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lecture+7_complete

# lecture+7_complete - The Simple Regression Model(cont y =...

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y = b 0 + b 1 x + u The Simple Regression Model (cont.)

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2 OUTLINE 1. Algebraic Properties of OLS 2. Goodness of Fit 3. Statistical properties of OLS
3 Property 1: The sum of the OLS residuals is zero. Thus, the sample average of the OLS residuals is zero as well. Property 2: The sample covariance between the regressors and the OLS residuals is zero. Property 3: The OLS regression line always goes through the mean of the sample. 1. Algebraic properties of OLS

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4 x y u x n u u n i i i n i i n i i 1 0 1 1 1 ˆ ˆ ) 3 ( 0 ˆ ) 2 ( 0 ˆ thus, and 0 ˆ ) 1 ( b b 1. Algebraic properties of OLS
Question Example: Wage equation wage = b 0 + b 1 educ + u Given a sample of data, you estimate the above model by OLS i.e. run the regression of hourly wages on years of education. You get the following estimates: ?𝒂?? = −?. ?? + ?. ?????? You’ve been given that the sample average of educ is 12.56. What is the sample average of hourly wages?

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Question Example: Wage equation wage = b 0 + b 1 educ + u Given a sample of data, you estimate the above model by OLS i.e. run the regression of hourly wages on years of education. You get the following estimates: ?𝒂?? = −?. ?? + ?. ?????? You’ve been given that the sample average of educ is 12.56 years What is the sample average of hourly wages? Make use of the third algebraic property. ?𝒂?? = −?. ?? + ?. ?? ??. ?? = ?. ? 𝒂?????.
7 We can decompose each observation of y in a explained part and an unexplained or residual part . 2. GOODNESS OF FIT SSR SSE SST : that show to possible is It (SSR) squares of sum residual the is ˆ (SSE) squares of sum explained the is ˆ (SST) squares of sum total the is : following the define then We ˆ ˆ 2 2 2 i i i i i i u y y y y u y y

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8 Proof that SST = SSE + SSR 0 ˆ ˆ that know we and SSE ˆ ˆ 2 SSR ˆ ˆ ˆ 2 ˆ ˆ ˆ ˆ ˆ 2 2 2 2 2 y y u y y u y y y y u u y y u y y y y y y i i i i i i i i i i i i i i
9 How do we think about how well our sample regression line fits our sample data?

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lecture+7_complete - The Simple Regression Model(cont y =...

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