Article

# Identifying Restrictions for Finite Parameter Continuous Time Models with Discrete Time Data

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## Abstract

This paper revisits the question of parameter identification when a linear continuous time model is sampled only at equispaced points in time. Following the framework and assumptions of Phillips (1973), we consider models characterized by first-order, linear systems of stochastic differential equations and use a priori restrictions on the model parameters as identifying restrictions. A practical rank condition is derived to test whether any particular collection of at least $\left\lfloor {n/2} \right\rfloor$ general linear restrictions on the parameter matrix is sufficient for identification. We then consider extensions to incorporate prior restrictions on the covariance matrix of the disturbances, to identify the covariance matrix itself, and to address identification in models with cointegration.

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... In the case of continuous-time data the transition rate matrix is identified without any extra restrictions. To identify the transition rate matrix using discrete-time data with arbitrary time intervals, we invoke insights from Blevins (2017Blevins ( , 2018. This latter result exploits the fact that the transition rate matrix in our model is rather parsimonious. ...
... The second one occurs when the researcher can observe the dynamic system at two different length intervals ∆ 1 and ∆ 2 that are not multiple of each other. (See, e.g., Blevins (2017) and the literature therein.) ...
... Proof This proof builds on Theorem 1 of Blevins (2017) and Theorem 3 of Blevins (2018). For the present case, it follows from the last two theorems, that the transition rate matrix M is generically identified if, in addition to the conditions in Proposition 3.3, we have that ...
Preprint
This paper develops a dynamic model of discrete choice that incorporates peer effects into consideration sets. We characterize equilibrium behavior and study the empirical content of the dynamic model we offer. In our set-up, the choices of friends act as exclusion restrictions in the stochastic variation of the subset of alternatives that each person considers at the moment of picking an option. They allow us to recover (from a sequence of observed choices) the ranking of preferences of each person, the attention mechanism, and the set of connections or nodes between the people in the network. The identification strategy we offer does not rely on the variation of the set of available options (or menus) which remain the same across all the observations.
... Unfortunately, both restrictions are not generally appropriate for economic time series: they rule out plausible cyclical behaviour resulting from complex eigenvalues and plausible trend behavior resulting from multiple unit roots (multiple zero eigenvalues of A). In the complex eigenvalue case, several authors achieve identification through additional restrictions: Phillips (1973) uses Cowles Commission type restrictions (see also Blevins, 2017) and Hansen and Sargent (1983) show there are restrictions inherent in the requirement that Ω ǫ be positive semidefinite. Hansen and Sargent (1991) use cross-equation restrictions implied by the rational expectations hypothesis. ...
Chapter
This chapter provides a survey of methods of continuous time modelling based on an exact discrete time representation. It begins by highlighting the techniques involved with the derivation of an exact discrete time representation of an underlying continuous time model, providing specific details for a second-order linear system of stochastic differential equations. Issues of parameter identification, Granger causality, nonstationarity and mixed frequency data are addressed, all being important considerations in applications in economics and other disciplines. Although the focus is on Gaussian estimation of the exact discrete time model, alternative time domain (state space) and frequency domain approaches are also discussed. Computational issues are explored, and two new empirical applications are included along with a discussion of applications in the field of macroeconometric modelling.
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I am very grateful to the Executive Editor, Edward George, for organizing this stimulating discussion. I would like to take this opportunity to thank Pro- fessors Peter Phillips, Jun Yu, Michael Sørensen, Per Mykland and Lan Zhang for their insightful and stimu- lating comments, touching both practical, methodolog- ical and theoretical aspects of financial econometrics and their applications in asset pricing, portfolio alloca- tion and risk management. They have made valuable contributions to the understanding of various financial econometric problems. The last two decades have witnessed an explosion of developments of data-analytic techniques in statis- tical modeling and analysis of complex systems. At the same time, statistical techniques have been widely employed to confront various complex problems aris- ing from financial and economic activities. While the discipline has grown rapidly over the last two decades and has rich and challenging statistical problems, the number of statisticians involved in studying financial econometric problems is still limited. In comparison with statisticians working on problems in biological sciences and medicine, the group working on finan- cial and econometric problems is dismally small. It is my hope that this article will provide statisticians with quick access to some important and interesting prob- lems in financial econometrics and to catalyze the ro- mance between statistics and finance. A similar effort was made by Cai and Hong (12), where various aspects of nonparametric methods in continuous-time finance are reviewed. It is my intention to connect financial econometric problems as closely to statistical problems as possible so that familiar statistical tools can be em- ployed. With this in mind, I sometimes oversimplify the problems and techniques so that key features can be highlighted.
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Estimating a dynamic game of spatial competition: e case of the U.K. supermarket industry. Working paper, London School of Economics Discrete approximations to continuous time distributed lags in econometrics
• Springer
• P Schiraldi
• H Smith
• Y Takahashi
Springer. Schiraldi, P., H. Smith, and Y. Takahashi (). Estimating a dynamic game of spatial competition: e case of the U.K. supermarket industry. Working paper, London School of Economics. Sims, C. A. (). Discrete approximations to continuous time distributed lags in econometrics. Econometrica , –.
Estimating continuous-time models using discretely sampled data Ellickson () Estimation of dynamic discrete choice models in continuous time Nonstationary continuous-time processes Financial Calculus: An Introduction to Derivative Pricing
• Y Aït-Sahalia
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• P Bayer
• J R Blevins
Aït-Sahalia, Y. (). Estimating continuous-time models using discretely sampled data. In T. P. R. Blundell and W. K. Newey (Eds.), Advances in Economics and Econometrics: eory and Applications, Ninth World Congress. Cambridge University Press.  Arcidiacono, P., P. Bayer, J. R. Blevins, and P. B. Ellickson (). Estimation of dynamic discrete choice models in continuous time. Working Paper , National Bureau of Economic Research. Bandi, F. M. and P. C. B. Phillips (). Nonstationary continuous-time processes. In Y. Aït-Sahalia and L. P. Hansen (Eds.), Handbook of Financial Econometrics, Volume , Chapter . Amsterdam: North Holland. Baxter, M. and A. Rennie (). Financial Calculus: An Introduction to Derivative Pricing. Cambridge University Press. Bergstrom, A. R. (). e history of continuous-time econometric models. Econometric eory , –.
Estimating a dynamic game of spatial competition: The case of the U.K. supermarket industry. Working paper
• P Schiraldi
• H Smith
• Y Takahashi
Schiraldi, P., H. Smith, and Y. Takahashi (). Estimating a dynamic game of spatial competition: e case of the U.K. supermarket industry. Working paper, London School of Economics.
Ellickson () Estimation of dynamic discrete choice models in continuous time Working Paper  Nonstationary continuous-time processes
• P Arcidiacono
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Arcidiacono, P., P. Bayer, J. R. Blevins, and P. B. Ellickson (). Estimation of dynamic discrete choice models in continuous time. Working Paper , National Bureau of Economic Research.  Bandi, F. M. and P. C. B. Phillips (). Nonstationary continuous-time processes. In Y. Aït-Sahalia and L. P. Hansen (Eds.), Handbook of Financial Econometrics, Volume , Chapter . Amsterdam: North Holland.
). e eory of Matrices
• F R Gantmacher
Gantmacher, F. R. (). e eory of Matrices, Volume . New York: Chelsea.
A selective overview of nonparametric methods in financial econometrics
• Fan