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The article studies a class of state-space models where the state equation is a local-to-unity process. The parameter of interest is the persistence parameter of the latent process. The large sample theory for the least squares (LS) estimator and an instrumental variable (IV) estimator of the persistent parameter in the autoregressive (AR) representation of the model is developed under two sets of conditions. In the first set of conditions, the measurement error is independent and identically distributed, and the error term in the state equation is stationary and fractionally integrated with memory parameter d ∈ (− 0.5 , 0.5) . For both estimators, the convergence rate and the asymptotic distribution crucially depend on d. The LS estimator has a severe downward bias, which is aggravated even more by the measurement error when d ≤ 0 . The IV estimator eliminates the effects of the measurement error and reduces the bias. In the second set of conditions, the measurement error is independent but not necessarily identically distributed, and the error term in the state equation is strongly mixing. In this case, the IV estimator still leads to a smaller bias than the LS estimator. Special cases of our models and results in relation to those in the literature are discussed. [ABSTRACT FROM AUTHOR]

This paper develops the asymptotic theory of the least squares estimator of the autoregressive (AR) coefficient in an AR(1) regression with intercept when data is generated from a polynomial trend model in different forms. It is shown that the commonly used right-tailed unit root tests tend to favour the explosive alternative. A new procedure, which implements the right-tailed unit root tests in an AR(2) regression, is proposed. It is shown that, when the data generating process has a polynomial trend, the test statistics based on the new procedure cannot find evidence of explosiveness. Whereas, when the data generating process is mildly explosive, the new procedure finds evidence of explosiveness. Hence, it enables robust bubble testing under polynomial trends. Empirical application of the proposed procedure using data from the US real estate market reveals some interesting findings. In particular, all the negative bubble episodes flagged by the traditional method are no longer regarded as bubbles by the new procedure. [ABSTRACT FROM AUTHOR]

This article obtains the exact distribution of the maximum likelihood estimator of structural break point in the Ornstein–Uhlenbeck process when a continuous record is available. The exact distribution is asymmetric, tri-modal, dependent on the initial condition. These three properties are also found in the finite sample distribution of the least squares (LS) estimator of structural break point in autoregressive (AR) models. Motivated by these observations, the article then develops an in-fill asymptotic theory for the LS estimator of structural break point in the AR(1) coefficient. The in-fill asymptotic distribution is also asymmetric, tri-modal, dependent on the initial condition, and delivers excellent approximations to the finite sample distribution. Unlike the long-span asymptotic theory, which depends on the underlying AR roots and hence is tailor-made but is only available in a rather limited number of cases, the in-fill asymptotic theory is continuous in the underlying roots. Monte Carlo studies show that the in-fill asymptotic theory performs better than the long-span asymptotic theory for cases where the long-span theory is available and performs very well for cases where no long-span theory is available. The article also proposes to use the highest density region to construct confidence intervals for structural break point. [ABSTRACT FROM AUTHOR]