A Gibbs Sampler for Ordinal IRT Models

This repo provides an efficient Gibbs Sampler for dynamic item response theory models with ordinal outcomes. An application can be found in Haosen Ge (2021) "Measuring Regulatory Barriers Using Annual Reports of Firms".

The model assumes the followiing data generating process:

$$U_{ijt} = \begin{cases} 1 \quad \text{if} \quad U_{ijt}^* \leq \alpha_{ij}^R \\ 2 \quad \text{if}\quad \alpha_{ij}^R < U_{ijt}^* \leq \alpha_{ij}^E \\ 3 \quad \text{if} \quad U_{ijt}^* > \alpha_{ij}^E \end{cases}$$

Denote the set of ${\theta_{j,t}}$ as $\Theta$ and the set of $\{\alpha_{ij}^E\}$ and $\{\alpha_{ij}^R\}$ as $\alpha^E$ and $\alpha^R$. Let $U$ denote the observed data and $U^*$ the augmented data. We can write the full data likelihood with the augmented data as:

$$ \mathcal{L}(\Theta, \alpha^E, \alpha^R | U, U^* ) = \prod_{t=1}^T \prod_{j = 1}^J \prod_{i = 1}^I \{ I(U_{ijt} = 1,U_{ijt}^* \leq \alpha_{ij}^R) + I(U_{ijt} = 2, \alpha_{ij}^R < U_{ijt}^* \leq \alpha_{ij}^E) + I(U_{ijt} = 3, U_{ijt}^* > \alpha_{ij}^E)\} \cdot \phi_{\theta_{jt}}(U_{ijt}^*) $$

where $\phi_{\theta_{jt}(\cdot)}$ denotes the probability density function of $\mathcal{N}(\theta_{jt}, 1)$.