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Notes4 - Lecture Notes 4 Econ 410 Introduction to...

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Lecture Notes 4 Econ 410 – Introduction to Econometrics 1 Linear Regression with One Regressor: Part II Regression When X is a Binary Variable Regression analysis can also be used when the regressor is a binary variable (that is, when it takes on only two values). In this case, the procedure for obtaining the OLS estimators 0 ˆ β and 1 ˆ β is the same as if the regressor is a continuous variable. However, the interpretation of the regression coefficients is different and it is related to the fact that a regression with a binary variable is equivalent to a difference of means analysis. Suppose that Y is the random variable “earnings” and X is a binary random variable defined as: = female a is person i the if male a is person i the if X th th i 0 1 The population regression model is: n i u X Y i i i , ..... , 1 1 0 = + + = β β In order to understand the meaning of the regression coefficients, we consider the two possible cases 0 = i X and 1 = i X one at the time. When the observed person is a female, the linear regression model can be written as: i i u Y + = 0 β From the conditional mean assumption, we know that ( ) 0 0 | β = = i i X Y E , so 0 β is the population mean value of earnings for females. Let now consider the case in which the observed person is a male. The linear regression model is: i i u Y + + = 1 0 β β From the conditional mean assumption, we know that ( ) 1 0 1 | β β + = = i i X Y E , so that 1 0 β β + is the population mean value of earnings for males. Since 1 0 β β + is the population mean value of earnings for males and 0 β is the population mean value of earnings for females, then 1 β is the difference in mean earnings in these two populations. This can be mathematically written as: ( ) ( ) 1 0 | 1 | β = = = i i i i X Y E X Y E .
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Lecture Notes 4 Econ 410 – Introduction to Econometrics 2 Hypothesis Testing and Confidence Intervals Hypothesis testing and confidence intervals use the sampling distribution of the OLS estimators 0 ˆ β and 1 ˆ β to infer about the true value of the coefficients in the linear regression model. In this part, we assume that the three least squares assumptions hold, so that in large samples 0 ˆ β and 1 ˆ β have a normal distribution.
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