BS2247 Introduction to Econometrics Lecture 4: The simple regression model OLS Unbiasedness, OLS Variances, Units of measurement, Nonlinearities Dr. Kai Sun Aston Business School 1 / 26
Unbiasedness of OLS SLR for “Simple Linear Regression” ◮ Assumption SLR.1: The population model is linear in parameters, i.e., y = β 1 0 + β 1 1 x + u ◮ Assumption SLR.2: Use a random sample of size n , { ( x i , y i ) : i = 1 , 2 , . . . , n } , so y i = β 0 + β 1 x i + u i ◮ Assumption SLR.3: There is sample variation in x , i.e., � x ) 2 > 0 (recall the formula for ˆ i ( x i − ¯ β 1 ) ◮ Assumption SLR.4: Zero conditional mean, i.e., E ( u i | x i ) = 0 2 / 26
Unbiasedness of OLS SLR for “Simple Linear Regression” ◮ Assumption SLR.1: The population model is linear in parameters, i.e., y = β 1 0 + β 1 1 x + u ◮ Assumption SLR.2: Use a random sample of size n , { ( x i , y i ) : i = 1 , 2 , . . . , n } , so y i = β 0 + β 1 x i + u i ◮ Assumption SLR.3: There is sample variation in x , i.e., � x ) 2 > 0 (recall the formula for ˆ i ( x i − ¯ β 1 ) ◮ Assumption SLR.4: Zero conditional mean, i.e., E ( u i | x i ) = 0 2 / 26
Unbiasedness of OLS SLR for “Simple Linear Regression” ◮ Assumption SLR.1: The population model is linear in parameters, i.e., y = β 1 0 + β 1 1 x + u ◮ Assumption SLR.2: Use a random sample of size n , { ( x i , y i ) : i = 1 , 2 , . . . , n } , so y i = β 0 + β 1 x i + u i ◮ Assumption SLR.3: There is sample variation in x , i.e., � x ) 2 > 0 (recall the formula for ˆ i ( x i − ¯ β 1 ) ◮ Assumption SLR.4: Zero conditional mean, i.e., E ( u i | x i ) = 0 2 / 26
Unbiasedness of OLS SLR for “Simple Linear Regression” ◮ Assumption SLR.1: The population model is linear in parameters, i.e., y = β 1 0 + β 1 1 x + u ◮ Assumption SLR.2: Use a random sample of size n , { ( x i , y i ) : i = 1 , 2 , . . . , n } , so y i = β 0 + β 1 x i + u i ◮ Assumption SLR.3: There is sample variation in x , i.e., � x ) 2 > 0 (recall the formula for ˆ i ( x i − ¯ β 1 ) ◮ Assumption SLR.4: Zero conditional mean, i.e., E ( u i | x i ) = 0 2 / 26
Unbiasedness of ˆ β 1 In order to think about unbiasedness, we need to rewrite our estimator, ˆ β ’s, in terms of the population parameter, β ’s. � i ( x i − ¯ x )( y i − ¯ y ) ˆ � β 1 = x ) 2 i ( x i − ¯ � i ( x i − ¯ x ) y i � = x ) 2 i ( x i − ¯ 3 / 26
Unbiasedness of ˆ β 1 Plug in y i = β 0 + β 1 x i + u i and define SST x = � x ) 2 , i ( x i − ¯ � i ( x i − ¯ x )( β 0 + β 1 x i + u i ) ˆ β 1 = SST x � i u i ( x i − ¯ x ) = 0 + β 1 + SST x using the fact that � i ( x i − ¯ x ) = 0 and � x ) = � x ) 2 = SST x . i x i ( x i − ¯ i ( x i − ¯ 4 / 26
Unbiasedness of ˆ β 1 ◮ Take expectation (conditional on x i ) for both sides: � E (ˆ 1 β 1 | x i ) = β 1 + i E ( u i | x i )( x i − ¯ x ) = β 1 SST x (using the Assumption SLR.4 that E ( u i | x i ) = 0) ◮ Finally, E (ˆ β 1 ) = E ( E (ˆ β 1 | x i )) = E ( β 1 ) = β 1 (first equality by law of iterated expectation). ◮ The fact that E (ˆ β 1 ) = β 1 means that ˆ β 1 is unbiased for β 1 . 5 / 26
Unbiasedness of ˆ β 1 ◮ Take expectation (conditional on x i ) for both sides: � E (ˆ 1 β 1 | x i ) = β 1 + i E ( u i | x i )( x i − ¯ x ) = β 1 SST x (using the Assumption SLR.4 that E ( u i | x i ) = 0) ◮ Finally, E (ˆ β 1 ) = E ( E (ˆ β 1 | x i )) = E ( β 1 ) = β 1 (first equality by law of iterated expectation). ◮ The fact that E (ˆ β 1 ) = β 1 means that ˆ β 1 is unbiased for β 1 . 5 / 26
Unbiasedness of ˆ β 1 ◮ Take expectation (conditional on x i ) for both sides: � E (ˆ 1 β 1 | x i ) = β 1 + i E ( u i | x i )( x i − ¯ x ) = β 1 SST x (using the Assumption SLR.4 that E ( u i | x i ) = 0) ◮ Finally, E (ˆ β 1 ) = E ( E (ˆ β 1 | x i )) = E ( β 1 ) = β 1 (first equality by law of iterated expectation). ◮ The fact that E (ˆ β 1 ) = β 1 means that ˆ β 1 is unbiased for β 1 . 5 / 26
Unbiasedness of ˆ β 0 Recall that ˆ y − ˆ u ) − ˆ x = β 0 + ( β 1 − ˆ β 0 = ¯ β 1 ¯ x = ( β 0 + β 1 ¯ x + ¯ β 1 ¯ β 1 )¯ x + ¯ u ◮ Take expectation for both sides: E (ˆ xE ( β 1 − ˆ β 0 ) = β 0 + ¯ β 1 ) + E (¯ u ) = β 0 using the fact that E (ˆ β 1 ) = β 1 and u ) = E (1 / n � i u i ) = 1 / n � E (¯ i E ( u i ) = 0 ◮ The fact that E (ˆ β 0 ) = β 0 means that ˆ β 0 is unbiased for β 0 . 6 / 26
Unbiasedness of ˆ β 0 Recall that ˆ y − ˆ u ) − ˆ x = β 0 + ( β 1 − ˆ β 0 = ¯ β 1 ¯ x = ( β 0 + β 1 ¯ x + ¯ β 1 ¯ β 1 )¯ x + ¯ u ◮ Take expectation for both sides: E (ˆ xE ( β 1 − ˆ β 0 ) = β 0 + ¯ β 1 ) + E (¯ u ) = β 0 using the fact that E (ˆ β 1 ) = β 1 and u ) = E (1 / n � i u i ) = 1 / n � E (¯ i E ( u i ) = 0 ◮ The fact that E (ˆ β 0 ) = β 0 means that ˆ β 0 is unbiased for β 0 . 6 / 26
Unbiasedness of ˆ β 0 Recall that ˆ y − ˆ u ) − ˆ x = β 0 + ( β 1 − ˆ β 0 = ¯ β 1 ¯ x = ( β 0 + β 1 ¯ x + ¯ β 1 ¯ β 1 )¯ x + ¯ u ◮ Take expectation for both sides: E (ˆ xE ( β 1 − ˆ β 0 ) = β 0 + ¯ β 1 ) + E (¯ u ) = β 0 using the fact that E (ˆ β 1 ) = β 1 and u ) = E (1 / n � i u i ) = 1 / n � E (¯ i E ( u i ) = 0 ◮ The fact that E (ˆ β 0 ) = β 0 means that ˆ β 0 is unbiased for β 0 . 6 / 26
Variance of OLS ◮ While unbiasedness means that the sampling distribution of our estimate is centered around the true parameter ◮ Want to think about how spread out this distribution is ◮ Much easier to think about this variance under an additional assumption: Assumption SLR.5: Var ( u | x ) = σ 2 (Homoskedasticity: the conditional variance of u is a constant) 7 / 26
Variance of OLS ◮ While unbiasedness means that the sampling distribution of our estimate is centered around the true parameter ◮ Want to think about how spread out this distribution is ◮ Much easier to think about this variance under an additional assumption: Assumption SLR.5: Var ( u | x ) = σ 2 (Homoskedasticity: the conditional variance of u is a constant) 7 / 26
Variance of OLS ◮ While unbiasedness means that the sampling distribution of our estimate is centered around the true parameter ◮ Want to think about how spread out this distribution is ◮ Much easier to think about this variance under an additional assumption: Assumption SLR.5: Var ( u | x ) = σ 2 (Homoskedasticity: the conditional variance of u is a constant) 7 / 26
Var ( u | x ) = E ( u 2 | x ) − [ E ( u | x )] 2 Since E ( u | x ) = 0, Var ( u | x ) = E ( u 2 | x ) = σ 2 ◮ By the law of iterated expectation, E ( u 2 ) = E ( E ( u 2 | x )) = E ( σ 2 ) = σ 2 . Var ( u ) = E ( u 2 ) = σ 2 means that the unconditional variance of u is a constant, too - σ 2 is also called error variance. ◮ If we take the conditional variance for both sides of y = β 0 + β 1 x + u , Var ( y | x ) = Var ( β 0 + β 1 x | x ) + Var ( u | x ) = Var ( u | x ) = σ 2 . Var ( β 0 + β 1 x | x ) = 0 because when we “conditional on x ”, we can view ( β 0 + β 1 x ) as a constant, whose variance = 0. 8 / 26
Var ( u | x ) = E ( u 2 | x ) − [ E ( u | x )] 2 Since E ( u | x ) = 0, Var ( u | x ) = E ( u 2 | x ) = σ 2 ◮ By the law of iterated expectation, E ( u 2 ) = E ( E ( u 2 | x )) = E ( σ 2 ) = σ 2 . Var ( u ) = E ( u 2 ) = σ 2 means that the unconditional variance of u is a constant, too - σ 2 is also called error variance. ◮ If we take the conditional variance for both sides of y = β 0 + β 1 x + u , Var ( y | x ) = Var ( β 0 + β 1 x | x ) + Var ( u | x ) = Var ( u | x ) = σ 2 . Var ( β 0 + β 1 x | x ) = 0 because when we “conditional on x ”, we can view ( β 0 + β 1 x ) as a constant, whose variance = 0. 8 / 26
Var ( u | x ) = E ( u 2 | x ) − [ E ( u | x )] 2 Since E ( u | x ) = 0, Var ( u | x ) = E ( u 2 | x ) = σ 2 ◮ By the law of iterated expectation, E ( u 2 ) = E ( E ( u 2 | x )) = E ( σ 2 ) = σ 2 . Var ( u ) = E ( u 2 ) = σ 2 means that the unconditional variance of u is a constant, too - σ 2 is also called error variance. ◮ If we take the conditional variance for both sides of y = β 0 + β 1 x + u , Var ( y | x ) = Var ( β 0 + β 1 x | x ) + Var ( u | x ) = Var ( u | x ) = σ 2 . Var ( β 0 + β 1 x | x ) = 0 because when we “conditional on x ”, we can view ( β 0 + β 1 x ) as a constant, whose variance = 0. 8 / 26
Var ( u | x ) = E ( u 2 | x ) − [ E ( u | x )] 2 Since E ( u | x ) = 0, Var ( u | x ) = E ( u 2 | x ) = σ 2 ◮ By the law of iterated expectation, E ( u 2 ) = E ( E ( u 2 | x )) = E ( σ 2 ) = σ 2 . Var ( u ) = E ( u 2 ) = σ 2 means that the unconditional variance of u is a constant, too - σ 2 is also called error variance. ◮ If we take the conditional variance for both sides of y = β 0 + β 1 x + u , Var ( y | x ) = Var ( β 0 + β 1 x | x ) + Var ( u | x ) = Var ( u | x ) = σ 2 . Var ( β 0 + β 1 x | x ) = 0 because when we “conditional on x ”, we can view ( β 0 + β 1 x ) as a constant, whose variance = 0. 8 / 26
Homoskedastic Case y f( y|x ) . E( y | x ) = 0 + 1 x . x 1 x 2 9 / 26
Heteroskedastic Case f( y|x ) . . E( y | x ) = 0 + 1 x . x 1 x 2 x 3 x 10 / 26
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