Victoria Zinde-Walsh

ON THE ROBUSTNESS OF LM, LR, AND W TESTS IN REGRESSION MODELS

In this paper we show that, in the small sample case, while the LR test is robust, the LM and W tests are not robust when the errors are Student t. For the large sample, however, all three tests are found to be robust. A variation of the W test is also considered and it turns out to be nonrobust. The implications of assuming the errors to be distributed as multivariate Student t, rather than multivariate normal, are also discussed. It is found that not all of Evans and Savins [4] results carry through when the errors are Student t.

Some Exact Formulae for Autoregressive Moving Average Processes

This paper demonstrates that for a finite stationary autoregressive moving average process the inverse of the covariance matrix differs from the matrix of the covariances of the inverse process by a matrix of low rank. The formula for the exact inverse of the covariance matrix of the scalar or multivariate process is provided. We obtain approximations based on this formula and evaluate some of the approximate results in the existing literature. Applications to computational algorithms and to the distributions of two-step estimators are discussed. In addition the paper contains the formula for the determinant of the covariance matrix which is useful in exact maximum likelihood estimation; it also lists the expressions for the autocovariances of scalar autoregressive moving average processes.

ESTIMATION OF THE VECTOR MOVING AVERAGE MODEL BY VECTOR AUTOREGRESSION

ABSTRACT We examine a simple estimator for the multivariate moving average model based on vector autoregressive approximation. In finite samples the estimator has a bias which is low where roots of the characteristic equation are well away from the unit circle, and more substantial where one or more roots have modulus near unity. We show that the representation estimated by this multivariate technique is consistent and asymptotically invertible. This estimator has significant computational advantages over Maximum Likelihood, and more importantly may be more robust than ML to mis-specification of the vector moving average model. The estimation method is applied to a VMA model of wholesale and retail inventories, using Canadian data on inventory investment, and allows us to examine the propagation of shocks between the two classes of inventory.

Estimation of a linear regression model with stationary ARMA(p, q) errors

Abstract It is well known that consistent estimation of a linear regression model with a stationary Gaussian ARMA process in the errors can be carried out by maximum likelihood or, alternatively, by two-stage procedures involving estimation of the nuisance parameters followed by feasible generalized least squares for the model parameters. We show that the estimators coincide up to O p (T − 3 2 ) and derive the variance to O(T−2), which up to terms of this order is the same for both estimators. Considering the form of the error covariance matrix for an ARMA(p,q) process allows us to examine a computationally convenient algorithm for estimation of the parameters of the regression model. Finally we provide a Monte Carlo comparison of the small-sample properties of OLS and two versions of the proposed estimator.

Asymptotic Theory For Some High Breakdown Point Estimators

High breakdown point estimators in regression are robust against gross contamination in the regressors and also in the errors; the least median of squares (LMS) estimator has the additional property of packing the majority of the sample most tightly around the estimated regression hyperplane in terms of absolute deviations of the residuals and thus is helpful in identifying outliers. Asymptotics for a class of high breakdown point smoothed LMS estimators are derived here under a variety of conditions that allow for time series applications; joint limit processes for several smoothed estimators are examined. The limit process for the LMS estimator is represented via a generalized Gaussian process that defines the generalized derivative of the Wiener process.

On the distributions of Augmented Dickey–Fuller statistics in processes with moving average components

Abstract This paper considers the distributions of augmented Dickey–Fuller statistics to test for a unit root in the presence of errors that have a moving average part. We examine the dependence of the augmented Dickey–Fuller statistic on the order, k, of the approximating autoregression, by deriving the asymptotic distribution of the statistic for finite k. The results provide a direct indication of the rate of growth of k with sample size that is necessary in order to avoid non-negligible size distortions, and are illustrated by simulations that provide empirical densities for various cases of interest. The size distortions are particularly difficult to control when the moving average part contains a root near unity; for such cases we examine the distribution of the ADF statistics in models estimated by feasible GLS, rather than OLS, to control for the moving-average component. We find that both size distortions and sensitivity of the test statistic to k are greatly reduced in this case.

The GLS Transformation Matrix and a Semi-recursive Estimator for the Linear Regression Model with ARMA Errors

For a general stationary ARMA( p,q ) process u we derive the exact form of the orthogonalizing matrix R such that R ′ R = Σ −1 , where Σ = E ( uu ′) is the covariance matrix of u , generalizing the known formulae for AR ( p ) processes. In a linear regression model with an ARMA( p,q ) error process, transforming the data by R yields a regression model with white-noise errors. We also consider an application to semi-recursive (being recursive for the model parameters, but not for the parameters of the error process) estimation.

KERNEL ESTIMATION WHEN DENSITY MAY NOT EXIST

Nonparametric kernel estimation of density and conditional mean is widely used, but many of the pointwise and global asymptotic results for the estimators are not available unless the density is continuous and appropriately smooth; in kernel estimation for discrete-continuous cases smoothness is required for the continuous variables. Nonsmooth density and mass points in distributions arise in various situations that are examined in empirical studies; some examples and explanations are discussed in the paper. Generally, any distribution function consists of absolutely continuous, discrete, and singular components, but only a few special cases of nonparametric estimation involving singularity have been examined in the literature, and asymptotic theory under the general setup has not been developed. In this paper the asymptotic process for the kernel estimator is examined by means of the generalized functions and generalized random processes approach; it provides a unified theory because density and its derivatives can be defined as generalized functions for any distribution, including cases with singular components. The limit process for the kernel estimator of density is fully characterized in terms of a generalized Gaussian process. Asymptotic results for the Nadaraya–Watson conditional mean estimator are also provided.

Robust kernel estimator for densities of unknown smoothness

Results on non-parametric kernel estimators of density differ according to the assumed degree of density smoothness. A kernel/bandwidth pair that was optimal for a twice differentiable function may not be suitable when the density is piecewise linear. If there is uncertainty about the degree of smoothness, an inappropriate choice may lead to under- or oversmoothing. To examine various possible outcomes we provide asymptotic results on kernel estimation of a continuous density for an arbitrary bandwidth/kernel pair and derive the limit joint distribution of kernel density estimators corresponding to different bandwidths and kernel functions. Using these results, we propose a combined estimator constructed as an optimal linear combination of several estimators with different bandwidth/kernel pairs. Its theoretical properties [Kotlyarova, Y. and Zinde-Walsh, V., 2006, Non- and semi-parametric estimation in models with unknown smoothness. Economics Letters, 93, 379–386] are such that it automatically attains the best possible rate without a priori knowledge of the degree of smoothness. Our Monte-Carlo results confirm the advantages of the combined estimator of density.

Transforming the error-components model for estimation with general ARMA disturbances

Abstract Baltagi and Li (1991) give a transformation which may be applied to certain autocorrelated disturbances in an error-components model to yield spherical disturbances. They derive the transformation for AR(1) and AR(2) cases. We show that the results of Galbraith and Zinde-Walsh (1992) on the general ARMA form of the transformation matrix can be applied to allow estimation of the error-components model with disturbances which follow any ARMA (p, q) process. We also show that, for models with a cross-sectional dimension which is large relative to the time dimension, it is possible to use this method to perform a general autocorrelation transformation without specifying a particular ARMA process or estimating ARMA parameters.