This dissertation studies estimation and inference on models with dyadic dependence, that is models for double indexed observations where observations are correlated whenever they share an index. Data exhibiting this form of dependence are commonplace: from international trade (e.g. \cite{Rose2004}) to sales on online platforms (e.g. \cite{Bajari2023}) or social networks (\cite{FafchampsGubet2007}). Because of the particular dependence structure, very little is known about efficiency in these models. For instance, for parametric models, only a handful of examples have likelihood functions or maximum likelihood estimators that can be expressed in closed form or that are computationally feasible. The analyst is forced to sacrifice efficiency for computational ease and tractability. Unfortunately, unlike cross-sectional models, efficiency losses in dyadic models can manifest as drops in rates of convergence rather than just asymptotic variance, immensely impacting the precision of estimation.
The dissertation explores new estimation methods for different dyadic models, with a particular attention to efficiency and computational feasibility. Each of The three chapters in this dissertation studies a set of dyadic models and estimators for those models. The first and last chapters present efficiency results.
In the first chapter I propose a two step rate optimal estimator for an undirected dyadic linear regression model with interactive unit-specific effects. The estimator remains consistent when the individual effects are additive rather than interactive. We observe that the unit-specific effects alter the eigenvalue distribution of the data's matrix representation in significant and distinctive ways. We offer a correction for the \textit{ordinary least squares}' objective function to attenuate the statistical noise that arises due to the individual effects, and in some cases, completely eliminate it. The new objective function is similar to the \textit{least squares} estimator's objective function from the large $N$ large $T$ panel data literature (\cite{Bai2009}). In general, the objective function is ill behaved and admits multiple local minima. Following a novel proof strategy, we show that in the presence of interactive effects, an iterative process in line with \cite{Bai2009}'s converges to a global minimizer and is asymptotically normal when initiated properly. The new proof strategy suggests a computationally more advantageous and asymptotically equivalent estimator. While the iterative process does not converge when the individual effects are additive, we show that the alternative estimator remains consistent for all slope parameters.
Chapter 2 proposes a general procedure to construct estimators for exchangeable network models. For any network model, consider an auxiliary i.i.d. model where each observation has the same distribution as any observation in the original model. The procedure returns estimators for the original model whenever valid estimators are known in the auxiliary i.i.d. model. The chapter then studies the asymptotic behavior of the ``the average MLE", the estimators returned by the procedure for parametric binomial network models. I show that the average MLE behaves asymptotically like the composite maximum likelihood estimator. Interestingly, the average MLE does not require the entire network to be observed. For instance, I show that for a balanced bipartite graph, observing almost any sub-graph with more than $N^{\frac{3}{2}+\epsilon}$ edges for some $\epsilon>0$ (out of the total $N^2$ edges) is enough for the asymptotic result to hold. These results are readily extendable beyond the binomial model.
The final chapter studies the properties of the maximum likelihood estimator (MLE) for exponential families of distributions on network data. I show that, under some conditions, the MLE is asymptotically normally distributed with an asymptotic variance equal to the inverse of the information matrix. I also show that under those same conditions, the MLE is efficient compared to regular estimators with the same rate of convergence. This extends well known results on MLE for $i.i.d.$ models.
