Sampling Distributions of OLS Estimators of β 0 and β 1. Monte Carlo Simulations
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1 Addendum to NOTE 4 Samplng Dstrbutons of OLS Estmators of β and β Monte Carlo Smulatons The True Model: s gven by the populaton regresson equaton (PRE) Y = β + β X + u = X + u () where β = 7. and β =.9; Y = weekly consumpton expendtures of the -th household; X = weekly dsposable ncome of the -th household; u = an d random error term that s assumed to be N(, σ ). Four Alternatve Models: specfy dfferent values for σ = Var( u X ). Model : sets σ = Var( u X ) =,6, σ = Var( u X ) se( u X ) = 4. = Model : sets σ = Var( u X ) = 6,4, σ = Var( u X ) se( u X ) = 8. = Model 3: sets σ = Var( u X ) = 5,6, σ = Var( u X ) se( u X ) = 6. = Model 4: sets σ = Var( u X ) = 6,, σ = Var( u X ) se( u X ) = 4. =... Page of 5 pages
2 The Monte Carlo Smulatons Four dfferent sample szes: N = 3, N = 6, N =, N = 3. Set populaton values of X, β and β, and σ = Var( u X ). Generate, ndependent random samples of Y and u values. For each of these, ndependent random samples, compute the values of the OLS coeffcent estmators: ˆ x y β = () x β ˆ = Y ˆ X (3) β where x X X, y Y Y, X = X, and Y = Y. N N Tabulate and plot the, estmates of β and the, estmates of β,.e., the, values of ˆβ and the, values of ˆβ. OLS estmates ˆβ of β are denoted as b. OLS estmates ˆβ of β are denoted as b.... Page of 5 pages
3 Smulaton Results for Sample Sze N = 3 Observatons (, Replcatons) N = 3: Smulaton of Model for whch. summarze σ =,6, σ = 4: Varable Obs Mean Std. Dev. Mn Max b b N = 3: Smulaton of Model for whch. summarze σ = 6,4, σ = 8: Varable Obs Mean Std. Dev. Mn Max b b N = 3: Smulaton of Model 3 for whch. summarze σ = 5,6, σ = 6: Varable Obs Mean Std. Dev. Mn Max b b N = 3: Smulaton of Model 4 for whch. summarze σ = 6,, σ = 4: Varable Obs Mean Std. Dev. Mn Max b b Page 3 of 5 pages
4 Smulaton Results for Sample Sze N = 6 Observatons (, Replcatons) N = 6: Smulaton of Model for whch. summarze σ =,6, σ = 4: Varable Obs Mean Std. Dev. Mn Max b b N = 6: Smulaton of Model for whch. summarze σ = 6,4, σ = 8: Varable Obs Mean Std. Dev. Mn Max b b N = 6: Smulaton of Model 3 for whch. summarze σ = 5,6, σ = 6: Varable Obs Mean Std. Dev. Mn Max b b N = 6: Smulaton of Model 4 for whch. summarze σ = 6,, σ = 4: Varable Obs Mean Std. Dev. Mn Max b b Page 4 of 5 pages
5 Smulaton Results for Sample Sze N = Observatons (, Replcatons) N = : Smulaton of Model for whch. summarze σ =,6, σ = 4: Varable Obs Mean Std. Dev. Mn Max b b N = : Smulaton of Model for whch. summarze σ = 6,4, σ = 8: Varable Obs Mean Std. Dev. Mn Max b b N = : Smulaton of Model 3 for whch. summarze σ = 5,6, σ = 6: Varable Obs Mean Std. Dev. Mn Max b b N = : Smulaton of Model 4 for whch. summarze σ = 6,, σ = 4: Varable Obs Mean Std. Dev. Mn Max b b Page 5 of 5 pages
6 Smulaton Results for Sample Sze N = 3 Observatons (, Replcatons) N = 3: Smulaton of Model for whch. summarze σ =,6, σ = 4: Varable Obs Mean Std. Dev. Mn Max b b N = 3: Smulaton of Model for whch. summarze σ = 6,4, σ = 8: Varable Obs Mean Std. Dev. Mn Max b b N = 3: Smulaton of Model 3 for whch. summarze σ = 5,6, σ = 6: Varable Obs Mean Std. Dev. Mn Max b b N = 3: Smulaton of Model 4 for whch. summarze σ = 6,, σ = 4: Varable Obs Mean Std. Dev. Mn Max b b Page 6 of 5 pages
7 Effect on Samplng Dstrbutons of ˆβ and ˆβ of Increasng the Error Varance σ Queston: What s the effect on the samplng dstrbutons of ˆβ and ˆβ of ncreasng the error varance σ = Var( u X ) whle holdng constant both sample sze N and sample varaton of X = N N X x (X X) = = TSS? Answer: Increasng the error varance σ = Var( u X ) ncreases the varances of the samplng dstrbutons of ˆβ and,.e., ncreases Var β ˆ Var β ˆ. and ( ) ˆβ ( )... Page 7 of 5 pages
8 Illustraton for Sample Sze N = 6 Observatons Model for whch σ =,6, σ = 4:. summarze Varable Obs Mean Std. Dev. Mn Max b b Model for whch σ = 6,4, σ = 8:. summarze Varable Obs Mean Std. Dev. Mn Max b b Model 3 for whch σ = 5,6, σ = 6:. summarze Varable Obs Mean Std. Dev. Mn Max b b Model 4 for whch σ = 6,, σ = 4:. summarze Varable Obs Mean Std. Dev. Mn Max b b Page 8 of 5 pages
9 Samplng Dstrbuton of OLS Estmator b MODEL : N = 6; Var(u X) = Sgma-Sq =,6 samplng densty of b (percentage ponts) OLS estmates b of beta Samplng Dstrbuton of OLS Estmator b MODEL : N = 6; Var(u X) = Sgma-Sq = 6,4 samplng densty of b (percentage ponts) OLS estmates b of beta... Page 9 of 5 pages
10 Samplng Dstrbuton of OLS Estmator b MODEL 3: N = 6; Var(u X) = Sgma-Sq = 5,6 samplng densty of b (percentage ponts) OLS estmates b of beta Samplng Dstrbuton of OLS Estmator b MODEL 4: N = 6; Var(u X) = Sgma-Sq = 6, samplng densty of b (percentage ponts) OLS estmates b of beta... Page of 5 pages
11 Effect on Samplng Dstrbutons of ˆβ and ˆβ of Increasng Sample Sze N Queston: What s the effect on the samplng dstrbutons of ˆβ and ˆβ of ncreasng the sample sze N whle holdng constant both the error varance σ = Var( u X ) and the sample varaton of X = TSS X x (X X)? Answer: Increasng the sample sze N decreases the varances of the samplng dstrbutons of ˆβ and ˆβ,.e., decreases Var( β ˆ ) and Var( β ˆ ). Illustraton for Model 3 for whch For N = 3:. summarze σ = 5,6, σ = 6: Varable Obs Mean Std. Dev. Mn Max b b For N = 6:. summarze Varable Obs Mean Std. Dev. Mn Max b b For N = :. summarze Varable Obs Mean Std. Dev. Mn Max b b For N = 3:. summarze Varable Obs Mean Std. Dev. Mn Max b b N = N =... Page of 5 pages
12 Model 3 for N = 3: Samplng Dstrbuton of OLS Estmator b MODEL 3: N = 3; Var(u X) = Sgma-Sq = 5,6 samplng densty of b (percentage ponts) OLS estmates b of beta... Page of 5 pages
13 Model 3 for N = 6: Samplng Dstrbuton of OLS Estmator b MODEL 3: N = 6; Var(u X) = Sgma-Sq = 5,6 samplng densty of b (percentage ponts) OLS estmates b of beta... Page 3 of 5 pages
14 Model 3 for N = : Samplng Dstrbuton of OLS Estmator b MODEL 3: N = ; Var(u X) = Sgma-Sq = 5,6 samplng densty of b (percentage ponts) OLS estmates b of beta... Page 4 of 5 pages
15 Model 3 for N = 3: Samplng Dstrbuton of OLS Estmator b MODEL 3: N = 3; Var(u X) = Sgma-Sq = 5,6 samplng densty of b (percentage ponts) OLS estmates b of beta... Page 5 of 5 pages
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