Unit Averaging for Heterogeneous Panels Christian BrownleesVladislav Morozov

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Unit Averaging
for Heterogeneous Panels
Christian Brownlees†Vladislav Morozov‡∗
May 10, 2024
Abstract
In this work we introduce a unit averaging procedure to efficiently recover unit-
specific parameters in a heterogeneous panel model. The procedure consists in
estimating the parameter of a given unit using a weighted average of all the unit-
specific parameter estimators in the panel. The weights of the average are determined
by minimizing an MSE criterion we derive. We analyze the properties of the resulting
minimum MSE unit averaging estimator in a local heterogeneity framework inspired
by the literature on frequentist model averaging, and we derive the local asymptotic
distribution of the estimator and the corresponding weights. The benefits of the
procedure are showcased with an application to forecasting unemployment rates for a
panel of German regions.
Keywords: heterogeneous panels, frequentist model averaging, prediction
JEL: C33, C52, C53
1 Introduction
Estimation of unit-specific parameters in panel data models with heterogeneous parameters
is a topic of active research in econometrics (Maddala, Trost, Li, and Joutz,1997;Pesaran,
Shin, and Smith,1999;Wang, Zhang, and Paap,2019;Liu, Moon, and Schorfheide,2020).
Estimation of unit-specific parameters is relevant, for instance, when interest lies in con-
structing forecasts for the individual units in the panel (Baltagi,2013;Zhang, Zou, and
∗†
Department of Economics and Business, Universitat Pompeu Fabra and Barcelona School of Economics;
e-mail: christian.brownlees@upf.edu;
‡
Department of Economics and Business, Universitat Pompeu Fabra and Barcelona School of Economics;
e-mail: vladislav.morozov@barcelonagse.eu.Corresponding author.
We thank Jan Ditzen, Kirill Evdokimov, Geert Mesters, Luca Neri, Katerina Petrova, Barbara Rossi,
Wendun Wang, and the participants at 26th IPDC, 7th RCEA Time Series Workshop, 9th SIDE WEEE,
the 2021 ERFIN workshop, and the 2023 EEA-ESEM conference for comments and discussion. Christian
Brownlees acknowledges support from the Spanish Ministry of Science and Technology (Grant MTM2012-
37195) and the Spanish Ministry of Economy and Competitiveness through the Severo Ochoa Programme
for Centres of Excellence in R&D (SEV-2011-0075).
1
arXiv:2210.14205v3 [econ.EM] 12 May 2024
Liang,2014;Wang et al.,2019;Liu et al.,2020), which typically arises in the analysis of
international panels of macroeconomic time series (Marcellino, Stock, and Watson,2003).
Other unit-specific parameters of interest include individual coefficients (Maddala et al.,
1997;Maddala, Li, and Srivastava,2001;Wang et al.,2019) and long-run effects of a change
in a covariate (Pesaran and Smith,1995;Pesaran et al.,1999).
There are three natural strategies for estimating unit-specific parameters (Baltagi,
Bresson, and Pirotte,2008). The simplest approach consists in estimating each unit-
specific parameter from its individual time series. While this strategy typically leads to
approximately unbiased estimation, such estimators suffer from large estimation variability
when the time dimension is small. In the second approach, an assumption of parameter
homogeneity is imposed and a common panel-wide estimator is used for all unit-specific
parameters. This strategy leads to small variability; however, it suffers from large bias in
the presence of heterogeneity. The third strategy is a compromise between the first two. It
uses panel-wide information to reduce the variability of the individual estimator to obtain
an estimator with favorable risk properties (Maddala et al.,2001;Wang et al.,2019;Liu
et al.,2020). This is appealing when the time dimension is moderate in the sense that there
is a nontrivial bias-variance trade-off between individual-specific and panel-wide estimation.
In this paper we propose a novel compromise estimator for unit-specific “focus” param-
eters — the unit averaging estimator. Focus parameters considered are smooth transfor-
mations of unit-specific parameters, including the examples mentioned above. The unit
averaging estimator for the unit-specific focus parameter is defined as a weighted average of
all the unit-specific focus parameter estimators in the panel. The weights are chosen by
minimizing one of the two unit-specific mean squared error (MSE) criteria we derive. One of
the criteria can leverage prior information about similarities between cross-sectional units in
terms of their parameters. The other criterion is agnostic and requires no prior information.
In both cases, the weights solve a straightforward quadratic optimization problem. The
estimator is fairly general and is designed for possibly nonlinear and dynamic panel models
estimated by M-estimation.
We analyze the theoretical properties of the our unit averaging methodology. We focus
on a moderate-
𝑇
setting — a setting in which the amount of information in each time
series is limited and the variance of individual estimators is of the same order of magnitude
as the coefficients. In this setting, we derive the leading terms of the MSE of the unit
averaging estimator. We do so using a limited information local asymptotic technique under
a local heterogeneity framework, in which the unit-specific coefficients are local in the time
dimension to a common mean. This theoretical device emulates a moderate-
𝑇
setting and
the trade-off between unit-specific and panel-wide information. It is inspired by the local
misspecification technique used in the frequentist model averaging literature for analyzing
2
finite-sample properties of estimators (Hjort and Claeskens,2003a;Liu,2015;Hansen,2016).
We propose and analyze minimum MSE weights that minimize an estimator of the
leading terms of the MSE. As we show, these minimum MSE weights minimize an appro-
priately defined notion of the population MSE contaminated by a noise component that
we characterize explicitly. We obtain the limiting distribution of the minimum MSE unit
averaging estimator in a local heterogeneity setting, similarly to Liu (2015). Finally, we
argue that the minimum MSE weights also have desirable properties a large-
𝑇
setting, in
which the amount of information in each time series grows without bound.
In a simulation study, we assess the finite sample properties of the our methodology.
We compare our minimum MSE unit averaging estimator against the unit-specific and
mean group estimators, along with AIC and BIC weighted averaging estimators (Buckland,
Burnham, and Augustin,1997). The proposed methodology performs favorably relative to
these benchmarks. Gains in the MSE are possible without prior information about unit
similarity. However, leveraging prior information may lead to stronger improvements.
An application to forecasting regional unemployment in Germany showcases the method-
ology (Schanne, Wapler, and Weyh,2010). Unemployment forecasting is a natural appli-
cation of the unit averaging methodology since the literature documents both evidence of
regional heterogeneity and the benefits of pooling data (Schanne et al.,2010;de Graaff,
Arribas-Bel, and Ozgen,2018). We find that unit averaging using minimum MSE weights
improves prediction accuracy. The gains in the MSE are larger for shorter panels.
This paper is related to two strands of the literature. First, it contributes to the literature
on estimation of unit-specific parameters. Important contributions in this area include
Zhang et al. (2014), Wang et al. (2019), Issler and Lima (2009) and Liu et al. (2020). In
contrast to these contributions, we focus on a setting where the time dimension is moderate
(as opposed to either large or small). Moreover, the existing literature largely focuses on
linear models under strict exogeneity (Baltagi et al.,2008;Wang et al.,2019) whereas our
framework allows for nonlinear and dynamic models. Second, our paper is related to the
literature on frequentist model averaging. Important contributions in this area include Hjort
and Claeskens (2003a), Hansen (2007), Hansen (2008), Wan, Zhang, and Zou (2010), Hansen
and Racine (2012), Liu (2015), and Gao, Zhang, Wang, and Zou (2016), among others. Gao
et al. (2016); Yin, Liu, and Lin (2021) deal with model averaging estimators specifically
tailored for panel models. The main difference with respect to these contributions is that
we focus on averaging different units with the same model whereas these papers average
different models for a given fixed unit or the pooled data.
The rest of the paper is structured as follows. Section 2introduces the unit averaging
methodology. Section 3studies the theoretical properties of the procedure. Section 4
contains the simulation study. Section 5contains the empirical application. Concluding
3
remarks follow in section 6. All proofs are collected in the proof appendix. Further
theoretical, numerical, and empirical results are collected in an online appendix.
2 Methodology
We introduce our unit averaging methodology within the framework of a fairly general
class of panel data models with heterogeneous parameters. Let
{𝑧𝑖 𝑡}
with
𝑖
= 1
, . . . , 𝑁
and
𝑡
= 1
, . . . , 𝑇
denote a panel where
𝑧𝑖 𝑡
denotes a random vector of observations taking values
in
𝒵 ⊂ R𝑑
. For each unit in the panel, we define the unit-specific parameter
𝜃𝑖∈
Θ
⊂R𝑝
as
𝜃𝑖= arg max
𝜃∈Θ
E1
𝑇
𝑇
𝑡=1
𝑚(𝜃,𝑧𝑖 𝑡),
where 𝑚: Θ × 𝒵 → Ris a smooth criterion function.
Our interest lies in estimating the unit-specific “focus” parameter
𝜇
(
𝜃𝑖
) for a fixed unit
𝑖
with minimal MSE, where
𝜇
: Θ
→R
is a smooth function (similarly to the setup in Hjort
and Claeskens (2003a)). For example,
𝜇
(
𝜃𝑖
) may denote a component of
𝜃𝑖
, the conditional
mean of a response variable given the covariates, or the long-run effect of a covariate. To
simplify exposition and without loss of generality, we focus on the problem of estimating
the focus parameter
𝜇
(
𝜃1
) for unit 1. In this paper we consider the case in which the focus
function
𝜇
is scalar-valued. It is straightforward to generalize the framework to a focus
function taking values in R𝑞for some 𝑞 > 1.
To estimate 𝜇(𝜃1) we consider the class of unit averaging estimators given by
^𝜇(𝑤) =
𝑁
𝑖=1
𝑤𝑖𝜇(^
𝜃𝑖),(1)
where
𝑤
= (
𝑤𝑖
) is a
𝑁
-vector such that
𝑤𝑖≥
0 for all
𝑖
and
𝑁
𝑖=1 𝑤𝑖
= 1, and
^
𝜃𝑖
is the
unit-specific estimator of unit 𝑖= 1, . . . , 𝑁, given by
^
𝜃𝑖= arg max
𝜃∈Θ
1
𝑇
𝑇
𝑡=1
𝑚(𝜃,𝑧𝑖 𝑡).(2)
The class of estimators in
(1)
is fairly broad and contains a number of important special
cases. It includes the individual estimator of unit 1
^𝜇1
=
𝜇
(
^
𝜃1
) and the mean group
estimator
^𝜇𝑀𝐺
=
𝑁−1𝑁
𝑖=1 𝜇
(
^
𝜃𝑖
). It also includes estimators based on smooth AIC/BIC
weights (Buckland et al.,1997) as well as Stein-type estimators (Maddala et al.,1997).
The class of estimators in
(1)
may be motivated by the following representation for the
4
individual parameters
𝜃𝑖
. Assume that
𝜃𝑖
can be written as
𝜃𝑖
=
𝜃0
+
𝜂𝑖
, where
𝜃0
is a
common mean component and
𝜂𝑖
is a zero-mean random component. All units in the panel
carry information on
𝜃0
, and so all units may be useful for estimating
𝜃1
=
𝜃0
+
𝜂1
. The
vector of weights
𝑤
controls the balance between the bias and the variance of estimator
(1)
. Assigning a large weight to unit 1 leads to low bias but may also lead to excessive
variability. Alternatively, assigning larger weights to units other than unit 1 induces bias
but may substantially reduce variability. This bias-variance trade-off is most relevant in
a moderate-
𝑇
setting, defined as the range of values of
𝑇
for which the variability of the
individual estimators
^
𝜃𝑖
is of the same order of magnitude as
𝜂𝑖
(see remark 1below for a
heuristic criterion for detecting a moderate-𝑇setting).
In this work we introduce two weighting schemes — the fixed-
𝑁
and the large-
𝑁
minimum-MSE unit averaging estimators. The key practical difference between the two is
that the large-
𝑁
estimator uses prior information about the similarity of cross-sectional
units in terms of the focus parameter. In contrast, the fixed-
𝑁
estimator requires no prior
information (see the discussion following eq.
(6)
explaining the names of the approaches)
These estimators seek to strike a balance between the bias and variance of the unit averaging
estimator. For both, the weights are chosen by minimizing an estimator of the local
approximation to the MSE (LA-MSE) of the unit averaging estimator. The LA-MSE
contains the leading terms of the the moderate-
𝑇
MSE of the unit averaging estimator and
is justified in detail in the next section.
The fixed-
𝑁
approach provides an agnostic way to determine the weights. It imposes no
structure on the weights. All of the weights are determined only by the data. Formally, let
¯
𝑁 < ∞be the number of units. Let 𝑤¯
𝑁= (𝑤¯
𝑁
𝑖) be a ¯
𝑁-vector such that 𝑤¯
𝑁
𝑖≥0 for all 𝑖
and ¯
𝑁
𝑖=1 𝑤¯
𝑁
𝑖= 1. The fixed-𝑁LA-MSE estimator associated with 𝑤¯
𝑁is given by
\
𝐿𝐴-𝑀𝑆𝐸 ¯
𝑁(𝑤¯
𝑁) =
¯
𝑁
𝑖=1
¯
𝑁
𝑗=1
𝑤¯
𝑁
𝑖[^
Ψ¯
𝑁]𝑖 𝑗𝑤¯
𝑁
𝑗,(3)
where
^
Ψ¯
𝑁∈R¯
𝑁ׯ
𝑁
with entries [
^
Ψ¯
𝑁
]
𝑖 𝑖
=
∇𝜇
(
^
𝜃1
)
′
(
𝑇
(
^
𝜃𝑖−^
𝜃1
)(
^
𝜃𝑖−^
𝜃1
)
′
+
^
𝑉𝑖
)
∇𝜇
(
^
𝜃1
) and
[
^
Ψ¯
𝑁
]
𝑖 𝑗
=
∇𝜇
(
^
𝜃1
)
′𝑇
(
^
𝜃𝑖−^
𝜃1
)(
^
𝜃𝑗−^
𝜃1
)
′∇𝜇
(
^
𝜃1
) when
𝑖̸
=
𝑗
. Here
^
𝑉𝑖
is an estimator of the
asymptotic variance of
^
𝜃𝑖
, and
∇𝜇
(
·
) is the gradient of
𝜇
. The terms
∇𝜇
(
^
𝜃1
)
′𝑇
(
^
𝜃𝑖−^
𝜃1
)(
^
𝜃𝑖−
^
𝜃1
)
′∇𝜇
(
^
𝜃1
) and
∇𝜇
(
^
𝜃1
)
′^
𝑉𝑖∇𝜇
(
^
𝜃1
) are estimators of, respectively, the squared bias and
variance of
𝜇
(
^
𝜃𝑖
) as estimators of
𝜇
(
𝜃1
). The fixed-
𝑁
minimum MSE weights are defined as
^
𝑤¯
𝑁= arg min
𝑤∈Δ¯
𝑁
\
𝐿𝐴-𝑀𝑆𝐸 ¯
𝑁(𝑤),(4)
where Δ ¯
𝑁={𝑤∈R¯
𝑁:¯
𝑁
𝑖=1 𝑤𝑖= 1, 𝑤𝑖≥0, 𝑖 = 1,..., ¯
𝑁}.
5
摘要：

UnitAveragingforHeterogeneousPanelsChristianBrownlees†VladislavMorozov‡∗May10,2024AbstractInthisworkweintroduceaunitaveragingproceduretoefficientlyrecoverunit-specificparametersinaheterogeneouspanelmodel.Theprocedureconsistsinestimatingtheparameterofagivenunitusingaweightedaverageofalltheunit-specif...

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