It is known that the least square estimator of the slope $\beta$ of the simple regression model $ Y_i = \alpha + \beta \, x_i + \epsilon_i$ $ (i=1, \ldots, n)$ (where $x_1, \ldots ,x_n$ are supposed to be known constants) can be obtained by setting to zero the sample covariance between the residuals $z_i =y_i - \beta x_i$ and the values $x_i,$ $(i=1, \ldots, n).$ It is similarly known that, when the error terms $\epsilon_i$ are not normally distributed, the least square estimator may not be optimal, being often inefficient and vulnerable to outliers. In these situations one can revert to other estimators of $\beta$ depending also on the ranks of the observations, rather than on their specific values; Cifarelli (1977) proposes a nonparametric estimator $B^*$ by choosing a more robust measure of association between the residuals $z_i$ and the values $x_i,$ that is Gini's cograduation index. However, even if the asymptotic properties of $B^*$ have been extensively studied, its behavior in small samples and its optimality under specific hypothesis for the distribution of the error terms need to be further investigated. This paper aims to fill this gap by a simulation study concerning the performance of Cifarelli's estimator in small samples. Different choices of the distribution of the error terms are tried, also reflecting the behavior of economic phenomena, which are often characterized by skewness and heavy tails. Under these conditions, the efficiency of $B^*$ is simulated and compared with the one of the least square estimator and of other nonparametric estimators of $\beta.$ In addition, the validity of Cifarelli's asymptotic conclusions about the estimator are discussed when the sample size is not large. The reported simulations show good results concerning $B^*$ and hence it seems clear that a deeper use of this estimator, especially in economic applications, can be worthwhile.
A simulation study about robust methods for regression analysis
BORRONI, CLAUDIO GIOVANNI;LUSCIA, FAUSTA
2006
Abstract
It is known that the least square estimator of the slope $\beta$ of the simple regression model $ Y_i = \alpha + \beta \, x_i + \epsilon_i$ $ (i=1, \ldots, n)$ (where $x_1, \ldots ,x_n$ are supposed to be known constants) can be obtained by setting to zero the sample covariance between the residuals $z_i =y_i - \beta x_i$ and the values $x_i,$ $(i=1, \ldots, n).$ It is similarly known that, when the error terms $\epsilon_i$ are not normally distributed, the least square estimator may not be optimal, being often inefficient and vulnerable to outliers. In these situations one can revert to other estimators of $\beta$ depending also on the ranks of the observations, rather than on their specific values; Cifarelli (1977) proposes a nonparametric estimator $B^*$ by choosing a more robust measure of association between the residuals $z_i$ and the values $x_i,$ that is Gini's cograduation index. However, even if the asymptotic properties of $B^*$ have been extensively studied, its behavior in small samples and its optimality under specific hypothesis for the distribution of the error terms need to be further investigated. This paper aims to fill this gap by a simulation study concerning the performance of Cifarelli's estimator in small samples. Different choices of the distribution of the error terms are tried, also reflecting the behavior of economic phenomena, which are often characterized by skewness and heavy tails. Under these conditions, the efficiency of $B^*$ is simulated and compared with the one of the least square estimator and of other nonparametric estimators of $\beta.$ In addition, the validity of Cifarelli's asymptotic conclusions about the estimator are discussed when the sample size is not large. The reported simulations show good results concerning $B^*$ and hence it seems clear that a deeper use of this estimator, especially in economic applications, can be worthwhile.
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