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chapter 1 [article] - Contents 1 Ordinary least squares(OLS...

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Contents 1 Ordinary least squares (OLS) 1 1.1 Simple linear model . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Least squares regression . . . . . . . . . . . . . . . . . . . . . . . 2 2 Interpretation and R 2 7 2.1 Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Goodness of fit: R 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1 Ordinary least squares (OLS) Contents 1.1 Simple linear model Contents We want to start looking for relationships between economic variables We’re going to start with the most straightforward possible such relation- ship one dependent variable Y that we are trying to explain one explanatory variable X that might help us explain Y any relationship between X and Y is assumed a priori to be linear This is known as ‘simple linear regression’ ‘simple’ because there’s only one explanatory variable X We will assume that the following is the ‘true’ process by which the depen- dent variable Y is generated: Y i = β 1 + β 2 X i + u i (1.1) β 1 and β 2 are parameters that we would like to estimate u is a disturbance term The i subscript refers to the particular observation of the variables for example, if we are examining the relationship between height and wages across people, then the i would refer to the people we only actually ‘observe’ the Y i and X i , not the u i
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Y X β 1 Y = β 1 + β 2 X u i The true, data-generating model Why is there a disturbance term? Why isn’t the relationship between Y and X ‘exact’? Omitted variables : something else also helps explain Y Aggregation : often the relationship is an aggregate one e.g. adding-up lots of little consumption functions aggregate con- sumption function unlikely to be exact Model misspecification : Y might depend on yesterday’s X , or the expectation of tomorrow’s X , rather than X itself relationship between Y and X will be close, but not exact Functional misspecification : maybe the relationship is nonlinear? Measurement error : Y or X might be mismeasurements of the ‘true’ variables 1.2 Least squares regression Contents Y i = β 1 + β 2 X i + u i (1.1) Suppose we have a sample of n observations of ( Y i , X i ) combinations How to actually estimate the parameters β 1 and β 2 , given that we only observe the Y i
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