# What's the difference between Random Intercepts Model and linear model with dummies?

I am investigating factors that may agricultural household's decision on whether or not will they participate in land circulation. One of the existed literature argues that individuals at the village level may be similar from one to the other hence they are going to use a random intercept model:

Here is the model specification:

$$logit(\pi_{ij})=log(\frac{\pi_{ij}}{1-\pi_{ij}})=\beta_{0j}+\beta_{1j}X_{1ij}+\beta_{2j}X_{2ij}+...+\beta_{hij}X_{hij} +\epsilon_{ij}$$ $$\beta_{0j}=\gamma_{00}+\gamma_{01}G_{1j}+\gamma_{02}G_{2j}+...+\gamma_{0k}G_{kj}+\mu_{0j}$$ $$\beta_{1j}=\gamma_{10}+\gamma_{11}G_{1j}+\gamma_{12}G_{2j}+...+\gamma_{1k}G_{kj}+\mu_{1j}$$ $$\beta_{hj}=\gamma_{h0}+\gamma_{h1}G_{1j}+\gamma_{h2}G_{2j}+...+\gamma_{hk}G_{kj}+\mu_{hj}$$

$\pi_{ij}$ represents the probability of household i in village j decides to participate in the land circulation program, h represents the number of the independent individual level variable while k represents the number of independent variable at village level.

Why can't I simply introduce village dummies and villege*individual interactive dummies and run the logit regression to get the results? What are the differences between two models?

• @MichaelM Do you mean using dummy variables would be a better option? – JoZ Nov 10 '17 at 19:50
• What do you mean with the village * individual interaction? Do you have more observations per individual? Otherwise, wouldn't this introduce more variables than you have observations (which would be an undefined model)? – Gijs Nov 10 '17 at 19:56
• @MichaelM sorry I misunderstood ... – JoZ Nov 10 '17 at 20:51

Random intercept models (and multi-level models in general) allow us to relax the assumption of independent errors. They do this by having two sorts of errors (labeled $\epsilon$ and $\mu$ in your question).