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