In regressions involving integrable functions we examine the limit properties of IV estimators that utilise integrable transformations of lagged regressors as instruments. The regressors can be either I(0) or nearly integrated (NI) processes. We show that this kind of nonlinearity in the regression function can significantly affect the relevance of the instruments. In particular, such instruments become weak when the signal of the regressor is strong, as it is in the NI case. Instruments based on integrable functions of lagged NI regressors display long range dependence and so remain relevant even at long lags, continuing to contribute to variance reduction in IV estimation. However, simulations show that OLS is generally superior to IV estimation in terms of MSE, even in the presence of endogeneity. Estimation precision is also reduced when the regressor is nonstationary.
Nielsen (2009) shows that vector autoregression is inconsistent when there are common explosive roots with geometric multiplicity greater than unity. This paper discusses that result, provides a co-explosive system extension and an illustrative example that helps to explain the finding, gives a consistent instrumental variable procedure, and reports some simulations. Some exact limit distribution theory is derived and a useful new reverse martingale central limit theorem is proved.
A limit theory is established for autoregressive time series that smooths the transition between local and moderate deviations from unity and provides a transitional form that links conventional unit root distributions and the standard normal. Edgeworth expansions of the limit theory are given. These expansions show that the limit theory that holds for values of the autoregressive coefficient that are closer to stationarity than local (i.e. deviations of the form = 1 + (c/n), where n is the sample size and c < 0) holds up to the second order. Similar expansions around the limiting Cauchy density are provided for the mildly explosive case.
It is well known that unit root limit distributions are sensitive to initial conditions in the distant past. If the distant past initialization is extended to the infinite past, the initial condition dominates the limit theory producing a faster rate of convergence, a limiting Cauchy distribution for the least squares coefficient and a limit normal distribution for the t ratio. This amounts to the tail of the unit root process wagging the dog of the unit root limit theory. These simple results apply in the case of a univariate autoregression with no intercept. The limit theory for vector unit root regression and cointegrating regression is affected but is no longer dominated by infinite past initializations. The latter contribute to the limiting distribution of the least squares estimator and produce a singularity in the limit theory, but do not change the principal rate of convergence. Usual cointegrating regression theory and inference continues to hold in spite of the degeneracy in the limit theory and is therefore robust to initial conditions that extend to the infinite past.
A limit theory is developed for multivariate regression in an explosive cointegrated system. The asymptotic behavior of the least squares estimator of the cointegrating coefficients is found to depend upon the precise relationship between the explosive regressors. When the eigenvalues of the autoregressive matrix are distinct, the centered least squares estimator has an exponential rate of convergence and a mixed normal limit distribution. No central limit theory is applicable here and Gaussian innovations are assumed. On the other hand, when some regressors exhibit common explosive behavior, a different mixed normal limiting distribution is derived with rate of convergence reduced to n^0.5. In the latter case, mixed normality applies without any distributional assumptions on the innovation errors by virtue of a Lindeberg type central limit theorem. Conventional statistical inference procedures are valid in this case, the stationary convergence rate dominating the behavior of the least squares estimator.
Keywords: Central limit theory, Exposive cointegration, Explosive process, Mixed normality
An asymptotic theory is given for autoregressive time series with weakly dependent innovations and a root of the form ρn = 1 + c/nα, involving moderate deviations from unity when α in (0,1) and c in R are constant parameters. The limit theory combines a functional law to a diffusion on D[0,∞) and a central limit theorem. For c > 0, the limit theory of the first-order serial correlation coefficient is Cauchy and is invariant to both the distribution and the dependence structure of the innovations. To our knowledge, this is the first invariance principle of its kind for explosive processes. The rate of convergence is found to be nαρnn, which bridges asymptotic rate results for conventional local to unity cases (n) and explosive autoregressions ((1 + c)n). For c < 0, we provide results for α in (0,1) that give an n(1+α)/2 rate of convergence and lead to asymptotic normality for the first order serial correlation, bridging the /n and n convergence rates for the stationary and conventional local to unity cases. Weakly dependent errors are shown to induce a bias in the limit distribution, analogous to that of the local to unity case. Linkages to the limit theory in the stationary and explosive cases are established.
Keywords: Central limit theory; Diffusion; Explosive autoregression, Local to unity; Moderate deviations, Unit root distribution, Weak dependence
An asymptotic theory is given for autoregressive time series with a root of the form ρn = 1 + c/nα, which represents moderate deviations from unity when α in (0,1). The limit theory is obtained using a combination of a functional law to a diffusion on D[0,∞) and a central limit law to a scalar normal variate. For c > 0, the results provide a n(1+α)/2 rate of convergence and asymptotic normality for the first order serial correlation, partially bridging the squareroot of n and n convergence rates for the stationary (α = 0) and conventional (α = 1) local to unity cases. For c > 0, the serial correlation coefficient is shown to have a nαρnn convergence rate and a Cauchy limit distribution without assuming Gaussian errors, so an invariance principle applies when ρn > 1. This result links moderate deviation asymptotics to earlier results on the explosive autoregression proved under Gaussian errors for α = 0, where the convergence rate of the serial correlation coefficient is (1 + c)n and no invariance principle applies.
Keywords: Central limit theory; Diffusion, Explosive autoregression, Local to unity, Moderate deviations, Unit root distribution