Narrow Results

As an example of econometric and statistical methods, production function estimation has several useful features as it brings together economic theory and statistical methods. This chapter will exploit those features to introduce (i) elementary econometric methods of thinking about economics, (ii) some basic statistical tools, and (iii) some basics of the R software system. We assume that the reader already has some familiarity with all three topics. The URLs (http://www.r-project.org/, http://wiki.r-project.org/rwiki/doku.php, and http://gking.harvard.edu/zelig/) are suitable for understanding the basics of R…

The following sections are included:

• Econometric Univariate Time Series are Ubiquitous
  • Boosting the Hodrick–Prescott filter
• Stochastic Difference Equations
• Second-Order Stochastic Difference Equation and Business Cycles
  • Complex number solution of the stochastic AR(2) difference equation
  • General solution to ARMA (p, p − 1) stochastic difference equations
• Properties of ARIMA Models
  • Identification of the lag order
  • ARIMA estimation
  • ARIMA diagnostic checking
• Stochastic Process and Stationarity
  • Stochastic process and underlying probability space
  • Autocovariance of a stochastic process and ergodicity
  • Stationary process
  • Detrending and differencing to achieve stationarity
• Mean Reversion
• Autocovariance Generating Functions and the Power Spectrum
  • How to get the power spectrum from the AGF?
• Explicit Modeling of Variance (ARCH, GARCH Models)
  • Advanced GARCH-type models in R
• Tests of Independence, Neglected Nonlinearity, Turning Points
• Long Memory Models and Fractional Differencing
• Forecasting
• Concluding Remarks and Examples

The following sections are included:

• Autoregressive Distributed Lag Models
• Economic Interpretations of ARDL(1, 1) Model
  • Description of M1 to M11 model specifications
  • ARDL(0,q) as M12 model, impact and long-run multipliers
  • Adaptive expectations model to test the rational expectations hypothesis
  • Statistical inference and estimation with lagged dependent variables
  • Identification problems involving expectational variables (I. Fisher example)
  • Impulse response, mean lag, and insights from a polynomials in L
  • Choice between M1 to M11 models using R
• Stochastic Diffusion Models for Asset Prices
• Cointegration between Non-stationary Series and Spurious Regression (R² > Durbin Watson)
  • Definition: Integrated process of order d, I(d)
  • Cointegration definition and discussion
  • Error correction models of cointegration
  • Economic equilibria and error reductions through learning
  • Signs and significance of coefficients on past errors while agents learn
• Granger Causality Testing
• Co-movement of Related Series

The following sections are included:

• Utility Theory
  • Expected utility theory
  • Arrow–Pratt coefficient of absolute risk aversion
  • Risk premium needed to encourage risky investments
  • Taylor series links EUT, moments of f (x), and derivatives of U(x)
• Non-Expected Utility Theory
  • Lorenz curve scaling over the unit square
  • Mapping from EUT to non-EUT within the unit square to get decision weights
• Incorporating Utility Theory into Risk Measurement and Stochastic Dominance
  • Class D1 of utility functions and investors
  • Class D2 of utility functions and investors
  • Explicit utility functions and Arrow–Pratt measures of risk aversion
  • Class D3 of utility functions and investors
  • Class D4 of utility functions and investors
  • First-order stochastic dominance
  • Second-order stochastic dominance
  • Third-order stochastic dominance
  • Fourth-order stochastic dominance
  • Empirical checking of stochastic dominance using matrix multiplications and incorporation of 4DPs of non-EUT

The following sections are included:

• Introduction and Vector Autoregression Models
  • Some R packages for vector modeling
  • Vector autoregression or VAR models
  • Data collection tips using R
  • VAR estimation of Sims’ model
  • Granger-causality analysis in VAR models
  • Forecasting out-of-sample in VAR models
  • Impulse response analysis in VAR models
• Multivariate Regressions: Canonical Correlations
  • Why canonical correlation popularity has lagged?
• VAR Estimation and Cointegration Testing Using Canonical Correlations
• Structural VAR or SVAR estimation
• Final Remarks: Multivariate Statistics Using R

The following sections are included:

• Introduction
  • Simultaneous equation notation system with stars and subscripts
  • Simultaneous equations bias and the reduced form
  • Successively weaker assumptions regarding the nature of the Z_j matrix of regressors
  • Reduced form estimation and other alternatives to OLS
  • Assumptions of simultaneous equations models
• Instrumental Variables and Generalized Least Squares
  • The instrumental variables and generalized IV estimator
  • Choice between OLS and IV by using Wu–Hausman specification test
  • Checking nonlinear endogeneity in R
• Limited Information and Two-Stage Least Squares
  • Two-stage least squares
  • The k-class estimator
  • Limited information maximum likelihood estimator
• Identification of Simultaneous Equation Models
  • Identification is uniquely going from the reduced form to the structure
• Full Information and Three-Stage Least Squares
  • Full information maximum likelihood
• Potential of Simultaneous Equations Beyond Econometrics

In this chapter, we discuss regression type models when the dependent variable is categorical, in the sense that it can assume a limited number of values. Dummy variables are known to take only a limited number of values (0, 1, −1, etc.). Binary variables are a special kind of dummy variables used when the number of values is further limited to only two, (such as 0 and 1, or accept and reject). Modeling for proportions of two things (living or dying) naturally leads to binary dependent variables…

Empirical analysis of consumer behavior has long inspired economists to develop new theories consistent with data. This chapter begins with the classic example of Friedman’s development of the permanent income hypothesis (PIH). Next, we show how Euler’s dynamic optimization leads to Hall’s random walk in consumption, supported by unit root tests in R. We show how Hall’s derivation implies a non-linear heteroscedastic (NLHC) errors model. We also show how to use R to maximize its complicated likelihood function. Next, we report a bivariate study of consumption and income, employing Granger causality and vector autoregressions (VAR). We describe how one can estimate non-parametric and nonlinear kernel regressions between economic variables by using the R package “np” to estimate unanticipated consumption having important implications for macroeconomic policy and stock markets. Wiener–Hopf target-seeking models developed during World War II provide linear solutions providing estimable equations from data. All five “puzzles” of consumer theory can be solved by an estimable Wiener–Hopf using R…

The following sections are included:

• The Motivation and Background Behind Bootstrapping
  • Pivotal quantity and p-value
  • Uncertainty regarding proper density for regression errors illustrated
  • The delta method for standard error of functions
• Description of Parametric iid Bootstrap
  • Simulated sampling distribution for statistical inference using OLS residuals
  • Steps in a parametric approximation
  • Percentile confidence intervals
  • Reflected percentile confidence interval for bias correction
  • Significance tests as duals to confidence intervals
• Description of Non-parametric iid Bootstrap
  • Map data from time-domain to (numerical magnitudes) values-domain
  • Wild bootstrap for well-behaved bootstrap residuals
• Double Bootstrap Illustrated with a Nonlinear Model
  • A digression on the size of resamples
  • Double bootstrap theory involving roots and uniform density
  • GNR implementation of nonlinear regression for metals data
• Maximum Entropy Density Bootstrap for Time Series Data
  • Wiener, Kolmogorov, Khintchine ensemble of time series
  • Avoiding unrealistic properties of iid bootstrap
  • Maximum entropy density is uniform when limits are known
  • Quantiles of the patchwork of the ME density
  • Numerical illustration of “meboot” package in R
  • Simple and size-corrected confidence bounds
  • Extensions of meboot algorithm and better Monte Carlo

The following sections are included:

• Feasible Generalized Least Squares to Adjust for Autocorrelated Errors and/or Heteroscedasticity
  • Consequences of ignoring non-spherical errors Ω ≠ I_T
  • Derivation of the GLS and efficiency comparison
  • Computation of the GLS and feasible GLS
  • Improved OLS inference for non-spherical errors
  • Efficient estimation of β coefficients
  • An illustration using Fisher’s model for interest rates
• Vector ARMA Estimation for Rational Expectations Models
  • Greater realism of VARMA(p, q) models
  • Expectational variables from conditional forecasts in a general model
  • A rational expectation model using VARMA
  • Further forecasts, transfer function gains, and response analysis
• Optimal Estimating Function (OptEF) and Generalized Method of Moments (GMM)
  • Derivation of optimal estimating functions for regressions
  • Finite sample optimality of OptEF
  • Introduction to the GMM
  • Cases where OptEF viewpoint dominates GMM
  • Advantages and disadvantages of GMM and OptEF
• Godambe Pivot Functions and Statistical Inference
  • Application of the Frisch–Waugh theorem to constructing CI95
  • Steps in application of GPF to feasible GLS estimation
• Consistent Estimation of New Keynesian Phillips Curve using R packages

The following sections are included:

• Further R Tools for Studying Nonlinear Relations
• Box–Cox Transformation
  • Logarithmic and square root transformations
• Scatterplot Smoothing and Loess Regressions
  • Improved fit (forecasts) by loess smoothing
• Projection Pursuit Methods
• Quantile Regression
• Ridge Regularization in Data Science
• Threshold Cointegration and Asymmetric Reactions
• Remarks on Nonlinear Econometrics

This chapter deals with several topics newly added in the second edition of the book. We include

• modeling of nonlinear dependence,
• nonlinear correlations,
• information theory and entropy,
• causality from passively observed data,
• panel data models, and
• computing exact stochastic dominance

The following sections are included:

• Appendix
• References
• Index

The following sections are included:

• Dedication
• Foreword
• Preface
• Preface to the Second Edition
• About the Author
• Contents

The following sections are included:

• 𝑅 and Microeconometric Preliminaries
  • Data on Metals Production Available in 𝑅
  • Descriptive Statistics Using 𝑅
  • Writing Skewness and Kurtosis Functions in 𝑅
  • Units of Measurement and Numerical Reliability of Regressions
  • Basic Graphics in 𝑅
  • The Isoquant
  • Total Productivity of an Input
  • The Marginal Productivity (MP) of an Input
  • Slope of the Isoquant and MRTS
  • Scale Elasticity as the Returns to Scale Parameter
  • Elasticity of Substitution
  • Typical Steps in Empirical Work

• Preliminary Regression Theory: Results Using 𝑅
  • Regression as an Object ‘reg1’ in 𝑅
  • Accessing Objects Within an 𝑅 Object by Using the Dollar Symbol

• Deeper Regression Theory: Diagonals of the Hat Matrix

• Discussion of Four Diagnostic Plots by 𝑅

• Testing Constant Returns and 3D Scatter Plots

• Homothetic Production and Cost Functions
  • • The Constant Elasticity of Substitution (CES) Production Function
    • Hidden Restrictions of Homogeneous/Homethetic Functional Forms
    • Cost Function Lagrangian
    • Cost Minimizing Solution
  • Euler Theorem and Duality Theorem
    • Expansion Path
    • Duality Theorem
  • Profit Maximizing Solutions
  • Elasticity of Total Cost w.r.t. Output

• Miscellaneous Microeconomic Topics
  • Analytic Input Demand Function for the Cobb–Douglas Form
  • Separability in the Presence of Three or More Inputs
  • Two or More Outputs as Joint Outputs
  • Economies of Scope
    • Bell System Example on Economies of Scope

• Nonhomogeneous Production Functions
  • Three-Input Production Function for Widgets
    • Creating a Text File of Data on Your C Drive for Reading in 𝑅
    • Reading Data and MNH Estimation in 𝑅
  • Isoquant Plotting for a Bell System Production Function

• Collinearity Problem, Singular Value Decomposition (SVD), and Ridge Regression
  • What is Collinearity?
    • Rank Deficiency m
    • Near Collinearity
  • Consequences of Near Collinearity
    • Collinearity and Ill-Conditioned Matrices
    • Rule of Thumb for a Large Condition Number
    • Collinearity as a Nonproblem
  • Regression Theory Using the Singular Value Decomposition
    • MSE(b) and Extreme Unreliability of OLS Under Near Collinearity
    • REMARK 1 (Extreme Unreliability of OLS)
    • Forecasting and Collinearity

• Near Collinearity Solutions by Coefficient Shrinkage
  • Ridge Regression
    • Derivation of Shrinkage Factors Δ
    • Declining Deltas and Justification for Shrinking to Zero Prior
    • Choice of k in Ridge Regression
  • Principal Components Regression

• Bell System Production Function in Anti-Trust Trial
  • Collinearity Diagnostics for Bell Data Trans-Log
  • Shrinkage Solution and Ridge Regression for Bell Data
  • Ridge Regression from Existing 𝑅 Packages

• Comments on Wrong Signs, Collinearity, and Ridge Scaling
  • Concluding Comments on the 1982 Bell System Breakup

• Data Appendix

The following sections are included:

• Econometric Univariate Time Series are Ubiquitous

• Stochastic Difference Equations

• Second-Order Stochastic Difference Equation and Business Cycles
  • Complex Number Solution of the Stochastic A𝑅(2) Difference Equation
  • General Solution to ARMA (p, p − 1) Stochastic Difference Equations

• Properties of ARIMA Models
  • Identification of the Lag Order
  • ARIMA Estimation
  • ARIMA Diagnostic Checking

• Stochastic Process and Stationarity
  • Stochastic Process and Underlying Probability Space
  • Autocovariance of a Stochastic Process and Ergodicity
  • Stationary Process
  • Detrending and Differencing to Achieve Stationarity

• Mean Reversion

• Autocovariance Generating Functions (AGF) and the Power Spectrum
  • How to Get the Power Spectrum from the AGF?

• Explicit Modeling of Variance (ARCH, GARCH Models)

• Tests of Independence, Neglected Nonlinearity, Turning Points

• Long Memory Models and Fractional Differencing

• Forecasting

• Concluding Remarks and Examples

The following sections are included:

• Autoregressive Distributed Lag (ARDL) Models

• Economic Interpretations of ARDL(1,1) Model
  • Description of M1 to M11 Model Specifications
  • ARDL(0,q) as M12 Model, Impact and Long-Run Multipliers
  • Adaptive Expectations Model to Test Rational Expectations Hypothesis
  • Statistical Inference and Estimation with Lagged-Dependent Variables
  • Identification Problems Involving Expectational Variables (I. Fisher Example)
  • Impulse Response, Mean Lag and Insights from a Polynomials in L
  • Choice Between M1 to M11 Models Using 𝑅

• Stochastic Diffusion Models for Asset Prices

• Spurious Regression (R² > Durbin Watson) and Cointegration
  • Definition of a Process Integrated of Order d, I(d)
  • Cointegration Definition and Discussion
  • Error Correction Models of Cointegration
  • Economic Equilibria and Error Reductions through Learning
  • Signs and Significance of Coefficients on Past Errors while Agents Learn

• Granger Causality Testing

The following sections are included:

• Utility Theory
  • Expected Utility Theory (EUT)
  • Arrrow–Pratt Coefficient of Absolute Risk Aversion (CARA)
  • Risk Premium Needed to Encourage Risky Investments
  • Taylor Series Links EUT, Moments of f(x) and Derivatives of U(x)

• Non-Expected Utility Theory
  • Lorenz Curve Scaling over the Unit Square
  • Mapping From EUT to Non-EUT within the Unit Square to Get Decision Weights

• Incorporating Utility Theory into Risk Measurement and Stochastic Dominance
  • Class D1 of Utility Functions and Investors
  • Class D2 of Utility Functions and Investors
  • Explicit Utility Functions and Arrow–Pratt Measures of Risk Aversion
  • Class D3 of Utility Functions and Investors
  • Class D4 of Utility Functions and Investors
  • First-Order Stochastic Dominance (1SD)
  • Second-Order Stochastic Dominance (2SD)
  • Third-Order Stochastic Dominance (3SD)
  • Fourth-Order Stochastic Dominance (4SD)
  • Empirical Checking of Stochastic Dominance Using Matrix Multiplications and Incorporation of 4DPs of Non-EUT

The following sections are included:

• Introduction and VAR Models
  • Some 𝑅 Packages for Vector Modeling
  • Vector Autoregression or VAR Models
  • Data Collection Tips Using 𝑅
  • VAR Estimation of Sims' Model
  • Granger-Causality Analysis in VAR Models
  • Forecasting Out-of-Sample in VAR Models
  • Impulse Response Analysis in VAR Models

• Multivariate Regressions: Canonical Correlations
  • Why Canonical Correlation is Not Popular So Far

• VAR Estimation and Cointegration Testing Using Canonical Correlations

• Final Remarks: Multivariate Statisics Using 𝑅

The following sections are included:

• Introduction
  • Simultaneous Equation Notation System with Stars and Subscripts
  • Simultaneous Equations Bias and the Reduced Form
  • Successively Weaker Assumptions Regarding the Nature of the Z_j Matrix of Regressors
  • Reduced Form Estimation and Other Alternatives to OLS
  • Assumptions of Simultaneous Equations Models

• Instrumental Variables and Generalized Least Squares
  • The Instrumental Variables (IV) and Generalized IV (GIV) Estimator
  • Choice Between OLS and IV by Using Wu–Hausman Specification Test

• Limited Information and Two-Stage Least Squares
  • Two-Stage Least Squares
  • The k-class Estimator
  • Limited Information Maximum Likelihood (LIML) Estimator

• Identification of Simultaneous Equation Models
  • Identification is Uniquely Going from the Reduced Form to the Structure

• Full Information and Three-Stage Least Squares (3SLS)
  • Full Information Maximum Likelihood

• Potential of Simultaneous Equations Beyond Econometrics