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This chapter derives asymptotic properties of the least squares (LS) estimator of the autoregressive (AR) parameter in local to unity processes with errors being fractional Gaussian noise (FGN) with the Hurst parameter H∈(0,1). It is shown that the estimator is consistent for all values of H∈(0,1). Moreover, the rate of convergence is n−1 when H∈[0.5,1). The rate of convergence is n−2H when H∈(0,0.5). Furthermore, the limiting distribution of the centered LS estimator depends on H. When H=0.5, the limiting distribution is the same as that obtained in Phillips (1987a) for the local to unity model with errors for which the standard functional central limit theorem is applicable. When H > 0.5 or when H < 0.5, the limiting distributions are new to the literature. The asymptotic properties of the LS estimator with fitted intercept are also derived. Simulation studies are performed to check the reliability of the asymptotic approximation for different values of sample size.

The discrete Fourier transform (dft) of a fractional process is studied. An exact representation of the dft is given in terms of the component data, leading to the frequency domain form of the model for a fractional process. This representation is particularly useful in analyzing the asymptotic behavior of the dft and periodogram in the nonstationary case when the memory parameter d≥12. Various asymptotic approximations are established including some new hypergeometric function representations that are of independent interest. It is shown that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d < 1. When d = 1, the spectral estimates are inconsistent and converge weakly to random variates. Applications of the theory to log periodogram regression and local Whittle estimation of the memory parameter are discussed and some modified versions of these procedures are suggested for nonstationary cases.

In this study, the authors investigate methods of sequential analysis to test prospectively for the existence of a unit root against stationary or explosive states in a p-th order autoregressive (AR) process monitored over time. Our sequential sampling schemes use stopping times based on the observed Fisher information of a local-to-unity parameter. In contrast to the Dickey–Fuller (DF) test statistic, the sequential test statistic has asymptotic normality. The authors derive the joint limit of the test statistic and the stopping time, which can be characterized using a 3/2-dimensional Bessel process driven by a time-changed Brownian motion. The authors obtain their limiting joint Laplace transform and density function under the null and local alternatives. In addition, simulations are conducted to show that the theoretical results are valid.

The invariance principle known as the functional central limit theorem is the fundamental knowledge for understanding the convergence of a Markov chain to a diffusion process and has been applied to many areas of financial economics. The derivation of that principle from the properties of stochastic process is nasty (Billingsley, 1999). Following the approach of Gikhman and Skorokhod (2000) this study uses the properties of probability measures to simplify the proof of the principle. If the sequences of finite distributions on weakly converge then they are tight. Using this properties, the invariance principle was directly proved. As applications of this principle, I derived the statistic of unit root process and the convergence of the Cox et al. (1979) binomial option pricing model to the continuous time Black Scholes (1972) option pricing model.

It is widely documented that while contemporaneous spot and forward financial prices trace each other extremely closely, their difference is often highly persistent and the conventional cointegration tests may suggest lack of cointegration. This chapter studies the possibility of having cointegrated errors that are characterized simultaneously by high persistence (near-unit root behavior) and very small (near zero) variance. The proposed dual parameterization induces the cointegration error process to be stochastically bounded which prevents the variables in the cointegrating system from drifting apart over a reasonably long horizon. More specifically, this chapter develops the appropriate asymptotic theory (rate of convergence and asymptotic distribution) for the estimators in unconditional and conditional vector error correction models (VECM) when the error correction term is parameterized as a dampened near-unit root process (local-to-unity process with local-to-zero variance). The important differences in the limiting behavior of the estimators and their implications for empirical analysis are discussed. Simulation results and an empirical analysis of the forward premium regressions are also provided.

Classical unit root tests are known to suffer from potentially crippling size distortions, and a range of procedures have been proposed to attenuate this problem, including the use of bootstrap procedures. It is also known that the estimating equation’s functional form can affect the outcome of the test, and various model selection procedures have been proposed to overcome this limitation. In this chapter, the authors adopt a model averaging procedure to deal with model uncertainty at the testing stage. In addition, the authors leverage an automatic model-free dependent bootstrap procedure where the null is imposed by simple differencing (the block length is automatically determined using recent developments for bootstrapping dependent processes). Monte Carlo simulations indicate that this approach exhibits the lowest size distortions among its peers in settings that confound existing approaches, while it has superior power relative to those peers whose size distortions do not preclude their general use. The proposed approach is fully automatic, and there are no nuisance parameters that have to be set by the user, which ought to appeal to practitioners.

The Laplace-type estimator (LTE) is a simulation-based alternative to the classical extremum estimator that has gained popularity in applied research. We show that even though the estimator has desirable asymptotic properties, in small samples the point estimate provided by LTE may not necessarily converge to the extremum of the sample objective function. Furthermore, we suggest a simple test to verify if the estimator converges. We illustrate these results by estimating a prototype dynamic stochastic general equilibrium model widely used in macroeconomics research.

We provide straightforward new nonparametric methods for testing conditional independence using local polynomial quantile regression, allowing weakly dependent data. Inspired by Hausman's (1978) specification testing ideas, our methods essentially compare two collections of estimators that converge to the same limits under correct specification (conditional independence) and that diverge under the alternative. To establish the properties of our estimators, we generalize the existing nonparametric quantile literature not only by allowing for dependent heterogeneous data but also by establishing a weak consistency rate for the local Bahadur representation that is uniform in both the conditioning variables and the quantile index. We also show that, despite our nonparametric approach, our tests can detect local alternatives to conditional independence that decay to zero at the parametric rate. Our approach gives the first nonparametric tests for time-series conditional independence that can detect local alternatives at the parametric rate. Monte Carlo simulations suggest that our tests perform well in finite samples. We apply our test to test for a key identifying assumption in the literature on nonparametric, nonseparable models by studying the returns to schooling.

The author develops and extends the asymptotic F- and t-test theory in linear regression models where the regressors could be deterministic trends, unit-root processes, near-unit-root processes, among others. The author considers both the exogenous case where the regressors and the regression error are independent and the endogenous case where they are correlated. In the former case, the author designs a new set of basis functions that are invariant to the parameter estimation uncertainty and uses them to construct a new series long-run variance estimator. The author shows that the F-test version of the Wald statistic and the t-statistic are asymptotically F and t distributed, respectively. In the latter case, the author shows that the asymptotic F and t theory is still possible, but one has to develop it in a pseudo-frequency domain. The F and t approximations are more accurate than the more commonly used chi-squared and normal approximations. The resulting F and t tests are also easy to implement – they can be implemented in exactly the same way as the F and t tests in a classical normal linear regression.

This paper proposes a new approach to testing in the generalized method of moments (GMM) framework. The new tests are constructed using heteroskedasticity autocorrelation (HAC) robust standard errors computed using nonparametric spectral density estimators without truncation. While such standard errors are not consistent, a new asymptotic theory shows that they lead to valid tests nonetheless. In an over-identified linear instrumental variables model, simulations suggest that the new tests and the associated limiting distribution theory provide a more accurate first order asymptotic null approximation than both standard nonparametric HAC robust tests and VAR based parametric HAC robust tests. Finite sample power of the new tests is shown to be comparable to standard tests.