Absolute Deviation Estimator Research Articles

Bootstrap tests for unit roots based on LAD estimation

In this paper we propose a new bootstrap test for unit roots in first-order autoregressive models based on least absolute deviation (LAD) estimators. It is well known that the behaviour of this estimator when the distribution is heavy tailed is very good compared with least-squares estimation. The innovations distribution dependence of the LAD asymptotic law is overcome using bootstrap, which automatically approaches the target distribution. Our strategy avoids the usual problem of estimating the variance matrix and the density at zero, and makes also unnecessary the construction of distribution free statistics through linear combinations with the least-squares estimator. We provide the bootstrap functional limit theory necessary to prove the asymptotic validity of the procedure. Moreover, a large simulation study shows that our test has very good power behaviour compared with others proposed in the literature.

Asymptotics of Quantiles and Rank Scores in Nonlinear Time Series

This paper extends the concept of regression and autoregression quantiles and rank scores to a very general nonlinear time series model. The asymptotic linearizations of these nonlinear quantiles are then used to obtain the limiting distributions of a class of L‐estimators of the parameters. In particular, the limiting distributions of the least absolute deviation estimator and trimmed estimators are obtained. These estimators turn out to be asymptotically more efficient than the widely used conditional least squares estimator for heavy‐tailed error distributions. The results are applicable to linear and nonlinear regression and autoregressive models including self‐exciting threshold autoregressive models with known threshold.

Estimation of multiple-regime regressions with least absolutes deviation

This paper considers least absolute deviations estimation of a regression model with multiple change points occurring at unknown times. Some asymptotic results, including rates of convergence and asymptotic distributions, for the estimated change points and the estimated regression coefficient are derived. Results are obtained without assuming that each regime spans a positive fraction of the sample size. In addition, the number of change points is allowed to grow as the sample size increases. Estimation of the number of change points is also considered. A feasible computational algorithm is developed. An application is also given, along with some Monte Carlo simulations.

A remark on approximate M-estimators

An approximate M-estimator is defined as a value that minimizes certain random function up to a ε n , where { ε n } is a sequence of real numbers converging to zero. We determine the rate of ε n so that the approximate M-estimator is asymptotically normal with rate n 1/2. Our results apply to common M-estimators such as the least absolute deviations estimator for the linear model.

Second Order Representations of the Least Absolute Deviation Regression Estimator

We consider exact weak and strong Bahadur-Kiefer representations of the least absolute deviation estimator for the linear regression model. The precise behavior of these representations is obtained under minimal conditions.

On the robustness of two alternatives to least squares: A Monte Carlo study

This note provides Monte Carlo evidence on the finite sample behavior of two alternatives to least squares—the least absolute deviations estimator and a partially adaptive estimator—when the regression errors are conditionally heteroscedastic.

Least Absolute Deviation Estimation for Regression with ARMA Errors

The asymptotic normality for least absolute deviation estimates of the parameters in a linear regression model with autoregressive moving average errors is established under very general conditions. The method of proof is based on a functional limit theorem for the LAD objective function.

Minimum distance estimators for random coefficient autoregressive models

This paper discusses a class of minimum distance (MD) estimators for a class of pth-order random coefficient autoregressive (RCAR( p)) models. These estimators are defined via certain weighted empiricals as in Koul (1986). The class of estimators considered includes the least absolute deviation estimator and an analogue of the Hodges-Lehmann estimator. The paper contains a proof of the asymptotic normality of these estimators and a simulation study. It is observed that the RCAR(2) model with the Hodges-Lehmann type estimator fits the Canadian lynx data at least as well as with the least square estimator.

Permutation Tests for Least Absolute Deviation Regression

A permutation test based on proportionate reduction in sums of absolute deviations when passing from reduced to full parameter models is developed for testing hypotheses about least absolute deviation (LAD) estimates of conditional medians in linear regression models. Sampling simulations demonstrated that the permutation test on full model LAD estimates had greater relative power (1.06-1.43) than normal theory tests on least squares estimates for asymmetric, chi-square error distributions and symmetric, double exponential error distributions for models with one (n = 35 and n = 63) and three (n 63) independent variables. Relative power was .77 -.84 for normal error distributions. Power simulations demonstrated the low sensitivity of LAD estimates and permutation tests to outlier contamination and heteroscedasticity that was a linear function of X, and increased sensitivity to heteroscedasticity that was a function of X2 for simple regression models. Three permutation procedures for testing partial models in multiple regression were compared: permuting residuals from the reduced model, permuting residuals from the full model, and permuting the dependent variable. Permuting residuals from the reduced model maintained nominal error rates best under the null hypothesis for all error distributions and for correlated and uncorrelated independent variables. Power under the alternative hypotheses for this partial testing procedure was similar to full LAD regression model tests. Four example applications of LAD regression are provided.

Asymptotic Theory of LAD Estimation in a Unit Root Process with Finite Variance Errors

In this paper we derive the asymptotic distribution of the least absolute deviations (LAD) estimator of the autoregressive parameter under the unit root hypothesis, when the errors are assumed to have finite variances, and present LAD-based unit root tests, which, under heavy-tailed errors, are expected to be more powerful than tests based on least squares. The limiting distribution of the LAD estimator is that of a functional of a bivariate Brownian motion, similar to those encountered in cointegrating regressions. By appropriately correcting for serial correlation and other distributional parameters, the test statistics introduced here are found to have either conditional or unconditional normal limiting distributions. The results of the paper complement similar ones obtained by Knight (1991, Canadian Journal of Statistics 17, 261-278) for infinite variance errors. A simulation study is conducted to investigate the finite sample properties of our tests.

Robust Nonstationary Regression

This paper provides a robust statistical approach to nonstationary time series regression and inference. Fully modified extensions of traditional robust statistical procedures are developed that allow for endogeneities in the nonstationary regressors and serial dependence in the shocks that drive the regressors and the errors that appear in the equation being estimated. The suggested estimators involve semiparametric corrections to accommodate these possibilities, and they belong to the same family as the fully modified least-squares (FM-OLS) estimator of Phillips and Hansen (1990, Review of Economic Studies 57,99–125). Specific attention is given to fully modified least absolute deviation (FM-LAD) estimation and fully modified M (FM-M) estimation. The criterion function for LAD and some M-estimators is not always smooth, and this paper develops generalized function methods to cope with this difficulty in the asymptotics. The results given here include a strong law of large numbers and some weak convergence theory for partial sums of generalized functions of random variables. The limit distribution theory for FM-LAD and FM-M estimators that is developed includes the case of finite variance errors and the case of heavytailed (infinite variance) errors. Some simulations and a brief empirical illustration are reported.

Least Absolute Deviation Estimation of a Shift

This paper develops the asymptotic theory for least absolute deviation estimation of a shift in linear regressions. Rates of convergence and asymptotic distributions for the estimated regression parameters and the estimated shift point are derived. The asymptotic theory is developed both for fixed magnitude of shift and for shift with magnitude converging to zero as the sample size increases. Asymptotic distributions are also obtained for trending regressors and for dependent disturbances. The analysis is carried out in the framework of partial structural change, allowing some parameters not to be influenced by the shift. Efficiency relative to least-squares estimation is also discussed. Monte Carlo analysis is performed to assess how informative the asymptotic distributions are.

Can Markets Value Air Quality? A Meta-Analysis of Hedonic Property Value Models

This paper reports the results of a statistical summary of estimates of the marginal willingness to pay (MWTP) for reducing particulate matter from hedonic property value models developed between 1967 and 1988. Results using both ordinary least squares and minimum absolute deviation estimators suggest that market conditions and the procedures used to implement the hedonic models were important to the resulting MWTP estimates. The interquartile range for these estimated marginal values (measured as a change in asset prices) lies between zero and $98.52 (in 1982-84 dollars) for a one-unit reduction in total suspended particulates (in micrograms per cubic meter). The mean MWTP is nearly five times the median ($109.90 vs. $22.40), suggesting that outliers are important influences to any summary statistics for these estimates.

On Curve Estimation by Minimizing Mean Absolute Deviation and Its Implications

The local median regression method has long been known as a robustified alternative to methods such as local mean regression. Yet, its optimal statistical properties are largely unknown. In this paper, we show via decision-theoretic arguments that a local weighted median estimator is the best least absolute deviation estimator in an asymptotic minimax sense, under $L_1$-loss. We also study asymptotic efficiency of the local median estimator in the class of all possible estimators. From a practical viewpoint our results show that local weighted medians are preferable to histogram estimators, since they enjoy optimality properties which the latter do not, under virtually identical smoothness assumptions on the underlying curve. Among smoothing methods that are adapted to functions with only one derivative, little is to be gained by using an estimator other than one based on the local median.

The influence functions for the least trimmed squares and the least trimmed absolute deviations estimators

The influence functions for Rousseeuw's (1987) least trimmed squares (LTS) estimator and for Tableman's (1994) least trimmed absolute deviations (LTAD) estimator are derived in the univariate case. The half-sample estimators which possess, by construction, the 50% breakdown point property satisfy three of the four robustness criteria defined by Hampel (1986). They have bounded influence functions, finite gross-error sensitivity, and finite rejection point. However, they have infinite local-shift sensitivity. Hence, these estimates can be highly sensitive to small perturbations in the data. Small shifts in centrally located data (inliers) can cause their values to change by relatively large (though bounded) amounts.

Weak Convergence of Randomly Weighted Dependent Residual Empiricals with Applications to Autoregression

This paper establishes the uniform closeness of a randomly weighted residual empirical process to its natural estimator via weak convergence techniques. The weights need not be independent, bounded or even square integrable. This result is used to yield the asymptotic uniform linearity of a class of rank statistics in $p$th-order autoregression models. The latter result, in turn, yields the asymptotic distributions of a class of robust and Jaeckel-type rank estimators. The main result is also used to obtain the asymptotic distributions of the least absolute deviation and certain other robust minimum distance estimators of the autoregression parameter vector.

Least absolute deviation estimation of stationary time series models

This paper is concerned with practical aspects of least absolute deviation (LAD) estimation of Box-Jenkins models for single time series. Formulation of the LAD method as a non-linear programming problem is described. The method is applied to a multiplicative seasonal moving average model for monthly rice sales data and to US airline passenger data.

Some aspects of measurement error in a censored regression model

When a linear model with a censored dependent variable is written as a regression model, the model is nonlinear in the explanatory variables. This means that when the explanatory variables are measured with error, estimation by instrumental variables will not lead to consistent estimates of the parameters. My approach to the problem involves assuming that the variables measured with error can be represented by a reduced form equation and estimating the parameters of the censored regression model by the least absolute deviations estimator. This estimator is based on the median of the distribution of the dependent variable rather than the mean, and the median is not affected by censoring in the lower tail. Hence the estimator is consistent under general conditions. I also consider the problem of testing for the presence of measurement error.

A comparison of some robust, adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based ...

M-estimation for autoregressions with infinite variance

We study the problem of estimating autoregressive parameters when the observations are from an AR process with innovations in the domain of attraction of a stable law. We show that non-degenerate limit laws exist for M-estimates if the loss function is sufficiently smooth; these results remain valid if location and scale are also estimated. For least absolute deviation (LAD) estimates, similar results hold under conditions on the innovations distribution near 0. We also discuss, under moment conditions on the innovations, consistency properties for M-estimators corresponding to the class of loss functions, ϱ( x) = | x | γ for some γ > 0.