Shrinkage Estimation of Linear Panel Data Models Under Nearly Singular Design

Shrinkage Estimation of LinearPanel Data Models UnderNearly 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

Introduction Motivation

This Thesis

We 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 issmall affect the standard asymptotic distribution of the FE estimator assumingthat the cross sectional dimension, n, is large and the longitudinal dimension, T , issmall.

To propose an alternative way of estimating the parameter of interest in thepresence of high degrees of Low Longitudinal Variation (LLV henceforth) withcorrelated, time invariant, individual effects.



This Thesis









This Thesis









This Thesis









This Thesis









This Thesis








Introduction Contributions

Main Findings

Relative to the first objective:

We notice that when a variable is exhibiting high degrees of LLV, the sample withinvariation may become indistinguishable from 0 as n→∞ with T fixed. This typeof asymptotic behavior is defined in the thesis as the “The LLV Problem”.

By relating the LLV problem with the nearly singular design, we provide atheorem that shows what is the correct asymptotic distribution of the FE estimatorunder the LLV problem.

The most important consequence of this result is that now the rate of convergenceof the FE estimator is unknown as it crucially depends on the severity of the LLVproblem.

By extending the subsampling methodology for the linear panel data context, wepropose a method to estimate this unknown rate of convergence.



Main Findings








Main Findings








Main Findings








Main Findings








Main Findings

Relative to the second objective:

We evaluate shrinkage estimation techniques for the FE linear panel data modelthat allow to trade the unbiasedness of the FE estimator with an estimator that hassmaller variance.

More specifically, we demonstrate how it is possible to obtain a shrinkageestimator whose Mean Square Error always dominates, under appropriateconditions, the Mean Square Error of the FE estimator.

We document the importance of this result with a specific empirical exampledrawn from the twin study of Ashenfelter and Krueger (1994).

In particular, we show that our proposed shrinkage estimator is more reliable thanthe FE estimator of Ashenfelter and Krueger (1994) in estimating the returns ofeducation.



Main Findings








Main Findings








Main Findings








Main Findings







The LLV Problem The Model

Asymptotic properties of the FE estimator

Our 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 convergencerate for the sample longitudinal variation in order to have β̂

p→ β.






Consistency.




p→ β.






Consistency.




p→ β.






Consistency.




p→ β.






Consistency.




p→ β.






Consistency.




p→ β.


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

plim n−1nX

i=1

γ̂iT = limn→∞

n−1nX

i=1

γiT = γT > 0, (1)

where γ̂iT = T−1Pt (Xit − X̄i )

2 and E(γ̂iT ) = γiT .

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 variationmay 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 LLVproblem.






plim n−1nX

i=1

γ̂iT = limn→∞

n−1nX

i=1

γiT = γT > 0, (1)











plim n−1nX

i=1

γ̂iT = limn→∞

n−1nX

i=1

γiT = γT > 0, (1)











plim n−1nX

i=1

γ̂iT = limn→∞

n−1nX

i=1

γiT = γT > 0, (1)











plim n−1nX

i=1

γ̂iT = limn→∞

n−1nX

i=1

γiT = γT > 0, (1)











plim n−1nX

i=1

γ̂iT = limn→∞

n−1nX

i=1

γiT = γT > 0, (1)







The LLV Problem Example

A 2x2 Example

Take 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

1n

nXi=1

γ̂i2 =mn

4n, (2)

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 convergingto 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 asymptoticallyincreases too slowly relative to the total increase in the cross sectionaldimension, n.



A 2x2 Example



1n

nXi=1

γ̂i2 =mn

4n, (2)







A 2x2 Example



1n

nXi=1

γ̂i2 =mn

4n, (2)







A 2x2 Example



1n

nXi=1

γ̂i2 =mn

4n, (2)







A 2x2 Example



1n

nXi=1

γ̂i2 =mn

4n, (2)







A 2x2 Example



1n

nXi=1

γ̂i2 =mn

4n, (2)







A 2x2 Example



1n

nXi=1

γ̂i2 =mn

4n, (2)






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 distributionof the FE estimator under the LLV problem.

In fact, assuming that

limn→∞

1n1−κ

nXi=1

γiT = γ∗T > 0 (3)

where 0 ≤ κ < 1, it is possible to prove the following Theorem






limn→∞

1n1−κ

nXi=1

γiT = γ∗T > 0 (3)







limn→∞

1n1−κ

nXi=1

γiT = γ∗T > 0 (3)




Theorem

Under Assumption (3), via the Lindeberg-Feller Central Limit Theorem for unequalvariances, as n→∞ with T fixed

cn(β̂ − β)d→ N

“0; AVT (β̂)

”. (4)

where cn = n1−κ

2 and AVT (β̂) is the asymptotic variance of the FE estimator.


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 constructinference, practitioners need now to estimate two objects: the asymptoticvariance of the FE estimator, AVT (β̂), and the unknown rate of convergence of theFE 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 consistentestimate for the unknown rate of convergence.

This type of procedure provides us with a diagnostic tool to evaluate the severityof the LLV problem.

Using the subsampling methodology, in fact, we are able to estimate and to testwhether the LLV problem actually decreases the rate of convergence of the FEestimator (i.e. κ > 0) or if the standard asymptotic result applies (i.e. κ = 0).























Shrinkage Estimation RE estimator vs. FE estimator

Shrinkage Estimation

We now turn our attention to the second objective of the thesis: to discussalternative ways of estimating the parameter of interest when the explanatoryvariable is characterized by LLV and the underlined model implies the presence ofcorrelated, time invariant, unobservables effects.

Under this framework, it is well-known that the FE estimator is unbiased but, dueto the LLV of the explanatory variable, it has also a large variance.

Consequently, we start to discuss estimation procedures that allow to trade theunbiasedness of the FE estimator with an estimator that has smaller variance.

We first highlight how traditional linear panel data estimators, such as the RandomEffect (RE) estimator, can theoretically have smaller MSE than the FE estimator,especially in contexts that involve LLV.

























The problem in using the RE estimator in place of the FE estimator is that thedifference between the MSEs of these estimators crucially depends, along with thewithin/between variation ratio, φ, on the squared bias of the RE estimator, π2.

∆MSE = MSE(β̂)−MSE(β̂r ) =σ2

u θ̂

φ“φ+

pθ̂” − θ̂2π2“

θ̂ + φ”2 . (5)

Since this bias cannot be precisely estimated in situations of LLV (see Table 2 inthe thesis), equation (5) is not able to suggest to practitioners whether the REestimator has lower MSE than the FE estimator.

Therefore, we begin to evaluate a particular shrinkage estimation technique thatcan guarantee us dominance in MSE.






u θ̂

φ“φ+

pθ̂” − θ̂2π2“

θ̂ + φ”2 . (5)








u θ̂

φ“φ+

pθ̂” − θ̂2π2“

θ̂ + φ”2 . (5)








u θ̂

φ“φ+

pθ̂” − θ̂2π2“

θ̂ + φ”2 . (5)




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 ashrinkage estimator whose MSE always dominates, under appropriateconditions, the MSE of the FE estimator.

In order to fully understand this last result, recall that when evaluating the ORRestimator, Hoerl and Kennard (1970a, pp. 84) write: “[...] it would appear to beimpossible to choose a value of k 6= 0 (i.e. the ridge constant) and thus to achievea 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 doesprovide, under appropriate conditions, this upper bound on β.

This boundedness assumption can be derived from the alternative linear paneldata estimators (i.e. the RE estimator, the BG estimator and the POLS estimator).
























Estimate of the Largest Biasing Factor

It is easy to see in fact that, under the assumption that the impact of the regressoron the dependent variable (that is, β) has the same direction of the impact of theregressor on the unobservables fixed effects (that is, π), the RE estimator providesan upper bound on the coefficient of interest.

By exploiting this prior bound on β, we can derive a consistent estimate of thelargest biasing factor for the ridge estimator, kmax , that ensures the existence of ashrinkage estimator whose MSE is always lower than the MSE of the FEestimator. That is,

0 < k < kmax =2σ2

u

β2 (6)






0 < k < kmax =2σ2

u

β2 (6)






0 < k < kmax =2σ2

u

β2 (6)






0 < k̂ ≤ k̂max =2s2

β̂2r

(7)






0 < k̂ ≤ k̂max < kmax =2σ2

u

β2 (8)






0 < k̂ ≤ k̂max < kmax =2σ2

u

β2 (9)

Using the Slutsky’s theorem, we prove that our estimate k̂max is consistentlylower than the largest biasing factor, kmax .


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) whichestimates the returns of education using a sample of identical twins.

In this study, the within-twin estimate of the return to schooling is surprisinglylarger than the comparable cross sectional estimates, suggesting therefore anegative correlation between omitted ability and level of education.

Many authors have consequently tried to link this relatively higher FE estimate to aproblem 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 andKruger (1994) is due to a generic problem of sampling variability.











































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 errorproblem is actually a problem of LLV, that is, twins tend to report identicalschooling levels (in particular, half of the twins in the sample of Ashenfelter andKruger (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 overcomeproblems 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 thatidentical twins tend to report similar educational levels.

Clearly, having the possibility to collect additional data so to increase significantlythe original sample size is something quite unusual in empirical works.

Therefore, the crucial question for us is the following: given the original sample ofAshenfelter and Krueger (1994), is it possible to obtain an estimate of the returnsof education that is more reliable than the usual FE estimate?













































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Shrinkage Estimation of Linear Panel Data Models Under Nearly Singular Design

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