Shrinkage Estimation of Linear Panel Data Models Under Nearly Singular Design

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In this thesis we evaluate fixed effects linear panel data models where the sample within variation of the explanatory variable is significantly small. We notice that when a covariate is exhibiting high degrees of low longitudinal variation (LLV), this indicates a nearly singular design.
Consistently with the related literature, we demonstrate that this design affects

the rate of convergence of the FE estimator. The new rate is slower than n and depends critically on the severity of the LLV problem. By extending the subsampling methodology for the linear panel data context, we show how to estimate this unknown rate of convergence.
We subsequently begin to evaluate alternative ways of estimating the parameter of interest when the explanatory variable is characterized by LLV. In particular, we demonstrate how it is possible to obtain a shrinkage estimator whose Mean Square Error always dominates, under appropriate conditions, the Mean Square Error of the fixed effect estimator. We demonstrate the importance of this result with a specific empirical example.

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Shrinkage Estimation of Linear Panel Data Models Under Nearly Singular Design

  1. 1. Shrinkage Estimation of Linear Panel Data Models Under Nearly Singular Design Raffaele Saggio University of Tor Vergata Graduation SessionRaffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 1 / 20
  2. 2. Introduction MotivationThis ThesisWe evaluate fixed effects (FE) linear panel data models where the sample withinvariation of the explanatory variable is significantly small.This type of outcome is often observed in microeconometrics and it imposes seriouschallenges when inferencing with FE models.The main problem is that - whenever the longitudinal variation is small - the FEestimator may provide very little information about the parameter of interest no matterhow large the cross sectional dimension might be.We have two objectives: To formalize how situations where the longitudinal variation of the regressor is small affect the standard asymptotic distribution of the FE estimator assuming that the cross sectional dimension, n, is large and the longitudinal dimension, T , is small. To propose an alternative way of estimating the parameter of interest in the presence of high degrees of Low Longitudinal Variation (LLV henceforth) with correlated, time invariant, individual effects.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 2 / 20
  3. 3. Introduction MotivationThis ThesisWe evaluate fixed effects (FE) linear panel data models where the sample withinvariation of the explanatory variable is significantly small.This type of outcome is often observed in microeconometrics and it imposes seriouschallenges when inferencing with FE models.The main problem is that - whenever the longitudinal variation is small - the FEestimator may provide very little information about the parameter of interest no matterhow large the cross sectional dimension might be.We have two objectives: To formalize how situations where the longitudinal variation of the regressor is small affect the standard asymptotic distribution of the FE estimator assuming that the cross sectional dimension, n, is large and the longitudinal dimension, T , is small. To propose an alternative way of estimating the parameter of interest in the presence of high degrees of Low Longitudinal Variation (LLV henceforth) with correlated, time invariant, individual effects.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 2 / 20
  4. 4. Introduction MotivationThis ThesisWe evaluate fixed effects (FE) linear panel data models where the sample withinvariation of the explanatory variable is significantly small.This type of outcome is often observed in microeconometrics and it imposes seriouschallenges when inferencing with FE models.The main problem is that - whenever the longitudinal variation is small - the FEestimator may provide very little information about the parameter of interest no matterhow large the cross sectional dimension might be.We have two objectives: To formalize how situations where the longitudinal variation of the regressor is small affect the standard asymptotic distribution of the FE estimator assuming that the cross sectional dimension, n, is large and the longitudinal dimension, T , is small. To propose an alternative way of estimating the parameter of interest in the presence of high degrees of Low Longitudinal Variation (LLV henceforth) with correlated, time invariant, individual effects.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 2 / 20
  5. 5. Introduction MotivationThis ThesisWe evaluate fixed effects (FE) linear panel data models where the sample withinvariation of the explanatory variable is significantly small.This type of outcome is often observed in microeconometrics and it imposes seriouschallenges when inferencing with FE models.The main problem is that - whenever the longitudinal variation is small - the FEestimator may provide very little information about the parameter of interest no matterhow large the cross sectional dimension might be.We have two objectives: To formalize how situations where the longitudinal variation of the regressor is small affect the standard asymptotic distribution of the FE estimator assuming that the cross sectional dimension, n, is large and the longitudinal dimension, T , is small. To propose an alternative way of estimating the parameter of interest in the presence of high degrees of Low Longitudinal Variation (LLV henceforth) with correlated, time invariant, individual effects.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 2 / 20
  6. 6. Introduction MotivationThis ThesisWe evaluate fixed effects (FE) linear panel data models where the sample withinvariation of the explanatory variable is significantly small.This type of outcome is often observed in microeconometrics and it imposes seriouschallenges when inferencing with FE models.The main problem is that - whenever the longitudinal variation is small - the FEestimator may provide very little information about the parameter of interest no matterhow large the cross sectional dimension might be.We have two objectives: To formalize how situations where the longitudinal variation of the regressor is small affect the standard asymptotic distribution of the FE estimator assuming that the cross sectional dimension, n, is large and the longitudinal dimension, T , is small. To propose an alternative way of estimating the parameter of interest in the presence of high degrees of Low Longitudinal Variation (LLV henceforth) with correlated, time invariant, individual effects.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 2 / 20
  7. 7. Introduction MotivationThis ThesisWe evaluate fixed effects (FE) linear panel data models where the sample withinvariation of the explanatory variable is significantly small.This type of outcome is often observed in microeconometrics and it imposes seriouschallenges when inferencing with FE models.The main problem is that - whenever the longitudinal variation is small - the FEestimator may provide very little information about the parameter of interest no matterhow large the cross sectional dimension might be.We have two objectives: To formalize how situations where the longitudinal variation of the regressor is small affect the standard asymptotic distribution of the FE estimator assuming that the cross sectional dimension, n, is large and the longitudinal dimension, T , is small. To propose an alternative way of estimating the parameter of interest in the presence of high degrees of Low Longitudinal Variation (LLV henceforth) with correlated, time invariant, individual effects.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 2 / 20
  8. 8. Introduction ContributionsMain FindingsRelative to the first objective: We notice that when a variable is exhibiting high degrees of LLV, the sample within variation may become indistinguishable from 0 as n → ∞ with T fixed. This type of asymptotic behavior is defined in the thesis as the “The LLV Problem”. By relating the LLV problem with the nearly singular design, we provide a theorem that shows what is the correct asymptotic distribution of the FE estimator under the LLV problem. The most important consequence of this result is that now the rate of convergence of the FE estimator is unknown as it crucially depends on the severity of the LLV problem. By extending the subsampling methodology for the linear panel data context, we propose a method to estimate this unknown rate of convergence.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 3 / 20
  9. 9. Introduction ContributionsMain FindingsRelative to the first objective: We notice that when a variable is exhibiting high degrees of LLV, the sample within variation may become indistinguishable from 0 as n → ∞ with T fixed. This type of asymptotic behavior is defined in the thesis as the “The LLV Problem”. By relating the LLV problem with the nearly singular design, we provide a theorem that shows what is the correct asymptotic distribution of the FE estimator under the LLV problem. The most important consequence of this result is that now the rate of convergence of the FE estimator is unknown as it crucially depends on the severity of the LLV problem. By extending the subsampling methodology for the linear panel data context, we propose a method to estimate this unknown rate of convergence.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 3 / 20
  10. 10. Introduction ContributionsMain FindingsRelative to the first objective: We notice that when a variable is exhibiting high degrees of LLV, the sample within variation may become indistinguishable from 0 as n → ∞ with T fixed. This type of asymptotic behavior is defined in the thesis as the “The LLV Problem”. By relating the LLV problem with the nearly singular design, we provide a theorem that shows what is the correct asymptotic distribution of the FE estimator under the LLV problem. The most important consequence of this result is that now the rate of convergence of the FE estimator is unknown as it crucially depends on the severity of the LLV problem. By extending the subsampling methodology for the linear panel data context, we propose a method to estimate this unknown rate of convergence.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 3 / 20
  11. 11. Introduction ContributionsMain FindingsRelative to the first objective: We notice that when a variable is exhibiting high degrees of LLV, the sample within variation may become indistinguishable from 0 as n → ∞ with T fixed. This type of asymptotic behavior is defined in the thesis as the “The LLV Problem”. By relating the LLV problem with the nearly singular design, we provide a theorem that shows what is the correct asymptotic distribution of the FE estimator under the LLV problem. The most important consequence of this result is that now the rate of convergence of the FE estimator is unknown as it crucially depends on the severity of the LLV problem. By extending the subsampling methodology for the linear panel data context, we propose a method to estimate this unknown rate of convergence.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 3 / 20
  12. 12. Introduction ContributionsMain FindingsRelative to the first objective: We notice that when a variable is exhibiting high degrees of LLV, the sample within variation may become indistinguishable from 0 as n → ∞ with T fixed. This type of asymptotic behavior is defined in the thesis as the “The LLV Problem”. By relating the LLV problem with the nearly singular design, we provide a theorem that shows what is the correct asymptotic distribution of the FE estimator under the LLV problem. The most important consequence of this result is that now the rate of convergence of the FE estimator is unknown as it crucially depends on the severity of the LLV problem. By extending the subsampling methodology for the linear panel data context, we propose a method to estimate this unknown rate of convergence.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 3 / 20
  13. 13. Introduction ContributionsMain FindingsRelative to the second objective: We evaluate shrinkage estimation techniques for the FE linear panel data model that allow to trade the unbiasedness of the FE estimator with an estimator that has smaller variance. More specifically, we demonstrate how it is possible to obtain a shrinkage estimator whose Mean Square Error always dominates, under appropriate conditions, the Mean Square Error of the FE estimator. We document the importance of this result with a specific empirical example drawn from the twin study of Ashenfelter and Krueger (1994). In particular, we show that our proposed shrinkage estimator is more reliable than the FE estimator of Ashenfelter and Krueger (1994) in estimating the returns of education.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 4 / 20
  14. 14. Introduction ContributionsMain FindingsRelative to the second objective: We evaluate shrinkage estimation techniques for the FE linear panel data model that allow to trade the unbiasedness of the FE estimator with an estimator that has smaller variance. More specifically, we demonstrate how it is possible to obtain a shrinkage estimator whose Mean Square Error always dominates, under appropriate conditions, the Mean Square Error of the FE estimator. We document the importance of this result with a specific empirical example drawn from the twin study of Ashenfelter and Krueger (1994). In particular, we show that our proposed shrinkage estimator is more reliable than the FE estimator of Ashenfelter and Krueger (1994) in estimating the returns of education.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 4 / 20
  15. 15. Introduction ContributionsMain FindingsRelative to the second objective: We evaluate shrinkage estimation techniques for the FE linear panel data model that allow to trade the unbiasedness of the FE estimator with an estimator that has smaller variance. More specifically, we demonstrate how it is possible to obtain a shrinkage estimator whose Mean Square Error always dominates, under appropriate conditions, the Mean Square Error of the FE estimator. We document the importance of this result with a specific empirical example drawn from the twin study of Ashenfelter and Krueger (1994). In particular, we show that our proposed shrinkage estimator is more reliable than the FE estimator of Ashenfelter and Krueger (1994) in estimating the returns of education.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 4 / 20
  16. 16. Introduction ContributionsMain FindingsRelative to the second objective: We evaluate shrinkage estimation techniques for the FE linear panel data model that allow to trade the unbiasedness of the FE estimator with an estimator that has smaller variance. More specifically, we demonstrate how it is possible to obtain a shrinkage estimator whose Mean Square Error always dominates, under appropriate conditions, the Mean Square Error of the FE estimator. We document the importance of this result with a specific empirical example drawn from the twin study of Ashenfelter and Krueger (1994). In particular, we show that our proposed shrinkage estimator is more reliable than the FE estimator of Ashenfelter and Krueger (1994) in estimating the returns of education.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 4 / 20
  17. 17. Introduction ContributionsMain FindingsRelative to the second objective: We evaluate shrinkage estimation techniques for the FE linear panel data model that allow to trade the unbiasedness of the FE estimator with an estimator that has smaller variance. More specifically, we demonstrate how it is possible to obtain a shrinkage estimator whose Mean Square Error always dominates, under appropriate conditions, the Mean Square Error of the FE estimator. We document the importance of this result with a specific empirical example drawn from the twin study of Ashenfelter and Krueger (1994). In particular, we show that our proposed shrinkage estimator is more reliable than the FE estimator of Ashenfelter and Krueger (1994) in estimating the returns of education.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 4 / 20
  18. 18. The LLV Problem The ModelAsymptotic properties of the FE estimatorOur results are based on a standard FE model with correlated, time invariant, fixedeffects. Without loss of generality, we consider the case when there is only oneregressor.We are interested in evaluating whether the fact of having a regressor that shows highdegrees of LLV affects the following two properties of the FE estimator: Consistency. Asymptotic Normality.We first prove that the FE estimator is always consistent, as n → ∞ with T fixed,regardless of any possible problem of LLV.This occurs because, intuitively, we do not need to specify a particular convergence ˆ prate for the sample longitudinal variation in order to have β → β.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 5 / 20
  19. 19. The LLV Problem The ModelAsymptotic properties of the FE estimatorOur results are based on a standard FE model with correlated, time invariant, fixedeffects. Without loss of generality, we consider the case when there is only oneregressor.We are interested in evaluating whether the fact of having a regressor that shows highdegrees of LLV affects the following two properties of the FE estimator: Consistency. Asymptotic Normality.We first prove that the FE estimator is always consistent, as n → ∞ with T fixed,regardless of any possible problem of LLV.This occurs because, intuitively, we do not need to specify a particular convergence ˆ prate for the sample longitudinal variation in order to have β → β.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 5 / 20
  20. 20. The LLV Problem The ModelAsymptotic properties of the FE estimatorOur results are based on a standard FE model with correlated, time invariant, fixedeffects. Without loss of generality, we consider the case when there is only oneregressor.We are interested in evaluating whether the fact of having a regressor that shows highdegrees of LLV affects the following two properties of the FE estimator: Consistency. Asymptotic Normality.We first prove that the FE estimator is always consistent, as n → ∞ with T fixed,regardless of any possible problem of LLV.This occurs because, intuitively, we do not need to specify a particular convergence ˆ prate for the sample longitudinal variation in order to have β → β.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 5 / 20
  21. 21. The LLV Problem The ModelAsymptotic properties of the FE estimatorOur results are based on a standard FE model with correlated, time invariant, fixedeffects. Without loss of generality, we consider the case when there is only oneregressor.We are interested in evaluating whether the fact of having a regressor that shows highdegrees of LLV affects the following two properties of the FE estimator: Consistency. Asymptotic Normality.We first prove that the FE estimator is always consistent, as n → ∞ with T fixed,regardless of any possible problem of LLV.This occurs because, intuitively, we do not need to specify a particular convergence ˆ prate for the sample longitudinal variation in order to have β → β.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 5 / 20
  22. 22. The LLV Problem The ModelAsymptotic properties of the FE estimatorOur results are based on a standard FE model with correlated, time invariant, fixedeffects. Without loss of generality, we consider the case when there is only oneregressor.We are interested in evaluating whether the fact of having a regressor that shows highdegrees of LLV affects the following two properties of the FE estimator: Consistency. Asymptotic Normality.We first prove that the FE estimator is always consistent, as n → ∞ with T fixed,regardless of any possible problem of LLV.This occurs because, intuitively, we do not need to specify a particular convergence ˆ prate for the sample longitudinal variation in order to have β → β.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 5 / 20
  23. 23. The LLV Problem The ModelAsymptotic properties of the FE estimatorOur results are based on a standard FE model with correlated, time invariant, fixedeffects. Without loss of generality, we consider the case when there is only oneregressor.We are interested in evaluating whether the fact of having a regressor that shows highdegrees of LLV affects the following two properties of the FE estimator: Consistency. Asymptotic Normality.We first prove that the FE estimator is always consistent, as n → ∞ with T fixed,regardless of any possible problem of LLV.This occurs because, intuitively, we do not need to specify a particular convergence ˆ prate for the sample longitudinal variation in order to have β → β.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 5 / 20
  24. 24. The LLV Problem ConvergenceThe Problem of ConvergenceThe problem of convergence of the sample within variation becomes fundamental,however, when deriving the asymptotic distribution of the FE estimator.This last result is usually derived assuming that, as n → ∞ with T fixed n X n X plim n−1 γiT = lim n−1 ˆ γiT = γT > 0, (1) n→∞ i=1 i=1where γiT = T −1 ˆ P ¯ − Xi )2 and E(ˆiT ) = γiT . t (Xit γSuppose now that the explanatory variable exhibits high degrees of LLV.Is assumption (1) still valid? The answer is: not necessarily. The average amount of the longitudinal variation may become indistinguishable from 0 as n → ∞ with T fixed. If that is the case, then we say that our variable of interest is affected by the LLV problem.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 6 / 20
  25. 25. The LLV Problem ConvergenceThe Problem of ConvergenceThe problem of convergence of the sample within variation becomes fundamental,however, when deriving the asymptotic distribution of the FE estimator.This last result is usually derived assuming that, as n → ∞ with T fixed n X n X plim n−1 γiT = lim n−1 ˆ γiT = γT > 0, (1) n→∞ i=1 i=1where γiT = T −1 ˆ P ¯ − Xi )2 and E(ˆiT ) = γiT . t (Xit γSuppose now that the explanatory variable exhibits high degrees of LLV.Is assumption (1) still valid? The answer is: not necessarily. The average amount of the longitudinal variation may become indistinguishable from 0 as n → ∞ with T fixed. If that is the case, then we say that our variable of interest is affected by the LLV problem.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 6 / 20
  26. 26. The LLV Problem ConvergenceThe Problem of ConvergenceThe problem of convergence of the sample within variation becomes fundamental,however, when deriving the asymptotic distribution of the FE estimator.This last result is usually derived assuming that, as n → ∞ with T fixed n X n X plim n−1 γiT = lim n−1 ˆ γiT = γT > 0, (1) n→∞ i=1 i=1where γiT = T −1 ˆ P ¯ − Xi )2 and E(ˆiT ) = γiT . t (Xit γSuppose now that the explanatory variable exhibits high degrees of LLV.Is assumption (1) still valid? The answer is: not necessarily. The average amount of the longitudinal variation may become indistinguishable from 0 as n → ∞ with T fixed. If that is the case, then we say that our variable of interest is affected by the LLV problem.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 6 / 20
  27. 27. The LLV Problem ConvergenceThe Problem of ConvergenceThe problem of convergence of the sample within variation becomes fundamental,however, when deriving the asymptotic distribution of the FE estimator.This last result is usually derived assuming that, as n → ∞ with T fixed n X n X plim n−1 γiT = lim n−1 ˆ γiT = γT > 0, (1) n→∞ i=1 i=1where γiT = T −1 ˆ P ¯ − Xi )2 and E(ˆiT ) = γiT . t (Xit γSuppose now that the explanatory variable exhibits high degrees of LLV.Is assumption (1) still valid? The answer is: not necessarily. The average amount of the longitudinal variation may become indistinguishable from 0 as n → ∞ with T fixed. If that is the case, then we say that our variable of interest is affected by the LLV problem.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 6 / 20
  28. 28. The LLV Problem ConvergenceThe Problem of ConvergenceThe problem of convergence of the sample within variation becomes fundamental,however, when deriving the asymptotic distribution of the FE estimator.This last result is usually derived assuming that, as n → ∞ with T fixed n X n X plim n−1 γiT = lim n−1 ˆ γiT = γT > 0, (1) n→∞ i=1 i=1where γiT = T −1 ˆ P ¯ − Xi )2 and E(ˆiT ) = γiT . t (Xit γSuppose now that the explanatory variable exhibits high degrees of LLV.Is assumption (1) still valid? The answer is: not necessarily. The average amount of the longitudinal variation may become indistinguishable from 0 as n → ∞ with T fixed. If that is the case, then we say that our variable of interest is affected by the LLV problem.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 6 / 20
  29. 29. The LLV Problem ConvergenceThe Problem of ConvergenceThe problem of convergence of the sample within variation becomes fundamental,however, when deriving the asymptotic distribution of the FE estimator.This last result is usually derived assuming that, as n → ∞ with T fixed n X n X plim n−1 γiT = lim n−1 ˆ γiT = γT > 0, (1) n→∞ i=1 i=1where γiT = T −1 ˆ P ¯ − Xi )2 and E(ˆiT ) = γiT . t (Xit γSuppose now that the explanatory variable exhibits high degrees of LLV.Is assumption (1) still valid? The answer is: not necessarily. The average amount of the longitudinal variation may become indistinguishable from 0 as n → ∞ with T fixed. If that is the case, then we say that our variable of interest is affected by the LLV problem.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 6 / 20
  30. 30. The LLV Problem ExampleA 2x2 ExampleTake T = 2 and assume that Xit is a dummy variable that indicates whether anindividual has received treatment between t = 1 and t = 2 (DID framework).In this case, it is easy to show that the sample longitudinal variation is simply n 1X mn γi2 = ˆ , (2) n 4n i=1where mn represents the number of treated individuals collected in the sample.Notice that: Empirical works always assume, at least implicitly, that equation (2) is converging to a term that is well distinguishable from 0 as n → ∞ with T fixed. However, if the relative number of policy changes observed in the data is small, equation (2) may converge to a term that is NOT bounded away from 0. This happens because the number of treated individuals asymptotically increases too slowly relative to the total increase in the cross sectional dimension, n.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 7 / 20
  31. 31. The LLV Problem ExampleA 2x2 ExampleTake T = 2 and assume that Xit is a dummy variable that indicates whether anindividual has received treatment between t = 1 and t = 2 (DID framework).In this case, it is easy to show that the sample longitudinal variation is simply n 1X mn γi2 = ˆ , (2) n 4n i=1where mn represents the number of treated individuals collected in the sample.Notice that: Empirical works always assume, at least implicitly, that equation (2) is converging to a term that is well distinguishable from 0 as n → ∞ with T fixed. However, if the relative number of policy changes observed in the data is small, equation (2) may converge to a term that is NOT bounded away from 0. This happens because the number of treated individuals asymptotically increases too slowly relative to the total increase in the cross sectional dimension, n.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 7 / 20
  32. 32. The LLV Problem ExampleA 2x2 ExampleTake T = 2 and assume that Xit is a dummy variable that indicates whether anindividual has received treatment between t = 1 and t = 2 (DID framework).In this case, it is easy to show that the sample longitudinal variation is simply n 1X mn γi2 = ˆ , (2) n 4n i=1where mn represents the number of treated individuals collected in the sample.Notice that: Empirical works always assume, at least implicitly, that equation (2) is converging to a term that is well distinguishable from 0 as n → ∞ with T fixed. However, if the relative number of policy changes observed in the data is small, equation (2) may converge to a term that is NOT bounded away from 0. This happens because the number of treated individuals asymptotically increases too slowly relative to the total increase in the cross sectional dimension, n.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 7 / 20
  33. 33. The LLV Problem ExampleA 2x2 ExampleTake T = 2 and assume that Xit is a dummy variable that indicates whether anindividual has received treatment between t = 1 and t = 2 (DID framework).In this case, it is easy to show that the sample longitudinal variation is simply n 1X mn γi2 = ˆ , (2) n 4n i=1where mn represents the number of treated individuals collected in the sample.Notice that: Empirical works always assume, at least implicitly, that equation (2) is converging to a term that is well distinguishable from 0 as n → ∞ with T fixed. However, if the relative number of policy changes observed in the data is small, equation (2) may converge to a term that is NOT bounded away from 0. This happens because the number of treated individuals asymptotically increases too slowly relative to the total increase in the cross sectional dimension, n.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 7 / 20
  34. 34. The LLV Problem ExampleA 2x2 ExampleTake T = 2 and assume that Xit is a dummy variable that indicates whether anindividual has received treatment between t = 1 and t = 2 (DID framework).In this case, it is easy to show that the sample longitudinal variation is simply n 1X mn γi2 = ˆ , (2) n 4n i=1where mn represents the number of treated individuals collected in the sample.Notice that: Empirical works always assume, at least implicitly, that equation (2) is converging to a term that is well distinguishable from 0 as n → ∞ with T fixed. However, if the relative number of policy changes observed in the data is small, equation (2) may converge to a term that is NOT bounded away from 0. This happens because the number of treated individuals asymptotically increases too slowly relative to the total increase in the cross sectional dimension, n.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 7 / 20
  35. 35. The LLV Problem ExampleA 2x2 ExampleTake T = 2 and assume that Xit is a dummy variable that indicates whether anindividual has received treatment between t = 1 and t = 2 (DID framework).In this case, it is easy to show that the sample longitudinal variation is simply n 1X mn γi2 = ˆ , (2) n 4n i=1where mn represents the number of treated individuals collected in the sample.Notice that: Empirical works always assume, at least implicitly, that equation (2) is converging to a term that is well distinguishable from 0 as n → ∞ with T fixed. However, if the relative number of policy changes observed in the data is small, equation (2) may converge to a term that is NOT bounded away from 0. This happens because the number of treated individuals asymptotically increases too slowly relative to the total increase in the cross sectional dimension, n.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 7 / 20
  36. 36. The LLV Problem ExampleA 2x2 ExampleTake T = 2 and assume that Xit is a dummy variable that indicates whether anindividual has received treatment between t = 1 and t = 2 (DID framework).In this case, it is easy to show that the sample longitudinal variation is simply n 1X mn γi2 = ˆ , (2) n 4n i=1where mn represents the number of treated individuals collected in the sample.Notice that: Empirical works always assume, at least implicitly, that equation (2) is converging to a term that is well distinguishable from 0 as n → ∞ with T fixed. However, if the relative number of policy changes observed in the data is small, equation (2) may converge to a term that is NOT bounded away from 0. This happens because the number of treated individuals asymptotically increases too slowly relative to the total increase in the cross sectional dimension, n.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 7 / 20
  37. 37. Asymptotics under the LLV Problem Nearly Singular DesignThe Nearly Singular DesignBy relating the LLV problem to the nearly singular design (Knight and Fu, 2000,JASA; Caner, 2008, JoE), we show how to derive the correct asymptotic distributionof the FE estimator under the LLV problem.In fact, assuming that n 1 X ∗ lim γiT = γT > 0 (3) n→∞ n1−κ i=1where 0 ≤ κ < 1, it is possible to prove the following TheoremRaffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 8 / 20
  38. 38. Asymptotics under the LLV Problem Nearly Singular DesignThe Nearly Singular DesignBy relating the LLV problem to the nearly singular design (Knight and Fu, 2000,JASA; Caner, 2008, JoE), we show how to derive the correct asymptotic distributionof the FE estimator under the LLV problem.In fact, assuming that n 1 X ∗ lim γiT = γT > 0 (3) n→∞ n1−κ i=1where 0 ≤ κ < 1, it is possible to prove the following TheoremRaffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 8 / 20
  39. 39. Asymptotics under the LLV Problem Nearly Singular DesignThe Nearly Singular DesignBy relating the LLV problem to the nearly singular design (Knight and Fu, 2000,JASA; Caner, 2008, JoE), we show how to derive the correct asymptotic distributionof the FE estimator under the LLV problem.In fact, assuming that n 1 X ∗ lim γiT = γT > 0 (3) n→∞ n1−κ i=1where 0 ≤ κ < 1, it is possible to prove the following TheoremRaffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 8 / 20
  40. 40. Asymptotics under the LLV Problem Nearly Singular DesignTheoremUnder Assumption (3), via the Lindeberg-Feller Central Limit Theorem for unequalvariances, as n → ∞ with T fixed “ ” ˆ d ˆ cn (β − β) → N 0; AVT (β) . (4) 1−κwhere cn = n 2 ˆ and AVT (β) is the asymptotic variance of the FE estimator.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 9 / 20
  41. 41. Asymptotics under the LLV Problem Inference under the LLV problemInference under the LLV problem The most important consequence of result (4) is that, in order to construct inference, practitioners need now to estimate two objects: the asymptotic ˆ variance of the FE estimator, AVT (β), and the unknown rate of convergence of the FE estimator, cn . We show that the subsampling methodology (Bertail, Politis and Romano, 1999, JASA) can be applied in our panel data context in order to derive a consistent estimate for the unknown rate of convergence. This type of procedure provides us with a diagnostic tool to evaluate the severity of the LLV problem. Using the subsampling methodology, in fact, we are able to estimate and to test whether the LLV problem actually decreases the rate of convergence of the FE estimator (i.e. κ > 0) or if the standard asymptotic result applies (i.e. κ = 0).Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 10 / 20
  42. 42. Asymptotics under the LLV Problem Inference under the LLV problemInference under the LLV problem The most important consequence of result (4) is that, in order to construct inference, practitioners need now to estimate two objects: the asymptotic ˆ variance of the FE estimator, AVT (β), and the unknown rate of convergence of the FE estimator, cn . We show that the subsampling methodology (Bertail, Politis and Romano, 1999, JASA) can be applied in our panel data context in order to derive a consistent estimate for the unknown rate of convergence. This type of procedure provides us with a diagnostic tool to evaluate the severity of the LLV problem. Using the subsampling methodology, in fact, we are able to estimate and to test whether the LLV problem actually decreases the rate of convergence of the FE estimator (i.e. κ > 0) or if the standard asymptotic result applies (i.e. κ = 0).Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 10 / 20
  43. 43. Asymptotics under the LLV Problem Inference under the LLV problemInference under the LLV problem The most important consequence of result (4) is that, in order to construct inference, practitioners need now to estimate two objects: the asymptotic ˆ variance of the FE estimator, AVT (β), and the unknown rate of convergence of the FE estimator, cn . We show that the subsampling methodology (Bertail, Politis and Romano, 1999, JASA) can be applied in our panel data context in order to derive a consistent estimate for the unknown rate of convergence. This type of procedure provides us with a diagnostic tool to evaluate the severity of the LLV problem. Using the subsampling methodology, in fact, we are able to estimate and to test whether the LLV problem actually decreases the rate of convergence of the FE estimator (i.e. κ > 0) or if the standard asymptotic result applies (i.e. κ = 0).Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 10 / 20
  44. 44. Asymptotics under the LLV Problem Inference under the LLV problemInference under the LLV problem The most important consequence of result (4) is that, in order to construct inference, practitioners need now to estimate two objects: the asymptotic ˆ variance of the FE estimator, AVT (β), and the unknown rate of convergence of the FE estimator, cn . We show that the subsampling methodology (Bertail, Politis and Romano, 1999, JASA) can be applied in our panel data context in order to derive a consistent estimate for the unknown rate of convergence. This type of procedure provides us with a diagnostic tool to evaluate the severity of the LLV problem. Using the subsampling methodology, in fact, we are able to estimate and to test whether the LLV problem actually decreases the rate of convergence of the FE estimator (i.e. κ > 0) or if the standard asymptotic result applies (i.e. κ = 0).Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 10 / 20
  45. 45. Shrinkage Estimation RE estimator vs. FE estimatorShrinkage Estimation We now turn our attention to the second objective of the thesis: to discuss alternative ways of estimating the parameter of interest when the explanatory variable is characterized by LLV and the underlined model implies the presence of correlated, time invariant, unobservables effects. Under this framework, it is well-known that the FE estimator is unbiased but, due to the LLV of the explanatory variable, it has also a large variance. Consequently, we start to discuss estimation procedures that allow to trade the unbiasedness of the FE estimator with an estimator that has smaller variance. We first highlight how traditional linear panel data estimators, such as the Random Effect (RE) estimator, can theoretically have smaller MSE than the FE estimator, especially in contexts that involve LLV.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 11 / 20
  46. 46. Shrinkage Estimation RE estimator vs. FE estimatorShrinkage Estimation We now turn our attention to the second objective of the thesis: to discuss alternative ways of estimating the parameter of interest when the explanatory variable is characterized by LLV and the underlined model implies the presence of correlated, time invariant, unobservables effects. Under this framework, it is well-known that the FE estimator is unbiased but, due to the LLV of the explanatory variable, it has also a large variance. Consequently, we start to discuss estimation procedures that allow to trade the unbiasedness of the FE estimator with an estimator that has smaller variance. We first highlight how traditional linear panel data estimators, such as the Random Effect (RE) estimator, can theoretically have smaller MSE than the FE estimator, especially in contexts that involve LLV.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 11 / 20
  47. 47. Shrinkage Estimation RE estimator vs. FE estimatorShrinkage Estimation We now turn our attention to the second objective of the thesis: to discuss alternative ways of estimating the parameter of interest when the explanatory variable is characterized by LLV and the underlined model implies the presence of correlated, time invariant, unobservables effects. Under this framework, it is well-known that the FE estimator is unbiased but, due to the LLV of the explanatory variable, it has also a large variance. Consequently, we start to discuss estimation procedures that allow to trade the unbiasedness of the FE estimator with an estimator that has smaller variance. We first highlight how traditional linear panel data estimators, such as the Random Effect (RE) estimator, can theoretically have smaller MSE than the FE estimator, especially in contexts that involve LLV.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 11 / 20
  48. 48. Shrinkage Estimation RE estimator vs. FE estimatorShrinkage Estimation We now turn our attention to the second objective of the thesis: to discuss alternative ways of estimating the parameter of interest when the explanatory variable is characterized by LLV and the underlined model implies the presence of correlated, time invariant, unobservables effects. Under this framework, it is well-known that the FE estimator is unbiased but, due to the LLV of the explanatory variable, it has also a large variance. Consequently, we start to discuss estimation procedures that allow to trade the unbiasedness of the FE estimator with an estimator that has smaller variance. We first highlight how traditional linear panel data estimators, such as the Random Effect (RE) estimator, can theoretically have smaller MSE than the FE estimator, especially in contexts that involve LLV.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 11 / 20
  49. 49. Shrinkage Estimation RE estimator vs. FE estimatorShrinkage Estimation The problem in using the RE estimator in place of the FE estimator is that the difference between the MSEs of these estimators crucially depends, along with the within/between variation ratio, φ, on the squared bias of the RE estimator, π 2 . 2ˆ ˆ ˆ ˆ σu θ θ2 π2 ∆MSE = MSE(β) − MSE(βr ) = “ p ”−“ ”2 . (5) φ φ+ θ ˆ ˆ θ+φ Since this bias cannot be precisely estimated in situations of LLV (see Table 2 in the thesis), equation (5) is not able to suggest to practitioners whether the RE estimator has lower MSE than the FE estimator. Therefore, we begin to evaluate a particular shrinkage estimation technique that can guarantee us dominance in MSE.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 12 / 20
  50. 50. Shrinkage Estimation RE estimator vs. FE estimatorShrinkage Estimation The problem in using the RE estimator in place of the FE estimator is that the difference between the MSEs of these estimators crucially depends, along with the within/between variation ratio, φ, on the squared bias of the RE estimator, π 2 . 2ˆ ˆ ˆ ˆ σu θ θ2 π2 ∆MSE = MSE(β) − MSE(βr ) = “ p ”−“ ”2 . (5) φ φ+ θ ˆ ˆ θ+φ Since this bias cannot be precisely estimated in situations of LLV (see Table 2 in the thesis), equation (5) is not able to suggest to practitioners whether the RE estimator has lower MSE than the FE estimator. Therefore, we begin to evaluate a particular shrinkage estimation technique that can guarantee us dominance in MSE.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 12 / 20
  51. 51. Shrinkage Estimation RE estimator vs. FE estimatorShrinkage Estimation The problem in using the RE estimator in place of the FE estimator is that the difference between the MSEs of these estimators crucially depends, along with the within/between variation ratio, φ, on the squared bias of the RE estimator, π 2 . 2ˆ ˆ ˆ ˆ σu θ θ2 π2 ∆MSE = MSE(β) − MSE(βr ) = “ p ”−“ ”2 . (5) φ φ+ θ ˆ ˆ θ+φ Since this bias cannot be precisely estimated in situations of LLV (see Table 2 in the thesis), equation (5) is not able to suggest to practitioners whether the RE estimator has lower MSE than the FE estimator. Therefore, we begin to evaluate a particular shrinkage estimation technique that can guarantee us dominance in MSE.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 12 / 20
  52. 52. Shrinkage Estimation RE estimator vs. FE estimatorShrinkage Estimation The problem in using the RE estimator in place of the FE estimator is that the difference between the MSEs of these estimators crucially depends, along with the within/between variation ratio, φ, on the squared bias of the RE estimator, π 2 . 2ˆ ˆ ˆ ˆ σu θ θ2 π2 ∆MSE = MSE(β) − MSE(βr ) = “ p ”−“ ”2 . (5) φ φ+ θ ˆ ˆ θ+φ Since this bias cannot be precisely estimated in situations of LLV (see Table 2 in the thesis), equation (5) is not able to suggest to practitioners whether the RE estimator has lower MSE than the FE estimator. Therefore, we begin to evaluate a particular shrinkage estimation technique that can guarantee us dominance in MSE.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 12 / 20
  53. 53. Shrinkage Estimation Ridge regressionThe Ordinary Ridge Estimator for FE Linear Panel Data Models We demonstrate how, by applying the Ordinary Ridge Regression (ORR) framework to our FE linear panel data context, it is possible to construct a shrinkage estimator whose MSE always dominates, under appropriate conditions, the MSE of the FE estimator. In order to fully understand this last result, recall that when evaluating the ORR estimator, Hoerl and Kennard (1970a, pp. 84) write: “[...] it would appear to be impossible to choose a value of k = 0 (i.e. the ridge constant) and thus to achieve a smaller mean square error without being able to assign an upper bound to β". The crucial remark of this thesis is that the linear panel data framework does provide, under appropriate conditions, this upper bound on β. This boundedness assumption can be derived from the alternative linear panel data estimators (i.e. the RE estimator, the BG estimator and the POLS estimator).Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 13 / 20
  54. 54. Shrinkage Estimation Ridge regressionThe Ordinary Ridge Estimator for FE Linear Panel Data Models We demonstrate how, by applying the Ordinary Ridge Regression (ORR) framework to our FE linear panel data context, it is possible to construct a shrinkage estimator whose MSE always dominates, under appropriate conditions, the MSE of the FE estimator. In order to fully understand this last result, recall that when evaluating the ORR estimator, Hoerl and Kennard (1970a, pp. 84) write: “[...] it would appear to be impossible to choose a value of k = 0 (i.e. the ridge constant) and thus to achieve a smaller mean square error without being able to assign an upper bound to β". The crucial remark of this thesis is that the linear panel data framework does provide, under appropriate conditions, this upper bound on β. This boundedness assumption can be derived from the alternative linear panel data estimators (i.e. the RE estimator, the BG estimator and the POLS estimator).Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 13 / 20
  55. 55. Shrinkage Estimation Ridge regressionThe Ordinary Ridge Estimator for FE Linear Panel Data Models We demonstrate how, by applying the Ordinary Ridge Regression (ORR) framework to our FE linear panel data context, it is possible to construct a shrinkage estimator whose MSE always dominates, under appropriate conditions, the MSE of the FE estimator. In order to fully understand this last result, recall that when evaluating the ORR estimator, Hoerl and Kennard (1970a, pp. 84) write: “[...] it would appear to be impossible to choose a value of k = 0 (i.e. the ridge constant) and thus to achieve a smaller mean square error without being able to assign an upper bound to β". The crucial remark of this thesis is that the linear panel data framework does provide, under appropriate conditions, this upper bound on β. This boundedness assumption can be derived from the alternative linear panel data estimators (i.e. the RE estimator, the BG estimator and the POLS estimator).Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 13 / 20
  56. 56. Shrinkage Estimation Ridge regressionThe Ordinary Ridge Estimator for FE Linear Panel Data Models We demonstrate how, by applying the Ordinary Ridge Regression (ORR) framework to our FE linear panel data context, it is possible to construct a shrinkage estimator whose MSE always dominates, under appropriate conditions, the MSE of the FE estimator. In order to fully understand this last result, recall that when evaluating the ORR estimator, Hoerl and Kennard (1970a, pp. 84) write: “[...] it would appear to be impossible to choose a value of k = 0 (i.e. the ridge constant) and thus to achieve a smaller mean square error without being able to assign an upper bound to β". The crucial remark of this thesis is that the linear panel data framework does provide, under appropriate conditions, this upper bound on β. This boundedness assumption can be derived from the alternative linear panel data estimators (i.e. the RE estimator, the BG estimator and the POLS estimator).Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 13 / 20
  57. 57. Shrinkage Estimation Ridge regressionEstimate of the Largest Biasing Factor It is easy to see in fact that, under the assumption that the impact of the regressor on the dependent variable (that is, β) has the same direction of the impact of the regressor on the unobservables fixed effects (that is, π), the RE estimator provides an upper bound on the coefficient of interest. By exploiting this prior bound on β, we can derive a consistent estimate of the largest biasing factor for the ridge estimator, k max , that ensures the existence of a shrinkage estimator whose MSE is always lower than the MSE of the FE estimator. That is, 2σ 2 0 < k < k max = 2u (6) βRaffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 14 / 20
  58. 58. Shrinkage Estimation Ridge regressionEstimate of the Largest Biasing Factor It is easy to see in fact that, under the assumption that the impact of the regressor on the dependent variable (that is, β) has the same direction of the impact of the regressor on the unobservables fixed effects (that is, π), the RE estimator provides an upper bound on the coefficient of interest. By exploiting this prior bound on β, we can derive a consistent estimate of the largest biasing factor for the ridge estimator, k max , that ensures the existence of a shrinkage estimator whose MSE is always lower than the MSE of the FE estimator. That is, 2σ 2 0 < k < k max = 2u (6) βRaffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 14 / 20
  59. 59. Shrinkage Estimation Ridge regressionEstimate of the Largest Biasing Factor It is easy to see in fact that, under the assumption that the impact of the regressor on the dependent variable (that is, β) has the same direction of the impact of the regressor on the unobservables fixed effects (that is, π), the RE estimator provides an upper bound on the coefficient of interest. By exploiting this prior bound on β, we can derive a consistent estimate of the largest biasing factor for the ridge estimator, k max , that ensures the existence of a shrinkage estimator whose MSE is always lower than the MSE of the FE estimator. That is, 2σ 2 0 < k < k max = 2u (6) βRaffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 14 / 20
  60. 60. Shrinkage Estimation Ridge regressionEstimate of the Largest Biasing Factor It is easy to see in fact that, under the assumption that the impact of the regressor on the dependent variable (that is, β) has the same direction of the impact of the regressor on the unobservables fixed effects (that is, π), the RE estimator provides an upper bound on the coefficient of interest. By exploiting this prior bound on β, we can derive a consistent estimate of the largest biasing factor for the ridge estimator, k max , that ensures the existence of a shrinkage estimator whose MSE is always lower than the MSE of the FE estimator. That is, ˆ ˆ 2s2 0 < k ≤ k max = 2 (7) ˆ βrRaffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 15 / 20
  61. 61. Shrinkage Estimation Ridge regressionEstimate of the Largest Biasing Factor It is easy to see in fact that, under the assumption that the impact of the regressor on the dependent variable (that is, β) has the same direction of the impact of the regressor on the unobservables fixed effects (that is, π), the RE estimator provides an upper bound on the coefficient of interest. By exploiting this prior bound on β, we can derive a consistent estimate of the largest biasing factor for the ridge estimator, k max , that ensures the existence of a shrinkage estimator whose MSE is always lower than the MSE of the FE estimator. That is, ˆ ˆ 2σ 2 0 < k ≤ k max < k max = 2u (8) βRaffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 16 / 20
  62. 62. Shrinkage Estimation Ridge regressionEstimate of the Largest Biasing Factor It is easy to see in fact that, under the assumption that the impact of the regressor on the dependent variable (that is, β) has the same direction of the impact of the regressor on the unobservables fixed effects (that is, π), the RE estimator provides an upper bound on the coefficient of interest. By exploiting this prior bound on β, we can derive a consistent estimate of the largest biasing factor for the ridge estimator, k max , that ensures the existence of a shrinkage estimator whose MSE is always lower than the MSE of the FE estimator. That is, ˆ ˆ 2σ 2 0 < k ≤ k max < k max = 2u (9) β ˆ Using the Slutsky’s theorem, we prove that our estimate k max is consistently lower than the largest biasing factor, k max .Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 17 / 20
  63. 63. Empirical Example Ashenfelter and Krueger (1994)The Empirical Framework of Ashenfelter and Kruger (1994)We show the importance of this result by focusing on a specific empirical application We analyze the widely cited paper of Ashenfelter and Kruger (1994) which estimates the returns of education using a sample of identical twins. In this study, the within-twin estimate of the return to schooling is surprisingly larger than the comparable cross sectional estimates, suggesting therefore a negative correlation between omitted ability and level of education. Many authors have consequently tried to link this relatively higher FE estimate to a problem of unobservable ability differences within-twin pairs. By extending the original data set of Ashenfelter and Kruger (1994), Rouse (1999) demonstrates, however, that the unusual result obtained by Ashenfelter and Kruger (1994) is due to a generic problem of sampling variability.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 18 / 20
  64. 64. Empirical Example Ashenfelter and Krueger (1994)The Empirical Framework of Ashenfelter and Kruger (1994)We show the importance of this result by focusing on a specific empirical application We analyze the widely cited paper of Ashenfelter and Kruger (1994) which estimates the returns of education using a sample of identical twins. In this study, the within-twin estimate of the return to schooling is surprisingly larger than the comparable cross sectional estimates, suggesting therefore a negative correlation between omitted ability and level of education. Many authors have consequently tried to link this relatively higher FE estimate to a problem of unobservable ability differences within-twin pairs. By extending the original data set of Ashenfelter and Kruger (1994), Rouse (1999) demonstrates, however, that the unusual result obtained by Ashenfelter and Kruger (1994) is due to a generic problem of sampling variability.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 18 / 20
  65. 65. Empirical Example Ashenfelter and Krueger (1994)The Empirical Framework of Ashenfelter and Kruger (1994)We show the importance of this result by focusing on a specific empirical application We analyze the widely cited paper of Ashenfelter and Kruger (1994) which estimates the returns of education using a sample of identical twins. In this study, the within-twin estimate of the return to schooling is surprisingly larger than the comparable cross sectional estimates, suggesting therefore a negative correlation between omitted ability and level of education. Many authors have consequently tried to link this relatively higher FE estimate to a problem of unobservable ability differences within-twin pairs. By extending the original data set of Ashenfelter and Kruger (1994), Rouse (1999) demonstrates, however, that the unusual result obtained by Ashenfelter and Kruger (1994) is due to a generic problem of sampling variability.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 18 / 20
  66. 66. Empirical Example Ashenfelter and Krueger (1994)The Empirical Framework of Ashenfelter and Kruger (1994)We show the importance of this result by focusing on a specific empirical application We analyze the widely cited paper of Ashenfelter and Kruger (1994) which estimates the returns of education using a sample of identical twins. In this study, the within-twin estimate of the return to schooling is surprisingly larger than the comparable cross sectional estimates, suggesting therefore a negative correlation between omitted ability and level of education. Many authors have consequently tried to link this relatively higher FE estimate to a problem of unobservable ability differences within-twin pairs. By extending the original data set of Ashenfelter and Kruger (1994), Rouse (1999) demonstrates, however, that the unusual result obtained by Ashenfelter and Kruger (1994) is due to a generic problem of sampling variability.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 18 / 20
  67. 67. Empirical Example Ashenfelter and Krueger (1994)The Empirical Framework of Ashenfelter and Kruger (1994)We show the importance of this result by focusing on a specific empirical application We analyze the widely cited paper of Ashenfelter and Kruger (1994) which estimates the returns of education using a sample of identical twins. In this study, the within-twin estimate of the return to schooling is surprisingly larger than the comparable cross sectional estimates, suggesting therefore a negative correlation between omitted ability and level of education. Many authors have consequently tried to link this relatively higher FE estimate to a problem of unobservable ability differences within-twin pairs. By extending the original data set of Ashenfelter and Kruger (1994), Rouse (1999) demonstrates, however, that the unusual result obtained by Ashenfelter and Kruger (1994) is due to a generic problem of sampling variability.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 18 / 20
  68. 68. Empirical Example Ashenfelter and Krueger (1994)The Empirical Framework of Ashenfelter and Kruger (1994)We show the importance of this result by focusing on a specific empirical application We analyze the widely cited paper of Ashenfelter and Kruger (1994) which estimates the returns of education using a sample of identical twins. In this study, the within-twin estimate of the return to schooling is surprisingly larger than the comparable cross sectional estimates, suggesting therefore a negative correlation between omitted ability and level of education. Many authors have consequently tried to link this relatively higher FE estimate to a problem of unobservable ability differences within-twin pairs. By extending the original data set of Ashenfelter and Kruger (1994), Rouse (1999) demonstrates, however, that the unusual result obtained by Ashenfelter and Kruger (1994) is due to a generic problem of sampling variability.Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 18 / 20
  69. 69. Empirical Example Ashenfelter and Krueger (1994)Shrinkage Estimation in Ashenfelter and Kruger (1994)Starting from this fundamental conclusion of Rouse (1999), we provide the followingremarks We argue that what Rouse (1999) generically describes as a sampling error problem is actually a problem of LLV, that is, twins tend to report identical schooling levels (in particular, half of the twins in the sample of Ashenfelter and Kruger (1994) report to have attained exactly the same level of education). As formally shown in the first part of this thesis, a natural way to overcome problems of LLV is to increase the sample size. This is exactly what Rouse (1999) proposes. She collects more data in order to counterbalance the fact that identical twins tend to report similar educational levels. Clearly, having the possibility to collect additional data so to increase significantly the original sample size is something quite unusual in empirical works. Therefore, the crucial question for us is the following: given the original sample of Ashenfelter and Krueger (1994), is it possible to obtain an estimate of the returns of education that is more reliable than the usual FE estimate?Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 19 / 20
  70. 70. Empirical Example Ashenfelter and Krueger (1994)Shrinkage Estimation in Ashenfelter and Kruger (1994)Starting from this fundamental conclusion of Rouse (1999), we provide the followingremarks We argue that what Rouse (1999) generically describes as a sampling error problem is actually a problem of LLV, that is, twins tend to report identical schooling levels (in particular, half of the twins in the sample of Ashenfelter and Kruger (1994) report to have attained exactly the same level of education). As formally shown in the first part of this thesis, a natural way to overcome problems of LLV is to increase the sample size. This is exactly what Rouse (1999) proposes. She collects more data in order to counterbalance the fact that identical twins tend to report similar educational levels. Clearly, having the possibility to collect additional data so to increase significantly the original sample size is something quite unusual in empirical works. Therefore, the crucial question for us is the following: given the original sample of Ashenfelter and Krueger (1994), is it possible to obtain an estimate of the returns of education that is more reliable than the usual FE estimate?Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 19 / 20
  71. 71. Empirical Example Ashenfelter and Krueger (1994)Shrinkage Estimation in Ashenfelter and Kruger (1994)Starting from this fundamental conclusion of Rouse (1999), we provide the followingremarks We argue that what Rouse (1999) generically describes as a sampling error problem is actually a problem of LLV, that is, twins tend to report identical schooling levels (in particular, half of the twins in the sample of Ashenfelter and Kruger (1994) report to have attained exactly the same level of education). As formally shown in the first part of this thesis, a natural way to overcome problems of LLV is to increase the sample size. This is exactly what Rouse (1999) proposes. She collects more data in order to counterbalance the fact that identical twins tend to report similar educational levels. Clearly, having the possibility to collect additional data so to increase significantly the original sample size is something quite unusual in empirical works. Therefore, the crucial question for us is the following: given the original sample of Ashenfelter and Krueger (1994), is it possible to obtain an estimate of the returns of education that is more reliable than the usual FE estimate?Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 19 / 20
  72. 72. Empirical Example Ashenfelter and Krueger (1994)Shrinkage Estimation in Ashenfelter and Kruger (1994)Starting from this fundamental conclusion of Rouse (1999), we provide the followingremarks We argue that what Rouse (1999) generically describes as a sampling error problem is actually a problem of LLV, that is, twins tend to report identical schooling levels (in particular, half of the twins in the sample of Ashenfelter and Kruger (1994) report to have attained exactly the same level of education). As formally shown in the first part of this thesis, a natural way to overcome problems of LLV is to increase the sample size. This is exactly what Rouse (1999) proposes. She collects more data in order to counterbalance the fact that identical twins tend to report similar educational levels. Clearly, having the possibility to collect additional data so to increase significantly the original sample size is something quite unusual in empirical works. Therefore, the crucial question for us is the following: given the original sample of Ashenfelter and Krueger (1994), is it possible to obtain an estimate of the returns of education that is more reliable than the usual FE estimate?Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 19 / 20
  73. 73. Empirical Example Ashenfelter and Krueger (1994)Shrinkage Estimation in Ashenfelter and Kruger (1994)Starting from this fundamental conclusion of Rouse (1999), we provide the followingremarks We argue that what Rouse (1999) generically describes as a sampling error problem is actually a problem of LLV, that is, twins tend to report identical schooling levels (in particular, half of the twins in the sample of Ashenfelter and Kruger (1994) report to have attained exactly the same level of education). As formally shown in the first part of this thesis, a natural way to overcome problems of LLV is to increase the sample size. This is exactly what Rouse (1999) proposes. She collects more data in order to counterbalance the fact that identical twins tend to report similar educational levels. Clearly, having the possibility to collect additional data so to increase significantly the original sample size is something quite unusual in empirical works. Therefore, the crucial question for us is the following: given the original sample of Ashenfelter and Krueger (1994), is it possible to obtain an estimate of the returns of education that is more reliable than the usual FE estimate?Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 19 / 20
  74. 74. Empirical Example Ashenfelter and Krueger (1994)Shrinkage Estimation in Ashenfelter and Kruger (1994)Starting from this fundamental conclusion of Rouse (1999), we provide the followingremarks We argue that what Rouse (1999) generically describes as a sampling error problem is actually a problem of LLV, that is, twins tend to report identical schooling levels (in particular, half of the twins in the sample of Ashenfelter and Kruger (1994) report to have attained exactly the same level of education). As formally shown in the first part of this thesis, a natural way to overcome problems of LLV is to increase the sample size. This is exactly what Rouse (1999) proposes. She collects more data in order to counterbalance the fact that identical twins tend to report similar educational levels. Clearly, having the possibility to collect additional data so to increase significantly the original sample size is something quite unusual in empirical works. Therefore, the crucial question for us is the following: given the original sample of Ashenfelter and Krueger (1994), is it possible to obtain an estimate of the returns of education that is more reliable than the usual FE estimate?Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 19 / 20
  75. 75. Empirical Example Ashenfelter and Krueger (1994)Shrinkage Estimation in Ashenfelter and Kruger (1994)Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 20 / 20

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