# Fixed Effects Estimation and Inconsistency

Consider estimation of the following population regression function: $$G{Y_{it}} = {\beta _0} + {\beta _1}IN{F_{it}} + {\beta _2}DE{M_i} + {\beta _3}POI{L_t} + {\varepsilon _{it}}$$

Where: $$GY_{it}$$ = percentage growth in real GDP per capita in country $$i$$ in year $$t$$

$$INF_{it}$$ = percentage rate of inflation in country $$i$$ in year $$t$$

$$DEM_i$$ = 1 if country $$i$$ is a democracy, 0 otherwise

$$POIL_t$$ = a price index of crude oil on the world market.

If you were to explore estimation with fixed effects both for countries and for years of observation. Which coefficients in the above model could be estimated? Suppose that INF and POIL are correlated over time, and INF is correlated with DEM across countries. Do the omitted variables in the fixed effects estimate result in inconsistent estimates? Why?

My guess is that the estimates are biased and consistent but I haven't been able to figure out why.

• Omitted variable bias doesn't lead to inconsistent estimates in this case, because we are separating out the effects of the omitted variable with respect to time and across countries. If that is the case then suppose that the omitted variable is only correlated across time. Is fixed effects inappropriate in this case? And how should the regression be estimated? – Jaffar Jan 2 at 4:14

## 1 Answer

• You can't identify the effect of oil price when Year FE are applied, since the world oil price is perfectly correlated with year Fixed Effects.
• You can't identify the democracy indicator if you country does not change its value in your observed period.

For the other variables, it should be possible to obtain estimates. You should apply a fairly large sample to avoid collinearity issues (sth like 100 countries, 50 years). Further problems with your model are reverse causality, i.e. does growth cause inflation?

• Yes but multicollinearity doesn't lead to inconsistent estimates, rather the standard deviations of the coefficients increase. Does omitted variable bias lead to consistent estimates in this case, because we are separating out the effects of the omitted variable with respect to time and across countries? If that is the case then suppose that the omitted variable is only correlated across time. Is fixed effects inappropriate in this case? And how should the regression be estimated? – Jaffar Jan 2 at 4:14
• It depends on which omitted variables you have in mind. This kind of macro-level regressions are rarely done nowadays because they are prone to a zillion of endogeneity problems and you will have a very hard time convincing anyone that you show some sort of causality here. In growth regressions, one usually adds growth from the period before as a control. But these dynamic models come with problems of their own. My personal experience is that these kind of models are incredibly sensitive to changing the time period or the number of countries. – E. Sommer Jan 2 at 17:59