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Static Models of Continuous Variables

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Raimundo Soto - Catholic University of Chile

ERF Training on Advanced Panel Data Techniques Applied to Economic Modelling

29 -31 October, 2018
Cairo, Egypt

Published in: Government & Nonprofit
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Static Models of Continuous Variables

  1. 1. ERF Training Workshop Panel Data 2 Raimundo Soto Instituto de Economรญa, PUC-Chile
  2. 2. STATIC MODELS OF CONTINUOUS VARIABLES 2
  3. 3. MODEL STRUCTURE โ€ข Canonical Model ๐‘ฆ๐‘–๐‘ก = ๐›ผ๐‘–๐‘ก + ๐‘ฅ๐‘–๐‘ก ๐›ฝ + ๐œ€๐‘–๐‘ก where โ€ข ๐‘ฆ๐‘–๐‘ก is the phenomenon of interest to be modelled, โ€ข ๐‘ฅ๐‘–๐‘ก represents all observed controls (regressors) โ€ข ๐›ผ๐‘–๐‘ก are the individual effects โ€ข ๐œ€๐‘–๐‘ก is the non-systematic part (what we choose not to model) 3
  4. 4. DATA STRUCTURE โ€ข Let us stack the data in the it structure โ€ข ๐‘ฆ11 ๐‘ฆ12 โ‹ฎ ๐‘ฆ1๐‘‡ ๐‘ฆ21 ๐‘ฆ22 โ‹ฎ ๐‘ฆ2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ฆ ๐‘1 โ‹ฎ ๐‘ฆ ๐‘๐‘‡ 4
  5. 5. DATA STRUCTURE EXAMPLE 5
  6. 6. DATA STRUCTURE โ€ข Let us stack the data in the it structure โ€ข ๐‘ฆ11 ๐‘ฆ12 โ‹ฎ ๐‘ฆ1๐‘‡ ๐‘ฆ21 ๐‘ฆ22 โ‹ฎ ๐‘ฆ2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ฆ ๐‘1 โ‹ฎ ๐‘ฆ ๐‘๐‘‡ = ๐›ผ11 ๐›ผ12 โ‹ฎ ๐›ผ1๐‘‡ ๐›ผ21 ๐›ผ22 โ‹ฎ ๐›ผ2๐‘‡ โ‹ฎ โ‹ฎ ๐›ผ ๐‘1 โ‹ฎ ๐›ผ ๐‘๐‘‡ + ๐‘ค11 ๐‘ค12 โ‹ฎ ๐‘ค1๐‘‡ ๐‘ค21 ๐‘ค22 โ‹ฎ ๐‘ค2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ค ๐‘1 โ‹ฎ ๐‘ค ๐‘๐‘‡ ๐‘ฃ11 ๐‘ฃ12 โ‹ฎ ๐‘ฃ1๐‘‡ ๐‘ฃ21 ๐‘ฃ22 โ‹ฎ ๐‘ฃ2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ฃ ๐‘1 โ‹ฎ ๐‘ฃ ๐‘๐‘‡ โ€ฆ ๐‘ง11 ๐‘ง12 โ‹ฎ ๐‘ง1๐‘‡ ๐‘ง21 ๐‘ง22 โ‹ฎ ๐‘ง2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ง ๐‘1 โ‹ฎ ๐‘ง ๐‘๐‘‡ ๐›ฝ + ๐œ€11 ๐œ€12 โ‹ฎ ๐œ€1๐‘‡ ๐œ€21 ๐œ€22 โ‹ฎ ๐œ€2๐‘‡ โ‹ฎ โ‹ฎ ๐œ€ ๐‘1 โ‹ฎ ๐œ€ ๐‘๐‘‡ 6
  7. 7. DATA STRUCTURE โ€ข Let us stack the data in the it structure โ€ข ๐‘ฆ11 ๐‘ฆ12 โ‹ฎ ๐‘ฆ1๐‘‡ ๐‘ฆ21 ๐‘ฆ22 โ‹ฎ ๐‘ฆ2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ฆ ๐‘1 โ‹ฎ ๐‘ฆ ๐‘๐‘‡ = ๐›ผ11 ๐›ผ12 โ‹ฎ ๐›ผ1๐‘‡ ๐›ผ21 ๐›ผ22 โ‹ฎ ๐›ผ2๐‘‡ โ‹ฎ โ‹ฎ ๐›ผ ๐‘1 โ‹ฎ ๐›ผ ๐‘๐‘‡ + ๐‘ค11 ๐‘ค12 โ‹ฎ ๐‘ค1๐‘‡ ๐‘ค21 ๐‘ค22 โ‹ฎ ๐‘ค2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ค ๐‘1 โ‹ฎ ๐‘ค ๐‘๐‘‡ ๐‘ฃ11 ๐‘ฃ12 โ‹ฎ ๐‘ฃ1๐‘‡ ๐‘ฃ21 ๐‘ฃ22 โ‹ฎ ๐‘ฃ2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ฃ ๐‘1 โ‹ฎ ๐‘ฃ ๐‘๐‘‡ โ€ฆ ๐‘ง11 ๐‘ง12 โ‹ฎ ๐‘ง1๐‘‡ ๐‘ง21 ๐‘ง22 โ‹ฎ ๐‘ง2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ง ๐‘1 โ‹ฎ ๐‘ง ๐‘๐‘‡ ๐›ฝ + ๐œ€11 ๐œ€12 โ‹ฎ ๐œ€1๐‘‡ ๐œ€21 ๐œ€22 โ‹ฎ ๐œ€2๐‘‡ โ‹ฎ โ‹ฎ ๐œ€ ๐‘1 โ‹ฎ ๐œ€ ๐‘๐‘‡ = ๐›ผ + ๐‘ฅ๐‘–๐‘ก ๐›ฝ + ๐œ€๐‘–๐‘ก 7
  8. 8. ESTIMATION IS IMPOSSIBLE โ€ข If we allow for ๐›ผ ๐‘๐‘‡ there will be ๐‘๐‘‡ constants (and, at most, NT observations), not enough degrees of freedom to estimate the parameters. โ€ข We will restrict ourselves to: โ€“ ๐›ผ๐‘– , i.e., N constants (that do not change in time) โ€“ ๐œ† ๐‘ก , i.e., T constants (that do not change by individual) 8
  9. 9. POOLED ESTIMATOR โ€ข Let us ignore all heterogeneity ๐‘ฆ๐‘–๐‘ก = ๐›ผ + ๐‘ฅ๐‘–๐‘ก ๐›ฝ + ๐œ€๐‘–๐‘ก โ€ข The OLS estimator is: ๐›ฝ = ๐‘ฅ๐‘–๐‘ก โ€ฒ ๐‘ฅ๐‘–๐‘ก โˆ’1 ๐‘ฅ๐‘–๐‘ก โ€ฒ ๐‘ฆ๐‘–๐‘ก โ€ข The variance estimator is ๐‘‰๐‘Ž๐‘Ÿ ๐›ฝ = ๐œŽ๐œ€ 2 ๐‘ฅ๐‘–๐‘ก โ€ฒ ๐‘ฅ๐‘–๐‘ก โˆ’1 = ๐œŽ๐œ€ 2 ๐‘‰(๐‘ฅ๐‘–๐‘ก) โ€ข Note the increase in precision (๐‘๐‘ฅ๐‘‡) 9
  10. 10. FIXED EFFECTS ESTIMATOR โ€ข Consider that the heterogeneity is only among individuals ๐‘ฆ๐‘–๐‘ก = ๐›ผ๐‘– + ๐‘ฅ๐‘–๐‘ก ๐›ฝ + ๐œ€๐‘–๐‘ก โ€ข ๐›ผ๐‘– represents individual characteristics that are fixed โ€ข We could use binary (dummy) variables to represent fixed characteristics 10
  11. 11. FIXED EFFECTS ESTIMATOR ๐‘ฆ11 ๐‘ฆ12 โ‹ฎ ๐‘ฆ1๐‘‡ ๐‘ฆ21 ๐‘ฆ22 โ‹ฎ ๐‘ฆ2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ฆ ๐‘1 โ‹ฎ ๐‘ฆ ๐‘๐‘‡ = ๐›ผ1 ๐›ผ1 โ‹ฎ ๐›ผ1 0 0 โ‹ฎ 0 โ‹ฎ โ‹ฎ 0 โ‹ฎ 0 0 0 โ‹ฎ 0 ๐›ผ2 ๐›ผ2 โ‹ฎ 0 โ‹ฎ โ‹ฎ 0 โ‹ฎ 0 โ€ฆ 0 0 โ‹ฎ 0 0 0 โ‹ฎ 0 โ‹ฎ โ‹ฎ ๐›ผ ๐‘ โ‹ฎ ๐›ผ ๐‘ + ๐‘ค11 ๐‘ค12 โ‹ฎ ๐‘ค1๐‘‡ ๐‘ค21 ๐‘ค22 โ‹ฎ ๐‘ค2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ค ๐‘1 โ‹ฎ ๐‘ค ๐‘๐‘‡ ๐‘ฃ11 ๐‘ฃ12 โ‹ฎ ๐‘ฃ1๐‘‡ ๐‘ฃ21 ๐‘ฃ22 โ‹ฎ ๐‘ฃ2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ฃ ๐‘1 โ‹ฎ ๐‘ฃ ๐‘๐‘‡ โ€ฆ ๐‘ง11 ๐‘ง12 โ‹ฎ ๐‘ง1๐‘‡ ๐‘ง21 ๐‘ง22 โ‹ฎ ๐‘ง2๐‘‡ โ‹ฎ โ‹ฎ ๐‘ง ๐‘1 โ‹ฎ ๐‘ง ๐‘๐‘‡ ๐›ฝ + ๐œ€11 ๐œ€12 โ‹ฎ ๐œ€1๐‘‡ ๐œ€21 ๐œ€22 โ‹ฎ ๐œ€2๐‘‡ โ‹ฎ โ‹ฎ ๐œ€ ๐‘1 โ‹ฎ ๐œ€ ๐‘๐‘‡ 11
  12. 12. FIXED EFFECTS ESTIMATOR โ€ข ๐‘ฆ๐‘–๐‘ก = ๐›ผ๐ท + ๐‘ฅ๐‘–๐‘ก ๐›ฝ + ๐œ€๐‘–๐‘ก โ€ข Note: same slope, different intercept (constant) โ€ข All classic results on econometric estimation techniques hold: nature of the OLS estimator, optimality, goodness of fit, and asymptotic distributions of estimators and tests. โ€ข This estimator is called LSDV least squares dummy variables. 12
  13. 13. FIXED EFFECTS ESTIMATOR Group 2 Group 1 ๐‘ฅ ๐‘ฆ 13
  14. 14. FIXED EFFECTS ESTIMATOR Group 2 Group 1 ๐‘ฅ ๐‘ฆ ๐›ผ2 ๐›ผ1 ๐‘ฆ2 = ๐›ผ2 + ๐‘ฅ2ฮฒ ๐‘ฆ1 = ๐›ผ1 + ๐‘ฅ1ฮฒ 14
  15. 15. FIXED EFFECTS ESTIMATOR Group 2 Group 1 ๐‘ฅ ๐‘ฆ ๐›ผ2 ๐›ผ1 ๐‘ฆ2 = ๐›ผ2 + ๐‘ฅ2ฮฒ ๐‘ฆ1 = ๐›ผ1 + ๐‘ฅ1ฮฒ ๐‘ฆ = ๐›ผ + ๐‘ฅฮฒ 15
  16. 16. FIXED EFFECTS ESTIMATOR โ€ข Example: โ€“ Vial y Soto (2002) revise the opinion that โ€œuniversity selection tests (PSU) do not predict student performance (R) in their faculties, only secondary-school marks are importantโ€. โ€“ When running the pooled regression: ๐‘…๐‘–๐‘ก = ๐›ผ + ๐›ฝ๐‘ƒ๐‘†๐‘ˆ๐‘–๐‘ก + ๐œ‡๐‘–๐‘ก The estimated ๐›ฝ is small, not significant or displays the โ€œwrongโ€ sign (negative). 16
  17. 17. PREDICTED EFFECT OF SELECTION TESTS ON STUDENTSโ€™ PERFORMANCE Note: * significant at 10% size
  18. 18. PREDICTED EFFECT OF SELECTION TESTS ON STUDENTSโ€™ PERFORMANCE Note: * significant at 10% size
  19. 19. POOLED ESTIMATOR ๐‘ƒ๐‘†๐‘ˆ ๐‘… 10 1 19 โ€œLow-qualityโ€ faculties โ€œHigh-qualityโ€ faculties
  20. 20. FIXED EFFECTS ESTIMATOR ๐‘ƒ๐‘†๐‘ˆ ๐‘… 10 1 20 โ€œLow-qualityโ€ faculties โ€œHigh-qualityโ€ faculties
  21. 21. POOLED VS FIXED EFFECTS ESTIMATORS ๐‘ƒ๐‘†๐‘ˆ ๐‘… 21 โ€œLow-qualityโ€ faculties โ€œHigh-qualityโ€ faculties
  22. 22. POOLED VS FIXED EFFECTS ESTIMATORS ๐‘ƒ๐‘†๐‘ˆ ๐‘… 22 โ€œLow-qualityโ€ faculties โ€œHigh-qualityโ€ faculties
  23. 23. FIXED EFFECTS ESTIMATOR โ€ข LSDV estimator is unfeasible if N is too large โ€“ HIECS has 24,000 households โ€ข Recall that constants in regressions only take away the means of the variables โ€ข It would be much simpler to eliminate the means of the variables and avoid specifying 24,000 dummy variables 23
  24. 24. FIXED EFFECTS ESTIMATOR ๐‘ฆ๐‘–๐‘ก = ๐›ผ๐‘– + ๐‘ฅ๐‘–๐‘ก ๐›ฝ + ๐œ€๐‘–๐‘ก โ€ข Let us take expected value for each individual โ€œiโ€ in time: ๐ธ๐‘– ๐‘ฆ๐‘–๐‘ก = ๐ธ๐‘– ๐›ผ๐‘– + ๐‘ฅ๐‘–๐‘ก ๐›ฝ + ๐œ€๐‘–๐‘ก ๐‘ฆ๐‘– = ๐›ผ๐‘– + ๐‘ฅ๐‘– ๐›ฝ โ€ข and subtract from the original model to eliminate ๐›ผ๐‘–: ๐‘ฆ๐‘–๐‘ก โˆ’ ๐‘ฆ๐‘– = ๐›ผ๐‘– + ๐‘ฅ๐‘–๐‘ก ๐›ฝ + ๐œ€๐‘–๐‘ก โˆ’ ๐›ผ๐‘– โˆ’ ๐‘ฅ๐‘– ๐›ฝ ๐‘ฆ๐‘–๐‘ก โˆ’ ๐‘ฆ๐‘– = ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ๐‘– ๐›ฝ + ๐œ€๐‘–๐‘ก 24
  25. 25. FIXED EFFECTS ESTIMATOR ๐‘ฆ๐‘–๐‘ก โˆ’ ๐‘ฆ๐‘– = ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ๐‘– ๐›ฝ + ๐œ€๐‘–๐‘ก โ€ข This is a very simple estimation, without the problems derived from dimensionality. โ€ข Obviously, we cannot estimate ๐›ผ๐‘–, but they are easily recovered as: ๐›ผ๐‘– = ๐‘ฆ๐‘– โˆ’ ๐‘ฅ๐‘– ๐›ฝ 25
  26. 26. FIXED EFFECTS ESTIMATOR Group 2 Group 1 ๐‘ฅ ๐‘ฆ ๐›ผ2 ๐›ผ1 26
  27. 27. FIXED EFFECTS ESTIMATOR Group 2 ๐‘ฅ ๐‘ฆ Group 1 w/o means 27
  28. 28. FIXED EFFECTS ESTIMATOR Groups 1 & 2 w/o means ๐‘ฅ ๐‘ฆ 28
  29. 29. FIXED EFFECTS ESTIMATOR ๐‘ฅ ๐‘ฆ 29 Groups 1 & 2 w/o means
  30. 30. FIXED EFFECTS ESTIMATOR ๐‘ฆ๐‘–๐‘ก โˆ’ ๐‘ฆ๐‘– = ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ๐‘– ๐›ฝ + ๐œ€๐‘–๐‘ก โ€ข This estimator uses only information within each group and it is therefore called within-groups estimator โ€ข Let us obtain certain useful โ€œsumsโ€ in order to better understand the nature of estimators. 30
  31. 31. POOLED ESTIMATOR ๐‘† ๐‘ ๐‘ฅ๐‘ฅ = ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ โ€ฒ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ ๐‘† ๐‘ ๐‘ฅ๐‘ฆ = ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ โ€ฒ ๐‘ฆ๐‘–๐‘ก โˆ’ ๐‘ฆ ๐›ฝ ๐‘ = ๐‘† ๐‘ ๐‘ฅ๐‘ฆ ๐‘† ๐‘ ๐‘ฅ๐‘ฅ 31
  32. 32. WITHIN-GROUPS ESTIMATOR ๐‘† ๐‘ค ๐‘ฅ๐‘ฅ = ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ๐‘– โ€ฒ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ๐‘– ๐‘† ๐‘ค ๐‘ฅ๐‘ฆ = ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ๐‘– โ€ฒ ๐‘ฆ๐‘–๐‘ก โˆ’ ๐‘ฆ๐‘– ๐›ฝ ๐‘ค = ๐‘† ๐‘ค ๐‘ฅ๐‘ฆ ๐‘† ๐‘ค ๐‘ฅ๐‘ฅ 32
  33. 33. WITHIN-GROUPS ESTIMATOR โ€ข From the pooled estimator ๐‘† ๐‘ ๐‘ฅ๐‘ฅ = ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ โ€ฒ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ๐‘– + ๐‘ฅ๐‘– โˆ’ ๐‘ฅ โ€ฒ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ๐‘– + ๐‘ฅ๐‘– โˆ’ ๐‘ฅ ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ๐‘– + ๐‘ฅ๐‘– โˆ’ ๐‘ฅ โ€ฒ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ๐‘– + ๐‘ฅ๐‘– โˆ’ ๐‘ฅ ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ๐‘– โ€ฒ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ + ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ โ€ฒ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ ๐‘† ๐‘ ๐‘ฅ๐‘ฅ = ๐‘† ๐‘ค ๐‘ฅ๐‘ฅ + ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ โ€ฒ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ 33
  34. 34. WITHIN-GROUPS ESTIMATOR โ€ข Thus ๐‘† ๐‘ ๐‘ฅ๐‘ฅ = ๐‘† ๐‘ค ๐‘ฅ๐‘ฅ + ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ โ€ฒ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ ๐‘† ๐‘ค ๐‘ฅ๐‘ฅ = ๐‘† ๐‘ ๐‘ฅ๐‘ฅ โˆ’ ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ โ€ฒ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ โ€ข Double sums are either zero or positive (these are squares) hence ๐‘† ๐‘ค ๐‘ฅ๐‘ฅ โ‰ค ๐‘† ๐‘ ๐‘ฅ๐‘ฅ 34
  35. 35. WITHIN-GROUPS ESTIMATOR โ€ข The variance of the within-groups estimator is ๐‘‰๐‘Ž๐‘Ÿ ๐›ฝ ๐‘ค = ๐œŽ2 ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘–๐‘ก โˆ’ ๐‘ฅ๐‘– โ€ฒ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ + ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ โ€ฒ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ ๐‘‰๐‘Ž๐‘Ÿ ๐›ฝ ๐‘ค = ๐œŽ2 ๐‘† ๐‘ ๐‘ฅ๐‘ฅ โˆ’ ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ โ€ฒ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ โ€ข Therefore, this variance is larger than that of the pooled estimator โ€ข The within-groups estimator is less precise than the pooled estimator 35
  36. 36. LET US SEE THIS IN PRACTICE โ€ข Open Stata โ€ข Open file ERF_Continuous Static.do โ€“ Declare Panel Data and Variables โ€ข xtset โ€“ Panel Data Analysis: commands xt โ€ข xtdes โ€ข xtsum โ€“ Panel Data Regression โ€ข xtreg โ€ข Let us check the estimation results 36
  37. 37. F test that all u_i=0: F(162, 5270) = 98.59 Prob > F = 0.0000 rho .95557807 (fraction of variance due to u_i) sigma_e .32607847 sigma_u 1.5123647 _cons -7.844568 .2477214 -31.67 0.000 -8.330205 -7.358932 l_popt .0731808 .0285325 2.56 0.010 .0172453 .1291163 l_infl2 -.0281481 .004758 -5.92 0.000 -.0374757 -.0188204 l_realgdp .3846652 .0153326 25.09 0.000 .354607 .4147234 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] corr(u_i, Xb) = -0.9200 Prob > F = 0.0000 F(3,5270) = 859.53 overall = 0.0036 max = 55 between = 0.0140 avg = 33.3 R-sq: within = 0.3285 Obs per group: min = 4 Group variable: idwbcode Number of groups = 163 Fixed-effects (within) regression Number of obs = 5436 _cons 3.315492 .0848041 39.10 0.000 3.149242 3.481742 l_popt .0014429 .0061381 0.24 0.814 -.0105903 .0134762 l_infl2 -.1650409 .0078825 -20.94 0.000 -.1804937 -.149588 l_realgdp -.0086728 .0036681 -2.36 0.018 -.0158636 -.0014819 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 2454.44502 5435 .451599819 Root MSE = .64482 Adj R-squared = 0.0793 Residual 2258.61346 5432 .415797764 R-squared = 0.0798 Model 195.831563 3 65.2771875 Prob > F = 0.0000 F( 3, 5432) = 156.99 Source SS df MS Number of obs = 5436 37
  38. 38. F test that all u_i=0: F(162, 5270) = 98.59 Prob > F = 0.0000 rho .95557807 (fraction of variance due to u_i) sigma_e .32607847 sigma_u 1.5123647 _cons -7.844568 .2477214 -31.67 0.000 -8.330205 -7.358932 l_popt .0731808 .0285325 2.56 0.010 .0172453 .1291163 l_infl2 -.0281481 .004758 -5.92 0.000 -.0374757 -.0188204 l_realgdp .3846652 .0153326 25.09 0.000 .354607 .4147234 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] corr(u_i, Xb) = -0.9200 Prob > F = 0.0000 F(3,5270) = 859.53 overall = 0.0036 max = 55 between = 0.0140 avg = 33.3 R-sq: within = 0.3285 Obs per group: min = 4 Group variable: idwbcode Number of groups = 163 Fixed-effects (within) regression Number of obs = 5436 _cons 3.315492 .0848041 39.10 0.000 3.149242 3.481742 l_popt .0014429 .0061381 0.24 0.814 -.0105903 .0134762 l_infl2 -.1650409 .0078825 -20.94 0.000 -.1804937 -.149588 l_realgdp -.0086728 .0036681 -2.36 0.018 -.0158636 -.0014819 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 2454.44502 5435 .451599819 Root MSE = .64482 Adj R-squared = 0.0793 Residual 2258.61346 5432 .415797764 R-squared = 0.0798 Model 195.831563 3 65.2771875 Prob > F = 0.0000 F( 3, 5432) = 156.99 Source SS df MS Number of obs = 5436 Total Observations 38
  39. 39. F test that all u_i=0: F(162, 5270) = 98.59 Prob > F = 0.0000 rho .95557807 (fraction of variance due to u_i) sigma_e .32607847 sigma_u 1.5123647 _cons -7.844568 .2477214 -31.67 0.000 -8.330205 -7.358932 l_popt .0731808 .0285325 2.56 0.010 .0172453 .1291163 l_infl2 -.0281481 .004758 -5.92 0.000 -.0374757 -.0188204 l_realgdp .3846652 .0153326 25.09 0.000 .354607 .4147234 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] corr(u_i, Xb) = -0.9200 Prob > F = 0.0000 F(3,5270) = 859.53 overall = 0.0036 max = 55 between = 0.0140 avg = 33.3 R-sq: within = 0.3285 Obs per group: min = 4 Group variable: idwbcode Number of groups = 163 Fixed-effects (within) regression Number of obs = 5436 _cons 3.315492 .0848041 39.10 0.000 3.149242 3.481742 l_popt .0014429 .0061381 0.24 0.814 -.0105903 .0134762 l_infl2 -.1650409 .0078825 -20.94 0.000 -.1804937 -.149588 l_realgdp -.0086728 .0036681 -2.36 0.018 -.0158636 -.0014819 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 2454.44502 5435 .451599819 Root MSE = .64482 Adj R-squared = 0.0793 Residual 2258.61346 5432 .415797764 R-squared = 0.0798 Model 195.831563 3 65.2771875 Prob > F = 0.0000 F( 3, 5432) = 156.99 Source SS df MS Number of obs = 5436 Total Groups 39
  40. 40. F test that all u_i=0: F(162, 5270) = 98.59 Prob > F = 0.0000 rho .95557807 (fraction of variance due to u_i) sigma_e .32607847 sigma_u 1.5123647 _cons -7.844568 .2477214 -31.67 0.000 -8.330205 -7.358932 l_popt .0731808 .0285325 2.56 0.010 .0172453 .1291163 l_infl2 -.0281481 .004758 -5.92 0.000 -.0374757 -.0188204 l_realgdp .3846652 .0153326 25.09 0.000 .354607 .4147234 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] corr(u_i, Xb) = -0.9200 Prob > F = 0.0000 F(3,5270) = 859.53 overall = 0.0036 max = 55 between = 0.0140 avg = 33.3 R-sq: within = 0.3285 Obs per group: min = 4 Group variable: idwbcode Number of groups = 163 Fixed-effects (within) regression Number of obs = 5436 _cons 3.315492 .0848041 39.10 0.000 3.149242 3.481742 l_popt .0014429 .0061381 0.24 0.814 -.0105903 .0134762 l_infl2 -.1650409 .0078825 -20.94 0.000 -.1804937 -.149588 l_realgdp -.0086728 .0036681 -2.36 0.018 -.0158636 -.0014819 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 2454.44502 5435 .451599819 Root MSE = .64482 Adj R-squared = 0.0793 Residual 2258.61346 5432 .415797764 R-squared = 0.0798 Model 195.831563 3 65.2771875 Prob > F = 0.0000 F( 3, 5432) = 156.99 Source SS df MS Number of obs = 5436 Group Characteristics 40
  41. 41. F test that all u_i=0: F(162, 5270) = 98.59 Prob > F = 0.0000 rho .95557807 (fraction of variance due to u_i) sigma_e .32607847 sigma_u 1.5123647 _cons -7.844568 .2477214 -31.67 0.000 -8.330205 -7.358932 l_popt .0731808 .0285325 2.56 0.010 .0172453 .1291163 l_infl2 -.0281481 .004758 -5.92 0.000 -.0374757 -.0188204 l_realgdp .3846652 .0153326 25.09 0.000 .354607 .4147234 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] corr(u_i, Xb) = -0.9200 Prob > F = 0.0000 F(3,5270) = 859.53 overall = 0.0036 max = 55 between = 0.0140 avg = 33.3 R-sq: within = 0.3285 Obs per group: min = 4 Group variable: idwbcode Number of groups = 163 Fixed-effects (within) regression Number of obs = 5436 _cons 3.315492 .0848041 39.10 0.000 3.149242 3.481742 l_popt .0014429 .0061381 0.24 0.814 -.0105903 .0134762 l_infl2 -.1650409 .0078825 -20.94 0.000 -.1804937 -.149588 l_realgdp -.0086728 .0036681 -2.36 0.018 -.0158636 -.0014819 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 2454.44502 5435 .451599819 Root MSE = .64482 Adj R-squared = 0.0793 Residual 2258.61346 5432 .415797764 R-squared = 0.0798 Model 195.831563 3 65.2771875 Prob > F = 0.0000 F( 3, 5432) = 156.99 Source SS df MS Number of obs = 5436 Different estimates 41
  42. 42. F test that all u_i=0: F(162, 5270) = 98.59 Prob > F = 0.0000 rho .95557807 (fraction of variance due to u_i) sigma_e .32607847 sigma_u 1.5123647 _cons -7.844568 .2477214 -31.67 0.000 -8.330205 -7.358932 l_popt .0731808 .0285325 2.56 0.010 .0172453 .1291163 l_infl2 -.0281481 .004758 -5.92 0.000 -.0374757 -.0188204 l_realgdp .3846652 .0153326 25.09 0.000 .354607 .4147234 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] corr(u_i, Xb) = -0.9200 Prob > F = 0.0000 F(3,5270) = 859.53 overall = 0.0036 max = 55 between = 0.0140 avg = 33.3 R-sq: within = 0.3285 Obs per group: min = 4 Group variable: idwbcode Number of groups = 163 Fixed-effects (within) regression Number of obs = 5436 _cons 3.315492 .0848041 39.10 0.000 3.149242 3.481742 l_popt .0014429 .0061381 0.24 0.814 -.0105903 .0134762 l_infl2 -.1650409 .0078825 -20.94 0.000 -.1804937 -.149588 l_realgdp -.0086728 .0036681 -2.36 0.018 -.0158636 -.0014819 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 2454.44502 5435 .451599819 Root MSE = .64482 Adj R-squared = 0.0793 Residual 2258.61346 5432 .415797764 R-squared = 0.0798 Model 195.831563 3 65.2771875 Prob > F = 0.0000 F( 3, 5432) = 156.99 Source SS df MS Number of obs = 5436 Different Fit 42
  43. 43. BETWEEN-GROUPS ESTIMATOR โ€ข Recall that the regression model goes through the averages (mean) of variables ๐ธ๐‘– ๐‘ฆ๐‘–๐‘ก = ๐ธ๐‘– ๐›ผ๐‘– + ๐‘ฅ๐‘–๐‘ก ๐›ฝ + ๐œ€๐‘–๐‘ก โ€ข We can run a regression on the means of each group ๐‘ฆ๐‘– = ๐›ผ + ๐‘ฅ๐‘– ๐›ฝ 43
  44. 44. BETWEEN-GROUPS ESTIMATOR Group 2 Group 1 ๐‘ฅ ๐‘ฆ 44
  45. 45. BETWEEN-GROUPS ESTIMATOR Group 2 Group 1 ๐‘ฅ ๐‘ฆ 45
  46. 46. BETWEEN-GROUPS ESTIMATOR Group 2 Group 1 ๐‘ฅ ๐‘ฆ 46 ๐‘ฆ๐‘– = ๐›ผ + ๐‘ฅ๐‘– ๐›ฝ
  47. 47. BETWEEN-GROUPS ESTIMATOR โ€ข Letยดs look at the estimator using sums ๐‘† ๐‘ ๐‘ฅ๐‘ฅ = ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ โ€ฒ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ ๐‘† ๐‘ ๐‘ฅ๐‘ฆ = ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฅ๐‘– โˆ’ ๐‘ฅ โ€ฒ ๐‘ฆ๐‘– โˆ’ ๐‘ฆ ๐›ฝ ๐‘ = ๐‘† ๐‘ ๐‘ฅ๐‘ฆ ๐‘† ๐‘ ๐‘ฅ๐‘ฅ 47
  48. 48. BETWEEN-GROUPS ESTIMATOR โ€ข Note that ๐‘† ๐‘ ๐‘ฅ๐‘ฅ = ๐‘† ๐‘ค ๐‘ฅ๐‘ฅ + ๐‘† ๐‘ ๐‘ฅ๐‘ฅ ๐‘† ๐‘ ๐‘ฅ๐‘ฆ = ๐‘† ๐‘ค ๐‘ฅ๐‘ฆ + ๐‘† ๐‘ ๐‘ฅ๐‘ฆ โ€ข Hence ๐›ฝ ๐‘ = ๐น ๐‘ค ๐›ฝ ๐‘ค + ๐ผ โˆ’ ๐น ๐‘ค ๐›ฝ ๐‘ ๐น ๐‘ค = ๐‘† ๐‘ค ๐‘ฅ๐‘ฅ ๐‘† ๐‘ค ๐‘ฅ๐‘ฅ + ๐‘† ๐‘ ๐‘ฅ๐‘ฅ 48
  49. 49. BETWEEN-GROUPS ESTIMATOR ๐›ฝ ๐‘ = ๐น ๐‘ค ๐›ฝ ๐‘ค + ๐ผ โˆ’ ๐น ๐‘ค ๐›ฝ ๐‘ โ€ข The pooled estimator is a weighted average of the between and within-group estimators โ€ข Weights depend on the information content of the data: โ€“ If groups are very similar, information comes from individuals within groups โ€“ If groups are very different, information comes from differences between groups 49
  50. 50. RESULTS BETWEEN-GROUPS ESTIMATOR 50 _cons 2.450055 .4769026 5.14 0.000 1.508174 3.391936 l_popt .008753 .029578 0.30 0.768 -.0496633 .0671694 l_infl2 -.4064124 .0650977 -6.24 0.000 -.53498 -.2778448 l_realgdp -.0062972 .0168565 -0.37 0.709 -.0395887 .0269942 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] sd(u_i + avg(e_i.))= .5167678 Prob > F = 0.0000 F(3,159) = 14.81 overall = 0.0787 max = 55 between = 0.2185 avg = 33.3 R-sq: within = 0.0155 Obs per group: min = 4 Group variable: idwbcode Number of groups = 163 Between regression (regression on group means) Number of obs = 5436 . xtreg l_money l_realgdp l_infl2 l_popt, be
  51. 51. ESTIMATING THE VARIANCE OF RESIDUALS โ€ข Compute the sample residuals as: ๐œ€๐‘–๐‘ก = ๐‘ฆ๐‘–๐‘ก โˆ’ ๐›ผ๐‘– โˆ’ ๐‘ฅ๐‘–๐‘ก ๐›ฝ โ€ข The residual variance estimator is simply: ๐œŽ2 = ๐‘–=1 ๐‘ ๐‘ก=1 ๐‘‡ ๐‘ฆ๐‘–๐‘ก โˆ’ ๐›ผ๐‘– โˆ’ ๐‘ฅ๐‘–๐‘ก ๐›ฝ 2 ๐‘๐‘‡ โˆ’ ๐‘ โˆ’ ๐พ 51
  52. 52. HYPOTHESIS TESTING โ€ข Having the estimated parameters and the residual variance estimator hypotheses testing is straightforward โ€“ Individual parameter tests distribute t in small samples and Normal in large samples โ€“ Multiple parameter tests distribute ๐œ’2 or ๐น(๐‘š, ๐‘›) 52
  53. 53. TWO-WAY FIXED EFFECTS ESTIMATOR โ€ข Model with fixed individual effects and fixed time effects ๐‘ฆ๐‘–๐‘ก = ๐›ผ๐‘– + ๐œ† ๐‘ก + ๐‘ฅ๐‘–๐‘ก ๐›ฝ + ๐œ€๐‘–๐‘ก where โ€ข ๐œ† ๐‘ก is a time effect affecting equally all individuals โ€ข ๐›ผ๐‘– is, again, an individual effect for all times 53
  54. 54. RESULTS OF THE TWO-WAY ESTIMATOR 54 1966 .1359131 .0705045 1.93 0.054 -.0023053 .2741315 1965 .1098147 .0730354 1.50 0.133 -.0333653 .2529947 1964 .1550872 .0736255 2.11 0.035 .0107504 .2994241 1963 .1448196 .0739148 1.96 0.050 -.0000844 .2897236 1962 .0419273 .0757742 0.55 0.580 -.1066219 .1904766 1961 -.0976766 .0752323 -1.30 0.194 -.2451635 .0498103 year l_popt -.1974149 .0380194 -5.19 0.000 -.2719489 -.1228809 l_infl2 -.0301174 .0054168 -5.56 0.000 -.0407367 -.0194982 l_realgdp .2112165 .0187937 11.24 0.000 .1743731 .24806 l_money Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 2454.44502 5435 .451599819 Root MSE = .31621 Adj R-squared = 0.7786 Residual 521.554775 5216 .09999133 R-squared = 0.7875 Model 1932.89024 219 8.82598285 Prob > F = 0.0000 F(219, 5216) = 88.27 Source SS df MS Number of obs = 5436 โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎโ‹ฎ โ‹ฎ

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