Identifying Restrictions for Finite Parameter Continuous Time Models With Discrete Time Data

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 n/2 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. IDENTIFICATION AND ALIASING IN CONTINUOUS TIME MODELSThis paper develops identification results for first-order, linear systems of stochastic differential equations of the formin the case where observations of y(t) are only available at discrete, equispaced points in time separated by intervals of length h. Here, y(t) is a stationary n × 1 random vector, A is a parameter matrix whose elements are real numbers, and ζ(dt) is a vector of white noise innovations with covariance matrix dt. The exact discrete time process y t = y(th) corresponding to the continuous time system in (1) is the vector autoregressive (VAR) process

Peer Effects in Random Consideration Sets

“…This proof builds on Theorem 1 of Blevins (2017) and Theorem 3 of Blevins (2018). For the present case, it follows from these two theorems, that the transition rate matrix M is generically identified if, in addition to the conditions in Proposition 3.8, we have that…”

“…The advantage of this fact for model identification is that there are fewer terms to recover. Blevins (2017Blevins ( , 2018 offer a nice discussion of this feature and its advantage over discrete time models. (In Section 4.5, we extend the idea to coordinated clocks.)…”

“…The second case requires the researcher to observe the dynamic system at two different intervals ∆ 1 and ∆ 2 that are not multiples of each other. (See, e.g., Blevins, 2017, and the literature therein. )…”

“…Adding an extra mild condition, we show that the CCPs are also uniquely identified. For this last result, we invoke insights from Blevins (2017Blevins ( , 2018.…”

Quasi-maximum likelihood estimation for cointegrated continuous-time linear state space models observed at low frequencies

“…Due to Fasen-Hartmann and Scholz [21, Theorem 3.1] a canonical form for cointegrated state space processes already exists and can be used to construct a model class satisfying Assumption C and Assumption D. Further details are presented in Fasen-Hartmann and Scholz [20]. Moreover, criteria to overcome the aliasing effect (see Blevins [10], Hansen and Sargent [25], McCrorie [40,41], Phillips [42,43], Schlemm and Stelzer [51]) are given there.…”