MathWorks Econometrics Toolbox™ User's Guide

4164 Pages·2022·27.308 MB·English

by  The MathWorks, Inc.

Econometrics Toolbox™
User's Guide
R2022b
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Econometrics Toolbox™ User's Guide
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Revision History
October 2008 Online only Version 1.0 (Release 2008b)
March 2009 Online only Revised for Version 1.1 (Release 2009a)
September 2009 Online only Revised for Version 1.2 (Release 2009b)
March 2010 Online only Revised for Version 1.3 (Release 2010a)
September 2010 Online only Revised for Version 1.4 (Release 2010b)
April 2011 Online only Revised for Version 2.0 (Release 2011a)
September 2011 Online only Revised for Version 2.0.1 (Release 2011b)
March 2012 Online only Revised for Version 2.1 (Release 2012a)
September 2012 Online only Revised for Version 2.2 (Release 2012b)
March 2013 Online only Revised for Version 2.3 (Release 2013a)
September 2013 Online only Revised for Version 2.4 (Release 2013b)
March 2014 Online Only Revised for Version 3.0 (Release 2014a)
October 2014 Online Only Revised for Version 3.1 (Release 2014b)
March 2015 Online Only Revised for Version 3.2 (Release 2015a)
September 2015 Online Only Revised for Version 3.3 (Release 2015b)
March 2016 Online Only Revised for Version 3.4 (Release 2016a)
September 2016 Online Only Revised for Version 3.5 (Release 2016b)
March 2017 Online Only Revised for Version 4.0 (Release 2017a)
September 2017 Online Only Revised for Version 4.1 (Release 2017b)
March 2018 Online Only Revised for Version 5.0 (Release 2018a)
September 2018 Online Only Revised for Version 5.1 (Release 2018b)
March 2019 Online Only Revised for Version 5.2 (Release 2019a)
September 2019 Online Only Revised for Version 5.3 (Release 2019b)
March 2020 Online Only Revised for Version 5.4 (Release 2020a)
September 2020 Online Only Revised for Version 5.5 (Release 2020b)
March 2021 Online Only Revised for Version 5.6 (Release 2021a)
September 2021 Online Only Revised for Version 5.7 (Release 2021b)
March 2022 Online Only Revised for Version 6.0 (Release 2022a)
September 2022 Online Only Revised for Version 6.1 (Release 2022b)

Contents
Getting Started
1
Econometrics Toolbox Product Description . . . . . . . . . . . . . . . . . . . . . . . . 1-2
Econometric Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3
Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3
Econometrics Toolbox Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3
Represent Time Series Models Using Econometrics Toolbox Objects . . . . 1-7
Model Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7
Model Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10
Create Model Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11
Retrieve Model Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-15
Modify Model Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-16
Object Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-17
Stochastic Process Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-18
What Is a Stochastic Process? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-18
Stationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-19
Linear Time Series Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-19
Unit Root Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-20
Lag Operator Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21
Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-22
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-24
Data Preprocessing
2
Data Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2
Why Transform? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2
Common Data Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2
Trend-Stationary vs. Difference-Stationary Processes . . . . . . . . . . . . . . . . 2-6
Nonstationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6
Trend Stationary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7
Difference Stationary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-7
Specify Lag Operator Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9
Lag Operator Polynomial of Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9
Difference Lag Operator Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11
Nonseasonal Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13
v
Nonseasonal and Seasonal Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16
Time Series Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19
Moving Average Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21
Moving Average Trend Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22
Parametric Trend Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-24
Hodrick-Prescott Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-29
Use Hodrick-Prescott Filter to Reproduce Original Result . . . . . . . . . . . 2-30
Seasonal Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-34
What Is a Seasonal Filter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-34
Stable Seasonal Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-34
Sn × m seasonal filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-35
Seasonal Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-37
What Is Seasonal Adjustment? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-37
Deseasonalized Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-37
Seasonal Adjustment Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-37
Seasonal Adjustment Using a Stable Seasonal Filter . . . . . . . . . . . . . . . . 2-39
Seasonal Adjustment Using S(n,m) Seasonal Filters . . . . . . . . . . . . . . . . 2-45
Model Selection
3
Box-Jenkins Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2
Select ARIMA Model for Time Series Using Box-Jenkins Methodology . . 3-4
Autocorrelation and Partial Autocorrelation . . . . . . . . . . . . . . . . . . . . . . 3-11
What Are Autocorrelation and Partial Autocorrelation? . . . . . . . . . . . . . . 3-11
Theoretical ACF and PACF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11
Sample ACF and PACF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11
Compute Sample ACF and PACF in MATLAB® . . . . . . . . . . . . . . . . . . . . 3-12
Ljung-Box Q-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-18
Detect Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-20
Compute Sample ACF and PACF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-20
Conduct the Ljung-Box Q-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-22
Engle’s ARCH Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-26
Detect ARCH Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-28
Test Autocorrelation of Squared Residuals . . . . . . . . . . . . . . . . . . . . . . . 3-28
vi Contents
Conduct Engle's ARCH Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-30
Unit Root Nonstationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-33
What Is a Unit Root Test? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-33
Modeling Unit Root Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-33
Available Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-37
Testing for Unit Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-38
Unit Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-41
Test Simulated Data for a Unit Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-41
Test Time Series Data for Unit Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-45
Test Stock Data for a Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-48
Assess Stationarity of a Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-51
Information Criteria for Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . 3-54
Compute Information Criteria Using aicbic . . . . . . . . . . . . . . . . . . . . . . . 3-54
Model Comparison Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-58
Available Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-58
Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-60
Lagrange Multiplier Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-60
Wald Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-60
Covariance Matrix Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-61
Conduct Lagrange Multiplier Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-62
Conduct Wald Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-65
Compare GARCH Models Using Likelihood Ratio Test . . . . . . . . . . . . . . . 3-67
Classical Model Misspecification Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-70
Check Fit of Multiplicative ARIMA Model . . . . . . . . . . . . . . . . . . . . . . . . . 3-81
Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-86
Residual Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-87
Check Residuals for Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-87
Check Residuals for Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-87
Check Residuals for Conditional Heteroscedasticity . . . . . . . . . . . . . . . . 3-87
Assess Predictive Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-89
Nonspherical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-90
What Are Nonspherical Models? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-90
Plot a Confidence Band Using HAC Estimates . . . . . . . . . . . . . . . . . . . . . 3-91
Change the Bandwidth of a HAC Estimator . . . . . . . . . . . . . . . . . . . . . . . 3-98
Check Model Assumptions for Chow Test . . . . . . . . . . . . . . . . . . . . . . . . 3-104
Power of the Chow Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-110
vii
Econometric Modeler
4
Analyze Time Series Data Using Econometric Modeler . . . . . . . . . . . . . . . 4-2
Prepare Data for Econometric Modeler App . . . . . . . . . . . . . . . . . . . . . . . 4-3
Import Time Series Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4
Perform Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6
Fitting Models to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15
Conducting Goodness-of-Fit Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-30
Finding Model with Best In-Sample Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 4-36
Export Session Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-38
Specifying Univariate Lag Operator Polynomials Interactively . . . . . . . . 4-44
Specify Lag Structure Using Lag Order Tab . . . . . . . . . . . . . . . . . . . . . . 4-45
Specify Lag Structure Using Lag Vector Tab . . . . . . . . . . . . . . . . . . . . . . 4-47
Specifying Multivariate Lag Operator Polynomials and Coefficient
Constraints Interactively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-50
Specify Lag Structure Using Lag Order Tab . . . . . . . . . . . . . . . . . . . . . . 4-51
Specify Lag Structure Using Lag Vector Tab . . . . . . . . . . . . . . . . . . . . . . 4-52
Specify Coefficient Matrix Equality Constraints for Estimation . . . . . . . . 4-54
Prepare Time Series Data for Econometric Modeler App . . . . . . . . . . . . 4-59
Prepare Table of Multivariate Data for Import . . . . . . . . . . . . . . . . . . . . . 4-59
Prepare Numeric Vector for Import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-60
Import Time Series Data into Econometric Modeler App . . . . . . . . . . . . 4-62
Import Data from MATLAB Workspace . . . . . . . . . . . . . . . . . . . . . . . . . . 4-62
Import Data from MAT-File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-63
Plot Time Series Data Using Econometric Modeler App . . . . . . . . . . . . . 4-66
Plot Univariate Time Series Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-66
Plot Multivariate Time Series and Correlations . . . . . . . . . . . . . . . . . . . . 4-67
Detect Serial Correlation Using Econometric Modeler App . . . . . . . . . . 4-71
Plot ACF and PACF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-71
Conduct Ljung-Box Q-Test for Significant Autocorrelation . . . . . . . . . . . . 4-73
Detect ARCH Effects Using Econometric Modeler App . . . . . . . . . . . . . . 4-77
Inspect Correlograms of Squared Residuals for ARCH Effects . . . . . . . . . 4-77
Conduct Ljung-Box Q-Test on Squared Residuals . . . . . . . . . . . . . . . . . . 4-80
Conduct Engle's ARCH Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-82
Assess Stationarity of Time Series Using Econometric Modeler . . . . . . . 4-84
Test Assuming Unit Root Null Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-84
Test Assuming Stationary Null Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-87
Test Assuming Random Walk Null Model . . . . . . . . . . . . . . . . . . . . . . . . 4-90
Assess Collinearity Among Multiple Series Using Econometric Modeler
App . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-94
Transform Time Series Using Econometric Modeler App . . . . . . . . . . . . 4-97
Apply Log Transformation to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-97
Stabilize Time Series Using Nonseasonal Differencing . . . . . . . . . . . . . 4-101
viii Contents
Convert Prices to Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-104
Remove Seasonal Trend from Time Series Using Seasonal Difference . . 4-107
Remove Deterministic Trend from Time Series . . . . . . . . . . . . . . . . . . . 4-109
Implement Box-Jenkins Model Selection and Estimation Using
Econometric Modeler App . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-112
Select ARCH Lags for GARCH Model Using Econometric Modeler App
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-122
Estimate Multiplicative ARIMA Model Using Econometric Modeler App
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-131
Perform ARIMA Model Residual Diagnostics Using Econometric Modeler
App . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-141
Specify t Innovation Distribution Using Econometric Modeler App . . . 4-150
Estimate Vector Autoregression Model Using Econometric Modeler . . 4-155
Conduct Cointegration Test Using Econometric Modeler . . . . . . . . . . . 4-170
Estimate Vector Error-Correction Model Using Econometric Modeler 4-180
Compare Predictive Performance After Creating Models Using
Econometric Modeler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-193
Estimate ARIMAX Model Using Econometric Modeler App . . . . . . . . . . 4-200
Estimate Regression Model with ARMA Errors Using Econometric
Modeler App . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-208
Compare Conditional Variance Model Fit Statistics Using Econometric
Modeler App . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-221
Perform GARCH Model Residual Diagnostics Using Econometric Modeler
App . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-230
Share Results of Econometric Modeler App Session . . . . . . . . . . . . . . . 4-237
Time Series Regression Models
5
Time Series Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3
Regression Models with Time Series Errors . . . . . . . . . . . . . . . . . . . . . . . . 5-5
What Are Regression Models with Time Series Errors? . . . . . . . . . . . . . . . 5-5
Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5
Create Regression Models with ARIMA Errors . . . . . . . . . . . . . . . . . . . . . . 5-8
Default Regression Model with ARIMA Errors Specifications . . . . . . . . . . 5-8
ix
Specify regARIMA Models Using Name-Value Pair Arguments . . . . . . . . . . 5-9
Specify Linear Regression Models Using Econometric Modeler App . . . . 5-15
Specify the Default Regression Model with ARIMA Errors . . . . . . . . . . . 5-19
Modify regARIMA Model Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21
Modify Properties Using Dot Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21
Nonmodifiable Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-23
Create Regression Models with AR Errors . . . . . . . . . . . . . . . . . . . . . . . . 5-26
Default Regression Model with AR Errors . . . . . . . . . . . . . . . . . . . . . . . . 5-26
AR Error Model Without an Intercept . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-27
AR Error Model with Nonconsecutive Lags . . . . . . . . . . . . . . . . . . . . . . . 5-27
Known Parameter Values for a Regression Model with AR Errors . . . . . . 5-28
Regression Model with AR Errors and t Innovations . . . . . . . . . . . . . . . . 5-29
Create Regression Models with MA Errors . . . . . . . . . . . . . . . . . . . . . . . . 5-31
Default Regression Model with MA Errors . . . . . . . . . . . . . . . . . . . . . . . 5-31
MA Error Model Without an Intercept . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-32
MA Error Model with Nonconsecutive Lags . . . . . . . . . . . . . . . . . . . . . . 5-32
Known Parameter Values for a Regression Model with MA Errors . . . . . . 5-33
Regression Model with MA Errors and t Innovations . . . . . . . . . . . . . . . . 5-34
Create Regression Models with ARMA Errors . . . . . . . . . . . . . . . . . . . . . . 5-36
Default Regression Model with ARMA Errors . . . . . . . . . . . . . . . . . . . . . 5-36
ARMA Error Model Without an Intercept . . . . . . . . . . . . . . . . . . . . . . . . 5-37
ARMA Error Model with Nonconsecutive Lags . . . . . . . . . . . . . . . . . . . . 5-37
Known Parameter Values for a Regression Model with ARMA Errors . . . . 5-38
Regression Model with ARMA Errors and t Innovations . . . . . . . . . . . . . 5-38
Specify Regression Model with ARMA Errors Using Econometric Modeler
App . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-40
Create Regression Models with ARIMA Errors . . . . . . . . . . . . . . . . . . . . . 5-44
Default Regression Model with ARIMA Errors . . . . . . . . . . . . . . . . . . . . . 5-44
ARIMA Error Model Without an Intercept . . . . . . . . . . . . . . . . . . . . . . . . 5-45
ARIMA Error Model with Nonconsecutive Lags . . . . . . . . . . . . . . . . . . . . 5-45
Known Parameter Values for a Regression Model with ARIMA Errors . . . 5-46
Regression Model with ARIMA Errors and t Innovations . . . . . . . . . . . . . 5-47
Create Regression Models with SARIMA Errors . . . . . . . . . . . . . . . . . . . . 5-49
SARMA Error Model Without an Intercept . . . . . . . . . . . . . . . . . . . . . . . 5-49
Known Parameter Values for a Regression Model with SARIMA Errors . . 5-50
Regression Model with SARIMA Errors and t Innovations . . . . . . . . . . . . 5-50
Specify Regression Model with SARIMA Errors . . . . . . . . . . . . . . . . . . . . 5-53
Specify ARIMA Error Model Innovation Distribution . . . . . . . . . . . . . . . . 5-59
About the Innovation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-59
Innovation Distribution Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-60
Specify Innovation Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-60
Impulse Response of Regression Models with ARIMA Errors . . . . . . . . . 5-64
Plot Impulse Response of Regression Model with ARIMA Errors . . . . . . 5-65
Regression Model with AR Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-65
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