Kernel Density Estimation for Undirected Dyadic Data

Kernel Density Estimation for Undirected Dyadic Data

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Introduction

Many important social and economic variables are naturally defined for pairs of agents (or dyads). Examples include trade between pairs of countries Tinbergen_SWE62, input purchases and sales between pairs of firms Atalay_et_al_PNAS11, research and development (R&D) partnerships across firms Konig_et_al_RESTAT18 and friendships between individuals Christakis_et_al_NBER10. Dyadic data arises frequently in the analysis of social and economic networks. In economics such analyses are predominant in, for example, the analysis of international trade flows. See Graham_HBE18 for many other examples and references.

While the statistical analysis of network data began almost a century ago, rigorously justified methods of inference for network statistics are only now emerging Goldenberg_etal_FTML09. In this paper we study nonparametric estimation of the density function of a (continuously-valued) dyadic random variable. Examples included the density of migration across states, trade across nations, liabilities across banks, or minutes of telephone conversation among individuals. While nonparametric density estimation using independent and identically distributed random samples, henceforth “monadic” data, is well-understood, its dyadic counterpart has, to our knowledge, not yet been studied.

Holland_Leinhardt_SM76 derived the sampling variance of the link frequency in a simple network (and of other low order subgraph counts). A general asymptotic distribution theory for subgraph counts, exploiting recent ideas from the probability literature on dense graph limits Diaconis_Janson_RM08,Lovasz_AMS12, was presented in Bickel_et_al_AS11.\footnote{See Nowicki_SN91 for a summary of earlier research in this area.} Menzel_arXiv17 presents bootstrap procedures for inference on the mean of a dyadic random variable. Our focus on nonparametric density estimation appears to be novel. Density estimation is, of course, a topic of intrinsic interest to econometricians and statisticians, but it also provides a relatively simple and canonical starting point for understanding nonparametric estimation more generally. In the conclusion of this paper we discuss ongoing work on other non- and semi-parametric estimation problems using dyadic data.

We show that an (obvious) adaptation of the Rosenblatt_AMS56 and Parzen_AMS62 kernel density estimator is applicable to dyadic data. While our dyadic density estimator is straightforward to define, its rate-of-convergence and asymptotic sampling properties, depart significantly from its monadic counterpart. Let $N$ be the number of sampled agents and $n=\tbinom{N}{2}$ the corresponding number of dyads. Estimation is based upon the $n$ dyadic outcomes. Due to dependence across dyads sharing an agent in common, the rate of convergence of our density estimate is (generally) much slower than it would be with $n$ i.i.d. outcomes. This rate-of-convergence is also invariant across a wide range of bandwidth sequences. This property is familiar from the econometric literature on semiparametric estimation Powell_HBE94. Indeed, from a certain perspective, our nonparametric dyadic density estimate can be viewed as a semiparametric estimator (in the sense that it can be thought of as an average of nonparametrically estimated densities). We also explore the impact of “degeneracy” -- which arises when dependence across dyads vanishes -- on our sampling theory; such degeneracy features prominently in Menzel's Menzel_arXiv17 innovative analysis of inference on dyadic means. We expect that many of our findings generalize to other non- and semi-parametric network estimation problems.

In the next section we present our maintained data/network generating process and proposed kernel density estimator. Section (ref) explores the mean square error properties of this estimator, while Section (ref) outlines asymptotic distribution theory. Section (ref) presents a consistent variance estimator, which can be used to construct Wald statistics and Wald-based confidence intervals. We summarize the results of a small simulation study in Section (ref). In Section (ref) we discuss various extensions and ongoing work. Calculations not presented in the main text are collected in Appendix (ref).

It what follows we interchangeably use unit, node, vertex, agent and individual all to refer to the $i=1,\ldots,N$ vertices of the sampled network or graph. We denote random variables by capital Roman letters, specific realizations by lower case Roman letters and their support by blackboard bold Roman letters. That is $Y$, $y$ and $\mathbb{Y}$ respectively denote a generic random draw of, a specific value of, and the support of, $Y$. For $W_{ij}$ a dyadic outcome, or weighted edge, associated with agents $i$ and $j$, we use the notation $\mathbf{W}=\left[W_{ij}\right]$ to denote the $N\times N$ adjacency matrix of all such outcomes/edges. Additional notation is defined in the sections which follow.

Model and estimator

Model

Let $i=1,\ldots,N$ index a simple random sample of $N$ agents from some large (infinite) network of interest. A pair of agents constitutes a dyad. For each of the $n=\tbinom{N}{2}$ sampled dyads, that is for $i=1,...,N-1$ and $j=i+1,\ldots,N$, we observe the (scalar) random variable $W_{ij}$, generated according to

where $A_{i}$ is a node-specific random vector of attributes (of arbitrary dimension, not necessarily observable), and $V_{ij}=V_{ji}$ is an unobservable scalar random variable which is continuously distributed on $\mathbb{R}$ with density function $f_{V}(v)$.\footnote{In words we observe the weighted subgraph induced by the randomly sampled agents.} Observe that the function $W(a_{1},a_{2},v_{12})$ is symmetric in its first two arguments, ensuring that $W_{ij}=W_{ji}$ is undirected.

In what follows we directly maintain (ref), however, it also a consequence of assuming that the infinite graph sampled from is jointly exchangeable Aldous_JMA81,Hoover_WP79. Joint exchangeability of the sampled graph $\mathbf{W}=\left[W_{ij}\right]$ implies that

for every $\pi\in\Pi$ where $\pi:\left\{ 1,\ldots,N\right\} \rightarrow\left\{ 1,\ldots,N\right\} $ is a permutation of the node indices. Put differently, when node labels have no meaning we have that the “likelihood” of any simultaneous row and column permutation of $\mathbf{W}$ is the same as that of $\mathbf{W}$ itself.\footnote{For $\mathbf{W}=\left[W_{ij}\right]$ the $N\times N$ weighted adjacency matrix and $\mathbf{P}$ any conformable permutation matrix \[ \Pr\left(\mathbf{W}\leq\mathbf{w}\right)=\Pr\left(\mathbf{P}\mathbf{W}\mathbf{P}\leq\mathbf{w}\right) \] for all $\mathbf{w}\in\mathbb{W=R}^{\tbinom{N}{2}}.$} See Menzel_arXiv17 for a related discussion.

Our target object of estimation is the marginal density function $f_{W}(w)$ of $W_{ij}$, defined as the derivative of the cumulative distribution function (c.d.f.) of $W_{ij},$ \[ \Pr\{W_{ij}\leq w\}\overset{def}{\equiv}F_{W}(w)=\int_{-\infty}^{w}f_{W}(u)\mathrm{d}u. \] To ensure this density function is well-defined on the support of $W_{ij},$ we assume that the unknown function $W(a_{1},a_{2},v)$ is strictly increasing and continuously differentiable in its third argument $v$, and we also assume that $A_{i}$ and $A_{j}$ are statistically independent of the “error term” $V_{ij}$ for all $i$ and $j.$ Under these assumptions, by the usual change-of-variables formula, the conditional density of $W_{ij}$ given $A_{i}=a_{1}$ and $A_{j}=a_{2}$ takes the form \[ f_{Y|AA}(w|a_{1},a_{2})=f_{V}(W^{-1}(a_{1},a_{2},w))\cdot\left\vert \frac{\partial W(a_{1},a_{2},W^{-1}(a_{1},a_{2},w))}{\partial v}\right\vert ^{-1}. \] In the derivations below we will assume this density function is bounded and twice continuously differentiable at $w$ with bounded second derivative for all $a_{1}$ and $a_{2}$; this will follow from the similar smoothness conditions imposed on the primitives $W^{-1}(\cdot,\cdot,w)$ and $f_{V}(v).$

To derive the marginal density of $W_{ij}$ note that, by random sampling, the $\{A_{i}\}$ sequence is independently and identically distributed (i.i.d.), as is the $\{V_{ij}\}$ sequence. Under these conditions, we can define the conditional densities of $W_{ij}$ given $A_{i}=a$ or $A_{j}=a$ alone as \[ f_{W|A}(w|a)\equiv\mathbb{E}[f_{W|AA}(w|a,A_{j})]=\mathbb{E}[f_{W|AA}(w|A_{i},a)], \] and, averaging, the marginal density of interest as \[ f_{W}(w)\overset{def}{\equiv}\mathbb{E}[f_{W|AA}(w|A_{i},A_{j})]=\mathbb{E}[f_{W|A}(w|A_{i})]. \]

Let $i,j,k$ and $l$ index distinct agents. The assumption that $\{A_{i}\}$ and $\{V_{ij}\}$ are i.i.d. implies that while $W_{ij}$ varies independently of $W_{kl}$ (since the $\left\{ i,j\right\} $ and $\left\{ k,l\right\} $ dyads share no agents in common), $W_{ij}$ will not vary independently of $W_{ik}$ as both vary with $A_{i}$ (since the $\left\{ i,j\right\} $ and $\left\{ i,k\right\} $ dyads both include agent $i$). This type of dependence structure is sometimes referred to as “dyadic clustering” in empirical social science research Fafchamp_Gubert_JDE07,Cameron_Miller_WP14,Aronow_et_al_PA17. The implications of this dependence structure for density estimation and -- especially -- inference is a key area of focus in what follows.

Estimator

Given this construction of the marginal density $f_{W}(w)$ of $W_{ij},$ it can be estimated using an immediate extension of the kernel density estimator for monadic data first proposed by Rosenblatt_AMS56 and Parzen_AMS62:

where \[ K_{ij}\overset{def}{\equiv}\frac{1}{h}K\left(\frac{w-W_{ij}}{h}\right). \] Here $K(\cdot)$ is a kernel function assumed to be (i) bounded ($K(u)\leq\bar{K}$ for all $u$), (ii) symmetric ($K(u)=K(-u)$), (ii) , and zero outside a bounded interval ($K(u)=0$ if $\left\vert u\right\vert >\bar{u}$); we also require that it (iv) integrates to one ($\int K(u)du=1$). The bandwidth $h=h_{N}$ is assumed to be a positive, deterministic sequence (indexed by the number of nodes $N$) that tends to zero as $N\rightarrow\infty,$ and will satisfy other conditions imposed below. A discussion of the motivation for the kernel estimator $\hat{f}_{W}(w)$ and its statistical properties under random sampling (of monadic variables) can be found in Silverman_DESDA86.

Rate of convergence analysis

To formulate conditions for consistency of $\hat{f}_{W}(w),$ we will evaluate its expectation and variance, which will yield conditions on the bandwidth sequence $h_{N}$ for its mean squared error to converge to zero.

A standard calculation yields a bias of $\hat{f}_{W}(w)$ equal to (see Appendix (ref))

with \[ B\left(w\right)\overset{def}{\equiv}\frac{1}{2}\frac{\partial^{2}f_{W}(w)}{\partial w{}^{2}}\int u^{2}K\left(u\right)\mathrm{d}u. \] Equation (ref) coincides with the bias of the kernel density estimate based upon a random (“monadic”) sample.

The expression for the variance of $\hat{f}_{W}(w)$, in contrast to that for bias, does differ from the monadic (i.i.d.) case due to the (possibly) nonzero covariance between $K_{ij}$ and $K_{ik}$ for $j\neq k$:

The third line of this expression uses the fact that, in the summation in the second line, there are $n=\frac{1}{2}N\left(N-1\right)$ terms with $(i,j)=(k,l)$ and $N(N-1)(N-2)=2n(N-2)$ terms with one subscript in common; as noted earlier, when $W_{ij}$ and $W_{kl}$ have no subscripts in common they are independent (and thus uncorrelated).

To calculate the dependence of this variance on the number of nodes $N,$ we analyze $\mathbb{V}(K_{12})$ and $\mathbb{C}(K_{12},K_{13}).$ Beginning with the former,

where \[ \Omega_{2}(w)\overset{def}{\equiv}f_{W}(w)\cdot\int[K\left(u\right)]^{2}\mathrm{d}u. \] Like the expected value, this own variance term is of the same order of magnitude as in the monadic case, \[ \mathbb{V}(K_{12})=O\left(\frac{1}{h}\right). \] However, the covariance term $\mathbb{C}(K_{ij},K_{il}),$ which would be absent for i.i.d. monadic data, is generally nonzero. Since

(where the second line uses the change of variables $s_{1}=w-hu_{1}$ and $s_{2}=w-hu_{2}$ and mutual independence of $A_{1},A_{2},$ and $A_{3}$). It follows that

with \[ \Omega_{1}(w)\overset{def}{\equiv}\mathbb{V}(f_{W|A}(w|A_{1})). \] Therefore,

and the mean-squared error of $\hat{f}_{W}(w)$ is, using (ref) and (ref),

Provided that $\Omega_{1}(w)\neq0$ and the bandwidth sequence $h_{N}$ is chosen such that

as $N\rightarrow\infty,$ we get that

and hence that \[ \sqrt{N}(\hat{f}_{W}(w)-f_{W}(w))=O_{p}(1). \] In fact, the rate of convergence of $\hat{f}_{W}(w)$ to $f_{W}(w)$ will be $\sqrt{N}$ as long as $Nh^{4}\leq C\leq Nh$ for some $C>0$ as $N\rightarrow\infty,$ although the mean-squared error will include an additional bias or variance term of $O(N^{-1})$ if either $Nh$ or $(Nh^{4})^{-1}$ does not diverge to infinity.

To derive the MSE-optimal bandwidth sequence we minimize (ref) with respect to its first and third terms, this yields an optimal bandwidth sequence of

This sequence satisfies condition (ref) above.

Interestingly, the rate of convergence of $\hat{f}_{W}(w)$ to $f_{W}(w)$ under condition (ref) is the same as the rate of convergence of the sample mean

to its expectation $\mu_{W}\overset{def}{\equiv}\mathbb{E}[W_{ij}]$ when $\mathbb{E}[W_{ij}^{2}]<\infty.$ Similar variance calculations to those for $\hat{f}_{w}(w)$ yield (see also Holland_Leinhardt_SM76 and Menzel_arXiv17)

provided $\mathbb{E}[W_{ij}|A_{i}]$ is non-degenerate, yielding \[ \sqrt{N}(\bar{W}-\mu)=O_{p}(1). \] Thus, in contrast to the case of i.i.d monadic data, there is no convergence-rate “cost” associated with nonparametric estimation of $f_{W}(w).$ The presence of dyadic dependence, due to its impact on estimation variance, does slow down the feasible rate of convergence substantially. With iid data the relevant rate for density estimation would be $n^{2/5}$ when the MSE-optimal bandwidth sequence is used. Recalling that $n=O\left(N^{2}\right)$, the $\sqrt{N}$ rate we find here corresponds to an $n^{1/4}$ rate. The slowdown from $n^{2/5}$ to $n^{1/4}$ captures the rate of convergence costs of dyadic dependence on the variance of our density estimate.

The lack of dependence of the convergence rate of $\hat{f}_{W}(w)$ to $f_{W}(w)$ on the precise bandwidth sequence chosen is analogous to that for semiparametric estimators defined as averages over nonparametrically-estimated components Newey_ET94b,Powell_HBE94. Defining $K_{ji}\overset{def}{\equiv}K_{ij},$ the estimator $\hat{f}_{W}(w)$ can be expressed as \[ \hat{f}_{W}(w)=\frac{1}{N}\sum_{i=1}^{N}\hat{f}_{W|A}(w|A_{i}), \] where \[ \hat{f}_{W|A}(w|A_{i})\overset{def}{\equiv}\frac{1}{N-1}\sum_{j\neq i,j=1}^{N}K_{ij}. \] Holding $i$ fixed, the estimator $\hat{f}_{W|A}(W|A_{i})$ can be shown to converge to $f_{W|A}(w|A_{i})$ at the nonparametric rate $\sqrt{Nh},$ but the average of this nonparametric estimator over $A_{i}$ converges at the faster (“parametric”) rate $\sqrt{N}.$ In comparison, while \[ \bar{W}=\frac{1}{N}\sum_{i=1}^{N}\hat{\mathbb{E}}\left[\left.W_{ij}\right|A_{i}\right], \] for \[ \hat{\mathbb{E}}\left[\left.W_{ij}\right|A_{i}\right]\overset{def}{\equiv}\frac{1}{N-1}\sum_{j\neq i,j=1}^{N}W_{ij}, \] the latter converges at the parametric rate $\sqrt{N},$ and the additional averaging to obtain $\bar{W}$ does not improve upon that rate.

Asymptotic distribution theory

To derive conditions under which $\hat{f}_{W}(w)$ is approximately normally distributed it is helpful to decompose the difference between $\hat{f}_{W}(w)$ and $f_{W}(w)$ into four terms:

To understand this decomposition observe that the projection of $\hat{f}_{W}(w)=\frac{1}{n}\sum_{i<j}K_{ij}$ onto $\{A_{i}\}_{i=1}^{N}$ equals, by the independence assumptions imposed on $\{A_{i}\}$ and $\{V_{ij}\},$ the U-statistic $\tbinom{N}{2}^{-1}\sum_{i<j}\mathbb{E}[K_{ij}|A_{i},A_{j}]$. This U-Statistic is defined in terms of the latent i.i.d. random variables $\{A_{i}\}_{i=1}^{N}$.

The first term in this expression, line (ref), is $\hat{f}_{W}(w)$ minus the projection/U-Statistic described above. Each term in this summation has conditional expectation zero given the remaining terms (i.e., the terms form a martingale difference sequence).

The second term in the decomposition, line (ref), is the difference between the second-order U-statistic $\frac{1}{n}\sum_{i<j}\mathbb{E}[K_{ij}|A_{i},A_{j}]$ and its H�jek projection vanderVaart_ASBook00\footnote{That is the projection of $\frac{1}{n}\sum_{i<j}\mathbb{E}[K_{ij}|A_{i},A_{j}]$ onto the linear subspace consisting of all functions of the form $\sum_{i=1}^{N}g_{i}\left(A_{i}\right)$.}, the third term, line (ref), is a centered version of that H�jek projection, and the final term, line (ref), is the bias of $\hat{f}_{W}(w).$ A similar “double projection” argument was used by Graham_EM17 to analyze the large sample properties of the Tetrad Logit estimator.

If the bandwidth sequence $h=h_{N}$ satisfies the conditions $Nh\rightarrow\infty$ and $Nh^{4}\rightarrow0,$ the calculations in the previous section can be used to show that the first, second, and fourth terms of this decomposition (i.e., $T_{1},$ $T_{2,}$ and $T_{4}$) will all converge to zero when normalized by $\sqrt{N}$. In this case, $T_{3}$, which is an average of i.i.d. random variables, will be the leading term asymptotically such that \[ \sqrt{N}(\hat{f}_{W}(w)-f_{W}(w))\overset{D}{\rightarrow}\mathcal{N}(0,4\Omega_{1}(w)), \] assuming $\Omega_{1}(w)=\mathbb{V}(f_{W|A}(w|A_{i}))>0$.

If, however, the bandwidth sequence $h$ has $Nh\rightarrow C<\infty$ (a “knife-edge” undersmoothing condition similar to one considered by Cattaneo_et_al_ET14 in a different context), then both $T_{1}$ and $T_{3}$ will be asymptotically normal when normalized by $\sqrt{N}.$ To accommodate both of these cases in a single result, we will show that a standardized version of the sum $T_{1}+T_{3}$ will have a standard normal limit distribution, although the first, $T_{1}$, term may be degenerate in the limit.

In Appendix (ref) we show that both $T_{2}$ and $T_{4}$ will be asymptotically negligible when normalized by the convergence rate of $T_{1}+T_{3},$ such that the asymptotic distribution of $\hat{f}_{W}(w)$ will only depend on the $T_{1}$ and $T_{3}$ terms.

We start by rewriting the sum of terms $T_{1}$ and $T_{3}$ as

where \[ T(N)\equiv N+n \] and the triangular array $X_{Nt}$ is defined as

That is, $\{X_{Nt}\}$ is the collection of terms of the form \[ \frac{2}{N}(\mathbb{E}[K_{ij}|A_{i}]-\mathbb{E}[K_{ij}]) \] for $i=1,...,N$ (with $j\neq i$) and \[ \frac{1}{n}(K_{ij}-\mathbb{E}[K_{ij}|A_{i},A_{j}]) \] for $i=1,...,N-1$ and $j=i+1,...,N.$ Using the independence assumptions on $\{A_{i}\}_{i=1}^{N}$ and $\{V_{ij}\}_{i<j}$, as well as iterated expectations, it is tedious but straightforward to verify that \[ \mathbb{E}[X_{Nt}|\{X_{Ns},s\neq t\}]=0, \] that is, $X_{NT}$ is a martingale difference sequence (MDS).

Defining the variance of this MDS as

we can demonstrate asymptotic normality of its standardized sum -- $\frac{1}{\sigma_{N}}\sum_{t=1}^{T(N)}X_{Nt}$ -- by a central limit theorem for martingale difference triangular arrays (see, for example, Hall_Heyde_Bk1980, Theorem 3.2 and Corollary 3.1 and White_Bk01, Theorem 5.24 and Corollary 5.26). Specifically, if the Lyapunov condition

holds for some $r>2,$ and also the stability condition

holds then

From the calculations used in the MSE analysis of Section (ref) we have that

so, taking $r=3$, \[ \frac{1}{\sigma_{N}^{2}}=O(N) \] assuming $\Omega_{1}(w)>0$ and $Nh\geq C>0.$ In the degenerate case, where $\mathbb{V}(\mathbb{E}[K_{ij}|A_{i}])=\Omega_{1}(w)=0$, we will still have $(\sigma_{N})^{-2}=O(nh)=O(N)$ as long as the “knife-edge” $h\propto N^{-1}$ undersmoothing bandwidth sequence is chosen.

To verify the Lyapunov condition ((ref)), note that

and

Putting things together we get that

when $Nh\geq C>0$ for all $N.$ Therefore the Lyapunov condition ((ref)) is satisfied for $r=3,$ since

To verify the stability condition ((ref)), we first rewrite that condition as

where

and \[ R_{2}\equiv N\sum_{i<j}\left[\mathbb{E}\left[\left.\left(\frac{1}{n}(K_{ij}-\mathbb{E}[K_{ij}|A_{i},A_{j}])\right)^{2}\right\vert A_{i},A_{j}\right]-\mathbb{E}\left[\left(\frac{1}{n}(K_{ij}-\mathbb{E}[K_{ij}|A_{i},A_{j}])\right)^{2}\right]\right]. \] Since $1/N\sigma_{N}^{2}=O(1),$ the stability condition ((ref)) will hold if $R_{1}$ and $R_{2}$ both converge to zero in probability.

By the independence restrictions on $\{U_{ij}\}$ and $\{A_{i}\},$ the (mean zero) summands in $R_{1}$ are mutually uncorrelated, so

But, using analogous arguments to ((ref)) and (((ref)), \[ \mathbb{E}\left[\mathbb{E}[K_{ij}|A_{i}]^{4}\right]=O\left(1\right) \] and \[ \mathbb{E}\left[K_{ij}^{4}\right]=O\left(\frac{1}{h^{3}}\right), \] so

under the bandwidth condition that $1/nh=O(1/N).$ So $R_{1}$ converges in probability to zero. Moreover, $R_{2}$ is proportional to a (mean zero) second-order U-statistic,

with kernel having second moment

again imposing the bandwidth restriction $1/nh=O(1/N)$. Thus by Lemma 3.1 of Powell_Stock_Stoker_EM89, $R_{2}$ converges in probability to its (zero) expected value.

Since conditions ((ref)) and ((ref)) both hold, a central limit theorem for martingale difference triangular arrays implies \[ \frac{1}{\sigma_{N}}(T_{1}+T_{3})\overset{D}{\rightarrow}\mathcal{N}(0,1). \] A final step is to used this result to obtain the asymptotic distribution of $\hat{f}_{W}(w).$ Because \[ \frac{1}{\sigma_{N}}=O\left(\sqrt{N}\right), \] we have that $T_{2}$ and $T_{4}$ are asymptotically negligible after standardization with $\sigma_{N}^{-1}$ (see Appendix (ref)), \[ \frac{T_{2}}{\sigma_{N}}=O_{p}\left(\sqrt{\frac{N}{n}}\right)=o_{p}(1) \] and \[ \frac{T_{4}}{\sigma_{N}}=O\left(\sqrt{N}h^{2}\right)=o(1), \] so that

When $Nh^{4}\rightarrow0$ and $Nh\rightarrow\infty,$ \[ N\sigma_{N}^{2}\rightarrow4\Omega_{1}(w) \] and \[ \sqrt{N}\left(\hat{f}_{W}(w)-f_{W}(w)\right)\overset{D}{\rightarrow}\mathcal{N}(0,4\Omega_{1}(w)) \] as long as $\mathbb{V}(\mathbb{E}[K_{ij}|A_{i}])>0.$

Under “knife-edge” bandwidth sequences, such that $Nh\rightarrow C>0,$ we have instead that \[ N\sigma_{N}^{2}\rightarrow4\Omega_{1}(w)+C^{-1}\Omega_{2}(w) \] and \[ \sqrt{N}(\hat{f}_{W}(w)-f_{W}(w))\overset{D}{\rightarrow}\mathcal{N}(0,4\Omega_{1}(w)+C^{-1}\Omega_{2}(w)). \]

Degeneracy

Degeneracy arises when $\mathbb{V}(\mathbb{E}[K_{ij}|A_{i}])=\Omega_{1}\left(w\right)=0.$ In terms of the underlying network generating process (NGP), degeneracy arises when the conditional density of $W_{ij}$ at $w$ given $A_{i}=a$ is constant in $a$ (i.e., when $\mathbb{V}\left(f_{W|A}\left(\left.w\right|A_{i}\right)\right)=0$).

As a simple example of such an NGP, let $A_{i}$ equal $-1$ with probability $\pi$ and $1$ otherwise; next set \[ W_{ij}=A_{i}A_{j}+V_{ij} \] with $V_{ij}$ standard normal. In this case the conditional density $f_{W|A}\left(\left.w\right|A_{i}\right)$ is the mixture \[ f_{W|A}\left(\left.w\right|A_{i}\right)=\pi\phi\left(w+A_{i}\right)+\left(1-\pi\right)\phi\left(w-A_{i}\right) \] with $\phi\left(\cdot\right)$ the standard normal density function. Unconditionally the density is \[ f_{W}\left(w\right)=\left[\pi^{2}+\left(1-\pi\right)^{2}\right]\phi\left(w-1\right)+2\pi\left(1-\pi\right)\phi\left(w+1\right). \] Observe that, if $\pi=1/2$, then $f_{W|A}\left(\left.w\right|A_{i}=1\right)=f_{W|A}\left(\left.w\right|A_{i}=-1\right)=f_{W}\left(w\right)$ and hence that $\mathbb{V}\left(f_{W|A}\left(\left.w\right|A_{i}\right)\right)=0$.\footnote{Degeneracy also arises when $w=1$.} Degeneracy arises in this case, even though there is non-trivial dependence across dyads sharing an agent in common. If $\pi\neq1/2$, then $\mathbb{V}\left(f_{W|A}\left(\left.w\right|A_{i}\right)\right)>0$, but one still might worry about “near degeneracy” when $\pi$ is close to $1/2$.

Menzel_arXiv17 shows that under degeneracy, the limit distribution of the sample mean, $\bar{W}$, equation (ref) on \vpageref{eq: sample_mean} above, may be non-Gaussian. This occurs because (i) the $T_{1}$ and $T_{2}$ terms in a double projection decomposition of $\bar{W}$ analogous to the one used here for $\hat{f}_{W}\left(w\right)$ will be of equal order and $T_{2}$, the H�jek Projection error, may be non-Gaussian (as is familiar from the theory of U-Statistics, e.g., Chapter 12 of vanderVaart_ASBook00).

The situation is both more complicated and simpler here. In the case of the estimated density $\hat{f}_{W}\left(w\right)$, if the bandwidth sequence $h=h_{N}$ satisfies the conditions $Nh\rightarrow\infty$ and $Nh^{4}\rightarrow0,$ then $T_{2}$ will be of smaller order than $T_{1}$ and hence not contribute to the limit distribution irrespective of whether the NGP is degenerate or not. In particular, under degeneracy the Liaponuv condition (ref) continues to hold for $r=3$ since \[ \sum_{t=1}^{T(N)}E\left(\frac{X_{Nt}}{\sigma_{N}}\right)^{3}=O\left(\frac{1}{\sqrt{nh}}\right) \] and it follows straightforwardly that $\frac{1}{\sigma_{N}}\left(\hat{f}_{W}\left(w\right)-f_{W}\left(w\right)\right)$ continues to be normal in the limit.

The “knife-edge” undersmoothing bandwidth sequence is primarily of interest because it results in a sequence where both $T_{1}$ and $T_{3}$ contribute to the limit distribution. In practice this does not mean that the researcher should set $h=h_{N}\propto N^{-1}$. Based on the theoretical analysis sketched above, we recommend choosing a sequence that tends to zero slightly faster than mean squared error optimal sequence where $h=h_{N}\propto n^{-1/5}$.\footnote{In practice “plug-in” bandwidths that would be appropriate in the absence of any dyadic dependence across the $\left\{ W_{ij}\right\} _{i<j}$ might work well; although this remains an unexplored conjecture.}

Under such a sequence we will have \[ \sqrt{N}(\hat{f}_{W}(w)-f_{W}(w))\overset{D}{\rightarrow}\mathcal{N}(0,4\Omega_{1}(w)) \] under non-degeneracy and \[ \sqrt{nh}(\hat{f}_{W}(w)-f_{W}(w))\overset{D}{\rightarrow}\mathcal{N}(0,\Omega_{2}(w)) \] under degeneracy. Although the rate of convergence of $\hat{f}_{W}(w)$ to $f_{W}(w)$ is faster in the case of degeneracy this will not affect inference in practice as long as an appropriate estimate of $\sigma_{N}$ is used; that is working directly with $(\hat{f}_{W}(w)-f_{W}(w))/\sigma_{N}$ ensures rate-adaptivity. Note also that, in the absence of degeneracy, the MSE optimal bandwidth sequence could be used. By slightly undersmoothing relative to this sequence, we ensure that the limit distribution remains unbiased in case of degeneracy.

Asymptotic variance estimation

To construct Wald-based confidence intervals for $\hat{f}_{W}(w),$ a consistent estimator of its asymptotic variance is needed. When $Nh\rightarrow C<\infty,$ the asymptotic variance depends on both \[ \Omega_{2}(w)\overset{def}{\equiv}f_{W}(w)\cdot\int[K\left(u\right)]^{2}\mathrm{d}u \] and \[ \Omega_{1}(w)\overset{def}{\equiv}\mathbb{V}\left(f_{W|A}(w|A_{i})\right). \] In this section we present consistent estimators for both of these terms.

A simple estimator of $\Omega_{2}(w)$ is

the consistency of which we demonstrate in Appendix (ref):

The estimator $\tilde{\Omega}_{2}(w)$ uses the second moment of $K_{ij}$ instead of its sample variance to estimate $\Omega_{2}(w);$ in practice we recommend, similar to Newey_ET94b in the context of monadic kernel-based estimation, the less conservative alternative:

We next turn to estimation of \[ \Omega_{1}(w)=\mathbb{V}\left(f_{W|A}(w|A_{1})\right)=\lim_{N\rightarrow\infty}\mathbb{C}(K_{ij},K_{ij}) \] where $i\neq k.$ A natural sample analog estimator, following a suggestion by Graham_HBE18 in the context of parametric dyadic regression, involves an average over the three indices $i,$ $j,$ and $k$:

for $S_{ijk}=\frac{1}{3}\left(K_{ij}K_{ik}+K_{ij}K_{jk}+K_{ik}K_{jk}\right).$\footnote{See also the variance estimator for density presented in Holland_Leinhardt_SM76.} In Appendix (ref) we show that

Inserting these estimators, $\hat{\Omega}_{1}(w)$ and $\hat{\Omega}_{2}(w)$, into the formula for the variance of $\hat{f}_{W}(w)$ yields a variance estimate of

We end this section by observing that the following equality holds

where \[ d_{ijkl}=1\{i=j,k=l,i=l,\text{or }j=k\}. \] As Graham_HBE18 notes, this coincides with the estimator for \[ \mathbb{V}(\bar{W})=\mathbb{V}\left(\frac{1}{n}\sum_{i<j}W_{ij}\right) \] proposed by Fafchamp_Gubert_JDE07, replacing “$W_{ij}-\bar{W}$” with “$K_{ij}-\bar{K}$”, with $\bar{K}\overset{def}{\equiv}\hat{f}_{W}(w)$ (see also Holland_Leinhardt_SM76, Cameron_Miller_WP14 and Aronow_et_al_PA17). Our variance estimator can also be viewed as a dyadic generalization of the variance estimate proposed by Newey_ET94b for “monadic” kernel estimates.

Simulation study

Our simulations design is based upon the example used to discuss degeneracy in Section (ref). As there we let $A_{i}$ equal $-1$ with probability $\pi$ and $1$ otherwise. We generate $W_{ij}$ \[ W_{ij}=A_{i}A_{j}+V_{ij} \] with $V_{ij}\sim\mathcal{N}(0,1)$. We set $\pi=1/3$ and estimate the density $f_{W}\left(w\right)$ at $w=1.645$.

We present results for three sample sizes: $N=100,400$ and $1,600$. These sample sizes are such that, for a “sufficiently non-degenerate” NGP, the standard error of $\hat{f}_{W}\left(w\right)$ would be expected to decline by a factor of $1/2$ for each increase in sample size (if the bandwidth is large enough to ensure that the $\frac{\Omega_{2}\left(w\right)}{nh}$ variance term is negligible relative to the $\frac{2\Omega_{1}\left(w\right)\left(N-2\right)}{n}\approx\frac{4\Omega_{1}\left(w\right)}{N}$ one). We set the bandwidth equal to the MSE-optimal one presented in equation (ref) above. This is an `oracle' bandwidth choice. Developing feasible data-based methods of bandwidth selection would be an interesting topic for future research.

Table (ref) presents the main elements of each simulation design. Panel B of the table lists “pencil and paper” bias and asymptotic standard error calculations based upon the expressions presented in Section (ref) above. Panel B also presents analytic estimates of the standard deviations of the $T_{1}$ and $T_{3}$ terms in the decomposition of $\hat{f}_{W}\left(w\right)$ used to derive its limit distribution. In the given designs both terms of are similar magnitude despite the fact that the contribution of the $T_{1}$ term is asymptotically negligible in theory.

Table (ref) summarizes the results of 1,000 Monte Carlo simulations. The median bias and standard deviation of our density estimates across the Monte Carlo replications closely track our theoretical predictions (compare rows 1 and 2 of Table (ref) with Rows 1 and 2 of Panel B of Table (ref). Row 3 of the table reports the median “Fafchamps and Gubert” asymptotic standard error estimate. This standard error estimate is generally larger than its asymptotic counterpart. Consequently the coverage of confidence intervals based upon it is conservative (Row 5). The degree of conservatism is declining in sample size, suggesting that -- as expected -- the “Fafchamps and Gubert” asymptotic standard error estimate is closer to its theoretical counterpart as $N$ grows. Row 4 of the table reports the coverage of confidence intervals based upon standard errors which ignore the presence of dyadic dependence; these intervals -- as expected -- fail to cover the true density frequently enough.

The simulations suggest, for the designs considered, that the asymptotic theory presented in Sections (ref) and (ref) provides an accurate approximation of finite sample behavior. Our variance estimate is a bit conservative for the designs considered; whether this is peculiar to the specific design considered or a generic feature of the estimate is unknown.\footnote{We observe that our variance estimate implicitly includes an estimate of the variance of $T_{2}$, which is negligible in the limit.} As with bandwidth selection, further exploration of methods of variance estimation in the presence of dyadic dependence is warranted.

Extensions

There are a number of avenues for extension or modification of the simple results for scalar density estimation presented above. One variant of these results would apply when the dyadic variable $W_{ij}$ lacks the idiosyncratic component $V_{ij},$ i.e., when \[ W_{ij}=W(A_{i},A_{j}), \] for $\{A_{i}\}$ an i.i.d. sequence. This case arises when $W_{ij}$ is a measure of “distance” between the attributes of nodes $i$ and $j,$ for example, \[ W_{ij}=\sqrt{\left(A_{i}-A_{j}\right)^{2}}, \] for $A_{i}$ a scalar measure of “location” for agent $i.$ The asymptotic distribution of $\hat{f}_{W}(w)$ derived above should be applicable to this case as long as the conditional density function $f_{W|A}(w|a)$ of $W_{ij}$ given $A_{i}$ is well-defined, which would be implied if $A_{i}$ has a continuously-distributed component given its remaining component (if any) and the function $W(\cdot)$ is continuously differentiable in that component. In the decomposition of $\hat{f}_{W}(w)-f_{W}(w)$ for this case, the term corresponding to $T_{1}$ would be identically zero (as would $\Omega_{2}(w)$), but the $T_{2}$ term could still be shown to be asymptotically negligible using Lemma 3.1 of Powell_Stock_Stoker_EM89 as long as $Nh\rightarrow\infty.$

Another straightforward extension of this analysis would be to directed dyadic data, where $W_{ij}$ is observed for all pairs of indices with $i\neq j$ and $W_{ij}\neq W_{ji}$ with positive probability. The natural generalization of the data generation process would be \[ W_{ij}=W(A_{i},B_{j},V_{ij}), \] with $\{A_{i}\},$ $\{B_{j}\},$ and $\{V_{ij}\}$ mutually independent and i.i.d. with $V_{ij}\neq V_{ji}$ in general. Here the conditional densities \[ f_{W|A}(w|a)=\mathbb{E}[f_{W|AB}(w|A_{i}=a,B_{j})] \] and \[ f_{W|B}(w|b)=\mathbb{E}[f_{W|AB}(w|A_{i},B_{j}=b)] \] will differ, and the asymptotic variance of $\hat{f}_{W}(w)$ will depend upon \[ \Omega_{1}(w)=\mathbb{V}\left(\frac{1}{2}\left(f_{W|A}(w|A_{i})+f_{W|B}(w|B_{i})\right)\right) \] in a way analogous to how $\Omega_{1}(w)$, defined earlier, does in the undirected case analyzed in this paper.

Yet another generalization of the results would allow $W_{ij}$ to be a $p$-dimensional jointly-continuous $W_{ij}$ random vector. The estimator \[ \hat{f}_{W}(w)=\frac{1}{n}\sum_{i=1}^{N-1}\sum_{j=1+1}^{N}\frac{1}{h^{p}}K\left(\frac{w-W_{ij}}{h}\right) \] of the $p$-dimensional density function $f_{W}(w)$ will continue to have the same form as derived in the scalar case, provided $Nh^{p}\rightarrow\infty$ (or $Nh^{p}\rightarrow C>0$) as long as the relevant bias term $T_{4}$ is negligible. If the density is sufficiently smooth and $K(\cdot)$ is a "higher-order kernel" with

then the bias term $T_{4}$ will satisfy

As long as $q$ can be chosen large enough so that $Nh^{2q}\rightarrow0$ while $Nh^{p}\geq C>0,$ the bias term $T_{4}$ will be asymptotically negligible and the density estimator $\hat{f}_{W}(w)$ should still be asymptotically normal with asymptotic distribution of the same form derived above.

Finally, a particularly useful extension of the kernel estimation approach for dyadic data would be to estimation of the conditional expectation of one dyadic variable $Y_{ij}$ conditional on the value $w$ of another dyadic variable $W_{ij},$ i.e., estimation of \[ g(w)\equiv\mathbb{E}[Y_{ij}|W_{ij}=w] \] when the vector $W_{ij}$ has $p$ jointly-continuously distributed components conditional upon any remaining components. Here the Nadaraya-Watson kernel regression estimator Nadaraya_TPA64,Watson_Sankhya64 would be defined as \[ \hat{g}(w)\equiv\frac{\sum_{i\neq j}K\left(\frac{w-W_{ij}}{h}\right)Y_{ij}}{\sum_{i\neq j}K\left(\frac{w-W_{ij}}{h}\right)}, \] and the model for the dependent variable $Y_{ij}$ would be analogous to that for $W_{ij},$ with

in the directed case (and $B_{j}\equiv A_{j}$ for undirected data), with $\{A_{i}\},$ $\{B_{j}\},$ and $\{(U_{i},V_{ij})\}$ assumed mutually independent and i.i.d. The large-sample theory would treat the numerator of $\hat{g}(w)$ similarly to that for the denominator (which is proportional to the kernel density estimator $\hat{f}_{W}(w)$); our initial calculations for undirected data with a scalar, continuously-distributed regressor $W_{ij}$ yield \[ \sqrt{N}\left(\hat{g}(w)-g(w)\right)\overset{D}{\rightarrow}\mathcal{N}(0,4\Gamma_{1}(w)), \] when $Nh^{p}\rightarrow\infty$ and $Nh^{4}\rightarrow0$, where \[ \Gamma_{1}(w)\equiv\mathbb{V}\left(\frac{\mathbb{E}[Y_{ij}|A_{i},W_{ij}=w]\cdot f_{W|A}(w|A_{i})}{f_{W}(w)}\right). \] If this calculation is correct, then, like the density estimator $\hat{f}_{W}(w)$ the rate of convergence for the estimator $\hat{g}(w)$ of the conditional mean $g(w)$ would be the same as the rate for the estimator $\hat{\mu}_{Y}=\bar{Y}$ of the unconditional expectation $\mu_{y}=\mathbb{E}[Y_{ij}]=\mathbb{E}[g(W_{ij})],$ in contrast to the estimation using i.i.d. (monadic) data. We intend to verify these calculations and derive the other extensions in future work.