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Unformatted text preview: The TwoVariable Model ECON 399 Neil Hepburn Contents 1 Introduction 1 2 The Two Variable Model 1 3 Assumptions 4 4 The PRF and the SRF 6 5 Estimating the SRF 8 6 Assessing Goodness of Fit 13 7 Functional Form 16 8 Properties of OLS Estimators 19 9 Recap 24 1 Introduction Introduction We begin our study of econometrics by looking at the simplest model  the twovariable model While this model is very simple and easy to work with conceptually, the results that we derive here are easily extended to the kvariable case 2 The Two Variable Model The Two Variable Model In the simple twovariable model we have two variables an explanatory variable and a response variable ( x and y ) 1 You will encounter many different terms for these two variables in different branches of the literature They are all telling us the same thing we think that one variable influ ences the other Motivation Imagine that we have a data set that contains ordered pairs ( x,y ) The x s have values of 0 , 5 , 10 ,..., 50 A scatter plot is shown on the next slide In your intro stats course you learned how to compute the mean of y The Conditional Mean 10 20 30 40 50100 100 200 300 400 500 x y The overall mean of y is 207.2409 This isnt very informative We can see from the graph that as x increases, the values of y tend to increase We can find conditional means the mean of y for each value of x 2 x y22.43 5 35.22 10 87.05 15 109.68 20 160.19 25 214.18 30 271.66 35 305.39 40 320.45 45 351.65 50 446.61 The Conditional Mean The conditional means for our data set are: For x values of 0, the mean of y is 22.43 For x values of 15, the mean of y is 109.68 These are the mean values of y conditional on some value of x The Regression Line The regression line gives us a formula for the conditional mean of y given some value of x In the previous example, things were pretty straight forward since the values of x were discrete This can be generalized to the more common case where the values of x are continuous. The figure on the next slide shows a scatter plot of a sample dataset. The Regression Line y = 9 . 840 + 2 . 715 x We plug in any value of x the result will be the expected value of y conditional on that value of x If we plug in x = 50, the expected value of y is 125.9205 3 40 50 60 70 80 90 100 110 100 150 200 250 300 x y...
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 Spring '11
 Neil.H
 Econometrics

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