Adjusted Confidence Intervals for The Slope of A Simple Regression Line
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Abstract
Nanayakkara and Cressie [1] have presented a confidence interval for the slope (β1) in the regression model (1) when the error term has non constant variance. For the case of a single predictor and the error terms are normally distributed with variance which is a function of the predictor variable. Their simulation results show that the proposed confidence interval, is preferable to the confidence interval based on the ordinary least squares, OLS method. Later, Wilcox [2] found that confidence intervals based on bootstrap techniques can improve Nanayakkara and Cressie’s (NC) method. However, the simulation results show only a comparison of coverage probabilities of confidence intervals for the slope. Unlike Nanayakkara and Cressie and Wilcox, this paper presents a comparison of confidence intervals for the slope of a simple regression using OLS and NC methods when these confidence intervals have minimum coverage probability 1-α. The ratio of the expected lengths of these confidence intervals are compared using Monte Carlo simulation. The conclusion is that the confidence interval for β1 based on the NC method is less efficient than the confidence interval for β1 based on the OLS method for some cases. These results are not similar to those of Nanayakkara and Cressie and Wilcox.
Keywords: Simple linear regression model, Heteroscedasticity, Ordinary least-squares method, Coverage probability, Expected length
Corresponding author: E-mail: snw@kmitnb.ac.th and chanaphun@yahoo.com
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References
[2] Wilcox, R. 1996 Confidence Intervals for the Slope of a Regression Line when the error term has nonconstant variance, Computational Statistics & Data Analysis, 22, 89-98.
[3] Kabaila, P. 1995 The effect of model selection on confidence region and prediction region, Econometric Theory, 11, 537-549.
[4] Kabaila, P. 1998 Valid confidence intervals in regression after variable selection, Econometric Theory, 14, 463-482.
[5] Hoaglin, D.C., 1985 Summarizing shape numerically: the g and h distributions, in: D. Hoaglin, F. Mosteller and J. Tukey Eds, Exploring data tables, trends and shapes., New York, Wiley.
