This thesis evaluates fixed effects linear panel data models, focusing on cases of low longitudinal variation of explanatory variables, which complicates the inference processes. It aims to formalize the impact of low longitudinal variation on the fixed effects estimator's asymptotic distribution and proposes a shrinkage estimator with reduced variance for better reliability in parameter estimation. The study highlights that while the fixed effects estimator remains consistent, the model's asymptotic properties can become complex due to low longitudinal variation.

1. Shrinkage Estimation of Linear
Panel Data Models Under
Nearly Singular Design
Raffaele Saggio
University of Tor Vergata
Graduation Session
Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 1 / 20

2. Introduction Motivation
This Thesis
We evaluate fixed effects (FE) linear panel data models where the sample within
variation of the explanatory variable is significantly small.
This type of outcome is often observed in microeconometrics and it imposes serious
challenges when inferencing with FE models.
The main problem is that - whenever the longitudinal variation is small - the FE
estimator may provide very little information about the parameter of interest no matter
how 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. Introduction Motivation
This Thesis
We evaluate fixed effects (FE) linear panel data models where the sample within
variation of the explanatory variable is significantly small.
This type of outcome is often observed in microeconometrics and it imposes serious
challenges when inferencing with FE models.
The main problem is that - whenever the longitudinal variation is small - the FE
estimator may provide very little information about the parameter of interest no matter
how 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. Introduction Motivation
This Thesis
We evaluate fixed effects (FE) linear panel data models where the sample within
variation of the explanatory variable is significantly small.
This type of outcome is often observed in microeconometrics and it imposes serious
challenges when inferencing with FE models.
The main problem is that - whenever the longitudinal variation is small - the FE
estimator may provide very little information about the parameter of interest no matter
how 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. Introduction Motivation
This Thesis
We evaluate fixed effects (FE) linear panel data models where the sample within
variation of the explanatory variable is significantly small.
This type of outcome is often observed in microeconometrics and it imposes serious
challenges when inferencing with FE models.
The main problem is that - whenever the longitudinal variation is small - the FE
estimator may provide very little information about the parameter of interest no matter
how 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. Introduction Motivation
This Thesis
We evaluate fixed effects (FE) linear panel data models where the sample within
variation of the explanatory variable is significantly small.
This type of outcome is often observed in microeconometrics and it imposes serious
challenges when inferencing with FE models.
The main problem is that - whenever the longitudinal variation is small - the FE
estimator may provide very little information about the parameter of interest no matter
how 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. Introduction Motivation
This Thesis
We evaluate fixed effects (FE) linear panel data models where the sample within
variation of the explanatory variable is significantly small.
This type of outcome is often observed in microeconometrics and it imposes serious
challenges when inferencing with FE models.
The main problem is that - whenever the longitudinal variation is small - the FE
estimator may provide very little information about the parameter of interest no matter
how 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. Introduction Contributions
Main Findings
Relative 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. Introduction Contributions
Main Findings
Relative 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. Introduction Contributions
Main Findings
Relative 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. Introduction Contributions
Main Findings
Relative 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. Introduction Contributions
Main Findings
Relative 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. Introduction Contributions
Main Findings
Relative 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. Introduction Contributions
Main Findings
Relative 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. Introduction Contributions
Main Findings
Relative 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. Introduction Contributions
Main Findings
Relative 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. Introduction Contributions
Main Findings
Relative 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. The LLV Problem The Model
Asymptotic properties of the FE estimator
Our results are based on a standard FE model with correlated, time invariant, fixed
effects. Without loss of generality, we consider the case when there is only one
regressor.
We are interested in evaluating whether the fact of having a regressor that shows high
degrees 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 p
rate 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. The LLV Problem The Model
Asymptotic properties of the FE estimator
Our results are based on a standard FE model with correlated, time invariant, fixed
effects. Without loss of generality, we consider the case when there is only one
regressor.
We are interested in evaluating whether the fact of having a regressor that shows high
degrees 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 p
rate 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. The LLV Problem The Model
Asymptotic properties of the FE estimator
Our results are based on a standard FE model with correlated, time invariant, fixed
effects. Without loss of generality, we consider the case when there is only one
regressor.
We are interested in evaluating whether the fact of having a regressor that shows high
degrees 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 p
rate 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. The LLV Problem The Model
Asymptotic properties of the FE estimator
Our results are based on a standard FE model with correlated, time invariant, fixed
effects. Without loss of generality, we consider the case when there is only one
regressor.
We are interested in evaluating whether the fact of having a regressor that shows high
degrees 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 p
rate 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. The LLV Problem The Model
Asymptotic properties of the FE estimator
Our results are based on a standard FE model with correlated, time invariant, fixed
effects. Without loss of generality, we consider the case when there is only one
regressor.
We are interested in evaluating whether the fact of having a regressor that shows high
degrees 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 p
rate 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. The LLV Problem The Model
Asymptotic properties of the FE estimator
Our results are based on a standard FE model with correlated, time invariant, fixed
effects. Without loss of generality, we consider the case when there is only one
regressor.
We are interested in evaluating whether the fact of having a regressor that shows high
degrees 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 p
rate 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. The LLV Problem Convergence
The Problem of Convergence
The 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=1
where γiT = T −1
ˆ
P ¯
− Xi )2 and E(ˆiT ) = γiT .
t (Xit γ
Suppose 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. The LLV Problem Convergence
The Problem of Convergence
The 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=1
where γiT = T −1
ˆ
P ¯
− Xi )2 and E(ˆiT ) = γiT .
t (Xit γ
Suppose 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. The LLV Problem Convergence
The Problem of Convergence
The 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=1
where γiT = T −1
ˆ
P ¯
− Xi )2 and E(ˆiT ) = γiT .
t (Xit γ
Suppose 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. The LLV Problem Convergence
The Problem of Convergence
The 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=1
where γiT = T −1
ˆ
P ¯
− Xi )2 and E(ˆiT ) = γiT .
t (Xit γ
Suppose 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. The LLV Problem Convergence
The Problem of Convergence
The 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=1
where γiT = T −1
ˆ
P ¯
− Xi )2 and E(ˆiT ) = γiT .
t (Xit γ
Suppose 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. The LLV Problem Convergence
The Problem of Convergence
The 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=1
where γiT = T −1
ˆ
P ¯
− Xi )2 and E(ˆiT ) = γiT .
t (Xit γ
Suppose 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. The LLV Problem Example
A 2x2 Example
Take T = 2 and assume that Xit is a dummy variable that indicates whether an
individual 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=1
where 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. The LLV Problem Example
A 2x2 Example
Take T = 2 and assume that Xit is a dummy variable that indicates whether an
individual 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=1
where 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. The LLV Problem Example
A 2x2 Example
Take T = 2 and assume that Xit is a dummy variable that indicates whether an
individual 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=1
where 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. The LLV Problem Example
A 2x2 Example
Take T = 2 and assume that Xit is a dummy variable that indicates whether an
individual 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=1
where 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. The LLV Problem Example
A 2x2 Example
Take T = 2 and assume that Xit is a dummy variable that indicates whether an
individual 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=1
where 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. The LLV Problem Example
A 2x2 Example
Take T = 2 and assume that Xit is a dummy variable that indicates whether an
individual 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=1
where 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. The LLV Problem Example
A 2x2 Example
Take T = 2 and assume that Xit is a dummy variable that indicates whether an
individual 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=1
where 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. Asymptotics under the LLV Problem Nearly Singular Design
The Nearly Singular Design
By 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 distribution
of the FE estimator under the LLV problem.
In fact, assuming that
n
1 X ∗
lim γiT = γT > 0 (3)
n→∞ n1−κ
i=1
where 0 ≤ κ < 1, it is possible to prove the following Theorem
Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 8 / 20

38. Asymptotics under the LLV Problem Nearly Singular Design
The Nearly Singular Design
By 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 distribution
of the FE estimator under the LLV problem.
In fact, assuming that
n
1 X ∗
lim γiT = γT > 0 (3)
n→∞ n1−κ
i=1
where 0 ≤ κ < 1, it is possible to prove the following Theorem
Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 8 / 20

39. Asymptotics under the LLV Problem Nearly Singular Design
The Nearly Singular Design
By 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 distribution
of the FE estimator under the LLV problem.
In fact, assuming that
n
1 X ∗
lim γiT = γT > 0 (3)
n→∞ n1−κ
i=1
where 0 ≤ κ < 1, it is possible to prove the following Theorem
Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 8 / 20

40. Asymptotics under the LLV Problem Nearly Singular Design
Theorem
Under Assumption (3), via the Lindeberg-Feller Central Limit Theorem for unequal
variances, as n → ∞ with T fixed
“ ”
ˆ d ˆ
cn (β − β) → N 0; AVT (β) . (4)
1−κ
where 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. Asymptotics under the LLV Problem Inference under the LLV problem
Inference 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. Asymptotics under the LLV Problem Inference under the LLV problem
Inference 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. Asymptotics under the LLV Problem Inference under the LLV problem
Inference 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. Asymptotics under the LLV Problem Inference under the LLV problem
Inference 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. Shrinkage Estimation RE estimator vs. FE estimator
Shrinkage 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. Shrinkage Estimation RE estimator vs. FE estimator
Shrinkage 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. Shrinkage Estimation RE estimator vs. FE estimator
Shrinkage 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. Shrinkage Estimation RE estimator vs. FE estimator
Shrinkage 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. Shrinkage Estimation RE estimator vs. FE estimator
Shrinkage 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. Shrinkage Estimation RE estimator vs. FE estimator
Shrinkage 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. Shrinkage Estimation RE estimator vs. FE estimator
Shrinkage 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. Shrinkage Estimation RE estimator vs. FE estimator
Shrinkage 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. Shrinkage Estimation Ridge regression
The 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. Shrinkage Estimation Ridge regression
The 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. Shrinkage Estimation Ridge regression
The 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. Shrinkage Estimation Ridge regression
The 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. Shrinkage Estimation Ridge regression
Estimate 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)
β
Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 14 / 20

58. Shrinkage Estimation Ridge regression
Estimate 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)
β
Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 14 / 20

59. Shrinkage Estimation Ridge regression
Estimate 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)
β
Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 14 / 20

60. Shrinkage Estimation Ridge regression
Estimate 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
βr
Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 15 / 20

61. Shrinkage Estimation Ridge regression
Estimate 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)
β
Raffaele Saggio (University of Tor Vergata) Master of Science in Economics Graduate Session - 29th of September 16 / 20

62. Shrinkage Estimation Ridge regression
Estimate 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. 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. 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. 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. 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. 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. 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. Empirical Example Ashenfelter and Krueger (1994)
Shrinkage Estimation in Ashenfelter and Kruger (1994)
Starting from this fundamental conclusion of Rouse (1999), we provide the following
remarks
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. Empirical Example Ashenfelter and Krueger (1994)
Shrinkage Estimation in Ashenfelter and Kruger (1994)
Starting from this fundamental conclusion of Rouse (1999), we provide the following
remarks
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. Empirical Example Ashenfelter and Krueger (1994)
Shrinkage Estimation in Ashenfelter and Kruger (1994)
Starting from this fundamental conclusion of Rouse (1999), we provide the following
remarks
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. Empirical Example Ashenfelter and Krueger (1994)
Shrinkage Estimation in Ashenfelter and Kruger (1994)
Starting from this fundamental conclusion of Rouse (1999), we provide the following
remarks
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. Empirical Example Ashenfelter and Krueger (1994)
Shrinkage Estimation in Ashenfelter and Kruger (1994)
Starting from this fundamental conclusion of Rouse (1999), we provide the following
remarks
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. Empirical Example Ashenfelter and Krueger (1994)
Shrinkage Estimation in Ashenfelter and Kruger (1994)
Starting from this fundamental conclusion of Rouse (1999), we provide the following
remarks
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. 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