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A Course on Statistics for Finance

Name: A Course on Statistics for Finance
Author: Stanley L. Sclove
ISBN: 9781439892558

by Stanley L. Sclove

December 2012

Beginner

269 pages

5h 31m

English

Chapman and Hall/CRC

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Preliminaries
Preface
About the Author
Part I Introductory Concepts and Definitions
Chapter 1 Review of Basic Statistics
1.1 What Is Statistics?1.1.1 Data Are Observations1.1.2 Statistics Descriptions; Statistics Methods1.1.3 Origins of Data1.1.4 Philosophy of Data and Information1.1.4.1 Data versus Information1.1.4.2 Decisions1.2 Characterizing Data1.2.1 Types of Data1.2.1.1 Modes and Ways1.2.1.2 Types of Variables1.2.1.3 Cross-Sectional Data versus Time Series Data1.2.2 Raw Data versus Derived Data1.2.2.1 Ratios1.2.2.2 Indices1.3 Measures of Central Tendency1.3.1 Mode1.3.2 Measuring the Center of a Set of Numbers1.3.2.1 Median1.3.2.2 Quartiles1.3.2.3 Percentiles1.3.2.4 Section Exercises1.3.2.5 Mean1.3.2.6 Other Properties of the Ordinary Arithmetic Average1.3.2.7 Mean of a Distribution1.3.3 Other Kinds of Averages1.3.3.1 Root Mean Square1.3.3.2 Other Averages1.3.4 Section Exercises1.4 Measures of Variability1.4.1 Measuring Spread1.4.1.1 Positional Measures of Spread1.4.1.2 Range1.4.1.3 IQR1.4.2 Distance-Based Measures of Spread1.4.2.1 Deviations from the Mean1.4.2.2 Mean Absolute Deviation1.4.2.3 Root Mean Square Deviation1.4.2.4 Standard Deviation1.4.2.5 Variance of a Distribution1.5 Higher Moments1.6 Summarizing Distributions1.6.1 Partitioning Distributions1.6.2 Moment-Preservation Method1.7 Bivariate Data1.7.1 Covariance and Correlation1.7.1.1 Computational Formulas1.7.1.2 Covariance, Regression Cooefficient, and Correlation Coefficient1.7.2 Covariance of a Bivariate Distribution1.8 Three Variables1.8.1 Pairwise Correlations1.8.2 Partial Correlation1.9 Two-Way Tables1.9.1 Two-Way Tables of Counts1.9.2 Turnover Tables1.9.3 Seasonal Data1.9.3.1 Data Aggregation1.9.3.2 Stable Seasonal Pattern1.10 Summary1.11 Chapter Exercises1.11.1 Applied Exercises1.11.2 Mathematical Exercises1.12 BibliographyTable 1.1Table 1.2Table 1.3Table 1.4Table 1.5
Chapter 2 Stock Price Series and Rates of Return
2.1 Introduction2.1.1 Price Series2.1.2 Rates of Return2.1.2.1 Continuous ROR and Ordinary ROR2.1.2.2 Advantages of Continuous ROR2.1.2.3 Modeling Price Series2.1.3 Review of Mean, Variance, and Standard Deviation2.1.3.1 Mean2.1.3.2 Variance2.1.3.3 Standard Deviation2.2 Ratios of Mean and Standard Deviation2.2.1 Coefficient of Variation2.2.2 Sharpe Ratio2.3 Value-at-Risk2.3.1 VaR for Normal Distributions2.3.2 Conditional VaR2.4 Distributions for RORs2.4.1 t Distribution as a Scale-Mixture of Normals2.4.2 Another Example of Averaging over a Population2.4.3 Section Exercises2.5 Summary2.6 Chapter Exercises2.7 Bibliography2.8 Further ReadingTable 2.1Table 2.2
Chapter 3 Several Stocks and Their Rates of Return
3.1 Introduction3.2 Review of Covariance and Correlation3.3 Two Stocks3.3.1 RORs of Two Stocks3.3.2 Section Exercises3.4 Three Stocks3.4.1 RORs of Three Stocks3.4.2 Section Exercises3.5 m Stocks3.5.1 RORs for m Stocks3.5.2 Parameters and Statistics for m Stocks3.6 Summary3.7 Chapter Exercises3.8 Bibliography3.9 Further ReadingTable 3.1Table 3.2Table 3.3Table 3.4
Part II Regression
Chapter 4 Simple Linear Regression; CAPM and Beta
4.1 Introduction4.2 Simple Linear Regression4.2.1 Data4.2.2 An Introductory Example4.3 Estimation4.3.1 Method of Least Squares4.3.1.1 Least Squares Criterion4.3.1.2 Least Squares Estimator4.3.2 Maximum Likelihood Estimator under the Assumption of Normality*4.3.3 A Heuristic Approach4.3.3.1 Observational Equations4.3.3.2 Method of Reduction of Observations4.3.4 Means and Variances of Estimators4.3.4.1 Means of Estimators4.3.4.2 Unbiasedness4.3.4.3 Variance of the Least Squares Estimator4.3.4.4 Nonlinear and Biased Estimators4.3.5 Estimating the Error Variance4.3.5.1 Computational Formulas4.3.5.2 Decomposition of Sum of Squares4.4 Inference Concerning the Slope4.4.1 Testing a Hypothesis Concerning the Slope4.4.2 Confidence Interval4.5 Testing Equality of Slopes of Two Lines through the Origin4.6 Linear Parametric Functions4.7 Variances Dependent upon X *4.8 A Financial Application: CAPM and “Beta”4.8.1 CAPM4.8.2 “Beta”4.9 Slope and Intercept4.9.1 Model with Slope and Intercept4.9.2 CAPM with Differential Return4.10 Appendix 4A: Optimality of the Least Squares Estimator4.11 Summary4.12 Chapter Exercises4.12.1 Applied Exercises4.12.2 Mathematical Exercises4.13 Bibliography4.14 Further ReadingFigure 4.1Table 4.1Table 4.2Table 4.3Table 4.4Table 4.5
Chapter 5 Multiple Regression and Market Models
5.1 Multiple Regression Models5.1.1 Regression Function5.1.2 Method of Least Squares5.1.3 Types of Explanatory Variables5.2 Market Models5.2.1 Fama/French Three-Factor Model5.2.2 Four-Factor Model5.3 Models with Numerical and Dummy Explanatory Variables5.3.1 Two-Group Models5.3.2 Other Market Models5.3.2.1 Two Betas5.3.2.2 More Advanced Models5.4 Model Building5.4.1 Principle of Parsimony5.4.2 Model-Selection Criteria5.4.2.1 Residual Mean Square5.4.2.2 Adjusted R-Square5.4.3 Testing a Reduced Model against a Full Model5.4.4 Comparing Several Models5.4.5 Combining Results from Several Models5.5 Chapter Summary5.6 Chapter Exercises5.6.1 Exercises for Two Explanatory Variables5.6.2 Mathematical Exercises: Two Explanatory Variables5.6.3 Mathematical Exercises: Three Explanatory Variables5.6.4 Exercises on Subset Regression5.6.5 Mathematical Exercises: Subset Regression5.7 BibliographyTable 5.1Table 5.2Table 5.3

Part III Portfolio Analysis
Chapter 6 Mean-Variance Portfolio Analysis
6.1 Introduction6.1.1 Mean-Variance Portfolio Analysis6.1.2 Single-Criterion Analysis6.2 Two Stocks6.2.1 Mean6.2.2 Variance6.2.3 Covariance and Correlation6.2.4 Portfolio Variance6.2.4.1 Variance of a Sum; Variance of a Difference.6.2.4.2 Portfolio Variance6.2.5 Minimum Variance Portfolio6.3 Three Stocks6.4 m Stocks6.5 m Stocks and a Risk-Free Asset6.5.1 Admissible Points6.5.2 Capital Allocation Lines6.6 Value-at-Risk6.6.1 VaR for Normal Distributions6.6.2 Conditional VaR6.7 Selling Short6.8 Market Models and Beta6.8.1 CAPM6.8.2 Computation of Covariances under the CAPM6.8.3 Section Exercises6.9 Summary6.9.1 Rate of Return6.9.2 Bi-Criterion Analysis6.9.3 Market Models6.10 Chapter Exercises6.10.1 Exercises on Covariance and Correlation6.10.2 Exercises on Portfolio ROR6.10.3 Exercises on Three Stocks6.10.4 Exercises on Correlation and Regression6.11 Appendix 6A: Some Results in Terms of Vectors and Matrices (Optional)*6.11.1 Variates6.11.2 Vector Differentiation6.11.2.1 Some Rules for Vector Differentiation6.11.2.2 Minimum-Variance Portfolio6.11.2.3 Maximum Sharpe Ratio6.11.3 Section Exercises6.12 Appendix 6B: Some Results for the Family of Normal Distributions6.12.1 Moment Generating Function; Moments6.12.2 Section Exercises6.13 Bibliography6.14 Further ReadingFigure 6.1Table 6.1Table 6.2Table 6.3Table 6.4Table 6.5Table 6.6Table 6.7
Chapter 7 Utility-Based Portfolio Analysis
7.1 Introduction7.1.1 Background7.1.2 Types of Portfolio Analysis7.2 Single-Criterion Analysis7.2.1 Mean versus Variance Plot7.2.2 Weights on the Risk-Free and Risky Parts of the Portfolio7.2.3 Separation7.3 Summary7.4 Chapter Exercises7.5 BibliographyFigure 7.1Table 7.1
Part IV Time Series Analysis
Chapter 8 Introduction to Time Series Analysis
8.1 Introduction8.2 Control Charts8.3 Moving Averages8.3.1 Running Median8.3.2 Various Moving Averages8.3.3 Exponentially Weighted Moving Averages8.3.4 Using a Moving Average for Prediction8.3.4.1 Smoothed Value as a Predictor of the Next Value8.3.4.2 A Predictor-Corrector Formula8.3.4.3 MACD8.4 Need for Modeling8.5 Trend, Seasonality, and Randomness8.6 Models with Lagged Variables8.6.1 Lagged Variables8.6.2 Autoregressive Models8.7 Moving-Average Models8.7.1 Integrated Moving-Average Model8.7.2 Preliminary Estimate of θ8.7.3 Estimate of θ8.7.4 Integrated Moving-Average with a Constant8.8 Identification of ARIMA Models8.8.1 Pre-Processing8.8.1.1 Transformation8.8.1.2 Differencing8.8.2 ARIMA Parameters p, d, q8.8.3 Autocorrelation Function; Partial Autocorrelation Function8.9 Seasonal Data8.9.1 Seasonal ARIMA Models8.9.2 Stable Seasonal Pattern8.10 Dynamic Regression Models8.11 Simultaneous Equations Models8.12 Appendix 8A: Growth Rates and Rates of Return8.12.1 Compound Interest8.12.2 Geometric Brownian Motion8.12.3 Average Rates of Return8.12.4 Section Exercises: Exponential and Log Functions8.13 Appendix 8B: Prediction after Data Transformation8.13.1 Prediction8.13.2 Prediction after Transformation8.13.3 Unbiasing8.13.4 Application to the Log Transform8.13.5 Generalized Linear Models8.14 Appendix 8C: Representation of Time Series8.14.1 Operators8.14.2 White Noise8.14.3 Stationarity8.14.4 AR8.14.4.1 Variance8.14.4.2 Covariances and Correlations8.14.4.3 Higher-Order AR8.14.5 MA8.14.5.1 Variance8.14.5.2 Correlation8.14.5.3 Representing the Error Variables in Terms of the Observations8.14.6 ARMA8.15 Summary8.16 Chapter Exercises8.16.1 Applied Exercises8.16.2 Mathematical Exercises8.17 Bibliography8.18 Further ReadingFigure 8.1Table 8.1Table 8.2Table 8.3Table 8.4Table 8.5
Chapter 9 Regime Switching Models
9.1 Introduction9.2 Bull and Bear Markets9.2.1 Definitions of Bull and Bear Markets9.2.2 Regressions on Bull39.2.2.1 Two Betas, No Alpha9.2.2.2 Two Betas, One Alpha9.2.2.3 Two Betas, Two Alphas9.2.3 Other Models for Bull/Bear9.2.3.1 Two Means and Two Variances9.2.3.2 Mixture Model9.2.3.3 Hidden Markov Model9.2.4 Bull and Bear Portfolios9.3 Summary9.4 Chapter Exercises9.4.1 Applied Exercises9.4.2 Mathematical Exercises9.5 Bibliography9.6 Further ReadingTable 9.1Table 9.2
Appendix A Vectors and Matrices
A.1 IntroductionA.2 VectorsA.2.1 Inner Product of Two VectorsA.2.2 Orthogonal VectorsA.2.3 VariatesA.2.4 Section ExercisesA.3 MatricesA.3.1 Entries of a MatrixA.3.2 Transpose of a MatrixA.3.3 Matrix MultiplicationA.3.4 Section ExercisesA.3.5 Identity MatrixA.3.6 InverseA.3.6.1 Inverse of a MatrixA.3.6.2 Inverse of a Product of MatricesA.3.7 DeterminantA.4 Vector DifferentiationA.5 PathsA.6 Quadratic FormsA.7 EigensystemA.8 Transformation to Uncorrelated VariablesA.8.1 Covariance Matrix of a Linear Transformation of a Random VectorA.8.2 Transformation to Uncorrelated VariablesA.8.3 Transformation to Uncorrelated Variables with Variances Equal to OneA.9 Statistical DistanceA.10 Appendix ExercisesA.11 BibliographyA.12 Further Reading
Appendix B Normal Distributions
B.1 Some Results for Univariate Normal DistributionsB.1.1 DefinitionsB.1.2 Conditional ExpectationB.1.3 Tail Probability ApproximationB.2 Family of Multivariate Normal DistributionsB.3 Role of D-SquareB.4 Bivariate Normal DistributionsB.4.1 Shape of the p.d.fB.4.2 Conditional Distribution of Y Given XB.4.3 Regression FunctionB.5 Other Multivariate DistributionsB.6 SummaryB.6.1 ConceptsB.6.2 MathematicsB.7 Appendix B ExercisesB.7.1 Applied ExercisesB.7.2 Mathematical ExercisesB.8 BibliographyB.9 Further Reading
Appendix C Lagrange Multipliers
C.1 NotationC.2 Optimization ProblemC.3 BibliographyC.4 Further Reading
Appendix D Abbreviations and Symbols
D.1 AbbreviationsD.1.1 StatisticsD.1.2 GeneralD.1.3 FinanceD.2 SymbolsD.2.1 StatisticsD.2.2 Finance

Content preview from A Course on Statistics for Finance

Chapter 9

Regime Switching Models

9.1 Introduction

(Parts of this chapter are more advanced than most of the rest of the book.) This chapter discusses states or phases of the market and the economy. The states of the market are conventionally called Bull and Bear. The states of the economy are referred to as recession, recovery, expansion, contraction. Questions arise as to how many such states to use, and whether the nature of the states detected by statistical methods corresponds to conventional notions about the phases of the market or the economy.

The models considered involve a series {Yt, St}, where Y is observed and S is the state, or label. For example, Yt may be the rate of return of the S&P500 index in month t, and St may be 1 or 0, ...

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A Course on Statistics for Finance

by Stanley L. Sclove

Regime Switching Models

9.1 Introduction

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