Abstract
A state space model with an unobserved multivariate random walk and a linear observation equation is studied. The purpose is to find out when the extracted trend cointegrates with its estimator, in the sense that a linear combination is asymptotically stationary. It is found that this result holds for the linear combination of the trend that appears in the observation equation. If identifying restrictions are imposed on either the trend or its coefficients in the linear observation equation, it is shown that there is cointegration between the identified trend and its estimator, if and only if the estimators of the coefficients in the observation equations are consistent at a faster rate than the square root of sample size. The same results are found if the observations from the state space model are analysed using a cointegrated vector autoregressive model. The findings are illustrated by a small simulation study.
Highlights
Introduction and SummaryThis paper is inspired by a study on long-run causality, see Hoover et al (2014)
The paper analyses a sample of n observations from a common trend model, where the state is an unobserved multivariate random walk and the observation is a linear combination of the lagged state variable and a noise term
The model leads naturally to cointegration between observations, trends, and the extracted trends. Using simulations it was discovered, that the extracted trends do not necessarily cointegrate with their estimators. This problem is investigated, and it is found to be related to the identification of the trends and their coefficients in the observation equation
Summary
This paper is inspired by a study on long-run causality, see Hoover et al (2014). Causality is usually studied for a sequence of multivariate i.i.d. variables using conditional independence, see Spirtes et al (2000) or Pearl (2009). In Hoover et al (2014), the concept is formulated in terms of independent common trends and their causal impact coefficients on the nonstationary observations. Et yt+1 = BEt Tt , Vart (yt+1 ) = BVt B0 + Ωε In this model it is clear that yt and Tt cointegrate, that is, yt+1 − BTt+1 = ε t+1 − Bηt+1 is stationary, and the same holds for Tt and the extracted trend Et Tt = E( Tt |y1 , . If the trends and their coefficients are identified by the trends being independent, the trend extracted by the state space model does not cointegrate with its estimator. The trends are identified by restrictions on the coefficients alone, they do cointegrate
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