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Lecture Notes on Regression

Lecture Notes on Regression - 1 Linear Regression Suppose...

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1 Linear Regression Suppose that ( x i , y i ) i = 1 , . . . , n denote the height in inches of adult male students and their fathers. If we plot ( x i , y i ) i = 1 , . . . , n we would see a linear pattern with taller fathers having taller sons. This linear pattern is not perfect, since there are factors other than father’s height that determine the height of a son. Yet we feel that to some extent, the height of the sons is a least in part determine by the fathers height. Thus our model for the height of the sons is of the form: y i = β o + β 1 x i + ² i where x i is the height of father i, β o and β 1 are unknown constants, ² i is a random error with mean zero, and y i is the height of son i. Given data ( x i , y i ) i = 1 , . . . , n the problem is to estimate β o and β 1 and then to test or validate the linear model. Once we obtain estimates of β o and β 1 and validate the model, we can use it to estimate the average height of sons whose fathers are 62 inches tall, or to find a confidence interval of the actual height of a son whose father is 62 inches tall. In this example the height of the father is the predictor, or independent, variable and the height of the son the response, or independent, variable As a second example consider a model to predict the performance of college students in first year calculus. The prediction may be based on a placement exam score, high school GPA and SAT scores. In this case the model may be of the form y i = β o + β 1 x i 1 + β 2 x i 2 + β 3 x i 3 + ² i where x i 1 , x i 2 , and x i 3 represent the placement exam score, the high school GPA and the SAT score of student i and y i represents the first year calculus grade of student i. In this example, the placement exam score, the GPA, and the SAT score are predictor, or independent variables, and the first year calculus grade is the response, or independent, variable. As a third example, consider the situation y i = β o + β 1 x i + β 2 x 2 i + . . . + β k x k i + ² i . Here the response variable y i depends on the predictor variable x i and its first k powers. There are also some non-linear models that can be transformed into a linear model. For example, if y i = e β o + β 1 x i + ² i then ln y = β o + β 1 x i + ² i . All of the above models are of the form y i = f ( x 1 i , . . . , x i,p - 1 ) + ² i where f ( x 1 , . . . , x p - 1 ) = β o + β 1 x 1 + . . . + β p - 1 x p - 1 and ² i is a mean zero random variable. For the most part we will assume that the ² i s are independent and have a common variance σ 2 .
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