Mathematical Statistics for Applied Econometrics covers the basics of statistical inference in support of a subsequent course on classical econometrics The book shows how mathematical statistics concepts form the basis of econometric formulations It also helps you think about statistics as more than a toolbox of techniques The text explores the unifying themes involved in quantifying sample information to make inferences After developing the necessary probability theory, it presents the concepts of estimation, such as convergence, point estimators, confidence intervals, and hypothesis tests The text then shifts from a general development of mathematical statistics to focus on applications particularly popular in economics It delves into matrix analysis, linear models, and nonlinear econometric techniques Features • Shows how mathematical statistics is useful in the analysis of economic decisions under risk and uncertainty • Describes statistical tools for inference, explaining the “why” behind statistical estimators, tests, and results • Provides an introduction to the symbolic computer programs Maxima and Mathematica®, which can be used to reduce the mathematical and numerical complexity of some formulations • Gives the R code for several applications • Includes summaries, review questions, and numerical exercises at the end of each chapter K20635 w w w c rc p r e s s c o m MOSS Avoiding a cookbook approach to econometrics, this book develops your theoretical understanding of statistical tools and econometric applications It provides you with the foundation for further econometric studies MATHEMATICAL STATISTICS FOR APPLIED ECONOMETRICS Statistics MATHEMATICAL STATISTICS FOR APPLIED ECONOMETRICS K20635_FM.indd 9/5/14 12:14 PM K20635_FM.indd 9/5/14 12:14 PM MATHEMATICAL STATISTICS FOR APPLIED ECONOMETRICS CHARLES B MOSS University of Florida Gainesville, USA K20635_FM.indd 9/5/14 12:14 PM CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20140805 International Standard Book Number-13: 978-1-4665-9410-4 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, 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http://www.crcpress.com Dedication This book is dedicated to the memory of Henri Theil Over the course of my training in econometrics, I had several motivated instructors However, I really did not understand econometrics until I collaborated with Hans Contents List of Figures xiii List of Tables xv Preface Defining Mathematical Statistics 1.1 Mathematical Statistics and Econometrics 1.1.1 Econometrics and Scientific Discovery 1.1.2 Econometrics and Planning 1.2 Mathematical Statistics and Modeling Economic Decisions 1.3 Chapter Summary 1.4 Review Questions I xix Defining Random Variables Introduction to Statistics, Probability, and Econometrics 2.1 Two Definitions of Probability for Econometrics 2.1.1 Counting Techniques 2.1.2 Axiomatic Foundations 2.2 What Is Statistics? 2.3 Chapter Summary 2.4 Review Questions 2.5 Numerical Exercises 12 14 17 18 19 Random Variables and Probability Distributions 3.1 Uniform Probability Measure 3.2 Random Variables and Distributions 3.2.1 Discrete Random Variables 3.2.2 Continuous Random Variables 3.3 Conditional Probability and Independence 3.3.1 Conditional Probability and Independence for Discrete Random Variables 3.3.2 Conditional Probability and Independence for Continuous Random Variables 3.4 Cumulative Distribution Function 21 27 28 32 38 39 40 40 43 44 50 51 52 55 57 61 71 vii viii Contents 3.5 3.6 3.7 3.8 3.9 3.10 3.11 Some Useful Distributions Change of Variables Derivation of the Normal Distribution Function An Applied Sabbatical Chapter Summary Review Questions Numerical Exercises Moments and Moment-Generating Functions 4.1 Expected Values 4.2 Moments 4.3 Covariance and Correlation 4.4 Conditional Mean and Variance 4.5 Moment-Generating Functions 4.5.1 Moment-Generating Functions for Specific Distributions 4.6 Chapter Summary 4.7 Review Questions 4.8 Numerical Exercises 71 75 76 81 83 84 85 87 87 96 97 103 105 106 109 110 110 Binomial and Normal Random Variables 5.1 Bernoulli and Binomial Random Variables 5.2 Univariate Normal Distribution 5.3 Linking the Normal Distribution to the Binomial 5.4 Bivariate and Multivariate Normal Random Variables 5.4.1 Bivariate Normal Random Variables 5.4.2 Multivariate Normal Distribution 5.5 Chapter Summary 5.6 Review Questions 5.7 Numerical Exercises 113 114 117 120 122 124 127 129 129 130 II Estimation Large Sample Theory 6.1 Convergence of Statistics 6.2 Modes of Convergence 6.2.1 Almost Sure Convergence 6.2.2 Convergence in Probability 6.2.3 Convergence in the rth Mean 6.3 Laws of Large Numbers 6.4 Asymptotic Normality 6.5 Wrapping Up Loose Ends 6.5.1 Application of Holder’s Inequality 6.5.2 Application of Chebychev’s Inequality 6.5.3 Normal Approximation of the Binomial 131 133 133 137 140 142 142 144 145 149 149 149 150 Contents 6.6 6.7 6.8 ix Chapter Summary Review Questions Numerical Exercises 150 151 151 Point Estimation 7.1 Sampling and Sample Image 7.2 Familiar Estimators 7.2.1 Estimators in General 7.2.2 Nonparametric Estimation 7.3 Properties of Estimators 7.3.1 Measures of Closeness 7.3.2 Mean Squared Error 7.3.3 Strategies for Choosing an Estimator 7.3.4 Best Linear Unbiased Estimator 7.3.5 Asymptotic Properties 7.3.6 Maximum Likelihood 7.4 Sufficient Statistics 7.4.1 Data Reduction 7.4.2 Sufficiency Principle 7.5 Concentrated Likelihood Functions 7.6 Normal Equations 7.7 Properties of Maximum Likelihood Estimators 7.8 Chapter Summary 7.9 Review Questions 7.10 Numerical Exercises 153 154 160 161 164 165 166 166 169 169 172 172 173 173 174 176 177 178 180 181 181 Interval Estimation 8.1 Confidence Intervals 8.2 Bayesian Estimation 8.3 Bayesian Confidence Intervals 8.4 Chapter Summary 8.5 Review Questions 8.6 Numerical Exercises 183 183 192 195 195 196 196 Testing Hypotheses 9.1 Type I and Type II Errors 9.2 Neyman–Pearson Lemma 9.3 Simple Tests against a Composite 9.4 Composite against a Composite 9.5 Testing Hypotheses about Vectors 9.6 Delta Method 9.7 Chapter Summary 9.8 Review Questions 9.9 Numerical Exercises 199 201 203 205 207 210 211 212 213 213 288 12.3.1 Mathematical Statistics for Applied Econometrics Least Absolute Deviation Following the absolute value formulation of ρ ( i ) in Equation 12.69, we can formulate the Least Absolute Deviation Estimator (LAD) as β N N ρ (yi − β0 − β1 xi ) = i=1 N N |yi − β0 − β1 xi | (12.72) i=1 where yi are observed values for the dependent variable, xi are observed values of the independent variables, and β is the parameters to be estimated To develop the concept of the different weighting structures, consider the firstorder conditions for the general formulation in Equation 12.72 ∂L ( y, x| β, ρ) = ∂β0 N ∂L ( y, x| β, ρ) = ∂β1 N N ρ (yi − β0 − β1 xi ) (−1) i=1 N (12.73) ρ (yi − β0 − β1 xi ) (−xi ) i=1 If ρ ( i ) is the standard squared error, Equation 12.73 yields the standard set of normal equations ∂ρ (yi − β0 − β1 xi ) = (yi − β0 − β1 xi ) xi ∂β1 (12.74) However, if ρ ( i ) is the absolute value, the derivative becomes ∂ρ (yi − β0 − β1 xi ) = ∂β1 xi for yi < β0 + β1 xi −xi for yi > β0 + β1 xi (12.75) which cannot be solved using the standard calculus (i.e., assuming a smooth derivative) Bassett and Koenker [3] develop the asymptotic distribution of the parameters as √ LD −1 N (βn∗ − β) → N 0, ω [Xn Xn ] (12.76) N N where ω is an asymptotic estimator of the variance (i.e., ω = 1/N i=1 (yi − Xi β) ) In order to demonstrate the applications of the LAD estimator, consider the effect of gasoline and corn prices on ethanol pet = β0 + β1 pgt + β2 pct + t (12.77) where pet is the price of ethanol at time t, pgt is the price of gasoline at time t, pct is the price of corn at time t, t is the residual, and β0 , β1 , and β2 are the parameters we want to estimate Essentially, the question is whether gasoline or corn prices determine the price of ethanol The data for 1982 through 2013 are presented in Table 12.5 The parameter estimates using OLS, LAD, quantile regression with τ = 0.50 discussed in the next section, and two different Huber weighting functions are presented in Table 12.6 Survey of Nonlinear Econometric Applications 289 TABLE 12.5 Ethanol, Gasoline, and Corn Prices 1982–2013 Year 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Nominal Prices Ethanol Gasoline Corn 1.71 1.00 2.55 1.68 0.91 3.21 1.55 0.85 2.63 1.60 0.85 2.23 1.07 0.51 1.50 1.21 0.57 1.94 1.13 0.54 2.54 1.23 0.61 2.36 1.35 0.75 2.28 1.27 0.69 2.37 1.33 0.64 2.07 1.16 0.59 2.50 1.19 0.56 2.26 1.15 0.59 3.24 1.35 0.69 2.71 1.15 0.55 2.43 1.05 0.43 1.94 0.98 0.59 1.82 1.35 0.93 1.85 1.48 0.88 1.97 1.12 0.81 2.32 1.35 0.98 2.42 1.69 1.25 2.06 1.80 1.66 2.00 2.58 1.94 3.04 2.24 2.23 4.20 2.47 2.57 4.06 1.79 1.76 3.55 1.93 2.17 5.18 2.70 2.90 6.22 2.37 2.95 6.89 2.47 2.90 4.50 12.3.2 Quantile Regression Real Prices PCE Ethanol Gasoline 50.479 3.631 2.123 52.653 3.420 1.853 54.645 3.040 1.667 56.581 3.031 1.610 57.805 1.984 0.946 59.649 2.174 1.024 61.973 1.954 0.934 64.640 2.040 1.012 67.439 2.146 1.192 69.651 1.954 1.062 71.493 1.994 0.960 73.278 1.697 0.863 74.802 1.705 0.802 76.354 1.614 0.828 77.980 1.856 0.948 79.326 1.554 0.743 79.934 1.408 0.577 81.109 1.295 0.780 83.128 1.741 1.199 84.731 1.872 1.113 85.872 1.398 1.011 87.573 1.652 1.199 89.703 2.019 1.494 92.260 2.091 1.929 94.728 2.919 2.195 97.099 2.473 2.462 100.063 2.646 2.753 100.000 1.919 1.886 101.654 2.035 2.288 104.086 2.780 2.986 106.009 2.396 2.983 107.187 2.470 2.900 Corn 5.415 6.535 5.159 4.225 2.781 3.486 4.393 3.913 3.624 3.647 3.103 3.657 3.238 4.548 3.725 3.283 2.601 2.405 2.385 2.492 2.896 2.962 2.462 2.324 3.440 4.636 4.349 3.805 5.462 6.405 6.967 4.500 The least absolute deviation estimator in Equation 12.72 provides a transition to the Quantile Regression estimator Specifically, following Koenker and Bassett [27] we can rewrite the estimator in Equation 12.72 as β i∈i:yi ≥xi β θ |yi − xi β| + (1 − θ) |yi − xi β| i∈i:yi