lecture2 - ISYE6414 Summer 2010 The Linear Model: OLS...

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ISYE6414 Summer 2010 The Linear Model: OLS Regression Lecture 2 Dr. Kobi Abayomi May 27, 2010 You can always infer which sections in the book correspond to the lecture by the Exercises at the end of the lecture - do the exercises! Introduction Regression Analysis is one of the simplest ways we have in statistics to investigate the relationship between two or more variables related in a non-deterministic way. Examples of a deterministic relationship: F = ma,C = 2 πr . If we know exactly everything on the right hand side of the equations, we know exactly everything on the left hand side Examples of a probabilistic relationship: X Bern ( p ) ,Y ∼ N ( n,p ). If we know exactly the parameters on the right hand sides of the relationship, there are still probabilistic (or non-definite) values on the left hand side of the relationship. The Normal Model , for example, is implicitly a probabilistic or relationship. In Regression Analysis - we will explicitly state a relationship between two variables, develop procedures for making inferences about the parameters of the model, and obtain a quantita- tive measure (the correlation coefficient) of the extent to which the two variables are related. I’ll just say it right out: the attractiveness of the linear model is the interpretability of the parameters - not the plausibility of a linear relationship. 1
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1 The simple linear regression model 1.1 The linear deterministic model A simple deterministic relationship between two variables (here x and y ) is a linear relation- ship y = β 0 + β 1 x You remember from algebra and geometry that this is the equation of line with a slope β 1 and a y-intercept β 0 . What is the value of y when x = 0? How much does the value of y change for each unit change x . Remember all of my comments on populations and parameters? If, say, we all were God (a belief not in contrast with some religions! 1 ), we would know exactly the values of y (because we knew exactly the values of β 0 1 ) for each given value of x. Conventionally, we call x the independent variable and y the dependent variable . For example, let y the selling price for a house and x the size of the house - in square feet (are you getting a feel for how we assign x and y ?). And you know that the value of y when x = 0 is 25 , 000. And you know that the change in y for each square foot increase is 75. Then you know β 0 = 25000 and β 1 = 75 and you can write y = 25000 + 75 x as the deterministic relationship between size of house and its selling price Here is a remark: A scatterplot of house and selling price if this relationship is true should look like? In R housesize<-c(1000,1000,1500,1500,1700,1250,2000,2150,2150) #the house sizes in square feet sellingprice<-c(25000000,27000000,38000000,37299999, 42500000,31350003,50000000,53800000,53700000) #the selling prices in dollars 1 This is meant to be funny 2
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plot(housesize,sellingprice,cex=1.5) 1.2 The linear model - with error
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lecture2 - ISYE6414 Summer 2010 The Linear Model: OLS...

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