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Unformatted text preview: Econ 139: Midterm 1 Review Andrew Sweeting 1 Department of Economics Duke University Spring 2010 Econ 139 Review 1 (Duke) Review 1 Spring 2010 1 / 30 1 Basic Statistics 2 Properties of Estimators (Bias, Consistency, Asymptotic Normality) 3 Properties of OLS 4 Hypothesis Testing Econ 139 Review 1 (Duke) Review 1 Spring 2010 2 / 30 Key Statistical Concepts If X is a random variable then we can characterize it using a cumulative distribution function (CDF) F ( x ) = Pr ( X & x ) or a probability function (discrete) f ( x ) = Pr ( X = x ) or probability density function (pdf, continuous) F ( y ) = dF ( y ) dy = f ( y ) Econ 139 Review 1 (Duke) Review 1 Spring 2010 3 / 30 Key Statistical Concepts for random variables X and Y Population Sample Mean,E() μ X = R ∞ & ∞ xf ( x ) dx ¯ X = 1 n ∑ n i = 1 X i Variance σ 2 X = E h ( X & μ X ) 2 i s 2 X = ∑ n i = 1 ( X i & ¯ X ) 2 n & 1 Standard Deviation p Var ( X ) s X Covariance E [( X & μ X )( Y & μ Y )] s XY = ∑ n i = 1 ( X i & ¯ X )( Y i & Y ) n & 1 Econ 139 Review 1 (Duke) Review 1 Spring 2010 4 / 30 Key Statistical Concepts conditional distribution where P ( X = x ) is the marginal distribution of X P ( Y = y j X = x ) = P ( X = x , Y = y ) P ( X = x ) conditional expectation E [ Y j X = x ] = n ∑ i = 1 y i P ( Y = y i j X = x ) conditional variance Var ( Y j X = x ) = n ∑ i = 1 [ y i & E [ Y j X = x ]] 2 P ( Y = y i j X = x ) Econ 139 Review 1 (Duke) Review 1 Spring 2010 5 / 30 Key Statistical Concepts random variables X and Y are independent if for all values x and y P ( Y = y j X = x ) = P ( Y = y ) equivalent P ( X = x , Y = y ) = P ( X = x ) P ( Y = y ) 8 x , y independent random variables will have zero covariance and zero correlation coe¢ cients zero covariance/correlation does not imply indepenence Econ 139 Review 1 (Duke) Review 1 Spring 2010 6 / 30 Key Statistical Concepts Var ( aX + bY ) = a 2 Var ( X ) + 2 abCov ( X , Y ) + b 2 Var ( Y ) if X and Y are independent then Var ( aX + bY ) = a 2 Var ( X ) + b 2 Var ( Y ) Var ( aX & bY ) = a 2 Var ( X ) + b 2 Var ( Y ) Econ 139 Review 1 (Duke) Review 1 Spring 2010 7 / 30 Key Statistical Concepts A sample is an iid (independent and identically distributed) random sample if it is a simple random sample (SRS, each observation equally likely to be drawn from the population) and each of the n observations in the sample, X 1 , X 2 , ..., X n , are independent (e.g. no relatives), and each observation is drawn from the same overall population distribution....
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 Spring '08
 ALESSANDROTAROZZI
 Economics, Econometrics, Normal Distribution, Standard Deviation, Variance, Duke

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