Linear regression in continuous time

Abstract

We consider a regression relation of the from wherein y(t) and x(t) are real (column) vectors of q and p components and e(t) is real and is generated by a stationary generalised vector process of q components with zero mean and covariance function (a q rowed matrix) Γ(t–s) = E{x(s)x(t)′}. (See Hannan (1970; pages 23–26, 91–94) and references therein for definitions of terms used.) We assume e(t) to be independent of x(s) for all s, t. Thus we may regard x(t) as a fixed time function and not stochastic and we shall henceforth do that. We take Γ(t) to be continuous and to correspond to an absolutely continuous spectral function with spectral density which is uniformly bounded and continuous. Then we have We do not exclude the possibility that for theyth diagonal element, fjj, of fwe have

Estimating the quadratic covariation matrix for asynchronously observed high frequency stock returns corrupted by additive measurement error

“…Practitioners often encounter a problem of running a regression between variables that are asynchronously observed; for example, we might be interested in the effect of returns and order book information of one asset on another asset. Hannan (1975) and Robinson (1975) are the earlier literature on using frequency domain to solve such problems. Mykland and Zhang (2006) discussed a general setup of the analysis of variance for continuous time regression.…”

Regression for randomly sampled spatial series: the trigonometric case

“…Trigonometric regression with unknown frequencies ('hidden periodicities') was considered in Whittle (1952), Hannan (1971Hannan ( ), (1973 and Walker (1971), (1973). The case of linear regression for continuous time has been examined in Chiang (1959), Kholevo (1969), (1971) and Hannan (1975). Again, least squares estimates are found to be asymptotically efficient for trigonometric 276 DAVID R. BRILLINGER regression with known frequencies.…”

The BLUE in continuous-time regression models with correlated errors

“…For a general regression model with m ≥ 1 regression functions (and q = 0), the BLUE was extensively discussed in Grenander (1954) and Rosenblatt (1956) who considered stationary processes in discrete time, where the spectral representation of the error process was heavily used for the construction of the estimators. In this and many other papers including Kholevo (1969) and Hannan (1975) the subject of the study was concentrated around the spectral representation of the estimators and hence the results in these references are only applicable to very specific models. A more direct investigation of the BLUE in the location scale model (with q = 0) can be found in Hájek (1956), where equation (1.3) for the BLUE was solved for a few simple kernels.…”