Robust and Clustered Standard Errors - Harvard University

1 Robust and Clustered Standard ErrorsMolly RobertsMarch 6, 2013 Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 20131 / 35An Introduction to Robust and Clustered Standard ErrorsOutline1An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceGLM s and Non-constant VarianceCluster- Robust Standard Errors2 Replicating in RMolly RobertsRobust and Clustered Standard ErrorsMarch 6, 20132 / 35An Introduction to Robust and Clustered Standard ErrorsOutline1An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceGLM s and Non-constant VarianceCluster- Robust Standard Errors2 Replicating in RMolly RobertsRobust and Clustered Standard ErrorsMarch 6, 20133 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceReview: Errors and ResidualsErrors are the vertical distances between observations and theunknown Conditional Expectation Function.

2 Therefore, they are the vertical distances between observations and theestimated regression function. Therefore, they are RobertsRobust and Clustered Standard ErrorsMarch 6, 20134 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceReview: Errors and ResidualsErrors are the vertical distances between observations and theunknown Conditional Expectation Function. Therefore, they are the vertical distances between observations and theestimated regression function. Therefore, they are RobertsRobust and Clustered Standard ErrorsMarch 6, 20134 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceReview: Errors and ResidualsErrors are the vertical distances between observations and theunknown Conditional Expectation Function.

3 Therefore, they are the vertical distances between observations and theestimated regression function. Therefore, they are RobertsRobust and Clustered Standard ErrorsMarch 6, 20134 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceNotationErrors represent the difference between the outcome and the +uu=y X Residuals represent the difference between the outcome and theestimated + u u=y X Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 20135 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceNotationErrors represent the difference between the outcome and the +uu=y X Residuals represent the difference between the outcome and theestimated + u u=y X Molly RobertsRobust and Clustered Standard ErrorsMarch 6.

4 20135 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceNotationErrors represent the difference between the outcome and the +uu=y X Residuals represent the difference between the outcome and theestimated + u u=y X Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 20135 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceNotationErrors represent the difference between the outcome and the +uu=y X Residuals represent the difference between the outcome and theestimated + u u=y X Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 20135 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceNotationErrors represent the difference between the outcome and the +uu=y X Residuals represent the difference between the outcome and theestimated + u u=y X Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 20135 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceVariance of depends on the Errors =(X X) 1X y=(X X) 1X (X +u)= +(X X) 1X uMolly RobertsRobust and Clustered Standard ErrorsMarch 6, 20136 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceVariance of depends on the errorsV[ ] =V[ ] +V[(X X) 1X u]=0+V[(X X)]

5 1X u]=E[(X X) 1X uu X(X X) 1] E[(X X) 1X u]E[(X X) 1X u] =E[(X X) 1X uu X(X X) 1] 0 Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 20137 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceVariance of depends on the Errors (continued)V[ ] =V[ ] +V[(X X) 1X u]=0+V[(X X) 1X u]=E[(X X) 1X uu X(X X) 1] E[(X X) 1X u]E[(X X) 1X u] =E[(X X) 1X uu X(X X) 1] 0=(X X) 1X E[uu ]X(X X) 1=(X X) 1X X(X X) 1 Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 20138 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceConstant Error Variance and DependenceUnder Standard OLS assumptions,u Nn(0, ) =Var(u) =E[uu ] = 2 What does this mean graphically for a CEF with one explanatoryvariable?

6 Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 20139 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceConstant Error Variance and DependenceUnder Standard OLS assumptions,u Nn(0, ) =Var(u) =E[uu ] = 2 What does this mean graphically for a CEF with one explanatoryvariable?Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 20139 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceConstant Error Variance and DependenceUnder Standard OLS assumptions,u Nn(0, ) =Var(u) =E[uu ] = 2 What does this mean graphically for a CEF with one explanatoryvariable?Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 20139 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceConstant Error Variance and DependenceUnder Standard OLS assumptions,u Nn(0, ) =Var(u) =E[uu ] = 2 What does this mean graphically for a CEF with one explanatoryvariable?

7 Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 20139 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceEvidence of Non-constant Error Variance (4 examples) RobertsRobust and Clustered Standard ErrorsMarch 6, 201310 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceNotationThe constant error variance assumption sometimes calledhomoskedasiticity states thatVar(u) =E[uu ] = 2 In this section we will allow violations of this assumption in thefollowing heteroskedastic (u) =E[uu ] = 2n Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 201311 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceNotationThe constant error variance assumption sometimes calledhomoskedasiticity states thatVar(u) =E[uu ] = 2 In this section we will allow violations of this assumption in thefollowing heteroskedastic (u) =E[uu ] = 2n Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 201311 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceNotationThe constant error variance assumption sometimes calledhomoskedasiticity states thatVar(u)

8 =E[uu ] = 2 In this section we will allow violations of this assumption in thefollowing heteroskedastic (u) =E[uu ] = 2n Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 201311 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceNotationThe constant error variance assumption sometimes calledhomoskedasiticity states thatVar(u) =E[uu ] = 2 In this section we will allow violations of this assumption in thefollowing heteroskedastic (u) =E[uu ] = 2n Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 201311 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceConsequences of non-constant error varianceIThe 2will not be unbiased for level tests, probability of Type I error will not be.

9 I 1 confidence intervals will not have 1 LS estimator is no longer ,IThe degree of the problem depends on the amount is still unbiased for Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 201312 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceConsequences of non-constant error varianceIThe 2will not be unbiased for level tests, probability of Type I error will not be .I 1 confidence intervals will not have 1 LS estimator is no longer ,IThe degree of the problem depends on the amount is still unbiased for Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 201312 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceConsequences of non-constant error varianceIThe 2will not be unbiased for level tests, probability of Type I error will not be.

10 I 1 confidence intervals will not have 1 LS estimator is no longer ,IThe degree of the problem depends on the amount is still unbiased for Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 201312 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceConsequences of non-constant error varianceIThe 2will not be unbiased for level tests, probability of Type I error will not be .I 1 confidence intervals will not have 1 LS estimator is no longer ,IThe degree of the problem depends on the amount is still unbiased for Molly RobertsRobust and Clustered Standard ErrorsMarch 6, 201312 / 35An Introduction to Robust and Clustered Standard ErrorsLinear Regression with Non-constant VarianceConsequences of non-constant error varianceIThe 2will not be unbiased for level tests, probability of Type I error will not be.