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notes02 - The Two-Variable Model ECON 399 Neil Hepburn...

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The Two-Variable 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 two-variable model While this model is very simple and easy to work with conceptually, the results that we derive here are easily extended to the k -variable case 2 The Two Variable Model The Two Variable Model In the simple two-variable model we have two variables – an explanatory variable and a response variable ( x and y ) 1
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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 0 10 20 30 40 50 -100 0 100 200 300 400 500 x y The overall mean of y is 207.2409 This isn’t 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
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x ¯ y 0 -22.43 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 ˆ 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
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40 50 60 70 80 90 100 110 100 150 200 250 300 x y ●●
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