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ESTIMATING CONTINUOUS-TIME MODELS ON THE BASIS OF DISCRETE DATA VIA AN EXACT DISCRETE ANALOG

Published online by Cambridge University Press:  01 August 2009

J. Roderick McCrorie*
Affiliation:
University of St. Andrews and CORE, Université catholique de Louvain
*
*Address correspondence to J.R. McCrorie, School of Economics and Finance, University of St Andrews, Castlecliffe, The Scores, St Andrews KY16 9AL, U.K.; e-mail: mccrorie@st-andrews.ac.uk.

Abstract

This paper offers a perspective on A.R. Bergstrom’s contribution to continuous-time modeling, focusing on his preferred method of estimating the parameters of a structural continuous-time model using an exact discrete-time analog. Some inherent difficulties in this approach are discussed, which help to explain why, in spite of his prescience, the methods around his time were not universally adopted as he had hoped. Even so, it is argued that Bergstrom’s contribution and legacy is secure and retains some relevance today for the analysis of macroeconomic and financial time series.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Aït-Sahalia, Y. (1996) Nonparametric pricing of interest rate derivative securities. Econometrica 64, 527560.CrossRefGoogle Scholar
Aït-Sahalia, Y. (2007) Estimating continuous-time models with discretely sampled data. In Blundell, R., Persson, T., & Newey, W. (eds.), Advances in Econometrics, Theory and Applications: Ninth World Congress of the Econometric Society, vol. 3, pp. 261327. Cambridge University Press.CrossRefGoogle Scholar
Aït-Sahalia, Y. & Mykland, P.A. (2003) The effects of random sampling and discrete sampling when estimating continuous-time diffusions. Econometrica 71, 483549.CrossRefGoogle Scholar
Aït-Sahalia, Y., Mykland, P.A., & Zhang, L. (2005) How often to sample a continuous-time process in the presence of market microstructure noise. Review of Financial Studies 18, 351416.CrossRefGoogle Scholar
Babbs, S.H. & Nowman, K.B. (1999) Kalman filtering of generalized Vasicek term structure models. Journal of Financial and Quantitative Analysis 34, 115130.CrossRefGoogle Scholar
Bandi, F.M. & Phillips, P.C.B. (2003) Fully nonparametric estimation of scalar diffusion models. Econometrica 71, 241283.CrossRefGoogle Scholar
Belcher, J., Hampton, J.S., & Tunnicliffe Wilson, G. (1994) Parametrization of continuous time autoregressive models for irregularly sampled time series data. Journal of the Royal Statistical Society, Series B 56, 141155.Google Scholar
Bergstrom, A.R. (1966) Non-recursive models as discrete approximations to systems of stochastic differential equations. Econometrica 34, 173182.CrossRefGoogle Scholar
Bergstrom, A.R. (1983) Gaussian estimation of structural parameters in higher-order continuous time dynamic models. Econometrica 51, 117152.CrossRefGoogle Scholar
Bergstrom, A.R. (1984) Continuous time stochastic models and issues of aggregation over time. In Griliches, Z. & Intriligator, M.D. (eds.), Handbook of Econometrics, vol. 2, pp. 11451221. North-Holland.CrossRefGoogle Scholar
Bergstrom, A.R. (1985) The estimation of parameters in nonstationary higher-order continuous time dynamic models. Econometric Theory 1, 369385.CrossRefGoogle Scholar
Bergstrom, A.R. (1986) The estimation of open higher-order continuous time dynamic models with mixed stock and flow data. Econometric Theory 2, 350373.CrossRefGoogle Scholar
Bergstrom, A.R. (1988) The history of econometric models. Econometric Theory 4, 365383.CrossRefGoogle Scholar
Bergstrom, A.R. (1990) Continuous Time Econometric Modelling. Oxford University Press.Google Scholar
Bergstrom, A.R. (1997) Gaussian estimation of mixed order continuous time dynamic models with unobservable stochastic trends from mixed stock and flow data. Econometric Theory 13, 467505.CrossRefGoogle Scholar
Bergstrom, A.R. (2009) The effects of differencing on the Gaussian likelihood of models with unobservable stochastic trends: A simple example. Econometric Theory 25 (this issue), 903913.CrossRefGoogle Scholar
Bergstrom, A.R. & Nowman, K.B. (1999) Gaussian estimation of a two-factor continuous time model of the short-term interest rate. Economic Notes 28, 2541.CrossRefGoogle Scholar
Bergstrom, A.R. & Nowman, K.B. (2007) A Continuous Time Econometric Model of the United Kingdom with Stochastic Trends. Cambridge University Press.CrossRefGoogle Scholar
Bergstrom, A.R., Nowman, K.B., & Wymer, C.R. (1992) Gaussian estimation of a second order continuous time macroeconometric model of the United Kingdom. Economic Modelling 9, 313351.CrossRefGoogle Scholar
Bergstrom, A.R. & Wymer, C.R. (1976) A model of disequilibrium neoclassical growth and its application to the United Kingdom. In Bergstrom, A.R. (ed.), Statistical Inference in Continuous Time Economic Models, pp. 267327. North-Holland.Google Scholar
Bernstein, D.S. & So, W. (1993) Some explicit formulas for the matrix exponential. IEEE Transactions on Automatic Control 38, 12281232.CrossRefGoogle Scholar
Bourgeois, G. (2007) On commuting exponentials in low dimensions. Linear Algebra and its Applications 423, 277286.CrossRefGoogle Scholar
Campbell, J.Y., Chacko, G., Rodriguez, J., & Viceira, L.M. (2004) Strategic asset allocation in a continuous-time VAR model. Journal of Economic Dynamics and Control 28, 21952214.CrossRefGoogle Scholar
Chambers, M.J. (1999) Discrete time representation of stationary and nonstationary continuous time systems. Journal of Economic Dynamics and Control 23, 619639.CrossRefGoogle Scholar
Chambers, M.J. (2003) The asymptotic efficiency of cointegration estimators under temporal aggregation. Econometric Theory 19, 4977.CrossRefGoogle Scholar
Chambers, M.J. (2004) Testing for unit roots with flow data and varying sampling frequency. Journal of Econometrics 119, 118. (Corrigendum: Journal of Econometrics 144 (2008), 524–525)CrossRefGoogle Scholar
Chambers, M.J. (2006) On Excess Differencing in Discrete Time Representations of Cointegrated Continuous Time Models with Mixed Sample. Mimeo, University of Essex.Google Scholar
Chambers, M.J. (2009) Discrete time representations of cointegrated continuous time models with mixed sample data. Econometric Theory 25 (this issue), 10301049.CrossRefGoogle Scholar
Chambers, M.J. & McCrorie, J.R. (2006) Identification and estimation of exchange rate models with unobservable fundamentals. International Economic Review 47, 573582.CrossRefGoogle Scholar
Chambers, M.J. & McCrorie, J.R. (2007) Frequency domain estimation of temporally aggregated Gaussian cointegrated systems. Journal of Econometrics 136, 129.CrossRefGoogle Scholar
Chen, B. & Zadrozny, P. (2001) Analytic derivatives for the matrix exponential for estimation of linear continuous-time models. Journal of Economic Dynamics and Control 25, 18671879.CrossRefGoogle Scholar
Chen, R. & Scott, L. (1992) Pricing interest rate options in a two-factor Cox-Ingersoll-Ross model of the term structure. Review of Financial Studies 5, 613636.CrossRefGoogle Scholar
Christiano, L.J. (2005) A method for estimating the timing interval in a linear econometric model with an application to Taylor’s model of staggered contracts. Journal of Economic Dynamics and Control 9, 363404.CrossRefGoogle Scholar
Christiano, L.J. & Eichenbaum, M. (1987) Temporal aggregation and structural inference in macroeconomics. Carnegie-Rochester Conference Series on Public Policy 26, 63130.CrossRefGoogle Scholar
Culver, W.J. (1966) On the existence and uniqueness of the real logarithm of a matrix. Proceedings of the American Mathematical Society 17, 11461151.CrossRefGoogle Scholar
Duffie, D. & Glynn, P. (2004) Estimation of continuous-time Markov processes sampled at random time intervals. Econometrica 7, 17731808.CrossRefGoogle Scholar
Engel, C, Mark, N.C., & West, K.D. (2007) Exchange models are not as bad as you think. NBER Macroeconomics Annual 22, 381441.CrossRefGoogle Scholar
Ercolani, J.S. (2009) Cyclical trends in continuous time models. Econometric Theory 25 (this issue), 11121119.CrossRefGoogle Scholar
Erickson, R.V. (1971) Constant coefficient linear differential equations driven by white noise. Annals of Mathematical Statistics 42, 820823.CrossRefGoogle Scholar
Evans, M.D. & Lyons, R.K. (2002) Order flow and exchange rate dynamics. Journal of Political Economy 110, 170180.CrossRefGoogle Scholar
Franchi, M. (2007) The integration order of vector autoregressive processes. Econometric Theory 23, 546553.CrossRefGoogle Scholar
Geweke, J.B. (1978) Temporal aggregation in the multivariate regression model. Econometrica 46, 643662.CrossRefGoogle Scholar
Hansen, L.P. & Sargent, T.J. (1981) The Dimensionality of the Aliasing Problem in Models with Rational Spectral Densities. Research Department Staff Report 72, Federal Reserve Bank of Minneapolis.CrossRefGoogle Scholar
Hansen, L.P. & Sargent, T.J. (1983) The dimensionality of the aliasing problem in models with rational spectral densities. Econometrica 51, 377387.CrossRefGoogle Scholar
Hansen, L.P. & Sargent, T.J. (1991) Identification of continuous time rational expectations from discrete data. In Hansen, L.P. and Sargent, T.J. (eds.), Rational Expectations Econometrics, pp. 219235. Westview Press.Google Scholar
Hansen, L.P. & Scheinkman, J.A. (1995) Back to the future: generating moment implications for continuous-time Markov processes. Econometrica 63, 767804.CrossRefGoogle Scholar
Hansen, L.P., Scheinkman, J.A., & Touzi, N. (1998) Spectral methods for identifying scalar diffusions. Journal of Econometrics 86, 132.CrossRefGoogle Scholar
Harvey, A.C. & Stock, J.H. (1985) The estimation of higher-order continuous time autoregressive models. Econometric Theory 1, 97117.CrossRefGoogle Scholar
Harvey, A.C. & Stock, J.H. (1988) Continuous time autoregressive models with common stochastic trends. Journal of Economic Dynamics and Control 12, 365384.CrossRefGoogle Scholar
Harvey, A.C. & Stock, J.H. (1989) Estimating integrated higher-order continuous time autoregressions with an application to money-income causality. Journal of Econometrics 42, 319336.CrossRefGoogle Scholar
Harvey, A.C. & Stock, J.H. (1993) Estimation, smoothing, interpolation and distribution for structural time series models in continuous time. In Phillips, P.C.B. (ed.), Models, Methods and Applications of Econometrics, pp. 5570. Blackwell.Google Scholar
Heaton, J. (1983) The interaction between time-nonseparable preferences and time aggregation. Econometrica 61, 353386.CrossRefGoogle Scholar
Hille, E. (1958) On roots and logarithms of elements of a complex Banach algebra. Mathematische Annalen 136, 4657.CrossRefGoogle Scholar
Jordà, O. & Marcellino, M. (2003) Modeling high-frequency foreign exchange data dynamics. Macroeconomic Dynamics 7, 618635.CrossRefGoogle Scholar
Kessler, M. & Rahbek, A. (2004) Identification and inference for multivariate cointegrated and ergodic Gaussian diffusions. Statistical Inference for Stochastic Processes 7, 137151.CrossRefGoogle Scholar
Kwakernaak, H. & Sivan, R. (1972) Linear Optimal Control Systems. Wiley.Google Scholar
Lii, K.-S. & Masry, E. (1992) Model fitting for continuous-time processes from discrete time data. Journal of Multivariate Analysis 30, 5679.CrossRefGoogle Scholar
Lii, K.-S. & Masry, E. (1994) Spectral estimation of continuous-time stationary processes from random sampling. Stochastic Processes and their Applications 52, 3964.CrossRefGoogle Scholar
Lyons, R. (2001) The Microstructure Approach to Exchange Rates. MIT Press.CrossRefGoogle Scholar
Marcet, A. (1991) Temporal aggregation of economic time series. In Hansen, L.P. and Sargent, T.J. (eds.), Rational Expectations Econometrics, pp. 237282. Westview Press.Google Scholar
McCrorie, J.R. (2000) Deriving the exact discrete analog of a continuous time system. Econometric Theory 16, 9981015.CrossRefGoogle Scholar
McCrorie, J.R. (2001) Interpolating exogenous variables in open continuous time dynamic models. Journal of Economic Dynamics and Control 25, 13991427.CrossRefGoogle Scholar
McCrorie, J.R. (2002) The likelihood of the parameters of a continuous time vector autoregressive model. Statistical Inference for Stochastic Processes 5, 273286.CrossRefGoogle Scholar
McCrorie, J.R. (2003) The problem of aliasing in identifying finite parameter continuous time models. Acta Applicandae Mathematicae 79, 916.CrossRefGoogle Scholar
McCrorie, J.R. (2009) The Integration Order of Vector Autoregressive Processes. Mimeo, University of St. Andrews.Google Scholar
McCrorie, J.R. & Chambers, M.J. (2006) Granger causality and the sampling of economic processes. Journal of Econometrics 132, 311326.CrossRefGoogle Scholar
McGarry, J.S. (2003) The exact discrete time representation of a system of fourth-order differential equations. Computers and Mathematics with Applications 46, 213230.CrossRefGoogle Scholar
Morinaga, K. & Nono, T. (1954) On the non-commutative solution of the exponential equation exey = ex +y, II. Journal of Science of the Hiroshima University, Series A 18, 137178.Google Scholar
Nowman, K.B. (1997) Gaussian estimation of single-factor continuous time models of the term structure of interest rates. Journal of Finance 52, 16951703.Google Scholar
Nowman, K.B. (2009) Rex Bergstrom’s contributions to continuous-time macroeconometric modeling. Econometric Theory 25 (this issue), 10871098.CrossRefGoogle Scholar
Phillips, P.C.B. (1972) The structural estimation of a stochastic differential-equation system. Econometrica 40, 10211041.CrossRefGoogle Scholar
Phillips, P.C.B. (1973) The problem of identification in finite parameter continuous time models. Journal of Econometrics 1, 351362.CrossRefGoogle Scholar
Phillips, P.C.B. (1974). The estimation of some continuous time models, Econometrica, 42, 803824.CrossRefGoogle Scholar
Phillips, P.C.B. (1978) The treatment of flow data in the estimation of continuous time systems. In Bergstrom, A.R., Catt, A.J.L., Peston, M.H., & Silverstone, B.D.J. (eds.), Stability and Inflation, pp. 257274. Wiley.Google Scholar
Phillips, P.C.B. (1988) The ET interview: Professor A.R. Bergstrom. Econometric Theory 4, 301327.CrossRefGoogle Scholar
Phillips, P.C.B. (1991) Error correction and long-run equilibrium in continuous time. Econometrica 59, 967980.CrossRefGoogle Scholar
Phillips, P.C.B. (2005) Albert Rex Bergstrom 1925–2005. New Zealand Economic Papers 39, 129152.CrossRefGoogle Scholar
Phillips, P.C.B. & Yu, J. (2005) Jackknifing bond option prices. Review of Financial Studies 18, 707742.CrossRefGoogle Scholar
Phillips, P.C.B. & Yu, J. (2009) Maximum likelihood and Gaussian estimation of continuous time models in finance. In Anderson, T.G., Davis, R.A., Kreiss, J.-P., & Mikosch, T. (eds.), Handbook of Financial Time Series. Springer, forthcoming.Google Scholar
Robinson, P.M. (1976) Fourier estimation of continuous time models. In Bergstrom, A.R. (ed.), Statistical Inference in Continuous Time Economic Models, pp. 215266. North-Holland.Google Scholar
Robinson, P.M. (1977) The construction and estimation of continuous time models and discrete approximations in econometrics. Journal of Econometrics 6, 173198.CrossRefGoogle Scholar
Robinson, P.M. (1980) Continuous model fitting from discrete data. In Brillinger, D.R. and Tiao, G.C. (eds.), Directions in Time Series, pp. 263278. Institute of Mathematical Statistics.Google Scholar
Robinson, P.M. (1992) Review of A.R. Bergstrom’s “Continuous Time Econometric Modelling.” Econometric Theory 8, 571579.CrossRefGoogle Scholar
Robinson, P.M. (1993) Continuous time models in econometrics: Closed and open systems, stocks and flows. In Phillips, P.C.B. (ed.),Models, Methods and Applications of Econometrics, pp. 7190. Blackwell.Google Scholar
Robinson, P.M. (2009) On discrete sampling of time-varying continuous-time systems. Econometric Theory 25 (this issue), 985994.CrossRefGoogle Scholar
Sargan, J.D. (1974) Some discrete approximations to continuous time stochastic models. Journal of the Royal Statistical Society, Series B 36, 7490.Google Scholar
Sims, C.A. (1971) Discrete approximations to continuous time lag distributions in econometrics. Econometrica 39, 545564.CrossRefGoogle Scholar
Stock, J.H. (1987) Temporal aggregation and structural inference in macroeconomics: A comment. Carnegie-Rochester Conference Series in Public Policy 26, 131140.CrossRefGoogle Scholar
Van Loan, C.F. (1978) Computing integrals involving the matrix exponential. IEEE Transactions on Automatic Control AC-23, 395404.CrossRefGoogle Scholar
Wymer, C. (1972) Econometric estimation of stochastic differential-equation systems. Econometrica 40, 565577.CrossRefGoogle Scholar
Yu, J. (2008) Bias in the Estimation of Mean Reversion Parameter in Continuous Time Models. Mimeo, Singapore Management University.Google Scholar
Yu, J. & Phillips, P.C.B. (2001) A Gaussian approach for continuous time models of the short-term interest rate. Econometrics Journal 4, 221225.CrossRefGoogle Scholar
Zadrozny, P. (1988) Gaussian likelihood of continuous-time ARMAX models when data are stocks and flows at different frequencies. Econometric Theory 4, 108124.CrossRefGoogle Scholar
Zhang, L, Mykland, P.A., & Aït-Sahalia, Y. (2005) A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association 100, 13941411.CrossRefGoogle Scholar
Zwanziger, D. (1964) Simple theorem on Hermitian matrices and an application to the polarization of vector particles. Physical Review 136, B558B562.CrossRefGoogle Scholar