panel data models with heterogeneous marginal effects

Panel Data Models with Heterogeneous Marginal Effects

Kirill Evdokimov

Princeton University

The NES 20th Anniversary ConferenceDecember 14, 2012

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 1 / 16

Based on two papers of mine:

“Identification and Estimation of a Nonparametric Panel Data Modelwith Unobserved Heterogeneity”

“Nonparametric Quasi-Differencing with Applications”

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 2 / 16

Model

Yit = m (Xit , αi ) + Uit ; i = 1, . . . , n; t = 1, . . . ,T

i - individual, t - time period

Observed: Yit and XitUnobserved: αi - scalar heterogeneity, Uit - idiosyncratic shocks

function m (x , α) is not known

∂m (Xit , αi ) /∂x depends on αi ⇒ heterogeneous marginal effects

will ID m(x , α) and Fαi (α|Xit )distributions of the outcomes, counterfactuals

"what percentage of the treated would be better/worse off?"

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 3 / 16

Model

Yit = m (Xit , αi ) + Uit ; i = 1, . . . , n; t = 1, . . . ,T

i - individual, t - time period

Observed: Yit and XitUnobserved: αi - scalar heterogeneity, Uit - idiosyncratic shocks

function m (x , α) is not known

∂m (Xit , αi ) /∂x depends on αi ⇒ heterogeneous marginal effects

will ID m(x , α) and Fαi (α|Xit )distributions of the outcomes, counterfactuals

"what percentage of the treated would be better/worse off?"

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 3 / 16

Model

Yit = m (Xit , αi ) + Uit ; i = 1, . . . , n; t = 1, . . . ,T

i - individual, t - time period

Observed: Yit and XitUnobserved: αi - scalar heterogeneity, Uit - idiosyncratic shocks

function m (x , α) is not known

∂m (Xit , αi ) /∂x depends on αi ⇒ heterogeneous marginal effects

will ID m(x , α) and Fαi (α|Xit )distributions of the outcomes, counterfactuals

"what percentage of the treated would be better/worse off?"

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 3 / 16

Model

Yit = m (Xit , αi ) + Uit ; i = 1, . . . , n; t = 1, . . . ,T

i - individual, t - time period

Observed: Yit and XitUnobserved: αi - scalar heterogeneity, Uit - idiosyncratic shocks

function m (x , α) is not known

Some assumptions:

E [Uit |Xit ] = 0 and Uit ⊥ αi |Xitno restrictions on Fαi (α|Xit = x) ⇒ fixed effects

no parametric assumptions; T = 2 is suffi cient

Yit must be cont.; also, no lagged dep. variables

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 4 / 16

Motivation

Union Wage Premium

Yit = m (Xit , αi ) + Uit

Yit is (log)-wage,

Xit is 1 if member of a union, 0 o/w,

αi is skill and Uit is luck

m(1,αi)−m(0,αi) is union wage premium

When m (Xit , αi ) = Xitβ+ αi , it becomes

m (1, αi )−m (0, αi ) = β = const !

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 5 / 16

Motivation

Union Wage Premium

Yit = m (Xit , αi ) + Uit

Yit is (log)-wage,

Xit is 1 if member of a union, 0 o/w,

αi is skill and Uit is luck

m(1,αi)−m(0,αi) is union wage premium may be NOT monotone in αi

When m (Xit , αi ) = Xitβ+ αi , it becomes

m (1, αi )−m (0, αi ) = β = const !

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 5 / 16

Motivation (Yit = m (Xit , αi) + Uit)

Life Cycle Models of Consumption and Labor SupplyHeckman and MaCurdy (1980), MaCurdy (1981). Under theassumptions of these papers and ρ = r :

Cit = C (Wit ,λi )︸ ︷︷ ︸unknown fn.

+ Uit︸︷︷︸meas .err .

Cit and Wit - (log) consumption and hourly wageλi - Lagrange multiplier

Parametric specification of utility ⇒ additively separable λiInstead note that:

C (w ,λ) is unknownC (w ,λ) is strictly increasing in λ(unobserved) λi is likely to be correlated with Witcan handle this nonparametrically

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 6 / 16

Motivation (Yit = m (Xit , αi) + Uit)

Life Cycle Models of Consumption and Labor SupplyHeckman and MaCurdy (1980), MaCurdy (1981). Under theassumptions of these papers and ρ = r :

Cit = C (Wit ,λi )︸ ︷︷ ︸unknown fn.

+ Uit︸︷︷︸meas .err .

Cit and Wit - (log) consumption and hourly wageλi - Lagrange multiplier

Parametric specification of utility ⇒ additively separable λiInstead note that:

C (w ,λ) is unknownC (w ,λ) is strictly increasing in λ(unobserved) λi is likely to be correlated with Witcan handle this nonparametrically

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 6 / 16

Motivation (Yit = m (Xit , αi) + Uit)

Life Cycle Models of Consumption and Labor SupplyHeckman and MaCurdy (1980), MaCurdy (1981). Under theassumptions of these papers and ρ = r :

Cit = C (Wit ,λi )︸ ︷︷ ︸unknown fn.

+ Uit︸︷︷︸meas .err .

Cit and Wit - (log) consumption and hourly wageλi - Lagrange multiplier

Parametric specification of utility ⇒ additively separable λiInstead note that:

C (w ,λ) is unknownC (w ,λ) is strictly increasing in λ(unobserved) λi is likely to be correlated with Witcan handle this nonparametrically

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 6 / 16

Motivation (Yit = m (Xit , αi) + Uit)

Life Cycle Models of Consumption and Labor SupplyHeckman and MaCurdy (1980), MaCurdy (1981). Under theassumptions of these papers and ρ = r :

Cit = C (Wit ,λi )︸ ︷︷ ︸unknown fn.

+ Uit︸︷︷︸meas .err .

Cit and Wit - (log) consumption and hourly wageλi - Lagrange multiplier

Parametric specification of utility ⇒ additively separable λiInstead note that:

C (w ,λ) is unknownC (w ,λ) is strictly increasing in λ(unobserved) λi is likely to be correlated with Witcan handle this nonparametrically

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 6 / 16

Identification, Yit = m (Xit , αi) + Uit

1 Subpopulation Xi1 = Xi2 = x : thenm(Xi1, αi ) = m(Xi2, αi ) = m(x , αi )

Yi1 = m(x , αi ) + Ui1Yi2 = m(x , αi ) + Ui2

Yi1 − Yi2 = Ui1 − Ui2 =⇒ Obtain distr L (Uit |Xit = x)2 Deconvolve: Yit = m(Xit , αi )+Uit , αi ⊥Uit given Xit = x , thus

L (Yit |Xit = x)︸ ︷︷ ︸from data

= L (m (x , αi ) |Xit = x) ∗L (Uit |Xit = x)︸ ︷︷ ︸from step 1

,

where "∗" means "convolution" and "L" means "probability Law".Then obtain L (m (x , αi ) |Xit = x) by deconvolution

3 Use L (m (x , αi ) |Xit = x), eg. quantiles Qm(x ,αi )|Xit (q|x) [fornonparametric between- or within- variation] more

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16

Identification, Yit = m (Xit , αi) + Uit

1 Subpopulation Xi1 = Xi2 = x : thenm(Xi1, αi ) = m(Xi2, αi ) = m(x , αi )

Yi1 = m(x , αi ) + Ui1Yi2 = m(x , αi ) + Ui2

Yi1 − Yi2 = Ui1 − Ui2 =⇒ Obtain distr L (Uit |Xit = x)2 Deconvolve: Yit = m(Xit , αi )+Uit , αi ⊥Uit given Xit = x , thus

L (Yit |Xit = x)︸ ︷︷ ︸from data

= L (m (x , αi ) |Xit = x) ∗L (Uit |Xit = x)︸ ︷︷ ︸from step 1

,

where "∗" means "convolution" and "L" means "probability Law".Then obtain L (m (x , αi ) |Xit = x) by deconvolution

3 Use L (m (x , αi ) |Xit = x), eg. quantiles Qm(x ,αi )|Xit (q|x) [fornonparametric between- or within- variation] more

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16

Identification, Yit = m (Xit , αi) + Uit

1 Subpopulation Xi1 = Xi2 = x : thenm(Xi1, αi ) = m(Xi2, αi ) = m(x , αi )

Yi1 = m(x , αi ) + Ui1Yi2 = m(x , αi ) + Ui2

Yi1 − Yi2 = Ui1 − Ui2 =⇒ Obtain distr L (Uit |Xit = x)2 Deconvolve: Yit = m(Xit , αi )+Uit , αi ⊥Uit given Xit = x , thus

L (Yit |Xit = x)︸ ︷︷ ︸from data

= L (m (x , αi ) |Xit = x) ∗L (Uit |Xit = x)︸ ︷︷ ︸from step 1

,

where "∗" means "convolution" and "L" means "probability Law".Then obtain L (m (x , αi ) |Xit = x) by deconvolution

3 Use L (m (x , αi ) |Xit = x), eg. quantiles Qm(x ,αi )|Xit (q|x) [fornonparametric between- or within- variation] more

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16

Identification, Yit = m (Xit , αi) + Uit

1 Subpopulation Xi1 = Xi2 = x : thenm(Xi1, αi ) = m(Xi2, αi ) = m(x , αi )

Yi1 = m(x , αi ) + Ui1Yi2 = m(x , αi ) + Ui2

Yi1 − Yi2 = Ui1 − Ui2 =⇒ Obtain distr L (Uit |Xit = x)2 Deconvolve: Yit = m(Xit , αi )+Uit , αi ⊥Uit given Xit = x , thus

L (Yit |Xit = x)︸ ︷︷ ︸from data

= L (m (x , αi ) |Xit = x) ∗L (Uit |Xit = x)︸ ︷︷ ︸from step 1

,

where "∗" means "convolution" and "L" means "probability Law".Then obtain L (m (x , αi ) |Xit = x) by deconvolution

3 Use L (m (x , αi ) |Xit = x), eg. quantiles Qm(x ,αi )|Xit (q|x) [fornonparametric between- or within- variation] more

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16

Identification, Yit = m (Xit , αi) + Uit

1 Subpopulation Xi1 = Xi2 = x : thenm(Xi1, αi ) = m(Xi2, αi ) = m(x , αi )

Yi1 = m(x , αi ) + Ui1Yi2 = m(x , αi ) + Ui2

Yi1 − Yi2 = Ui1 − Ui2 =⇒ Obtain distr L (Uit |Xit = x)2 Deconvolve: Yit = m(Xit , αi )+Uit , αi ⊥Uit given Xit = x , thus

L (Yit |Xit = x)︸ ︷︷ ︸from data

= L (m (x , αi ) |Xit = x) ∗L (Uit |Xit = x)︸ ︷︷ ︸from step 1

,

where "∗" means "convolution" and "L" means "probability Law".Then obtain L (m (x , αi ) |Xit = x) by deconvolution

3 Use L (m (x , αi ) |Xit = x), eg. quantiles Qm(x ,αi )|Xit (q|x) [fornonparametric between- or within- variation] more

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16

Identification, Yit = m (Xit , αi) + Uit

1 Subpopulation Xi1 = Xi2 = x : thenm(Xi1, αi ) = m(Xi2, αi ) = m(x , αi )

Yi1 = m(x , αi ) + Ui1Yi2 = m(x , αi ) + Ui2

Yi1 − Yi2 = Ui1 − Ui2 =⇒ Obtain distr L (Uit |Xit = x)2 Deconvolve: Yit = m(Xit , αi )+Uit , αi ⊥Uit given Xit = x , thus

L (Yit |Xit = x)︸ ︷︷ ︸from data

= L (m (x , αi ) |Xit = x) ∗L (Uit |Xit = x)︸ ︷︷ ︸from step 1

,

where "∗" means "convolution" and "L" means "probability Law".Then obtain L (m (x , αi ) |Xit = x) by deconvolution

3 Use L (m (x , αi ) |Xit = x), eg. quantiles Qm(x ,αi )|Xit (q|x) [fornonparametric between- or within- variation] more

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16

Identification, Yit = m (Xit , αi) + Uit

1 Subpopulation Xi1 = Xi2 = x : thenm(Xi1, αi ) = m(Xi2, αi ) = m(x , αi )

Yi1 = m(x , αi ) + Ui1Yi2 = m(x , αi ) + Ui2

Yi1 − Yi2 = Ui1 − Ui2 =⇒ Obtain distr L (Uit |Xit = x)2 Deconvolve: Yit = m(Xit , αi )+Uit , αi ⊥Uit given Xit = x , thus

L (Yit |Xit = x)︸ ︷︷ ︸from data

= L (m (x , αi ) |Xit = x) ∗L (Uit |Xit = x)︸ ︷︷ ︸from step 1

,

where "∗" means "convolution" and "L" means "probability Law".Then obtain L (m (x , αi ) |Xit = x) by deconvolution

3 Use L (m (x , αi ) |Xit = x), eg. quantiles Qm(x ,αi )|Xit (q|x) [fornonparametric between- or within- variation] more

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16

Identification, Yit = m (Xit , αi) + Uit

1 Subpopulation Xi1 = Xi2 = x : thenm(Xi1, αi ) = m(Xi2, αi ) = m(x , αi )

Yi1 = m(x , αi ) + Ui1Yi2 = m(x , αi ) + Ui2

Yi1 − Yi2 = Ui1 − Ui2 =⇒ Obtain distr L (Uit |Xit = x)2 Deconvolve: Yit = m(Xit , αi )+Uit , αi ⊥Uit given Xit = x , thus

L (Yit |Xit = x)︸ ︷︷ ︸from data

= L (m (x , αi ) |Xit = x) ∗L (Uit |Xit = x)︸ ︷︷ ︸from step 1

,

where "∗" means "convolution" and "L" means "probability Law".Then obtain L (m (x , αi ) |Xit = x) by deconvolution

3 Use L (m (x , αi ) |Xit = x), eg. quantiles Qm(x ,αi )|Xit (q|x) [fornonparametric between- or within- variation] more

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16

Identification, Yit = m (Xit , αi) + Uit

1 Subpopulation Xi1 = Xi2 = x : thenm(Xi1, αi ) = m(Xi2, αi ) = m(x , αi )

Yi1 = m(x , αi ) + Ui1Yi2 = m(x , αi ) + Ui2

Yi1 − Yi2 = Ui1 − Ui2 =⇒ Obtain distr L (Uit |Xit = x)2 Deconvolve: Yit = m(Xit , αi )+Uit , αi ⊥Uit given Xit = x , thus

L (Yit |Xit = x)︸ ︷︷ ︸from data

= L (m (x , αi ) |Xit = x) ∗L (Uit |Xit = x)︸ ︷︷ ︸from step 1

,

where "∗" means "convolution" and "L" means "probability Law".Then obtain L (m (x , αi ) |Xit = x) by deconvolution

3 Use L (m (x , αi ) |Xit = x), eg. quantiles Qm(x ,αi )|Xit (q|x) [fornonparametric between- or within- variation] more

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16

Identification, Yit = m (Xit , αi) + Uit

1 Subpopulation Xi1 = Xi2 = x : thenm(Xi1, αi ) = m(Xi2, αi ) = m(x , αi )

Yi1 = m(x , αi ) + Ui1Yi2 = m(x , αi ) + Ui2

Yi1 − Yi2 = Ui1 − Ui2 =⇒ Obtain distr L (Uit |Xit = x)2 Deconvolve: Yit = m(Xit , αi )+Uit , αi ⊥Uit given Xit = x , thus

L (Yit |Xit = x)︸ ︷︷ ︸from data

= L (m (x , αi ) |Xit = x) ∗L (Uit |Xit = x)︸ ︷︷ ︸from step 1

,

where "∗" means "convolution" and "L" means "probability Law".Then obtain L (m (x , αi ) |Xit = x) by deconvolution

3 Use L (m (x , αi ) |Xit = x), eg. quantiles Qm(x ,αi )|Xit (q|x) [fornonparametric between- or within- variation] more

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16

Identification, Yit = m (Xit , αi) + Uit

1 Subpopulation Xi1 = Xi2 = x : thenm(Xi1, αi ) = m(Xi2, αi ) = m(x , αi )

Yi1 = m(x , αi ) + Ui1Yi2 = m(x , αi ) + Ui2

Yi1 − Yi2 = Ui1 − Ui2 =⇒ Obtain distr L (Uit |Xit = x)2 Deconvolve: Yit = m(Xit , αi )+Uit , αi ⊥Uit given Xit = x , thus

L (Yit |Xit = x)︸ ︷︷ ︸from data

= L (m (x , αi ) |Xit = x) ∗L (Uit |Xit = x)︸ ︷︷ ︸from step 1

,

where "∗" means "convolution" and "L" means "probability Law".Then obtain L (m (x , αi ) |Xit = x) by deconvolution

3 Use L (m (x , αi ) |Xit = x), eg. quantiles Qm(x ,αi )|Xit (q|x) [fornonparametric between- or within- variation] more

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 7 / 16

More things in the paper

Time Effects: Yit = m (Xit , αi ) + ηt (Xit ) + Uit , η1 (x) ≡ 0Dynamic models, e.g., Yit = m (Xit , αi ) + Uit , Uit = ρUit−1 + εit

Measurement error in covariates

Multivariate unobserved heterogeneity

Yit = m(Xit , β

′iWit

)+ Uit

Estimation - conditional deconvolution

Simple. No optimization. Just run many kernel regressions andcompute a one-dimensional integral

Rates of convergence

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 8 / 16

Empirical Illustration - Union Wage Premium

Data: Current Population Survey (CPS), T = 2matchingmigrationmeasurement errorlimited set of variables1987/1988, n = 17, 756 (replicating Card, 1996)

Most related papers: Card (1996) and Lemieux (1998)

Union Wage Premium

Yit = m (Xit , αi ) + Uit

Yit is (log)-wage,

Xit is 1 if member of a union, 0 o/w,

αi is skill and Uit is luck.

∆(α) = m (1, α)−m (0, α) is the union wage premium.Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 9 / 16

Empirical Illustration - Union Wage Premium

Data: Current Population Survey (CPS), T = 2matchingmigrationmeasurement errorlimited set of variables1987/1988, n = 17, 756 (replicating Card, 1996)

Most related papers: Card (1996) and Lemieux (1998)

Union Wage Premium

Yit = m (Xit , αi ) + Uit

Yit is (log)-wage,

Xit is 1 if member of a union, 0 o/w,

αi is skill and Uit is luck.

∆(α) = m (1, α)−m (0, α) is the union wage premium.Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 9 / 16

Empirical Illustration - Union Wage Premium

Data: Current Population Survey (CPS), T = 2matchingmigrationmeasurement errorlimited set of variables1987/1988, n = 17, 756 (replicating Card, 1996)

Most related papers: Card (1996) and Lemieux (1998)

Union Wage Premium

Yit = m (Xit , αi ) + Uit

Yit is (log)-wage,

Xit is 1 if member of a union, 0 o/w,

αi is skill and Uit is luck.

∆(α) = m (1, α)−m (0, α) is the union wage premium.Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 9 / 16

Empirical Illustration - Union Wage Premium

Data: Current Population Survey (CPS), T = 2matchingmigrationmeasurement errorlimited set of variables1987/1988, n = 17, 756 (replicating Card, 1996)

Most related papers: Card (1996) and Lemieux (1998)

Union Wage Premium

Yit = m (Xit , αi ) + Uit

Yit is (log)-wage,

Xit is 1 if member of a union, 0 o/w,

αi is skill and Uit is luck.

∆(α) = m (1, α)−m (0, α) is the union wage premium.Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 9 / 16

Empirical Illustration - Union Wage Premium

∆ (Qαi (0.25)) = 7.0%, ∆ (Qαi (0.5)) = 3.2%, ∆ (Qαi (0.75)) = 0.7%Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 10 / 16

Empirical Illustration - Union Wage Premium

∆ (Qαi (0.25)) = 7.0%, ∆ (Qαi (0.5)) = 3.2%, ∆ (Qαi (0.75)) = 0.7%Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 11 / 16

Empirical Illustration - Propensity Score

Also interested in P (Xit = 1|αi = a). Examples:

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 12 / 16

Empirical Illustration - Propensity Score

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 13 / 16

Alternative Model, Quasi-Differencing

We had

Yit = m (Xit , αi ) + Uit (1)

Identification is based on the NP first-differencing

Now consider

Yit = gt (Xit , αi + Uit ) (2)

Identification is based on the NP quasi-differencing

Applications:

Nonparametric panel (transformation) models

Duration model with multiple spells/states

A class of static games of incomplete information (e.g., Common ValueAuctions, Cournout with common demand parameter)

Also:

Yit = gt (Xit , βiWit + Uit ) (2′)

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 14 / 16

Related Literature (incomplete list)

Linear and Semiparametric models:Chamberlain(1984,1992), Horowitz & Markatou (1996),Wooldridge(1997), Blundell, Griffi th, and Windmeijer (2002), Graham andPowell(2008), Graham, Hahn, and Powell (2009), Bonhomme(2010),Honore and Tamer (2006)

Nonparametric Panel Models:Altonji and Matzkin (2005), Arellano and Bonhomme (2011a,gb), Besterand Hansen (2007, 2009), Chernozhukov, Fernandez-Val, Hahn, andNewey (2009), Cunha, Heckman, and Schennach (2010), Hoderlein andWhite (2009), Hu and Shum (2009)

Transformation and Multiple Spell Duration Models:

Abbring and Ridder (2003, 2010), Abrevaya (1999), van den Berg (2001),Flinn and Heckman (1982,1983), Honoré (1993), Horowitz and Lee(2004), Lee (2008), Khan and Tamer (2009), Ridder (1990), and a list ofempirical applications

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 15 / 16

Summary

Generalization of first-differencing and quasi-differencing approachesto nonlinear panel data models. Nonparametric identification

Allow identification of distributional, policy, and counterfactual effects

Are useful in a range of applied models

Estimation:

First-Differencing: a simple estimation procedure

Quasi-Differencing: sieve estimation

More details in the papers

Kirill Evdokimov (Princeton U) Nonlinear Panel Data Models 12/14/12 16 / 16