Statistical and econometric methods for transportation data analysis. PDF
Preview Statistical and econometric methods for transportation data analysis.
Statistical and Econometric Methods for Transportation Data Analysis Chapman & HALL/CRC Interdisciplinary Statistics Series Series editors: N. Keiding, B. J. T. Morgan, C. K. Wikle, P. van der Heijden Recently Published Titles Correspondence Analysis in Practice, Third Edition M. Greenacre Statistics of Medical Imaging T. Lei Capture-Recapture Methods for the Social and Medical Sciences D. Böhning, P. G. M. van der Heijden, and J. Bunge The Data Book Collection and Management of Research Data Meredith Zozus Modern Directional Statistics C. Ley and T. Verdebout Survival Analysis with Interval-Censored Data A Practical Approach with Examples in R, SAS, and Bugs K. Bogaerts, A. Komarek, and E. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface........................................................................................................................................... xiii Authors........................................................................................................................................ xvii Section I Fundamentals 1. Statistical Inference I: Descriptive Statistics....................................................................3 1.1 Measures of Relative Standing .....................................................................................3 1.2 Measures of Central Tendency......................................................................................4 1.3 Measures of Variability..................................................................................................5 1.4 Skewness and Kurtosis..................................................................................................8 1.5 Measures of Association.............................................................................................. 10 1.6 Properties of Estimators............................................................................................... 12 1.6.1 Unbiasedness..................................................................................................... 13 1.6.2 Efficiency............................................................................................................ 13 1.6.3 Consistency........................................................................................................ 14 1.6.4 Sufficiency.......................................................................................................... 15 1.7 Methods of Displaying Data ....................................................................................... 15 1.7.1 Histograms ........................................................................................................ 15 1.7.2 Ogives................................................................................................................. 16 1.7.3 Box Plots............................................................................................................. 16 1.7.4 Scatter Diagrams............................................................................................... 17 1.7.5 Bar and Line Charts.......................................................................................... 17 2. Statistical Inference II: Interval Estimation, Hypothesis Testing, and Population Comparisons Descriptive Statistics.............................................................21 2.1 Confidence Intervals..................................................................................................... 21 2.1.1 Confidence Interval for μ with Known σ2.....................................................22 2.1.2 Confidence Interval for the Mean with Unknown Variance......................23 2.1.3 Confidence Interval for a Population Proportion......................................... 24 2.1.4 Confidence Interval for the Population Variance.........................................25 2.2 Hypothesis Testing.......................................................................................................25 2.2.1 Mechanics of Hypothesis Testing................................................................... 26 2.2.2 Formulating One- and Two-Tailed Hypothesis Tests..................................28 2.2.3 The p-Value of a Hypothesis Test....................................................................30 2.3 Inferences Regarding a Single Population ................................................................ 31 2.3.1 Testing the Population Mean with Unknown Variance.............................. 31 2.3.2 Testing the Population Variance..................................................................... 32 2.3.3 Testing for a Population Proportion............................................................... 32 2.4 Comparing Two Populations ......................................................................................33 2.4.1 Testing Differences between Two Means: Independent Samples..............33 2.4.2 Testing Differences between Two Means: Paired Observations................36 2.4.3 Testing Differences between Two Population Proportions ........................ 37 2.4.4 Testing the Equality of Two Population Variances......................................38 v vi Contents 2.5 Nonparametric Methods ........................................................................................... 39 2.5.1 The Sign Test....................................................................................................40 2.5.2 The Median Test..............................................................................................43 2.5.3 The Mann–Whitney U Test ...........................................................................44 2.5.4 The Wilcoxon-Signed Rank Test for Matched Pairs .................................. 47 2.5.5 The Kruskal–Wallis Test................................................................................48 2.5.6 The Chi-Square Goodness-of-Fit Test.......................................................... 49 Section II Continuous Dependent Variable Models 3. Linear Regression.................................................................................................................55 3.1 Assumptions of the Linear Regression Model.......................................................55 3.1.1 Continuous Dependent Variable Y...............................................................56 3.1.2 Linear-in-Parameters Relationship between Y and X...............................56 3.1.3 Observations Independently and Randomly Sampled.............................56 3.1.4 Uncertain Relationship between Variables................................................. 57 3.1.5 Disturbance Term Independent of X and Expected Value Zero.............. 57 3.1.6 Disturbance Terms Not Autocorrelated...................................................... 57 3.1.7 Regressors and Disturbances Uncorrelated................................................ 57 3.1.8 Disturbances Approximately Normally Distributed ................................58 3.1.9 Summary..........................................................................................................58 3.2 Regression Fundamentals ......................................................................................... 59 3.2.1 Least Squares Estimation...............................................................................60 3.2.2 Maximum Likelihood Estimation................................................................63 3.2.3 Properties of OLS and MLE Estimators.......................................................64 3.2.4 Inference in Regression Analysis.................................................................65 3.3 Manipulating Variables in Regression..................................................................... 69 3.3.1 Standardized Regression Models................................................................. 69 3.3.2 Transformations .............................................................................................. 70 3.3.3 Indicator Variables.......................................................................................... 71 3.4 Estimate a Single Beta Parameter.............................................................................72 3.5 Estimate Beta Parameter for Ranges of the Variable.............................................72 3.6 Estimate a Single Beta Parameter for m − 1 of the m Levels of the Variable......73 3.6.1 Interactions in Regression Models...............................................................73 3.7 Checking Regression Assumptions ......................................................................... 76 3.7.1 Linearity........................................................................................................... 76 3.7.2 Homoscedastic Disturbances........................................................................ 79 3.7.3 Uncorrelated Disturbances............................................................................ 81 3.7.4 Exogenous Independent Variables ............................................................... 81 3.7.5 Normally Distributed Disturbances............................................................83 3.8 Regression Outliers ....................................................................................................86 3.8.1 The Hat Matrix for Identifying Outlying Observations...........................86 3.8.2 Standard Measures for Quantifying Outlier Influence............................. 87 3.8.3 Removing Influential Data Points from the Regression............................88 3.9 Regression Model Goodness-of-Fit Measures........................................................92 3.10 Multicollinearity in the Regression.......................................................................... 96 Contents vii 3.11 Regression Model-Building Strategies..................................................................... 97 3.11.1 Stepwise Regression ..................................................................................... 97 3.11.2 Best Subsets Regression................................................................................ 98 3.11.3 Iteratively Specified Tree-Based Regression.............................................. 98 3.12 Estimating Elasticities................................................................................................ 98 3.13 Censored Dependent Variables—Tobit Model...................................................... 101 3.14 Box–Cox Regression.................................................................................................. 104 4. Violations of Regression Assumptions..........................................................................105 4.1 Zero Mean of the Disturbances Assumption........................................................ 105 4.2 Normality of the Disturbances Assumption ........................................................ 106 4.3 Uncorrelatedness of Regressors and Disturbances Assumption....................... 106 4.4 Homoscedasticity of the Disturbances Assumption ........................................... 109 4.4.1 Detecting Heteroscedasticity..................................................................... 110 4.4.2 Correcting for Heteroscedasticity............................................................. 111 4.5 No Serial Correlation in the Disturbances Assumption ..................................... 114 4.5.1 Detecting Serial Correlation ...................................................................... 116 4.5.2 Correcting for Serial Correlation .............................................................. 117 4.6 Model Specification Errors ...................................................................................... 120 5. Simultaneous Equation Models ......................................................................................123 5.1 Overview of the Simultaneous Equations Problem............................................. 123 5.2 Reduced Form and the Identification Problem..................................................... 124 5.3 Simultaneous Equation Estimation........................................................................ 125 5.3.1 Single Equation Methods ........................................................................... 126 5.3.2 System Equation Methods ......................................................................... 126 5.4 Seemingly Unrelated Equations ............................................................................. 130 5.5 Applications of Simultaneous Equations to Transportation Data ..................... 133 Appendix 5A: A Note on Generalized Least Squares Estimation ................................ 133 6. Panel Data Analysis ...........................................................................................................135 6.1 Issues in Panel Data Analysis.................................................................................. 135 6.2 One-Way Error Component Models....................................................................... 136 6.2.1 Heteroscedasticity and Serial Correlation............................................... 139 6.3 Two-Way Error Component Models....................................................................... 140 6.4 Variable Parameter Models...................................................................................... 144 6.5 Additional Topics and Extensions .......................................................................... 145 7. Background and Exploration in Time Series................................................................147 7.1 Exploring a Time Series ........................................................................................... 148 7.1.1 The Trend Component................................................................................ 148 7.1.2 The Seasonal Component........................................................................... 149 7.1.3 The Irregular (Random) Component ....................................................... 150 7.1.4 Filtering of Time Series............................................................................... 150 7.1.5 Curve Fitting................................................................................................ 150 7.1.6 Linear Filters and Simple Moving Averages ........................................... 151 7.1.7 Exponential Smoothing Filters.................................................................. 152 7.1.8 Difference Filter........................................................................................... 154 viii Contents 7.2 Basic Concepts: Stationarity and Dependence ..................................................... 158 7.2.1 Stationarity..................................................................................................... 158 7.2.2 Dependence ................................................................................................... 159 7.2.3 Addressing Nonstationarity........................................................................ 160 7.2.4 Differencing and Unit-Root Testing........................................................... 162 7.2.5 Fractional Integration and Long Memory................................................. 162 7.3 Time Series in Regression........................................................................................ 165 7.3.1 Serial Correlation .......................................................................................... 165 7.3.2 Dynamic Dependence.................................................................................. 165 7.3.3 Volatility......................................................................................................... 166 7.3.4 Spurious Regression and Cointegration.................................................... 167 7.3.5 Causality ........................................................................................................ 169 8. Forecasting in Time Series: Autoregressive Integrated Moving Average (ARIMA) Models and Extensions...................................................................................173 8.1 Autoregressive Integrated Moving Average Models........................................... 173 8.2 The Box–Jenkins Approach ..................................................................................... 176 8.2.1 Order Selection.............................................................................................. 176 8.2.2 Parameter Estimation................................................................................... 179 8.2.3 Diagnostic Checking .................................................................................... 179 8.2.4 Forecasting..................................................................................................... 180 8.3 Autoregressive Integrated Moving Average Model Extensions ........................ 186 8.3.1 Random Parameter Autoregressive (RPA) Models.................................. 186 8.3.2 Stochastic Volatility (SV) Models................................................................ 186 8.3.3 Autoregressive Conditional Duration (ACD) Models ............................. 187 8.3.4 Integer-Valued ARMA (INARMA) Models .............................................. 187 8.4 Multivariate Models ................................................................................................. 188 8.5 Nonlinear Models..................................................................................................... 189 8.5.1 Testing for Nonlinearity .............................................................................. 190 8.5.2 Bilinear Models ............................................................................................. 191 8.5.3 Threshold Autoregressive Models ............................................................. 191 8.5.4 Functional Parameter Autoregressive Models ......................................... 192 8.5.5 Neural Networks .......................................................................................... 193 9. Latent Variable Models .....................................................................................................197 9.1 Principal Components Analysis ............................................................................. 197 9.2 Factor Analysis.......................................................................................................... 203 9.3 Structural Equation Modeling................................................................................205 9.3.1 Basic Concepts in Structural Equation Modeling....................................206 9.3.2 Fundamentals of Structural Equation Modeling.....................................208 9.3.3 Nonideal Conditions in the Structural Equation Model......................... 210 9.3.4 Model Goodness-of-Fit Measures............................................................... 211 9.3.5 Guidelines for Structural Equation Modeling.......................................... 213 10. Duration Models.................................................................................................................217 10.1 Hazard-Based Duration Models............................................................................. 217 10.2 Characteristics of Duration Data............................................................................220 10.3 Nonparametric Models............................................................................................ 221 10.4 Semi-Parametric Models..........................................................................................222 Contents ix 10.5 Fully Parametric Models........................................................................................225 10.6 Comparisons of Nonparametric, Semi-Parametric, and Fully Parametric Models......................................................................................................................228 10.7 Heterogeneity..........................................................................................................229 10.8 State Dependence.................................................................................................... 232 10.9 Time-Varying Explanatory Variables................................................................... 232 10.10 Discrete-Time Hazard Models..............................................................................233 10.11 Competing Risk Models ........................................................................................234 Section III Count and Discrete-Dependent Variable Models 11. Count Data Models ............................................................................................................237 11.1 Poisson Regression Model..................................................................................... 237 11.2 Interpretation of Variables in the Poisson Regression Model..........................238 11.3 Poisson Regression Model Goodness-of-Fit Measures...................................... 239 11.4 Truncated Poisson Regression Model.................................................................. 243 11.5 Negative Binomial Regression Model.................................................................. 245 11.6 Zero-Inflated Poisson and Negative Binomial Regression Models................. 247 11.7 Random Effects Count Models ............................................................................. 251 12. Logistic Regression ............................................................................................................253 12.1 Principles of Logistic Regression..........................................................................253 12.2 The Logistic Regression Model.............................................................................254 13. Discrete Outcome Models ................................................................................................259 13.1 Models of Discrete Data......................................................................................... 259 13.2 Binary and Multinomial Probit Models .............................................................. 260 13.3 Multinomial Logit Model ...................................................................................... 261 13.4 Discrete Data and Utility Theory.........................................................................265 13.5 Properties and Estimation of Multinomial Logit Models................................. 266 13.5.1 Statistical Evaluation ................................................................................ 270 13.5.2 Interpretation of Findings ....................................................................... 271 13.5.3 Specification Errors................................................................................... 273 13.5.4 Data Sampling...........................................................................................277 13.5.5 Forecasting and Aggregation Bias ......................................................... 278 13.5.6 Transferability ...........................................................................................280 13.6 The Nested Logit Model (Generalized Extreme Value Models) ...................... 281 13.7 Special Properties of Logit Models ...................................................................... 287 14. Ordered Probability Models ............................................................................................289 14.1 Models for Ordered Discrete Data....................................................................... 289 14.2 Ordered Probability Models with Random Effects ........................................... 296 14.3 Limitations of Ordered Probability Models........................................................ 299
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