INSTRUCTOR’S RESOURCE GUIDE TO
ACCOMPANY
APPLIED ECONOMETRIC TIME SERIES
(2nd edition)
Walter Enders
University of Alabama
Prepared by
Pin Chung
American Reinsurance Company and Iowa State University
Walter Enders
Ling Shao
Jingan Yuan
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple
of business exams, case studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically
and make strategic decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in
Business ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-
making skills, as well as the ability to evaluate various business alternatives.Quantitative Analysis: For subjects like finance or economics, business exams often require students to perform calculations and
interpret data. The ability to analyze financial statements, project revenues, and manage budgets is essential.Communication and
, CONTENTS
1. Difference Equations
Lecture Suggestions 1
Answers to Questions 2
2. Stationary Time-Series Models
Lecture Suggestions 17
Answers to Questions 20
3. Modeling Volatility
Lecture Suggestions 41
Answers to Questions 43
4. Models With Trend
Lecture Suggestions 59
Answers to Questions 60
5. Multiequation Time-Series Models
Lecture Suggestions 81
Answers to Questions 83
6. Cointegration and Error-Correction Models
Lecture Suggestions 107
Answers to Questions 110
7. Nonlinear Tine-Series Models
Lecture Suggestions 127
Answers to Questions 128
Semester Project 137
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business exams,
case studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic
decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business
ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-making
skills, as well as the ability to evaluate various business alternatives.Quantitative Analysis: For subjects like finance or economics, business exams often require students to perform calculations and interpret
data. The ability to analyze financial statements, project revenues, and manage budgets is essential.Communication and
, CHAPTER 1
DIFFERENCE EQUATIONS
1. Time-Series Models 1
2. Difference Equations and Their Solutions 6
3. Solution by Iteration 9
4. An Alternative Solution Methodology 14
5. The Cobweb Model 17
6. Solving Homogeneous Difference Equations 22
7. Particular Solutions for Deterministic Processes 30
8. The Method of Undetermined Coefficients 33
9. Lag Operators 38
10. Summary 41
Questions and Exercises 42
APPENDIX 1.1 Imaginary Roots and de Moivre’s Theorem 44
APPENDIX 1.2 Characteristic Roots in Higher-Order Equations 46
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business exams,
case studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic
decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business
ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-making
skills, as well as the ability to evaluate various business alternatives.Quantitative Analysis: For subjects like finance or economics, business exams often require students to perform calculations and interpret
data. The ability to analyze financial statements, project revenues, and manage budgets is essential.Communication and
Lecture Suggestions
Nearly all students will have some familiarity the concepts developed in the chapter. A first
course in integral calculus makes reference to convergent versus divergent solutions. I draw the
analogy between the particular solution to a difference equation and indefinite integrals.
It is important to stress the distinction between convergent and divergent solutions. Be sure to
emphasize the relationship between characteristic roots and the convergence or divergence of a
sequence. Much of the current time-series literature focuses on the issue of unit roots. It is wise to
introduce students to the properties of difference equations with unitary characteristic roots at this
early stage in the course. Question 5 at the end of this chapter is designed to preview this important
issue. The tools to emphasize are the method of undetermined coefficients and lag operators.
Few students will have been exposed to these methods in other classes.
I use overheads to show the students several data series and ask them to discuss the type of
difference equation model that might capture the properties of each. The figure below shows three of
the real exchange rate series used in Chapter 4. Some students see a tendency for the series to revert
to a long-run mean value. The classroom discussion might center on the appropriate way to model
the tendency for the levels to meander. At this stage, the precise models are not important. The
objective is for students to conceptualize economic data in terms of difference equations.
Page 1: Difference Equations
, Real Exchange Rates (Panel.xls)
180
160
(1996 = 100)
140
120
100
80
60
1980 1983 1986 1989 1992 1995 1998 2001
U.S. Canada Germany
Answers to Questions
1. Consider the difference equation: yt = a0+ a1yt-1 with the initial condition y0. Jill solved the
difference equation by iterating backwards:
yt = a0 + a1yt-1
= a0 + a1[a0 + a1yt-2 ]
= a0 + a0a1 + a0(a1)2 + ..... + a0(a1)t-1 + (a1)ty0
Bill added the homogeneous and particular solutions to obtain: yt = a0/(1 - a1) + (a1)t[y0 - a0/(1 -
a1)].
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business exams,
case studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic
decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business
ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-making
skills, as well as the ability to evaluate various business alternatives.Quantitative Analysis: For subjects like finance or economics, business exams often require students to perform calculations and interpret
data. The ability to analyze financial statements, project revenues, and manage budgets is essential.Communication and
A. Show that the two solutions are identical for a1 < 1.
Answer: The key is to demonstrate:
a0 + a0a1 + a0(a1)2 + ..... + a0(a1)t-1 + (a1)ty0 = a0/(1 - a1) + (a1)t[y0 - a0/(1 - a1)]
First, cancel (a1)ty0 from each side and then divide by a0. The two sides of the equation are
identical if:
1 + a1 + (a1)2 + .... + (a1)t-1 = 1/(1 - a1) - (a1)t/(1 - a1)
Page 2: Difference Equations
ACCOMPANY
APPLIED ECONOMETRIC TIME SERIES
(2nd edition)
Walter Enders
University of Alabama
Prepared by
Pin Chung
American Reinsurance Company and Iowa State University
Walter Enders
Ling Shao
Jingan Yuan
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple
of business exams, case studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically
and make strategic decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in
Business ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-
making skills, as well as the ability to evaluate various business alternatives.Quantitative Analysis: For subjects like finance or economics, business exams often require students to perform calculations and
interpret data. The ability to analyze financial statements, project revenues, and manage budgets is essential.Communication and
, CONTENTS
1. Difference Equations
Lecture Suggestions 1
Answers to Questions 2
2. Stationary Time-Series Models
Lecture Suggestions 17
Answers to Questions 20
3. Modeling Volatility
Lecture Suggestions 41
Answers to Questions 43
4. Models With Trend
Lecture Suggestions 59
Answers to Questions 60
5. Multiequation Time-Series Models
Lecture Suggestions 81
Answers to Questions 83
6. Cointegration and Error-Correction Models
Lecture Suggestions 107
Answers to Questions 110
7. Nonlinear Tine-Series Models
Lecture Suggestions 127
Answers to Questions 128
Semester Project 137
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business exams,
case studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic
decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business
ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-making
skills, as well as the ability to evaluate various business alternatives.Quantitative Analysis: For subjects like finance or economics, business exams often require students to perform calculations and interpret
data. The ability to analyze financial statements, project revenues, and manage budgets is essential.Communication and
, CHAPTER 1
DIFFERENCE EQUATIONS
1. Time-Series Models 1
2. Difference Equations and Their Solutions 6
3. Solution by Iteration 9
4. An Alternative Solution Methodology 14
5. The Cobweb Model 17
6. Solving Homogeneous Difference Equations 22
7. Particular Solutions for Deterministic Processes 30
8. The Method of Undetermined Coefficients 33
9. Lag Operators 38
10. Summary 41
Questions and Exercises 42
APPENDIX 1.1 Imaginary Roots and de Moivre’s Theorem 44
APPENDIX 1.2 Characteristic Roots in Higher-Order Equations 46
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business exams,
case studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic
decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business
ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-making
skills, as well as the ability to evaluate various business alternatives.Quantitative Analysis: For subjects like finance or economics, business exams often require students to perform calculations and interpret
data. The ability to analyze financial statements, project revenues, and manage budgets is essential.Communication and
Lecture Suggestions
Nearly all students will have some familiarity the concepts developed in the chapter. A first
course in integral calculus makes reference to convergent versus divergent solutions. I draw the
analogy between the particular solution to a difference equation and indefinite integrals.
It is important to stress the distinction between convergent and divergent solutions. Be sure to
emphasize the relationship between characteristic roots and the convergence or divergence of a
sequence. Much of the current time-series literature focuses on the issue of unit roots. It is wise to
introduce students to the properties of difference equations with unitary characteristic roots at this
early stage in the course. Question 5 at the end of this chapter is designed to preview this important
issue. The tools to emphasize are the method of undetermined coefficients and lag operators.
Few students will have been exposed to these methods in other classes.
I use overheads to show the students several data series and ask them to discuss the type of
difference equation model that might capture the properties of each. The figure below shows three of
the real exchange rate series used in Chapter 4. Some students see a tendency for the series to revert
to a long-run mean value. The classroom discussion might center on the appropriate way to model
the tendency for the levels to meander. At this stage, the precise models are not important. The
objective is for students to conceptualize economic data in terms of difference equations.
Page 1: Difference Equations
, Real Exchange Rates (Panel.xls)
180
160
(1996 = 100)
140
120
100
80
60
1980 1983 1986 1989 1992 1995 1998 2001
U.S. Canada Germany
Answers to Questions
1. Consider the difference equation: yt = a0+ a1yt-1 with the initial condition y0. Jill solved the
difference equation by iterating backwards:
yt = a0 + a1yt-1
= a0 + a1[a0 + a1yt-2 ]
= a0 + a0a1 + a0(a1)2 + ..... + a0(a1)t-1 + (a1)ty0
Bill added the homogeneous and particular solutions to obtain: yt = a0/(1 - a1) + (a1)t[y0 - a0/(1 -
a1)].
exams include:Multiple Choice Questions (MCQs): These are frequently used to assess students’ understanding of business terminology, theories, and principles.Case Studies: A staple of business exams,
case studies present students with real-world business scenarios and ask them to apply their knowledge to solve complex problems. Case studies evaluate students' ability to think critically and make strategic
decisions.Essay/Short Answer Questions: These types of questions test the student’s ability to explain and analyze business concepts in a detailed and coherent manner.1.3. Skills Tested in Business
ExamsCritical Thinking and Problem-Solving: Business exams often include case studies that challenge students to apply theoretical knowledge to real-life situations. These tests assess decision-making
skills, as well as the ability to evaluate various business alternatives.Quantitative Analysis: For subjects like finance or economics, business exams often require students to perform calculations and interpret
data. The ability to analyze financial statements, project revenues, and manage budgets is essential.Communication and
A. Show that the two solutions are identical for a1 < 1.
Answer: The key is to demonstrate:
a0 + a0a1 + a0(a1)2 + ..... + a0(a1)t-1 + (a1)ty0 = a0/(1 - a1) + (a1)t[y0 - a0/(1 - a1)]
First, cancel (a1)ty0 from each side and then divide by a0. The two sides of the equation are
identical if:
1 + a1 + (a1)2 + .... + (a1)t-1 = 1/(1 - a1) - (a1)t/(1 - a1)
Page 2: Difference Equations
