When is a Time Series I(0)?

This paper surveys the extensive recent literature on the problems of deciding what is meant by an I(0) process, and then deciding how to test for the property. A formidable difficulty exists in the construction of consistent and asymptotically correctly sized tests for the I(0) hypothesis, and this may appear to place a question mark over the validity of a large area of econometric theory and practice. To overcome these difficulties in practical applications, the paper proposes that a slightly different question needs to be posed, relating to the adequacy of approximation to asymptotic inference criteria in finite samples. A simulation-based test, aimed at discriminating between data sets on this basis, is examined in a Monte Carlo experiment.

Type I and Type II Fractional Brownian Motions: a Reconsideration

The so-called type I and type II fractional Brownian motions are limit distributions associated with the fractional integration model in which pre-sample shocks are either included in the lag structure, or suppressed. There can be substantial differences between the distributions of these two processes and of functionals derived from them, so that it becomes an important issue to decide which model to use as a basis for inference. Alternative methods for simulating the type I case are contrasted, and for models close to the nonstationarity boundary, truncating infinite sums is shown to result in a significant distortion of the distribution. A simple simulation method that overcomes this problem is described and implemented. The approach also has implications for the estimation of type I ARFIMA models, and a new conditional ML estimator is proposed, using the annual Nile minima series for illustration.Fractional Brownian motion, long memory, ARFIMA, simulation.

Wild bootstrap tests for IV regression

We propose a wild bootstrap procedure for linear regression models estimated by instrumental variables. Like other bootstrap procedures that we have proposed elsewhere, it uses efficient estimates of the reduced-form equation(s). Unlike them, it takes account of possible heteroskedasticity of unknown form. We apply this procedure to t tests, including heteroskedasticity-robust t tests, and to the Anderson-Rubin test. We provide simulation evidence that it works far better than older methods, such as the pairs bootstrap. We also show how to obtain reliable confidence intervals by inverting bootstrap tests. An empirical example illustrates the utility of these procedures.Instrumental variables estimation, two-stage least squares, weak instruments, wild bootstrap, pairs bootstrap, residual bootstrap, confidence intervals, Anderson-Rubin test

Representation and Weak Convergence of Stochastic Integrals with Fractional Integrator Processes

This paper considers the asymptotic distribution of the covariance of a nonstationary fractionally integrated process with the stationary increments of another such process - possibly, itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analysed in a previous paper, and the construction derived from moving average representations in the time domain. The limiting integrals are shown to be expressible in terms of functionals of Itô integrals with respect to two distinct Brownian motions. Their mean is nonetheless shown to match that of the harmonic representation, and they satisfy the required integration by parts rule. The advantages of our approach over the harmonic analysis include the facts that our formulae are valid for the full range of the long memory parameters, and extend to non-Gaussian processes.Stochastic integral, weak convergence, fractional Brownian motion.

Improving the reliability of bootstrap tests with the fast double bootstrap

Two procedures are proposed for estimating the rejection probabilities of bootstrap tests in Monte Carlo experiments without actually computing a bootstrap test for each replication. These procedures are only about twice as expensive (per replication) as estimating rejection probabilities for asymptotic tests. Then a new procedure is proposed for computing bootstrap P values that will often be more accurate than ordinary ones. This “fast double bootstrap” is closely related to the double bootstrap, but it is far less computationally demanding. Simulation results for three different cases suggest that the fast double bootstrap can be very useful in practice.Bootstrap

Tests for Cointegration with Structural Breaks Based on Subsamples

This paper considers tests for cointegration with allowance for structural breaks, using the extrema of residual-based tests over subsamples of the data. One motivation for the approach is to formalize the practice of data snooping by practitioners, who may examine subsamples after failing to find a predicted cointegrating relationship. Valid critical values for such multiple testing situations may be useful. The methods also have the advantage of not imposing a form for the alternative hypothesis, in particular slope vs. intercept shifts and single versus multiple breaks, and being comparatively easy to compute. A range of alternative subsampling procedures, including sample splits, incremental and rolling samples are tabulated and compared experimentally. Shiller's annual stock prices and dividends series provide an illustration.Level shift, Regime shift, Cointegration, Brownian motion

Tests of Bias in Log-Periodogram Regression

This paper proposes simple Hausman-type tests to check for bias in the log-periodogram regression of a time series believed to be long memory. The statistics are asymptotically standard normal on the null hypothesis that no bias is present, and the tests are consistent.Long memory, log periodogram regression, Hausman test.

Bootstrap inference in a linear equation estimated by instrumental variables

We study several tests for the coefficient of the single right-hand-side endogenous variable in a linear equation estimated by instrumental variables. We show that writing all the test statistics—Student's t, Anderson-Rubin, the LM statistic of Kleibergen and Moreira (K), and likelihood ratio (LR)—as functions of six random quantities leads to a number of interesting results about the properties of the tests under weakinstrument asymptotics. We then propose several new procedures for bootstrapping the three non-exact test statistics and also a new conditional bootstrap version of the LR test. These use more efficient estimates of the parameters of the reduced-form equation than existing procedures. When the best of these new procedures is used, both the K and conditional bootstrap LR tests have excellent performance under the null. However, power considerations suggest that the latter is probably the method of choice.bootstrap, weak instruments, IV estimation

Moments of IV and JIVE estimators

We develop a method based on the use of polar coordinates to investigate the existence of moments for instrumental variables and related estimators in the linear regression model. For generalized IV estimators, we obtain familiar results. For JIVE, we obtain the new result that this estimator has no moments at all. Simulation results illustrate the consequences of its lack of moments.instrumental variables, JIVE, moments of estimators