On some definitions in matrix algebra

We derive test regressions whose structure provides a link between tests for a unit root and tests on the nullity of the parameters associated with the regression's trend function. These test regressions turn out to be equivalent to those proposed by Perron (1989). Using these regression equations, we extend Perron's (1989) asymptotic results by deriving limiting distributions of the deterministic components for all the models considered. The asymptotic representations of these distributions show that there is no conflict between testing for unit roots and for structural breaks: acceptance of a unit root rules out acceptance of a structural break, as modelled by a dummy variable.

Economics Letters, 2007

It is well known that the Durbin-Watson and several other tests for first-order autocorrelation have limiting power of either zero or one in a linear regression model without an intercept, and a constant lying strictly between these values when an intercept term is present. This paper considers the limiting power of these tests in models with possibly incorrect restrictions on the coefficients. It is found that with linear restrictions on the coefficients, the limiting power can still drop to zero even with the inclusion of an intercept in the regression. Our results also accommodate the situation of a possibly mis-specified linear model.

Statistics & Probability Letters, 2006

This paper considers the asymptotic distribution of the OLS estimator in a simple, Gaussian unit-root AR(1) with fixed, non-zero startup. All asymptotic possibilities are considered. The approach is new, relatively simple, and relies on observing and determining the asymptotic/limiting behavior of the underlying finite sample distribution. It does not rely on inversion of joint moment generating or characteristic functions to derive limiting distributions. The paper introduces small-sigma/parameter-based asymptotic theory and re-examines large-sample asymptotic theory. In addition, combinations of these asymptotic approaches are considered explicitly. The analysis provides a set of very interesting and sometimes surprising results. r

Econometric Theory, 2005

This paper studies the asymptotic properties of a nonstationary partially linear regression model+ In particular, we allow for covariates to enter the unit root~or near unit root! model in a nonparametric fashion, so that our model is an extension of the semiparametric model analyzed in Robinson~1988, Econometrica 56, 931-954!+ It is proved that the autoregressive parameter can be estimated at rate N even though part of the model is estimated nonparametrically+ Unit root tests based on the semiparametric estimate of the autoregressive parameter have a limiting distribution that is a mixture of a standard normal and the Dickey-Fuller distribution+ A Monte Carlo experiment is conducted to evaluate the performance of the tests for various linear and nonlinear specifications+ We thank Bruce Hansen, Roger Koenker, Helmut Lütkepohl, Peter Phillips, three referees, and participants of the 8th World Congress of the Econometric Society and the 10th Midwest Econometrics Group Meeting for helpful comments on an earlier version of this paper+ This investigation was supported by the University of

Journal of Empirical Finance, 2014

We propose a discrete-time multivariate model where lagged levels of the process enter both the conditional mean and the conditional variance. This way we allow for the empirically observed persistence in time series such as interest rates, often implying unit-roots, while at the same time maintain stationarity despite such unit-roots. Specifically, the model bridges vector autoregressions and multivariate ARCH models in which residuals are replaced by levels lagged. An empirical illustration using recent US term structure data is given in which the individual interest rates have unit roots, have no finite first-order moments, but remain strictly stationary and ergodic, while they co-move in the sense that their spread has no unit root. The model thus allows for volatility induced stationarity, and the paper shows conditions under which the multivariate process is strictly stationary and geometrically ergodic. Interestingly, these conditions include the case of unit roots and a reduced rank structure in the conditional mean, known from linear co-integration to imply non-stationarity. Asymptotic theory of the maximum likelihood estimators for a particular structured case (so-called self-exciting) is provided, and it is shown that √  −convergence to Gaussian distributions apply despite unit roots as well as absence of finite first and higher order moments. Monte Carlo simulations confirm the usefulness of the asymptotics in finite samples.

Economics Letters, 2008

This paper investigates the performance of the OLS estimator in the context of a cointegrating system, which exhibits a single variance shift. It is shown that the limiting distribution of OLS and that of the associated t-statistic depend on the time, the size and the direction of the break.

Economic Modelling, 1998

In this article we show that sufficient conditions for the unit roots found in the AR representations of time series to persist in bivariate or trivariate VARs amount to long-run non-causality restrictions among the variables involved. Furthermore, in first-order models long-run non-causality is also a necessary and sufficient condition for the autoregressive coefficients in the AR representations to be equal to the corresponding coefficients in the VAR. We also discuss causality inference in the presence of cointegration and show that the omission of an 'important' variable results in invalid inference about the causality structure of the system, unless causality runs to the omitted variable but not vice-versa. These theoretical results can account for our empirical findings on causality between output and financial variables: whilst the bivariate analysis misleadingly suggests that causality runs primarily from M1 to output, the trivariate model implies that interest rates are a better predictor of output. ᮊ 1998 Elsevier Science B.V.

Journal of Computational and Applied Mathematics, 1995

For a first-order autoregressive AR(l) model with zero initial value, x, = xx,-r + E,, we provide closed-form analytical expressions for the asymptotic bias and variance of the maximum likelihood (ML) estimator oi = 1; x,x,-r/C;-' x: when ltll = 1. For the bias, numerical accuracy of up to six significant digits is achieved for sample sizes n > 100.

To find approximations for bias, variance and mean-squared error of least-squares estimators for all coefficients in a linear dynamic regression model with a unit root, we derive asymptotic expansions and examine their accuracy by simulation. It is found that in this particular context useful expansions exist only when the autoregressive model contains at least one non-redundant exogenous explanatory variable. Surprisingly, the large-sample and smalldisturbance asymptotic techniques give closely related results, which is not the case in stable dynamic regression models. We specialize our general expressions for moment approximations to the case of the random walk with drift model and find that they are unsatisfactory when the drift is small. Therefore, we develop what we call small-drift asymptotics which proves to be very accurate, especially when the sample size is very small. and , and extended by to the case of included exogenous variables under normality assumptions, while Peters (1987) analysed the same ARX(1) model with non-normal disturbances. These papers did not provide explicit expressions for moments that allow further analysis, but they did yield useful numerical results for different disturbance structures and specific exogenous data series. Such numerical results, however, can also be obtained to a high degree of accuracy by straightforward simulation.

A new parameters' encompassing test is proposed for deciding between the deterministic unit root processes with a structural break and the bilinear unit root model without such break. The test consists in testing three sets of hypotheses regarding parameters in a simple regression model. The test uses the t-ratio and F-statistics, of non-trivial distributions under the null. The finite sample distributions for the relevant statistics are tabulated and the asymptotic distribution of the F-test is derived. The test has been applied for the daily stock price indices for 66 countries, for the period 1992-2001. The results support the conjecture that the bilinear model dominates the structural break model more often than the other way around. Also, it is likely that in practical applications the bilinear unit root process might be mistaken for the deterministic unit root process with a structural break.