Hypothesis Testing Part III: Pretest Bias James J. Heckman - - PowerPoint PPT Presentation

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Hypothesis Testing Part III: Pretest Bias James J. Heckman - - PowerPoint PPT Presentation

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Hypothesis Testing Part III: Pretest Bias James J. Heckman University of Chicago Based on T.A. Bancroft, The Annals of Mathematical Statistics , Vol. 15, No. 2. (June, 1944), pp. 190-204. Econ 312 Spring 2019 1 A

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Hypothesis Testing

Part III: Pretest Bias

James J. Heckman University of Chicago Based on T.A. Bancroft, The Annals of Mathematical Statistics,

  • Vol. 15, No. 2. (June, 1944), pp. 190-204.

Econ 312 Spring 2019

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A common procedure almost universal in economics is to select a model on the basis of a preliminary test. Example:

  • =

+ ( | ) = 0

  • =

[1 2]

  • =

(1 2) Include 2 if a “” statistic on ˆ 2 is big enough.

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ˆ ˆ ˜ 1,pretest = 11(2 ) + 11(2 ) ˆ 1 comes from unrestricted model; ˜ 1 comes from model deleting 2.

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For a fixed size test, say , % of time we reject when null 0 : 2 = 0 is true. Estimator is inconsistent unless we change with sample size (e.g. 0). Bayesians show it’s inadmissible under quadratic loss. Examples of Bias:

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Model : Yi = Xiβ + i; i = 1,...,N X ≡ X1 X2 ∼ N 0, 1 ρ ρ 1 ; iid  ∼ N0,σ2; iid Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.9 σ =10 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.9 σ =10 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.9; σ2 = 10 Distribution of β2 N = 50, ρ = 0.9; σ2 = 10

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.75 σ =10 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.75 σ =10 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.75; σ2 = 10 Distribution of β2 N = 50, ρ = 0.75; σ2 = 10

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.5 σ =10 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.5 σ =10 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.5; σ2 = 10 Distribution of β2 N = 50, ρ = 0.5; σ2 = 10

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.25 σ =10 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.25 σ =10 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.25; σ2 = 10 Distribution of β2 N = 50, ρ = 0.25; σ2 = 10

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.9 σ =5 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.9 σ =5 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.9; σ2 = 5 Distribution of β2 N = 50, ρ = 0.9; σ2 = 5

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.75 σ =5 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.75 σ =5 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

Distribution of β1 N = 50, ρ = 0.75; σ2 = 5 Distribution of β2 N = 50, ρ = 0.75; σ2 = 5

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.5 σ =5 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.5 σ =5 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.5; σ2 = 5 Distribution of β2 N = 50, ρ = 0.5; σ2 = 5

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.25 σ =5 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.25 σ =5 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.25; σ2 = 5 Distribution of β2 N = 50, ρ = 0.25; σ2 = 5

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.9 σ =2 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.9 σ =2 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.9; σ2 = 2 Distribution of β2 N = 50, ρ = 0.9; σ2 = 2

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.75 σ =2 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.75 σ =2 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.75; σ2 = 2 Distribution of β2 N = 50, ρ = 0.75; σ2 = 2

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
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1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.5 σ =2 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.5 σ =2 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.5; σ2 = 2 Distribution of β2 N = 50, ρ = 0.5; σ2 = 2

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
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1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.25 σ =2 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.25 σ =2 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.25; σ2 = 2 Distribution of β2 N = 50, ρ = 0.25; σ2 = 2

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
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1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.9 σ =1 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.9 σ =1 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.9; σ2 = 1 Distribution of β2 N = 50, ρ = 0.9; σ2 = 1

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.75 σ =1 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.75 σ =1 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.75; σ2 = 1 Distribution of β2 N = 50, ρ = 0.75; σ2 = 1

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.5 σ =1 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.5 σ =1 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.5; σ2 = 1 Distribution of β2 N = 50, ρ = 0.5; σ2 = 1

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Pre-Test Estimator Density Function Pre-Test Estimator Density Function

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β1, ρ =0.25 σ =1 Probability Density Function of estimated β1 N = 50 β1 density without drop of β2 β1 density with drop of β2

  • 2
  • 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 β2, ρ =0.25 σ =1 Probability Density Function of estimated β1 N = 50 β2 density without drop of β2 β2 density with drop of β2

Distribution of β1 N = 50, ρ = 0.25; σ2 = 1 Distribution of β2 N = 50, ρ = 0.25; σ2 = 1

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