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I'm seeking clarification on the distinction between fixed effects and random effects in the context of panel data analysis. My understanding is as follows:

  1. Fixed Effects: In a fixed effects model, individual characteristics that could influence the dependent variable, $Y$, are considered. For instance, with panel data comprising time and country, the model includes $C(country)$ as a categorical variable. This variation in the intercept aims to capture the relationship between $Y$ and $X$ that isn't solely accounted for by $X$. The model can be represented as:

    $$ Y_{it} = \alpha_i + \beta X_{it} + \epsilon_{it} $$

    where $i$ indexes countries, $t$ indexes time, $\alpha_i$ is the country-specific intercept, and $\epsilon_{it}$ is the error term. This is equivalent to OLS with C(country) as dummy but different p-val since OLS thinks all $\epsilon$ is iid.

  2. Random Effects: In a random effects model, it is assumed that $Y$ varies linearly with $X$ across all countries, but the error term may have a different distribution. This model can be written as:

    $$ Y_{it} = \beta X_{it} + u + \epsilon_{it} $$

    Here, $\epsilon_{it}$ is the idiosyncratic error, and $u_i$ captures the random effect associated with each country. It's assumed that the moments of $Y$ are linearly related to the moments of $X$ across countries.

  3. Ordinary Least Squares (OLS): A standard OLS model, disregarding the country factor, would take this a step further, assuming that the error term is independently and identically distributed (iid) across all observations:

    $$ Y_{it} = \beta X_{it} + \mu + \epsilon $$

    with $\epsilon$ being iid.

In summary, my intuition is that the fixed effects model accounts for the most complex dynamics between $Y$ and $X$, followed by the random effects model, and then the simplistic OLS model. Could someone correct me if my understanding is not aligned with standard econometric principles?

The One
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  • This has been discussed at length already (this is probably the best reference), but your intuition is not correct in that random - or particularly mixed - effects usually allow much more flexibility in modeling (co)variance. – PBulls Dec 15 '23 at 18:11
  • @PBulls I assume when you make forecasts with random effects model you don't have the constant term changing for different countries? – The One Dec 15 '23 at 18:23
  • You would if you condition on the country-specific random effect(s), not in the marginal (population-level, fixed part only) model. – PBulls Dec 15 '23 at 18:33
  • Your $u$ in your random effects equation is also missing an index. Not sure what $u$ is doing in your OLS model: if it is a single number, then it is simply part of the intercept, if it is not a single number, and is just missing an index, then that's not OLS. – Alexis Dec 15 '23 at 18:33

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