Rolf Larsson

Likelihood-Based Cointegration Tests in Heterogeneous Panels

This paper presents a likelihood-based panel test of cointegrating rank in heterogeneous panel models based on the mean of the individual rank trace statistics. The existence of the first two moments of the asymptotic distribution of the individual trace statistic is established. Based on this, the asymptotic distribution of the test statistic is shown to be normal. The small-sample size and power properties are investigated using Monte Carlo simulations. An empirical example for a consumption model including consumption, income and inflation is estimated for 23 OECD countries over the period 1960-1994. The results indicate that two cointegrating relations exist in the system: One containing consumption and income and one inflation only.

Bartlett Corrections for Unit Root Test Statistics

Bartlett corrections of the log likelihood ratio test for a unit root in an AR(1) process, as well as for some asymptotically equivalent tests, are studied. The corrections are obtained by an analytical method. The numerical performance of the results is checked in a simulation study.

The finite sample distribution of the KPSS test

This paper gives numerical approximations of the finite sample distribution of the KPSS test statistic. We use two types of approximations depending on whether we estimate the long-run variance or not. In the known variance case we apply a very simple Laplace inversion formula, while in the unknown case we use saddlepoint approximation to calculate the right-hand tail of the distribution of the KPSS test statistic. In the unknown variance case, we also investigate the robustness of the test to non i.i.d. innovations.

The Joint Moment Generating Function of Quadratic Forms in Multivariate Autoregressive Series

Let Xt be a discrete multivariate autoregressive process of order 1. The paper derives the joint moment generating function (mgf) of the two quadratic forms that are used to define statistics relating to the parameters of this process. The formula is then specialized to some cases of interest, including the mgf of functionals of multivariate Ornstein-Uhlenbeck processes.

A distance measure between cointegration spaces

Abstract A distinguishing feature of cointegration models, and many other multivariate models, is that only spaces spanned by parameter vectors are identified. We point out that traditional distance measures, such as the Euclidean measure, are not reasonable to use when measuring distances between spaces. This point has been either missed or ignored in many simulation studies where inappropriate distance measures have been used. We propose a simple measure based on the idea that the space spanned by the orthogonal complement of a matrix lies as far away as possible from the space spanned by the matrix itself. Several properties of this measure are derived.

Barlett corrections in cointegration testing

When testing for cointegration, the asymptotic inference typically in use can be plagued by size distortion due to an inadequate first order approximation. Hence, for practical purposes the inference can be completely misleading and result in false conclusions regarding the presence of long-run relationships in the data. Which, of course, in many applications is a key issue. We explore the potentials of Bartlett correction of two cointegration test statistics. The idea is to multiply the test statistic by a correcting factor derived from an asymptotic expansion of its expectation. As a consequence, the reference distribution should then provide a closer approximation to the resulting adjusted statistic in comparison with the unadjusted statistic. In a simple bivariate framework we derive a likelihood ratio test, as well as a first order approximation thereof, for testing the null hypothesis of no cointegration. Suitable Bartlett corrections for the two tests are suggested and using Monte Carlo simulation we evaluate the effectiveness of the proposed methods.

Approximation Of The Asymptotic Distribution Of The Log Likelihood Ratio Test For Cointegration

Tail approximation of the asymptotic distribution of the log likelihood ratio test for cointegration in a vector autoregressive process is studied. In dimension 2, an approximation of weighted I‡2 type is derived by applying multivariate saddlepoint approximation techniques to a Fourier inversion integral.

Distribution approximation of unit root tests in autoregressive models

The present work applies saddlepoint approximation to calculate the left-hand tail of the distribution of the unit root t test and an asymptotic equivalent test under the null hypothesis of a unit root. (This is the tail of interest when testing against a stationary alternative.) The saddlepoint equation is solved numerically. Distribution approximations are obtained both in the asymptotic and finite-sample cases. In the finite-sample case, two slightly different methods are suggested and compared.

The Order of the Error Term for Moments of the Log Likelihood Ratio Unit Root Test in an Autoregressive Process

This paper investigates the asymptotics of the log likelihood ratio test for a unit root in an autoregressive (AR) process of general order. The main result is that the expectation and variance (in fact, all moments) of the test statistic may, to the order of T-1, where T is the number of observations, be approximated by the expectation and variance of the corresponding test in an AR(1) process. This result has obvious implications for the asymptotics of unit root tests for panels. An explicit formula for the approximation error of a test in an AR(2) process is also given.

How close is a fractional process to a random walk with drift

Abstract In this paper, we investigate how close a fractional process can be to a random walk with drift in terms of the sample path. Given the innovation sequence, we calculate the distance to the closest random walk with drift in the sum of squares sense. We also derive the expected distance between the processes under the assumption of white noise normal innovations. A local approximation formula for this distance is given in terms of the sample size, showing that it increases with the sample size more rapidly than the square of the number of observations. Two empirical examples illustrate the results.