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Multivariate Simultaneous Generalized ARCH
- Robert F. Engle, Kenneth F. Kroner
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- Journal:
- Econometric Theory / Volume 11 / Issue 1 / February 1995
- Published online by Cambridge University Press:
- 11 February 2009, pp. 122-150
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This paper presents theoretical results on the formulation and estimation of multivariate generalized ARCH models within simultaneous equations systems. A new parameterization of the multivariate ARCH process is proposed, and equivalence relations are discussed for the various ARCH parameterizations. Constraints sufficient to guarantee the positive definiteness of the conditional covariance matrices are developed, and necessary and sufficient conditions for covariance stationarity are presented. Identification and maximum likelihood estimation of the parameters in the simultaneous equations context are also covered.
19 - A Long Memory Property of Stock Market Returns and a New Model
- Clive W. J. Granger
- Edited by Eric Ghysels, University of North Carolina, Chapel Hill, Norman R. Swanson, Rutgers University, New Jersey, Mark W. Watson, Princeton University, New Jersey
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- Book:
- Essays in Econometrics
- Published online:
- 06 July 2010
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- 23 July 2001, pp 349-372
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Summary
Abstract
A “long memory” property of stock market returns is investigated in this paper. It is found that not only there is substantially more correlation between absolute returns than returns themselves, but the power transformation of the absolute turn |rt|d also has quite high autocorrelation for long lags. It is possible to characterize |rt|d to be “long memory” and this property is strongest when d is around 1. This result appears to argue against ARCH type specifications based upon squared returns. But our Monte-Carlo study shows that both ARCH type models based on squared returns and those based on absolute return can produce this property. A new general class of models is proposed which allows the power δ of the heteroskedasticity equation to be estimated from the data.
INTRODUCTION
If rt is the return from a speculative asset such as a bond or stock, this paper considers the temporal properties of the functions |rt|d for positive values of d. It is well known that the returns themselves contain little serial correlation, in agreement with the efficient market theory. However, Taylor (1986) found that |rt| has significant positive serial correlation over long lags. This property is examined on long daily stock market price series. It is possible to characterize |rt|d to be “longmemory”, with quite high autocorrelations for long lags. It is also found, as an empirical fact, that this property is strongest for d = 1 or near 1 compared to both smaller and larger positive values of d.
8 - Co-Integration and Error-Correction: Representation, Estimation, and Testing
- Clive W. J. Granger
- Edited by Eric Ghysels, University of North Carolina, Chapel Hill, Norman R. Swanson, Rutgers University, New Jersey, Mark W. Watson, Princeton University, New Jersey
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- Book:
- Essays in Econometrics
- Published online:
- 06 July 2010
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- 23 July 2001, pp 145-172
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Summary
The relationship between co-integration and error correction models, first suggested in Granger (1981), is here extended and used to develop estimation procedures, tests, and empirical examples.
If each element of a vector of time series xt first achieves stationarity after differencing, but a linear combination α′xt is already stationary, the time series xt are said to be co-integrated with co-integrating vector α. There may be several such co-integrating vectors so that α becomes a matrix. Interpreting α′xt = 0 as a long run equilibrium, co-integration implies that deviations from equilibrium are stationary, with finite variance, even though the series themselves are nonstationary and have infinite variance.
The paper presents a representation theorem based on Granger (1983), which connects the moving average, autoregressive, and error correction representations for co-integrated systems. A vector autoregression in differenced variables is incompatible with these representations. Estimation of these models is discussed and a simple but asymptotically efficient two-step estimator is proposed. Testing for co-integration combines the problems of unit root tests and tests with parameters unidentified under the null. Seven statistics are formulated and analyzed. The critical values of these statistics are calculated based on a Monte Carlo simulation. Using these critical values, the power properties of the tests are examined and one test procedure is recommended for application.
In a series of examples it is found that consumption and income are co-integrated, wages and prices are not, short and long interest rates are, and nominal GNP is co-integrated with M2, but not M1, M3, or aggregate liquid assets.
Looking Backward, Looking Forward: MLA Members Speak
- April Alliston, Elizabeth Ammons, Jean Arnold, Nina Baym, Sandra L. Beckett, Peter G. Beidler, Roger A. Berger, Sandra Bermann, J.J. Wilson, Troy Boone, Alison Booth, Wayne C. Booth, James Phelan, Marie Borroff, Ihab Hassan, Ulrich Weisstein, Zack Bowen, Jill Campbell, Dan Campion, Jay Caplan, Maurice Charney, Beverly Lyon Clark, Robert A. Colby, Thomas C. Coleman III, Nicole Cooley, Richard Dellamora, Morris Dickstein, Terrell Dixon, Emory Elliott, Caryl Emerson, Ann W. Engar, Lars Engle, Kai Hammermeister, N. N. Feltes, Mary Anne Ferguson, Annie Finch, Shelley Fisher Fishkin, Jerry Aline Flieger, Norman Friedman, Rosemarie Garland-Thomson, Sandra M. Gilbert, Laurie Grobman, George Guida, Liselotte Gumpel, R. K. Gupta, Florence Howe, Cathy L. Jrade, Richard A. Kaye, Calhoun Winton, Murray Krieger, Robert Langbaum, Richard A. Lanham, Marilee Lindemann, Paul Michael Lützeler, Thomas J. Lynn, Juliet Flower MacCannell, Michelle A. Massé, Irving Massey, Georges May, Christian W. Hallstein, Gita May, Lucy McDiarmid, Ellen Messer-Davidow, Koritha Mitchell, Robin Smiles, Kenyatta Albeny, George Monteiro, Joel Myerson, Alan Nadel, Ashton Nichols, Jeffrey Nishimura, Neal Oxenhandler, David Palumbo-Liu, Vincent P. Pecora, David Porter, Nancy Potter, Ronald C. Rosbottom, Elias L. Rivers, Gerhard F. Strasser, J. L. Styan, Marianna De Marco Torgovnick, Gary Totten, David van Leer, Asha Varadharajan, Orrin N. C. Wang, Sharon Willis, Louise E. Wright, Donald A. Yates, Takayuki Yokota-Murakami, Richard E. Zeikowitz, Angelika Bammer, Dale Bauer, Karl Beckson, Betsy A. Bowen, Stacey Donohue, Sheila Emerson, Gwendolyn Audrey Foster, Jay L. Halio, Karl Kroeber, Terence Hawkes, William B. Hunter, Mary Jambus, Willard F. King, Nancy K. Miller, Jody Norton, Ann Pellegrini, S. P. Rosenbaum, Lorie Roth, Robert Scholes, Joanne Shattock, Rosemary T. VanArsdel, Alfred Bendixen, Alarma Kathleen Brown, Michael J. Kiskis, Debra A. Castillo, Rey Chow, John F. Crossen, Robert F. Fleissner, Regenia Gagnier, Nicholas Howe, M. Thomas Inge, Frank Mehring, Hyungji Park, Jahan Ramazani, Kenneth M. Roemer, Deborah D. Rogers, A. LaVonne Brown Ruoff, Regina M. Schwartz, John T. Shawcross, Brenda R. Silver, Andrew von Hendy, Virginia Wright Wexman, Britta Zangen, A. Owen Aldridge, Paula R. Backscheider, Roland Bartel, E. M. Forster, Milton Birnbaum, Jonathan Bishop, Crystal Downing, Frank H. Ellis, Roberto Forns-Broggi, James R. Giles, Mary E. Giles, Susan Blair Green, Madelyn Gutwirth, Constance B. Hieatt, Titi Adepitan, Edgar C. Knowlton, Jr., Emanuel Mussman, Sally Todd Nelson, Robert O. Preyer, David Diego Rodriguez, Guy Stern, James Thorpe, Robert J. Wilson, Rebecca S. Beal, Joyce Simutis, Betsy Bowden, Sara Cooper, Wheeler Winston Dixon, Tarek el Ariss, Richard Jewell, John W. Kronik, Wendy Martin, Stuart Y. McDougal, Hugo Méndez-Ramírez, Ivy Schweitzer, Armand E. Singer, G. Thomas Tanselle, Tom Bishop, Mary Ann Caws, Marcel Gutwirth, Christophe Ippolito, Lawrence D. Kritzman, James Longenbach, Tim McCracken, Wolfe S. Molitor, Diane Quantic, Gregory Rabassa, Ellen M. Tsagaris, Anthony C. Yu, Betty Jean Craige, Wendell V. Harris, J. Hillis Miller, Jesse G. Swan, Helene Zimmer-Loew, Peter Berek, James Chandler, Hanna K. Charney, Philip Cohen, Judith Fetterley, Herbert Lindenberger, Julia Reinhard Lupton, Maximillian E. Novak, Richard Ohmann, Marjorie Perloff, Mark Reynolds, James Sledd, Harriet Turner, Marie Umeh, Flavia Aloya, Regina Barreca, Konrad Bieber, Ellis Hanson, William J. Hyde, Holly A. Laird, David Leverenz, Allen Michie, J. Wesley Miller, Marvin Rosenberg, Daniel R. Schwarz, Elizabeth Welt Trahan, Jean Fagan Yellin
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- Journal:
- PMLA / Publications of the Modern Language Association of America / Volume 115 / Issue 7 / December 2000
- Published online by Cambridge University Press:
- 23 October 2020, pp. 1986-2078
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- December 2000
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7 - The Kalman filter: applications to forecasting and rational-expectations models
- Edited by Truman F. Bewley, Yale University, Connecticut
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- Book:
- Advances in Econometrics
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- 05 January 2013
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- 27 November 1987, pp 245-284
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Summary
Introduction
Economics is not engineering; yet, perhaps, we can track the economy using the same tools used to track a spacecraft, an oil tanker, or a chemical reaction. In the 25 years since the publication of the original Kalman (1960) and Kalman and Bucy (1961) papers that introduced digital filters for nonstationary problems, economists have been studying these possibilities, and the presence of the August 1985 session of the World Congress of the Econometric Society suggests that it is still a question of great interest.
The initial attempts to apply these methods to economic problems immediately faced a major difficulty. Engineers usually had quantitative theories that described the equations of motion of physical systems and were primarily interested in estimates of the “state” of the system obtained from noisy measurements. The extraction of estimates of such signals from noise was called the estimation, or “state estimation,” problem. Economists, however, knew far less about the fundamental laws of motion of economic systems and were therefore particularly interested in discovering such laws of motion from the noisy data rather than in merely estimating the state of the economy. Since the Kalman filter takes the parameters of the process as given in estimating the state, it appeared that there would be little possibility to apply such methods in economics.