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Robust Inference in Dynamic Regression Models with Persistent Regressors
Abstract
Correct inference on the coefficients of dynamic regression models depends in a crucial manner on the degree of persistence of the explanatory variables. While standard asymptotic inference applies in OLS regressions with stationary regressors, the limiting distributions are nonstandard in the presence of integrated regressors. The paper studies test procedures that are robust in the sense that their asymptotic null distributions are the same irrespective of whether the regressors are stationary or (nearly/fractionally) integrated. Two alternative approaches are considered. We first propose an extension of the variable addition method with improved asymptotic power. Second, inference based on instrumental variables may further improve the (local) power of the test. We give primitive conditions under which the suggested procedures are robust.All statistics proposed here are asymptotically standard nor-mal or χ 2 distributed irrespective of the degree of persistence of the regressors. Monte Carlo experiments show that tests based on simple combinations of instru-ments perform quite promising relative to existing tests.
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An asymptotic theory is developed for multivariate regression in cointegrated systems whose variables are moderately integrated or moderately explosive in the sense that they have autoregressive roots of the form , involving moderate deviations from unity when 130). Moreover, the limit theory applies without any distributional assumptions and for weakly dependent errors under conventional moment conditions, so an invariance principle holds, unlike the well-known case of an explosive autoregression. This theory validates inference in cointegrating regression with mildly explosive regressors. The special case in which the regressors themselves have a common explosive component is also considered.
We consider weak convergence of sample averages of nonlinearly transformed stochastic triangular arrays satisfying a functional invariance principle. A fundamental paradigm for such processes is constituted by integrated processes. The results obtained are extensions of recent work in the literature to the multivariate and non-Gaussian case. As admissible nonlinear transformation, a new class of functionals (so-called locally p-integrable functions) is introduced that adapts the concept of locally integrable functions in P 22) to the multidimensional setting.
It is well known that the distribution of statistics testing restrictions on the coefficients in time series regressions can depend on the order of integration of the regressors. In practice the order of integration is rarely blown. This paper examines two conventional approaches to this problem, finds them unsatisfactory, and proposes a new procedure. The two conventional approaches- simply to ignore unit root problems or to use unit root pretests to determine the critical values for second-stage inference - both often induce substantial size distortions. In the case of unit root pretests, this arises because type I and II pretest errors produce incorrect second-stage critical values and because, in many empirically plausible situations, the first stage test (the unit root test) and the second stage test (the exclusion restriction test) are dependent. Monte Carlo simulations reveal size distortions even if the regressor is stationary but has a large autoregressive root, a case that might arise for example in a regression of excess stock returns against the dividend yield. In the proposed alternative procedure, the second-stage test is conditional on a first-stage "unit root" statistic developed in Stock (1992); the second-stage critical values vary continuously with the value of the first-stage statistic. The procedure is shown to have the correct size asymptotically and to have good local asymptotic power against Granger-causality alternatives.
This paper develops an asymptotic theory for a first-order autoregression with a root near unity. Deviations from the unit
root theory are measured through a noncentrality parameter. When this parameter is negative we have a local alternative that
is stationary; when it is positive the local alternative is explosive; and when it is zero we have the standard unit root
theory. Our asymptotic theory accommodates these possibilities and helps to unify earlier theory in which the unit root case
appears as a singularity of the asymptotics. The general theory is expressed in terms of functionals of a simple diffusion
process. The theory has applications to continuous time estimation and to the analysis of the asymptotic power of tests for
a unit root under a sequence of local alternatives.
This paper develops a general asymptotic theory of regression for processes which are integrated of order one. The theory
includes vector autoregressions and multivariate regressions amongst integrated processes that are driven by innovation sequences
which allow for a wide class of weak dependence and heterogeneity. The models studied cover cointegrated systems such as those
advanced recently by Granger and Engle and quite general linear simultaneous equations systems with contemporaneous regressor
error correlation and serially correlated errors. Problems of statistical testing in vector autoregressions and multivariate
regressions with integrated processes are also studied. It is shown that the asympotic theory for conventional tests involves
major departures from classical theory and raises new and important issues of the presence of nuisance parameters in the limiting
distribution theory.
It is pointed out that two contradictory definitions of fractional Brownian motion are well-established, one prevailing in the probabilistic literature, the other in the econometric literature. Each is associated with a different definition of nonstationary fractional time series, arising in functional limit theorems based on such series. These various definitions have occasionally led to some confusion. The paper discusses the definitions and attempts a clarification.
Weak convergence to a form of fractional Brownian motion is established for a wide class of nonstationary fractionally integrated multivariate processes. Instrumental for the main argument is a result of some independent interest on approximations for partial sums of stationary linear vector sequences. A functional central limit theorem for smoothed processes is established under more general assumptions.
This paper examines regression tests of whether x forecasts y when the largest autoregressive root of the regressor is unknown. It is shown that previously proposed two-step procedures, with first stages that consistently classify x as I(1) or I(0), exhibit large size distortions when regressors have local-to-unit roots, because of asymptotic dependence on a nuisance parameter that cannot be estimated consistently. Several alternative procedures, based on Bonferroni and Scheffe methods, are therefore proposed and investigated. For many parameter values, the power loss from using these conservative tests is small.
This paper shows how we can estimate VAR's formulated in levels and test general restrictions on the parameter matrices even if the processes may be integrated or cointegrated of an arbitrary order. We can apply a usual lag selection procedure to a possibly integrated or cointegrated VAR since the standard asymptotic theory is valid (as far as the order of integration of the process does not exceed the true lag length of the model). Having determined a lag length k, we then estimate a (k + dmax)th-order VAR where dmax is the maximal order of integration that we suspect might occur in the process. The coefficient matrices of the last dmax lagged vectors in the model are ignored (since these are regarded as zeros), and we can test linear or nonlinear restrictions on the first k coefficient matrices using the standard asymptotic theory.
An asymptotic theory is given for autoregressive time series with a root of the form ρn=1+c/kn, which represents moderate deviations from unity when (kn)n∈N is a deterministic sequence increasing to infinity at a rate slower than n, so that kn=o(n) as n→∞. For c<0, the results provide a rate of convergence and asymptotic normality for the first order serial correlation, partially bridging the and n convergence rates for the stationary (kn=1) and conventional local to unity (kn=n) cases. For c>0, the serial correlation coefficient is shown to have a 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 kn=1, where the convergence rate of the serial correlation coefficient is (1+c)n and no invariance principle applies.
To estimate the parameters of a stationary univariate fractionally integrated time series, the unconditional exact likelihood function is derived. This allows the simultaneous estimation of all the parameters of the model by exact maximum likelihood. Issues involved in obtaining maximum likelihood estimates are discussed. Particular attention is given to efficient procedures to evaluate the likelihood function, obtaining starting values, and the small sample properties of the estimators. Limitations of previous estimation procedures are also discussed.
Wald tests of restrictions on the coefficients of vector autoregressive (VAR) processes are known to have nonstandard asymptotic properties for 1(1) and cointegrated systems of variables. A simple device is proposed which guarantees that Wald tests have asymptotic X2-distributions under general conditions. If the true generation process is a VAR(p) it is proposed to fit a VAR(p+l) to the data and perform a Wald test on the coefficients of the first p lags only. The power properties of the modified tests are studied both analytically and numerically by means of simple illustrative examples.
We consider the fractional analogue of the Ornstein–Uhlenbeck process, that is, the solution of a one-dimensional homogeneous
linear stochastic differential equation driven by a fractional Brownian motion in place of the usual Brownian motion. The
statistical problem of estimation of the drift and variance parameters is investigated on the basis of a semimartingale which
generates the same filtration as the observed process. The asymptotic behaviour of the maximum likelihood estimator of the
drift parameter is analyzed. Strong consistency is proved and explicit formulas for the asymptotic bias and mean square error
are derived. Preparing for the analysis, a change of probability method is developed to compute the Laplace transform of a
quadratic functional of some auxiliary process.
For autoregressive processes, we propose new estimators whose pivotal statistics have the standard normal limiting distribution for all ranges of the autoregressive parameters. The proposed estimators are approximately median unbiased. For seasonal time series, the new estimators give us unit root tests that have limiting normal distribution regardless of period of the seasonality. Using the estimators, confidence intervals of the autoregressive parameters are constructed. A Monte-Carlo simulation for first-order autoregressions shows that the proposed tests for unit roots are locally more powerful than the tests based on the ordinary least squares estimators. It also shows that the proposed confidence intervals have shorter average lengths than those of Andrews (1993, Econometrica 61, 139 165) based on the ordinary least squares estimators when the autoregressive coefficient is close to one.
This paper presents a new test for fractionally integrated ("FI") processes. In particular, we propose a testing procedure in the time domain that extends the well-known Dickey-Fuller approach, originally designed for the "I"(1) versus "I"(0) case, to the more general setup of "FI"("d"-sub-0) versus "FI"("d"-sub-1), with "d"-sub-1<"d"-sub-0. When "d"-sub-0=1, the proposed test statistics are based on the OLS estimator, or its "t"-ratio, of the coefficient on Δ-super-"d"-sub-1"y"-sub-"t" - 1 in a regression of Δ"y-sub-t" on Δ-super-"d"-sub-1"y"-sub-"t" - 1 and, possibly, some lags of Δ"y-sub-t". When "d"-sub-1 is not taken to be known a priori, a pre-estimation of "d"-sub-1 is needed to implement the test. We show that the choice of any "T"-super-1/2-consistent estimator of "d"-sub-1 is an element of [0 ,1) suffices to make the test feasible, while achieving asymptotic normality. Monte-Carlo simulations support the analytical results derived in the paper and show that proposed tests fare very well, both in terms of power and size, when compared with others available in the literature. The paper ends with two empirical applications. Copyright The Econometric Society 2002.
This paper considers estimation and hypothesis testing in linear time series when some or all of the variables have (possibly multiple) unit roots. The motivating example is a vector autoregression with some unit roots in the companion matrix, which might include polynomials in time as regressors. Parameters that can be written as coefficients on mean zero, nonintegrated regressors have jointly normal asymptotic distribution, converging at the rate of T(superscript "one-half") In general, the other coefficients (including the coefficient on polynomials in time), and associated t and F test statistics, have nonstandard asymptotic distributions. Copyright 1990 by The Econometric Society.
This paper considers the problem of conducting inference on the regression coefficient in a bivariate regression model with a highly persistent regressor. Gaussian asymptotic power envelopes are obtained for a class of testing procedures that satisfy a conditionality restriction. In addition, the paper proposes testing procedures that attain these power envelopes whether or not the innovations of the regression model are normally distributed. Copyright The Econometric Society 2006.
Some new tools for analyzing spurious regressions are presented. The theory utilizes the general representation of a stochastic process in terms of an orthonormal system and provides an extension of the Weierstrass theorem to include the approximation of continuous functions and stochastic processes by Wiener processes. The theory is applied to two classic examples of spurious regressions: regressions of stochastic trends on time polynomials and regressions among independent random walks. It is shown that such regressions reproduce in part and in whole the underlying orthonormal representations.
Using the asymptotic normality of the least-squares estimates for the autoregressive (AR) process with real, positive unit roots and at least one stable root, we consider the asymptotic distributions of the Wald and t ratio tests on AR coefficients. In addition, we propose a method of constructing confidence intervals for the sum of AR coefficients possibly in the presence of a unit root. Using simulation methods, we compare the finite-sample cumulative distributions of the t ratios for individual autoregressive coefficients with those of standard normal distributions, and investigate the finite-sample performance of our confidence intervals and t ratios. Our simulation results show that the t ratios for nonstationary processes converge to a standard normal distribution more slowly than those for stationary processes. Further, the confidence intervals are shown to work reasonably well in moderately large samples, but they display unsatisfactory performance at small sample sizes.
This paper provides conditions to establish the weak convergence of stochastic integrals. The theorems are proved under the assumption that the innovations are strong mixing with uniformly bounded 2-h moments. Several applications of the results are given, relevant for the theories of estimation with I(1) processes, I(2) processes, processes with nonstationary variances, near-integrated processes, and continuous time approximations.