# What is the intuition on fixed and random effects models? [duplicate]

Now I’m having a hard time having a grasp on the difference between fixed and random effects of regression models. I believe I understand it’s recommended to use random effects if you consider heterogeneity of slopes, when the data is nested among hierarchical levels, etc.

But here’s the question.

1. Why don’t we just put moderating variable(interaction term) if we want to reflect the changing effect among different groups? for example, if the effect of study time on GPA differs among different classrooms, then why not just make a dummy variable for classroom variable, and put an interaction term? I cannot understand what the point is here.

2. What is an overall intuition on the grand assumption of random effects model? what is the main idea that can penetrate the logic of random effects model? I don’t want any mathematical or statistical explanation, I want to draw some hypothetical picture in my head.

One way to think about fixed-effects vs. random effects is by examining how the fixed-effects estimator works in comparison to the random effects estimator.

Let’s say I have panel data on firms. Let $y_{i,t}$ be dividends for firm $i$ at time $t$. Let $x_{i,t}$ be something we’re looking at like free cash flow.

Imagine our model is:

So dividends for firm $i$ at time $t$ are the sum of $\beta$ times free cash flow plus a firm specific effect $u_i$ and a firm, time specific error-term $\epsilon_{i,t}$. Now let’s imagine two different estimators:

• The within estimator. $\beta$ is estimated using only time-series variation within each firm.
• The between estimator. $\beta$ is estimated using only the variation between different firms. (The between estimator is $\beta$ from the cross-sectional regression $\bar{y}_i = \beta \bar{x}_i + v_i$.)

The within estimator is the fixed-effect estimator. It takes off the mean from each group and the only variation leftover to estimate $\beta$ is time series variation within each firm. If the fixed effects can be anything, this is what you have to do.

The random effects estimator is a weighted average of the within estimator and the between estimator. If the effects $u_i$ are random and mean zero, then variation between firms also contains information about $\beta$ and the between estimator is also a consistent estimator. Rather than tossing out the between firm variation (as occurs in the fixed effect estimator), the between firm variation is given some weight in the random effects estimator of $\beta$.