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Heterogeneity-robust granular instruments
\onehalfspacing \justifying
\noindentKeywords: internal instrument, generalized method of moments, granular instrumental variables, spillovers. JEL codes: C33, C36.
In many macroeconomic settings, researchers wish to estimate the spillover of an idiosyncratic shock to one unit onto other units. Examples include estimating the contagion effects of financial market distress, aggregate demand externalities on individual consumption, strategic complementarities in price setting, and the price elasticity of demand of the stock market Allen2009,Auclert2023,Alvarez2022,Gabaix2021. Identifying plausibly exogenous variation in these settings however is notoriously difficult and contributes to the continued challenges of conducting empirical work.
Gabaix2020 tackles this difficulty with a novel technique, called granular instrumental variables (GIV). They consider environments where an individual unit's outcome is partially determined by the size-weighted outcome. Thus, idiosyncratic shocks to a single unit spill over to all other units in equilibrium. To overcome the resulting endogeneity bias, their instrument for the size-weighted outcome is constructed as the difference between size- and equal-weighted outcomes. Their instrument is “granular” in that idiosyncratic shocks to large units are the primary source of identifying variation as they disproportionately contribute to movements in the size-weighted outcome variable. The broad adoption of GIV in applied macroeconomics and finance is indicative of its usefulness Chodorow-Reich2021a,Adrian2022,Camanho2022,Gabaix2021.
The baseline GIV estimator of Gabaix2020, however, requires strong assumptions to satisfy the instrumental variables relevance condition and exclusion restriction and its extensions require further conditions. The baseline estimator requires homogeneous spillovers across units, homogeneous shock variances, skewed unit size, and idiosyncratic shocks (uncorrelated shocks between units after accounting for a known factor structure). In particular, there is typically no ex-ante rationale for there to be homogeneous spillovers and shock variances across units, contributing to the gap between theory and practice. Furthermore, I show that there is no general guarantee that the GIV spillover estimand admits a positive-weighted average of unit-specific spillovers. To address these concerns, Gabaix2020 generalize their procedure to allow for heterogeneity that is a function of observables, but the empirically important question of how to best tackle unobserved heterogeneity remains open.
This paper's main contribution is to establish global identification for a GIV model with unit-specific spillovers and unknown shock variances. I propose an estimator, called robust granular instrumental variables (RGIV), that is robust in the sense that it is applicable to a wider set of environments than the baseline Gabaix2020 GIV estimator. Ultimately, RGIV brings the study of unit-level heterogeneity---a feature that is of substantial importance in other areas of macroeconomics{\interfootnotelinepenalty10000 \footnote{For instance, take household-level differences in the marginal propensity to consume akin to Fagereng2021,Lewis2021,Fuster2021.}}---to the estimation of spillovers.
RGIV uses internally estimated idiosyncratic shocks as instruments for the size-weighted outcome variable. Informally, the procedure can be described sequentially. An initial guess of the first unit's spillover gives an estimate of the first unit's idiosyncratic shock. Using this estimated shock as an instrument for the size-weighted outcome variable, the remaining units' spillover coefficients and estimated idiosyncratic shocks can be computed. If one or more pairs of estimated idiosyncratic shocks are correlated, we can guess a new spillover for the first unit and repeat the procedure until the estimated idiosyncratic shocks are uncorrelated. RGIV makes use of granularity by exploiting the contribution of individual shocks (even to relatively small units) to movements in the size-weighted outcome variable. This approach differs from the baseline GIV estimator in that skewness in the size distribution of units and homogeneity in shock variances are not required.
I prove that there is an inevitable trade-off in assumptions between allowing for general spillover heterogeneity and a general shock covariance structure. Like GIV, the RGIV estimator can easily accommodate cross-unit correlations that are due to observed covariates (both time-varying and unit-specific) or a known factor structure in the residuals. However, I prove that a model with an unknown factor structure and unrestricted spillover heterogeneity is not identified. Hence, researchers must restrict either spillover heterogeneity or the correlation structure of the shocks.
I develop tests that evaluate the appropriateness of (1) the RGIV framework and (2) the homogeneous spillovers assumptions featured in Gabaix2020. These hypothesis tests are applications of standard GMM results Newey1994. The first test is the Sargan--Hansen test, where the null hypothesis of correct specification is rejected when one or more pairs of estimated idiosyncratic shocks are correlated. The second test is the distance metric test, where the null hypothesis is rejected when the constraint of spillover homogeneity binds.
Building on the analysis of a working paper version of Gabaix2020, I apply RGIV to sovereign yield spillovers in the Euro area and find strong evidence of spillover heterogeneity among Euro area members. I control for correlations induced by heterogeneous loadings on unobserved aggregate shocks by including a rich set of observed explanatory variables. In addition, I account for the well-documented correlation of shocks among Euro area countries by size-aggregating countries into larger “core” and “periphery” blocks Bayoumi1992. Under the preferred specification, I fail to reject the null hypothesis of correct specification. Meanwhile, the null hypothesis of spillover coefficient homogeneity is strongly rejected. I find that the size-weighted spillovers of countries in the western-periphery block (Portugal, Spain, and Ireland) are approximately twice that of the core country block.
In a simulation study, I also show that RGIV confidence intervals have good finite sample coverage properties under a DGP taken from the empirical application. GIV confidence intervals in contrast can undercover under spillover heterogeneity and when shock variances are taken to be unknown. \\
Literature My model generalizes the baseline environment of Gabaix2020 to allow for spillover heterogeneity, shock variance heterogeneity, and equal-sized units. For estimation, RGIV fully exploits second moment information through GMM and can also be adapted to the case in which correlations in shocks are induced by factors with known loadings. In contrast, the spillover coefficient homogeneity, skewed size, and known shock variance conditions are used to justify the validity of the GIV instrument. Gabaix2020 also propose extensions that separately estimate spillover coefficient heterogeneity\footnote{Gabaix2020 Proposition 6 considers the case where unit-level spillover heterogeneity is determined by observables, Section D.9.1 proposes unit-specific “leave-one-out” granular instruments for when shock variances are known, and Section D.9.2 lists moment conditions for adapting GIV to heterogeneous spillovers though without the analysis of its econometric properties.} and unknown shock variances\footnote{Gabaix2020 Section D.3 appends shock variance moments after assuming homogeneous spillovers and global identification.}. In contrast, RGIV doesn't require skewness of the size distribution, heterogeneity to depend on observables, or for shock variances to be known; uncorrelated shocks is sufficient for jointly allowing unit-level spillovers and unknown shock variances.
Baumeister2023a applies the insights of GIV to show that a rich, structural model of the oil market (with unobserved explanatory variables and inventories) can be estimated using full information maximum likelihood. Their estimator similarly exploits shock orthogonality (in their case orthogonality of supply and demand shocks) and also allows for elasticity heterogeneity. In contrast, I extend the simple setting originally considered by Gabaix2020 to transparently illustrate the conditions needed for identification, for non-identification with unobserved explanatory variables, and to characterize the RGIV estimator's limit behavior. Rather than focusing on modeling assumptions that are specific to particular applications, this paper seeks to provide researchers with a parsimonious framework that could be adapted to their specific applications.
Banafti2022 propose a refinement to the GIV estimator, allowing for unobserved explanatory variables but require the number of economic units to be large and the size distribution to be skewed, unlike RGIV. Here, the principal challenge is to prevent the law of large numbers from averaging away the granularity of units. The authors overcome this challenge by requiring the right tail of the unit size distribution to be highly skewed, with a Pareto tail index of less than 1. In contrast, RGIV is consistent even when the size distribution is uniform.
GIV procedures are closely linked to models found in the spatial panel econometric literature Su2023,Aquaro2021,Chen2022. GIV's spillover network structure can be viewed as a restricted form of spatial autocorrelation. Unlike these papers, I allow the number of units to be small (finite), show that the GIV spillovers under unit-level heterogeneity are globally identified, and estimate spillovers without the use of external instruments. More broadly, my procedure can also be viewed as a special case of a simultaneous equation model with covariance restrictions, which is closely related to impact matrix identification in the structural VAR setting Sims1980, Hausman1983a. \\
Outline Section (ref) illustrates the properties of GIV and RGIV in a simple three unit setting. Section (ref) presents the main consistency and asymptotic normality results for the RGIV estimator. Section (ref) describes the RGIV specification test and parameter homogeneity tests. Section (ref) extends RGIV to include observable explanatory variables and discusses non-identification under unobserved explanatory variables. Section (ref) applies the RGIV estimator to investigate sovereign yield spillovers in the Euro area. Section (ref) examines finite-sample coverage accuracy of RGIV and GIV through simulations. Section (ref) concludes.
The remainder of this paper will consider the sovereign yield spillovers application as a running example. An idiosyncratic shock to one Euro area country raises yields in that country, as well as those of other Euro area countries since losses from default are partially shared. Country-specific spillovers are permitted.
In this section, I use a simple three-country setting to illustrate the construction and properties of GIV and RGIV. Section (ref) formalizes the discussion and generalizes to $n$ countries. \\
Notation Throughout this paper, I will follow the notation of Gabaix2020 for convenience. For a vector $X = (X_i)_{i=1,...,n}$ and size $S_i$ satisfying $\sum_{i=1}^n S_i = 1$, the equal-weighted sum $X_E$ and size-weighted sum $X_S$ are defined below:
Similarly, the equal-weighted and size-weighted cross-sectional averages for time series data $X_t = (X_{it})_{i=1,...,n}$ are defined as $X_{Et} \equiv \frac{1}{n} \sum_{i=1}^n X_{it}$ and $X_{St} \equiv \sum_{i=1}^n S_i X_{it}$ respectively.
I will outline my “baseline” setting for the three country case using sovereign yield spillovers in the Euro area as a running example. For countries $i=1,2,3$, let $y_{it}$ be the yield spread relative to some comparison country. Yield spread growth is $r_{it} = \frac{ y_{it}- y_{i,t-1}}{y_{i,t-1}}$. Size $S_i$ is observed and corresponds to a country's “debt at risk.” Here, $S_i \in (0,1)$ is taken to be time-invariant and sums to 1.
Suppose the researcher is interested in estimating elasticity $\phi_i$, which will be called the “spillover coefficient.” Specifically, yield spread growth $r_{it}$ is determined by a country-specific spillover coefficient $\phi_i$, size-aggregated yield spread $r_{St}$, and idiosyncratic shock $u_{it}$:
The size-weighted spillover coefficient $\phi_S$ is taken to be less than 1. Moreover, shocks $u_{it}$ are mutually and serially independent, mean zero, and have country-specific variance ${\mathbb E}(u_{it}^2) = \sigma^2_i > 0$.
The propagation of an idiosyncratic shock depends on the size of the shock's origin country, the recipient country's spillover coefficient, and the size-weighted average spillover coefficient. To illustrate, consider a unit idiosyncratic shock to country 1 ($u_{1t} = 1$) holding the idiosyncratic shocks of countries 2 and 3 at zero ($u_{2t} = u_{3t} =0$). Then, the size-weighted yield spread growth $r_{St}$ increases by $S_1 \times \frac{1}{1-\phi_S}$, which is computed from taking a size-weighted average of Equation (ref):
The increase in $r_{St}$ is larger when the size of country 1 is large and when countries are, on average, sensitive to spillovers (from multiplier $\frac{1}{1-\phi_S}$). Then, idiosyncratic shock $u_{1t}$ spills over to $r_{2t}$ and $r_{3t}$, giving rise to increases of $\phi_j r_{St} = \phi_j\frac{S_1}{1-\phi_S}$ for $j=2,3$. The total increase in $r_{1t}$ is $1+\phi_1 \frac{S_1}{1-\phi_S}$, a composition of the direct effect of idiosyncratic shock $u_{1t}$ and a spillover effect.
Granularity rules out the ordinary least squares regression of $r_{it}$ on $r_{St}$ as a method for estimating $\phi_i$. Equation (ref) highlights granularity in the baseline setting, showing that idiosyncratic shocks are responsible for movements in the aggregated yield spread $r_{St}$ since $S_i>0$ and $\sigma^2_i>0$. Therefore, regressing $r_{it}$ on $r_{St}$ to estimate $\phi_i$ as $T \to \infty$ suffers from endogeneity bias since ${\mathbb E}(r_{St} u_{1t}) = \frac{S_1 \sigma^2_1}{1-\phi_S} \neq 0$. An alternative estimator is needed.
In this section, I review the baseline GIV estimator of Gabaix2020 applied to this simple setting and discuss the conditions for its validity. I show that a skewed size distribution is necessary for the relevance condition to hold. Shock orthogonality and homogeneous idiosyncratic shock variances are necessary for the instrumental variables exclusion restriction to hold. Moreover, the exclusion restriction fails under heterogeneity of spillover coefficients across countries. As discussed later, there are GIV extensions that individually accommodate spillover coefficient heterogeneity and heterogeneous (and unknown) shock variances, but these require additional conditions.
Take the setting of Section (ref) and further assume homogeneous spillover coefficients ($\phi_1=\phi_2=\phi_3 = \phi$), homogeneous shock variances ($\sigma^2_1= \sigma^2_2 = \sigma^2_3 = \sigma^2$), and skewed unit sizes (ruling out $S_1 = S_2 = S_3$). Just as in Section (ref), granularity induces endogeneity bias in the regression of $r_{it}$ on $r_{St}$ for regression coefficient $\widehat{\phi}^i_{OLS}$
Gabaix2020 propose estimating $\phi$ using instrumental variables. An equal-weighted aggregation of the country-level yield spread growth gives rise to the following IV regression model
The equal-weighted yield spread growth responds to the size-weighted yield spread growth $r_{St}$ (through spillover coefficient $\phi$) and equal-weighted idiosyncratic shocks $u_{Et}$. For the IV regression model in the above display, “granular instrument” $z_t$ is constructed as the difference between the size- and equal-weighted yield spread growth
From the homogeneous spillovers assumption, the endogeneity from the size-weighted yield spread growth $r_{St}$ is differenced away.
Skewness in country sizes is crucial for the instrumental variables relevance condition to hold and $z_t$ reflects variation coming from idiosyncratic shocks to relatively large countries. To illustrate, first consider the extreme case of no skewness ($S_1 = S_2 = S_3$). Then the size- and equal-weighted yield spread growths are identical, producing a granular instrument that is identically zero $z_t = 0$. More generally, the instrumental variables relevance condition can be explicitly computed:
When the size of country 1 is large relative to other countries (when $S_1 \gg S_2,S_3$), the relevance condition is further from zero reflected by the $S_1(S_1 - 1/3)$ term.
The shock orthogonality and homogeneous (or known) shock variance assumptions are crucial for the exclusion restriction to hold. Explicitly, the covariance between the instrument and IV regression model error $u_{Et}$ is
In the first line of the above display, the uncorrelated shock assumption ensures that covariance terms ${\mathbb E}(u_{it} u_{jt})$ are zero. The homogeneous shock variance assumption ensures that the difference between the first line's bracketed terms is zero. Note that any instrument of the form $z_t' = W_1 r_{1t} + W_2 r_{2t} + W_3 r_{3t} - \frac{1}{3}r_{1t} - \frac{1}{3}r_{2t} - \frac{1}{3} r_{3t}$ where $W_1 + W_2+W_3= 1$ satisfies the instrumental variables exclusion restriction. Gabaix2020 show that weighting by size gives a variance-minimizing estimator for the IV regression model. The spillover coefficient can also be consistently estimated when a time fixed effect is included, as the time fixed effect is differenced away in the construction of $z_t$.
The exclusion restriction argument described in the preceding paragraph can easily be extended to settings in which the shock variances are known to the econometrician rather than being homogeneous across units. Gabaix2020 show that identical computations hold after replacing the equal weights in $z_t = r_{St} - r_{Et}$ with inverse variance weights.
Spillover coefficient heterogeneity leads to a failure in the instrumental variables exclusion restriction. Allowing for unit-specific spillover coefficients, the granular instrument contains variation from $r_{St}$:
From the $(\phi_S - \phi_E)$ term, the magnitude of the exclusion restriction's violation is greater when spillover coefficients vary systematically by size. Moreover, I show later in Proposition (ref) that there is no guarantee that the GIV spillover coefficient estimand is a non-negative weighted average of spillover coefficients. In this sense, the GIV spillover coefficient estimand could be far from the potentially heterogeneous true spillover coefficients.
This subsection introduces the estimator proposed in this paper, robust granular instrumental variables (RGIV). RGIV exploits the uncorrelatedness of idiosyncratic shocks through the generalized method of moments. The estimator's identifying variation comes from individual country-level idiosyncratic shocks.
Returning to the baseline environment described in Section (ref)---which again features unit-specific spillover coefficients $\phi_i$ and unit-specific shock variances $\sigma^2_i$---the RGIV estimator encodes the uncorrelatedness of idiosyncratic shocks through GMM moment conditions. Taking sizes $S_1, S_2, S_3$ as given, store data in the vector $\mathbf{r}_t = [r_{1t}, r_{2t}, r_{3t}]'$ and parameters in $\boldsymbol{\phi} = [\phi_1, \phi_2, \phi_3]'$. Letting $u_{i}(\mathbf{r}_t, \boldsymbol{\phi}) = r_{it} - \phi_i r_{St}$, moment function $g(\mathbf{r}_t, \boldsymbol{ \phi})$ encodes the condition that idiosyncratic shocks are uncorrelated
where ${\mathbb E}[g(\mathbf{r}_t, \boldsymbol{ \phi}_0)] = 0$ for true parameter $\boldsymbol{ \phi}_0$. The parameter-dependent weight matrix is $\widehat{W}(\boldsymbol{\phi}) = \textrm{diag}\big(\frac{1}{\widehat{\sigma}^2_1(\boldsymbol{\phi})\widehat{\sigma}^2_2(\boldsymbol{\phi})}, \frac{1}{\widehat{\sigma}^2_1(\boldsymbol{\phi})\widehat{\sigma}^2_3(\boldsymbol{\phi})}, \frac{1}{\widehat{\sigma}^2_2(\boldsymbol{\phi})\widehat{\sigma}^2_3(\boldsymbol{\phi})}\big)$ for $\widehat{\sigma}^2_i(\boldsymbol{\phi}) = \frac{1}{T} \sum_{t=1}^T u_{i}(\mathbf{r}_t, \boldsymbol{\phi}) ^2$. Then, the robust granular instrumental variables (RGIV) estimator is defined as a continuously updating GMM estimator:
$\widehat{W}(\boldsymbol{ \phi})$ is an efficient GMM weight matrix when idiosyncratic shocks are independent. Avoiding inversion of a potentially non-diagonal matrix, $\widehat{W}(\boldsymbol{ \phi})$ also ensures numerical stability if the initialization of $\boldsymbol{\phi}$ is far from the GMM objective function's minimum.
Equivalently, the optimization problem characterized in Equation (ref) minimizes the average squared correlation coefficients between pairs of estimated idiosyncratic shocks. To see this, define the estimated correlation coefficient of idiosyncratic shocks as $\widehat{\rho}_{ij}(\boldsymbol{\phi}) = \frac{\frac{1}{T} \sum_{t=1}^T u_{i}(\mathbf{r}_t, \boldsymbol{\phi})u_{j}(\mathbf{r}_t, \boldsymbol{\phi}) }{\sqrt{ \widehat{\sigma}_i^2(\boldsymbol{\phi})\widehat{\sigma}_j^2(\boldsymbol{\phi})} }$. Then, the RGIV estimator is
after multiplying the objective function in Equation (ref) by $\frac{1}{3}$. Intuitively, RGIV chooses the spillover coefficient vector $\boldsymbol{\phi}$ that makes the estimated shocks the least correlated.
RGIV also admits an instrumental variables interpretation, as a country's spillover coefficient is estimated using information from internally estimated idiosyncratic shocks to other countries. The asymptotic variance matrix of the RGIV estimator is $V = (G' \Sigma^{-1}G)^{-1}$ for Jacobian matrix $G = {\mathbb E}[\nabla_{\phi} g(\mathbf{r}_t, \boldsymbol{ \phi}_0)]$ and moment covariance matrix $\Sigma = {\mathbb E}[g(\mathbf{r}_t, \boldsymbol{ \phi}_0) g(\mathbf{r}_t, \boldsymbol{ \phi}_0)']$. Then the diagonal entries of $V$ are
The above display illustrates that the identifying variation of the RGIV estimator doesn't require skewness in the unit size distribution, as the identifying variation comes from individual idiosyncratic shocks. Estimates for $\widehat{\phi}_i^{RGIV}$ are more precise when idiosyncratic shocks to countries $j \neq i$ have higher variance. Intuitively, the estimated idiosyncratic shocks can be viewed as sequentially estimated internal instruments. Guessing the spillover coefficient of country 1, the resulting estimated idiosyncratic shock to country 1 can be used as an instrument for the estimation of the spillover coefficients for countries 2 and 3. If one or more pairs of estimated idiosyncratic shocks are too correlated, the procedure is repeated for a new spillover coefficient.
This section describes the assumptions needed for the robust GIV estimator for $n$ countries.
In Assumption (ref), the outcome variable $r_{it}$ responds to the size-aggregated outcome $r_{St}$ according to spillover coefficient $\phi_i$ and idiosyncratic shock $u_{it}$. Through $\phi_S< 1$, positive idiosyncratic shocks increase the size-weighted outcome $r_{St}$. As will be outlined below, the RGIV estimator is just-identified for $n=3$ and over-identified for $n>3$. Moreover, size $S_i$ is taken to be known by the econometrician. Hence, the model presented in (ref) can be be modified to allow for time-varying size (for $S_{it} \in (0,1)$ and $\sum_{i = 1}^n S_{it}=1$) without changing the proofs to follow. In Assumption (ref), the parameter space is assumed to be compact and restricted to encode the sign restriction of $\phi_S< 1$. Compactness is a standard technical assumption for the consistency of extremum estimators Newey1994.
The GMM estimator established in Definition (ref) below (called RGIV) exploits the uncorrelated unit-specific shock condition established in Assumption (ref)(ii). RGIV is constructed as a continuously updating GMM estimator Hansen1996. The moment function $g(\mathbf{r}_t, \boldsymbol{ \phi})$ contains the pairwise products of each of the estimated shocks, and the weight matrix inversely weights each moment by the product of the respective estimated shock variances. Exploiting shock uncorrelatedness can be best understood as being consistent with the tradition of exploiting second moment information in the traditional SVAR identification setting Kilian2017,Leeper1996.
The RGIV estimator is robust in that the estimator allows for unit-specific spillover coefficient heterogeneity while also allowing for unknown shock variances and equal unit sizes. When the researcher is a priori certain of homogeneous spillover coefficients $\phi_i = \phi$, the RGIV moment vector can accommodate this parameter restriction by restricting the elements of parameter vector $\boldsymbol{ \phi}$ to be homogeneous across units yielding possible gains in efficiency. Moreover, the assumption of parameter homogeneity is formally testable as will be discussed in Section (ref).
For Lemma (ref), the parameter space restriction $\widetilde{\phi}_S < 1$ in Assumption (ref) rules out the false solution to the population moment condition. The proof of Lemma (ref) shows that there is a second parameter $\check{\boldsymbol{ \phi}}$ such that $g_0(\check{\boldsymbol{ \phi}}) = 0$. However, $\check{\boldsymbol{ \phi}}$ is not in the parameter space because $\check{\phi}_S = 1 + (1-\phi_S) > 1$ and thus is not a candidate solution. In economic terms, Assumption (ref) represents the researcher's knowledge of the sign of an idiosyncratic shock's effect on the size-weighted outcome $r_{St}$. This knowledge can come from an application's institutional details; for the case of the running example of Eurozone yield spread spillovers, a positive idiosyncratic shock to one country gives rise to an increase in aggregated yield spreads since losses from the default of government debt are partially shared. The identification lemma can also be adapted for the opposite case $\phi_S>1$ by reversing the inequality specified in Assumption (ref) to $\widetilde{\phi}_S > 1$.
Mimicking Assumption 1 of Gabaix2020, the Lemma is derived under the weaker condition that the correlation structure of shocks is known (see Condition (ii') in Appendix (ref) for details). Theorems (ref) and (ref) (below), which establish consistency and asymptotic normality of the RGIV estimator, are shown under the assumption of uncorrelated shocks but can analagously be adapted to the known shock correlation case. These results follow from a standard application of arguments provided in Pakes1989.
\noindentRemarks.
In general, the baseline Gabaix2020 GIV estimand $\phi^{GIV}$ under spillover coefficient heterogeneity cannot be interpreted as a weighted average of unit-specific spillover coefficients. From Equation (ref), the spillover coefficient homogeneity assumption is necessary for the instrumental variable exclusion restriction to hold, as it ensures the endogenous term $r_{St}$ is differenced away. When shock variances are homogeneous across countries, Proposition (ref) (below) decomposes the GIV estimand into an equal-weighted spillover coefficient term and a term that depends on $(\phi_S - \phi_E)$. The bias is larger when the spillover coefficient varies systematically with the size distribution, giving rise to a larger gap between $\phi_S$ and $\phi_E$. Moreover, as also discussed in Appendix D.8 of Gabaix2020, the bias is smaller when the number of units is large.
Proposition (ref) implies that the GIV estimand doesn't admit a weighted average interpretation of unit-specific spillover coefficients. To see this, consider the following example. Suppose $n=3, \, \mathbf{S} =
',$ and $\boldsymbol{ \phi} =
'$. Applying Proposition \ref{prop:GIV_het_elasticities}, $ \phi^{GIV} =-0.18 \not\in [0.3,0.6]$, so $\phi^{GIV}$ is not a positive weighted average of individual spillover coefficients. Section (ref) investigates the practical implications of GIV under spillover coefficient heterogeneity using an empirically relevant DGP.
While Proposition (ref) highlights the potential pitfalls from mistakenly applying the baseline GIV estimator, Gabaix2020 also give guidance for estimating unit-specific spillover coefficients under additional restrictions. Their Proposition 6 and Appendix D.9 present alternative procedures that require heterogeneity to depend on observables and for the shock variance to be known respectively. In contrast, these restrictions are unnecessary for RGIV.
In this section, I propose two tests derived from standard GMM results Newey1994: a test of over-identifying restrictions that evaluates the uncorrelatedness of idiosyncratic shocks and a test that evaluates the homogeneous spillover coefficient condition of Gabaix2020. For what follows, impose Assumption (ref).
The uncorrelatedness of idiosyncratic shocks is directly testable when there are four or more units, as the number of moment conditions exceeds the number of unit-specific spillover coefficients. Recall that the RGIV estimator is a GMM estimator for moment function $g(\mathbf{r}_t, \boldsymbol{ \phi})$, which encodes the pairwise uncorrelatedness of idiosyncratic shocks. For $n\geq 4$ countries, the number of moments exceeds the number of estimated parameters allowing for the use of the Sargan--Hansen test. The null hypothesis $H_0{:}\, \, {\mathbb E}[g(\mathbf{r}_t, \boldsymbol{ \phi}_0)] = \mathbf{0}$ for true parameter $\boldsymbol{\phi}_0$ is rejected for large values of the $J$-statistic $J_T =T\cdot \widehat{Q}_T(\widehat{\boldsymbol{ \phi}}^{RGIV})$. Intuitively, $J_T$ is large when one or more pairs of estimated idiosyncratic shocks are correlated.
Spillover coefficient homogeneity across units is not only a subject of potential substantive interest---particularly given its role in the Gabaix2020 GIV estimator---but also can be formally evaluated within the framework of RGIV. Since the null hypothesis of coefficient homogeneity $H_0{:}\,\, \phi_1=\phi_2=...=\phi_n$ is a special case of RGIV's unit-specific spillover coefficients, $H_0$ can be tested with the distance metric test. Here, estimator $\overline{\boldsymbol{\phi}}$ minimizes $\widehat{Q}_T(\boldsymbol{\phi})$ subject to the constraints of null hypothesis $H_0$. Then, spillover coefficient homogeneity is rejected when the distance metric test statistic $DM_T = T(\widehat{Q}_T(\overline{\boldsymbol{\phi}}) - \widehat{Q}_T(\widehat{\boldsymbol{\phi}}^{RGIV}) )$ is large since $DM_T \xrightarrow{d} \chi^2_{n-1}$ under the null hypothesis. A large value of $DM_T$ indicates that the constraints of null hypothesis $H_0$ bind.
Motivated by the requirements of empirical applications, this section discusses two extensions that relax the condition of uncorrelated idiosyncratic shocks discussed in Section (ref) through observed and unobserved explanatory variables. I show that when observable time-varying explanatory variables determine the shocks' correlation structure, RGIV can be used after residualizing the outcome variables with respect to these observables. When the correlation structure is instead determined by unobserved factors with unknown loadings, the global identification condition fails.
In practice, observable characteristics can determine the correlation structure among shocks as described by the following two situations. First, the outcome variable could have unit-specific exposures to a particular observed variable---take country-specific exposures to the USD-EUR exchange rate in the Euro area sovereign yields example. Second, observable characteristics could also be used to account for a correlation structure driven by unobserved explanatory variables---like country-specific exposures to a “global financial conditions” factor---so long as such unobserved factors are in the span of the observed explanatory variables.
Observed variable $\mathbf{x}_t$ ($k \times 1$) affects outcome variable $r_{it}$ through a direct effect and an indirect effect. Modifying Assumption (ref), unit-specific coefficients $\boldsymbol{\beta}_i$ determine the cross-sectional correlation of $v_{it}$
where $u_{it}$ is still idiosyncratic in the sense that $u_{it}$ is independent of $u_{jt}$ for $i \neq j$. The orthogonality condition $u_{it} \perp \!\!\! \perp \mathbf{x}_t$ can be interpreted as a “selection-on-observables” assumption; the cross-sectional correlation of $v_{it}$ is entirely determined by the observed explanatory variables. Holding $r_{St}$ and $u_{it}$ constant, $\boldsymbol{{\beta}}_i$ can be interpreted as the direct effect of $\mathbf{x}_t$ on $r_{it}$. Additionally, mediated through changes in $r_{St} = \frac{1}{1-\phi_S}[\boldsymbol{\beta}'_S \mathbf{x}_t + u_{St}]$, $\frac{\phi_i}{1-\phi_S} \beta_S$ determines the indirect effect of $\mathbf{x}_t$ on $r_{it}$. Analogous to the discussion in Section (ref), the indirect effect of a change of $\mathbf{x}_t$ on unit $i$ is larger when units are on average sensitive to $\mathbf{x}_t$ (through $\boldsymbol{ \beta}_S$), unit $i$ is sensitive to spillovers (through $\phi_i$), and when units are on average sensitive to spillovers (through $\phi_S$).
The spillover coefficients in in Equation (ref) can be estimated by using RGIV after residualizing the outcome variable $r_{it}$ with respect to observed variables $\mathbf{x}_t$. Residualizing purges $r_{it}$ of the variation induced by the direct and indirect effects of $\mathbf{x}_t$ on $r_{it}$. Concretely, the procedure has two steps:
Conveniently, estimation uncertainty of the first step's regression coefficients has no effect on the asymptotic variance of the RGIV estimator in the second step. Thus, treating $\dot{r}_{it}$ as data in the second step produces valid standard errors for $\widehat{\boldsymbol{\phi}}^{RGIV}$. See Appendix (ref) for formal results.
In this section, I describe the tradeoff in assumptions between allowing spillover coefficient heterogeneity/unknown shock variances and a factor structure for the shocks. I show that global identification of unit-specific spillover coefficients is lost when a single unobserved factor is included in the error term. Such a case is empirically relevant because there is typically no a priori reason to believe that spillover coefficients are homogeneous across units and because estimated latent factors are commonly used as control variables in the applied GIV literature.
GIV regressions with estimated latent factors as control variables are ubiquitous in the applied GIV literature (Flynn2022, Gabaix2020,Gabaix2021, Camanho2022,Baumeister2023a,Adrian2022 among others). Typically, latent factors are estimated using principal components on the demeaned outcome variable before being included as control variables in the GIV regression. Such an approach is attractive because it enables practitioners to apply GIV to applications where the correlations between unit shocks are driven by a small number of latent factors. These estimated factors however are subject to measurement error, so their inclusion as control variables in subsequent GIV regressions give rise to attenuation bias.
Banafti2022 address this concern by extending GIV with homogeneous spillover coefficients to a large time and panel dimension framework. When the size distribution of units is very skewed (more skewed than Zipf's law), the sampling uncertainty arising from latent factors and loadings is negligible under their procedure. Homogeneity of spillover coefficients across units is crucial for the procedure's validity. When spillover coefficients are heterogeneous, cross-sectionally demeaning each unit no longer differences away the (no longer constant) contribution of spillovers. As a result, PCA estimates of the latent factors are polluted by the presence of spillovers.
Given the empirical relevance of unit-level heterogeneity and latent factors, I consider a heterogeneous spillover coefficient latent factor model. Taking $n$ to be fixed, restrictions on the skewness of the unit size distribution are unneeded. Investigating identification, I augment Assumption (ref) to include a single latent factor $f_t$ with unknown unit-specific loading $\lambda_i$
where the number of units $n$ is fixed and time $T\to\infty$. Factor $f_t$ is normalized so that ${\mathbb E}(f_t) = 0$, ${\mathbb E}(f_t^2) =1$, and $\lambda_1 > 0$. Shocks $u_{it}$ are idiosyncratic in that they are independent of latent factor $f_t$ and $u_{it} \perp \!\!\! \perp u_{jt}$ for $i \neq j$. Then, extending the logic of the RGIV estimator to the single factor case, the moment function between units $i \neq j$ is
where $\boldsymbol{\theta} = [\boldsymbol{ \phi}', \boldsymbol{\lambda}']'$ for $\boldsymbol{\lambda} = [\lambda_1,\dots,\lambda_n]'$. $g_{ij}^\text{factor}(\mathbf{r}_t, \boldsymbol{\theta})$ can then be stored in moment vector $g^\text{factor}(\mathbf{r}_t, \boldsymbol{\theta})$. When $n \geq 5$, the number of moments (a total of $n(n-1)/2$) is greater than or equal to the number of parameters to be estimated (a total of $2n$).
Lemma (ref) (below) however shows that the population moment condition $g_0^\text{factor}(\boldsymbol{\theta}) = {\mathbb E}[g(\mathbf{r}, \boldsymbol{\theta})]$ has multiple roots, so the model described in Equation (ref) is not identified.
The failure in the global identification condition comes from the assumptions of unknown shock variances and unknown factor loadings. If instead shock variances and factor loadings were taken to be known---mirroring Assumption 1 of Gabaix2020 and as is the case for the proof of Lemma (ref)---then the unit-specific spillover coefficients are identified.\footnote{In a related case, Appendix (ref) shows that differencing RGIV moments can account for factor loadings determined by unit-specific observables.} In the proof of Lemma (ref), I show that factor loadings can compensate for incorrect guesses for the spillover coefficient. To see this, let $\widetilde{\boldsymbol{\theta}} = [\widetilde{\boldsymbol{\phi}}', \, \widetilde{\boldsymbol{\lambda}}']'$ be a candidate root to the population moment condition. In the proof, I consider the set of solutions where the first unit's spillover coefficient and loading equal their true values ($\widetilde{\phi}_1 = \phi_1$ and $\widetilde{\lambda}_1 = \lambda_1$). I then show that a subset of the population moment conditions imply that
In words, $\widetilde{\lambda}_k$ can compensate for an incorrect spillover coefficient $\widetilde{\phi}_k \neq \phi_k$. Formally, the proof shows that there is at least one $\widetilde{\boldsymbol{\theta}}=\boldsymbol{\theta}_0$ such that $g_0^\textrm{factor}(\widetilde{\boldsymbol\theta})=0$. Moreover, $\widetilde{\boldsymbol{\theta}}$ need not be “close” to the true parameter, as there is no guarantee that $\max_i \widetilde{\phi}_i \geq \min_i \phi_i$ or $\min_i \widetilde{\phi}_i \leq \max_i \phi_i$. In this sense, $\widetilde{\boldsymbol{\phi}}$ is potentially far from the true spillover coefficient $\boldsymbol{\phi}_0$.
Lemma (ref) also implies a tradeoff between modeling unit-level heterogeneity and allowing for correlated shocks. Recall that Proposition 7 of Gabaix2020 shows that a homogeneous spillover coefficient is identified when shocks admit a factor structure with unknown loadings and if shock variances are homogeneous across units. In contrast, Lemma (ref) shows that global identification is lost under unrestricted heterogeneity on spillover coefficients and shock variances. Taken together, these results suggest that practitioners face a tradeoff between two empirically-relevant models.
Applying the robust granular instrumental variables (RGIV) methodology to the Euro area sovereign yield spillovers application of a working paper version of Gabaix2020, this section finds strong evidence of country-level heterogeneity in the spillovers of idiosyncratic shocks.
Just as in Gabaix2020, the sample consists of daily data on 10-year zero coupon yields from Bloomberg from September 1, 2009 to May 31, 2018 giving a total of 2283 observations. The included countries are Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Netherlands, Portugal, Slovenia, and Spain. For the yield spread of country $i$ (relative to Germany) $y_{it}$, the outcome variable $r_{it}$ is defined as $r_{it} = \frac{y_{it} - y_{it-1}}{0.01 + y_{i,t-1}}$ just as in Gabaix2020. Since shocks are linearly related to the outcome variable as outlined in Assumption (ref), $r_{it}$ is winsorized (over time) at the 0.5 and 99.5th percentiles. Following Gabaix2020, size is time-varying and computed as “debt-at-risk” $S_{i,t-1} = \frac{B_{i,t-1} y_{i,t-1}}{\sum_j B_{j,t-1} y_{j,t-1}}$ where $B_{i,t-1}$ is the outstanding government debt of country $i$.
Observed explanatory variables are included to account for unit-specific exposures to latent aggregate shocks. In particular I include the STOXX 50 Volatility Index (differences), the STOXX Europe 600 Index (growth), the EUR-USD exchange rate (growth), the United States 10 Year Treasury yield (growth), BBB/Baa-10Y spread (differences), and the European Fama-French 5 factors Fama2015. These observed explanatory variables account for differential exposure of countries to uncertainty, exchange rates, equity prices, and risk. All data are downloaded from Bloomberg.
Even with observed explanatory variables, well-documented regional correlations among Europe's core and periphery countries likely still drive correlations between shocks Bayoumi1992. To address, I size-aggregate $r_{it}$ to form larger country blocks; even if shocks within blocks are correlated, RGIV is still valid so long as shocks between blocks are uncorrelated. Hence, uncorrelatedness between blocks is a weaker condition than uncorrelatedness between countries. Specifically, I consider the following four blocks:
Block 1 includes countries typically classified as being members of the EU “core.” Blocks 2 and 3 contain countries typically classified as being members of the EU “periphery.” Block 4 contains Slovenia, which is typically left uncategorized on account of its distinct institutional structure as a former member of Yugoslavia.
I report RGIV point estimates and standard errors for individual spillover coefficients. Standard errors are computed using a HAC GMM weight matrix to account for the possible serial correlation of idiosyncratic shocks.\footnote{Specifically, a Newey-West kernel with the Lazarus2018 truncation parameter rule of $1.3 \sqrt{T}$.} Size-weighted spillover coefficients are constructed using the Delta method, using the average block size over the estimation sample.
I also present GIV results based on the shock variance approximation and factor estimation procedures detailed in Section 5.3 of the July 2021 working paper version of Gabaix2020. I include this procedure for comparison, as it is used in the original application. This procedure approximates the variance of the shocks with $\text{Var}(r_{it})$, valid when spillovers are small. As a diagnostic for GIV instrument strength, the first stage $F$-statistic is also reported.
I find strong evidence of spillovers in the aggregate and of spillover heterogeneity across countries. In the preferred specification (Column 1 of Table (ref)), the null hypothesis of uncorrelated idiosyncratic shocks isn't rejected suggesting that the RGIV moment conditions are consistent with the data. Evidence of spillovers in the aggregate, the size-weighted spillover coefficient is 0.54 (with a standard error of 0.08). Moreover, with a $p$-value of $<0.001$ for the spillover coefficient homogeneity test, the null hypothesis of spillover coefficient homogeneity is rejected at conventional significance levels. Turning to individual spillover coefficients, the spillover coefficient for the western periphery block (0.83) is more than twice that of the core countries (0.40).
The qualitative features of the preferred specification are robust to omitting observed explanatory variables and using an estimator that is efficient under higher-order dependence among shocks. Omitting explanatory variables, Column 2 of Table (ref) shows spillover coefficient heterogeneity across blocks. Relative to the preferred specification, the estimated size-weighted spillover coefficient is slightly larger (at 0.63). Recall that the RGIV estimator is efficient under the independence of idiosyncratic shocks, but isn't guaranteed to be efficient under the weaker condition of idiosyncratic shock uncorrelatedness.\footnote{For example, a shared volatility term would induce dependence of higher order moments.} To address, Column 3 shows the results of a “higher-order efficient” estimator, which uses the procedure outlined in Remark 5 of Section (ref). The point estimates and standard errors are nearly identical, suggesting that higher order dependence of idiosyncratic shocks play a minor role in this application. The spillover coefficient estimated with GIV procedure featured in a working paper version of Gabaix2020 is comparable to the size-weighted spillover coefficient computed with RGIV, but is unable to speak to country-level heterogeneity. Column 4 of Table (ref) presents results of the baseline GIV methodology applied to this section's core-periphery-aggregated panel (distinct from the country-level panel of Gabaix2020). The spillover coefficient of 0.57 is close to the size-aggregated spillover coefficient computed in the preferred RGIV specification.
Section (ref) gives additional RGIV results under alternative winsorizations, the more widely-used Andrews1991 truncation parameter, omitting the Fama-French factors as observed explanatory variables, and alternative choices of country blocking. Section (ref) also contains results for the 0-factor and 2-factor GIV specifications.
Summarizing, RGIV finds strong evidence of country-level differences in sovereign yield spillovers. In response to a 1% increase in the size-weighted relative yield spread, the relative yield spread of “core” countries increases by 0.4% compared to an increase of 0.8% for countries in the western “periphery.” Substantively, these estimates point to the importance of understanding the role of country-level characteristics in the heterogeneous propagation of idiosyncratic shocks during sovereign debt crises.
I close by showing that RGIV has good finite sample performance in simulation using four DGPs based on the preferred specification studied in Section (ref). In contrast, I show that two versions of the GIV procedures---the “feasible” GIV procedure (featured in this paper's application application) and “oracle” GIV procedure that takes idiosyncratic shock variances $\sigma^2_i$ to be known---can severely under-cover under spillover coefficient heterogeneity. \\
\noindentSetup The “homogeneous spillovers” DGP is loosely based on the preferred specification of Section (ref). Under spillover coefficient homogeneity across units, the DGP serves as a baseline, as both the oracle GIV and RGIV estimators are correctly specified. For $n=4$ units and a sample length of $T = 2283$, the model parameters are below:
Having established an environment for which both the oracle GIV and RGIV estimators are valid, I study three additional DGPs that deviate from the homogeneous spillovers DGP. These illustrate the finite sample properties of RGIV and GIV. First, the “coefficient outlier” DGP takes the homogeneous spillovers specification and sets the spillover coefficient of the fourth unit to $0.75$. Second, the “shock variance outlier” DGP takes the homogeneous spillovers specification and sets the shock standard deviation of the first unit to $0.03$. Third, the “application” DGP allows for heterogeneous spillover coefficients and shock variances by assigning them to be the estimated values from the application's preferred specification.
For each DGP, I consider results from the RGIV estimator, “feasible” GIV estimator, and “oracle” estimator. For the RGIV estimator, I compute confidence intervals for the spillover coefficients of individual units, size-weighted spillover coefficients, and equal-weighted spillover coefficients. These confidence intervals highlight how RGIV can be used for researchers interested in individual spillover coefficients and for those interested in a single, aggregated parameter. The “feasible” GIV estimator is computed using the $\text{Var}(r_{it}) \approx \text{Var}(u_{it})$ approximation (valid when spillovers are small) while the “oracle” GIV estimator is computed as if the true idiosyncratic shock variance $\text{Var}(u_{it})$ were known. Both GIV estimators are included to show the practical implications of mistakenly assuming spillover coefficient homogeneity across units. The feasible GIV estimator, in particular, is included to evaluate the approximation $\text{Var}(r_{it}) \approx \text{Var}(u_{it})$ and for completeness (it is featured in the application section).
A conservative notion of coverage is used for the GIV estimators. Since the GIV estimators require spillover coefficient homogeneity across units, I report the proportion of GIV confidence intervals that contain any positive-weighted average of potentially heterogeneous spillover coefficients---measured as the proportion of GIV confidence intervals with a nonempty intersection with the closed interval $[\min_{1\leq i \leq n} \phi_i, \max_{1\leq i \leq n} \phi_i]$. \\
\noindentResults While the empirical coverage is near the nominal level for RGIV, feasible RGIV can severely under-cover. Table (ref) shows that nearly 95% of the confidence intervals for $\widehat{\phi}_S$ and $\widehat{\phi}_E$ contain the true estimands $\phi_S$ and $\phi_E$ in all four DGPs. In contrast, the true spillover coefficient is contained in none of the feasible GIV confidence intervals in the homogeneous spillovers DGP. Considering that the true coefficient is contained in 95% of the oracle estimator confidence intervals, together these results suggest that the approximation $\text{Var}(r_{it}) \approx \text{Var}(u_{it})$ is inappropriate for this DGP.
Under spillover coefficient heterogeneity, the GIV estimators can substantially under-cover---even with the conservative notion of coverage featured in this simulation study. For the coefficient outlier DGP, none of the feasible GIV confidence intervals contain any positive weighted average of heterogeneous spillovers, compared to 15% for the oracle estimator. Hence, for certain DGPs, mistakenly assuming spillover coefficient homogeneity and applying versions of the baseline GIV estimator can result in unreliable inference.
Evaluating the properties of the testing procedures featured in Section (ref), the empirical size of the RGIV specification and coefficient homogeneity tests are near their nominal level. The empirical rejection rate of the coefficient homogeneity test is 4.2% and 5.2% for the homogeneous spillovers and variance outlier DGPs respectively, where the null hypothesis of spillover coefficient homogeneity across units is true. When spillover coefficients are not homogeneous, as is the case under the elasticity outlier and application DGPs, the null hypothesis of homogeneous coefficients is rejected in more than 99% of Monte Carlo replications. Turning to the specification test, note that the RGIV framework is correctly specified in all four DGPs. At worst, the null hypothesis of idiosyncratic shock uncorrelatedness is rejected in 6.4% of Monte Carlo replications under the application DGP.
The RGIV estimator presents a modest (if any) power tradeoff relative to the oracle GIV estimator, and unit-specific RGIV spillover coefficients have good coverage properties despite applying to a more general environment. In the homogeneous spillovers application, the equal-weighted RGIV spillover coefficient has a comparable median confidence interval length (0.038) to that of GIV (0.046).\footnote{In general, researchers interested in any positive-weighted average of potentially heterogeneous spillover coefficients could estimate unit-specific spillover coefficients using RGIV and compute the weights that minimize the confidence interval length of the resulting aggregated estimator.} Turning to unit-specific spillover coefficients, Table (ref) shows that the empirical coverage across all four DGPs and RGIV spillover coefficients is near $0.95$.
Gabaix2020 introduces granularity-based identification, a substantial step forward for credible spillover estimates in macroeconomics and finance. Its baseline granular instrumental variables procedure however requires strong assumptions---namely homogeneous spillovers across units, homogeneous shock variances, skewed unit size, and idiosyncratic shocks (after accounting for common factors with known loadings). I build on this innovative approach by showing that unit-specific spillover coefficient heterogeneity with heterogeneous (and unknown) shock variances can jointly be accounted for without further restrictions. My estimator, called robust granular instrumental variables, also allows for homogeneous unit sizes unlike GIV. Intuitively, my approach uses internally estimated individual idiosyncratic shocks as instruments. I give results on identification and inference, showing that the required GIV assumption of coefficient homogeneity is directly testable. Relaxing the idiosyncratic shock assumption, I highlight an tradeoff: practitioners must choose between allowing unrestricted unit-level heterogeneity and a general shock covariance structure. Studying Euro zone sovereign yields, I find strong evidence of country-level heterogeneity in spillovers. I find that my proposed estimator has good finite sample properties through simulation.
There are several directions for future work. First, my approach builds on the baseline framework proposed by Gabaix2020. Here, an individual unit's outcome is determined by the size-weighted aggregate outcome. A structure where spillover responses are allowed to differ by the shock's source could be of interest for empirical work. Second, while the core-periphery structure of Euro area countries serves as a natural basis for grouping units in the empirical application, grouping could be automated. Third, as discussed in Section (ref), practitioners face a tradeoff between unrestricted unit-level heterogeneity in spillover coefficients and correlated shocks. Further work on alternative economically-motivated conditions that preserve global identification would be of considerable interest to applied users.