M. Qamarul Islam

Multiple Linear Regression Model Under Nonnormality

Abstract We consider multiple linear regression models under nonnormality. We derive modified maximum likelihood estimators (MMLEs) of the parameters and show that they are efficient and robust. We show that the least squares esimators are considerably less efficient. We compare the efficiencies of the MMLEs and the M estimators for symmetric distributions and show that, for plausible alternatives to an assumed distribution, the former are more efficient. We provide real-life examples.

NONNORMAL REGRESSION. II. SYMMETRIC DISTRIBUTIONS

Salient features of a family of short-tailed symmetric distributions, introduced recently by Tiku and Vaughan [1], are enunciated. Assuming the error distribution to be one of this family, the methodology of modified likelihood is used to derive MML estimators of parameters in a linear regression model. The estimators are shown to be efficient, and robust to inliers. This paper is essentially the first to achieve robustness to inliers. The methodology is extended to long-tailed symmetric distributions and the resulting estimators are shown to be efficient, and robust to outliers. This paper should be read in conjunction with Islam et al. [2]who develop modified likelihood methodology for skew distributions in the context of linear regression.

NONNORMAL REGRESSION. I. SKEW DISTRIBUTIONS

In a linear regression model of the type y= θ X+e, it is often assumed that the random error eis normally distributed. In numerous situations, e.g., when ymeasures life times or reaction times, etypically has a skew distribution. We consider two important families of skew distributions, (a) Weibull with support IR: (0, ∞) on the real line, and (b) generalised logistic with support IR: (−∞, ∞). Since the maximum likelihood estimators are intractable in these situations, we derive modified likelihood estimators which have explicit algebraic forms and are, therefore, easy to compute. We show that these estimators are remarkably efficient, and robust. We develop hypothesis testing procedures and give a real life example. Symmetric families of distributions, both long and short tailed, will be considered in a future paper.

Estimation in bivariate nonnormal distributions with stochastic variance functions

Data sets in numerous areas of application can be modelled by symmetric bivariate nonnormal distributions. Estimation of parameters in such situations is considered when the mean and variance of one variable is a linear and a positive function of the other variable. This is typically true of bivariate t distribution. The resulting estimators are found to be remarkably efficient. Hypothesis testing procedures are developed and shown to be robust and powerful. Real life examples are given.

Multiple linear regression model with stochastic design variables

In a simple multiple linear regression model, the design variables have traditionally been assumed to be non-stochastic. In numerous real-life situations, however, they are stochastic and non-normal. Estimators of parameters applicable to such situations are developed. It is shown that these estimators are efficient and robust. A real-life example is given.

Estimation in multivariate nonnormal distributions with stochastic variance function

In this paper the problem of estimation of location and scatter of multivariate nonnormal distributions is considered. Estimators are derived under a maximum likelihood setup by expressing the non-linear likelihood equations in the linear form. The resulting estimators are analytical expressions in terms of sample values and, hence, are easily computable and can also be manipulated analytically. These estimators are found to be remarkably more efficient and robust as compared to the least square estimators. They also provide more powerful tests in testing various relevant statistical hypotheses.

Impact of exchange rate and customs union on trade balance at commodity level of Turkey with EU(15)

Abstract This paper investigates the short-run and long-run impact of exchange rate and customs union on the trade balance at commodity-group level of Turkey with EU (15). Bounds testing approach is employed where a new strategy in the model selection phase is adopted ensuring that optimal model is selected from those models satisfying both diagnostics and cointegration. Results indicate that in the short-run exchange rate matters in determination of trade balance of 13 commodity groups out of 21 and customs union in 8 cases. Pattern of response of trade balance to exchange rate does not suggest a J-curve effect in any of cases. As for the long-run effect, neither exchange rate nor customs union has a statistically significant effect on trade balance of any of commodity groups, suggesting that those significant short-run effects don’t last into long-run.

Mahalanobis distance under non-normality

We give a novel estimator of Mahalanobis distance D 2 between two non-normal populations. We show that it is enormously more efficient and robust than the traditional estimator based on least squares estimators. We give a test statistic for testing that D 2=0 and study its power and robustness properties.

Firm size and job creation: evidence from Turkey

Abstract This study examines the relationship between firm size and job creation by using an extensive data set covering all non-farm Turkish businesses with 20 or more employees from 2003 to 2010. We find that small firms (firms with employees between 20 and 100 employees) have higher mean job flow rates (job creation, job destruction and net job creation rates) than large firms. Firm size and job flow rates are inversely related, and this relationship is especially prominent for firms with 50 employees or more. Although the overall pattern observed is also observed in both sectors, job creation rates in services are higher than the ones in manufacturing. The magnitudes of job destruction rates are comparable across sectors. Higher job creation rate in services but comparable job destruction rate results in higher net job creation rate in services. As for shares, only for smaller firms (20–49 and 50–99 size categories), job creation shares are greater than their shares in employment. But these firms have disproportionate job destruction shares as well. We also find that only the 20–49 category firms contribute to net job creation more than their share in employment. The smaller firms have high disproportionate shares in job creation and destruction in manufacturing and services as well.

Inference in multivariate linear regression models with elliptically distributed errors

In this study we investigate the problem of estimation and testing of hypotheses in multivariate linear regression models when the errors involved are assumed to be non-normally distributed. We consider the class of heavy-tailed distributions for this purpose. Although our method is applicable for any distribution in this class, we take the multivariate t-distribution for illustration. This distribution has applications in many fields of applied research such as Economics, Business, and Finance. For estimation purpose, we use the modified maximum likelihood method in order to get the so-called modified maximum likelihood estimates that are obtained in a closed form. We show that these estimates are substantially more efficient than least-square estimates. They are also found to be robust to reasonable deviations from the assumed distribution and also many data anomalies such as the presence of outliers in the sample, etc. We further provide test statistics for testing the relevant hypothesis regarding the regression coefficients.