INSEAD Library

Includes bibliographical references and index

Digitized

Modern Linear and Nonlinear Econometrics Contents I Linear and Nonlinear Econometric Inference: Estimation and Testing 1 1 Estimation in Linear and Nonlinear Models 5 1.1 Introduction to linear and nonlinear optimization ............................................ 5 1.1.1 Nonlinear least squares ......................................................................... 6 1.1.2 Nonlinear maximum likelihood ............................................................. 7 1.2 Econometric methods of nonlinear estimation ............................................... 10 1.2.1 Gauss method ...................................................................................... 10 1.2.2 Method of scoring ............................................................................... 13 1.2.3 Quasi-Gauss method ........................................................................... 13 1.2.4 Marquardt method ............................................................................... 14 1.2.5 Quadratic hill-climbing method .......................................................... 14 1.2.6 Concluding remarks ............................................................................ 14 2 Generalized Method of Moments 19 2.1 Endogeneity bias: two examples ...................................................................... 19 2.2 The method of moments ................................................................................... 21 2.2.1 Some examples .................................................................................... 22 2.2.2 MoM encompasses many estimation methods .................................... 23 2.3 Generalized method of moments ..................................................................... 27 2.3.1 Example: GMM estimators based on Euler equations . ...................... 28 2.3.2 Three empirical examples ................................................................... 30 2.3.3 Exercises .............................................................................................. 31 2.4 Some concluding remarks ................................................................................ 32 3 Testing in Linear and Nonlinear Models 33 3.1 Nested model tests ............................................................................................ 34 3.1.1 Lagrange multiplier test ...................................................................... 34 3.1.2 Wald test .............................................................................................. 35 3.1.3 Likelihood ratio test ............................................................................ 35 3.1.4 Confidence intervals and hypotheses tests .......................................... 36 3.1.5 Recapitulating and extending nested hypotheses ................................ 38 3.1.6 Examples of nested model tests .......................................................... 39 3.2 Nonnested model tests ...................................................................................... 47 ix x CONTENTS 3.2.1 Cox test for two different linear models ....................................... 48 3.2.2 Cox test for two different nonlinear models ................................ 50 3.3 Theoretical and empirical exercises ........................................................... 53 II Time Series Analysis 4 A Typology of Dynamic Models 4.1 Autoregressive distributed lag models ....................................................... 4.2 The partial adjustment model ...................................................................... 4.3 The error correction mechanism ................................................................. 55 59 59 60 63 5 Univariate ARIMA Models 65 5.1 Stationary processes ..................................................................................... 66 5.2 Autoregressive (time series) processes ...................................................... 68 5.2.1 Stationary autoregressive processes .............................................. 68 5.2.2 Estimation and identification of AR processes ............................ 69 5.3 Moving average (time series) processes .................................................... 72 5.3.1 Definition ......................................................................................... 73 5.3.2 Identification of an MA process .................................................... 74 5.3.3 Parameter estimation of an MA(q) process ................................. 75 5.4 ARMA models .............................................................................................. 75 5.4.1 Stationarity restrictions .................................................................. 75 5.4.2 The ARMA (1,1) model ................................................................ 76 5.5 Testing for unit roots .................................................................................... 79 5.5.1 Testing for unit roots in a first order autoregressive model 79 5.5.2 Testing for unit roots in higher order AR models ....................... 83 5.5.3 Multiple unit roots ........................................................................... 85 5.5.4 Seasonal unit roots .......................................................................... 88 5.6 ARIMA models ............................................................................................. 89 5.7 Box-Jenkins approach for ARIMA models................................................ 90 5.7.1 Identification of a tentative time series model ............................ 90 5.7.2 Estimation of a time series model.................................................. 91 5.7.3 Diagnostic Checking ....................................................................... 94 5.8 Time series with aberrant observations ..................................................... 98 5.8.1 Definition and problems ................................................................. 98 5.8.2 Testing for and dealing with aberrant observations . . . ........... 101 5.8.3 Detecting and estimating aberrant observations ........................ 103 5.9 Forecasting of ARIMA models ................................................................ 104 5.9.1 Forecasting AR processes 105 5.9.2 Forecasting MA processes ........................................................... 108 5.9.3 Forecasting ARMA processes ..................................................... 109 5.9.4 Forecasting an ARIMA process .................................................. 111 5.9.5 Ad hoc forecasting methods ....................................................... 113 5.9.6 Forecasts with density functions ................................................. 117 5.10 Cases and exercises about ARIMA models .......................................... 121 CONTENTS 5.10.1 5.10.2 5.10.3 5.10.4 x i Theoretical exercises ................................................................. 121 Solved (empirical) cases ............................................................ 129 Empirical exercises .................................................................... 148 Unsolved cases ............................................................................ 151 153 153 153 154 156 157 157 159 161 165 166 166 170 178 181 181 185 186 187 188 188 189 191 191 6 Cointegration and Transfer Functions 6.1 Cointegration ............................................................................................ 6.1.1 A simple example ......................................................................... 6.1.2 Definition and properties ............................................................. 6 . 2 Causality ................................................................................................... 6.3 Transfer function modeling ...................................................................... 6.3.1 Single output single input models .............................................. 6.3.2 Intervention analysis .................................................................... 6.3.3 Theoretical and empirical examples and exercises .................. 7 Multivariate Time Series 7.1 Vector autoregressive models .................................................................. 7.1.1 A simple bivariate VAR(1) ........................................................... 7.1.2 V AR(p) models ............................................................................. 7.1.3 Structural VAR models ................................................................. 7.2 V ARM models ........................................................................................... 7.2.1 Multivariate cointegration ........................................................... 7.2.2 Multivariate causality .................................................................. 7.2.3 Dynamic simultaneous equations models .................................. 7.2.4 D S E M s and SVARs ..................................................................... 7.3 Exercises and cases ................................................................................... 7.3.1 Solved theoretical exercises ........................................................ 7.3.2 Solved empirical exercise ............................................................ 7.3.3 Solved empirical cases ................................................................. 7.3.4 Unsolved exercises ....................................................................... 8 Varying Parameter Models 193 8.1 Regime switching models ............................................................................ 193 8.1.1 Specification and estimation of regime switching models based on observables ................................................................ 194 8.1.2 Specification and estimation of regime switching models based on unobservables ............................................................ 195 8.1.3 Solved empirical case for regime switching models . . . . 198 8.2 Volatility modeling .................................................................................... 199 8.2.1 Univariate A R C H models ................................................... 203 8.2.2 The symmetric LARCH class of models ..................................... 204 8.2.3 Other functional forms of GARCH models ................................ 216 8.2.4 Multivariate GARCH processes ................................................... 221 8.2.5 Exercises and cases ...................................................................... 229 xii CONTENTS III Categorical and Limited Dependent Variables 237 9 Discrete Choice Models 241 9.1 Binary choice models ................................................................................. 241 9.1.1 Regression approach ..................................................................... 242 9.1.2 Repeated observations for discrete choice ................................. 243 9.1.3 Marginal effects (or `slopes') ....................................................... 245 9.1.4 Forecasts ......................................................................................... 246 9.1.5 No repeated observations ............................................................. 246 9.1.6 Index function models: latent variables representation . .......... 247 9.2 Multiple response models .......................................................................... 248 9.2.1 Ordered response models ............................................................. 248 9.2.2 Multinomial models ...................................................................... 249 9.3 Cases and exercises .................................................................................... 251 9.3.1 Solved cases ................................................................................... 251 9.3.2 Exercises ........................................................................................ 255 10 Limited Responses, Duration and Count Data 257 10.1 Censoring and truncation ......................................................................... 257 10.1.1 Problems raised by censoring .................................................... 257 10.1.2 Tobit models for censoring......................................................... 258 10.2 Models for duration data ......................................................................... 259 10.2.1 Parametric models of duration .................................................. 260 10.3 Count data ................................................................................................. 262 10.3.1 Poisson model .............................................................................. 263 10.3.2 Negative binomial model ............................................................ 265 10.3.3 Zero-inflated count models ........................................................ 265 10.4 Exercises .................................................................................................... 266 IV Panel Data Analysis 267 11 Linear Panel Data Models 271 11.1 Panel data models with constant coefficients ....................................... 273 11.2 Intercepts varying over individuals ........................................................ 274 11.2.1 pi fixed ........................................................................................... 275 11.2.2 µi random ....................................................................................... 278 11.2.3 Testing for random effects ......................................................... 282 11.2.4 Random effects versus fixed effects ......................................... 283 11.3 Intercepts varying over individuals and time ........................................ 284 11.3.1 µi and At fixed: dummy variables model .................................... 285 11.3.2 µi and At random: error components model ............................... 286 11.3.3 Specification tests ....................................................................... 288 11.3.4 Fixed or random effects? ............................................................ 288 11.4 All coefficients varying over individuals .............................................. 289 11.4.1 ßk, fixed: SUR .............................................................................. 289 CONTENTS xiii 11.4.2 ß k i random: Swamy's random coefficient model ..................... 293 11.5 All coefficients vary over individuals and time ................................... 295 11.5.1 Fixed coefficients ........................................................................ 296 11.5.2 Stochastic coefficients: Hsiao's model ....................................... 296 11.6 Advantages of panel data reconsidered ................................................ 297 11.7 Incomplete panels and selection bias .................................................... 300 11.7.1 Incomplete panels ........................................................................ 301 11.7.2 Attrition and selection bias ......................................................... 302 11.8 Dynamic linear panel data models ......................................................... 303 11.9 Empirical cases and exercises ................................................................ 309 11.9.1 Case 1. Investment and market value of firms ......................... 309 11.9.2 Case 2. Money flows and the performance of hedge funds 315 11.9.3 Exercises ....................................................................................... 319 12 Nonlinear Panel Data Models 321 12.1 FE estimation for logit and probit ......................................................... 321 12.1.1 Logit models and conditional MLE ........................................... 321 12.1.2 FEs in probit models and a semiparametric estimator . . 323 12.2 RE estimation ............................................................................................ 324 12.2.1 ML estimation .............................................................................. 324 12.3 Panel count data models ....................................................................... . 325 12.3.1 Fixed Effects Poisson models .................................................... 325 12.3.2 Random Effects Poisson models ................................................. 325 12.4 Cases and exercises ................................................................................. 326 12.4.1 Case 1. Firms' decision to report their RandD expenditures 326 12.4.2 Case 2. Count data model for patents-RandD relationship . 328 12.4.3 Exercise ........................................................................................ 328 A Nonlinear Optimization and Estimation A.1 General nonlinear optimization problem ............................................... A.2 Gradient methods of nonlinear estimation ............................................. A.2.1 Steepest descent ............................................................................ A.2.2 Newton (-Raphson) ....................................................................... A.2.3 Quasi-Newton procedures ............................................................ A.3 Constrained nonlinear optimization methods ........................................ A.3.1 Reparameterization ....................................................................... A.3.2 Transformation of the objective function .................................. A.3.3 Dual methods ................................................................................. A.3.4 Extended (augmented) Lagrange function method . . . ............ B Mathematical Formulation of GMM B.1 Redefining GMM ...................................................................................... B.2 Assumptions and properties ..................................................................... B.3 Nonlinear two stage least squares ........................................................... C Stability Criteria for AR(p) Models 333 333 334 335 338 339 343 343 344 345 345 349 349 350 351 353 xiv CONTENTS 355 D MLE of the RSM with Endogenous Prices E Volatility Modeling 359 E.1 Detection and reduction of additive outliers ................................................... 359 E.2 Forecasting in G ARC H(p, q) models ............................................................. 360 E.3 Generalized exponential distribution 360 E.4 Generalized Student t-distribution .................................................................. 361 E.5 Aggregation of GAR CH processes ................................................................. 361