Least Trimmed Squares: Cointegration and outliers

• Department of Economics, University of Oxford, Oxford, UK [vanessa.berenguer-rico@economics.ox.ac.uk]

• Department of Economics, University of Oxford, Oxford, UK [bent.nielsen@nuffield.ox.ac.uk]

Keywords: Asymptotic theory, boundedness, least trimmed squares estimator, non-stationary time series

1 Abstract

When applying the cointegrated autoregressive distributed lag model it is common to include indicator variables for outlying observations or residuals. This is done out of a concern that inference may be distorted if there are unmodelled outliers and an intuition that standard inference may be valid when outliers are modelled. We investigate this intuition through an asymptotic analysis of the Least Trimmed Squares (LTS) estimator. LTS estimation detects outlying residuals, it results in a classification of observations as good and outlying and it applies Ordinary Least Squares (OLS) to the estimated good observations. We formulate a cointegrated model with good and outlying observations and show that when the proportion of outliers is small then LTS inference is indeed the same as OLS inference applied infeasible to the actual good observations.

The least trimmed squares (LTS) estimator [Rousseeuw, 1984] is defined as follows. The investigator specifies that there are $h$ ‘good’ observations and $T-h$ ‘outliers’ in a sample of $T$ observations. The set of good observations is then estimated by the $h$-subsample with the smallest residual sum of squares. The LTS estimator is maximum likelihood in a regression model where $h$ good observations have normal innovations and $T-h$ outliers have innovations that are more extreme than the realized good observations [Berenguer-Rico et al., 2023]. Under regularity conditions, it can be shown that the estimator is asymptotically bounded in probability and it has the asymptotic distribution as the ordinary least squares (OLS) estimator applied infeasibly to the actual good observations [Berenguer-Rico and Nielsen, 2024].

In this paper we check the regularity conditions for an autoregressive distributed lag regression for data generated by a vector autoregression with cointegration. The consequence is that LTS estimator for the autoregressive distributed lag model has the same asymptotic theory as the infeasible OLS estimator on the good observations.