Bayesian inference using least median of squares and least trimmed squares in models with independent or correlated errors and outliers

Abstract

We provide Bayesian inference in the context of Least Median of Squares and Least Trimmed Squares, two well-known techniques that are highly robust to outliers. We apply the new Bayesian techniques to linear models whose errors are independent or AR and ARMA. Model comparison is performed using posterior model probabilities, and the new techniques are examined using Monte Carlo experiments as well as an application to four portfolios of asset returns.

KEYWORDS:

• Bayesian inference
• least Median of squares
• least trimmed squares
• stochastic volatility
• ARMA models
• outliers

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 Similar ideas have been proposed by Kato (2013) who, in the context of instrumental variables models, instead of assuming a distributional assumption, he proposed a quasi-likelihood induced from conditional moment restrictions. A similar approach has been used in Chernozhukov and Hong (2003).

2 All MCMC uses 1,500,000 passes the first 500,000 of which are omitted during the burn-in phase to mitigate possible start up effects. This is, of course, excessive, and in all cases, 15,000 omitting the first 5,000 produce the same results. Initial conditions for MCMC are provided by numerical minimization of the loss function. The standard Geweke (1992) numerical performance and MCMC convergence are inspected in all cases and they are available on request.

3 This is computed in two steps. In the first step we use OLS to obtain the residuals. From the residuals we obtain ρ by regressing the residuals on their lagged values. Finally, we use the estimated value of ρ to estimate (11)(11) $$y_t=\rho y_{t-1}+{(x_t-\rho x_{t-1})}^{\prime }\beta +e_t.$$(11) by OLS.

4 In this case, all parameters are estimated simultaneously using a standard Gibbs sampler. MCMC is based on 150,000 draws and the first 50,000 are omitted in the burn-in phase.

5 It is possible to treat e₀ as unknown parameter.

6 Since BMA for ARMA favors an ARMA(2,2) reported here for ARMA method are marginal posteriors of the first order coefficients.