Unit Averaging for Heterogeneous Panels

Unit Averaging for Heterogeneous Panels

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Introduction

Estimation of unit-specific parameters in panel data models with heterogeneous parameters is a topic of active research in econometrics Maddala1997,Pesaran1999,Wang2019,Liu2020. Estimation of unit-specific parameters is relevant, for instance, when interest lies in constructing forecasts for the individual units in the panel Baltagi2013,Zhang2014,Wang2019,Liu2020, which typically arises in the analysis of international panels of macroeconomic time series Marcellino2003. Other unit-specific parameters of interest include individual coefficients Maddala1997, Maddala2001, Wang2019 and long-run effects of a change in a covariate Pesaran1995, Pesaran1999.

There are three natural strategies for estimating unit-specific parameters Baltagi2008a. 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 Maddala2001, Wang2019, Liu2020. 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” parameters --- the unit averaging estimator. Focus parameters considered are smooth transformations 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 finite-sample properties of estimators Hjort2003,Liu2015, Hansen2016.

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 appropriately 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 Liu2015. 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 Buckland1997. 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 methodology Schanne2010. Unemployment forecasting is a natural application of the unit averaging methodology since the literature documents both evidence of regional heterogeneity and the benefits of pooling data Schanne2010, Graaff2018. 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 Zhang2014, Wang2019, Issler2009 and Liu2020. 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 Baltagi2008a, Wang2019 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 Hjort2003, Hansen2007, Hansen2008, Wan2010, Hansen2012, Liu2015, and Gao2016, among others. Gao2016, Yin2019 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 (ref) introduces the unit averaging methodology. Section (ref) studies the theoretical properties of the procedure. Section (ref) contains the simulation study. Section (ref) contains the empirical application. Concluding remarks follow in section (ref). All proofs are collected in the proof appendix. Further theoretical, numerical, and empirical results are collected in an online appendix.

Methodology

We introduce our unit averaging methodology within the framework of a fairly general class of panel data models with heterogeneous parameters. Let $\{ {\bm{z}_{i\,t}} \}$ with $i=1,\ldots, N$ and $t=1,\ldots, T$ denote a panel where $\bm{z}_{i\,t}$ denotes a random vector of observations taking values in $\mathcal Z \subset \mathbb R^d$. For each unit in the panel, we define the unit-specific parameter $\bm{\theta}_{i} \in \Theta \subset \mathbb R^p$ as

where $m : \Theta \times \mathcal Z \rightarrow \mathbb R $ is a smooth criterion function.

Our interest lies in estimating the unit-specific “focus” parameter $\mu(\bm{\theta}_{i})$ for a {fixed} unit $i$ with minimal MSE, where $\mu : \Theta \rightarrow \mathbb R$ is a smooth function (similarly to the setup in Hjort2003). For example, $\mu(\bm{\theta}_i)$ may denote a component of $\bm{\theta}_i$, 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 $\mu(\bm{\theta}_{1})$ for unit 1. In this paper we consider the case in which the focus function $\mu$ is scalar-valued. It is straightforward to generalize the framework to a focus function taking values in $\mathbb R^q$ for some $q >1$.

To estimate $\mu(\bm{\theta}_1)$ we consider the class of unit averaging estimators given by

where $\bm{w} = ( w_{i} ) $ is a $N$-vector such that $ w_{i} \geq 0$ for all $i$ and $\sum_{i=1}^N w_{i}=1$, and $\hat{\bm{\theta}}_i$ is the unit-specific estimator of unit $i=1, \dots, N$, given by

The class of estimators in (ref) is fairly broad and contains a number of important special cases. It includes the individual estimator of unit 1 \( \hat{\mu}_1 = \mu(\hat{\bm{\theta}}_1) \) and the mean group estimator $\hat \mu_{MG} = N^{-1} \sum_{i=1}^N \mu(\hat{\bm{\theta}}_i)$. It also includes estimators based on smooth AIC/BIC weights Buckland1997 as well as Stein-type estimators Maddala1997.

The class of estimators in (ref) may be motivated by the following representation for the individual parameters $\bm{\theta}_i$. Assume that $ \bm{\theta}_i$ can be written as $ \bm{\theta}_i = \bm{\theta}_0 + \bm{\eta}_i $, where $\bm{\theta}_0$ is a common mean component and $\bm{\eta}_i$ is a zero-mean random component. All units in the panel carry information on $\bm{\theta}_0$, and so all units may be useful for estimating $\bm{\theta}_1=\bm{\theta}_0+ \bm{\eta}_1$. The vector of weights $\bm{w}$ controls the balance between the bias and the variance of estimator (ref). 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 $\hat{\bm{\theta}}_i$ is of the same order of magnitude as $\bm{\eta}_i$ (see remark (ref) below for a heuristic criterion for detecting a moderate-$T$ setting).

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.\ (ref) 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 $\bar{N}<\infty$ be the number of units. Let $\bm{w}^{\bar{N}}=(w_i^{\bar{N}})$ be a $\bar{N}$-vector such that $w_i^{\bar{N}}\geq 0$ for all $i$ and $\sum_{i=1}^{\bar{N}} w_i^{\bar{N}}=1$. The fixed-$N$ LA-MSE estimator associated with $\bm{w}^{\bar{N}}$ is given by

where $\hat{\bm{\Psi}}_{\bar{N}} \in \mathbb R^{\bar{N} \times \bar{N}}$ with entries $[\hat{\bm{\Psi}}_{\bar{N}}]_{i\,i} = \nabla \mu(\hat{\bm{\theta}}_1)' ( T ( \hat{\bm{\theta}}_i -\hat{\bm{\theta}}_1 )( \hat{\bm{\theta}}_i -\hat{\bm{\theta}}_1 )' + \hat{\bm{V}}_i ) \nabla \mu(\hat{\bm{\theta}}_1)$ and $[\hat{\bm{\Psi}}_{\bar{N}}]_{i\,j} = \nabla \mu(\hat{\bm{\theta}}_1)' T ( \hat{\bm{\theta}}_i -\hat{\bm{\theta}}_1 )( \hat{\bm{\theta}}_j -\hat{\bm{\theta}}_1 )' \nabla \mu(\hat{\bm{\theta}}_1)$ when $i\neq j$. Here $\hat \bm{V}_i$ is an estimator of the asymptotic variance of $\hat \bm{\theta}_i$, and $\nabla \mu(\cdot)$ is the gradient of $\mu$. The terms \( \nabla \mu(\hat{\bm{\theta}}_1)'T (\hat{\bm{\theta}}_i -\hat{\bm{\theta}}_1 )( \hat{\bm{\theta}}_i -\hat{\bm{\theta}}_1 )' \nabla \mu(\hat{\bm{\theta}}_1)\) and \( \nabla \mu(\hat{\bm{\theta}}_1)'\hat{\bm{V}}_i \nabla \mu(\hat{\bm{\theta}}_1)\) are estimators of, respectively, the squared bias and variance of $\mu (\hat \bm{\theta}_i )$ as estimators of $\mu (\bm{\theta}_1 )$. The fixed-$N$ minimum MSE weights are defined as

where $\Delta^{\bar{N}} = \curl{\bm{w}\in \operatorname{\mathbb{R}}^{\bar{N}}: \sum_{i=1}^{\bar{N}} w_i=1, w_i\geq 0, i=1,\dots, \bar{N} }$.

Alternatively, the researcher may have prior information on which units are potentially more important for estimating $\mu(\bm{\theta}_1)$ (in terms of having a similar $\mu(\bm{\theta}_i)$ or being similar in observables, see below). Accordingly, units are partitioned into two sets -- a set of $\bar{N} \geq 0$ unrestricted potentially important units, and a set of the remaining $N-\bar{N}$ restricted units. The number of restricted units $N-\bar{N}$ is assumed to be at least somewhat large for the partition of units to have a meaningful impact on the resulting estimator.

The large-$N$ estimator leverages prior information expressed through these two sets. Intuitively, the weights of the unrestricted units are freely determined by the data. For the restricted units, the optimization problem determines only the total mass assigned to the whole restricted set. This mass is then equally split over its members, though we note that other weighting schemes are allowed for the restricted units; see theorem (ref) below. Formally, let $\bm{w}^{N,\infty}=(w_{i}^{N,\infty})$ be an $N$-vector and assume that the weights of the unrestricted units are placed in the first $\bar{N}$ positions. The vector of weights $\bm{w}^{N,\infty}$ is such that $w_i^{N,\infty}\geq 0$ for all $i$, $\sum_{i=1}^N w_i^{N,\infty}= 1$, and the weights of the restricted units ($i > \bar N$) are equal and given by \( w_i^{N,\infty} = ( 1 - \sum_{j=1}^{\bar N} w^{N,\infty}_j )/( N - \bar N) \). Let $\bm{w}^{\bar{N},\infty}=(w_i^{\bar{N},\infty})$ be a $\bar{N}$-vector such that $ w_i^{N,\infty}=w_i^{\bar{N},\infty} $ for $i=1,\ldots,\bar N$. These are the weights of the unrestricted units. The large-$N$ LA-MSE estimator associated with $\bm{w}^{N,\infty}$ is controlled by $\bm{w}^{\bar{N},\infty}$ and given by

The above approximation to the MSE assumes that the number $N-\bar{N}$ of restricted units is large. In this case the restricted units have an impact on the bias of the estimator, but only a negligible contribution to its variance (asymptotically as $N\to\infty$).

The large-$N$ minimum MSE weights $\hat{\bm{w}}^{N, \infty} = (\hat{w}^{N, \infty}_i)$ are given by

where \[ \hat{\bm{w}}^{\bar{N}, \infty} = \operatorname*{arg\,min }_{\bm{w}\in\tilde{\Delta}^{\bar{N}}} \widehat{LA\mhyphen MSE}_{\infty}(\bm{w}) \] with $\tilde \Delta^{\bar{N}}= \curl{\bm{w}\in \operatorname{\mathbb{R}}^{\bar{N}}: w_i\geq 0, \sum_{i=1}^{N} w_i\leq 1}$. Note that the optimization problem defining $\hat{\bm{w}}^{\bar{N}, \infty}$ is $\bar N$-dimensional and can be solved by standard quadratic programming methods.

Three comments are in order before we proceed. First, the names of the approaches come from the frameworks used to study their properties. The fixed-$N$ estimator is studied in a setting where the number of units $\bar{N}$ is held finite and fixed, regardless of whether $\bar{N}$ is small or large in practical terms. In contrast, the large-$N$ estimator is studied in a framework where the size of the restricted set grows without bound.

Second, using the large-$N$ estimator requires choosing the set of unrestricted units. In principle, this set may be chosen arbitrarily, with weights (ref) adapting to the choice. However, larger reductions in bias are possible if the unrestricted set contains units with $\mu(\bm{\theta}_i)$ similar to $\mu(\bm{\theta}_1)$. For example, when dealing with country-level, this similarity may be established by using previous country-level studies focusing on the parameter of interest or related parameters. We explore several ways of specifying this set in sections (ref)-(ref).

Last, the fixed- and large-$N$ LA-MSE estimators have the appealing property of being applicable both when the amount of time series information in the panel is moderate or large. When the amount of time series information is moderate, the LA-MSE approximates the infeasible population problem of minimizing the MSE, along with uncertainty about individual parameters (see the discussion following theorem (ref)). When the amount of time series information is large, the bias term in the MSE dominates. Then the unit averaging estimator based on the minimum MSE weights converges to the individual estimator $\mu(\hat{\bm{\theta}}_1)$, if the coefficients $\bm{\theta}_i$ are continuously distributed (see remark (ref) in the next section).

Theory

Assumptions

We focus on a moderate-$T$ setting --- in which the variance of the individual estimators is of the same order of magnitude as the individual components $\bm{\eta}_i$. In this case, the amount of information in each individual time series is limited. To emulate this and the trade-off between unit-specific and panel-wide information, we make a local heterogeneity assumption.

Scaling $\bm{\eta}_i$ by $\sqrt{T}$ is a mathematical device that allows us to approximate a limited-information moderate-$T$ setting using asymptotic techniques with $T\to\infty$. Intuitively, as $T$ becomes larger, the signal strength becomes proportionally weaker, so that the amount of information in each time series is unchanged and bounded even if $T\to\infty$. At the same time, this assumption will permit us to apply asymptotic techniques to characterize the leading terms of the bias and the variance of the unit averaging estimator. The local heterogeneity assumption is analogous to the local misspecification device used in the frequentist model averaging literature Hjort2003,Hjort2003rejoinder, Hansen2016, Yin2019. It is also similar to the techniques of weak instrument asymptotics Staiger1997 and local alternatives used in test evaluation Lehmann2022. Like in those settings, this assumption should not be interpreted literally as meaning that the true parameters change depending on time series length (see Raftery2003 and Hjort2003rejoinder for some important criticism of such an interpretation of locality).

Since the focus lies on recovering the realized individual parameter $\mu(\bm{\theta}_1)$, all probability statements are implicitly conditional on $\sigma(\bm{\eta}_1, \bm{\eta}_2, \dots)$. Such conditioning is typical when individual parameters are of interest Vaida2005,Donohue2011, Zhang2014. Importantly, all the results are shown to hold with $\bm{\eta}$-probability 1 (for almost any realization of $\{ \bm{\eta}_i \}$).

In this paper we assume that the cross-sectional units are independent.

Note that together (ref) and (ref) permit cross-sectional heterogeneity. In particular, $\bm{\eta}_i$ may be heterogeneously distributed, provided the coefficients $\bm{\theta}_i$ share a common mean $\bm{\theta}_0$.

The unit-specific estimators $\hat{\bm{\theta}}_i$ are assumed to satisfy a number of regularity conditions.

(ref) requires the unit-specific estimators to be consistent, asymptotically normal and to satisfy a number of regularity conditions. This assumption allows for a fair amount of dependence, heterogeneity, and non-stationarity in the unit-specific time series; we refer to ch.\ 11 of Potscher1997 for a catalog of low-level conditions. Assumption (ref)$(iii)$ states that the unit-specific estimator lies in the interior of the parameter space almost surely. If the problem is linear or defined by a convex smooth objective function and continuous covariates, the parameter space can be taken to be $\operatorname{\mathbb{R}}^p$, and the condition holds automatically. Assumption (ref)$(iv)$ is standard in the M-estimation literature, it requires the gradient of the objective function evaluated at $\bm{\theta}_i$ to satisfy a CLT. Assumption (ref)$(v)$ is a moment condition on the gradient of the objective function. In an i.i.d. setting such an assumption translates into a moment condition on the individual gradients. More generally, this would be implied by appropriate moment and dependence assumption on the individual gradients. Assumption (ref)$(vi)$ is also standard in the M-estimation literature; it requires the Hessian to satisfy a uniform law of large numbers. Assumption (ref)$(vii)$ effectively requires that the sample Hessian is nonsingular in a small enough neighborhood of $\bm{\theta}_i$. In a scalar problem, $(vii)$ restricts the possible range of the second derivative as $\bm{\theta}$ ranges over a shrinking interval around $\bm{\theta}_i$. In addition, $(vii)$ places an assumption on the moments of deviation from the population limit Hessian. In case of linear regression, the sample and population Hessians do not depend on the slope parameters and $(vii)$ is an assumption on moments of covariates. Assumption (ref)$(viii)$ implies a uniform restriction on the asymptotic variance $\bm{V}_i$ of the individual estimators. Assumption (ref)$(ix)$ states that there exists a sequence of nonsingular estimators $\{ \hat \bm{V}_i \}$ for the asymptotic variance-covariance matrix of the individual estimator. We remark that Assumptions (ref)$(iii)$ and $(vii)$ state that the sequence of unit-specific estimation problems satisfies approprite uniformity conditions. Such conditions allow us to distill the key arguments relevant to our averaging theory and, in a sense, should be intrepreted as a simplifying approximation. In general, $(iii)$ and $(vii)$ would hold with probability approaching one for each unit. In this case all our results would still hold, though under appropriate rate conditions on $(N, T)$ and trimming to ensure certain well-behavedness of individual estimators. We further note that assumptions $(iii)$ and $(vii)$ might hold in practice in certain special cases regardless (such as linear or nonlinear models with a convex and smooth objective function and continuous covariates).

Assumption (ref) requires that the bias of individual estimators for {their} own parameters is bounded uniformly in $i$. The order of the bias is consistent with the results obtained by Rilstone1996 and Bao2007. The higher order terms can be subsumed into the $T^{-1}$ term for a sufficiently large $C_{Bias}$. Assumption (ref) is satisfied for linear models under assumption (ref). For nonlinear models it is sufficient that for all $s$ and $i$ it holds that $ \operatorname{\mathbb{E}}(\left\lVert\nabla^s m(\bm{\theta}_i, z_{i\,t})\right\rVert^2)\leq C_s<\infty$ Bao2007.

Assumption (ref) lays out mild smoothness assumptions on $\mu$. For simplicity we assume that $\mu$ is a scalar focus parameter. However, all our results can be extended to the case in which $\mu$ is a vector focus parameter.

Properties of the Minimum MSE Unit Averaging Estimator

We begin with a lemma that establishes the properties of the unit-specific estimators ${\mu}(\hat{\bm{\theta}}_i)$ as estimators for the target parameter $\mu(\bm{\theta}_1)$ of unit 1 in limited-information local setting.

Lemma (ref) approximates the exact moderate-$T$ bias and variance of $\mu(\hat{\bm{\theta}}_i)$ with their leading terms, which appear as the mean and variance of $\Lambda_i$. This approximation relies on the locality assumption (ref): as $T\to\infty$, the amount of information in each individual time series remains limited (see the discussion after (ref)). Consequently, both the asymptotic mean and variance are non-negligible and of the same order.

We now establish a local asymptotic approximation to the MSE (LA-MSE) of the unit averaging estimator (ref). Let $\{ \bm{w}_1, \bm{w}_2 , \ldots \}$ be a (non-random) sequence where $\bm{w}_k$ is a $k$-vector of weights. Suppose that $\bm{w}_N$ converges to some $\bm{w} \in \operatorname{\mathbb{R}}^{\infty}$ in the sense defined below. In what follows we treat $\bm{w}_k = (w_{i\,k})$ as an element both in $\operatorname{\mathbb{R}}^{k}$ and in $\operatorname{\mathbb{R}}^{\infty}$ (with coordinates $i>k$ restricted to zero). Consider the unit averaging estimator $\hat \mu( \bm{w}_N )$ (ref).

Theorem (ref) provides a local approximation to the MSE (LA-MSE) of the averaging estimator. The LA-MSE consists of the leading terms of the moderate-$T$ bias and variance of the estimator. This result parallels local approximations for the finite-sample risk in the model averaging literature (e.g. Hjort2003, Hansen2016).

The LA-MSE highlights the bias-variance trade-off associated with the choice of the weights. The two extremes of the trade-off correspond to the individual estimator $\mu(\hat{\bm{\theta}}_1)$ of the first unit and the mean group estimator $\hat{\mu}_{MG} = N^{-1}\sum_{i=1}^N \mu(\hat{\bm{\theta}}_i)$. $\mu(\hat{\bm{\theta}}_1)$ is obtained by setting $w_{1\,N}=1$ for all $N$. It is asymptotically unbiased, and its LA-MSE is equal to $\bm{d}_0'\bm{V}_i\bm{d}_0$, the asymptotic variance of the individual estimator. The mean group estimator is obtained by setting $w_{i\,N} = (N)^{-1}\operatorname{\mathbb{I}}_{i\leq N}$ for $i=1,\ldots, N$ for all $N$. The variance term for $\hat{\mu}_{MG}$ is zero, and the LA-MSE is equal to $(\bm{d}_0'\bm{\eta}_1)^2$.

The weight convergence condition $(iii)$ characterizes the spaces of weights over which the MSE is validly approximated by the LA-MSE. $(iii)$ requires the sequence $\{ \bm{w}_1, \bm{w}_2, \dots \}$ of weight vectors to converge uniformly to some limit $\bm{w}$ as the cross-section grows. Note that the sum of the limit $\bm{w}$ can be less than one, as is the case for the mean group estimator.

We now specialize the LA-MSE expression to the fixed-$N$ and large-$N$ averaging approaches of section (ref). In the fixed-$N$ case, suppose that only the first $\bar{N}$ units are being averaged, where $\bar{N}$ is fixed and finite. Only these units affect the bias and the variance of the estimator, and both sums in eq.\ (ref) are finite sums. The LA-MSE is a quadratic function of the weights. Formally, for all $N\geq\bar{N}$, let $\bm{w}_N=(w_{i\,N})$ satisfy two conditions. First, set $w_{i\,N}=0$ for all $i>\bar{N}$. Second, let $\bm{w}^{\bar{N}}$ be a $\bar{N}$-vector that satisfies $\sum_{i=1}^{\bar{N}} w_i^{\bar{N}}=1, w_{i}^{\bar{N}}\geq 0$. Then let $w_{i\, N} = w_{i}^{\bar{N}}$. The condition that $N\to\infty$ becomes superfluous and condition $(iii)$ holds automatically. The LA-MSE is controlled by the $\bar{N}$-vector $\bm{w}^{\bar{N}}$ and can be written as

where $\bm{\Psi}_{\bar{N}}$ is an $\bar{N} \times \bar{N}$ matrix with elements $[\bm{\Psi}_{\bar{N}}]_{i\,i} =\bm{d}_0'\left((\bm{\eta}_i-\bm{\eta}_1)\left(\bm{\eta}_i-\bm{\eta}_1 \right)' + \bm{V}_i\right)\bm{d}_0 $ and $[\bm{\Psi}_{\bar{N}}]_{i\,j} = \bm{d}_0'(\bm{\eta}_i-\bm{\eta}_1)\left(\bm{\eta}_j-\bm{\eta}_1 \right)'\bm{d}_0$ when $ i\neq j$.

In the large-$N$ case, let the $\bar{N}$ unrestricted units be placed in the first $\bar{N}$ positions, with the $N-\bar{N}$ remaining units forming the restricted set. By eq.\ (ref), the individual weights of the restricted units converge to 0 uniformly and satisfy $(iii)$. The restricted units contribute only to the bias component of the LA-MSE. The LA-MSE itself is fully determined by the individual weights of the unrestricted units and the total mass assigned to the restricted set. Formally, let $\bm{w}^{\bar{N}, \infty}$ be a $\bar{N}$-vector that satisfies $\sum_{i=1}^{\bar{N}} w_{i}^{\bar{N}, \infty} \leq 1$, $w_{i}^{\bar{N}, \infty} \geq 0$; the vector $\bm{w}^{\bar{N}, \infty}$ holds the weights of the unrestricted units. Set $w_{iN} = w_i^{\bar{N}, \infty}$ for $i\leq \bar{N}$ and $w_{iN} = (1-\sum_{j=1}^{\bar{N}} w_{j}^{\bar{N}, \infty})/(N-\bar{N})$, $i \in\curl{\bar{N}+1, \dots, N}$. Let $\bm{w}=(w_i)$ where $w_i=w_i^{\bar{N}, \infty}$, $i\leq \bar{N}$ and $w_i =0$, $i>\bar{N}$. Then $\sup_{i} \abs*{w_{iN} - w_i} = O(N^{-1})$. Note that the mass of the restricted units $(1-\sum_{i=1}^{\bar{N}} w_{i}^{\bar{N}, \infty})$ may lie anywhere between 0 and 1 (the latter being the case for the mean group estimator). The LA-MSE is controlled by $\bm{w}^{\bar{N}, \infty}$ as

The same expression for the LA-MSE can be obtained with other weighting schemes for the restricted set. The weights in $\bm{w}_N$ beyond $\bar{N}$ can display strong variations in orders of magnitude, with some weights decaying like $N^{-1/2-\varepsilon}$, and some at a faster rate.

The above arguments also show that it is internally consistent to use the fixed-$N$ and large-$N$ approaches to minimize the MSE. These approaches minimize (an estimator) of the LA-MSE. The weights returned lie within the class of weights for which the LA-MSE provides a valid approximation to the MSE.

The quantities $\widehat{LA\mhyphen MSE}_{\bar{N}}$ and $\widehat{LA\mhyphen MSE}_{\infty}$ used to define the minimum MSE weights introduced in section (ref) are estimators of the population expressions for the LA-MSE given above. In the rest of the section we focus on the properties of these estimators as well as the optimal weights (ref) and (ref) associated with them.

We begin by noting that in our framework the population LA-MSE cannot be consistently estimated. Under local heterogeneity the idiosyncratic components $\bm{\eta}_i$ cannot be consistently estimated, as the amount of information in each time series is finite and bounded under (ref) Hjort2003. Instead, we form $\widehat{LA\mhyphen MSE}_{\bar{N}}$ and $\widehat{LA\mhyphen MSE}_{\infty}$ by plugging in asymptotically unbiased estimators for $\bm{\eta}_i-\bm{\eta}_1$ and $\bm{\eta}_1$ Hjort2003. Such estimators are provided by $\sqrt{T}(\hat{\bm{\theta}}_i-\hat{\bm{\theta}}_1)$ and $\sqrt{T}(\hat{\bm{\theta}}_1-N^{-1} \sum_{i=1}^N \hat{\bm{\theta}}_i)$, respectively:

The following two theorems establish the properties of our LA-MSE estimators and the associated minimum MSE weights (ref) and (ref). The theorem also characterizes the asymptotic distribution of the minimum MSE unit averaging estimators. First, we state a result for the fixed-$N$ estimator. Recall that $\Delta^{\bar{N}} = \curl{\bm{w}\in \operatorname{\mathbb{R}}^{\bar{N}}: \sum_{i=1}^{\bar{N}} w_i=1, w_i\geq 0, i=1,\dots, \bar{N} }$.

The quantity $\overline{LA\mhyphen MSE}_{\bar{N}}$ plays the same role to $\widehat{LA\mhyphen MSE}_{\bar{N}}$ as $\bm{Z}_i$ does to $\sqrt{T}(\hat{\bm{\theta}}_i-\bm{\theta}_1)$ in lemma (ref). $\overline{LA\mhyphen MSE}_{\bar{N}}$ uses a local approximation to express $\widehat{LA\mhyphen MSE}_{\bar{N}}$ in terms of the leading components of the MSE and the approximate distribution of the individual estimators. We can see that $\overline{LA\mhyphen MSE}_{\bar{N}}$ is composed of the population LA-MSE, a bias term, and a noise component. In fact, the entries of the matrix $\overline{\bm{\Psi}}_{\bar{N}}$ may be expressed as

where $\bm{e}_{i\,j} = (\bm{Z}_i-\bm{Z}_1)(\bm{Z}_j-\bm{Z}_1)' - \operatorname{\mathbb{E}}\left((\bm{Z}_i-\bm{Z}_1)(\bm{Z}_j-\bm{Z}_1)' \right)$. The noise terms $\bm{e}_{i\,j}$ may be interpreted as the result of the fact that in a moderate-$T$ setting there is limited information about the idiosyncratic components $\bm{\eta}_i$. These terms are mean zero and independent conditional on unit 1. The bias terms guarantee that $\overline{\bm{\Psi}}_{\bar{N}}$ is positive definite and arise as a consequence of using the biased positive definite estimator $\hat{\bm{\Psi}}_{\bar{N}}$ (see remark (ref) below). The bias can be split into two components. The $\bm{d}'_0\bm{V}_1\bm{d}_0$ is common for all elements of $\overline{\bm{\Psi}}_{\bar{N}}$ and does not affect the solution of the MSE minimization problem. The second component $\bm{d}_0'\bm{V}_i\bm{d}_0$ only affects the diagonal of $\overline{\bm{\Psi}}_{\bar{N}}$ and measures the individual variances. This component does not modify the ordering of the estimators in terms of their variances.

Result $(iii)$ shows that the minimum MSE unit averaging estimator has a nonstandard asymptotic distribution in the local heterogeneity framework. The limit distribution is a randomly weighted sum of independent normal random variables. This result is somewhat similar to the distributional results for model averaging estimators Liu2015. In the Online Appendix, we show how to construct confidence intervals based on theorem (ref).

The following theorem establishes an analogous result for the large-$N$ estimator.

Note that the estimator in equation (ref) is a valid averaging estimator, with weights summing to unity. The exact way $\bm{v}_N$ is picked does not matter, as long as the decay condition holds. All admissible choices lead to the same limit. In particular, we may pick equal weights $v_{i\,N} = 1/(N-\bar{N})$, as we do in eq. (ref). Also note that the convergence result $(ii)$ applies to the vector $\hat{\bm{w}}^{\bar{N}, \infty}$ of the weights of the unrestricted units, a vector of fixed length $\bar{N}$.

Simulation Study

In this section, we study the performance of our minimum MSE unit averaging estimator for a variety of sample sizes via a simulation exercise. We consider a model similar to the one we use in our empirical application -- a linear dynamic heterogeneous panel model defined as

The error $u_{i\,t}$ is cross-sectionally heteroskedastic, with variance $\sigma^2_i$ drawn independently from an exponential(1) distribution. $u_{i\, t}$ is independent from the coefficients and the covariates. The exogenous variable $x_{i\,t}$ is independently drawn from a $N(0, 1)$ distribution. The initial conditions $y_{i0}$ are drawn from a $N(0, (\beta_i^2+\sigma^2_i)/(1-\lambda_i^2))$ distribution to ensure that $\{ y_{it} \}_t$ is covariance stationary. The two components of the parameter $\bm{\theta}_i=(\beta_i,\lambda_i)'$ are independently drawn from a $N(0, 1)$ and a Beta$(5,5)$ distribution on $[0.2, 0.8]$, respectively. Note that, in order to measure the impact of increasing information and to compare results across $T$, we model the distribution of $\bm{\theta}_i$ as independent from $T$. Under this (fixed parameter) approach, the amount of information in each time series increases as $T$ grows.

We study both moderate-$T$ and large-$T$ settings for a variety of cross-sectional sample sizes $N$. Specifically, we consider $N=50, 150, 450$, and $T=50, 60, 600$. $T=30$ and $T=60$ are moderate values of $T$, according to the heuristic criterion of remark (ref): the average $t$-statistic of the parameter estimates is 2 for $T=30$ and 3.5 for $T=60$. In contrast, $T=600$ is a large value of $T$, with an average $t$-statistic value of 10. We also note that $N=150, T=60$ is one of the estimation sample sizes in our empirical application.

The measures of interest are the MSE, bias, and variance of the unit averaging estimators (see below) for the focus parameter $\mu(\bm{\theta}_1)=\lambda_1$. Specifically, we evaluate the MSE of the form $\operatorname{\mathbb{E}}\left[(\hat{\mu}(\bm{w}) - \mu(\bm{\theta}_1))^2|\lambda_1=c\right]$, where $c$ ranges through a grid of values in $[0.2, 0.8]$, and the expectation is over the distribution of data, $\beta_1$, and the parameters of units 2-$N$. The bias and variance of interest are defined similarly. We draw 10000 datasets for each value of $c$ and $(N, T)$. For each sample, we estimate eq.\ (ref) by OLS, compute the estimators, and record the estimates and estimation errors. The MSE is approximated with the average square Monte Carlo estimation error; we compute biases and variances similarly.

We estimate the focus parameter using the fixed-$N$ and large-$N$ minimum MSE estimators. We consider three specifications for the large-$N$ estimator.

Note that the latter specification is data-driven and thus not directly covered by theorem (ref). The corresponding tuning parameter (number of top units) matches the empirical application; in the Online Appendix we explore the impact of this choice.

The performance of the minimum MSE estimator is benchmarked against the individual estimator of unit 1, the mean group estimator, as well as the unit averaging estimator based on AIC/BIC weights Buckland1997,Vaida2005. AIC and BIC generate the same likelihood-based weights, as each unit has the same number of coefficients.

Our key result is that the minimum MSE estimators generally have lower MSE for both moderate and large-$T$. As fig.\ (ref) shows, all of the minimum MSE estimators (bar the Stein-like one) perform favorably throughout most of the parameter space for all $(N, T)$. Gains in the MSE are possible without prior information, as shown by the agnostic fixed-$N$ estimator, and the data-driven top unit large-$N$ specification. However, leveraging prior information may lead to stronger improvements for some parameter values (the “most similar” line).

Fig.\ (ref) shows a trade-off between stronger improvements for more typical values of $\lambda_1$ vs. for less typical ones (closer to $\operatorname{\mathbb{E}}\left[\lambda_1\right]=0.5$ vs. closer to the boundary of the support of $\lambda_1$). This trade-off is controlled by the flexibility of the estimator, determined by the number of free weights it has. Importantly, this trade-off is not identical to the bias variance trade-off (fig.\ (ref)). More flexible estimators (such as the fixed-$N$ estimator) have uniformly lower bias for all values of $\lambda_1$. However, more flexible estimators also have lower variance for more extreme values of $\lambda_1$, while less flexible estimators have lower variance for $\lambda_1$ close to $\operatorname{\mathbb{E}}[\lambda_1]$.

Increasing $N$ has a twofold effect. First, it strictly improves the performance of the similarity-based large-$N$ estimator. For larger $N$, more units will lie within any given neighborhood of $\lambda_1$ on average, reducing bias. Second, more flexible estimators offer a stronger gain for less typical $\lambda_1$, as larger cross-sections will have more units with similar $\lambda_i$. At the same time, the region around $\operatorname{\mathbb{E}}[\lambda_i]$ in which improvements are modest grows.

Gains in MSE are strongest for smaller values of $T$. The impact is not symmetric around $\operatorname{\mathbb{E}}[\lambda_1]=0.5$, with stronger improvements in the left tail than in the right one. This asymmetry is due to the increase in the convergence rate of the individual estimator as $\lambda_1$ moves into a near-unit root region. At the extreme, if $\lambda_1\approx0.8$, most of the other units will have smaller values of $\lambda_i$. Their own individual estimators will converge at a rate closer to $T^{-1/2}$. Accordingly, for larger values of $T$, the variance of the individual estimator of unit $1$ may be significantly smaller than the variance of most of the individual estimators. This effect has little impact for $T=30, 60$, but is more notable for $T=600$.

As $T$ increases, more weight is placed on the individual estimator of unit 1, in line with the discussion after theorem (ref) (fig.\ (ref)). This effect is more pronounced in smaller cross-sections, for less flexible estimators, more extreme values of $\lambda_1$, and values of $\lambda_1$ where the individual estimator is more efficient ($\lambda_1\approx 0.8$).

Additional simulation results are reported in the Online Appendix. We consider an additional data-driven large-$N$ specification, further focus parameters; perform simulations for the intermediate case $T=180$; analyze the choice of tuning parameters for large-$N$ estimators; and examine the estimated weights. The evidence emerging from these simulations is in line with the results presented above.

Empirical Application

We illustrate our averaging methodology with an application to forecasting monthly unemployment rates for a panel of German regions. This setting provides a natural application for two reasons. First, the unemployment dynamics of German regions are heterogeneous due to differences in sectoral composition, regional laws, and historical trends such as the East-West divide Graaff2018. At the same time, using data on other regions at least partially improves prediction. Schanne2010. Second, the performance of our methodology can be explicitly measured against realized unemployment rates. Our application contributes to the growing literature on forecasting regional unemployment Schanne2010, Patuelli2012, Wozniak2020, Aaronson2022.

The regions of interest are the 150 German labor market districts (Arbeitsagenturbezirke, AABs) of the German Federal Employment Agency. Each AAB is medium-size region, between a NUTS-2 and a NUTS-3 region in size. Together, the 150 AABs cover all of Germany. The AABs are grouped into 10 regional directorates (RDs). These RDs correspond either to German federal states or unions of two states (NUTS-2).

We make use of monthly AAB-, RD-, and Germany-wide seasonally adjusted unemployment data from May 2007 to February 2024 (a total of 202 time series observations). The resulting panel is balanced with $N=150$. All data is freely available from the Federal Employment Agency.

We model the AAB-level unemployment rate as a function of the past values of AAB-, RD-, and national-level unemployment rates. Specifically, let $y_{i\, t}^{AAB}$ be the unemployment rate in the $i$th AAB at month $t$. Let $y_{i\, t}^{RD}$ be the unemployment rate of the RD to which the $i$th AAB belongs. Finally, let $y_{t}^{DE}$ be the unemployment rate in Germany. Then $y_{i\, t}^{AAB}$ is modeled as:

In model (ref), we allow both idiosyncratic and regional dynamics to drive the AAB-level unemployment rate, following Schanne2010. These dynamics may be heterogeneous between AABs, and all coefficients are AAB-specific.

For each AAB, we forecast $y_{i\,t}^{AAB}$ with its conditional mean $\operatorname{\mathbb{E}}\left[y_{i\,t}^{AAB}|y_{i\, t-1}^{AAB}, y_{i\, t-1}^{RD}, y_{t-1}^{DE} \right]$ implied by eq.\ (ref). Formally, the target parameter for the $i$th AAB in month $t$ is $\mu(\bm{\theta}_i) = \theta_{i0} + \theta_{i1} y_{i\, t-1}^{AAB} + \theta_{i2} y_{i\, t-1}^{RD} + \theta_{i3} y_{t-1}^{DE} $. Observe that the period $(t-1)$ unemployment rates are treated as part of the parameter $\mu$.

The key measure of interest in our study is the out-of-sample forecasting MSE of our unit averaging approaches (see below). To estimate this MSE, we adopt a rolling-window approach. The data is split into all possible contiguous subsamples of window sizes $T=40, 60$, and $80$ months (between 3 and 7 years of data). On each window we estimate the individual parameters of eq.\ (ref) with OLS. We compute the one-step-ahead out-of-sample unit averaging forecasts and record the forecast error. These errors are used to estimate the MSE for each AAB and averaging approach. Note that estimating the MSE from rolling windows implicitly assumes that individual parameters are stable over time, see remark (ref) below for evidence in favor of this. We also note that the values of $T$ considered satisfy the heuristic criterion for moderate-$T$ of remark (ref). The average $t$-statistic across coefficients, AABs, and $T$s is approximately 2.

We estimate the conditional mean using our fixed-$N$ and large-$N$ minimum MSE estimators. For the large-$N$ approach, we consider two specifications. For the Stein-like specification, only the target AAB is unrestricted. For the top units specification, we first run the fixed-$N$ estimator. The 15 AABs (10% of total) with the largest weights are set as unrestricted units, and the rest are restricted, and the large-$N$ estimator is then ran (see also the discussion in section (ref)). The choice of the number of top units is explored in the Online Appendix. The pre-averaging fixed-$N$ procedure is done for every AAB in every window subsample. The performance of our minimum MSE unit averaging estimator is benchmarked against the individual, mean group, and AIC-weighted averaging estimators.

Figures (ref)-(ref) visualize our results for the MSE. Fig.\ (ref) provides a box plot for the MSE for all averaging approaches relative to the MSE of the individual estimator, along with a box plot of the (absolute) MSE of the individual estimator. Table (ref) complements fig.\ (ref) with the average relative MSEs. The underlying geographic distribution of the MSE is plotted on fig.\ (ref) for $T=40$. Finally, on fig.\ (ref) we compare the individual and the minimum MSE estimators, and depict the best performing approach for each AAB and each value of $T$. Maps for all of the averaging approaches and $T$ are provided in the Online Appendix.

Our key finding is that averaging with minimum MSE weights generally improves forecasting performance. For most AABs, at least one minimum MSE approach outperforms the individual estimator for all $T$, as can be seen on fig.\ (ref). The gain in MSE can be substantial, as fig.\ (ref) and table (ref) show. These gains are stronger for regions where the individual estimator does relatively poorly (fig.\ (ref)); these regions are predominantly concentrated in the former East Germany. The improvement is also stronger for smaller values of $T$, although it is also non-negligible even for $T=80$.

The fixed-$N$ and the top units large-$N$ minimal MSE estimators emerge as the leading averaging approaches, in line with the simulation evidence of section (ref). Both offer roughly similar gains in MSE (fig.\ (ref)). For $T =40$, greater flexibility makes the fixed-$N$ approach the overall best, as fig.\ (ref) shows. For $T=80$, the leading option is the top units large-$N$ estimator, which has only 15 unrestricted units.

The other averaging methods considered perform somewhat worse. Mean group and AIC weights do not improve forecasting performance on average, although they offer an improvement for a non-trivial share of AABs. The Stein-like large-$N$ performs similarly to the mean group estimator, but with smaller variation in the MSEs across AABs.

Conclusions

In this work we introduce a unit averaging estimator to recover unit-specific parameters in a general class of panel data models with heterogeneous parameters. 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. The paper studies the properties of the procedures using a local heterogeneity framework that builds upon the literature on frequentist model averaging Hjort2003,Hansen2008. An application to forecasting regional unemployment for a panel of German regions shows that the procedure performs favorably for prediction relative to a number of alternative procedures.

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\part*{Proofs of Results in the Main Text}

Under assumption (ref) we work conditional on $\curl{\bm{\eta}_1, \bm{\eta}_2, \dots}$. We use $\operatorname{\mathbb{E}}[\cdot]$ to denote the expectation operator conditional on $\curl{\bm{\eta}_1, \bm{\eta}_2, \dots}$, whereas $\operatorname{\mathbb{E}}_{\bm{\eta}}[\cdot]$ is the expectation taken with respect the distribution of $\bm{\eta}$. All results are shown to hold with probability one with respect to the distribution of $\bm{\eta}$ (denoted $\bm{\eta}$-a.s.).

Proof of Lemma (ref)

Recall that the data vector $\bm{z}_{i\,t}$ takes values in $\mathcal{Z}\subset \operatorname{\mathbb{R}}^d$ and define the data matrix $\bm{z}_i = (\bm{z}_{i\,1}',\ldots,\bm{z}_{i\,T}')'$ that takes values in $ \mathcal{Z}^T= \prod_{t=1}^{T}\mathcal{Z}$. Recall that that the parameter vector $\bm{\theta}= (\theta_1, \dots, \theta_p)$ takes values in $\Theta \subset \mathbb R^p$. We denote by $\nabla m(\bm{\theta}, \bm{z}_{i\,t})$ the gradient vector of $m$ with respect to $\bm{\theta}$, by $\nabla^2 m(\bm{\theta}, \bm{z}_{i\,t})$ the Hessian matrix of $m$ with respect to $\bm{\theta}$, by $\nabla_{\theta_k} m(\bm{\theta}, \bm{z}_{i\,t})$ the partial derivative of $m$ with respect to $\theta_k$, and by $\nabla^2_{\bm{\theta}\,\theta_k}$ the gradient vector of $\nabla_{\theta_k} m(\bm{\theta}, \bm{z}_{i\,t})$ with respect to $\bm{\theta}$.

We establish a mean value theorem that does not require compactness of $\Theta$.

The following lemma is needed to prove lemmas (ref) and (ref).

Proof of Theorem (ref)

Before presenting the proof of theorem (ref) we introduce a number of intermediate results.

We need an extension of a weighted law of large numbers due to Rohatgi1971.

Observe that the limit sequence of weights can be defective. If $w_{i\,N} = N^{-1}\operatorname{\mathbb{I}}_{i\leq N}$ (equal weights), the above result becomes a standard SLLN with a second moment assumption.

Finally, we present the proof of theorem (ref).

Proof of Lemma (ref)

Proof of Theorems (ref) and (ref)