ch4_2 - OLS regression: EVIEWS output Dependent Variable:...

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4-11 OLS regression: EVIEWS output Dependent Variable: TESTSCR Method: Least Squares Date: 02/27/06 Time: 18:23 Sample: 1 420 Included observations: 420 White Heteroskedasticity-Consistent Standard Errors & Covariance Variable Coefficient Std. Error t-Statistic Prob. C 698.9330 10.36436 67.43619 0.0000 STR -2.279808 0.519489 -4.388557 0.0000 R-squared 0.051240 Mean dependent var 654.1565 Adjusted R-squared 0.048970 S.D. dependent var 19.05335 S.E. of regression 18.58097 Akaike info criterion 8.686903 Sum squared resid 144315.5 Schwarz criterion 8.706143 Log likelihood -1822.250 F-statistic 22.57511 Durbin-Watson stat 0.129062 Prob(F-statistic) 0.000003 ± TestScore = 698.9 – 2.28 × STR (we’ll discuss the rest of this output later)
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4-12 The OLS regression line is an estimate, computed using our sample of data; a different sample would have given a different value of 1 ˆ β . How can we: quantify the sampling uncertainty associated with 1 ˆ ? use 1 ˆ to test hypotheses such as 1 = 0? construct a confidence interval for 1 ? Like estimation of the mean, we proceed in four steps: 1. The probability framework for linear regression 2. Estimation 3. Hypothesis Testing 4. Confidence intervals
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4-13 1. Probability Framework for Linear Regression Population population of interest (ex: all possible school districts) Random variables : Y , X Ex: ( Test Score, STR ) Joint distribution of ( Y , X ) The key feature is that we suppose there is a linear relation in the population that relates X and Y ; this linear relation is the “population linear regression”
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4-14 The Population Linear Regression Model (Section 4.3) Y i = β 0 + 1 X i + u i , i = 1,…, n X is the independent variable or regressor Y is the dependent variable • β 0 = intercept • β 1 = slope u i = “error term” The error term 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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4-15 Ex. : The population regression line and the error term What are some of the omitted factors in this example?
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4-16 Data and sampling The population objects (“ parameters ”) β 0 and 1 are unknown; so to draw inferences about these unknown parameters we must collect relevant data. Simple random sampling : Choose n entities at random from the population of interest, and observe (record) X and Y for each entity Simple random sampling implies that {( X i , Y i )}, i = 1,…, n , are independently and identically distributed (i.i.d.). ( Note : ( X i , Y i ) are distributed independently of ( X j , Y j ) for different observations i and j .)
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4-17 Task at hand: to characterize the sampling distribution of the OLS estimator. To do so, we make three assumptions: The Least Squares Assumptions 1. The conditional distribution of u given X has mean zero, that is, E ( u | X = x ) = 0. 2. ( X i ,Y i ), i =1,…, n , are i.i.d. 3. X and u have four moments, that is: E ( X 4 ) < and E ( u 4 ) < . We’ll discuss these assumptions in order.
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4-18 Least squares assumption #1: E ( u | X = x ) = 0.
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ch4_2 - OLS regression: EVIEWS output Dependent Variable:...

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