..... .... 100
Appendix 2: Term Structure of Interest Rate... .... ..... .... 100
Appendix 3: Partial Adjustment Dividend Behavior
Model ... .... .... ..... .... .... .... .... .... ..... .... 102
Appendix 4: Logistic Model and Probit Model. .... ..... .... 108
