JPART 23:395–428
Panel Data Analysis in Public
Administration: Substantive and
Statistical Considerations
Ling Zhu
University of Houston
Panel data analysis has become a popular tool for researchers in public policy and public
administration. Combining information from both spatial and temporal dimensions, panel
data allow researchers to use repeated observations of the same units (e.g., government
agencies, public organizations, public managers, etc.), and could increase both quantity
and quality of the empirical information. Nonetheless, practices of choosing different
panel model specifications are not always guided by substantive considerations. Using a
state-level panel data set related to public health administration as an example, I compare
four categories of panel model specifications: (1) the fixed effects model, (2) the random effects model, (3) the random coefficients (heterogeneous parameter) model, and
(4) ­linear dynamic models. I provide an overview of the substantive consideration relevant
to different statistical specifications. Furthermore, I compare estimation results and discuss
how these different model choices may lead to different substantive interpretations. Based
on model comparisons, I demonstrate several potential problems of different panel models. I conclude with a discussion on how to choose among different models based on
substantive and theoretical considerations.
Introduction
Empirical research in public administration has been enriched by the availability of
panel data (Eom, Ho, Lee, and Xu 2008).1 Longitudinal data, tracking a particular
sample of spatial units, increase both quantity and quality of the empirical information and have several advantages over cross-section or time-series data (Hsiao 2003).
An earlier version of this article was presented at the 2012 Fall Research Conference of the Association for
Public Policy Analysis and Management. I thank the panel participants for their feedback on the earlier draft.
I thank Solé Prillaman and Andrea Eckelman for research assistance. I am indebted to Kenneth J. Meier,
Andrew B. Whitford, and the anonymous reviewers for their comments and suggestions. All errors remain
mine. Address correspondence to the author at lzhu4@central.uh.edu.
1 Panel
data refer to data combining information of both spatial and temporal dimensions, which allow
researchers to use repeated observations of the same units over time (Baltagi 2008; Hsiao 2003). It is also
referred as cross-section-time-series (CSTS) data (Beck and Katz 1995), time-series-cross-section (TSCS) data,
or “pooled” data.
doi:10.1093/jopart/mus064
Advance Access publication December 10, 2012
© The Author 2012. Published by Oxford University Press on behalf of the Journal of Public Administration Research
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Abstract
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Journal of Public Administration Research and Theory
2 For
example, Cornwell and Kellough (1994) use panel data on federal civil service agencies to examine
what account for workforce diversity. Boyne, James, John, and Petrovsky (2010) use a panel data design to
study organizational performance and turnover in 148 English local governments in 4 years. Soss, Fording, and
Schram (2011) use a panel data set of 24 Florida Workforce Board regions across 30 months to test the effect
of the Florida Welfare Transition program on decisions to sanction clients. Meier and O’Toole have conducted
a series studies on public management and organizational performance using panel designs pooling data of
more than 1,000 Texas school districts across multiple years (Meier and O’Toole 2002, 2003, 2008).
3 A new development in the panel data literature is to estimate panel data models in a Bayesian context
(Gelman and Hill 2007; Pang 2010). The overview on panel data models in this article, however, primarily
focuses on the non-Bayesian context.
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Stimson (1985, 916) asserts “many of the possible threats to valid inferences are specific to either cross-sectional or time-series design, and many of them can be jointly
controlled by incorporating both space and time into the analysis.”
Students of public policy and public administration particularly benefit from
panel data analysis.2 Policy analysts often encounter data limitations when assessing
the impact of a policy, and only limited annual policy data are available (Stimson
1985). Panel data can increase the number of observations by pooling different timeseries together. With greater degrees of freedom, researchers are able to investigate
complex empirical associations among variables such as the interaction between
space and time. Public administration scholars often study governments, large organizations, social and institutional contexts, which do not vary much within a single
time-series. Panel data could outperform single-dimension data by providing the ability to deal with rarely changing variables and unobserved heterogeneity across units
(Allison 2009; Plümper and Troeger 2007).
The methodological sophistication of panel data analysis, moreover, has evolved
over the past two decades. The early work relies on ordinary least square (OLS) techniques (i.e., completely pooling) (Mundlak 1978; Stimson 1985; Taylor 1980). The
next generation of methodology focuses on the analysis of covariates associated
with spatial and temporal dimensions and statistical theories that are employed to
justify error assumptions (Arellano and Honore 1999; Wooldridge 2002). Various
dynamic and spatial econometric models are developed to handle complex panel data
structures (Baltagi 2008). A recent development of this methodology enterprise is
the effort of linking statistical models to substantive and theoretical considerations
(Granato and Scioli 2004; Plümper, Troeger, and Manow 2005; Primo, Jacobsmeier,
and Milyo 2007; Whitten and Williams 2011).3
Despite the increasing availability of data and statistical models, panel data analysis still remains to be an “art” because of several enduring issues. Firstly, there has
been a long-standing debate on the utility of the fixed effects (FE) model (Baltagi,
Bresson, and Pirotte 2003; Greene 2011a; Nickell 1981). Secondly, although pooling could increase the statistical power of an empirical data set, it could also make a
statistical model vulnerable to problems caused by both cross-sectional and temporal dimensions. Adjusting for correlated errors along both the spatial and temporal
dimensions could be quite challenging. Thirdly, when theory is weak (i.e., researchers
do not have a strong and explicit theory to guide the choice of explanatory variables), estimating a consistent and efficient panel model could be difficult. Beck (2011)
points out that panel data always suffer from potential omitted variable bias, thus
leaving a room for competing model choices and different specification errors. As a
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Panel Data Analysis in Public Administration
Panel Data Models: An Overview
In this and the subsequent section, panel models are illustrated with a data set of
public health administration. In this empirical example, the cross-sectional units are
50 American states. Annual data are available for all 50 states from 1990 to 2006.
The data set is used to investigate the characteristics of state health care systems
and their effects on citizens’ access to health insurance. Table 1 reports the descriptive statistics of all variables in this panel data set. Three institutional variables are
included in the data set: the public ownership of state health care systems, the public
source of financing state Medicaid programs, and the generosity of state Medicaid
eligibility rules. Although this empirical example is based on data related to public
health administration, it resembles a few common characteristics of a panel data set
in public administration.
Firstly, it is quite common to see a panel application with more cross-sectional
units than time units in public administration and public management.4 Secondly,
although data variation exists across spatial and temporal dimensions, some key theoretical variables (in the empirical example, institutional variables) do not vary substantially across time. For example, from 1990 to 2006, Delaware, New Hampshire,
Rhode Island, and Vermont did not experience any institutional change in the
ownership of their health care systems (within variation equals 0). States such as
Arizona, Maine, and New Jersey experienced incremental changes in the public ownership (within-state mean <10% public hospitals, standard deviation <2). Overall,
states do not change their Medicaid eligibility rules dramatically. Only a few states
experienced substantial policy changes in their Medicaid income eligibility rules,
such as Arizona (from 107% in 2003 to 200% after 2003) and New Jersey (from 100%
in 2004 to 115% in 2005). Thirdly, different theories and empirical research show
that many factors, combined, affect the dependent variable (state-level uninsured
rates). Markowitze, Gold, and Rice (1991) contend that people’s employment status
plays a direct role in determining their access to health insurance. They also find that
4 There
are many other panel data applications that have similar data structures; for example, a panel data
set of large-N organizations across multiple years, longitudinal data for a large number of public managers,
and so on.
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result, the “art” of panel data analysis hinges on conjectures about the theoretical
relationship, specific data structures, and particular assumptions about the source of
bias. Knowing the pros and cons of different types of panel analytic models, therefore, becomes essential.
This paper provides a synopsis of commonly used panel data models, reviews
advantages and disadvantages of each panel method, and illustrates the analytic
applications with a panel data set of public health administration across 50 states
and 17 years. After comparing how different substantive conclusions could be drawn
from different model applications, a brief discussion on extensions pertaining discrete
variables is presented. Overall, this article makes the argument that statistical models
are essentially tools for theory building. Researchers need to link their model choices
to substantive considerations, not simply to statistical assumptions.
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Table 1
Descriptive Statistics of the Panel Data Set: 50 States from 1990 to 2006
Variable
Mean
SD
15.56
4.46
24.21
18.64
Public finance
15.09
3.88
Eligibility
87.70
56.36
5.15
12.76
1.39
3.57
47.99
25.38
10.28
7.25
12.61
9.46
8.67
1.94
% Black population
% Latino population
% Population older than 65 years
18.53
4.25
3.99
1.37
% Population whose body mass index scores are
above 30
% Population who reported themselves as being
in poor health
Dependent variable
Uninsured
Institutional variables
Public ownership
Demographic variables
Black population
Latino population
Aged population
Health risk factors
Obesity rate
Perceived poor health
% Population without health insurance
% Community hospitals owned by
government
State Medicaid spending as the percentage of
total personal health care spending
State Medicaid income eligibility limits for
working adults, as a percentage of the federal
poverty level (FPL)
State unemployment rate
State poverty rate
Berry, Ringquist, Fording, and Hanson (1998)’s
measure of state liberalism
Notes: Data for the dependent variable are aggregated from the Current Population Survey (CPS)’s Annual Social and Economic
Supplement, the American Community Survey, and the Survey of Income and Program Participation. Data are archived in the
Census Bureau historical tables, Health Insurance Coverage Status and Type of Coverage by State, Persons Under 65. Data for
Public Ownership are drawn from the American Hospital Association (AHA)’s annual data on the number of community hospitals by type. Data for Medicaid spending are drawn from the US Department of Health and Human Services, the Center for
Medicaid and Medicare Services Expenditure Reports. Data for state Medicaid eligibility rules are drawn from the Kaiser Family
Foundation’s policy report on state Medicaid eligibility rules (Heberlein, Brooks, Guyer, Artiga, and Stephens 2011). Data for
the economic variables are drawn from the Bureau of Labor Statistics (BLS) annual estimates on state unemployment, and US
Census Bureau, Income, Poverty, and Health Insurance Coverage in the United States, Current Population Reports. Data for the
state government ideology are based on Berry, Ringquist, Fording, and Hanson (1998) and accessed from http://www.bama.
ua.edu/~rcfording/stateideology.html. Data for the demographic variables and health risk factors are drawn from the Centers for
Diseases Control and Prevention (CDC) WONDER Current Population Information, and Behavior Risk Factor Surveillance
System (BRFSS).
individual characteristics, such as race, ethnicity, and health risks, affect access to
health insurance. Nelson, Bolen, Wells, Smith, and Bland (2004) find evidence that
states’ fiscal responsibilities in health care affect citizens’ insurance take-up. Hacker
(2004) contends that the structure of welfare policies and the characteristics of the
American hybrid health care system are important determinants of access to care.
In sum, various factors need to be considered as substantively important variables
when analyzing state-level uninsured rates, including economic variables, the institutional characteristics of a state health care system, demographic variables, and
health risks.
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Economic variables
Unemployment
Poverty
Political variable
State liberalism
Measurement
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The generic panel model equation can be written as:
Yi,t = α + Xi,t β + ei,t(1)
The Unobserved Heterogeneity
One important assumption of the population average model is that all units are
homogenous. After all, researchers pool different spatial units across time in seeking
a commonality of these units. As for the empirical example, equation (1) implies unit
homogeneity across all state–year observations. This assumption is often applied to
work done in scientific laboratories, but rarely holds in nonrandomized observational
studies (Holland 1986; Rosenbaum 2005; Wilson and Bulter 2007). Unobserved unit
heterogeneity may bias statistical estimation and lead to invalid causal inferences.
Rosenbaum (2005, 148) contends “observational studies vary markedly in sensitivity
to unobserved biases.”
The Fixed Effects Model
The FE model deals with unobserved heterogeneity by using unit-specific intercepts
(Greene 2011b). Equation (2) is used to denote the unit-specific fixed effects, and fi is
defined based on a vector of spatial units, zi.
Yi,t = α + Xi,t β + fi + ei,t
f i = z iγ
(2)
The FE model can be estimated based on a full set of unit dummy variables or by
mean-centering the dependent variable and all the explanatory variables to “clear”
5 Essentially,
consistency and efficiency are two desirable properties of a good estimator. Here, I mainly
focus on threats to consistency. Various panel data techniques related to improving estimation efficiency will be
discussed in the subsequent section.
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In equation (1), i denotes the spatial dimension (i.e., states), t denotes time (i.e., year),
α is the intercept, β refers to the vector of slope coefficients, and Xi,t refers to the vector of regressors. This generic model equation refers to the application of complete
pooling. In other words, we do not speculate that state-specific effects or time-specific
effects exist with regard to how institutional characteristics of state health care systems, economic factors, and demographics affect state-level uninsured rates (Gelman
and Hill 2007). This is also referred as the population average model (Baltagi 2008).
In essence, the parameter of interest in a population average model is the mean
effect across time (within) and spatial (between) units. Although complete pooling
produces a simple and generic model, it could lead to biased estimation due to two
major threats: unobserved heterogeneity across spatial units (i.e., states) and temporal dependency across time units.5 There are various types of panel models developed
to handle one or both issues. Each panel model has its pros and cons and is linked to
different substantive relationships.
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The Random Effects Model
Different from the FE model, which conceptualizes each state as having its own
baseline, the random effects (RE) model assumes the intercept to be some random
deviation from the underlying mean intercept. Greene (2011b, 347) points out that if
“the unobserved individual heterogeneity, however formulated, can be assumed to be
uncorrelated with included variables, then the model may be formulated as”:
Yi,t = α +Xi,t β + ui + ei,t(3)
According to equation (3), the RE model specifies that ui is a state-specific random
element, such that “there is but a single draw that enters the regression identically in
each period” (Greene 2011b, 347). The RE model has several desirable properties.
One is that time-invariant variables can be included in the panel model (Wilson and
Bulter 2007). Furthermore, if we have a large-N panel data set and the random effect
is uncorrelated with the regressors, the RE effects model is more efficient than the
FE model. The estimation efficiency, in addition, increases as the number of crosssection units (N) increases. As for the state-level panel data example, the model based
on equation (3) will have more degrees of freedom than the model based on equation
(2), because equation (3) does not estimate state-specific intercepts. However, if the
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cross-unit heterogeneity (Allison 2009). Both the least square dummy variable
(LSDV) approach and the mean-centering approach account for within-variance and
eliminate between-variance. In other words, the FE model controls for cross-unit heterogeneity that is not captured by the conditional mean, Xi,t β.
When cross-unit heterogeneity (fi ) is correlated with the regressors (Xi,t), failure to account for the heterogeneity could lead to biased estimation of β. In such a
situation, the FE model is better than the OLS specification because it improves the
estimation consistency. The FE model, however, has a few limitations. The LSDV
specification includes a large number of dummy variables to account for each unit
as a specific source of unobserved heterogeneity. It may become problematic when
T is relatively small. Both LSDV approach and mean-centering approach have been
accused of absorbing a lot of cross-sectional variance. This can be a serious problem
if the panel data set contains much greater cross-sectional variation than cross-time
variation (Kennedy 1998). The LSDV approach is also deemed to be a crude method
to model unmeasured heterogeneity. In other words, unit dummy variables can only
tell researchers that there is cross-unit heterogeneity but cannot show what factors
explicitly cause the heterogeneity.
Considering the aforementioned empirical example, if the substantive interest is
to investigate how the institutional arrangements of state health care systems affect
state-level uninsured rates, it is reasonable to expect that these institutional variables
may capture some cross-state heterogeneity. Estimating the panel model by including a full set of state dummy variables may throw away cross-state variations that
are captured by the institutional variables and lead to artificial null findings on the
institutional variables.
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Panel Data Analysis in Public Administration
The Random Coefficients Model
Comparing equations (2) and (3), we can see that the key difference between the FE
model and the RE model is substantive. Equations (2) and (3) conceptualize different sources of cross-unit heterogeneity. Equation (2) implies part of the state-specific
effects is not captured by the vector of regressors and need to be controlled (the
covariates are completely correlated with unobserved state-specific effects). Equation
(3) implies that the substantive variables (i.e., institutional, economic, demographic,
and health risk variables) may well capture most state-specific effects, and thus including state dummy variables is unnecessary (the covariates are not correlated with the
unobserved state-specific effects). In equation (3), ui is theorized as a unit-specific random intercept. Greene (2011b, 347) contends that with a sufficiently rich data set, one
can also estimate models including both random intercepts and random coefficients,
expressed as equation (4):
Yi,t = α + Xi,t (β + hi) + ui + ei,t(4)
Based on equation (4), both the intercept and the slopes vary by states, i. This type of
random coefficients model (RCM) can also be estimated based on time, t. Beck and
Katz (2007) argue that panel data analysis essentially comes down to the question
about how much to pool observations across time and space. With a series of Monte
Carlo experiments, they show that RCM estimators obtained via the maximum-likelihood method outperform pooled OLS, unit-by-unit OLS (FE) with finite sample
panel data.6
6 The
aforementioned FE, RE, and RCM models all recognize the multi-level structure of the dataset.
Although I discuss these model considerations based on the nonnested state-level CSTS data example, similar
considerations can apply to hierarchical linear models that capture more complex multi-level structures. For
example, if a panel dataset contains both individual-level, organization-level, and state-level units (e.g., public
managers from different state agencies in different years), one may consider a three-level nested parameter
structure for fixed effects, random effects, or random coefficients [see Gelman and Hill (2007) for more detailed
discussion on complex/nested multi-level specifications].
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random intercept ui is correlated with Xi,t, the RE model could produce inconsistent
estimation. Hence, it is always critical to consider whether the random effects (ui) are
correlated with substantively important variables in the model.
In addition, the Hausman model specification test can be used to evaluate the
consistency and efficiency of the fixed effects and the random effects specification
(Hausman 1978). Large χ2 statistics produced by the Hausman test suggest that the
RE model is inconsistent and one may not consider the RE model as an appropriate
model choice. When both fixed effects and random effects specifications are consistent, model-fit statistics can be used to choose between the two different specifications.
For example, both mean squared error (MSE) and root mean squared error (RMSE)
reflect estimation bias, and the model associated with smaller MSE/RMSE statistics
would be preferred (James and Stein 1961).
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Journal of Public Administration Research and Theory
Time Dependence and Dynamic Panel Models
The second major threat to consistent panel estimation is the issue of temporal
dependence. Most public administration data feature first-order autoregressive processes, whereby the current status of organizations, governments, or policy networks
is a function of their own past (De Boef 2001). In other words, administrative processes are inertial and organizational memory persists. Formally, a first-order autoregressive process can be written as:7
Yi,t = ρYi,t−1 + ei,t(5)
7 Statistically,
we can also define a higher-order autoregressive process by changing the notation of Yi,t
into Yi,t−2,3,4 … k. A higher-order autoregressive process may occur if an administrative process is affected by
cyclical factors, such as budgetary and election cycles. The first-order autoregressive process, however, is most
commonly seen in public administration. In addition, if a panel data set only has very short T (for instance,
pooling hundreds of local government units in 4 or 5 years), estimating a higher-order autoregressive process
would be unfeasible. Box and Jenkins (1970) lay out the theoretical foundation of modeling an autoregressive
process. De Boef (2001) and De Boef and Granato (1997) provide nice discussion on how various political/
policy processes can be conceptualized by an autoregressive model.
8 With a CSTS setup, one can examine the common (mean) dependence across all included series (ρ) or
panel-specific dependence (ρi).
9 The Levin–Lin–Chu panel unit root test assumes that all units share the same AR(1) process but allows
specifying the different unit effects, time dynamics, and time trends. An alternative test to the aforementioned
tests is a multivariate augmented Dickey–Fuller test using the seemingly unrelated regression (SUR) estimator.
This test, however, requires that T must be larger than N (Taylor and Sarno 1998).
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In equation (5), ρ characterizes the feature of organizational memory.8 An administrative process carries permanent memory if ρ equals to 1 or −1 (i.e., the presence of
panel unit root).
If ρ ϵ (−1, 1), a process is stationary and the organizational memory is not permanent (De Boef 2001). According to De Boef (2001, 81), a stationary process is
strongly autoregressive when | ρ | is near 1 (formally ρ = 1 + ϵ, and ϵ is a small
negative fraction), and a process mainly carries short-term memory when | ρ | is near
0. When pooling different states, we assume that all these administrative processes are
bounded by a common set of relationships, which are constant over time (Pesaran
and Smith 1995). Dynamic data could violate this time-consistent assumption and
thus can lead to spurious regressions and biased coefficient estimation (Beck 2001;
Dickey and Fuller 1981; Im, Pesaran, and Shin 2003; Maddala and Wu 1999).
The first consideration of estimating panel dynamics is to examine the dynamic
nature of the dependent variable. Various statistical tests can be used to evaluate the
stationarity of the dependent variable. For example, both the augmented Dickey–
Fuller unit root approach and the Phillips–Perron approach can be applied to a panel
data set. According to Maddala and Wu (1999), both approaches assume that all
series are not stationary against the alternative that at least one series in the panel is
stationary. If a panel data set is balanced, one can also use the Im–Pesaran–Shin (Im,
Pesaran, and Shin 2003) test or the Levin–Lin–Chu panel unit root test (Levin, Lin,
and Chu 2002).9
Once it has been determined if the dependent variable is panel stationary, two
substantive questions need to be further considered. The first substantive question
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is how one could conceptualize temporal dynamics and model persistence (Wawro
2002). In the state health care example, the substantive question is whether the impact
of health care institutions persists through time. If so, how can the long-term and
short-term impacts be reflected by statistical models? Permanent memory, long-term
equilibrium relationships, and short-term dynamics represent substantively different
administrative/organizational processes and thus require different statistical specifications. The second question is whether to model data dynamics as a nuisance (ei,t),
or to incorporate the temporal dynamics in the conditional mean (Xi,t β). Dynamic
specifications differ based on different substantive considerations and different data
structures of temporal dependence.
When the dependent variable is panel stationary (| ρ | ≠ 1), one can specify an autoregressive distributed lag (ADL) model either by including a lagged dependent variable
(LDV) or by estimating a static model with serially correlated error terms (e.g., an
AR(1) error specification).10 In public administration, one common practice of modeling stationary panel data is to include a LDV as a control for organizational memory.
This dynamic specification is usually deemed partial adjustment of behavior over
time (Wawro 2002). The inclusion of a LDV is also justified by the substantive consideration that large organizations, institutional contexts, and government agencies
are normally inertial. The LDV approach is a parsimonious way to capture both the
long-term and short-term effects of regressors (Beck and Katz 1996; Wawro 2002).
An additional attraction of including lagged terms of the dependent variable
is that they are usually easy to estimate and interpret. In the state health care example, if the dynamic model is specified as equation (6), the autoregressive parameter,
ρ, captures how the long-term effects of health care institution variables (Xi,t) are
distributed over time. βk, the vector of coefficients corresponding to the health care
institutional variables (Xk), captures the short-term effects. One can also calculate the
total long-term effect based on the approximation, [βk / (1 − ρ)]. The size of βk and
ρ determines the magnitude of the short-term effect and the persistence of the longterm effect. When ρ is large, the short-term effect, βk, is slowly discounted over time
and the total long-term effect cumulates over a long time period. Conversely, if ρ is
small, the short-term effect will be substantially discounted in the future and the total
long-term effect will be relatively small.
Yi,t = ρYi,t−1 + Xi,t β + ei,t(6)
The inclusion of a LDV, nevertheless, has been accused of absorbing covariates associated with other regressors and could potentially wash out the explanatory power
of other exogenous regressors (Achen 2000; Keele and Kelly 2006). Substantively,
the LDV approach also makes a strict underlying assumption that the effects of all
state health care variables dissipate exponentially over the long run. Combining a
10 Beck
(1991) argues that when data are stationary, the LDV specification is not fundamentally different
from a static model with an AR(1) specification. Both specifications belong to the family of ADL model.
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Stationary Panel Data: Autoregressive Distributed Lag Models
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Journal of Public Administration Research and Theory
Dealing with Nonstationary Panel Data
If panel unit root presents, estimating a LDV model can lead to biased results.
There are alternative dynamic panel methods to the LDV approach, based on different substantive considerations of short-term and long-term relationships. The
class of error correction models (ECM) was firstly proposed to deal with integrated
and near-integrated dynamic data (De Boef and Granato 1997; Engle and Granger
1987; Westerlund 2007) and later applied to stationary autoregressive processes (De
Boef 2001; De Boef and Keele 2008). Despite the varying forms, the error correction
method deals with integrated or near-integrated dynamic data by differencing the
dependent variable (i.e., Δyi,t) and including changes and lagged values of explanatory
variables and lagged values of the dependent variable as the regressors.11 Following
De Boef (2001, 84), a generalized ECM model can be written as:
Δyi,t = α + β1Δxi,t − γ (yi,t−1 − xi,t−1) + β2 xi,t−1 + ei,t(7)
Using the state health care panel example, Δyi,t (the annual change in state uninsured
rates) is affected by Δxi,t (the annual change in government Medicaid spending) and
the long-run relationship between yi,t (uninsured rates) and xi,t (Medicaid spending).12
11 See
De Boef (2001) for the comparison between the Engle and Granger two-step ECM and the
generalized one-step ECM. De Boef and Keele (2008) provide a thorough overview of different variants of the
generalized one-step ECM. De Boef and Granato (1997) contend that integrated and near-integrated processes
have very similar statistical properties in finite samples. Hence, ECM is deemed a common statistical method to
characterize long-run relationships represented in both integrated and near-integrated processes.
12 For simplicity, I only included one explanatory variable in equation (7). To illustrate the substantive
relationships, I use xi,t to denote the Public Finance variable (i.e., state Medicaid spending). Similar lag and
difference terms may apply to other explanatory variables if the dynamic relationships are the same.
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LDV with fixed effects in panel data analysis, futhermore, could potentially lead to
biased estimation in some empirical contexts. Wawro (2002) points out that combining a LDV with fixed effects may produce finite sample autoregressive bias to the
coefficient of the LDV (i.e., the “Nickell bias” that attenuates the coefficient of the
LDV) when T is very small (Nickell 1981; Phillips and Sul 2007; Wawro 2002). Based
on their recent Monte Carlo experiments, Keele and Kelly (2006) report that the performance of a LDV model is still desirable if the dependent variable is very dynamic
and if serial autocorrelation in the error term is very small. If these two conditions
are met, the bias caused by LVD diminishes as T increases.
In the state health care example, the panel data set includes data for 17 years.
It grants some degree of freedom for estimating a simple dynamic relationship with
a LDV. The data may also permit a more complex dynamic specification than the
LDV specification. If T is very short, estimating complex dynamic relationships would
become unfeasible. Substantive relationships between state health care variables and
the uninsured rates, furthermore, will be different in a static model with a serially correlated error term. Such a model specification only captures the mean association
between the level of uninsured rates and the level of a particular state health care
variable.
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In particular, the long-run relationship between the level of uninsured rates and state
Medicaid spending is reflected by −γ (yi,t−1 − xi,t−1) + β2 xi,t−1, whereby γ represents
how disequilibrium between uninsured rates and spending is “corrected” over long
run. Approximately, the long-run effects of spending can be calculated based on
1 − (β2 / γ) (Alogoskoufis and Smith 1990; Campbell and Shiller 1988).
Alternatively, one may deal with panel unit root by differencing both the dependent variable and the key explanatory variables. Such a first-difference model is often
seen as a particular case in the ECM family, whereby the long-run relationship is
restricted to be 0 (Beck 1991; De Boef and Keele 2008). The first-difference model is
easy to estimate. Empirically, it may not perform well because it does not account for
the long-run relationship in the process (Beck 1991).
Most aforementioned dynamic specifications add a lagged term of the dependent
variable as a regressor. It violates the strict exogenous assumption of OLS due to
the inclusion of an endogenous lagged term. The issue is further complicated if the
dynamic specification includes error terms (such as panel-specific fixed effects), which
are correlated with the endogenous lagged term (Anderson and Hsiao 1981; Wawro
2002). Various instrument variable (IV) approaches have been proposed to deal with
this additional complication. Anderson and Hsiao (1981) proposed using a secondorder change measure of the dependent variable (Δyi,t − 2) or a second-order LDV
(yi,t − 2) as instruments to remove the correlation between panel-specific effects and the
endogenous lagged term. Arellano and Bond (1991) modified the Anderson–Hsiao
estimator and proposed the generalized method of movements (GMM). The GMM
dynamic panel estimators have become increasingly popular because of a few nice
properties. It is designed to be suitable for panel data with small T and large N. It
is flexible to deal with regressors that are not strictly exogenous by instrumenting
endogenous regressors with their own lags. The Arrellano–Bond method, moreover,
is flexible to model dynamics associated with both levels and changes. The major
drawbacks of the GMM method include: (1) researchers need to predetermine which
regressors are not strictly exogenous; and (2) it is usually an arbitrary decision on how
many instrument variables need to be included. Another caveat is that although the
GMM approach can handle endogenous variables in a flexible way, the IV procedure
is not a panacea to all the endogeneity issues.
Estimation Efficiency: Corrections for Nonspherical Errors
Aside from the aforementioned considerations on consistency, researchers also need
to consider the efficiency of a particular estimator that they choose. Several issues may
rise and affect estimation efficiency. Public administration scholars often model panel
data with large N and small T, whereby some key theoretical variables do not change
in a short time period. In the aforementioned empirical example, two institutional
variables tend to be very stable over time within each state: state Medicaid eligibility
rules and the public ownership of state health care systems. In this situation, if the
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Additional Considerations: Endogenous Lags and the Instrument Variable Approach
406
Journal of Public Administration Research and Theory
13 There
has been a debate, however, on the quality of standard errors produced by the FEVD method (Beck
2011). Scholars who are skeptical to the FEVD method contend that it often produces small standard errors,
leads to overconfidence in inferences, and thus caution needs to be warranted when using this method (Breusch,
Ward, Nguyen, and Kompas 2011; Greene 2011a).
14 In this section, I only provide a brief overview on available FGLS (feasible generalized least squares)
methods for improving efficiency. The relevant statistical tests for detecting different nonspherical errors will be
illustrated along various empirical models in the next section.
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within variation is substantively important, estimating a FE model could lose a substantial amount of efficiency, because of the rarely changing variables. Plümper and
Troeger (2007) modified the conventional FE model and developed the fixed effects
vector decomposition (FEVD) method to deal with time-invariant (or rarely varying)
variables. Despite the longish debate on the quality of standard errors produced by
the FEVD method (Beck 2011), Plümper and Troeger (2007)’s method could be useful to pooled data analysis if key theoretical variables are time-invariant and estimating within effects is substantively critical.13
Another potential challenge to efficiency is the heterogeneity in the error term, ei,t.
In theory, there are three sources of nonspherical errors: (1) heteroskedasticity across
units (or across panels), (2) contemporaneous correlation across panels, and (3) serial
autocorrelation. Various methods are available to detect and correct nonspherical errors.
If unit-specific heteroskedasticity occurs, one can estimate the White standard errors to
improve estimation efficiency. If heteroskedasticity only occurs across states, not within
states, clustered standard errors can be estimated to improve estimation efficiency. Beck
and Katz (1995, 1996) proposed to use the panel-corrected standard errors (PCSE) to
deal with the situation whereby both panel heteroskedasticity and spatial correlation
across panels occur. Serial autocorrelation can be corrected by implementing the Prais–
Winsten procedure, such as a generalized AR(1) correction or a panel-specific correction,
PSAR(1) (Beck 2001). The general AR(1) correction only estimates one common serial
autocorrelation parameter by averaging across all included series. PSAR(1), however, estimates a unique autocorrelation parameter for each series. PSAR(1) could yield a better
model fit than the general AR(1) if great heterogeneity is detected in the error structure.
Estimating PSAR(1) is less desirable, however, when a panel data set contains very short
T. Researchers also need to consider whether they have substantive reasons to justify a
heterogeneous structure of the serially correlated error terms (Beck and Katz 1995).14
Newer approaches for handling spatial dependence have also gained scholarly
attention. A few scholars have proposed spatial lag models to handle spatial dependence across panels. The spatial lag approach and the PCSE approach handle cross-unit
interdependence differently. The spatial lag approach treats spatial interdependence
as a substantively relevant term, whereas the PCSE approach models spatial interdependence as nuisance (Franzese and Hayes 2007).
Beside the aforementioned models, there are many mixed applications in the
practice of panel data analysis. Researchers often need to make the specification
decisions that are related to issues of modeling heterogeneity and time dynamics.
Because the error term (ei,t) is always an estimated quantity, any misspecification in
the conditional mean (Xi,t β) would lead to nonspherical errors. Hence, it is usually a
sound practice to use substantive considerations to guide decisions in specifying the
conditional mean first. In the next section, I use the aforementioned state-level panel
example to illustrate different panel model specifications.
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Panel Data Analysis in Public Administration
407
Panel Data Analysis: Empirical Illustrations
Modeling Heterogeneity: OLS, Fixed Effects, or Random Effects?
The nature of the dependent variable (state-level uninsured rates) demonstrates that
substantively interesting variation exists across 50 states. The three institutional variables also demonstrate that the characteristics of the health care system in a state may
not vary substantially across time. The important implication of these two substantive
Figure 1 State-Level Uninsured Rates, 1990–2006
The Percentage of State Residents Who Live without Health Insurance (Persons Under 65)
30
20
10
0
30
20
10
0
30
20
10
0
30
20
10
0
30
20
10
0
30
20
10
0
30
20
10
0
2005
2000
1995
1990
2005
2000
1995
1990
Year
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The first step of analyzing a panel data set is to examine the nature of the dependent variable along the two dimensions, T and N. Beck (2001) and Beck and Katz
(1995) emphasize that it is critical to evaluate the relative statistical powers along the
two dimensions, because TSCS data and CSTS data are suitable for different models. The aforementioned state health care panel data set contains more N (50 states)
than T (17 years). It is also necessary to examine the substantive characteristics of
the dependent variable. Figure 1 presents data variation by state and year. Figure 1
demonstrates that the cross-state difference in health insurance coverage exhibits a
persistent pattern across 17 years. Within most states, the uninsured rates vary incrementally over the 17-year period.
408
Journal of Public Administration Research and Theory
15 The
variable measuring public finance in health insurance is lagged by 2 years because budgetary decisions
are made based on fiscal cycles, and in some states, they are made biannually.
16 In table 2, the first FE model is estimated using the STATA 12 command “xtreg..., fe”. The same
estimation can be obtained using “areg..., absorb(state id)”. The second FE model is estimated based on the least
square specification with fixed effects dummies, “xi:xtreg...i.state”. These two procedures produce the same slope
coefficients as the ones obtained based on the mean-centering approach (i.e., removing group means from both
the dependent variable and the explanatory variables). The mean-centering approach, however, produces smaller
standard errors than the ones computed using “xtreg..., fe” or “xi:xtreg...i.state,” because the two STATA fixed
effects procedures do not adjust for changes in the degree of freedom when computing the standard errors.
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considerations is that the fitted panel model needs to capture the cross-state data
variation. As such, estimating an OLS model and a FE model including state dummies could be problematic. Table 2 reports the comparison of four models: OLS,
FE using the mean-centering approach, FE including a full set of state dummies,
and RE.15 Table 2 shows that estimation results are very sensitive to different model
specifications. The slope coefficients change substantially from the OLS specification
to the two FE specifications, indicating that the complete pooling strategy may not
be appropriate. The two FE models clearly demonstrate the problem of estimating
within effects when a large proportion of variance exists across space. The within and
between R2 statistics suggest that the fixed effects specification absorbs a lot of data
variation across states, and thus the two models capture very little data variation in
the panel data set.
The second FE model is particularly problematic because the between R2 becomes
1 after including a full set of state dummies. The between R2 indicates that the second
FE model is a saturated model. This is because state dummies fully control away any
contextual difference in states. This can be deemed to be double-counting data variation driven by the three state health care variables. If the theoretical expectation is
that the levels of all three state health care institutional variables determine the mean
level of state uninsured rates, one cannot estimate a FE model that absorbs all the
institutional differences across states.16
The fourth model in table 2 is a one-way RE model, which assumes that the
group effect (ui) is a random draw from the underlying population effect and is not
correlated with any time-invariant unit (i.e., state dummies). Is the RE model a more
appropriate specification than the OLS model and the two FE models? If ui is uncorrelated with any state-specific effect, the RE model is consistent and more efficient
than the OLS and the two FE models. Running the Hausman test based on comparing the first FE model and the RE model in table 2, the test produces a χ2 of 39.25
(df = 11, p = .000). The test result suggests that the RE model is not consistent and
produces biased estimations.
What does the comparison in table 2 mean? Both the OLS and the RE model are
not consistent, suggesting that one may take the risk of using predetermined factors
to explain the state uninsured rates without a good control for cross-unit heterogeneity. The two FE models, however, are also problematic. The two FE models purely rely
on within-state changes and exclude data variations across states, which are substantively relevant. Plümper, Troeger, and Manow (2005) contend that if the substantive
consideration is to predict level effects, one could choose not to include unit dummies.
They argue that the substantive tradeoff one needs to make is to choose between mild
Zhu
Panel Data Analysis in Public Administration
409
Table 2
Estimation Results: Comparing OLS, FE, and RE Models
(1) OLS
Variable
Coeff.
(T-score)
SE
0.002
0.031
0.006
0.004
0.085
0.041
0.012
0.012
0.060
0.026
0.082
1.058
Coeff.
(Z-score)
−0.005
(−1.04)
0.109*
(1.98)
0.0185
(0.61)
−0.007
(−1.65)
0.204*
(2.46)
0.279**
(5.22)
−0.366
(−1.60)
0.280**
(3.07)
0.702**
(3.02)
−0.062
(−1.70)
−0.118
(−1.66)
4.308
(1.02)
750
—
0.116
0.060
0.058
1.855
(3) FE 2
SE
0.005
0.055
0.031
0.004
0.083
0.053
0.229
0.091
0.233
0.036
0.070
4.234
Coeff.
(Z-score)
−0.005
(−1.04)
0.109*
(1.98)
0.0185
(0.61)
−0.007
(−1.65)
0.204*
(2.46)
0.279**
(5.22)
−0.366
(−1.60)
0.280**
(3.07)
0.702**
(3.02)
−0.062
(−1.70)
−0.118
(−1.66)
11.12
(1.58)
750
—
0.116
1.000
0.842
1.855
(4) RE
SE
0.005
0.055
0.031
0.004
0.083
0.053
0.229
0.091
0.233
0.036
0.070
7.098
Coeff.
(Z-score)
0.011**
(−2.93)
0.076
(1.66)
0.037*
(2.57)
−0.007
(−1.71)
0.186*
(2.33)
0.342**
(6.94)
0.080**
(2.59)
0.259**
(8.46)
0.170
(1.27)
−0.078**
(−3.07)
−0.051
(−0.74)
6.502**
(3.09)
750
—
0.097
0.744
0.627
1.888
SE
0.004
0.046
0.014
0.004
0.080
0.049
0.031
0.031
0.134
0.025
0.069
2.107
Notes: FE1 is the FE model estimated by using the mean-centering approach. FE2 is the FE model estimated including a full set
of state-specific dummies. T-statistics and Z-statistics are reported in parentheses.
Significance level: *p < .05, **p < .01, two-tailed test.
omitted variable bias caused by excluding unit dummies and the saturated model
including all unit dummies. This recommendation, however, is an arbitrary decision
on choosing between two biased models.
Considering Panel Dynamics
Despite that the state-level uninsured rates vary incrementally over time in most states,
data dynamics need to be considered when modeling such a state-level panel data set.
As aforementioned, it is essential to examine whether the pooled 50 time-series are
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−0.011**
(−5.11)
−0.064*
(−2.09)
Public ownership
0.0107
(1.76)
State liberalism
−0.010*
(−2.56)
Unemployment
0.220**
(2.60)
Poverty
0.539**
(13.13)
Black population 0.0494**
(4.17)
Latino population 0.228**
(18.92)
Aged population −0.210**
(−3.51)
Obesity rate
−0.052*
(−1.98)
Perceived poor
0.127
health
(1.55)
Intercept
10.82**
(10.22)
N
750
Adjusted R2
0.663
Within R2
—
Between R2
—
Overall R2
—
RMSE
2.590
Medicaid
eligibility
Public financet−2
(2) FE 1
410
Journal of Public Administration Research and Theory
17 If
the dependent variable is panel stationary and T is very small, dynamic specifications normally are
unnecessary. If panel unit root is detected, regardless the size of T, one should consider modeling the dynamics,
and the ECM approach would be an appropriate solution. If T is large and the dependent variable is highly
autoregressive (i.e., ρ takes a large or near 1 value), the LDV approach could be a parsimonious way to specify
the dynamics.
18 The RE model in table 3 and the LDV model in table 4 are estimated based on the same specification
except the inclusion of a LDV.
19 The only unaffected regressor is Unemployment.
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panel stationary. If so, how much uninsured rates in the current year are dependent
on those in the last year?
Table 3 reports the results of different panel unit root tests, with different number
of lags, a linear trend term, or a drift. Table 3 includes consistent evidence that the
dependent variable is panel stationary when considering a linear trend term, a drift, a
first-order lag, or a combination of these test specifications. If one lags the dependent
variable by 2 years and adds a linear trend term, the Phillip–Perron test still rejects
the null hypothesis of panel unit root. Therefore, one can conclude that the dependent
variable is panel stationary.
Figure 2 further examines the first-order correlation between Uninsured Ratest
and Uninsured Ratest−1 for each state. This descriptive figure is produced by running
50 separate bivariate models, regressing Uninsured Ratest on Uninsured Ratest−1 for
each state. We are interested in examining: (1) if the correlation for each state is near
1 (an integration or near-integration), and (2) if not near 1, if the correlation is statistically significant (an autoregressive process that captures some long-term memory).
Figure 2 shows that most states have mean correlations around 0.5. Statistically significant correlations are observed in 23 states. Insignificant correlations are observed
in 27 states. Table 3 and figure 2 together show that the dependent variable, state-level
uninsured rates, is panel stationary. The specific form of the autoregressive process,
moreover, slightly varies by states.
What would happen if one applied a LDV specification to stationary panel data
with some heterogeneity in the autoregressive parameter? Given that nearly half of
the states do not see a significant correlation between Uninsured Ratest and Uninsured
Ratest−1, including a LDV may wash out some of the level effects by controlling away
level variance without explaining if the long-term relationship between the dependent
variable and the key explanatory variables is common across all states. Similarly, running a first-differenced model or a GMM model becomes unnecessary.17
Table 4 reports the estimation results based on the five dynamic model specifications: the LDV approach, the first-difference specification, the Anderson–Hsiao specification, and two Arellano–Bond GMM specifications. First, the estimation results
of the LDV model show that the average correlation between the DV and the LDV is
0.678. Comparing the coefficients produced by the LDV model and the RE model in
table 2, one may find that including the LDV inflates the overall model fit and within
R2.18 The LDV model, however, does not increase the explanatory power of most
regressors. Instead, including the LDV attenuates the slope coefficients of most exogenous regressors in the model.19 For example, the coefficient of Medicaid Eligibility is
−0.011 in the RE model and becomes −0.004 after including the LDV. The coefficient
of Poverty is shrunk by about 50%, changing from 0.342 to 0.179.
Zhu
Panel Data Analysis in Public Administration
411
Table 3
Different Panel Unit Root Tests, State Uninsured Rates from 1990 to 2006
Test
df
252.269
225.483
189.549
179.152
152.546
117.507
440.559
255.269
254.386
249.366
179.152
179.228
171.487
—
—
—
—
—
—
—
—
100
100
100
100
100
100
100
100
100
100
100
100
100
—
—
—
—
—
—
—
—
T-statistics
—
—
—
—
—
—
—
—
—
—
—
—
—
−2.366
−1.986
−2.668
−2.140
−17.142
−15.264
−20.506
−17.937
p
.000
.000
.000
.000
.000
.111
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.242
.000
.000
.000
.457
Notes: ADF, augmented Dickey–Fuller test; PP, Phillips–Perron test; IPShin, Im–Pearson–Shin test; LLC, Levin–Lin–Chu test.
All the four tests have the null hypothesis of panel nonstationarity.
Table 4 also demonstrates a similar issue of applying a linear dynamic specification based on the GMM approach. Both Arellano–Bond’s models produce smaller
coefficients of the LDV than the LDV model. Using the Arellano–Bond method, the
average correlation between the DV and the LDV is less than 0.40, indicating that
the long-term effects are not very persistent. The dynamic specification, moreover,
reduces the coefficient size of many exogenous regressors (e.g., Medicaid Eligibility,
Unemployment, Poverty, Black Population, and Latino Population).
The first-difference approach and the Anderson–Hsiao specification, furthermore,
are also not good model choices given that the DV is panel stationary. Differencing
both the DV and the regressors removes a large proportion of variance and thus makes
both within R2 and between R2 near zero. Substantively, the two FE models in table 2
and the AH specification in table 4 can be viewed as two typical examples of misspecification: the saturated model and the null model. The unit-specific FE model specification produces a saturated model because it overfits the data. The AH specification
produces a null model that barely explains any variation of state uninsured rates.
Adjusting for Coefficient Stability and the RCM
Because a large proportion of data variation exists across states and the change of
uninsured rates is not very dynamic, one needs to consider a panel model that captures
cross-state variance. The RE model in table 2 produces biased coefficients because it
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ADF, no trend, lag = 0
ADF, no trend, lag = 1
ADF, no trend, lag = 2
ADF, with trend, lag = 0
ADF, with trend, lag = 1
ADF, with trend, lag = 2
ADF, with drift, lag = 0
PP, no trend, lag = 0
PP, no trend, lag = 1
PP, no trend, lag = 2
PP, with trend, lag = 0
PP, with trend, lag = 1
PP, with trend, lag = 2
IPShin, no trend, lag = 1
IPShin, no trend, lag = 2
IPShin, with trend, lag = 1
IPShin, with trend, lag = 2
LLC, no trend, lag = 1
LLC, no trend, lag = 2
LLC, with trend, lag = 1
LLC, with trend, lag = 2
χ2
412
Journal of Public Administration Research and Theory
Figure 2 Illustration of First-order Correlations between Uninsuredt and Uninsuredt−1
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does not capture the possible state-level heterogeneity caused by state-specific effects.
However, running a FE model including all state dummies is also problematic. Thus,
the substantive consideration becomes evaluating how unobserved heterogeneity is
caused by specific states and is not fully explained by the regressors in the model. In
other words, one needs to detect which states are “average states” explained by the
model, and which states are outlier states that require additional model considerations.
Outlier states may be detected by analyzing residuals after fitting a baseline
model (e.g., an OLS model). Evaluating the standardized residuals or the Cook’s
distance normally helps to detect outlier states. Another way to evaluate outlier cases
is to run a weighted (robust) regression analysis weighting observations based on the
absolute value of residuals.
Obesity rate
Aged population
Latino population
Black population
Poverty
Unemployment
State liberalism
Public ownership
Public financet−2
Medicaid eligibility
Uninsuredt−1
0.678**
(26.29)
−0.004*
(−2.33)
−0.0263
(−1.20)
0.002
(0.41)
−0.004
(−1.46)
0.182**
(2.98)
0.179**
(5.52)
0.011
(1.27)
−0.072**
(6.88)
−0.084
(−1.95)
−0.004
(−0.21)
Coeff.
(Z-score)
−0.003
(−0.40)
0.198*
(2.18)
−0.027
(−0.59)
0.002
(0.033)
0.216
(1.78)
0.129**
(2.62)
−1.435
(−1.85)
−0.139
(−0.34)
−0.199
(−0.43)
−0.105
(−1.58)
0.002
0.019
0.043
0.010
0.009
0.033
0.061
0.003
0.004
0.022
—
Coeff.
(Z-score)
0.026
SE
(2) FD
0.066
0.458
0.403
0.775
0.049
0.121
0.006
0.046
0.091
0.007
—
SE
0.143**
(2.99)
−0.003
(−0.47)
0.200*
(2.19)
−0.069
(−1.45)
0.004
(0.68)
0.190
(1.42)
0.112*
(2.15)
−1.446
(−1.79)
0.0867
(0.21)
−0.754
(−1.40)
−0.076
(−1.14)
Coeff.
(Z-score)
(3) AH
0.066
0.538
0.419
0.806
0.052
0.134
0.006
0.048
0.091
0.007
0.047
SE
0.384**
(8.21)
−0.008
(−1.10)
0.298**
(3.89)
−0.045
(−1.01)
0.005
(0.81)
0.150
(1.55)
0.201**
(3.35)
−0.892*
(−2.04)
0.0257
(0.18)
−0.425
(−0.94)
−0.026
(−0.52)
Coeff.
(Z-score)
(4) AB-1
0.050
0.453
0.145
0.436
0.060
0.097
0.006
0.045
0.077
0.007
0.046
SE
0.356**
(10.18)
−0.007
(−1.51)
0.277**
(6.96)
−0.044**
(2.71)
0.004*
(2.03)
0.213**
(5.25)
0.181**
(3.89)
−0.257
(−0.31)
−0.052
(−0.24)
0.305
(0.46)
−0.039
(−1.36)
Coeff.
(Z-score)
0.029
0.671
0.218
0.830
0.047
0.040
0.002
0.016
0.040
0.005
0.035
SE
Continued
(5) AB-2
Panel Data Analysis in Public Administration
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Variable
(1) LDV
Table 4
Estimation Results: Comparing the LDV, the First-difference, the Anderson−Hsiao, and the Arellano–Bond GMM Specifications
Zhu
413
3.233**
(3.97)
750
0.214
0.981
0.829
1.862
Intercept
0.814
0.059
SE
0.284
(1.84)
700
0.036
0.020
0.034
2.005
−0.076
(−1.20)
Coeff.
(Z-score)
(2) FD
0.152
0.064
SE
0.130
(0.82)
650
0.036
0.340
0.022
—
−0.078
(−1.23)
Coeff.
(Z-score)
(3) AH
0.158
0.064
SE
18.81**
(2.71)
700
—
—
—
—
−0.097
(−1.15)
Coeff.
(Z-score)
(4) AB-1
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−0.095*
(−2.05)
4.030
(0.36)
700
—
—
—
—
6.931
Coeff.
(Z-score)
0.084
SE
(5) AB-2
SE
11.256
0.046
Notes: LV refers to the RE model with a lagged dependent variable. FD refers to the first-difference specification by differencing both the dependent variable and all the explanatory variables. AH
refers to the Anderson–Hsiao specification that transforms variables by taking the first-order difference and using ΔYi,t−2 as the instrument. The lagged DV term in Model (3) is Uninsured Ratesi, t−2.
AB-1 refers to the Arellano–Bond GMM specification, using the one-step estimation approach. AB-2 refers to the Arellano–Bond GMM specification, using the two-step estimation approach. In
both AB models, the number of instruments equals to 131. Z-statistics are reported in parentheses.
Significance level: *p < .05, **p < .01, two-tailed test.
N
Within R2
Between R2
Overall R2
RMSE
0.018
(0.31)
Coeff.
(Z-score)
(1) LDV
Perceived poor
health
Variable
Table 4 (continued)
414
Journal of Public Administration Research and Theory
Zhu
Panel Data Analysis in Public Administration
The Beck–Katz Procedure and Beyond
Despite the considerations of modeling heterogeneity and panel dynamics, one also
needs to consider if heterogeneity is detected in the error term, ei,t. After estimating the
OLS model, it is necessary to test for heteroskedasticity, group-based heterogeneity,
cross-section dependence, and serial autocorrelation (usually testing for the first-order
auto-correlation).
After estimating an OLS model including dummy variables for outlier states, the
Breusch–Pagan/Cook–Weisberg test for heteroskedasticity reports a χ2 of 20.72 and a
p-value of .0000. This suggests that heteroskedasticity is detected. The Arellano–Bond
test for AR(1) reports a significant Z-score, suggesting that serial autocorrelation is
detected. After estimating a FE model including the same set of regressors and the full
set of year dummy variables, the modified Wald test reports a χ2 of 299.81(df = 50,
p = .0000), suggesting that panel-wise heteroskedasticity is detected. Lastly, the Frees
method (Frees 1995) is used to test spatial dependence for the panel data with small
T and large N. The spatial dependence test reports a Z-score of 1.057, which is larger
than the corresponding critical values and suggests spatial dependence. The LSDV
model reported in table 5, therefore, needs to be adjusted for correlated errors.
Beck and Katz (1995) develop a robust standard error approach adjusting for
cross-unit dependence. Table 6 reports five models estimated based on the Beck
and Katz approach, with different error term specifications. Model (1) is estimated
adjusting for spatial dependence, panel-wise heteroskedasticity, and panel-specific
serial autocorrelations. Model (2) only corrects for panel-specific serial autocorrelation and panel-wise heteroskedasticity without correcting for spatial dependence.
Model (3) specifies PSAR(1) but assumes independent error structure across panels.
20 Cases
with large residuals are always downweighted when running the robust regression analysis. The
weights are closely corresponding to the Cook’s distance.
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Approximately, as residuals increase, the estimated weights decrease. Hence,
states with relatively small weights can be considered as outlier states.20 Figure 3
illustrates the outlier states detected based on residual analysis, which need to be
controlled. Another possible source of unobserved heterogeneity might be the timespecific effects (i.e., heterogeneous parameters across time). A set of year dummies
can be included in the model to control for time-specific effects. In other words, I use
each year as its own baseline to adjust for unstable coefficient estimations across years.
An alternative way to think about heterogeneity is that parameters may vary
across years, but the variation is random based on a general mean baseline. The
RCM is estimated based on this alternative consideration and uses second-level
(year-level) parameters to capture heterogeneity across time. Table 5 reports the
comparison between the RCM model and the least square model with a set of year
dummies. Both models are estimated by controlling for the outlier states detected
based on residual analysis. The two models produce comparable results for Medicaid
Eligibility, Unemployment, Latino Population, and Obesity Rate; the intercepts differ
substantially across two models and so do the coefficients of Public Finance, Public
Ownership, Poverty, Aged Population, and Perceived Poor Health.
415
Journal of Public Administration Research and Theory
Figure 3 Illustration of Outlier States
25
30
Arkansas
Oklahoma
Florida Arkansas
Utah
20
15
10
Uninsured Rates
Uninsured Rates
30
25
Arkansas
Arkansas
20
15
10
Hawaii
Hawaii
5
100
80
60
Medicaid Eligibility
30
30
Arkansas
15
10
Hawaii
Hawaii
Uninsured Rates
Arkansas
Oklahoma
Florida
Utah
Nevada
20
5
40
20
50
40
30
20
10
Public Ownership
25
Oklahoma
Utah Florida
25
20
15
10
Hawaii
Hawaii
5
7
6
5
4
3
16
14
12
10
8
Public Finance
Arkansas
Oklahoma
Arkansas Nevada
Florida
Utah
Unemployment
Model (1) produces larger standard errors than those in Models (2) and (3) for
most variables, except for Medicaid Eligibility, Black Population and Perceived Poor
Health. In this empirical case, ignoring spatial dependence may lead to overconfidence in the statistical inference. Models (4) and (1) take the same error specification, but Model (4) only specifies a common AR(1). Hence, Model (1) and Model
(4) show the statistical difference between using AR(1) and PSAR(1). Given the
heterogeneous correlation pattern shown in figure 2, it is not surprising to see that
Model (1) produces a larger R2 than Model (4). The coefficients corresponding to
Public Finance, Public Ownership, and Obesity Rate become insignificant in Model
(4). The coefficient size of Aged Population is substantially attenuated. Model (5)
is estimated by including a LDV, without an AR(1) or PSAR(1) specification for
the error term. Model (5) also corrects for panel-wise heteroskedasticity and spatial autocorrelation. Again, including a LDV attenuates most coefficients. This is
because the LDV picks up some long-term effects and the other β coefficients only
carry short-term effects.
Downloaded from http://jpart.oxfordjournals.org/ at Rutgers University on December 1, 2015
Hawaii
Hawaii
5
Uninsured Rates
416
Zhu
Panel Data Analysis in Public Administration
417
Table 5
Estimation Results: Comparing the LSDV and the RCM Specifications
(1) LSDV
Variable
Medicaid eligibility
Public financet−2
Public ownership
State liberalism
Poverty
Black population
Latino population
Aged population
Obesity rate
Perceived poor health
Intercept
Within R2
Between R2
Overall R2
Random effects
SD (cons)
SD (residuals)
N
Log likelihood
Coeff. (Z-score)
SE
Coeff. (Z-score)
SE
−0.010**
(−3.05)
−0.0346
(−0.75)
0.0302*
(2.48)
−0.004
(−1.08)
0.303**
(3.10)
0.327**
(7.03)
0.090**
(3.34)
0.243**
(8.42)
0.110
(0.76)
−0.152**
(−2.59)
0.0319
(0.49)
10.898**
(4.36)
0.245
0.850
0.740
0.003
−0.012**
(−6.43)
−0.062*
(−2.05)
0.008
(1.45)
−0.001
(−0.30)
0.258**
(2.92)
0.564**
(15.47)
0.044**
(3.95)
0.196**
(16.14)
−0.330**
(−4.62)
−0.155**
(−3.97)
0.147*
(2.12)
13.433**
(12.00)
—
—
—
0.002
—
—
750
—
0.046
0.012
0.004
0.098
0.046
0.027
0.029
0.146
0.059
0.066
2.501
0.900
2.163
750
−1659.927
0.029
0.005
0.004
0.090
0.036
0.011
0.012
0.073
0.039
0.069
1.112
0.213
0.057
Notes: LSDV refers to the least square dummy variable specification. RCM refers to the random coefficients model. Z-statistics
are reported in parentheses.
Significance level: *p < .05, **p < .01, two-tailed test.
Comparing estimation results between Model (1) (the LSDV–PCSE model) in
table 6 and the RCM in table 5, RCM produces smaller population-averaged coefficients
than the LSDV–PCSE model for all three institutional variables and State Liberalism.
The coefficients for the two economic variables (Poverty and Unemployment), Latino
Population, and the two health risk variables are larger in the RCM model than those
in the LSDV–PCSE model. The RCM model reported in table 5 only captures random
intercepts across years and assumes that the regression lines for a regressor are parallel across different years. As for Perceived Poor Health, the big shift in coefficient size
indicates that there might be additional variance related to this variable, which is not
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Unemployment
(2) RCM
0.002
−0.017**
(−9.74)
0.136**
(3.11)
0.019*
(2.40)
−0.003
(−0.62)
0.160
(1.21)
0.403**
(8.08)
0.050**
(3.38)
0.171**
(10.98)
−0.435**
(−2.92)
−0.059
(−1.21)
−0.051
(−0.90)
13.47**
(6.67)
750
0.880
2.019
0.056
0.048
0.149
0.016
0.015
0.050
0.132
0.005
0.008
0.044
—
—
−0.017**
(−8.62)
0.136**
(3.51)
0.019**
(2.76)
−0.003
(−0.70)
0.160
(1.94)
0.403**
(10.34)
0.050**
(3.27)
0.171**
(9.42)
−0.435**
(−3.91)
−0.059*
(−2.23)
−0.051
(−0.86)
13.47**
(8.70)
750
0.880
—
1.549
0.059
0.026
0.111
0.018
0.015
0.039
0.083
0.004
0.007
0.039
0.002
—
(2) LSDV PSAR(1)
Coeff.
Robust
(Z-score)
SE
−0.016**
(−7.59)
0.136**
(3.45)
0.0189**
(2.70)
−0.003
(−0.71)
0.160
(1.89)
0.403**
(10.91)
0.050**
(3.31)
0.171**
(9.15)
−0.435**
(−3.63)
−0.059*
(−2.19)
−0.051
(−0.83)
13.47**
(8.02)
750
0.880
—
1.679
0.061
0.027
0.120
0.019
0.015
0.037
0.085
0.004
0.007
0.039
0.002
—
(3) LSDV PSAR(1)
Coeff.
(Z-score)
SE
−0.016**
(−7.34)
0.059
(1.40)
0.016
(1.69)
−0.002
(−0.56)
0.155
(1.07)
0.407**
(7.45)
0.057**
(3.21)
0.222**
(14.77)
−0.284*
(−2.36)
−0.055
(−0.95)
0.016
(0.24)
12.04**
(6.75)
750
0.633
—
1.783
0.067
0.058
0.120
0.015
0.018
0.055
0.146
0.005
0.009
0.042
0.002
—
(4) LSDV AR(1)
Coeff.
(Z-score)
PCSE
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0.603**
(10.05)
−0.005**
(−3.64)
−0.001
(−0.04)
0.001
(0.20)
−0.002
(−0.60)
0.153
(1.63)
0.225**
(5.52)
0.009
(0.96)
0.077**
(4.95)
−0.191**
(−2.88)
−0.016
(−0.42)
0.011
(0.19)
5.092**
(4.37)
750
0.837
1.165
0.059
0.037
0.067
0.016
0.009
0.041
0.094
0.003
0.005
0.005
0.001
0.060
(5) LDV
Coeff.
(Z-score)
PCSE
Notes: LSDV refers to the least square dummy variable specification. LDV refers to the lagged dependent variable specification. AR(1) and PSAR(1) refer to first-order serial autocorrelation and
panel-specific serial autocorrelation, respectively. Z-statistics are reported in parentheses.
Significance level: *p < .05, **p < .01, two-tailed test.
N
R2
Perceived poor
health
Intercept
Obesity rate
Aged population
Latino population
Black population
Poverty
Unemployment
State liberalism
Public ownership
Public financet−2
Medicaid eligibility
Uninsuredt−1
Variable
(1) LSDV PSAR(1)
Coeff.
(Z-score) PCSE
Table 6
Estimation Results: Comparing LSDV and LDV with Different Standard Error Specifications
418
Journal of Public Administration Research and Theory
Zhu
Panel Data Analysis in Public Administration
21 Figure
5 is generated based on a robust analysis incrementally dropping year panels. Both models are not
robust when the number of years is very small. When including all the years from 1998 to 2006, results based
on the RCM specification are quite robust.
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incorporated in the RCM. It is likely that citizens’ perceptions about their own health
status may vary across years. To adjust for possible parameter heterogeneity, one can
add a random slope term into the RCM for the variable Perceived Poor Health.
Model (1) in table 7 reports the estimation results of the RCM model with varying intercepts by year and varying slopes for Perceived Poor Health. Examining the
RCM model in table 7 and Model (1) (the LSDV–PCSE model) in table 6, substantive
findings are comparable based on these two models. Controlling for the institutional
characteristics of state health care systems and the macroeconomy (Unemployment
and Poverty), state liberalism does not predict the level of state uninsured rates. Both
institutional and economic factors matter substantively for explaining state uninsured
rates. The uninsured rates are slightly lower in states with generous Medicaid eligibility rules and more public finance in health care than in states with tight Medicaid
eligibility rules and less public finance in health care. Both unemployment and poverty
strongly predict the level of uninsured rates.
The RCM model in table 7 is preferred to the LSDV–PCSE model in table 6
because of two reasons. First, the RCM specification better captures the nature of
heterogeneity across state-year observations. Although the substantive findings for
many variables based on population-averaged coefficients in two models are similar,
the RCM specification explicitly models heterogeneity and is more flexible to capture
varying parameters across time. Figure 4 illustrates homogeneity and heterogeneity
in parameters. In this panel data set, the coefficients of the poverty measure do not
vary across different years. The coefficients of the variable for perceived poor health,
however, exhibit a heterogeneous pattern across different years. Although the LSDV–
PCSE specification also deals with heterogeneity across time by including a set of year
dummies, it does not accurately reflect the varying slopes for this particular variable.
When parameter heterogeneity is detected, it is better to model the heterogeneous
coefficients instead of simply assuming that all varying parameters are nested in the
year dummy variables. The RCM model reports findings on public ownership and
public finance that make more substantive sense. The LSDV–PCSE model reports
positive associations between public ownership, which is opposite to the substantive
expectation. The RCM model, however, reports negative associations between the two
institutional variables and the uninsured rates.
Second, the RCM specification performs better than the LSDV–PCSE model
based on coefficient stability at the population-averaged level. Figure 5 compares the
performances of the two models based on the population-averaged coefficients of
six variables: Medicaid Eligibility, Public Finance, Public Ownership, Unemployment,
Poverty, and Latino Population.21 All the six subfigures demonstrate that the LSDV–
PCSE model produces the slope coefficients that are more sensitive to changes in
year panels than those produced by the RCM model. Figure 5(a–c) shows that the
RCM model yields a more stable mean estimation for the three institutional variables:
Medicaid Eligibility, Public Finance, and Public Ownership when T is relatively large.
The coefficients of Poverty and Unemployment are both insensitive in two models,
419
420
Journal of Public Administration Research and Theory
Table 7
RCM with Random Intercept and Random Slope
(1) RCM
Variable
(2) RCM–EC
Coeff. (Z-score)
SE
Coeff. (Z-score)
SE
0.002
−0.005**
(−3.16)
−0.020
(-0.88)
0.002
(0.45)
−0.001
(−0.43)
—
0.002
Unemploymentt-1
−0.012**
(6.29)
−0.060*
(−2.03)
0.008
(1.45)
−0.001
(−0.36)
0.238**
(2.73)
—
∆Unemploymentt
—
—
Uninsuredt−1
− Unempt−1
Poverty
—
—
0.563**
(15.48)
0.044**
(3.94)
0.197**
(16.37)
−0.330**
(−4.63)
−0.149**
(−3.84)
0.136
(1.74)
13.453**
(12.10)
0.036
Medicaid eligibility
Public financet−2
Public ownership
State liberalism
Black population
Latino population
Aged population
Obesity rate
Perceived poor health
Intercept
Random effects
SD (poor health)
SD (cons)
SD (residuals)
N
Log likelihood
0.134
0.675
2.158
750
−1659.16
0.005
0.003
0.087
—
0.011
0.012
0.071
0.039
0.078
1.111
0.072
0.313
0.057
−0.276**
(−3.80)
0.010
(0.08)
−0.391**
(−13.71)
0.226**
(6.98)
0.013
(1.47)
0.075**
(6.78)
−0.171**
(−3.03)
−0.054
(1.83)
0.037
(0.63)
5.747**
(6.16)
0.089
0.484
1.696
750
−1476.76
0.023
0.004
0.003
—
0.073
0.126
0.029
0.032
0.009
0.011
0.057
0.030
0.060
0.934
0.050
0.187
0.044
Notes: Model (1) is estimated including random intercepts and random slopes for Perceived Poor Health. Model (2) is estimated
including random intercepts, random slopes for Perceived Poor Health and an EC specification for Unemployment. Z-statistics
are reported in parentheses.
Significance level: *p < .05, **p < .01, two-tailed test.
especially Unemployment. The RCM model produces slightly different populationaveraged coefficients than the LSDV–PCSE because the RCM model does not assume
that every year generates its unique baseline value.
In addition, figure 5(d) shows that the effects of Unemployment, estimated in both
the RCM and LSDV–PCSE model, may not be time-consistent, especially in years
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Unemployment
0.029
Zhu
Panel Data Analysis in Public Administration
421
Figure 4 Homogeneity and Heterogeneity in Parameter
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
30
20
10
0
30
20
10
20
10
0
30
20
10
0
25
20
15
10
5
25
20
15
10
5
(a) State Poverty Rate: Parameter Homogeneity
Fitted Values
Uninsured Rate
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
30
20
10
0
30
20
10
0
30
20
10
0
30
20
10
0
10
8
6
4
2
10
8
6
4
2
(b) Self−Reported Poor Health Status (% population): Parameter Heterogeneity
Fitted values
Uninsured Rate
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0
30
.1
.15
.2
.8
1
.25
1.2
−1.5 −1 −.5
−.04 −.02
0
0
.5
.02
1
.04
Year<1994
.6
Year<1995
.4
Year<1994
95% CIs
prior to 1998. This may point to the possibility that there is some long-term equilibrium
relationship between unemployment and uninsured rates. This might be a reasonable
substantive consideration given that most state health care systems are labor market–
driven. Model (2) in table 7 incorporates this substantive consideration and adds an
error correction (EC) specification for Unemployment. As Model (2) shows, with a
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Year<1994
Year<1996
Full Sample
Year<2006
Year<2005
Year<2004
Year<2003
Year<2002
Year<2001
Year<2000
Year<1999
Year<1998
Year<1997
Year<1996
Year<1995
Year<1994
Full Sample
Year<2006
Year<2005
Year<2004
Year<2003
Year<2002
Year<2001
Year<2000
Year<1999
Year<1998
Year<1997
Year<1995
Year<1999
Year<1998
Year<1997
Year<1996
Year<1995
Year<1994
Full Sample
Year<2006
Year<2005
Year<2004
Year<2003
Year<2002
Year<2001
Year<2000
Year<1999
Year<1998
Year<1997
Year<1995
Year<1997
Year<1998
Year<1999
Year<2000
Year<2001
Year<2002
Year<2003
Year<2004
Year<2005
Year<2006
Full Sample
Year<1994
Year<1995
Year<1996
Year<1997
Year<1998
Year<1999
Year<2000
Year<2001
Year<2002
Year<2003
Year<2004
Year<2005
Year<2006
Full Sample
(f) Sample: Latino Population
(e) Sample:Poverty
Year<1996
(d) Sample: Unemployment
(c) Sample: Ownership
Year<1996
(b) Sample: Public Finance
(a) Sample: Medicaid Eligibility
RCM
95% CIs
LSDV−PCSE
Year<2000
Year<2001
Year<2002
Year<2003
Year<2004
Year<2005
Year<2006
Full Sample
.2
−.3 −.2 −.1
0
−.03 −.02 −.01
.1
0
.2
.01
Journal of Public Administration Research and Theory
422
Figure 5 Comparing Coefficients’ Stability: LSDV–PCSE and RCM
Zhu
Panel Data Analysis in Public Administration
Discussion: Implications for Observational Studies Using Panel Data
In this article, I have discussed various panel models suitable for different substantive
considerations. Panel data can become powerful research designs for answering questions of interest in public administration. Panel data analysis can be useful in analyzing incremental changes in policies, comparing policy institutions, or generalizing
substantive relationships across organizational contexts. The empirical example discussed in this article, however, does not represent all types of panel data sets. Beside the
CSTS example discussed in this article (i.e., continuous dependent variable with large
N and small T), a panel data set may also have larger T than N, or contain hierarchical data structures (e.g., individual-level observations nested in social groups, organizations nested in different levels of government, and so on). Public administration
scholars may also be concerned with different substantive relationships that require
alternative panel data approaches. For example, the generalized estimating equation
(GEE) method would be appropriate if the substantive question concerns inferring
average groups effects instead of unit-specific effects. Statistically, the GEE method is
flexible in that it allows “for a range of substantively-motivated correlation patterns
within [data] clusters and offer the potential for valuable substantive insights into the
dynamics of that correlation” (Zorn 2001, 470).22 More complex multi-level specifications should be considered when the substantive questions focus on micro–macro connections, such as how institutional contexts are linked to micro-level administrative
decisions (Gelman and Hill 2007). Public administration scholars, furthermore, often
explore substantive policy outcomes pertaining to small-domain estimations, such as
public health outcomes for racial/ethnic minorities, unemployment in local communities, failing organizations, etc. Advanced statistical methods such as Bayesian analysis
22 See
Whitford and Yates (2009) for a recent example of applying GEE models in public policy and public
administration research. The GEE models estimate the population-averaged expectation of the dependent
variable instead of unit-specific or time-specific conditional means (Zorn 2001).
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dynamic specification for the Unemployment variable, the dependent variable has now
taken a differencing transformation. Added in the model are a change term, a lag
term of Unemployment, and the disequilibrium term (Uninsuredt−1 – Unemploymentt−1).
In Model (2), significant coefficients are observed for Unemploymentt−1 and (Uninsur
edt−1 –Unemploymentt−1), indicating that state uninsured rates and unemployment, on
average, are tracking each other from a long run. The two models in table 7 clearly
show some statistical tradeoffs. Adding EC as a dynamic specification shrinks coefficients for other regressors because Model (2) reflects how these variables affect changes
in uninsured rates. Without specifying the long-run dynamic relationship, Model (1)
slightly underestimates the effect of Unemployment on uninsured rates. Nevertheless,
coefficient signs remain the same in Models (1) and (2), pointing toward the same
direction of relationships. If the substantive focus is to examine how economic factors
(e.g., unemployment) affect people’s access to health care, one certainly needs to put
more efforts into thinking about the nuanced difference between long-term and shortterm effects.
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Journal of Public Administration Research and Theory
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(Gill and Meier 2000; Wagner and Gill 2005) would be called for making substantive
inferences based on small-domain statistics.
Beyond the aforementioned different substantive considerations, many public
administration data sets contain discrete policy variables. For example, public managers’ rating of their organizational performance can be recorded by ordered grade
scales. The frequency of networking activities can be coded as count data or ordinal scales. Statistical specifications suitable for linear probability models are needed
to model panel data with discrete dependent variables (Baltagi 2008; Honore and
Kryiazidou 2000). Although statistical considerations may vary in accordance with
different data structures, the preceding theoretical discussion on panel methods and
the analytic example led to several general implications for observational studies in
using panel data.
First, and most importantly, different models make different underlying
assumptions about the nature and the source of heterogeneity. Pooling data across
time and space, as Stimson (1985) warns, may raise challenges to valid inferences
because unobserved heterogeneity could be time-dependent and spatially dependent.
Although panel data have advantages over pure cross-sectional and time-series data,
they require scholars to consider how to model heterogeneity substantively. If heterogeneity mainly exists across the 50 states, the underlying data-generating process
may be applied to some states, but not others. If heterogeneity is time-dependent, one
needs to consider how to model heterogeneous dynamics. The state-level panel data
example shows that some conventional techniques may produce problematic results.
For example, the unit-specific FE model is widely used in observational studies to
control for the potential omitted variable bias. Running a FE model by states, however, produces artificial null findings for substantively interesting variables. The oneway RE model, furthermore, is inconsistent because it assumes unit homogeneity.
The preferred model choice, in this example, is the RCM specification that fits the
data with partial pooling.
Second, dummy variables are effective ways to control for unobserved heterogeneity, but they are not substantively helpful to explain why heterogeneity is state-specific
or year-specific. In the analytic example, a few states are identified to be outlier states
that do not fit the partial pooling model. Using dummy variables to control for these
outlier states improves estimation reliability, but tells nothing about why in Hawaii a
relatively large increase in unemployment did not lead to substantial changes in the
uninsured rates. Nor can the state dummy variables explain why in Arkansas, with
the same Medicaid eligibility rules and the same level of public finance, the uninsured
rate was around 28% in one year and around 25% in another year (see figure 3).
Further theorization on these substantively interested states is warranted. Could it
be possible that the macroeconomic forces interact with health care institutions to
affect citizens’ access to health care? If this conjecture were true, an interaction term
between unemployment and the institutional variables could be added into the model
to evaluate how institutional arrangements buffer the economic shocks. Could it also
be possible that Hawaii is a pioneering state mandating comprehensive health insurance, and thus the uninsured rate in Hawaii is less responsive to the macroeconomy
than that in a different state? To evaluate this conjecture, one would need to include
a new variable measuring state mandates on employment-based health insurance. In
Zhu
Panel Data Analysis in Public Administration
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