■ Financial Econometrics Chapter 6.Control Variable and Fixed Effect Jin Ling School of Finance,Zhongnan University of Economics and Law 1
Financial Econometrics Chapter 6. Control Variable and Fixed Effect Jin Ling School of Finance, Zhongnan University of Economics and Law 1
Outline Causal Inference and OLS Regression 。Control Variable ·Fixed Effect 2
• Causal Inference and OLS Regression • Control Variable • Fixed Effect 2 Outline
Causal Inference and OLS Regression The methodology for causal inference: Potential outcomes with or without treatment under same condition. Assignment mechanism. Experiment design:Randomized experiment. Testable implication:Mediation Effect and Moderating Effect. 3
• The methodology for causal inference: • Potential outcomes with or without treatment under same condition. • Assignment mechanism. • Experiment design: Randomized experiment. • Testable implication: Mediation Effect and Moderating Effect. 3 Causal Inference and OLS Regression
Causal Inference and OLS Regression The OLS regression for causal inference: Yi=DYL+(1-Di)Yoi=Yoi+(Yu-Yoi)D =Yo+6:D:=o十8D:+o =+(h一)D,+[%+D(yh:一%：)] Ho=E[Yo],voi=Yoi-E[Yoi],=E[Y:],v=Yu-E[Yii] ·Equivalently, Yi a+TATE D:+e E[Y;I D:1]=a+TATE +E[e;I X E[Y,I D:=0]=a+ECeX:] 4
• The OLS regression for causal inference: • Equivalently, 4 Causal Inference and OLS Regression
Causal Inference and OLS Regression The OLS regression for causal inference: x=E[Y,ID=1]-E[Y,|D,=0] =tE+E[e:ID:=1]-E[e|D,=0] =xAE+E[h:|D:=1]-E[o:ID,=0] =TATE +E[Yo D,=1]-E[Yor I D;=0] +(1-p)(E[YH-Y{D:=1]-E[Y:-Yo|D,=0]} 全期望公式E[Y]=E[EYIX]] t=EY:-Y]+E[Yo I D,=1]-E[Yo|D,=0]+{1-Pr[D,=1} ATE 选择偏艺 ·{EYi-Y1D,=1]-EY:-Y|D,=0]》 两组因果效应差舞 5
• The OLS regression for causal inference: 5 Causal Inference and OLS Regression
Causal Inference and OLS Regression The OLS regression for causal inference: t E[Y,D:=1]-ECY:D.=0] =ECY D,≈1]-ECY1D=0] =E[Y:-Y1D,=1]+E[Yi|D.=1]-E[Y:ID:=o] ATT 远择纳盐 E[Y;D;1]-ECY,I D=0] =E[Y D,1]-ECYor D,=0] ECYu:-Yoi I D:=0]+ECYu I D:1]-ECY1:I D.=0] ATC 进保编丝 6
• The OLS regression for causal inference: 6 Causal Inference and OLS Regression
Outline Causal Inference and OLS Regression 。Control Variable ·Fixed Effect 7
• Causal Inference and OLS Regression • Control Variable • Fixed Effect 7 Outline
Control Variable ·When we control for X: ·X is a dummy variable Y:=tD,+∑adr+e, dx=1(X,=x) -CovY.D2-EY.Dl(其中D.=D-ED,1X]) Var(D;) E[D:] E[D.E[Y:I D.,X:]]E[D.(ECY:I D:=0.X.]+rxD ) ELD] E[D] ELVar(D0 Var(D,Xi) xx=E[Y,IX,D,=1]-E[Y,|X:,D,=0] 8
• When we control for X: • X is a dummy variable. 8 Control Variable
Control Variable ·When we control for X: If ty indicates a causal effect,we can infer that Tols has a causal effect: cx=E[Y,|X,D,=1]-EY,|X,D,=0] =E[Y X,D;1]-E[Yoi Xi,D=0] =E[Yi -YoiI X.D 1]+E[Yo.I X,D.1]-ECYo I X,D =0] TATTIX,】 选轻编差 =E[Y-Yo X:]+E[Yo I X:,D:=1]-E[Yo I X,D =0] FATE(X) 选韩编忍 +(1-(X))(ECYu-Yo i X,D 1]-ECYi-Yoi I X,D=0]) 两国网果效应编左 E[Y:|X,D=1]=E[Yu|X,D,=0] (Y,Yi)ILD,Xi
• When we control for X: • If 𝜏𝑋 indicates a causal effect, we can infer that 𝜏𝑜𝑙𝑠 has a causal effect: 9 Control Variable
Control Variable When we control for X: Selection bias equals 0: EYo Xi,D;=1]=E[Yoi Xi,D:=0 The difference in treatment effect is 0: E[Y:IX,D:=1]=E[Y|X:,D,=0] ·Unconfoundedness(非混杂性) (Yoi,Yi)IL D,X: 10
• When we control for X: • Selection bias equals 0: • The difference in treatment effect is 0: • Unconfoundedness (非混杂性): 10 Control Variable