Chapter 3 of 4

Interpreting the Results

Fixed Effects Regression

You've run a fixed effects regression and got a coefficient. What does it actually mean?

Let's say we run fixed effects on our football data — training sessions per week on points per game, with club fixed effects — and get:

Points per Gameit=β1⋅Trainingit+αi+εit\text{Points per Game}_{it} = \beta_1 \cdot \text{Training}_{it} + \alpha_i + \varepsilon_{it}

Estimated: β^1=0.15\hat{\beta}_1 = 0.15

This means: when the same club trains one more session per week than its own average, it earns about 0.15 more points per game than its own average.

Key word: the same club. The coefficient is entirely about within-unit changes. It does not say "clubs that train more earn more points" — it says "when a club trains more than usual, it earns more points than usual."

How does the fixed effects estimate compare to naive OLS (pooled OLS)?

Remember from Chapter 2: the naive OLS on our raw data showed a negative relationship between training and points. That's (maybe) because smaller clubs have more time to train — the between-club variation was confounding things.

Naive OLS

Negative slope

Mixes between + within variation

Fixed Effects

Positive slope

Uses only within variation

We can see the difference. The FE estimate flipped the sign because it removed the confounding between-club differences. When we compare each club only to itself, more training clearly helps.

Fixed effects controls for everything that doesn't change over time within a unit. But only that.

Controlled by FE

  • Club's city / location
  • Stadium capacity
  • Historical prestige
  • Fan base size
  • Any other time-invariant trait

NOT controlled by FE

  • New manager appointment
  • Transfer window spending
  • Player injuries (that season)
  • League rule changes
  • Any other time-varying confounder

The left column disappears through demeaning — these things are constant for each club across seasons, so they cancel out. The right column still varies over time, and fixed effects alone can't help with those.

Implication: if a time-varying confounder (like transfer spending) is correlated with both training and performance, your FE estimate will still be biased.

One remark: you cannot estimate the effect of anything that's fixed over time.

Want to know whether playing in a bigger stadium causes better results? Fixed effects can't tell you — stadium size doesn't change across seasons (for most clubs), so it gets swept away with the demeaning.

αi absorbs stadium size, city, prestige, ...⏟all time-invariant characteristics\alpha_i \text{ absorbs } \underbrace{\text{stadium size, city, prestige, ...}}_{\text{all time-invariant characteristics}}

This is the trade-off. By removing all between variation to control for unobserved confounders, we also lose the ability to study anything that lives in that between variation.

If the effect of interest is itself time-invariant (e.g., "does being a London club help?"), fixed effects is the wrong tool. You'd need a different identification strategy.

One more practical detail: standard errors matter as much as the point estimate.

With panel data, observations within the same club are likely correlated — Arsenal's points this season are related to last season's. Standard OLS standard errors ignore this and tend to be too small, making you overconfident.

The solution: cluster your standard errors at the entity level (by club). This allows for arbitrary correlation of errors within each club across time.

Without clustering, you might think your estimate is statistically significant when it isn't. This is one of the most common approach when working with panel data.

A checklist for interpreting fixed effects results:

1

Read the coefficient as a within-unit effect

"When this club trains one more session than its own average, it earns 0.15 more points than its own average."

2

Check what's controlled

All time-invariant confounders are gone. But time-varying confounders remain — list them and assess whether they're a problem.

3

Cluster your standard errors

Cluster at the entity level (club) to account for within-unit correlation over time.

4

Verify sufficient within variation

If your key variable barely changes within units, the estimate will be imprecise. Check before committing to FE.

The key takeaway:

The fixed effects coefficient tells you the effect of a within-unit change. It controls for all time-invariant confounders, but you must still worry about time-varying ones. Cluster your standard errors, and remember: FE can't estimate effects of things that don't change.

Fixed effects is one of the most widely used tools in empirical research precisely because of this powerful property: it handles unobserved heterogeneity without us needing to measure it. But interpreting the results correctly requires understanding that you're working with within variation only.

But what about shocks that hit all units at the same time? In the next chapter, we'll talk about time fixed effects and two-way fixed effects to handle exactly that.

Built with SvelteKit + D3.js