Estimation of impulse response functions in two-output systems

On cross-correlogram IRF’s estimators of two-output systems in spaces of continuous functions

“…Our study of weak convergence of a centred estimator H in Banach spaces of continuous functions leads to a situation where two types of processes appear (Blazhievska and Zaiats 2018): a limiting non-stationary Gaussian process and a pre-limiting process which is stationary and square-Gaussian, belonging to an Orlicz space. Almost sure (a. s.) continuity is proved by means of the entropy approach related to these classes of processes, see (Dudley 1973), (Fernique 1975), (Lifshits 1995, Chapter 14), (Buldygin and Kozachenko 2000, Chapter 4) and (Kozachenko et al 2017, Chapter 6), while the classical Prokhorov theorem is employed in proving a CLT.…”

“…In this paper, we assume that H is unknown while g is known (controlled), and our aim is to estimate H after observations of the outputs X and Y . For a detailed survey of deterministic and statistical approaches to the mentioned problem including correlogram and periodogram methods, we refer the reader to (Blazhievska and Zaiats 2018). Our main idea is based on the fact that H can be obtained by cross-correlating the X and Y given that g is "close" to Dirac's delta:…”

“…A similar problem was considered in (Li 2005) where the centred estimator H was proved to be asymptotically normal. A recent paper (Blazhievska and Zaiats 2018) has removed extra assumptions on H reducing them to the one and only condition of H ∈ L 2 (R). An extension to SIDO-systems has been new.…”