The Ordinary Least Squares (OLS) Estimator

[Pages:30]The Ordinary Least Squares (OLS) Estimator

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Regression Analysis

? Regression Analysis: a statistical technique for investigating and modeling the relationship between variables.

? Applications: Engineering, the physical and chemical science, economics, management, life and biological science, and the social science

? Regression analysis may be the most widely used statistical technique

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? Example 1: delivery time v.s. delivery volume

? Suspect that the time required by a route deliveryman to load and service a machine is related to the number of cases of product delivered

? 25 randomly chosen retail outlet ? The in-outlet delivery time and the volume of

product delivery ? Scatter diagram: display a relationship between

delivery time and delivery volume

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? Y: delivery time, x: delivery volume Y = 0 + 1 x +

? Error, :

? The difference between y and 0 + 1 x ? A statistical error, i.e. a random variable ? The effects of the other variables on delivery

time, measurement errors, ...

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? Simple linear regression model: Y = 0 + 1 x +

? x: independent (predictor, regressor) variable ? Y: dependent (response) variable ? : error

? If x is fixed, Y is determined by . ? Suppose that E() = 0 and Var() = 2 .

Then E(Y|x) = E(0 + 1 x + ) = 0 + 1 x Var(Y|x) = Var(0 + 1 x + ) = 2

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? The true regression line is a line of mean values: the height of the regression line at any x is the expected value of Y for that x.

? The slope, 1: the change in the mean of Y for a unit change in x

? The variability of Y at x is determined by the variance of the error

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