Reproducible econometrics using R
Jeffrey S. Racine

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Contents
                                    Contents
List of Tables	xiii
List of Figures	xiv
Preface	xix
About the Companion Website	xxiii
I	Linear Time Series Methods	1
R and Time Series Analysis	3
Overview	3
Some Useful R Functions for Time Series Analysis	4
1 Introduction to Linear Time Series Models	7
1.1	Overview	7
1.2	Time Series Data	8
1.3	Patterns in Time Series	9
1.4	Stationary versus Non-Stationary Series	9
1.5	Examples of Univariate Random Processes	12
1.5.1	White Noise Processes	12
1.5.2	Random Walk Processes	13
1.6	Characterizing Time Series	14
1.6.1	The Autocorrelation Function	14
1.6.2	The Sample Autocorrelation Function	16
1.6.3	Non-Stationarity and Differencing	16
1.7	Tests for White Noise Processes	18
1.7.1	Individual Test for ?: pk = 0—Bartlett’s Test	18
1.7.2	Joint Test for #o : pi = 0 ? /92 = 0 ? • • • ? p*, = 0—
Ljung & Box’s Test	19
1.7.3	A Simulated Illustration—Testing for a White Noise
Process	19
1.7.4	A Simulated Illustration White Noise Tests when
the Series is a Random Walk Process	21
vi
CONTENTS
vii
2	Random Walks, Unit Roots, and Spurious Relationships 23
2.1	Overview	23
2.2	Properties of a Random Walk	24
2.3	The Autocorrelation Function for a Random Walk	25
2.4	Classical Least Squares Estimators and Random Walks	25
2.5	Classical Least Squares Inference and Random Walks	26
2.5.1	Cross-Section (I.I.D. Data) Monte Carlo	26
2.5.2	Time Series (Random Walk) Monte Carlo	27
2.5.3	Time Series (Random Walk with Drift) Monte Carlo 29
2.6	Unit Root Tests	30
2.6.1	Testing for a Unit Root in Spot Exchange Rates	32
2.7	Random Walks and Spurious Regression	33
3	Univariate Linear Time Series Models	37
3.1	Overview	37
3.2	Moving Average Models (MA(g))	38
3.2.1	Structure of MA(q) Processes	38
3.2.2	Example—Residential Electricity Sales	38
3.2.3	Properties of MA(q) Processes	39
3.2.4	Stationarity of MA(g) Processes	40
3.2.5	The Autocorrelation Function and Identification of
MA(q) Processes	40
3.2.6	Forecasting MA(q) Processes	41
3.2.7	Forecasting MA(q) Processes Assuming the ct-í and
the 6i are Known	41
3.2.8	Forecasting MA(g) Processes when the ??-i and the
di are Estimated	42
3.2.9	Forecasting MA(q) Processes in the Presence of a
Trend	43
3.3	Autoregressive Models (AR(p))	44
3.3.1	Structure of AR(p) Processes	44
3.3.2	Example—Residential Electricity Sales	45
3.3.3	Properties of AR(p) Processes	46
3.3.4	Stationarity of AR(p) Processes	48
3.3.5	Invertiblity of Stationary AR(p) Processes	50
3.3.6	Identification of AR(p) Processes—The Partial Autocorrelation Function	50
3.3.7	Forecasting AR.(p) Processes	51
3.3.8	Forecasting AR(p) Processes when the Parameters
?{ are Unknown	53
3.3.9	Forecasting AR(p) Processes in the Presence of a
Trend	54
3.4	Non-Seasonal Autoregressive Moving Average Models
(ARMA(p, <?))	55
CONTENTS
viii
3.4.1	Structure	55
3.5	Non-Seasonal Autoregressive Integrated Moving Average
Models (ARIMA(p? d, g))	55
3.5.1	Structure	55
3.5.2	Stationarity of ARIMA(p, d. q) Models	56
3.5.3 Identification of ARIMA(p, d, q) Processes	57
3.5.4 Estimation of A RIM A (p, d, q) Processes	57
3.5.5	Forecasting ARIMA(p, d, q)	Processes	58
3.5.6	Trends, Constants, and ARIMA(p, d, <7) Models	59
3.5.7	Model Selection Criteria, Trends, and Stationaritv	62
3.5.8	Model Selection via auto.arima0	65
3.5.9	Diagnostics for ARIMA(p, d, q) Models	67
3.6	Seasonal Autoregressive Integrated Moving Average Models
(ARIMA(p, d, q)(P,D, Q)m)	67
3.6.1	Example—Modelling and Forecasting European
Quarterly Retail Trade	71
3.6.2	Example—Modelling Monthly Cortecosteroid Drug
Sales	72
3.7 ARLMA(p,d,q)(Pi D,Q)m Models with External Predictors 74
3.8	Assessing Model Accuracy on Hold-Out Data	77
Problem Set	81
II	Robust Parametric Inference	85
R, The Bootstrap and the Jackknife	87
Overview	87
Some Useful R Functions for Data-Driven Inference	87
4	Robust Parametric Inference	89
4.1	Overview	89
4.2	Analytical Versus Numerical, i.e., Data-Driven,	Procedures 90
4.2.1	Drawbacks of the Analytical Approach	91
4.2.2	An Illustrative Example—Testing for a	Unit	Root	91
4.3	Alternatives to Analytical Approaches	92
4.3.1	Motivating Example—Compute the Standard Error
of X	92
4.4	An Introduction to Efron's Bootstrap	93
4.4.1	Bootstrapping a Standard Error for the	Sample	Mean 93
4.4.2	Bootstrap Implementations in R	94
4.5	Jackknifing—Background and Motivating Example	95
4.6	Jackknife and Bootstrap Estimates of Bias	97
4.7	To Bootstrap or Jackknife?	99
CONTENTS	ix
4.8	Data-Driven Covariance Matrices	99
4.8.1	Bootstrap Heteroskedasticity Consistent Covariance
Matrix Estimation	100
4.9	The Wild Bootstrap	102
4.10	Bootstrapping Dependent Processes	104
4.11	Bootstrap Confidence Intervals	105
4.11.1	Example—Nonparametric Confidence Intervals for
the Population Mean	106
4.12	Bootstrap Inference	108
4.12.1	How Many Bootstrap Replications?	109
4.12.2	Generating 0* Under the Null	110
4.12.3	Example—The Two-Sample Problem	111
4.12.4	Example—Regression-Based Bootstrap Inference	112
4.12.5	Example—Unit Root Testing	114
Problem Set	117
III	Robust Parametric Estimation	119
R and Robust Parametric Estimation	121
Overview	121
Some Useful R Functions for Robust Parametric Estimation	121
5	Robust Parametric Estimation	123
5.1	Overview	123
5.2	Robust Estimation Basics	124
5.2.1	Outlier	124
5.2.2	Breakdown Point	125
5.2.3	Sensitivity Curve	125
5.2.4	Contamination Neighborhoods	126
5.2.5	Influence Function	128
5.3	Unmasking Univariate Outliers	129
5.3.1	L\\ and Z,2-norm Estimators of Central Tendency	130
5.3.2	Robustness versus Efficiency	132
5.3.3	?-Estimator Methods	133
5.3.4	Optimal Robustness	135
5.3.5	Huber’s ?-Estimator of Location—A More Efficient
Robust Location Estimator than the Median	135
5.3.6	Rousseeuw and Croux’s Qn Estimator of Scale—A
More Efficient Robust Scale Estimator than MADn	137
5.3.7	Af-Estimators of Scale	139
5.3.8	Unmasking Univariate Outliers—The three-sigma
edit rule	142
X	CONTENTS
5.4	Unmasking Multivariate Outliers	142
5.5	Unmasking Regression Outliers	148
5.5.1	Outliers in the Y Direction	148
5.5.2	Outliers in the X Direction	150
5.5.3	Leverage Points	150
5.5.4	Dealing with Outlying Observations and Leverage
Points	152
5.6	Robust Regression	160
5.6.1	Robust Residuals and High Breakdown Diagnostics	164
5.7	Some Useful Points to Remember	165
Problem Set	167
IV	Model Uncertainty	171
R and Model Uncertainty	173
Overview	173
Some Useful R Functions for Model Uncertainty	173
6	Model Uncertainty	175
6.1	Overview	175
6.1.1	Model Selection References	176
6.1.2	Model Averaging References	176
6.1.3	Resources	177
6.2	A Reflection on Models and Data Generating	Processes	177
6.2.1	Model Selection and Averaging—A Simulation	181
6.2.2	Discussion	183
6.3	Kullback-Leibler Distance and Maximum	Likelihood	Estimation	184
6.4	Model Selection Methods	186
6.4.1	AIC, BIG, Cp and Cross-Validated Model Selection
Criteria	186
6.5	Model Averaging Methods	189
6.5.1	Solving for the Optimal Model Average Weights	190
6.5.2	Selecting Candidate Models	191
6.5.3	Pitfalls of Model Selection and Model Averaging	196
6.5.4	An Experimental Robust Regression M-Estimator
Model Averaging Procedure	196
Problem Set
201
CONTENTS
XI
V	Advanced Topics	207
R and Advanced Topics	209
Overview	209
Some Useful It Functions for Advanced Topics	209
7	Advanced Topics	211
7.1	Overview	211
7.2	Classification Analysis and Support Vector Machines	211
7.2.1	The Confusion Matrix	212
7.2.2	Support Vector Machines	213
7.3	Nonparametric Kernel Regression	220
Problem Set	225
VI	Appendix	227
A R, RStudio, TeX, and Git	229
A.l Installation of R and RStudio Desktop	229
A.2	What is R?	229
A.2.1 R in the News	230
A.2.2 Introduction to R	230
A.2.3 Econometrics in R	230
A.3	What is RStudio Desktop?	231
A.3.1 Introduction to RStudio	231
A.4	Installation of TeX	231
A.	5	Installation of Git	231
B R Markdown for Assignments	233
B.	l	Source Code (R Markdown) for this Document	233
B.2 R, RStudio, TeX and git	233
B.3	What is R Markdown?	233
B.4	Creating a New R Markdown Document in RStudio	234
B.5	Including R Results in your R Markdown Document	234
B.6 Reading Data from a URL	234
B.7	Including	Plots	235
B.8	Including	Bulleted	and	Numbered	lists	236
B.9	Including	Tables	237
B.10	Including	Verbatim,	i.e.,	Freeform,	Text	237
B.ll Typesetting Mathematics	237
B.12 Flexible Document Creation	238
B.13 Knitting your R Markdown Document	238
B.14 Printing Your Assignment for Submitting in Class	238
B.l5 Troubleshooting and Tips	239
CONTENTS
xii
C	Maximum Likelihood Estimation and Inference	243
C.l Maximum Likelihood Estimation	243
C.2	Properties of the Maximum Likelihood Estimators	244
C.3	Maximum Likelihood Estimation in Practice	246
C.4	A Simple Example Using Discrete Data	246
C.4.1 Example—	247
C.4.2 Example—	248
C.5 Maximum Likelihood Estimation of the Normal Linear
Multivariate Regression Model	249
C.6	Information and the Normal Linear Multivariate Model	252
C.6.1 Example—	253
C.7 Restricted Maximum Likelihood Estimates	254
C.	8	Hypothesis Testing in a Maximum Likelihood Framework	254
C.8.1 Example—	255
C.8.2 Example—	256
D	Solving a Quadratic Program Using R	259
D.	l Example	260
E	A Primer on Regression Splines	263
E.	l	Overview	263
E.2	Bezier curves	264
E.2.1 Example—A quadratic Bézier curve	264
E.2.2 The Bézier curve defined	265
E.2.3 Example—A quadratic Bézier curve as a linear interpolation between two linear Bézier curves	265
E.2.4 Example The quadratic Bézier curve basis functions	266
E.3	Derivatives of spline functions	267
E.4	B-splines	267
E.4.1 B-spline knots	267
E.4.2 The B-spline basis function	268
E.4.3 Example—A fourth-order B-spline basis function with three interior knots and its first derivative function	269
E.5	The B-spline function	269
E.6	Multivariate B-spline regression	269
E.6.1 Multivariate knots, intervals, and spline bases	271
E.7	Spline regression	272
Bibliography	273
Author	Index	281
Subject	Index	283
                                
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