The Econometrics of Financial Markets
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The Econometrics of Financial Markets
John Y. Campbell
Andrew W. Lo
A. Craig MacKinlay
Princeton University Press
Princeton, New Jersey
List of Figures xiii
List of Tables xv
Preface xvii
1 Introduction 3
1.1 Organization of the Book . . . . . . . . . . . . . . . . . . 4
1.2 Useful Background . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Mathematics Background . . . . . . . . . . . . . . 6
1.2.2 Probability and Statistics Background . . . . . . . . 6
1.2.3 Finance Theory Background . . . . . . . . . . . . . 7
1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Prices, Returns. and Compounding . . . . . . . . . . . . . 9
1.4.1 Definitions and Conventions . . . . . . . . . . . . . 9
1.4.2 The Marginal, Conditional. and Joint Distribution
of Returns . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Market Efficiency . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 Efficient Markets and the Law of Iterated
Expectations . . . . . . . . . . . . . . . . . . . . . . 22
1.5.2 Is Market Efficiency Testable? . . . . . . . . . . . . 24
2 The Predictability of Asset Returns 27
2.1 The Random Walk Hypotheses . . . . . . . . . . . . . . . 28
2.1.1 The Random Walk 1: IID Increments . . . . . . . . 31
2.1.2 The Random Walk 2: Independent Increments . . 32
2.1.3 The Random Walk 3: Uncorrelated Increments . . 33
2.2 Tests of Random Walk 1: IID Increments . . . . . . . . . . 33
2.2.1 Traditional Statistical Tests . . . . . . . . . . . . . . 33
2.2.2 Sequences and Reversals, and Runs . . . . . . . . . 34
2.3 Tests of Random Walk 2: Independent Increments . . . . 41
2.3.1 Filter Rules . . . . . . . . . . . . . . . . . . . . . . 42
2.3.2 Technical Analysis . . . . . . . . . . . . . . . . . . . 43
2.4 Tests of Random Walk 3: Uncorrelated Increments . . . . 44
2.4.1 Autocorrelation Coefficients . . . . . . . . . . . . . 44
2.4.2 Portmanteau Statistics . . . . . . . . . . . . . . . . 47
2.4.3 Variance Ratios . . . . . . . . . . . . . . . . . . . . 48
2.5 Long-Horizon Returns . . . . . . . . . . . . . . . . . . . . 55
2.5.1 Problems with Long-Horizon Inferences . . . . . . 57
2.6 Tests For Long-Range Dependence . . . . . . . . . . . . . 59
2.6.1 Examples of Long-Range Dependence . . . . . . . 59
2.6.2 The Hurst-Mandelbrot Rescaled Range Statistic . . 62
2.7 Unit Root Tests . . . . . . . . . . . . . . . . . . . . . . . . 64
2.8 Recent Empirical Evidence . . . . . . . . . . . . . . . . . . 65
2.8.1 Autocorrelations . . . . . . . . . . . . . . . . . . . 66
2.8.2 Variance Ratios . . . . . . . . . . . . . . . . . . . . 68
2.8.3 Cross-Autocorrelations and Lead-Lag Relations . . 74
2.8.4 Tests Using Long-Horizon Returns . . . . . . . . . 78
2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Market Microstructure 83
3.1 Nonsynchronous Trading . . . . . . . . . . . . . . . . . . 84
3.1.1 A Model of Nonsynchronous Trading . . . . . . . . 85
3.1.2 Extensions and Generalizations . . . . . . . . . . . 98
3.2 The Bid-Ask Spread . . . . . . . . . . . . . . . . . . . . . . 99
3.2.1 Bid-Ask Bounce . . . . . . . . . . . . . . . . . . . . 101
3.2.2 Components of the Bid-Ask Spread . . . . . . . . . 103
3.3 Modeling Transactions Data . . . . . . . . . . . . . . . . . 107
3.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . 108
3.3.2 Rounding and Barrier Models . . . . . . . . . . . . 114
3.3.3 The Ordered Probit Model . . . . . . . . . . . . . . 122
3.4 Recent Empirical Findings . . . . . . . . . . . . . . . . . . 128
3.4.1 Nonsynchronous Trading . . . . . . . . . . . . . . 128
3.4.2 Estimating the Effective Bid-Ask Spread . . . . . . . 134
3.4.3 Transactions Data . . . . . . . . . . . . . . . . . . . 136
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4 Event-Study Analysis 149
4.1 Outline of an Event Study . . . . . . . . . . . . . . . . . . 150
4.2 An Example of an Event Study . . . . . . . . . . . . . . . . 152
4.3 Models for Measuring Normal Performance . . . . . . . . 153
4.3.1 Constant-Mean-Return Model . . . . . . . . . . . . 154
4.3.2 Market Model . . . . . . . . . . . . . . . . . . . . . 155
4.3.3 Other Statistical Models . . . . . . . . . . . . . . . 155
4.3.4 Economic Models . . . . . . . . . . . . . . . . . . . 156
Measuring and Analyzing Abnormal Returns . . . . . . . . 157
4.4.1 Estimation of the Market Model . . . . . . . . . . . 158
4.4.2 Statistical Properties of Abnormal Returns . . . . . 159
4.4.3 Aggregation of Abnormal Returns . . . . . . . . . . 160
4.4.4 Sensitivity to Normal Return Model . . . . . . . . . 162
4.4.5 CARS for the Earnings-Announcement Example . . 163
4.4.6 Inferences with Clustering . . . . . . . . . . . . . . 166
Modifying the Null Hypothesis . . . . . . . . . . . . . . . 167
Analysis of Power . . . . . . . . . . . . . . . . . . . . . . . 168
Nonparametric Tests . . . . . . . . . . . . . . . . . . . . . 172
Cross-Sectional Models . . . . . . . . . . . . . . . . . . . . 173
Further Issues . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.9.1 Role of the Sampling Interval . . . . . . . . . . . . 175
4.9.2 Inferences with Event-Date Uncertainty . . . . . . . 176
4.9.3 Possible Biases . . . . . . . . . . . . . . . . . . . . . 177
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5 The Capital Asset Pricing Model 181
Reviewof the C A M. . P . . . . . . . . . . . . . . . . . 181
Results from Efficient-Set Mathematics . . . . . . . . . . . 184
Statistical Framework for Estimation and Testing . . . . . . 188
5.3.1 Sharpe-Lintner Version . . . . . . . . . . . . . . . . 189
5.3.2 Black Version . . . . . . . . . . . . . . . . . . . . . 196
Size of Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Power of Tests . . . . . . . . . . . . . . . . . . . . . . . . . 204
Nonnormal and Non-IID Returns . . . . . . . . . . . . . . 208
Implementation of Tests . . . . . . . . . . . . . . . . . . . 211
5.7.1 Summary of Empirical Evidence . . . . . . . . . . . 211
5.7.2 Illustrative Implementation . . . . . . . . . . . . . 212
5.7.3 Unobservability of the Market Portfolio . . . . . . . 213
Cross-Sectional Regressions . . . . . . . . . . . . . . . . . 215
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6 Multifactor Pricing Models 219
6.1 Theoretical Background . . . . . . . . . . . . . . . . . . . 219
6.2 Estimation and Testing . . . . . . . . . . . . . . . . . . . . 222
6.2.1 Portfolios as Factors with a Riskfree Asset . . . . . . 223
6.2.2 Portfolios as Factors without a Riskfree Asset . . . . 224
6.2.3 Macroeconomic Variables as Factors . . . . . . . . . 226
6.2.4 Factor Portfolios Spanning the Mean-Variance
Frontier . . . . . . . . . . . . . . . . . . . . . . . . . 228
Estimation of Risk Premia and Expected Returns . . . . . 231
Selection of Factors . . . . . . . . . . . . . . . . . . . . . . 233
6.4.1 Statistical Approaches . . . . . . . . . . . . . . . . . 233
6.4.2 Number of Factors . . . . . . . . . . . . . . . . . . 238
6.4.3 Theoretical Approaches . . . . . . . . . . . . . . . 239
Empirical Results . . . . . . . . . . . . . . . . . . . . . . . 240
Interpreting Deviations from Exact Factor Pricing . . . . . 242
6.6.1 Exact Factor Pricing Models, Mean-Variance Analysis.
and the Optimal Orthogonal Portfolio . . . . . 243
6.6.2 Squared Sharpe Ratios . . . . . . . . . . . . . . . . 245
6.6.3 Implications for Separating Alternative Theories . . 246
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 251
7 Present-Value Relations 253
The Relation between Prices. Dividends. and Returns . . . 254
7.1.1 The Linear Present-Value Relation with Constant
Expected Returns . . . . . . . . . . . . . . . . . . . 255
7.1.2 Rational Bubbles . . . . . . . . . . . . . . . . . . . 258
7.1.3 An Approximate Present-Value Relation with Time-
Varying Expected Returns . . . . . . . . . . . . . . . 260
7.1.4 Prices and Returns in a Simple Example . . . . . . 264
Present-Value Relations and US Stock Price Behavior . . . 267
7.2.1 Long-Horizon Regressions . . . . . . . . . . . . . . 267
7.2.2 Volatility Tests . . . . . . . . . . . . . . . . . . . . . 275
7.2.3 Vector Autoregressive Methods . . . . . . . . . . . 279
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 286
8 Intertemporal Equilibrium Models 29 1
The Stochastic Discount Factor . . . . . . . . . . . . . . . 293
8.1.1 Volatility Bounds . . . . . . . . . . . . . . . . . . . 296
Consumption-Based Asset Pricing with Power Utility . . . . 304
8.2.1 Power Utility in a Lognormal Model . . . . . . . . . 306
8.2.2 Power Utility and Generalized Method of
Moments . . . . . . . . . . . . . . . . . . . . . . . . 314
Market Frictions . . . . . . . . . . . . . . . . . . . . . . . 314
8.3.1 Market Frictions and Hansen-Jagannathan
Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 315
8.3.2 Market Frictions and Aggregate Consumption
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
More General Utility Functions . . . . . . . . . . . . . . . 326
8.4.1 HabitFormation . . . . . . . . . . . . . . . . . . . 326
8.4.2 Psychological Models of Preferences . . . . . . . . 332
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 334
Derivative Pricing Models 339
9.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . 341
9.1.1 Constructing Brownian Motion . . . . . . . . . . . 341
9.1.2 Stochastic Differential Equations . . . . . . . . . . 346
9.2 A Brief Review of Derivative Pricing Methods . . . . . . . . 349
9.2.1 The Black-Scholes and Merton Approach . . . . . . 350
9.2.2 The Martingale Approach . . . . . . . . . . . . . . 354
9.3 Implementing Parametric Option Pricing Models . . . . . 355
9.3.1 Parameter Estimation of Asset Price Dynamics . . . 356
9.3.2 Estimating0 in the Black-Scholes Model . . . . . . 361
9.3.3 Quantifying the Precision of Option Price
Estimators . . . . . . . . . . . . . . . . . . . . . . . . 367
9.3.4 The Effects of Asset Return Predictability . . . . . . 369
9.3.5 Implied Volatility Estimators . . . . . . . . . . . . . 377
9.3.6 Stochastic Volatility Models . : . . . . . . . . . . . . 379
9.4 Pricing Path-Dependent DerivativesVia Monte Carlo Simulation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
9.4.1 Discrete Versus Continuous Time . . . . . . . . . . 383
9.4.2 How Many Simulations to Perform . . . . . . . . . 384
9.4.3 Comparisons with a Closed-Form Solution . . . . . 384
9.4.4 Computational Efficiency . . . . . . . . . . . . . . 386
9.4.5 Extensions and Limitations . . . . . . . . . . . . . . 390
9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 391
10 Fixed-Income Securities 395
10.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 396
10.1.1 Discount Bonds . . . . . . . . . . . . . . . . . . . . 397
10.1.2 Coupon Bonds . . . . . . . . . . . . . . . . . . . . 401
10.1.3 Estimating the Zero-Coupon Term Structure . . . . 409
10.2 Interpreting the Term Structure of Interest Rates . . . . . 413
10.2.1 The Expectations Hypothesis . . . . . . . . . . . . 413
10.2.2 Yield Spreads and Interest Rate Forecasts . . . . . . 418
10.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 423
1 1 TermStructure Models 427
11.1 Affine-Yield Models . . . . . . . . . . . . . . . . . . . . . . 428
11 . 1. 1 A Homoskedastic Single-Factor Model . . . . . . . 429
1 1.1.2 A Square-Root Single-Factor Model . . . . . . . . . 435
11.1.3 A Two-Factor Model . . . . . . . . . . . . . . . . . . 438
11.1.4 Beyond Affine-Yield Models . . . . . . . . . . . . . 441
11.2 Fitting Term-Structure Models to the Data . . . . . . . . . 442
11 .2.1 Real Bonds, Nominal Bonds, and Inflation . . . . . 442
11.2.2 Empirical Evidence on Affine-Yield Models . . . . 445
11.3 Pricing Fixed-Income Derivative Securities . . . . . . . . . 455
11.3.1 Fitting the Current Term Structure Exactly . . . . . 456
11.3.2 Forwards and Futures . . . . . . . . . . . . . . . . . 458
11.3.3 Option Pricing in a Term-Structure Model . . . . . 461
11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 464
12 Nonlinearities in Financial Data 467
12.1 Nonlinear Structure in Univariate Time Series . . . . . . . 468
12.1.1 Some Parametric Models . . . . . . . . . . . . . . . 470
12.1.2 Univariate Tests for Nonlinear Structure . . . . . . 475
12.2 Models of Changing Volatility . . . . . . . . . . . . . . . . 479
12.2.1 Univariate Models . . . . . . . . . . . . . . . . . . . 481
12.2.2 Multivariate Models . . . . . . . . . . . . . . . . . . 490
12.2.3 Links between First and Second Moments . . . . . 494
12.3 Nonparametric Estimation . . . . . . . . . . . . . . . . . . 498
12.3.1 Kernel Regression . . . . . . . . . . . . . . . . . . . 500
12.3.2 Optimal Bandwidth Selection . . . . . . . . . . . . 502
12.3.3 Average Derivative Estimators . . . . . . . . . . . . 504
12.3.4 Application: Estimating State-Price Densities . . . . 507
12.4 Artificial Neural Networks . . . . . . . . . . . . . . . . . . 512
12.4.1 Multilayer Perceptrons . . . . . . . . . . . . . . . . 512
12.4.2 Radial Basis Functions . . . . . . . . . . . . . . . . 516
12.4.3 Projection Pursuit Regression . . . . . . . . . . . . 518
12.4.4 Limitations of Learning Networks . . . . . . . . . . 518
12.4.5 Application: Learning the Black-Scholes Formula . 519
12.5 Overfitting and Data-Snooping . . . . . . . . . . . . . . . 523
12.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 524
Appendix 527
A.l Linear Instrumental Variables . . . . . . . . . . . . . . . . 527
A.2 Generalized Method of Moments . . . . . . . . . . . . . . 532
A.3 Serially Correlated and Heteroskedastic Errors . . . . . . . 534
A.4 GMM and Maximum Likelihood . . . . . . . . . . . . . . 536
References 541
Author Index 587
Subject Index 597
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