# Likelihood Formulation of a Time Dynamic Bradley Terry Model with Random Effects

Let us assume a mixed logit model with a binary dependent variable $$y_{i, t }$$ that is explained by a fixed effect matrix X and a simple random intercept for each individual $$i$$

$$y_{i,t}^* = x_{i,t}'\beta +\alpha_i + \epsilon_{i,t} \hspace{35pt} \alpha \sim N(0, \sigma_{\alpha}) \hspace{17pt} where \hspace{17pt}\begin{array}{ll} 1 & \widetilde{y}_{ij,t}^{*} \geq 0 \\ 0 & \, \widetilde{y}_{ij,t}^{*} < 0 \\ \end{array}$$

As far as I know, when we want to estimate the fixed and random effects via maximum likelihood, we have to integrate out the random effects over every individual so that the log likelihood becomes

$$LL(\beta) = \sum_{i=1}^N \log \left(\int_{-\infty}^{\infty} \prod_{t=1}^{T_i}\left(\left(\frac{e^{x_{i,t}'\beta +\alpha_i}}{1 + e^{x_{i,t}'\beta +\alpha_i}}\right)^{y_{i,t}} \left(\frac{1}{1 + e^{x_{i,t}'\beta +\alpha_i}}\right)^{1 - y_{i,t}} \frac{1}{\sqrt{2\pi\sigma_\alpha^2}}e^{-\frac{\alpha_i^2}{2\sigma_\alpha^2}}\right) d \alpha_i \right)$$

Let us now assume a time-dynamic Bradley-Terry model. The probability that player $$i$$ wins a match against player $$j$$ at time $$t$$ is then given by

$$p_{ij,t} = .P(y_{ij,t} = 1| \lambda_{i,t},\lambda_{j,t}) = \frac{e^{\lambda_{i,t} - \lambda_{j,t}}}{1 + e^{\lambda_{i,t} - \lambda_{j,t}}} \label{eq:second_BTm}$$

where

$$\lambda_{i,t} = x_{i,t}'\beta$$

can be though of the skill level of player $$i$$ at time $$t$$, which is explained by a player-specific covariate vector $$\beta$$.

While this is the common formulation of the Bradley-Terry model, it is also possible to write this in the latent variable representation (as in the case of the first model)

$$\widetilde{y}_{ij,t} = \left\{ \begin{array}{ll} 1 & \widetilde{y}_{ij,t}^{*} \geq 0 \\ 0 & \, \widetilde{y}_{ij,t}^{*} < 0 \\ \end{array} \right. \\ \widetilde{y}_{ij,t}^{*} = \sum_{l=1}^{k}\beta_l x_{il,t} - \sum_{l=1}^{k}\beta_l x_{jl,t} + \epsilon_{i,t} - \epsilon_{j,t} \hspace{20pt} \epsilon_{q,t} \sim Logistic(0, \frac{\pi^2}{3}) \hspace{20pt} q = i,j$$

Since the model is time dynamic, the log likelihood is given by

$$LL(\Theta) = \sum_{t = 1}^{T}\sum_{(i,j)\in I_t}\log \left(y_{ij,t}p_{ij,t} + (1 - y_{ij,t})(1 - p_{ij,t}) \right) \\ I_t = \{(i,j): \text{a match between player} \hspace{3pt} i \hspace{3pt} \text{and} \hspace{3pt} j \hspace{3pt} \text{is played at time} \hspace{3pt} t \}$$

The reason the likelihood needs to be formulated in a time-dynamic fashion is that some of the covariate values are constantly updated in random-walk process, as proposed by Gorgi, Koopman and Lit (2018): https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3110555

Let us now add a player-specific time-constant random intercept $$\alpha_i$$ to account for player specific heterogeneity, so that we obtain

$$\lambda_{i,t} = x_{i,t}'\beta +\alpha_i \hspace{35pt} \alpha \sim N(0, \sigma_{\alpha})$$

This can also be written in terms of the latent variable representation

$$\widetilde{y}_{ij,t} = \left\{ \begin{array}{ll} 1 & \widetilde{y}_{ij,t}^{*} \geq 0 \\ 0 & \, \widetilde{y}_{ij,t}^{*} < 0 \\ \end{array} \right. \\ \widetilde{y}_{ij,t}^{*} = \sum_{l=1}^{k}\beta_l x_{il,t} - \sum_{l=1}^{k}\beta_l x_{jl,t} + u_{i,t} - u_{j,t} \hspace{35pt} u_{q,t} = \alpha_q + \epsilon_{q,t} \hspace{35pt} \\ \epsilon_{q,t} \sim Logistic(0, \frac{\pi^2}{3}) \hspace{35pt} q = i,j$$

What form does the log-likelihood now take? Do we still have to integrate out the random effects over every player $$i$$ or rather over every player pair $$i,j$$?

Thank you very much in advance!