Functional Network Autoregressive Models for Panel Data

Suppose that we have balanced panel data of size $(n,T)$: $\{(Y_{it},X_{it},w_{i,1},\ldots,w_{i,n}):i\in[n],\>t\in[T]\}$. The number of time periods $T$ can be either fixed or tending to infinity jointly with the sample size $n$. Here, $Y_{it}:[0,1]\to\mathbb{R}$ denotes a random outcome function of interest with the common support $[0,1]$, $X_{it}=(X_{it}^{1},\ldots,X_{it}^{d_{x}})^{\top}$ denotes a vector of covariates, and $w_{i,j}\in\mathbb{R}$ is the $(i,j)$-th element of an $n\times n$ time-invariant interaction matrix $W_{n}=(w_{i,j})$. The value of each $w_{i,j}$ is pre-determined non-randomly. In social network analysis, it is common to set $w_{i,j}=c_{i,j}\bm{1}\{\text{$i$ and $j$ are peers}\}$, where $c_{i,j}$ is some normalizing constant. Similarly, if each $i$ represents a spatial unit, one may use $w_{i,j}=c_{i,j}\bm{1}\{\Delta(i,j)\leq\overline{\Delta}\}$, where $\Delta(i,j)$ is the distance between $i$ and $j$, and $\overline{\Delta}$ is a given threshold. As is the convention, we set $w_{i,i}=0$ for all $i$ for normalization.

As shown in (1.2), our working model is given as follows: for $s\in[0,1]$,

$\displaystyle Y_{it}(s)=\alpha_{0}(s)A(\overline{Y}_{it},s)+X_{it}^{\top}\beta_{0}(s)+f_{0i}(s)+\varepsilon_{it}(s),\;\;i\in[n],\;\;t\in[T]$

(2.1)

where $\overline{Y}_{it}=\sum_{j=1}^{n}w_{i,j}Y_{jt}$. Recall that we throughout assume $A(\cdot,\cdot)$ is linear in its first argument so that we have $\sum_{j=1}^{n}w_{i,j}A(Y_{jt},\cdot)=A(\overline{Y}_{it},\cdot)$. For the structure of network interaction, Beyaztas et al. (2024) and Hoshino (2024) consider alternatively the following form: $\int_{0}^{1}\overline{Y}_{it}(u)\alpha_{0}(u,s)\,\text{d}u$, reflecting the usual functional linear form in the FDA literature. We cannot say which of this type of interaction structure or the proposed one is more general, but ours may offer some interpretational simplicity. We impose additional shape restrictions on $A(\cdot,\cdot)$ later.

The parameters of primary interest are the interaction effect function $\alpha_{0}(s)$ and the coefficient functions $\beta_{0}(s)=(\beta_{01}(s),\ldots,\beta_{0d_{x}}(s))$. The functional individual effects $f_{01}(s),\ldots,f_{0n}(s)$ are treated as nuisance parameters. Restricting the support of $s$ to be a unit interval is a normalization, which is harmless as long as the response functions have the identical interval support. For simplicity, we do not explicitly assume that $X_{it}$ is a function of $s$, which can be relaxed easily at the expense of more complicated notations and proofs. A constant term is not included in $X_{it}$. In the following, we assume that $Y_{it}\in L^{2}(0,1)$, $\varepsilon_{it}\in L^{2}(0,1)$, and that $\alpha_{0}$ and $\beta_{0}$ are continuous.

We discuss the stationarity of our model. Recall that our model is a system of simultaneous functional equations, which may not have a unique interior solution in general, depending on the parameter values. Thus, just like the stability condition for vector autoregressive models in the time series literature, we need to impose some conditions on the model to ensure that the outcome functions follow a unique stationary data-generating process and prevent explosive behavior.

Let $Y_{t}(s)=(Y_{1t}(s),\ldots,Y_{nt}(s))^{\top}$, $\bm{A}(Y_{t},s)=(A(Y_{1t},s),\ldots,A(Y_{nt},s))^{\top}$, $X_{t}=(X_{1t},\ldots,X_{nt})^{\top}$, $F_{0}(s)=(f_{01}(s),\ldots,f_{0n}(s))^{\top}$, and $\mathcal{E}_{t}(s)=(\varepsilon_{1t}(s),\ldots,\varepsilon_{nt}(s))^{\top}$. Then, we can re-write (1.2) in matrix form as

$\displaystyle Y_{t}(s)=\alpha_{0}(s)W_{n}\bm{A}(Y_{t},s)+X_{t}\beta_{0}(s)+F_{0}(s)+\mathcal{E}_{t}(s),\;\;t\in[T].$

(2.2)

This expression clearly indicates that our model is characterized as $T$ distinct systems of functional equations of size $n$. We introduce the following assumption to ensure that the model has a unique stationary solution.

Assumption 2.1(i) requires that the magnitude of the network interaction is not too strong. The existence of $\overline{\alpha}_{0}$ is guaranteed by the continuity of $\alpha_{0}$. With Assumption 2.1(ii), we have for any $h,h^{\prime}\in L^{2}(0,1)$ that $||A(h-h^{\prime},\cdot)||_{L^{2}}\leq||h-h^{\prime}||_{L^{2}}$, which indicates the contraction property of the operator $A$. This assumption still accommodates many empirically interesting interaction patterns. For example, in the case of point-evaluation functional $A(h,s)=h(s)$, it trivially satisfies $||A(h,\cdot)||_{L^{2}}=||h||_{L^{2}}$. For another example, suppose $A(h,s)=\int_{0}^{1}h(u)\nu(u,s)\text{d}u$ for some continuous $\nu$. Since

$\displaystyle||A(h,\cdot)||_{L^{2}}^{2}\leq\int_{0}^{1}\int_{0}^{1}\left|h(u)\nu(u,s)\right|^{2}\text{d}u\text{d}s\leq\overline{\nu}^{2}||h||_{L^{2}}^{2},$

(2.3)

where $\overline{\nu}\coloneqq\max_{u,s\in[0,1]^{2}}|\nu(u,s)|$, Assumption 2.1(ii) is implied if $\overline{\nu}\leq 1$ holds.

Now, denote $\mathcal{H}_{n,p}\coloneqq\{H=(h_{1},\ldots,h_{n}):h_{i}\in L^{p}(0,1)\;\text{for all}\;i\}$, and define a linear operator $\mathcal{A}$ as

$\displaystyle(\mathcal{A}H)(s)\coloneqq\alpha_{0}(s)W_{n}\bm{A}(H,s),\;\;H\in\mathcal{H}_{n,p}.$

(2.4)

Then, we can write our model symbolically as follows:

$\displaystyle Y_{t}=\mathcal{A}Y_{t}+X_{t}\beta_{0}+F_{0}+\mathcal{E}_{t},\;\;t\in[T].$

(2.5)

Further, denoting Id to be the identity operator, if the inverse operator $(\text{Id}-\mathcal{A})^{-1}$ exists, the solution $Y_{t}$ of the system can be uniquely determined up to an equivalence class in $\mathcal{H}_{n,2}$ as

$\displaystyle Y_{t}=(\text{Id}-\mathcal{A})^{-1}[X_{t}\beta_{0}+F_{0}+\mathcal{E}_{t}],\;\;t\in[T].$

(2.6)

The next proposition states that Assumption 2.1 is sufficient for the existence of $(\text{Id}-\mathcal{A})^{-1}$.

Note that the explicit form of the inverse operator $(\text{Id}-\mathcal{A})^{-1}$ cannot be derived in general, except for some simple cases such as $(\mathcal{A}Y_{t})(s)=\alpha_{0}(s)W_{n}Y_{t}(s)$. In this case, $(\text{Id}-\mathcal{A})^{-1}$ is obtained as $(I_{n}-\alpha_{0}(\cdot)W_{n})^{-1}$. However, in practice, we can approximate it with arbitrary precision by truncating the Neumann series expansion $(\text{Id}-\mathcal{A})^{-1}=\sum_{\ell=0}^{\infty}\mathcal{A}^{\ell}$ at a sufficiently large order (see, e.g., Kress, 2014). See Remark 1 in Zhu et al. (2022) for a related discussion.

To estimate the unknown functional parameters $\alpha_{0}(s)$ and $\beta_{0}(s)$, there are broadly two approaches. The first is a ”local” approach that estimates the values of these functions at specific $s$-values, repeating the estimation across different points to recover the full functional forms. The second is a ”global” approach that estimates the entire functional forms in a single step using a series approximation method. Although both approaches are theoretically valid, the local approach typically requires more computation time and often leads to larger variance (but smaller bias) because it does not exploit information from nearby evaluation points. This study adopts the global approach.

Let $\{\phi_{k}:k=1,2,\ldots\}$ be a series of orthonormal basis functions. We throughout assume that $\phi_{k}$’s are continuous on $[0,1]$. Then, if the functions $\alpha_{0}$ and $\beta_{0}$ are sufficiently smooth, we can approximate

	$\displaystyle\alpha_{0}(s)$	$\displaystyle\approx\sum_{k=1}^{K}\phi_{k}(s)\theta_{0\alpha,k},$		(3.1)
	$\displaystyle\beta_{0j}(s)$	$\displaystyle\approx\sum_{k=1}^{K}\phi_{k}(s)\theta_{0j,k},\;\;j\in[d_{x}],$		(3.2)

uniformly in $s\in[0,1]$, for some coefficient vectors $\theta_{0\alpha}=(\theta_{0\alpha,1},\ldots,\theta_{0\alpha,K})^{\top}$ and $\theta_{0j}=(\theta_{0j,1},\ldots,\theta_{0j,K})^{\top}$, $j\in[d_{x}]$. Here, $K\equiv K_{nT}$ is a sequence of positive integers tending to infinity as $nT$ increases. For simplicity of presentation, we use the same basis function $\phi_{k}$ and the same basis order $K$ to approximate both $\alpha_{0}$ and $\beta_{0}$. Define $\theta_{0}=(\theta_{0\alpha}^{\top},\theta_{01}^{\top},\ldots,\theta_{0d_{x}}^{\top})^{\top}$, $\phi^{K}(s)=(\phi_{1}(s),\ldots,\phi_{K}(s))^{\top}$,

$\displaystyle R_{it}(s)\coloneqq(A(\overline{Y}_{it},s),X_{it}^{\top})^{\top},\;\;H_{it}(s)\coloneqq R_{it}(s)\otimes\phi^{K}(s),\;\;\text{and}\;\;H_{t}(s)=(H_{1t}(s),\ldots,H_{nt}(s))^{\top}.$

(3.3)

Then, we can further re-write the model (1.2) as

$\displaystyle Y_{t}(s)=H_{t}(s)\theta_{0}+F_{0}(s)+V_{t}(s)+\mathcal{E}_{t}(s),\;\;t\in[T].$

(3.4)

Here, $V_{t}(s)=(v_{1t}(s),\ldots,v_{nt}(s))^{\top}$ is an $n\times 1$ vector of series approximation errors:

$\displaystyle v_{it}(s)\coloneqq A(\overline{Y}_{it},s)\{\alpha_{0}(s)-\phi^{K}(s)^{\top}\theta_{0\alpha}\}+\sum_{j=1}^{d_{x}}X_{it}^{j}\{\beta_{0j}(s)-\phi^{K}(s)^{\top}\theta_{0j}\}.$

(3.5)

Under the assumptions we will introduce, this approximation error diminishes to zero at a certain rate as $K$ goes to infinity. How to choose an appropriate $K$ will be discussed in Remark 3.2.

Further, let

$\displaystyle\bm{Y}(s)=\left(\begin{array}[]{c}Y_{1}(s)\\ \vdots\\ Y_{T}(s)\end{array}\right),\quad\bm{H}(s)=\left(\begin{array}[]{c}H_{1}(s)\\ \vdots\\ H_{T}(s)\end{array}\right),\quad\bm{V}(s)=\left(\begin{array}[]{c}V_{1}(s)\\ \vdots\\ V_{T}(s)\end{array}\right),\quad\bm{\mathcal{E}}(s)=\left(\begin{array}[]{c}\mathcal{E}_{1}(s)\\ \vdots\\ \mathcal{E}_{T}(s)\end{array}\right),$

(3.18)

and $\underset{n(T-1)\times nT}{\bm{D}}=(d_{ij})$ be the one-period lag operator, whose $(i,j)$-th element is defined as

$\displaystyle d_{ij}=\begin{cases}-1&\text{if}\;\;i=j\\ 1&\text{if}\;\;n+i=j\\ 0&\text{otherwise}\end{cases}$

(3.19)

Then, we can remove the unknown fixed effects from the model in the following manner:

$\displaystyle\bm{D}\bm{Y}(s)$

$\displaystyle=\bm{D}\bm{H}(s)\theta_{0}+\bm{D}\bm{V}(s)+\bm{D}\bm{\mathcal{E}}(s).$

(3.20)

We estimate $\theta_{0}$ based on this expression. In order to consistently estimate $\theta_{0}$, we need to address the endogeneity issue caused by the simultaneous interactions of the response functions; that is, since $A(\overline{Y}_{it},s)$ is correlated with the error term $\varepsilon_{it}(s)$ in general, simply regressing $\bm{D}\bm{Y}(s)$ on $\bm{D}\bm{H}(s)$ does not yield a consistent estimate of $\theta_{0}$. To tackle this issue, we employ an instrumental variable (IV) approach.

Suppose we have a $d_{q}\times 1$ vector of IVs $Q_{it}=(Q_{it}^{1},\ldots,Q_{it}^{d_{q}})^{\top}$ for $A(\overline{Y}_{it},s)$. For example, noting $W_{n}Y_{t}=W_{n}\mathcal{A}Y_{t}+W_{n}X_{t}\beta_{0}+W_{n}F_{0}+W_{n}\mathcal{E}_{t}$, we can find that the network lagged covariates $\overline{X}_{it}\coloneqq\sum_{j=1}^{n}w_{i,j}X_{jt}$ (and also their lags) are valid candidates for $Q_{it}$. Define

$\displaystyle B_{it}\coloneqq(Q_{it}^{\top},X_{it}^{\top})^{\top},\;\;Z_{it}(s)\coloneqq B_{it}\otimes\phi^{K}(s),\;\;Z_{t}(s)=(Z_{1t}(s),\ldots,Z_{nt}(s))^{\top},$

(3.21)

and $\bm{Z}(s)=(Z_{1}(s)^{\top},\ldots,Z_{T}(s)^{\top})^{\top}$. Then, we have $\mathbb{E}[\bm{Z}(s)^{\top}\bm{D}^{\top}\bm{D}\bm{\mathcal{E}}(s)]=\bm{0}_{(d_{q}+d_{x})K}$.

Now, although one can estimate $\theta_{0}$ based on the linear moment conditions $\mathbb{E}[\bm{Z}(s)^{\top}\bm{D}^{\top}\bm{D}\bm{\mathcal{E}}(s)]=\bm{0}_{(d_{q}+d_{x})K}$ only, which results in a two-stage least squares (2SLS) type estimator, we can utilize additionally the quadratic moment conditions to improve the efficiency of estimation (see, e.g., Lin and Lee, 2010). That is, under the independence assumption on the error terms $\{\varepsilon_{it}(s)\}_{i\in[n],t\in[T]}$ (see Assumption 3.3(i) below), for any $n(T-1)\times n(T-1)$ matrices $P_{m}\coloneqq I_{T-1}\otimes P_{m,1}$, where $P_{m,1}$ ($m=1,\ldots,M$) is an $n\times n$ matrix whose diagonal elements are all zero, we have

$\displaystyle\mathbb{E}[\bm{\mathcal{E}}(s)^{\top}\bm{D}^{\top}P_{m}\bm{D}\bm{\mathcal{E}}(s)]=0,\;\;m\in[M].$

(3.22)

Some examples of $P_{m,1}$ include $P_{m,1}=W_{n}$ and $P_{m,1}=W_{n}^{\top}W_{n}-\text{diag}(W_{n}^{\top}W_{n})$.

Combining the linear and quadratic moment conditions, we can construct our estimator based on the following $d_{g}\coloneqq(d_{q}+d_{x})K+M$ moment conditions: for $s\in[0,1]$,

$\displaystyle\frac{1}{n(T-1)}\left(\begin{array}[]{c}\mathbb{E}[\bm{Z}(s)^{\top}\bm{D}^{\top}\bm{D}\bm{\mathcal{E}}(s)]\\ \mathbb{E}[\bm{\mathcal{E}}(s)^{\top}\bm{D}^{\top}P_{1}\bm{D}\bm{\mathcal{E}}(s)]\\ \vdots\\ \mathbb{E}[\bm{\mathcal{E}}(s)^{\top}\bm{D}^{\top}P_{M}\bm{D}\bm{\mathcal{E}}(s)]\end{array}\right)=\bm{0}_{d_{g}}.$

(3.27)

As the empirical counterpart of these moment conditions, given a candidate value $\theta$ for $\theta_{0}$, we define

$\displaystyle\underset{d_{g}\times 1}{g_{nT}(s;\theta)}\coloneqq\frac{1}{n(T-1)}\left(\begin{array}[]{c}\bm{Z}(s)^{\top}\bm{D}^{\top}\bm{D}\bm{E}(s;\theta)\\ \bm{E}(s;\theta)^{\top}\bm{D}^{\top}P_{1}\bm{D}\bm{E}(s;\theta)\\ \vdots\\ \bm{E}(s;\theta)^{\top}\bm{D}^{\top}P_{M}\bm{D}\bm{E}(s;\theta)\end{array}\right),$

(3.32)

where $\bm{E}(s;\theta)\coloneqq\bm{Y}(s)-\bm{H}(s)\theta$. The estimator of $\theta_{0}$ is obtained by minimizing the norm of $g_{nT}(s;\theta)$ over $s\in[0,1]$. To this end, we pre-specify $L\equiv L_{nT}$ grid points in $[0,1]$, denoted by $0<s_{1}<\cdots<s_{L}<1$, and numerically integrate the moment functions across these points:

$\displaystyle\overline{g}_{nT}(\theta)\coloneqq\frac{1}{L}\sum_{l=1}^{L}g_{nT}(s_{l};\theta).$

(3.33)

Situations where the response functions are not fully observable are discussed in Remark 3.3.

Now, we are ready to introduce our estimator:

$\displaystyle\begin{split}&\widehat{\theta}_{nT}=(\widehat{\theta}_{nT,\alpha}^{\top},\widehat{\theta}_{nT,1}^{\top},\ldots,\widehat{\theta}_{nT,d_{x}}^{\top})^{\top}\coloneqq\operatorname*{argmin}_{\theta\in\Theta_{K}}\mathcal{Q}_{nT}(\theta)\\ &\text{where}\;\;\mathcal{Q}_{nT}(\theta)\coloneqq\overline{g}_{nT}(\theta)^{\top}\Omega_{nT}\overline{g}_{nT}(\theta),\end{split}$

(3.34)

$\Omega_{nT}$ is a $d_{g}\times d_{g}$ positive definite symmetric weight matrix, and $\Theta_{K}\subset\mathbb{R}^{(d_{x}+1)K}$ is a compact parameter space containing $\theta_{0}$ in its interior. For one example of the weight matrix, we can use $\Omega_{nT}=I_{d_{g}}$, or for another example,

$\displaystyle\Omega_{nT}=\left(\begin{array}[]{cc}\left(\sum_{l=1}^{L}\bm{Z}(s_{l})^{\top}\bm{D}^{\top}\bm{D}\bm{Z}(s_{l})/N\right)^{-1}&\bm{0}_{(d_{q}+d_{x})K\times M}\\ \bm{0}_{M\times(d_{q}+d_{x})K}&I_{M}\end{array}\right),$

(3.37)

where

$\displaystyle N\coloneqq nL(T-1).$

(3.38)

Once $\widehat{\theta}_{nT}$ is obtained, the estimators of $\alpha_{0}(s)$ and $\beta_{0}(s)$ are given as

	$\displaystyle\widehat{\alpha}_{nT}(s)$	$\displaystyle\coloneqq\phi^{K}(s)^{\top}\widehat{\theta}_{nT,\alpha},$		(3.39)
	$\displaystyle\widehat{\beta}_{nT,j}(s)$	$\displaystyle\coloneqq\phi^{K}(s)^{\top}\widehat{\theta}_{nT,j},\;\;j\in[d_{x}]$		(3.40)

which we refer to as the integrated-GMM estimators. Additionally, if one is interested in the estimation of individual fixed effect functions, the following estimator can be used:

$\displaystyle\widehat{f}_{ni}(s)\coloneqq\frac{1}{T}\sum_{t=1}^{T}\left(Y_{it}(s)-\widehat{\alpha}_{nT}(s)A(\overline{Y}_{it},s)-X_{it}^{\top}\widehat{\beta}_{nT}(s)\right),\;\;i\in[n]$

(3.41)

where $\widehat{\beta}_{nT}(s)=(\widehat{\beta}_{nT,1}(s),\ldots,\widehat{\beta}_{nT,d_{x}}(s))^{\top}$. Consistent estimation of $f_{0i}(s)$ by $\widehat{f}_{ni}(s)$ requires $T$ to increase to infinity, while $\alpha_{0}(s)$ and $\beta_{0}(s)$ can be consistently estimated even when $T$ is fixed. More specifically, noting that the error term $\varepsilon_{it}(s)$ has mean zero, to average out the errors applying the law of large numbers at each $i$, $T$ must increase to infinity. Unlike $\widehat{\alpha}_{nT}(s)$ and $\widehat{\beta}_{nT}(s)$, the estimator $\widehat{f}_{ni}(s)$ is not necessarily continuous, as we do not preclude cases where $Y_{it}(s)$ and $A(\overline{Y}_{it},s)$ are discontinuous in $s$.

To derive the asymptotic properties of our estimator, we first need to specify the structure of our sampling space. Following Jenish and Prucha (2012), let $\mathcal{D}\subset\mathbb{R}^{d}$, $1\leq d<\infty$ be a possibly uneven lattice, and $\mathcal{D}_{n}\subset\mathcal{D}$ be the set of observation locations. Once the observation locations are determined for a given sample of size $n$, we assume that they do not vary over time $t$. For spatial data, $\mathcal{D}$ would be defined by a geographical space with $d=2$. Note that $\mathcal{D}$ does not necessarily have to be exactly observable to us. For example, $\mathcal{D}$ is possibly a complex space of general social and economic characteristics. In this case, we can consider it to be an embedding of individuals in a latent space, rather than their physical locations.

We first derive the rates of convergence of our estimators under the following set of assumptions.

Assumptions 3.1(i) and (ii) together imply that the number of interacting partners for each unit is bounded (i.e., the network must be sparse). These assumptions play a crucial role in characterizing the stochastic process of the outcome functions. In Assumption 3.2, part (i) assumes that the covariates are non-stochastic and bounded. This type of assumption is frequently utilized in the spatial and network literature and can be interpreted as viewing the analysis conditional on the realized values of the covariates. Meanwhile, part (ii) is introduced to ensure some convergence results for the quadratic moments.

Assumption 3.3(i) allows the error terms to be fully heteroskedastic. Part (ii) should be standard. Part (iii) is a high-level condition, which plays an important role to obtain the parametric convergence rate for the GMM estimator. In general, if $\Gamma_{it}$ belongs to $L^{2}([0,1]^{2})$, it admits the following series expansion:

$\displaystyle\Gamma_{it}(s_{l},s_{l^{\prime}})=\sum_{k_{1},k_{2}=1}^{\infty}\kappa_{it,k_{1},k_{2}}\phi_{k_{1}}(s_{l})\phi_{k_{2}}(s_{l^{\prime}})$

(3.42)

for some sequence of constants $\{\kappa_{it,k_{1},k_{2}}\}$. By the orthonormality of $\phi_{k}$,

	$\displaystyle\int_{0}^{1}\int_{0}^{1}\Gamma_{it}(s_{l},s_{l^{\prime}})\phi_{k}(s_{l})\phi_{k}(s_{l^{\prime}})\,\text{d}s_{l}\,\text{d}s_{l^{\prime}}$	$\displaystyle=\sum_{k_{1},k_{2}=1}^{\infty}\kappa_{it,k_{1},k_{2}}\left(\int_{0}^{1}\phi_{k}(s_{l})\phi_{k_{1}}(s_{l})\,\text{d}s_{l}\right)\left(\int_{0}^{1}\phi_{k}(s_{l^{\prime}})\phi_{k_{2}}(s_{l^{\prime}})\,\text{d}s_{l^{\prime}}\right)$		(3.43)
		$\displaystyle=\kappa_{it,k,k}.$		(3.44)

Since $L^{-2}\sum_{l=1}^{L}\sum_{l^{\prime}=1}^{L}\Gamma_{it}(s_{l},s_{l^{\prime}})\phi_{k}(s_{l})\phi_{k}(s_{l^{\prime}})$ can be seen as a numerical approximation of the left-hand side of the above expression, Assumption 3.3 part (iii) essentially requires that $\sum_{k=1}^{K}\kappa_{it,k,k}\lesssim 1$ uniformly in $K$. In particular, if the $\kappa_{it,k,k}$’s are ordered in decreasing manner such that $\kappa_{it,1,1}\geq\kappa_{it,2,2}\geq\cdots$, this assumption can be interpreted in two ways: there exists a constant $a>1$ such that $\kappa_{it,k,k}\lesssim k^{-a}$, or there exists a fixed $b$ such that $\kappa_{it,k,k}=0$ for all $k>b$.