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The Econometrics of Utility Transferability in Dyadic Network Formation Models
Given a group of individuals, can we determine how much benefit one individual derives from connecting with another? And, by extension, can we predict who will connect with whom? These questions are central to the study of social and economic networks. For example, suppose that a new technology is invented and trade economists wish to predict which countries will become exporters of the technology and which will become importers. Or imagine a NGO tasked with running group cognitive behavioral therapy sessions to improve the mental health of struggling adolescents. If the NGO wishes for the adolescents to form friendships with people in their groups so that they can discuss their shared experiences, then the NGO needs to know which group designs are most conducive to friendship formation. Would it be better to have separate groups of boys and girls? Or would it be better to construct groups of adolescents with different characteristics? This paper contributes to the econometric models that seek to answer questions like these.
To understand the contribution of this paper, it is necessary to introduce two econometric models that are used to estimate individuals' taste for linking with one another. The first model assumes that utility is transferable between individuals; the second model does not. The primary goal of this paper is to compare the assumptions required for estimation of the non-transferable utility model relative to the transferable utility model. This is achieved by comparing the sufficient conditions for consistency and asymptotic normality of the maximum likelihood estimators corresponding to each model.
The simpler econometric model used to determine how much benefit two individuals derive from forming a connection is known as the transferable utility (TU) model. This is a binary choice model in which the outcome is an indicator for if the two individuals decide to form a link or not. The name of the model derives from the fact that utility is assumed to be transferable between individuals. If individual $i$ wishes to form a link with individual $j$, but $j$ does not wish to reciprocate, then this assumption means that $i$ can provide $j$ with transfers of some kind (e.g. money) such that forming the link becomes profitable for $j$. The practical implication of this assumption is that we only require a single indicator to summarize the joint linking decision of the two individuals.
Formally, suppose that individuals $i$ and $j$ form a link, denoted $Y_{ij}^{\text{TU}} = 1$, if the joint utility that they derive from the link is non-negative. The joint utility is comprised of three parts: 1) some continuous function $W_{ij} \coloneq w(X_i, X_j)$ from $\mathbb{R}^{k \times 2}$ to $\mathbb{R}^{k}$ that combines $i$'s random vector of characteristics, $X_i \in \mathbb{R}^k$, with $j$'s random vector of characteristics, $X_j \in \mathbb{R}^k$, component-wise; 2) a vector of “taste for linking parameters”, $\beta \in \Theta \subseteq \mathbb{R}^k$; and 3) a link-specific error term, $\varepsilon_{ij} \in \mathbb{R}$, that is assumed to follow a standard normal distribution. In sum, the TU model is given by:
$$ Y_{ij}^{\text{TU}} = \mathds{1} [W_{ij}^{'}\beta \geq \varepsilon_{ij}], $$ where $\varepsilon_{ij} \sim \mathcal{N}(0, 1)$.
For example, suppose that $\beta \in \mathbb{R}$ and $W_{ij} = |X_i - X_j|$ where $X_i$ is $i$'s income and $X_j$ is $j$'s income. In this case, $\beta$ is the effect of the distance between $i$ and $j$'s income on the likelihood that they become friends. We typically find that individuals who share similar characteristics are more likely to be friends, a phenomenon known as “homophily” in the networks literature jackson_social_2008. Therefore, we might expect $\beta$ to be negative here to capture the fact that people with far apart incomes are less likely to be friends. In this sense, (the negative of) $\beta$ can be thought of as a homophily parameter.
Not allowing for self-links and summing across unique links, we have the log-likelihood function for the TU model:\footnote{We have duplicate links because $Y_{ij}^{\text{TU}} = Y_{ji}^{\text{TU}}$, i.e. the TU model generates undirected networks. To see how the double summation ignores self and duplicate links, set $n = 3$. Then the links that are considered are $(1, 2)$, $(1, 3)$, and $(2, 3)$.}
$$ \frac{1}{N}\sum^{n-1}_{i=1} \sum^{n}_{j=i+1} \Big[Y_{ij}^{\text{TU}} \log \big(\Phi(W_{ij}^{'}\beta) \big) + (1 - Y_{ij}^{\text{TU}})\log \big(1 - \Phi(W_{ij}^{'}\beta) \big) \Big], $$ where $\Phi$ is the univariate standard normal CDF and $N = \binom{n}{2} = \frac{n(n-1)}{2}$ is the number of unique dyads excluding self-links. Compare this with the well-known probit log-likelihood function with non-dyadic data:
$$ \frac{1}{n}\sum^{n}_{i=1} \Big[Y_{i} \log \big(\Phi(X_{i}^{'}\beta) \big) + (1 - Y_{i})\log \big(1 - \Phi(X_{i}^{'}\beta) \big) \Big], $$ where $Y_{i} = \mathds{1} [X_{i}^{'}\beta \geq \varepsilon_{i}]$ and $\varepsilon_{i} \sim \mathcal{N}(0, 1)$. We know from Example 1.2 in newey_chapter_1994 that the maximum likelihood estimator for the probit model is consistent and asymptotically normal provided that $\mathbb{E} [X_iX_i']$ exists and is nonsingular and the data $(Y_i, X_i)$ is i.i.d. The log-likelihood function for the TU model is essentially the probit log-likelihood function with a double summation and distinct $i$ and $j$ indices to account for the fact that observations are at the dyad level. With dyadic models, it is not possible to assume that the data $(Y_{ij}, X_i, X_j)$ is i.i.d.; $Y_{12}$ and $Y_{13}$ both depend on $X_1$, so are clearly correlated. Instead, we can only assume that $X_i$ is i.i.d., which then requires us to make a stronger finite moment assumption than in newey_chapter_1994 to establish asymptotic normality. In particular, it can be shown that the maximum likelihood estimator for the TU model (TU-MLE) is consistent provided that $\mathbb{E} [W_{ij} W_{ij}']$ exists and is nonsingular and $X_i$ is i.i.d. (plus some standard regularity conditions on the parameter space $\Theta$), and is asymptotically normal provided additionally that $\mathbb{E} [\|W_{ij}\|^8]$ exists.\footnote{Simply replicate the proofs for probit in newey_chapter_1994, but account for the additional complications that arise with dyadic data exactly as is done in Section 2 below. See Section 2 for a discussion of these assumptions.}
If one is not willing to assume that utility is transferable between individuals, then one cannot express the link decision of the two individuals as a single indicator containing a joint utility term. Instead, for a link to form, denoted $Y_{ij}^{\text{NTU}} = 1$, we require that $i$ and $j$'s separate utilities from forming the link are simultaneously non-negative. Formally, we have the product of two indicators: an indicator containing $i$'s utility from forming the link and an indicator containing $j$'s utility from forming the link. $i$'s utility is comprised of $W_{ij} \coloneq w(X_i, X_j) \in \mathbb{R}^k$, $\beta \in \Theta \subseteq \mathbb{R}^k$, and $\varepsilon_{ij} \in \mathbb{R}$; $j$'s utility is comprised of $W_{ji} \coloneq w(X_j, X_i) \in \mathbb{R}^k$, $\beta \in \Theta \subseteq \mathbb{R}^k$, and $\varepsilon_{ji} \in \mathbb{R}$. We assume that $(\varepsilon_{ij}, \varepsilon_{ji})$ follows a bivariate normal distribution with $\mathrm{Var}(\varepsilon_{ij})$ and $\mathrm{Var}(\varepsilon_{ji})$ both normalised to 1 and $\mathrm{Cov}(\varepsilon_{ij}, \varepsilon_{ji}) = \rho \in (-1, 1)$. While the errors $(\varepsilon_{ij}, \varepsilon_{ji})$ can be correlated within dyads (governed by $\rho$), they are assumed to be independent across dyads. In sum, the NTU model is given by:
$$ Y_{ij}^{\text{NTU}} = \mathds{1} [W_{ij}^{'}\beta \geq \varepsilon_{ij}] \cdot \mathds{1} [W_{ji}^{'}\beta \geq \varepsilon_{ji}], $$ where
In the most general NTU model considered in this paper, we allow for pairs of individuals $ij$ such that $W_{ij} \neq W_{ji}$. Such regressors are called “asymmetric regressors”. Although in many applications regressors are symmetric, e.g. if $W_{ij} = |X_i - X_j| = |X_j - X_i| = W_{ji}$, there are some situations in which one may wish to allow for asymmetric regressors. One such example is if an individual only cares about the income of the other person, i.e. $W_{ij} = X_j \neq X_i = W_{ji}$ where $X_i$ is $i$'s income. We do however assume throughout that both individuals share a common $\beta$. An alternative formulation of the model would be to impose regressor symmetry while allowing for individuals $i$ and $j$ to have different tastes for linking, $\beta_1$ and $\beta_2$; this is the formulation that poirier_partial_1980 opts for.\footnote{Note that poirier_partial_1980 is concerned solely with the identification of their model and does not consider consistency nor asymptotic normality. We will see in Section 2.2 that their identification issues resurface in our model, but to a lesser extent.} Since the order of $i$ and $j$ is arbitrary, I consider it less natural to impose differences in $i$ and $j$'s tastes, relative to differences in their characteristics.
Irrespective of regressor (a)symmetry, the NTU model generates symmetric outcomes, $Y_{ij}^{\text{NTU}} = Y_{ji}^{\text{NTU}}$, and therefore undirected networks. This is immediate from the definition of the model: $Y_{ij}^{\text{NTU}} = \mathds{1} [W_{ij}^{'}\beta \geq \varepsilon_{ij}] \cdot \mathds{1} [W_{ji}^{'}\beta \geq \varepsilon_{ji}] = \mathds{1} [W_{ji}^{'}\beta \geq \varepsilon_{ji}] \cdot \mathds{1} [W_{ij}^{'}\beta \geq \varepsilon_{ij}] = Y_{ji}^{\text{NTU}}$.
Analogous to the log-likelihood function for the TU model, we sum across unique links (excluding self-links) to obtain the log-likelihood function for the NTU model:
\[
\] where $\Phi_{2}^{\rho}$ is the CDF of the bivariate normal distribution with covariance $\rho$. The theoretical results presented in this paper are all under the special case of $\rho = 0$, i.e. independence between $\varepsilon_{ij}$ and $\varepsilon_{ji}$.\footnote{Simulation results when $\rho \neq 0$ are provided in the Appendix.} When $\rho = 0$, we write $\hat{Q}_n(\beta)$ instead of $\hat{Q}_n(\beta, 0)$. It is simple to see that:
\[
\] This log-likelihood is nicer to work with because dealing with the univariate standard normal CDF, $\Phi$, is much easier than dealing with bivariate normal CDF, $\Phi_{2}^{\rho}$. As well as simplifying the analysis, $\rho = 0$ is a natural special case to consider. Indeed, in the first application in Section 4.1, I estimate $\rho$ to be very close to 0 using real-world data.
The central object of interest in this paper is the maximum likelihood estimator for the NTU model (NTU-MLE) with $\rho = 0$:
$$ \hat{\beta} \coloneq \argmax_{\beta \in \Theta} \hat{Q}_n(\beta). $$ Proving that the NTU-MLE, $\hat{\beta}$, is consistent and asymptotically normal under standard assumptions is harder than with the TU-MLE; this is the focus of Section 2 and the primary contribution of this paper. The cases of symmetric and asymmetric regressors are considered separately. This is because, somewhat surprisingly, when regressors are symmetric the sufficient conditions for consistency and asymptotic normality of the TU-MLE turn out to be sufficient for consistency and asymptotic normality of the NTU-MLE. However, when regressors are asymmetric, slightly stronger sufficient conditions are imposed.
I now outline how this paper is structured and the contributions it makes.
Within the econometrics literature on network formation, there is a dichotomy between “strategic” and “dyadic” models. Strategic models allow the decision of two agents to form a link to depend on the existence of other links in the network; dyadic models ignore the wider network and focus on the link decision of the two agents independent of the link decisions of others. While arguably more realistic in some settings, strategic models are plagued with issues of multiple equilibria, forcing researchers to either take a stance on the equilibrium selection mechanism as in christakis_empirical_2020 or resort to set identification as in sheng_structural_2020.\footnote{For excellent reviews of strategic models of network formation, see Section 4 of chandrasekhar_econometrics_2016 and Section 4 of de_paula_econometric_2020.}
The TU and NTU models introduced above are dyadic models. As well as circumventing issues of model incompleteness, dyadic models are more realistic in settings where agents do not consider the wider network when making their link decisions. For example, as an application of the tools developed in this paper, I study risk-sharing decisions among households in a small community in rural Tanzania. It seems rather implausible that households would know or be able to predict the entire lending network in the community and, consequently, they would be unable to account for the network structure when making their link decisions.
Most papers in the dyadic literature assume that utility is transferable between agents. The landmark paper on the TU dyadic network formation model is graham_econometric_2017, which adds unobserved individual heterogeneity terms for individual $i$, $A_i$, and individual $j$, $A_j$, to the TU link decision indicator:
$$ Y_{ij}^{\text{TU}} = \mathds{1} [W_{ij}^{'}\beta + A_i + A_j \geq \varepsilon_{ij}]. $$ The introduction of fixed effects leads to the incidental parameter problem neyman_consistent_1948. To eliminate the fixed effects, graham_econometric_2017 assumes that $\varepsilon_{ij}$ is a standard logistic random variable that is i.i.d. across dyads and then uses sufficient statistics that depend on $\varepsilon_{ij}$'s parametric distribution. candelaria_semiparametric_2020 and toth_semiparametric_2017 extend graham_econometric_2017 to a semi-parametric setting in which the distribution of $\varepsilon_{ij}$ is left unspecified; jochmans_semiparametric_2018 and dzemski_empirical_2019 consider related models that generate directed networks.
At the time of writing, two papers have been written on dyadic network formation models with non-transferable utility: gao_logical_2023 and li_ming_estimation_2024.\footnote{There are strategic network formation models with NTU, such as in goldsmith-pinkham_social_2013, but again these are not the focus of this paper.} Both papers include unobserved individual heterogeneity terms for $i$ and $j$ in their respective indicators, and initially assume that regressors are symmetric,\footnote{gao_logical_2023 extend their results to asymmetric regressors in their Appendix C, but li_ming_estimation_2024 do not.} such that their main model is given by:
$$ Y_{ij}^{\text{NTU}} = \mathds{1} [W_{ij}^{'}\beta + A_i \geq \varepsilon_{ij}] \cdot \mathds{1} [W_{ij}^{'}\beta + A_j \geq \varepsilon_{ji}]. $$
Since gao_logical_2023 and li_ming_estimation_2024 are the works that are most similar to this paper, it is beneficial to describe what exactly they do and how their contributions are distinct from this paper's contributions. li_ming_estimation_2024 is parametric, although the joint distribution of $(\varepsilon_{ij}, \varepsilon_{ji})$ is kept general enough to incorporate both the logistic and normal distributions as special cases (with $\varepsilon_{ij}$ and $\varepsilon_{ji}$ assumed to be independent). Their estimation procedure is fairly complex: first solve a high-dimensional system of moment equations, then plug this solution into a one-step estimator of le_cam_l_m_theorie_1969, then refine the estimator to achieve $\sqrt{N}$-consistency via split-network jackknife, and finally perform bootstrap aggregating to achieve Cramer-Rao efficiency. This procedure allows for estimation of the fixed effects as well as $\beta$.
On the other hand, gao_logical_2023 is semi-parametric in that they leave the joint distribution of $(\varepsilon_{ij}, \varepsilon_{ji})$ unspecified. They eliminate the fixed effects, which are not additively separable in the NTU model unlike in the TU model, via a procedure called “logical differencing”. Namely, consider an individual $i$ who is a) more popular than $j$ in some group with characteristics $X_k = \bar{x}$, but b) less popular than $j$ in some group with characteristics $X_k = \underaccent{\bar}{x}$. a) implies that either $w(X_i, \bar{x})'\beta > w(X_j, \bar{x})'\beta$ or $A_i > A_j$; b) implies that either $w(X_i, \underaccent{\bar}{x})'\beta < w(X_j, \underaccent{\bar}{x})'\beta$ or $A_i < A_j$. Since $A_i > A_j$ and $A_i < A_j$ cannot both be true, gao_logical_2023 conclude that either $w(X_i, \bar{x})'\beta > w(X_j, \bar{x})'\beta$ or $w(X_i, \underaccent{\bar}{x})'\beta < w(X_j, \underaccent{\bar}{x})'\beta$. These restrictions on $\beta$ are used to identify and then estimate it, similar to maximum score estimation. While the semi-parametric set-up is more general than the set-up in li_ming_estimation_2024, gao_logical_2023 lack inference results and cannot estimate the fixed effects because they eliminate them.
The literature developed from the TU model, to the TU model with fixed effects, to the NTU model with fixed effects (see Figure 1). To the best of my knowledge, no one has studied the NTU model without fixed effects given in Section 1.2 (the focus of this paper). Although unobserved individual heterogeneity is important in many applications, the obvious advantage of ignoring it is that we can avoid the incidental parameter problem and use maximum likelihood estimation, which is considerably easier to implement than the estimation procedures in gao_logical_2023 and li_ming_estimation_2024. One might worry that the results presented in this paper are special cases of the results in gao_logical_2023 and li_ming_estimation_2024 with the fixed effects set equal to 0. This is not the case because the logical differencing estimator in gao_logical_2023 and the moment condition-based estimator in li_ming_estimation_2024 are very different to the standard MLE.
Going directly from the TU model (without fixed effects) to the NTU model (without fixed effects) also reveals some novel theoretical insights. In particular, Theorems 1 and 2 in Section 2.1 establish that the standard assumptions for consistency and asymptotic normality of the TU-MLE are sufficient for consistency and asymptotic normality of the NTU-MLE with independent errors and symmetric regressors. That is, existence and nonsingularity of $\mathbb{E} [W_{ij} W_{ij}']$, i.i.d-ness of $X_i$, and some standard regularity conditions on the parameter space $\Theta$ are sufficient for consistency of the NTU-MLE (Theorem 1). And if one further assumes that $\mathbb{E} [\|W_{ij}\|^8]$ exists, then asymptotic normality of the NTU-MLE is guaranteed (Theorem 2). The proofs follow the probit example in newey_chapter_1994 closely, but have to deal with the additional complications that arise due to the dyadic structure of the data and the fact that the CDF of the standard normal distribution in the log-likelihood function is replaced with the CDF squared. Relative to gao_logical_2023 and li_ming_estimation_2024, these sufficient conditions are easier to interpret and make clear how much more needs to be assumed relative to the TU model -- which turns out to be nothing in the symmetric regressor case. This result stems primarily from the surprising fact that the log-likelihood function for the NTU model is concave in this case.
Unlike li_ming_estimation_2024, this paper also considers what happens with asymmetric regressors (theoretical results in Section 2.2) and dependent errors (simulation results in the Appendix). Although gao_logical_2023 also consider the case of asymmetric regressors, they do not have any results on the asymptotic distribution of their estimator, while this paper does (Theorem 4). Related to the work of poirier_partial_1980, identification is tricky to establish with asymmetric regressors, so I provide a proposition with an easy-to-verify condition that ensures identification (Proposition 1). Namely, as well as imposing standard nonsingularity conditions, one simply needs to check that either all asymmetric regressors take on at least three distinct values, or that there is a pair of individuals for which the regressors are symmetric. At least one of these conditions is almost always true in practice. Consistency is also trickier to establish with asymmetric regressors because the log-likelihood function is no longer globally concave. Without concavity, I establish consistency with asymmetric regressors by invoking a uniform law of large numbers for U-statistics (Theorem 3). Finally, for asymptotic normality, I impose a further nonsingularity condition that essentially rules out independence of $W_{ij}$ and $W_{ji}$ (Theorem 4). All conditions are discussed in due course. Even with asymmetric regressors, remarkably little extra is assumed relative to the sufficient conditions for consistency and asymptotic normality of the TU-MLE.
An additional contribution of this paper is to provide a specification test for whether a given dataset is better explained by the TU or NTU model. li_ming_estimation_2024 discuss what happens if, say, one believes that the errors follow a logistic distribution when in fact they follow a normal distribution. This form of misspecification is interesting, but arguably not as interesting as if one believes that the TU model best explains link formation when in fact the NTU model does; this is the focus of Section 3. I provide a likelihood ratio specification test that accounts for the fact that the parameter of interest is at its boundary in the spirit of self_asymptotic_1987 and present simulation-based evidence for the test's correct size and strong power.
In Section 4, I apply the estimation and testing procedures developed in this paper to two different network datasets. The first application focuses on the determinants of adolescent friendship at an all-girls school in California, as explored in goeree_1d_2010. I find that goeree_1d_2010 estimate the TU model when the TU assumption is rejected by my specification test, and that the NTU-MLEs suggest different determinants of friendship than their TU-MLEs. The second application focuses on the determinants of risk-sharing in rural Tanzania. This is the application also considered in gao_logical_2023 and li_ming_estimation_2024. I find that this dataset is best explained by the TU model, suggesting that the NTU-model-based estimators used in gao_logical_2023 and li_ming_estimation_2024 may be unnecessary, if not inappropriate, for this dataset. Without a testing procedure for TU vs NTU, other authors are at risk of applying NTU-model-based estimators to TU-model-based data incorrectly, and vice versa.
Finally, I conclude in Section 5. In the Appendix, I provide the proofs of all results and simulation-based evidence on the consistency of the NTU-MLE with $\rho \neq 0$ and the exceptional small-sample behavior of the NTU-MLE.
Consider again the NTU model without fixed effects and with independent errors that was introduced in detail in Section 1.2:\footnote{Now writing $Y_{ij} = Y_{ij}^{\text{NTU}}$ for notational convenience.}
$$ Y_{ij} = \mathds{1} [W_{ij}^{'}\beta \geq \varepsilon_{ij}] \cdot \mathds{1} [W_{ji}^{'}\beta \geq \varepsilon_{ji}], $$ where
$W_{ij} \coloneq w(X_i, X_j) \in \mathbb{R}^{k}$, $W_{ji} \coloneq w(X_j, X_i) \in \mathbb{R}^{k}$, and $\beta \in \Theta \subseteq \mathbb{R}^k$.
Also consider again the corresponding log-likelihood for the NTU model:
\[
\] where $\Phi$ is the univariate standard normal CDF, $N = \binom{n}{2} = \frac{n(n-1)}{2}$, and $n$ is the number of individuals in the network.
The NTU-MLE is given by:
$$ \hat{\beta} \coloneq \argmax_{\beta \in \Theta} \hat{Q}_n(\beta). $$ I now present the main theoretical contribution of this paper: conditions for identification of $\beta$ and for consistency and asymptotic normality of the NTU-MLE.
In this section, I simplify the analysis even further by assuming that the regressors are symmetric, i.e. $W_{ij} = W_{ji}$ for all pairs of individuals $ij$. For the asymmetric regressor case, see Section 2.2 below.
With symmetric regressors, the likelihood of observing a particular $Y_{ij}$ conditional on $X_i$, $X_j$, and $\beta$ is given by:
$$ f(Y_{ij} \mid X_i, X_j, \beta) = \left[\Phi (W_{ij}' \beta)^2\right]^{Y_{ij}} \left[1 - \Phi (W_{ij}' \beta)^2\right]^{1 - Y_{ij}}. $$
The first theoretical result to establish is identification of $\beta$, i.e. that there exists a pair of individuals $ij$ such that, for $\beta, \tilde{\beta} \in \Theta$:
Or equivalently, via the contrapositive, that there exists a pair of individuals $ij$ such that, for $\beta, \tilde{\beta} \in \Theta$, if $\beta \neq \tilde{\beta}$, then $f(Y_{ij} \mid X_i, X_j, \beta) \neq f(Y_{ij} \mid X_i, X_j, \tilde{\beta})$.
As discussed in Section 1.1, the key condition for identification of the probit model in newey_chapter_1994 (hereafter NM), where the (random) regressors are simply $X_i$ instead of $W_{ij}$, is that $\mathbb{E} [X_iX_i']$ exists and is nonsingular, and hence positive definite. This is a standard assumption that rules out perfect multicollinearity of the regressors. In particular, NM show that for all individuals $i$ and for all $\beta, \tilde{\beta} \in \Theta$, if $\beta \neq \tilde{\beta}$ and $\mathbb{E} [X_i X_i']$ exists and is nonsingular, then $\Phi(X_i' \beta) \neq \Phi(X_i' \tilde{\beta})$. The analogous result in our setting, where the regressors are $W_{ij}$ instead of simply $X_i$, is that: for all pairs of individuals $ij$ and for $\beta, \tilde{\beta} \in \Theta$, if $\beta \neq \tilde{\beta}$ and $\mathbb{E} [W_{ij} W_{ij}']$ exists and is nonsingular, then $\Phi(W_{ij}' \beta) \neq \Phi(W_{ij}' \tilde{\beta})$. Moreover, clearly $\Phi(W_{ij}' \beta) \neq - \Phi(W_{ij}' \tilde{\beta})$ since $\Phi(W_{ij}' \beta) > 0$ and $-\Phi(W_{ij}' \tilde{\beta}) < 0$. So, $\Phi(W_{ij}' \beta)^2 \neq \Phi(W_{ij}' \tilde{\beta})^2$, and therefore:
$$
$$ as is required for identification.
With identification guaranteed under $\rho = 0$, $W_{ij} = W_{ji}$, and existence and nonsingularity of $\mathbb{E} [W_{ij} W_{ij}']$, we now move towards consistency. We will continue to rely on results from NM; namely, we will rely on their consistency theorem with concave log-likelihood functions.\footnote{See Theorem 2.7 in newey_chapter_1994.} Consistency requires that (2.7.i) $Q_0 (\beta) \coloneq \mathbb{E} [ \log f(Y_{ij} \mid X_i, X_j, \beta) ]$ has a unique maximum at the true value of $\beta$, $\beta_0$, (2.7.ii) $\beta_0$ is an element of the interior of a convex set, (2.7.iii) $\hat{Q}_n(\beta) \coloneq \frac{1}{N} \sum^{n-1}_{i=1} \sum^{n}_{j=i+1} \big[Y_{ij} \log \big(\Phi(W_{ij}^{'}\beta)^2 \big) + (1 - Y_{ij})\log \big(1 - \Phi(W_{ij}^{'}\beta)^2 \big) \big]$ is concave, and (2.7.iv) $\hat{Q}_n(\beta) \overset{p}{\to} Q_0 (\beta)$ for all $\beta \in \Theta$.
Lemma 1, which can be found in the Appendix along with its proof (and all other proofs in this paper unless stated otherwise), guarantees that $Q_0 (\beta)$ is uniquely maximised at $\beta_0$ under existence and nonsingularity of $\mathbb{E} [W_{ij} W_{ij}']$ and boundedness of $\Theta$.\footnote{Technically, there is no formal requirement in NM for the parameter space to be bounded in this case. But boundedness aids the proof and will be assumed when the regressors are asymmetric in Section 2.2 anyway.}
The difficulty lies in establishing concavity of the log-likelihood function, $\hat{Q}_n(\beta)$; this is achieved through Lemma 2 in the Appendix. While the log-likelihood function for the TU model is known to be concave, it is somewhat surprising that the log-likelihood function for the NTU model (with $\rho = 0$ and $W_{ij} = W_{ji}$), where $\Phi$ is replaced with $\Phi^2$, is still concave. Concavity helps significantly with the theory; we will see in Section 2.2 that when we relax $W_{ij} = W_{ji}$ for all pairs $ij$ the log-likelihood function loses its concavity and the theory becomes more complicated.
Finally, to establish the high-level convergence condition, $\hat{Q}_n(\beta) \overset{p}{\to} Q_0 (\beta)$, we will assume that the characteristics, $X_i$, are drawn i.i.d. across dyads.\footnote{The same assumption is made in graham_econometric_2017, gao_logical_2023, and li_ming_estimation_2024.} However, this does not imply that $Y_{ij}$ is independent across individuals: $Y_{12}$ and $Y_{13}$ both depend on $X_1$, so are clearly correlated. Since $\hat{Q}_n(\beta)$ contains $Y_{ij}$, and $Y_{ij}$ is not i.i.d., we cannot simply invoke the usual weak law of large numbers to show that $\hat{Q}_n(\beta) \overset{p}{\to} Q_0 (\beta)$. Instead, we will need to use a LLN that accounts for the dependence between $Y_{12}$ and $Y_{13}$, for example.\footnote{Note that this is a fairly weak form of dependence (but not to be confused with the formal definition of weak dependence) since most pairs of dyads do not contain a common individual. Namely, there are $\frac{N(N-1)}{2}$ unordered pairs of distinct dyads and $\frac{n(n-1)(n-2)}{2}$ such pairs have an individual in common. So, given that $N = \frac{n(n-1)}{2}$, the proportion of pairs of dyads with an individual in common is equal to $\frac{4(n-2)}{n(n-1) - 2}$, which goes to zero as $n \to \infty$.} This can be achieved by recognizing that $\hat{Q}_n(\beta)$ is a U-statistic of degree 2 and then invoking the strong LLN for U-statistics in serfling_approximation_1980, courtesy of hoeffding_strong_1961. Specifically, Theorem 5.4.A in serfling_approximation_1980 applied to this setting states that if (5.4.A.i) $X_i$ is i.i.d., (5.4.A.ii) $h_1(X_i, X_j) \coloneq Y_{ij} \log \big(\Phi(W_{ij}^{'}\beta)^2 \big) + (1 - Y_{ij}) \log \big(1 - \Phi(W_{ij}^{'}\beta)^2 \big)$ is such that $h_1(X_i, X_j) = h_1(X_j, X_i)$, and (5.4.A.iii) $\mathbb{E} [ \|h_1(X_i, X_j)\|] < \infty $, then $\hat{Q}_n(\beta) \overset{a.s.}{\to} Q_0 (\beta)$. Condition (5.4.A.i) is assumed; Condition (5.4.A.ii) follows from the fact that $W_{ij} = W_{ji}$ in this case; and Condition (5.4.A.iii) was shown in the proof of Lemma 1 given existence of $\mathbb{E} [W_{ij} W_{ij}']$ and boundedness of $\Theta$. Consequently, under the same conditions as Lemma 1 plus i.i.d-ness of $X_i$, $\hat{Q}_n(\beta) \overset{a.s.}{\to} Q_0 (\beta)$, so, in particular, $\hat{Q}_n(\beta) \overset{p}{\to} Q_0 (\beta)$.
Combining Lemma 1 and Lemma 2 with convergence in probability of the sample log-likelihood to the population log-likelihood, we therefore arrive at the following theorem:
It is interesting that the conditions for consistency of the NTU-MLE (with symmetric regressors and independent errors) are the same as the conditions for consistency of the TU-MLE that were stated in Section 1.1.
With identification and consistency established, we now show asymptotic normality of the NTU-MLE.\footnote{Using Theorem 3.3 in NM.} Recall that the effective sample size is the number of unique dyads excluding self-links, $N = \binom{n}{2} = \frac{n(n-1)}{2}$. We will have $\sqrt{N}$-asymptotic-normality in the sense of $\sqrt{N}(\hat{\beta} - \beta_0) \overset{d}{\to} \mathcal{N}(0, J^{-1})$ as $n \to \infty$ under the following conditions: i.i.d-ness of $z_{ij} = (Y_{ij}, X_i, X_j)$, consistency of the MLE $\hat{\beta}$, (3.3.i) $\beta_0 \in \text{interior}(\Theta)$, (3.3.ii) $f(Y_{ij} \mid X_i, X_j, \beta)$ is twice continuously differentiable and $f(Y_{ij} \mid X_i, X_j, \beta) > 0$ in a neighborhood $\mathcal{N}_0$ around $\beta_0$, (3.3.iii) $\int \sup_{\beta \in \mathcal{N}_0} \|\nabla_{\beta} f(Y_{ij} \mid X_i, X_j, \beta)\| dz < \infty$ and $\int \sup_{\beta \in \mathcal{N}_0} \|\nabla_{\beta \beta} f(Y_{ij} \mid X_i, X_j, \beta)\| dz < \infty$, (3.3.iv) $J \coloneq \mathbb{E}[\nabla_{\beta} \log f( Y_{ij} \mid X_i, X_j, \beta_0) \nabla_{\beta} \log f( Y_{ij} \mid X_i, X_j, \beta_0)^\prime]$ exists and is nonsingular, and (3.3.v) $\mathbb{E}[\sup_{\beta \in \mathcal{N}_0} \|\nabla_{\beta\beta} \log f(Y_{ij} \mid X_i, X_j, \beta)\|] < \infty$.
As discussed in the previous section on consistency, we cannot assume that the data $(Y_{ij}, X_i, X_j)$ are i.i.d., only that the individual characteristics $X_i$ are i.i.d. The i.i.d-ness assumption is used “under the hood” in NM to apply the Lindeberg-Levy CLT to establish that $\sqrt{N} \nabla_{\beta} \hat{Q}_n(\beta_0)$ is asymptotically normal. We can still establish asymptotic normality of $\sqrt{N} \nabla_{\beta} \hat{Q}_n(\beta_0)$ despite dependence of $Y_{ij}$ by recognizing that $Y_{ij}$ are independent conditional on the individual characteristics (since the errors are independent across dyads), and then using a conditional version of the Lindeberg-Feller CLT.\footnote{Unlike for consistency, we cannot rely on limit theory for U-statistics because the U-statistic $\nabla_{\beta} \hat{Q}_n(\beta_0)$ is degenerate.} This is achieved via Lemma 3 in the Appendix, provided now $\mathbb{E} [\|W_{ij}\|^8] < \infty$ instead of just $\mathbb{E} [W_{ij} W_{ij}'] < \infty$ as was assumed for consistency. Finite eighth moments may seem like a strong assumption, but we are attempting to control a complicated form of dependence that arises in this network setting. Indeed, li_ming_estimation_2024 assume a bounded support for $X_i$ and therefore that all moments are finite, not just eighth moments. Furthermore, the same dependence structure is present in the TU model, and therefore finite eighth moments are also assumed for asymptotic normality of the TU-MLE.
Given Lemma 3 and consistency of the NTU-MLE, we verify conditions (3.3.i)-(3.3.v) to arrive at the following asymptotic normality theorem:\footnote{Where $\Phi$ is the CDF and $\phi$ is the PDF of the standard normal distribution.}
Again, the sufficient conditions for asymptotic normality of the NTU-MLE (with symmetric regressors and independent errors) are the same as the sufficient conditions for asymptotic normality of the TU-MLE. This means that if the TU-MLE is theoretically justified, then the NTU-MLE is too (with symmetric regressors and independent errors).
Now allow for the possibility that $W_{ij} \neq W_{ji}$ for some pairs $ij$ while maintaining that $\rho = 0$. We can try to follow the steps above to establish identification, consistency, and asymptotic normality in this case, but we run into difficulties very quickly.
The first issue is with identification. Identification here means that there exists a pair of individuals $ij$ such that, for $\beta, \tilde{\beta} \in \Theta$, if $\beta \neq \tilde{\beta}$, then:
\[
\]
The question becomes whether there are pairs of individuals with symmetric regressors. For example, suppose that individuals only care about the education of the other person when deciding to form a link with them. Say $W_{ij} = \text{education}_j$ and $W_{ji} = \text{education}_i$, where $\text{education}$ is a discrete random variable taking on four values: at least tertiary, at least secondary, at least primary, and less than primary. Provided there are two individuals $i$ and $j$ with at least tertiary education, for example, for this pair of individuals $W_{ij} = \text{education}_j = \text{education}_i = W_{ji}$. Then, we know from Section 2.1.1 that $\beta$ is identified assuming existence and nonsingularity of $\mathbb{E} [W_{ij} W_{ij}']$.
The difficulty with identification arises when there are no pairs of individuals with symmetric regressors, i.e., $W_{ij} \neq W_{ji} \ \forall ij$, so that we cannot rely on the identification result from Section 2.1.1. An example of such a scenario would be if we had a single regressor which equaled the income of the other individual in the pair, i.e., $W_{ij} = \text{income}_j$ and $W_{ji} = \text{income}_i$, where income is a continuous random variable such that it is possible for everyone in the sample to have different values of it. We can show identification if we can show that there exists a pair of individuals $ij$ such that for $\beta, \tilde{\beta} \in \Theta$, if $\beta \neq \tilde{\beta}$, then:
$$ \Phi (W_{ij}' \beta) \Phi (W_{ji}' \beta) \neq \Phi (W_{ij}' \tilde{\beta}) \Phi (W_{ji}' \tilde{\beta}). $$ The problem is that even if we assume existence and nonsingularity (and hence positive definiteness) of $\mathbb{E} [W_{ij} W_{ij}']$, $\mathbb{E} [W_{ji} W_{ji}']$, and $\mathbb{E} [W_{ij} W_{ji}']$, such that:
$$ W_{ij}' \beta \neq W_{ji}' \beta \neq W_{ij}' \tilde{\beta} \neq W_{ji}' \tilde{\beta}, $$ we do not have identification because it is still possible to have:
$$ \Phi (W_{ij}' \beta) \Phi (W_{ji}' \beta) = \Phi (W_{ij}' \tilde{\beta}) \Phi (W_{ji}' \tilde{\beta}). $$
I illustrate the lack of identification through the following example. Write $W_{ij}'\beta = \beta_0 + \beta_1 W_{1ij} + ... + \beta_{k-1} W_{k-1ij}$ and $W_{ji}'\beta = \beta_0 + \beta_1 W_{1ji} + ... + \beta_{k-1} W_{k-1ji}$ without loss of generality, where $W_{lij}$ is the $l$-th component of $W_{ij}$ and $W_{lji}$ is the $l$-th component of $W_{ji}$. Since $W_{ij} \neq W_{ji}$ we cannot have $W_{1ij} = W_{1ji}, ..., W_{k-1ij} = W_{k-1ji}$ simultaneously.
Example 1: $\beta, \tilde{\beta} \in \mathbb{R}^2$ and $W_{1ij}, W_{1ji} \in \{0, 1\}$ with $W_{1ij} \neq W_{1ji}$. Then identification requires that if $\beta \neq \tilde{\beta}$:
$$ \Phi (\beta_0 + \beta_1 W_{1ij}) \Phi (\beta_0 + \beta_1 W_{1ji}) \neq \Phi (\tilde{\beta}_0 + \tilde{\beta}_1 W_{1ij}) \Phi (\tilde{\beta}_0 + \tilde{\beta}_1 W_{1ji}). $$ WLOG,\footnote{Considering $W_{1ij} = 1$ and $W_{1ji} = 0$ instead will yield an identical condition.} when $W_{1ij} = 0$ and therefore $W_{1ji} = 1$, we require that if $\beta \neq \tilde{\beta}$:
$$ \Phi (\beta_0) \Phi (\beta_0 + \beta_1) \neq \Phi (\tilde{\beta}_0) \Phi (\tilde{\beta}_0 + \tilde{\beta}_1). $$ But $\beta_0 = 0, \beta_1 = 1, \tilde{\beta}_0 = 1, \tilde{\beta}_1 = -1$ is such that $\beta \neq \tilde{\beta}$ while $\Phi (\beta_0) \Phi (\beta_0 + \beta_1) = \Phi (\tilde{\beta}_0) \Phi (\tilde{\beta}_0 + \tilde{\beta}_1)$, thus violating identification in this case. Plus, since the regressors are binary, and we have asymmetric regressors for all pairs of individuals, there are no additional conditions that we can use to establish identification. Therefore, we follow the suggestion in poirier_partial_1980 and allow the regressors to take on a third value to establish identification in the following example.
Example 2: $\beta, \tilde{\beta} \in \mathbb{R}^2$ and $W_{1ij}, W_{1ji} \in \{0, 1, 2\}$ with $W_{1ij} \neq W_{1ji}$. As in Example 1 where we used the regressor configuration $(W_{1ij}, W_{1ji}) = (0, 1)$ to derive the restriction that $\Phi (\beta_0) \Phi (\beta_0 + \beta_1) \neq \Phi (\tilde{\beta}_0) \Phi (\tilde{\beta}_0 + \tilde{\beta}_1)$, we can use $(W_{1ij}, W_{1ji}) = (0, 2)$ to derive $\Phi(\beta_0) \Phi (\beta_0 + 2\beta_1) \neq \Phi (\tilde{\beta}_0) \Phi(\tilde{\beta}_0 + 2\tilde{\beta}_1)$, and $(W_{1ij}, W_{1ji}) = (1, 2)$ to derive $\Phi(\beta_0 + \beta_1) \Phi(\beta_0 + 2\beta_1) \neq \Phi (\tilde{\beta}_0 + \tilde{\beta}_1) \Phi (\tilde{\beta}_0 + 2\tilde{\beta}_1)$. Then, using the contrapositive of the definition of identification above, we can establish identification by showing that if:
then $\beta_0 = \tilde{\beta}_0$ and $\beta_1 = \tilde{\beta}_1$. First, divide equation 1. by equation 2. to get:
$$ \frac{\Phi (\beta_0 + \beta_1)}{\Phi (\beta_0 + 2\beta_1)} = \frac{\Phi (\tilde{\beta}_0 + \tilde{\beta}_1)}{\Phi (\tilde{\beta}_0 + 2\tilde{\beta}_1)}. $$ Then, substitute for $\Phi (\beta_0 + \beta_1)$ in equation 3. to get:
$$ \Phi (\beta_0 + 2\beta_1)^2 = \Phi (\tilde{\beta}_0 + 2\tilde{\beta}_1)^2. $$ Then, by monotonicity of $\Phi$, we get the condition $\beta_0 + 2\beta_1 = \tilde{\beta}_0 + 2\tilde{\beta}_1$. Plugging this into the ratio above and cancelling out the $\Phi \left(\beta_0 + 2\beta_1\right)$ term, we get the second condition $\beta_0 + \beta_1 = \tilde{\beta}_0 + \tilde{\beta}_1$. Together, these conditions imply that $\beta_0 = \tilde{\beta}_0$ and $\beta_1 = \tilde{\beta}_1$, as is required for identification.
poirier_partial_1980 conjectured that the key to identification is allowing the regressors to take on more values than there are unknown parameters, but they do not provide a proof of this claim. I show formally that $\beta$ is identified in the NTU model with $\rho = 0$ and a general number of asymmetric and symmetric regressors if the asymmetric regressors can take on at least 3 distinct values.\footnote{It is worth noting that identification is typically easier to establish here than in poirier_partial_1980. This is because poirier_partial_1980 considers asymmetry in tastes, i.e. different betas in the two indicators in the NTU model, instead of asymmetry in characteristics. Since characteristics depend on $i$ and $j$ (unlike $\beta$), we can use individuals with symmetric characteristics to establish identification. It is only in the fully asymmetric case with $W_{ij} \neq W_{ji} \ \forall ij$ that we are forced to rely on the “Poirier principle” of allowing the regressors to take on more values.} Of course, we also need that there is no perfect multicollinearity between the regressors, which is guaranteed by nonsingularity of $\mathbb{E} [W_{ij} W_{ij}']$ and $\mathbb{E} [W_{ji} W_{ji}']$. This is summarized in the following proposition:
Consider again the income example where $W_{ij} = \text{income}_j$, $W_{ji} = \text{income}_i$, and income is a continuous random variable such that it is possible to have $W_{ij} \neq W_{ji} \ \forall ij$. Since the asymmetric regressor is continuous and therefore takes on at least 3 distinct values, by Proposition 1 we know that $\beta$ is identified provided $\mathbb{E} [W_{ij} W_{ij}'] = \mathbb{E} [\text{income}_j^2]$ and $\mathbb{E} [W_{ji} W_{ji}'] = \mathbb{E} [\text{income}_i^2]$ exist and are nonsingular, i.e. there is finite and non-zero variation in the income variable. In practice, once one writes down their model, one must simply determine if they have any asymmetric regressors, and if they do, that either these regressors take on at least 3 distinct values, or that there are some individuals for which these regressors are symmetric; I provide further guidance on this in the application section. Assuming existence and nonsingularity of $\mathbb{E} [W_{ij} W_{ij}']$ and $\mathbb{E} [W_{ji} W_{ji}']$, the only situation in which $\beta$ is not identified is if one has a large number of binary, asymmetric regressors such that one cannot find a pair of individuals with the same values for all these regressors. This problem is alleviated as one collects a larger and richer sample of individuals.
Because identification is case-by-case in this sense, the consistency and asymptotic normality theorems with asymmetric regressors below include a generic identification condition.
Identification aside, the second issue with applying our previous proof strategy for consistency of the NTU-MLE is that the log-likelihood function is no longer globally concave. Therefore, we need to rely on a more general consistency theorem that does not require concavity of the log-likelihood function.\footnote{Namely, we rely on Theorem 2.1 in NM.} In particular, the NTU-MLE will be consistent if (2.1.i) $Q_0 (\beta) \coloneq \mathbb{E} [ \log f(Y_{ij} \mid X_i, X_j, \beta) ]$ has a unique maximum at $\beta_0$, (2.1.ii) $\Theta$ is compact, (2.1.iii) $Q_0 (\beta)$ is continuous, and (2.1.iv) $\hat{Q}_n(\beta) \coloneq \frac{1}{N}\sum^{n-1}_{i=1} \sum^{n}_{j=i+1} \big[Y_{ij} \log \big(\Phi(W_{ij}^{'}\beta) \Phi(W_{ji}^{'}\\\beta) \big) + (1 - Y_{ij})\log \big(1 - \Phi(W_{ij}^{'}\beta) \Phi(W_{ji}^{'}\beta) \big) \big]$ converges uniformly in probability to $Q_0 (\beta)$. The key difference relative to the symmetric regressor case is that, without concavity, we have to establish uniform convergence of the sample log-likelihood to the population log-likelihood, not just pointwise convergence.
Just as Lemma 1 provided conditions for $\beta_0$ to be the unique maximizer of $Q_0 (\beta)$ with symmetric regressors, Lemma 4 provides conditions for $\beta_0$ to be the unique maximizer of $Q_0 (\beta)$ with asymmetric regressors. Unsurprisingly, as well as assuming that $\mathbb{E} [W_{ij} W_{ij}'] < \infty$ as we did in the symmetric regressor case, we also assume that $\mathbb{E} [W_{ji} W_{ji}'] < \infty$ in the asymmetric regressor case.
We assume Condition (2.1.ii) to be true, i.e. that $\Theta$ is compact. If we had i.i.d. data $(Y_{ij}, X_i, X_j)$, then establishing Conditions (2.1.iii) and (2.1.iv) would be easy; all we would have to do is use the bound on $\mathbb{E} [| \log f(Y_{ij} \mid X_i, X_j, \beta) |]$ from the proof of Lemma 4 to invoke a standard uniform LLN. Since $Y_{ij}$ is not i.i.d., but $\hat{Q}_n(\beta)$ is a U-statistic of degree 2 again, we can use a uniform LLN for U-statistics instead. In particular, Theorem 7 in nolan_u-processes_1987 applied to this setting states that $\hat{Q}_n(\beta)$ will converge uniformly in probability to $Q_0 (\beta)$ if (7.i) $X_i$ is i.i.d., (7.ii) we can bound $| \log f(Y_{ij} \mid X_i, X_j, \beta) |$ by some dominating function $d(X_i, X_j)$ with a finite expectation, (7.iii) $\Theta$ is compact and finite-dimensional, and (7.iv) $\log f(Y_{ij} \mid X_i, X_j, \beta)$ is globally Lipschitz in $\beta$ with respect to some integrable envelope $G(X_i, X_j)$.\footnote{Technically, Theorem 7 in nolan_u-processes_1987 involves a series of “entropy” conditions, but these are implied by (7.iii) and (7.iv).} Lemma 5 in the Appendix shows that the existence of $\mathbb{E} [W_{ij} W_{ij}']$ and $\mathbb{E} [W_{ji} W_{ji}']$ (plus i.i.d-ness and compactness) are sufficient for the conditions of Theorem 7 in nolan_u-processes_1987, and therefore for Condition (2.1.iv). Lemma 5 also establishes Condition (2.1.iii) as a by-product.
Combining Lemma 4 and Lemma 5, we arrive at the following consistency theorem with asymmetric regressors:
It is worth comparing the sufficient conditions for consistency with and without regressor symmetry. Since we were willing to assume existence of $\mathbb{E} [W_{ij} W_{ij}']$ in the symmetric case, assuming existence of $\mathbb{E} [W_{ji} W_{ji}']$ too in the asymmetric case is natural. Non-concavity of the log-likelihood function means we need to rely on a uniform LLN instead of a pointwise LLN, but we do not need to make stronger assumptions for this, we just require more complicated asymptotic theory. Non-concavity also changes the assumptions we make about $\Theta$: with symmetric regressors, $\Theta$ is assumed to be bounded and convex; with asymmetric regressors, $\Theta$ is assumed to be compact. That is, we add closed-ness and drop convexity when regressors are asymmetric. If one wishes to impose unified assumptions on the parameter space for consistency irrespective of regressor symmetry, then simply assume that $\Theta$ is compact and convex throughout (and that $\beta_0$ is in the interior of $\Theta$).
Identification is assumed. However, as discussed above, identification can be established by existence and nonsingularity of $\mathbb{E} [W_{ij} W_{ij}']$ and $\mathbb{E} [W_{ji} W_{ji}']$ if there exists a pair of individuals with symmetric regressors, or by Proposition 1 if there is no such pair but all asymmetric regressors can take on at least 3 distinct values. Proposition 1 also imposes existence and nonsingularity of $\mathbb{E} [W_{ij} W_{ij}']$ and $\mathbb{E} [W_{ji} W_{ji}']$, so, although we do not formally assume nonsingularity in Theorem 3, it is practically always assumed to establish identification as a precursor to Theorem 3. This is in contrast with the asymptotic normality theorem below (Theorem 4) which imposes nonsingularity for reasons other than identification.
The bottom-line is that, although the asymptotic theory required to establish consistency is more complicated when regressors are asymmetric, the sufficient conditions are only marginally stronger. Practically speaking, the main difference is that identification (as a precursor to consistency) now requires all asymmetric regressors to take on at least 3 distinct values. This means we simply need to avoid pathological specifications with many binary regressors, as then it might be possible to find a pair of individuals with different values for all such regressors.
For asymptotic normality with asymmetric regressors, we can apply the same proof strategy as with symmetric regressors, but with more algebra and an additional nonsingularity condition. As in the symmetric regressor case, we begin by invoking the conditional Lindeberg-Feller CLT to show that $\sqrt{N} \nabla_{\beta} \hat{Q}_n(\beta_0) \coloneq \frac{1}{\sqrt{N}} \sum^{n-1}_{i=1} \sum^{n}_{j=i+1} \big[Y_{ij} \nabla_{\beta}\log \big(\Phi(W_{ij}^{'}\beta_0) \Phi(W_{ji}^{'}\beta_0)\big) + (1 - Y_{ij}) \nabla_{\beta}\log \big(1 - \Phi(W_{ij}^{'}\beta_0) \Phi(W_{ji}^{'}\beta_0) \big) \big]$ is asymptotically normal with mean zero. This is guaranteed by Lemma 6 in the Appendix assuming finite eighth moments again.
Given consistency, and Lemma 6 so that there is convergence in distribution “under the hood”, we have the following theorem:
Note that we do not need to assume existence of $\mathbb{E}[W_{ij} W_{ji}' + W_{ji} W_{ij}']$ because it is implied by the existence of $\mathbb{E} [W_{ij} W_{ij}']$ and $\mathbb{E} [W_{ji} W_{ji}']$, which is implied by the existence of $\mathbb{E} [\|W_{ij}\|^8]$ and $\mathbb{E} [\|W_{ji}\|^8]$ (use the Cauchy-Schwarz inequality). However, nonsingularity of $\mathbb{E} [W_{ij} W_{ij}']$ and $\mathbb{E} [W_{ji} W_{ji}']$ does not imply nonsingularity of $\mathbb{E}[W_{ij} W_{ji}' + W_{ji} W_{ij}']$. To see this, suppose $\mathbb{E} [W_{ij} W_{ij}']$ and $\mathbb{E} [W_{ji} W_{ji}']$ are nonsingular, and also that $W_{ij}$ and $W_{ji}$ are independent and zero mean. Then $\mathbb{E} [W_{ij} W_{ji}'] = \mathbb{E} [W_{ij}] \mathbb{E} [W_{ji}'] = 0$ (and likewise with $i$ and $j$ interchanged). So, $\mathbb{E}[W_{ij} W_{ji}' + W_{ji} W_{ij}'] = \mathbb{E}[W_{ij} W_{ji}'] + \mathbb{E}[W_{ji} W_{ij}'] = 0$, is the zero matrix which is clearly singular. Hence, we assume nonsingularity of $\mathbb{E}[W_{ij} W_{ji}' + W_{ji} W_{ij}']$ via assumption (4e) to satisfy Condition (3.3.iv) in the proof of Theorem 4. This assumption rules out situations exactly like those described above where $W_{ij}$ and $W_{ji}$ are independent and zero mean. It is interesting that some degree of dependence between $W_{ij}$ and $W_{ji}$ assists in establishing asymptotic normality.
As with consistency, although the theory required to establish asymptotic normality is more complicated when regressors are asymmetric, the sufficient conditions are only marginally stronger. Namely, on top of the conditions for consistency, we now impose non-zero dependence between $W_{ij}$ and $W_{ji}$. E.g., if $W_{ij} = \text{income}_j$ and $W_{ji} = \text{income}_i$ as in our running example, we would require some correlation in incomes across individuals, which would be natural in a setting with common shocks to income. Or e.g., if individuals cared about the weighted average of the pair's income, but gave more weight to their own income, say $W_{ij} = 0.7 \text{income}_i + 0.3 \text{income}_j$, then dependence between $W_{ij}$ and $W_{ji}$ would be structurally present.
As well as extending the model to allow for asymmetric regressors, one could allow for an arbitrary covariance between $\varepsilon_{ij}$ and $\varepsilon_{ji}$, $\rho \in (-1, 1)$. For reasons described in Section 1.2, we opt for simulation-based results only for the consistency of the NTU-MLEs, $\hat{\beta}$ and $\hat{\rho}$ -- the maximizers of the generic NTU log-likelihood $\hat{Q}_n(\beta, \rho)$. In the Appendix, the simulation results suggest that the NTU-MLEs are consistent for all values of $\rho \in (-1, 1)$ that we consider, irrespective of regressor (a)symmetry. Furthermore, $\hat{\beta}$ and $\hat{\rho}$ are close to the true values of $\beta$ and $\rho$ with very small $n = 20$.
In this section, we propose a specification test for whether a given dataset is better explained by the TU or NTU model. This test is useful because it can be shown that the TU-MLE will be inconsistent if a) the regressors are asymmetric or b) the regressors are not distributed symmetrically around 0.\footnote{a) is natural because the TU model has no way of incorporating asymmetric regressors. b) relies on a standard misspecified MLE argument à la white_maximum_1982.} The test relies on recognizing that the TU and NTU models can be nested in a more general model. In particular, both models can be nested in the general NTU model from Section 1.2 if we allow $\rho \in [-1, 1]$ instead of $\rho \in (-1, 1)$. The fact that this model nests the NTU model is obvious. To see that this model also nests the TU model, set $W_{ij} = W_{ji}$ and $\rho = 1$. If $\rho = 1$, then $\varepsilon_{ij} = \varepsilon_{ji}$, and therefore $Y_{ij} = \mathds{1} [W_{ij}^{'}\beta \geq \varepsilon_{ij}]^2 = \mathds{1} [W_{ij}^{'}\beta \geq \varepsilon_{ij}]$, which is the TU model. Thus, we can test if the DGP is the TU model vs the NTU model (with symmetric regressors) by testing:
$$ H_0 : \rho_0 = 1 \quad \text{vs} \quad H_A : \rho_0 < 1, $$ where $\rho_0$ is the true value of $\rho$.
Let $\hat{Q}_n(\beta, \rho)$ be the log-likelihood function of the general NTU model from Section 1.2, now allowing for $\rho \in [-1, 1]$, and let $(\hat{\beta}, \hat{\rho}) \coloneq \argmax_{(\beta, \rho)}\hat{Q}_n(\beta, \rho)$. A naive testing approach would be to form the Wald statistic, $W = \frac{1 - \hat{\rho}}{\text{SE}(\hat{\rho})}$, and reject $H_0$ at significance level $\alpha$ if $W > Z(1 - \frac{\alpha}{2})$, where $Z(1 - \frac{\alpha}{2})$ is the $1 - \frac{\alpha}{2}$-th quantile of the standard normal distribution. Alternatively, one could form the likelihood ratio statistic, $LR = -2 \left[\hat{Q}_n(\hat{\beta}, 1) - \hat{Q}_n(\hat{\beta}, \hat{\rho})\right]$, and reject $H_0$ at significance level $\alpha$ if $LR > \chi^2_1(1-\alpha)$, where $\chi^2_1(1-\alpha)$ is the $1 - \alpha$-th quantile of the chi-squared distribution with 1 degree of freedom.\footnote{Technically, the $\hat{Q}_n(\hat{\beta}, 1)$ in the LR statistic is usually $\hat{Q}_n(\beta_0, 1)$ instead. However, because for this specification test we are only interested in testing for the value of $\rho_0$ and not $\beta_0$, and because $\hat{\beta}$ is consistent such that $\hat{Q}_n(\hat{\beta}, 1)$ and $\hat{Q}_n(\beta_0, 1)$ are indistinguishable in large samples, I include $\hat{Q}_n(\hat{\beta}, 1)$ such that the $LR$ statistic is computable irrespective of the hypothesized value of $\beta_0$. If one wishes to test a joint hypothesis about both $\rho_0$ and $\beta_0 \in \mathbb{R}^k$, then simply replace $\hat{Q}_n(\hat{\beta}, 1)$ with $\hat{Q}_n(b, 1)$, where $b$ is the value of $\beta$ under the joint null of interest, and compare to the chi-squared distribution with $k+1$ degrees of freedom.} However, simulation evidence suggests that neither the naive Wald nor naive LR approach have correct size: the Wald test seriously over-rejects while the LR test under-rejects.
This is perhaps unsurprising because we are testing whether a parameter is equal to the boundary of the parameter space: $\rho_0 = 1$. Focusing on the LR test, self_asymptotic_1987 derive the asymptotic distributions of LR test statistics for tests of parameters at their boundaries. Thinking of $\beta$ as a nuisance parameter,\footnote{Again, if instead one wishes to test a joint hypothesis about both $\rho$ and $\beta \in \mathbb{R}^k$, then one would be in Case 6 of self_asymptotic_1987 and the relevant asymptotic distribution would be $\frac{1}{2}\chi^2_k + \frac{1}{2}\chi^2_{k+1}$ under $H_0$.} we have one parameter of interest at the boundary ($\rho$) and one nuisance parameter not at the boundary ($\beta$). This means that we are in Case 5 of self_asymptotic_1987 and thus the LR test statistic converges in distribution to a $50:50$ mixture of $\chi^2_0$ and $\chi^2_1$ under $H_0$. That is, the correct procedure to test $H_0: \rho_0 = 1$ vs $H_A: \rho_0 < 1 $ at significance level $\alpha$ is to reject $H_0$ if:
$$ -2 \left[\hat{Q}_n(\hat{\beta}, 1) - \hat{Q}_n(\hat{\beta}, \hat{\rho})\right] > \left(\frac{1}{2}\chi^2_0 + \frac{1}{2}\chi^2_{1}\right)(1 - \alpha), $$ where $\left(\frac{1}{2}\chi^2_0 + \frac{1}{2}\chi^2_{1}\right)(1 - \alpha)$ is the $(1 - \alpha)$-th quantile of a mixture distribution that is 50% $\chi^2_0$ and 50% $\chi^2_1$.
I verify that this testing procedure works via simulation. Namely, I plot power curves, i.e. the proportion of simulations for which the null $H_0: \rho_0 = 1$ is rejected against the true value of $\rho$ used to generate the networks, for various sample sizes in Figure 2. The empirical test sizes can be read off the figure by looking at the proportion of rejections when $\rho = 1$ (as indicated by the small dotted line). The true size is $\alpha = 0.05$; empirical sizes are 0.058 when $n=50$, 0.048 when $n=20$, and 0.058 when $n= 10$. As well as achieving size control, the test has strong power, with power jumping to 100% at $\rho = 0.8$ when $n=50$. The reason why we have such good finite-sample performance is because, due to the dyadic nature of the data, we have $\sqrt{N}$-convergence instead of $\sqrt{n}$-convergence in Theorems 2 and 4. The number of dyads, $N = \frac{n(n-1)}{2}$, is quadratic in the number of individuals, $n$. So, although $n=20$ is small, $N=\frac{20(20-1)}{2}=190$ is not. This result is echoed by the exceptional small-sample behavior of the NTU-MLE in the Appendix.
I now apply the estimation and testing procedures developed in this paper to two network datasets from different fields of economics. The first comes from goeree_1d_2010. goeree_1d_2010 use dictator games to show that 10-18-year-old students from an all-girls school in Pasadena, California are more likely to give to their friends. Furthermore, students are more likely to be friends with those that are more similar to themselves (homophily). In their Table 2, goeree_1d_2010 estimate a TU model with logistic errors via MLE to determine which shared characteristics explain the decision of two students to form a link (friendship). Specifically, they consider whether both students a) are the same race, b) have the same height, c) have the same amount of confidence, and d) have had the same boyfriend, as well as e) whether the student in the pair that was the recipient in the dictator game is shy and f) the height of the other student. When I estimate their TU model with logistic errors via MLE in Column 1 of Table 1 below, I confirm the finding in goeree_1d_2010 that all factors except for the shyness of the recipient and height of the partner are significantly correlated with the link decision. The same is true with normal errors as per Column 2 of Table 1.
However, the assumption of transferable utility is questionable here. If Abbie wants to be friends with Becca but Becca does not want to be friends with Abbie, is it reasonable to assume that Abbie will provide Becca with transfers of some kind (e.g. money) such that the friendship becomes mutually beneficial? This does not seem to be a very plausible model of how adolescent friendship works. More formally, we can test for the appropriateness of the TU model here by using the specification test developed in Section 3. The NTU-MLE of $\rho$ and the p-value for the test of $H_0: \rho_0 = 1$ vs $H_A: \rho_0 < 1$ using the LR specification test procedure from Section 3 are reported at the bottom of Column 3 of Table 1. We find that $\hat{\rho} = -0.031$ and that the p-value for the specification test is essentially $0$. That is, we reject the appropriateness of the TU model in favor of the NTU model with $\rho$ practically equal to 0. It is satisfying that we seem to be in the $\rho = 0$ case here as this was the case for which consistency and asymptotic normality of the NTU-MLE were proven formally in Section 2.
Column 3 of Table 1 also reports the estimates for the homophily parameters in the NTU model.\footnote{The p-values for the homophily parameters are derived from standard Wald tests of $H_0: \beta_0=0$ vs $H_A: \beta_0\neq0$. For justification that standard Wald/LR tests work for hypotheses involving $\beta$, see the simulation evidence in the Appendix. For example, when $\rho = 0$ as seems to be the case here, the Wald test has empirical size 0.052 and the standard LR test has empirical size 0.050.} While the TU estimates suggest that having had the same boyfriend matters for friendship formation and that the height of one's partner does not, the NTU estimates suggest the opposite. This application highlights that, when the validity of the TU assumption can be rejected, the more appropriate NTU-MLE can yield qualitatively different estimates to the TU-MLE, therefore underlining the value of the testing and estimation procedures developed in this paper.
Readers may have noticed that the final regressor in Table 1 -- the height of the other student -- is asymmetric. For example, the first individual in the dataset has a height of 54 inches and the second individual has a height of 58 inches. Nonetheless, $\beta$ in the NTU model is identified for two reasons here (assuming $\rho$ = 0 and existence and nonsingularity of $\mathbb{E} [W_{ij} W_{ij}']$ and $\mathbb{E} [W_{ji} W_{ji}']$). First, there are pairs of individuals that have the same height, e.g. the third and fourth individual in the dataset are both 56 inches tall. Second, even if there were not two individuals with the same height, $\beta$ would still be identified by Proposition 1 because the height variable takes on 24 ($>3$) distinct values. As we know from Theorems 3 and 4, the complications that arise due to regressor asymmetry can be dealt with by assuming compactness of the parameter space and an additional nonsingularity condition.\footnote{Strictly speaking, since we have an asymmetric regressor, the specification test developed in Section 3 that assumes symmetric regressors does not apply here. Nonetheless, given that $\hat{\rho} = 0$ is very far away from 1, and only one of the six regressors are asymmetric, it seems highly likely that a specification test generalized to asymmetric regressors would still reject the TU assumption. For a discussion of how one might generalize the specification test, see the end of Section 4.2.}
The second application is the same application considered in the two papers that are most similar to this paper: gao_logical_2023 and li_ming_estimation_2024. It concerns the determinants of risk-sharing across households in a small community called Nyakatoke in rural Tanzania.\footnote{de_weerdt_risk-sharing_2002 is to thank for the collection of this data.} In particular, gao_logical_2023 and li_ming_estimation_2024 attempt to explain the decision of two households to share cash, kind, or labor by a) the absolute value of the difference between the (log) wealth of the households, b) the (log) physical distance between the households, and c) the strength of the tie between the households in terms of blood relationships and shared religions.\footnote{All three regressors are symmetric.} Focusing on li_ming_estimation_2024's results since they provide statistical inference, not just point estimation as in gao_logical_2023, they use their more complicated NTU-model-based estimator to find a zero coefficient on the difference in wealth, a negative and significant coefficient on distance, and a positive and significant coefficient on the tie variable.
In Table 2, I compute the simpler NTU-MLEs from this paper and perform the specification test for the TU vs NTU model. I find coefficients that are in line with those in li_ming_estimation_2024: a zero coefficient on the difference in wealth, a negative and significant coefficient on distance, and a positive and significant coefficient on the tie variable. What is interesting is that we estimate $\hat{\rho}$ to be very close to 1 such that we cannot reject the null hypothesis that the TU model is appropriate for explaining risk-sharing decisions in Nyakatoke. This suggests that we might not need to worry about the additional identification and estimation complications that arise in NTU models here. Since neither gao_logical_2023 nor li_ming_estimation_2024 have a specification test for whether the TU model or NTU model is more appropriate, they apply their NTU-model-based estimators to this (seemingly) TU-model-based dataset when they may have been able to use a simpler TU-model-based estimator here, such as the one in graham_econometric_2017.
Although the specification test developed in this paper has proven useful here, skeptics of the results in Tables 1 and 2 are welcome to respond with the fact that the errors might not be normally distributed and there might unobserved individual heterogeneity that is not being accounted for. A specification test in a semi-parametric setting with fixed effects would be advantageous, but this is beyond the scope of this paper. Of course, then the specification testing procedure would not be as simple as testing for $\rho = 1$. I conjecture that the easiest way to implement a specification test in a more general setting would be via a non-nested likelihood ratio test vuong_likelihood_1989, like the one used in comola_testing_2014.\footnote{comola_testing_2014 use a non-nested LR test to determine if link formation is done unilaterally, i.e. one individual decides whether to form a link without the consent of the other individual, or bilaterally, as in all the models considered in this paper.} It is also not clear whether we actually need to be concerned about fixed effects here, especially in light of the fact that my estimates (which ignore fixed effects) are similar to li_ming_estimation_2024's estimates (which explicitly account for fixed effects). This motivates future work on a specification test for the NTU model without fixed effects vs the NTU model with fixed effects.
This paper has contributed to the literature on dyadic network formation models by providing sufficient conditions for the consistency and asymptotic normality of the NTU-MLE with and without symmetric regressors. It turns out that, when regressors are symmetric, the sufficient conditions for consistency and asymptotic normality of the TU-MLE are also sufficient for the NTU-MLE. That is, if the TU-MLE is theoretically justified, then the NTU-MLE is too. Furthermore, when regressors are asymmetric, remarkably little extra is assumed. Although this paper has produced simulation-based evidence for the consistency of the NTU-MLE with correlated errors (see the Appendix), it has not provided formal conditions, and this is an obvious next step for researchers to work on.
Furthermore, this paper has provided a specification test to determine whether the TU or NTU model is more appropriate in a given setting. Two applications have confirmed the value of this specification test, but I have been careful to state that this test assumes that a) the regressors are symmetric, b) the errors follow a normal distribution and c) there is no unobserved individual heterogeneity. I reiterate the merit of a more general specification test in a semi-parametric setting with asymmetric regressors and fixed effects.
From an applied perspective, this paper has shown that the choice between the TU-MLE and NTU-MLE matters, yet, with symmetric regressors, the NTU-MLE is no harder to justify than the TU-MLE. Furthermore, relative to other NTU-model-based estimators that allow for fixed effects, the NTU-MLE is very easy to compute using standard maximum likelihood routines. Given this ease of implementation and the possibility of substantive differences in empirical conclusions, researchers should treat the choice between TU and NTU as an empirical question rather than a default assumption. And the specification test developed in this paper provides an answer to this empirical question.
I thank Francis DiTraglia and Martin Weidner for insightful comments and guidance.
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