Dynamic Panel Data Models with Irregular Spacing SMU CORE Colloquium Series Daniel L. Millimet1 1 Southern
Ian McDonough2
Methodist University & IZA
2 Southern
Methodist University
February 2014
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The Situation Many interesting outcomes or behaviors of interest in social and physical sciences are dynamic in nature I I I I I I
Education: Cumulative process whereby current academic achievement depends on current and past accumulation of knowledge Health: Cumulative process whereby current health state depends on current inputs and past health Demography: Current population depends on past population and changes Policy: Partial adjustment implies current policy choice is a weighted average of past policy choice and desired adjustment Finance: Asset returns have signi…cant persistence Physical sciences: Biological growth models, ???
Analysis of dynamic processes in practice require either time series data or panel data I I
Time series: yt , t = 1, ..., T Panel: yit , i = 1, ..., N; t = 1, ..., T
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Dynamic models have a long history in applied time series I
Univariate models... Example: AR (1) yt = ρyt F
I
1
+ εt
Unit root testing is focused on Ho : ρ = 1 vs. Ha : ρ < 1
Multivariate models... Example: VAR (p ) Yt = ΠYt where Y is p
1, Π is p
1
+ εt
p, and ε is p
1
Dynamic panel data (DPD) originated with Balestra and Nerlove (1966) yit = γyit 1 + xit β + αi + εit where αi may be modeled as a random or …xed e¤ect
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The Problem Given access to typical time series data, estimation of univariate and multivariate dynamic models, testing for unit roots, etc. is (reasonably) straightforward Given access to typical panel data, estimation of DPD models is (reasonably) straightforward I I I I I I
First-di¤erenced IV (Anderson & Hsiao 1981) GMM (Arellano & Bond 1991) System GMM (Arellano & Bover 1995; Blundell & Bond 1998) Long-Di¤erenced IV (Hahn et al. 2007) Bias-Corrected LSDV (Kiviet 1995; Bun & Carree 2005) Orthogonal to Backward Mean Transformation (Everaert 2012)
Problem arises in all these contexts when typical data are unavailable In particular, we are concerned with the problem of irregularly spaced data Millimet & McDonough (SMU)
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De…nition A data set is said to be irregularly spaced if successive periods of observed data do not conform to successive periods as de…ned by the underlying data-generating process (DGP). Hypothetical Example... Actual periods: t = 0, 1, ..., 8 Observed periods: t = 0, 2, 4, 5, 8 Denote the observed periods as m = 0, 1, 2, 3, 4 Let t (m ) represent a mapping from the mth observed period to the corresponding t th actual period
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A bit of philosophy: 1
It is possible to assume the problem away I
I I 2
In time series, the distance between successive waves is referred to as the observation interval, whereas the unit period denotes the reference unit of time for the underlying process (Fuleky 2011) In discrete time models in time series, it is commonplace to set the unit period equal to the observation interval (Hamilton 1994) Whether this is convincing depends on the data structure and problem
Data collected at uniform intervals but with gaps are irregularly spaced according to our de…nition if the underlying DGP does not have such gaps; i.e., the unit period is smaller than the observation interval.
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Comment The same periods are assumed to be observed for all i I I
Not the typical missing data scenario (not observation-speci…c missing) Missingness is clearly ‘at random’
Actual Examples... Time series I I
Stock market closings on holidays, weekends Changes in historical data collection frequency (mixed frequency)
Longitudinal surveys I I
McKenzie (2001) provides developing country examples Table 1 in our paper lists developed country examples
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The Literature Time Series I
Univariate & multivariate models: Savin and White (1978); Jones (1980, 1985, 1986); Dunsmuir and Robinson (1981); Harvey and Pierse (1984); Palm and Nijman (1984); Robinson (1985); Dufour and Dagenais (1985); Kohn and Ansley (1986); Shively (1993); Chiu et al. (2012)
I
Unit root testing: Shin and Sarkar (1994a,b); Ryan & Giles (1998)
Panel I
Static models with AR(1) errors: Jones and Boadi-Boateng (1991); Baltagi & Wu (1999) yit = xit β + αi + εit ,
I
εit = ρεit
1
+ uit
DPD F
No unobserved e¤ects: Rosner & Munoz (1988) yit = γyit
F
1
+ xit β + εit ,
εit
iid
Pseudo-panels: McKenzie (2001)
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The Details
DGP given by yit = γyit
1
+ xit β + αi + εit ,
i = 1, ..., N; t = 1, ..., T ,
where I I I I I I
yit = outcome for observation i in period t γ is the autoregressive parameter (jγj < 1) x is a vector of covariates with associated parameter vector β αi is the unobserved e¤ect εit is the idiosyncratic, mean zero error term yi 0 is the initial condition
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Given a random sample, fyit , xit gi =1,..,N ;t =0,1,...,T , estimation is straightforward I
Anderson & Hsiao (1981) suggest F
First-di¤erence to eliminate αi ∆yit = γ∆yit
F F
I I I
1
+ ∆xit β + ∆εit ,
i = 1, ..., N; t = 2, ..., T
Instrument for ∆yit 1 (xit 1 or yit 2 are valid IVs) Additional restrictions can be incorporated in a GMM framework (Arellano & Bond 1991; Arellano & Bover 1995; Blundell & Bond 1998)
Other transformations to eliminate αi such as long-di¤erencing Bias-corrected …xed e¤ects Orthogonal to backward mean transformation
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Suppose we do not have a random sample, fyit , xit gi =1,..,N ;t =0,1,...,T Instead observe fyim , xim gi =1,..,N ;m =0,1,...,M With repeated substitution, the model de…ned over the observed periods is given by yim = γgm yim
1 + xim β +
"
|
gm 1
∑
xi ,t (m )
jγ
j
1
β+
j =1
gm 1
γg m 1 γ {z
αi +
∑
γj εi ,t (m )
j =0
Composite Error
where I i = 1, ..., N; m = 1, ..., M I gm , m = 1, ..., M, is the gap size or the number of periods between I I
j
# }
observed period m and m 1 t (m ) is the actual period re‡ected by observed period m Brackets contains all unobserved determinants of yim t:
0
m: gm:
0 0
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1 2
3
4
5
2 2
3 1
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Three estimation di¢ culties arise: 1
The coe¢ cient on the lagged dependent variable is not constant
2
The error term contains the covariates, x, and the idiosyncratic errors, ε, from the missing periods
3
The unobserved e¤ect, α, now has a period-speci…c factor loading (as in the interactive …xed e¤ects and time-varying ine¢ ciency literatures)
Note: If gm = g > 1 for all m, where g is any …nite constant, then the data are equally-spaced but with gaps ) FD will eliminate α, but issues (1) and (2) remain.
) All the standard DPD estimators are no longer valid!
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Re-write the model compactly as yim = γgm yim where θm
1 γg m 1 γ
1
+ xim β + θ m αi + eεim
gm 1
;
Prior literature re-visited: Rosner & Munoz (1988)
eεim
∑
j
xi ,t (m ) j γ β +
j =1
gm 1
∑
γj εi ,t (m )
j
j =0
I Non-linear least squares (NLS) I Linear interpolation of x I No unobserved e¤ects
McKenzie (2001) I Non-linear least squares (NLS) I No covariates (or available from alternative data source) I Pseudo-panel ) averaging within cohort-time cells to eliminate α
We o¤er two new, potential solutions 1 Quasi-di¤erencing 2 NLS-IV with and without imputation based on Everaert (2012) Millimet & McDonough (SMU)
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Proposed Solution #1: Quasi-Di¤erencing Quasi-di¤erencing (QD) entails yim
ϕm yim
1
Letting ϕm =
θm θm
= 1
8 ϕm
1 γg m 1 γg m 1
simpli…es to yim
ϕm yim
1
= γgm yim
1
ϕm γgm 1 yim
where e eεim = f
xi ,t (m
2
+ (xim
ϕm xim
e
εim 1 )β + e
2 )+1 , ..., xi ,t (m 1 ) 1 , xi ,t (m 1 )+1 , ..., xi ,t (m ) 1 , εi ,t (m 2 )+1 , ..., εi ,t (m )
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If ϕm were known, then a QD-NLS-IV estimator is feasible I I
xim 1 (and others) as IVs b, b γ β are consistent if x is strictly exogenous and serially uncorrelated
With ϕm unknown, we propose a QD-GMM estimator which requires x to be strictly exogenous and serially uncorrelated
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Monte Carlo Design
DGP yit = γyit
1
+ βxit + αi + εit ,
i = 1, ..., N; t =
99, ..., 0, 1, .., T
T = 17, M = 6, and observed periods include f0, 1, 2, 3, 7, 11, 17g
(γ, β) = (0.8, 0.8) N = 500; Simulations = 250 Two designs 1 2
Cov(xit , αi ) 6= 0, Cov(xit , xit Cov(xit , αi ) 6= 0, Cov(xit , xit
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=0 6= 0
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Monte Carlo Results
x strictly exogenous and serially uncorrelated I I I
QD-GMM performs well Everaert-type estimator with the Mundlak approach also performs well Traditional estimators (AH, AB, BB) perform reasonably well in some designs for β, but not γ
x strictly exogenous and serially correlated I I I
QD-GMM performs well Everaert-type estimator with the Mundlak approach and AR (1) imputation performs well QD-GMM is generally better when γ is relatively low; Everaert otherwise
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Applications Dynamics of Child Development
yit = γyit 1
+ xit β + αi + εit ,
i = 1, ..., N; t = 1, ..., T ,
Value-added models for student test scores, where I I I
2
1
y = student math or reading test score x = current school inputs (e.g., class size, class behavior, teacher experiece, teacher certi…cation status) α = student family background, innate ability
Dynamic models for child obesity, where I I I
y = child’s BMI x = current inputs (e.g., household characteristics, welfare/SNAP participation, school meals) α = genetics
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Applications Data
Early Childhood Longitudinal Survey – Kindergarten Cohort I I I I I
Collected by US Dept of Education Nationally representative sample of 20,000+ children entering kindergarten in Fall 1998 Dispersed across about 1,000 schools Waves in Fall K, Spring K, Spring 1, Spring 3, Spring 5, and Spring 8 Sample sizes F F
Achievement: N = 5, 977, NT = 29, 855 (math); N = 5, 564, NT = 27, 820 BMI: N = 9, 155, NT = 54, 930
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Applications Results
Estimator matters for degree of persistence in achievement E¤ects of other covariates are generally unaltered
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Conclusion Traditional estimators of DPD models fair poorly when data are irregularly spaced
Mind the gap!
QD-GMM performs well in many cases Does not perform particularly well when persistence is high and covariates are serially correlated and really matter Much work to be done given the popularity of irregularly spaced longitudinal surveys
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Thank you!
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