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Midterm_Review_Prof._Arkonac's_Slides_(Ch_1_-_Ch_9)

Midterm_Review_Prof._Arkonac's_Slides_(Ch_1_-_Ch_9) -...

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Review for Midterm Prof: Seyhan Erden Arkonac, PhD Midterm Exam is on Tuesday March 26 th in class at 9:10am. (1) You may bring one A4 size paper of formulas (2) Bring a simple calculator (IT83 or IT89). You may not use your cell phone as calculator in the exam. 1
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2 The Population Linear Regression Model – general notation Y i = 0 + 1 X i + u i , i = 1,…, n X is the independent variable or regressor Y is the dependent variable 0 = intercept 1 = slope u i = the regression error The regression error consists of omitted factors, or possibly measurement error in the measurement of Y . In general, these omitted factors are other factors that influence Y , other than the variable X
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3 This terminology in a picture : Observations on Y and X ; the population regression line; and the regression error (the “error term”):
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4 The Ordinary Least Squares Estimator (SW Section 4.2) How can we estimate 0 and 1 from data? Recall that Y was the least squares estimator of Y : Y solves, 2 1 min ( ) n mi i Ym By analogy, we will focus on the least squares (“ ordinary least squares ” or “ OLS ”) estimator of the unknown parameters 0 and 1 , which solves, 01 2 , 0 1 1 min [ ( )] n b b i i i Y b b X 
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5 Mechanics of OLS The population regression line: Test Score = 0 + 1 STR 1 = Test score STR = ??
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6 Interpretation of the estimated slope and intercept TestScore = 698.9 – 2.28 STR Districts with one more student per teacher on average have test scores that are 2.28 points lower. That is, Test score STR = –2.28 The intercept (taken literally) means that, according to this estimated line, districts with zero students per teacher would have a (predicted) test score of 698.9. This interpretation of the intercept makes no sense – it extrapolates the line outside the range of the data – here, the intercept is not economically meaningful.
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7 Predicted values & residuals: One of the districts in the data set is Antelope, CA, for which STR = 19.33 and Test Score = 657.8 predicted value: ˆ Antelope Y = 698.9 – 2.28 19.33 = 654.8 residual: ˆ Antelope u = 657.8 – 654.8 = 3.0
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8 OLS regression: STATA output regress testscr str, robust Regression with robust standard errors Number of obs = 420 F( 1, 418) = 19.26 Prob > F = 0.0000 R-squared = 0.0512 Root MSE = 18.581 ------------------------------------------------------------------------- | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] --------+---------------------------------------------------------------- str | -2.279808 .5194892 -4.39 0.000 -3.300945 -1.258671 _cons | 698.933 10.36436 67.44 0.000 678.5602 719.3057 ------------------------------------------------------------------------- TestScore = 698.9 – 2.28 STR (we’ll discuss the rest of this output later)
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9 Measures of Fit (Section 4.3) A natural question is how well the regression line “fits” or explains the data. There are two regression statistics that provide complementary measures of the quality of fit: The regression R 2 measures the fraction of the variance of Y that is explained by X ; it is unitless and ranges between zero (no fit) and one (perfect fit) The standard error of the regression ( SER ) measures the magnitude of a typical regression residual in the units of Y .
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10 The regression R 2 is the fraction of the sample variance of Y i “explained” by the regression.
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