ARE/ECN 240C
Department of Economics
University of California, Davis
Professor Òscar Jordà
ARE/ECN 240C
TIME SERIES
ANALYSIS
Winter 2002
Professor Òscar Jordà
Instructor:
1150 Social Sciences and Humanities Bldg.
Phone: 752 7021
e-mail: ojorda@ucdavis.edu
CLASS URL:
http://www.econ.ucdavis.edu/faculty/jorda/class/240c/240c.html
Class Meets:
T – R, 12:10 – 1:30pm. Room: WICKSON 1020
Office Hours:
Fridays 10-12, or by appointment
Course Goals: The course is intended to fulfill two needs: (1) to provide students with
applied interests with the most sophisticated and up to date techniques used in empirical
time series analysis, and (2) to introduce students with more theoretical inclinations to the
tools that are used to derive some of the more interesting results.
Textbook: Hamilton, J. D. (1994) Time Series Analysis, Princeton University Press, New
Jersey. I will follow Hamilton's book rather closely. Regardless, this is a great book,
worth having in your library.
Computer Assignments: I am hoping to have 5 computer assignments that will involve
the packages EViews and GAUSS. The home page for the course has a number of links
to brief introductions to these programs.
Resources: Mostly, you should check the home page for the course. Additional readings
for each topic can be found at the end of this document.
Grading: There will be three components to your grade, namely, assignments (30%),
midterm (30%) and final (40%).
Course Outline:
1. INTRODUCTION TO UNIVARIATE, STATIONARY, TIME SERIES ANALYSIS
1.1. Introduction
1.1.1.
1.1.2.
1.1.3.
Dynamic Multipliers, Impulse Response Functions, and Long-Run Propensity
Lag Operators
Difference Equations and Stability
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ARE/ECN 240C
Department of Economics
University of California, Davis
Professor Òscar Jordà
1.2. Stationary ARMA Processes
1.2.1.
Preliminary Concepts: Gaussian white noise, martingale difference sequences,
autocovariances and autocorrelations, stationarity, ergodicity
1.2.2. Basic ARMA Models: MA models, AR models, ARMA(p,q) models, invertibility
1.3. Identification of ARMA Models
1.3.1.
1.3.2.
1.3.3.
1.3.4.
Common Transformations
Sample ACF and PACF
State-Space Representation of ARMA Models
Wold Representation Theorem
2. ESTIMATION AND TESTING
2.1. Maximum Likelihood Estimation
2.1.1.
2.1.2.
2.1.3.
2.1.4.
Quasi MLE: Fundamental Theorems
The MLE of an AR(1)
The MLE of a MA(1)
The MLE of an ARMA(p,q)
2.2. Review of Numerical Optimization
2.2.1.
2.2.2.
General Concepts
Review of Algorithms: Newton-Raphson, Quadratic-Hill, Gauss-Newton/BHHH,
Marquardt, Davidon-Fletcher-Powell
2.3. Statistical Inference with MLE
2.3.1.
2.3.2.
2.3.3.
Asymptotic Standard Errors
QMLE Standard Errors
Hypothesis Testing: Wald, Likelihood Ratio and LM tests; Ljung-Box/QStatistics
3. FORECASTING
3.1. Cost Functions
3.2. Forecasting with ARMA Models: State-Space representation
3.3. Forecasting with Non-Linear Models: Naïve, Exact, Monte-Carlo and Bootstrap
forecasts
4. ASYMPTOTIC DISTRIBUTION THEORY
4.1. Preliminaries: Autocovariance generating function, laws of large numbers, and the
ergodic theorem
4.2. Central Limit Theorem: CLT for martingale difference sequences, and CLT for
dependent processes. The Beveridge-Nelson decomposition
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ARE/ECN 240C
Department of Economics
University of California, Davis
Professor Òscar Jordà
5. NON-STATIONARY UNIVARIATE TIME SERIES MODELS
5.1. Introduction: Stochastic vs. deterministic trends
5.2. Asymptotic Distribution Theory of the Non-Stochastic Trend Model
5.3. Unit Roots
5.3.1.
5.3.2.
5.3.3.
5.3.4.
5.3.5.
5.3.6.
5.3.7.
5.3.8.
Review: Little o, Big O, and motivation
Brownian Motion
Functional Central Limit Theorem
Convergence in Law for Random Functions
Convergence in Probability for Random Functions
Continuous Mapping Theorem
Phillips-Perron Test
Augmented Dickey-Fuller Test
6. COVARIANCE STATIONARY VECTOR TIME-SERIES
6.1. The VAR(p) Model
6.1.1.
6.1.2.
6.1.3.
6.1.4.
Rewriting a VAR(p) as a VAR(1)
Stationarity
Multivariate Wold Theorem
VMA() Representation of a VAR(p)
6.2. Autocovariances of Vector Processes
6.3. Heteroskedasticity, Autocorrelation, Consistent, Covariance Matrix Estimation
6.3.1.
6.3.2.
The Newey-West Estimator
Other Estimators
7. CAUSALITY, FEEDBACK AND EXOGENEITY
7.1. Granger Causality
7.1.1. Definitions and Concepts
7.1.2. Testing
7.1.3. Interpretation
7.2. Exogeneity
7.2.1.
7.2.2.
Definition and Concepts
Weak Exogeneity, Strict Exogeneity, and Super Exogeneity
8. MAXIMUM LIKELIHOOD ESTIMATION OF VECTOR PROCESSES
8.1. MLE of Unrestricted VARs
8.2. Structural Models
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ARE/ECN 240C
Department of Economics
University of California, Davis
Professor Òscar Jordà
8.3. Impulse Response Function Analysis
8.3.1.
8.3.2.
8.3.3.
8.3.4.
Calculating the IRF: Methods
Wold Causal Orderings
Variance Decomposition
Interpretation of IRFs and VDs
9. COINTEGRATION
9.1. Spurious Regressions
9.2. Cointegration
9.2.1.
9.2.2.
9.2.3.
9.2.4.
9.2.5.
Definition and Concepts
Error Correction Representation
Granger Representation Theorem
Phillips Triangular Representation
Stock and Watson Common Trends Representation
9.3. Testing
9.3.1.
9.3.2.
9.3.3.
Bivariate Analysis
Engle-Granger Test: Properties
Correcting Cointegration Tests for Serial Correlation
9.4. FIML Analysis of Cointegrated Systems
9.4.1.
9.4.2.
9.4.3.
Review of Canonical Correlation Analysis
Johansen's Test
Hypothesis Testing and Estimation
10. STATE-SPACE MODELS
10.1.
State Space Representation of a Linear Dynamic System
10.2.
Overview of the Kalman Filter
10.3.
ML Estimation
10.4.
Asymptotic Properties of MLE and QMLE
Additional Reading
Almost all the material for the class comes from Hamilton's book so you need not worry
about the reading list except when indicated in class. The references contained in Hamilton's
book are quite comprehensive if you ever need to go deeper into a topic. The references
below might be helpful if you have difficulty understanding the material. An asterisk
indicates references that I find particularly useful.
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ARE/ECN 240C
Department of Economics
University of California, Davis
Professor Òscar Jordà
1. INTRODUCTION TO UNIVARIATE, STATIONARY, TIME SERIES ANALYSIS



Box, G.E.P. and G.M. Jenkins (1976), Time Series Analysis: Forecasting and Control,
2nd ed. San Francisco: Holden Day.
Gourieroux C. and A. Monfort (1997), Time Series and Dynamic Models. Cambridge:
Cambridge University Press.
*Sargent, T. J. (1987), Macroeconomic Theory. Boston: Academic Press. (Chapters 9, 10,
and 11).
2. ESTIMATION AND TESTING




*Amemiya, T. (1985), Advanced Econometrics. Cambridge: Harvard University Press.
(Chapter 4).
*Davidson, R. and J. G. MacKinnon (1993) Estimation and Inference in Econometrics.
Oxford: Oxford University Press. (Chapter 8).
*Engle, R. F. (1984), "Wald, Likelihood Ratio, and Lagrange Multiplier Tests in
Econometrics," Ch. 13 in Handbook of Econometrics, Vol. II, eds. Z. Griliches and M.D.
Intrilligator, Amsterdam: North-Holland.
*Greene, W. H. (1997) Econometric Analysis, 4th Edition. New Jersey: Prentice Hall.
(Chapter 5).
3. FORECASTING

*Granger, C. W. J. and P. Newbold, (1986) Forecasting Economic Time Series.
Academic Press.
4. ASYMPTOTIC DISTRIBUTION THEORY



Davidson, J. (1994) Stochastic Limit Theory. Oxford; Oxford University Press.
*White, H. (1999) Asymptotic Theory for Econometricians. Revised Edition. San Diego:
Academic Press
van der Vaart, A. W. (1998) Asymptotic Statistics. Cambridge: Cambridge University
Press.
5. NON-STATIONARY UNIVARIATE TIME SERIES MODELS

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
Brockwell, P.J. and R. A. Davis. Time Series: Theory and Methods. Springer-Verlag.
* Davidson, J. (1994) Stochastic Limit Theory. Oxford; Oxford University Press.
Dickey, D. A. and W. A. Fuller (1979), " Distribution Estimators for Autoregressive
Time Series with a Unit Root," Journal of the American Statistical Association, 74, 437431.
Fuller, W. A. Intriduction to Statistical Time Series. Wiley series in Probability and
Statistics. John Wiley.
Phillips, P.C.B. (1986), "Time Series Regression with Unit Roots," Econometrica, 55,
227-302.
Phillips, P.C.B. (1998), "New Tools for Understanding Spurious Regressions,"
Econometrica, 66, 1299-1236.
*Stock, J. H. (1994), "Unit Roots, Structural Breaks, and Trends," in Handbook of
Econometrics, Vol. IV, eds. D. McFadden and R. F. Engle. Amsterdam: North-Holland.
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ARE/ECN 240C
Department of Economics

University of California, Davis
Professor Òscar Jordà
* Tanaka, Katsuto, (1996) Time Series Analysis. New York: John Wiley (Chapter 3 and
Chapter 9).
6. COVARIANCE STATIONARY VECTOR TIME-SERIES



Den Haan, W. J. and A. Levin, (1997), " A Practioner's Guide to Robust Covariance
Matrix Estimation," Handbook of Statistics 15 (Chapter 12, 291-341)
*Den Haan, W. J. and A. Levin, (1996), " Inferences from Parametric and NonParametric Covariance Matrix Estimation Procedures," NBER Technical Working Paper
195.
Both papers can be downloaded from: http://weber.ucsd.edu/~wdenhaan/papers.html
Newey W. N. and K. D. West (1987), "A Simple Positive Semi-Definite
Heteroskedasticity and Autocorrelation Consistent Covariance Matrix," Econometrica,
55, 703-708.
7. CAUSALITY, FEEDBACK AND EXOGENEITY

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
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
*Engle, R. F., D. F. Hendry, and J.-F. Richard, (1983), "Exogeneity," Econometrica, 51,
277-305.
*Granger, C. W. J. (1980), "Testing for Causality: A Personal Viewpoint," Journal of
Economic Dynamics and Control, 2, 329-352
Granger, C. W. J. (1989), Modelling Economic Series, Oxford: Oxford University Press.
*Hendry, D. F. (1995), Dynamic Econometrics, Oxford: Oxford University Press.
Hoover, K. D. and S. M. Sheffrin, (1992), " Causation, Spending, and Taxes: Sand in the
Sandbox or Tax Collector for the Welfare State?" American Economic-Review; 82(1),
225-48.
Swanson, N. R. and C. W. J. Granger, (1997), "Impulse Response Functions Based on a
Causal Approach to Residual Orthogonalization in Vector Autoregressions," Journal of
the American Statistical Association, 92(437), 357-367.
Demiralp S. and K. D. Hoover, (2000), “Structural VARs,” U.C. Davis, manuscript.
8. MAXIMUM LIKELIHOOD ESTIMATION OF VECTOR PROCESSES

*Reinsel, G. C. (1993), Elements of Multivariate Time Series Analysis, New York:
Springer-Verlag.
9. COINTEGRATION
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
*Engle, R. F. and C. W. J. Granger, (1991), Long-Run Economic Relationships, Oxford:
Oxford University Press.
*Johansen, S. (1995), Likelihood-Based Inference in Cointegrated Vector-Autoregressive
Models, Oxford: Oxford University Press.
*Sims, C. A., J. H. Stock and M. W. Watson, (1990), "Inference in Time Series Models
with some Unit Roots," Econometrica, 58, 113-44.
Watson, M. W. (1994), "Vector Autoregressions and Cointegration," in Handboof of
Econometrics, Vol. IV, eds. D. McFadden and R. F. Engle. Amsterdam: North-Holland.
10. STATE-SPACE MODELS
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ARE/ECN 240C
Department of Economics
University of California, Davis
Professor Òscar Jordà

*Hamilton, J. D. (1994), "State-Space Models," in Handbook of Econometrics, Vol. IV,
R. F. Engle and D. L. McFadden, eds. Amsterdam: North-Holland.
 Harvey, Andrew C. (1989), Forecasting Structural Time Series Models and the Kalman
Filter. Cambridge: Cambridge University Press.
TIME SERIES ECONOMETRICIANS IN DAVIS
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
Arthur M. Havenner, Professor, Department of Agricultural and Resource
Economics.
o Web-site: http://www.agecon.ucdavis.edu/Faculty/Art.H/Havenner.html
o Research Interests: Econometrics, both classical and time series analysis,
involving any aspects of hypothesis testing, policy simulation / optimal
control, and forecasting. If you have any questions about State Space
modeling and the Kalman filter, he is your man.
o Book: Applications of Computer Aided Time Series Modeling, M. Aoki and
A. Havenner, (eds.). Berlin: Springer-Verlag, 1997.
Prasad Naik, Assistant Professor, Graduate School of Management. Although
professor Naik is formally professor in Marketing, he has a lot of experience in time
series modeling, forecasting and control.
o Web-site: http://www.gsm.ucdavis.edu/Faculty/Profiles/Naik/
o Research Interests: advertising strategy, consumer choice
behavior, sales forecasting, dynamic market response models.
o Representative Publication: "Partial Least Squares Estimator for SingleIndex Models, with Chih-Ling Tsai, Journal of the Royal Statistical Society,
Series B, 2000, Vol. 62, No. 4
Aaron Smith, Assistant Professor, Department of Agricultural and Resource
Economics.
o Web-site: http://asmith.ucdavis.edu
o Research Interests: In econometrics, forecasting time series processes using
nonlinear models and measuring the distance between unknown distributions.
In finance, studying excess volatility and bubbles in equity markets.
o Representative Publication: “Stochastic Permanent Breaks” (with Robert
Engle), Review of Economics and Statistics, 81(4): 553-574, 1999.
Robert Shumway, Professor, Department of Statistics.
o Web-site: http://anson.ucdavis.edu/faculty/shumway.html
o Research Interests: applications of multivariate spectral analysis to high
dimensional problems in discrimination and clustering for multivariate time
series and to linear filter contrasts arising in a multivariate analysis of
variance. Analysis of incomplete data that can be modeled in terms of linear
and nonlinear state-space models.
o Books: Shumway, R.H. (1988). Applied Statistical Time Series Analysis.
Englewood Cliffs, New Jersey: Prentice Hall.
Shumway, R.H. & Stoffer, D. S. (2000). Time Series Analysis and Its
Applications. New York: Springer.
Chih-Ling Tsai, Professor, Graduate School of Business.
o Web-site: http://www.gsm.ucdavis.edu/Faculty/Profiles/tsai/
o Research Interests: application of statistics in business, regression
diagnostics, model selection, optimal design.
o Books: Regression and Time Series Model Selection, with Allan McQuarrie,
World Scientific, 1998.
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ARE/ECN 240C
Department of Economics
University of California, Davis
Professor Òscar Jordà
TIME SERIES TOPICS
Here is a potpourri of other topics that we will not discuss in class. There is no particular order
and were possible, I enclose relevant references.
1. Time Series Topics in Finance

Campbell, John Y., A. W. Lo, and A. C. MacKinlay, (1997), The Econometrics of
Financial Markets. Princeton University Press.

Mills, T. C. (1993), The Econometric Modelling of Financial Time Series. Cambridge
University Press.
1. ARCH Models and Stochastic Volatility Models

Engle, R. F. (1995), ARCH Selected Readings, Oxford University Press

Shepard, N., S. Kim., and S. Chib (1998), “Stochastic Volatility: Likelihood
Inference and Comparison with ARCH Models,” Review of Economic Studies,
65, 361-393.
2. Continuous Time Models

Bergstrom, A. R. (1990), Continuous Time Econometric Modelling, Oxford
University Press.
3. Neural Networks

White, H. (1992), Artificial Neural Networks: Approximation and Learning
Theory. Blackwell Publishers.
4. Time Series Duration Models

Engle, R. F. and J. R. Russell, (1998), “Autoregressive Conditional Duration: A
New Model for Irregularly Spaced Transaction Data,” Econometrica, V. 66, N. 5,
1127-1162.

Jorda, O. and M. Marcellino, (2000), “Stochastic Processes subject to Time Scale
Transformation: An Application to High Frequency FX Data,” UC Davis
working paper 00-02.
5. Long Memory Models

Journal of Econometrics, (1996), Volume 73, Issue 1, July. This issue is a
monograph on Long Memory Models and is edited by Richard T. Baillie and
Maxwell L. King.
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ARE/ECN 240C
Department of Economics
University of California, Davis
Professor Òscar Jordà
2. Time Series Topics in Macroeconomics
1. Nonlinear Time Series Models

Granger, C. W. J. and T. Terasvirta (1993), Modelling Nonlinear Economic
Relationships, Oxford University Press.
i.
Markov Switching Regimes Models

Hamilton, J. D. (1989), “A New Approach to the Economic
Analysis of Nonstationary Time Series and the Business
Cycle,” Econometrica, 57(2), 357-384

Kim, C. J. and C. R. Nelson, (1999), State Space Models with
Regime Switching, MIT Press.
ii.
Threshold Autoregressive and Smooth Transition Models

Tong, H. (1983), Threshold Models in Non-Linear Time
Series Analysis, Springer-Verlag

Terasvirta, T. (1994), “Specification, Estimation and
Evaluation of Smooth Transition Autoregressive Models,”
Journal of the American Statistical Association.
2. Discrete-Time, Time Series Duration Models: The ACH Model

Hamilton, J. D. and O. Jorda, (2000), “A Model for the Federal Funds Rate
Target.” NBER working paper 7847.

Demiralp, S. and O. Jorda, (2000), “The Pavlovian Response of Term Rates to
Fed Announcements,” UC Davis, working paper 99-06R.
3. Structural Break Testing

Bai, J. and P. Perron, (1998), “ Estimating and Testing Linear Models with
Multiple Structural Changes,” Econometrica, 66(6), 47-78.
4. Causality in Macroeconomics

Demiralp, S. (2000), The Structure of Monetary Policy and Transmission
Mechanism, UC Davis Thesis.

Hoover, K. D. (2001), Causality in Macroeconomics, Cambridge University
Press.
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ARE/ECN 240C
Department of Economics
University of California, Davis
Professor Òscar Jordà
3. Other Topics
1. Time Aggregation

Jorda, O. and M. Marcellino, (2000), “Stochastic Processes subject to Time Scale
Transformations: An Application to High-Frequency FX Data,” UC Davis,
working paper 00-02.
2. Simulation Methods for Time Series Models

Kim, C. J. and C. R. Nelson, (1999), State Space Models with Regime Switching,
MIT Press.

Gourieroux, C. and A. Monfort, (1996), Simulation Based Econometric Methods,
Cambridge University Press.
3. Bootstrap in Time Series

Berkowitz, J. and L. Kilian (2000), “Recent Developments in Bootstrapping
Time Series,” Econometric Reviews, 19(1), 1-48.
4. Spectral Analysis and Wavelets

Percival, D. B. and A. T. Walden, (2000), Wavelet Methods for Time Series
Analysis, Cambridge University Press
5. Dynamic Panel Data Models

Arellano, M. and B. Honore, (1999), “Panel Data Models: Some Recent
Developments,” prepared for the Handbook of Econometrics, Vol. 5.
Additional Resources
I recommend that you visit Eric Zivot’s home page. He has collected numerous interesting links if
you are interested in time series. This is his web-site:
http://faculty.washington.edu/ezivot/econ584/econ584.htm
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