GMM Estimation of Autoregressive Roots Near Unity with Panel Data

This paper investigates a generalized method of moments (GMM) approach to the estimation of autoregressive roots near unity with panel data. The two moment conditions studied are obtained by constructing bias corrections to the score functions under OLS and GLS detrending, respectively. It is shown that the moment condition under GLS detrending corresponds to taking the projected score on the Bhattacharya basis, linking the approach to recent work on projected score methods for models with infinite numbers of nuisance parameters (Waterman and Lindsay, 1998). Assuming that the localizing parameter makes a nonpositive value, we establish consistency of the GMM estimator and find its limiting distribution. A notable new finding is that the GMM estimator has convergence rate n^{1/6}, slower than root{n}, when the true localizing parameter is zero (i.e., when there is a panel unit root) and the deterministic trends in the panel are linear. These results, which rely on boundary point asymptotics, point to the continued difficulty of distinguishing unit roots from local alternatives, even when there is an infinity of additional data.Bias, boundary point asymptotics, GMM estimation, local to unity, moment conditions, nuisance parameters, panel data, pooled regression, projected score

Nonstationary Panel Data Analysis: An Overview of Some Recent Developments

This paper overviews some recent developments in panel data asymptotics, concentrating on the nonstationary panel case and gives a new result for models with individual effects. Underlying recent theory are asymptotics for multi-indexed processes in which both indexes may pass to infinity. We review some of the new limit theory that has been developed, show how it can be applied and give a new interpretation of individual effects in nonstationary panel data. Fundamental to the interpretation of much of the asymptotics is the concept of a panel regression coefficient which measures the long run average relation across a section of the panel. This concept is analogous to the statistical interpretation of the coefficient in a classical regression relation. A variety of nonstationary panel data models are discussed and the paper reviews the asymptotic properties of estimators in these various models. Some recent developments in panel unit root tests and stationary dynamic panel regression models are also reviewed.

Estimation of Autoregressive Roots Near Unity Using Panel Data

Time series data are often well modelled by using the device of an autoregressive root that is local to unity. Unfortunately, the localizing parameter (c) is not consistently estimable using existing time series econometric techniques and the lack of a consistent estimator complicates inference. This paper develops procedures for the estimation of a common localizing parameter using panel data. Pooling information across individuals in a panel aids the identification and estimation of the localising parameter and leads to consistent estimation in simple panel models. However, in the important case of models with concomitant deterministic trends, it is shown that pooled panel estimators of the localising parameter are asymptotically biased. Some techniques are developed to overcome this difficulty and consistent estimators of c in the region cBias, local to unity, panel data, pooled regression, subgroup testing

GMM Estimation of Autoregressive Roots Near Unity with Panel Data

This paper investigates a generalized method of moments (GMM) approach to the estimation of autoregressive roots near unity with panel data and incidental deterministic trends. Such models arise in empirical econometric studies of firm size and in dynamic panel data modeling with weak instruments. The two moment conditions in the GMM approach are obtained by constructing bias corrections to the score functions under OLS and GLS detrending, respectively. It is shown that the moment condition under GLS detrending corresponds to taking the projected score on the Bhattacharya basis, linking the approach to recent work on projected score methods for models with infinite numbers of nuisance parameters (Waterman and Lindsay, 1998). Assuming that the localizing parameter takes a nonpositive value, we establish consistency of the GMM estimator and find its limiting distribution. A notable new finding is that the GMM estimator has convergence rate n/{1/6}, slower than /n, when the true localizing parameter is zero (i.e., when there is a panel unit root) and the deterministic trends in the panel are linear. These results, which rely on boundary point asymptotics, point to the continued difficulty of distinguishing unit roots from local alternatives, even when there is an infinity of additional data.Bias, boundary point asymptotics, GMM estimation, local to unity, moment conditions, nuisance parameters, panel data, pooled regression, projected score

GMM Estimation of Autoregressive Roots Near Unity with Panel Data

This paper investigates a generalized method of moments (GMM) approach to the estimation of autoregressive roots near unity with panel data and incidental deterministic trends. Such models arise in empirical econometric studies of firm size and in dynamic panel data modeling with weak instruments. The two moment conditions in the GMM approach are obtained by constructing bias corrections to the score functions under OLS and GLS detrending, respectively. It is shown that the moment condition under GLS detrending corresponds to taking the projected score on the Bhattacharya basis, linking the approach to recent work on projected score methods for models with infinite numbers of nuisance parameters (Waterman and Lindsay, 1998). Assuming that the localizing parameter takes a nonpositive value, we establish consistency of the GMM estimator and find its limiting distribution. A notable new finding is that the GMM estimator has convergence rate n/{1/6}, slower than /n, when the true localizing parameter is zero (i.e., when there is a panel unit root) and the deterministic trends in the panel are linear. These results, which rely on boundary point asymptotics, point to the continued difficulty of distinguishing unit roots from local alternatives, even when there is an infinity of additional data

Properties of the Soliton-Lattice State in Double-Layer Quantum Hall Systems

Application of a sufficiently strong parallel magnetic field $B_\parallel > B_{c}$ produces a soliton-lattice (SL) ground state in a double-layer quantum Hall system. We calculate the ground-state properties of the SL state as a function of $B_\parallel$ for total filling factor $\nu_{T}=1$, and obtain the total energy, anisotropic SL stiffness, Kosterlitz-Thouless melting temperature, and SL magnetization. The SL magnetization might be experimentally measurable, and the magnetic susceptibility diverges as $|B_\parallel - B_{c}|^{-1}$.Comment: 4 pages LaTeX, 1 EPS figure. Proceedings of the 12th International Conference on the Electronic Properties of Two-Dimensional Electron Systems (EP2DS-12), to be published in Physica B (1998

Incidental Trends and the Power of Panel Unit Root Tests

The asymptotic local powers of various panel unit root tests are investigated. The power envelope is obtained under homogeneous and heterogeneous alternatives. It is compared with asymptotic power functions of the pooled t -test, the Ploberger–Phillips (2002) test, and a point optimal test in neighborhoods of unity that are of order n –1/ 4 T –1 and n –1/ 2 T –1 , depending on whether or not incidental trends are extracted from the panel data. In the latter case, when the alternative hypothesis is homogeneous across individuals, it is shown that the point optimal test and Ploberger–Phillips test both achieve the power envelope and are uniformly most powerful, in contrast to point optimal unit root tests for time series. Some simulations examining the finite sample performance of the tests are reported

How to Estimate Autoregressive Roots Near Unity

A new model of near integration is formulated in which the local to unity parameter is identifiable and consistently estimable with time series data. The properties of the model are investigated, new functional laws for near integrated time series are obtained, and consistent estimators of the localizing parameter are constructed. The model provides a more complete interface between I(0) and I(1) models than the traditional local to unity model and leads to autoregressive coefficient estimates with rates of convergence that vary continuously between the O(/n) rate of stationary autoregression, the O(n) rate of unit root regression and the power rate of explosive autoregression. Models with deterministic trends are also considered, least squares trend regression is shown to be efficient, and consistent estimates of the localising parameter are obtained for this case as well. Conventional unit root tests are shown to be consistent against local alternatives in the new class.

Incidental Trends and the Power of Panel Unit Root Tests

The asymptotic local powers of various panel unit root tests are investigated. The power envelope is obtained under homogeneous and heterogeneous alternatives. It is compared with asymptotic power functions of the pooled t-test, the Ploberger-Phillips (2002) test, and a point optimal test in neighborhoods of unity that are of order n^{1/4}T{-1} and n^{1/2}T^{-1}, depending on whether or not incidental trends are extracted from the panel data. In the latter case, when the alternative hypothesis is homogeneous across individuals, it is shown that the point optimal test and Ploberger-Phillips test both achieve the power envelope and are uniformly most powerful, in contrast to point optimal unit root tests for time series. Some simulations examining the finite sample performance of the tests are reported.Asymptotic power envelope, Common point optimal test, Heterogeneous alternatives, Incidental trends, Local to unity, Power function, Panel unit root test

GMM Estimation of Autoregressive Roots Near Unity with Panel Data

This paper investigates a generalized method of moments (GMM) approach to the estimation of autoregressive roots near unity with panel data. The two moment conditions studied are obtained by constructing bias corrections to the score functions under OLS and GLS detrending, respectively. It is shown that the moment condition under GLS detrending corresponds to taking the projected score on the Bhattacharya basis, linking the approach to recent work on projected score methods for models with infinite numbers of nuisance parameters (Waterman and Lindsay, 1998). Assuming that the localizing parameter makes a nonpositive value, we establish consistency of the GMM estimator and find its limiting distribution. A notable new finding is that the GMM estimator has convergence rate n 1 /6 , slower than √ n , when the true localizing parameter is zero (i.e., when there is a panel unit root) and the deterministic trends in the panel are linear. These results, which rely on boundary point asymptotics, point to the continued difficulty of distinguishing unit roots from local alternatives, even when there is an infinity of additional data