Fixed effect v.s. Clustered standard errors

 

	Fixed effect	Clustered standard errors
When to use it	A sample has a time-invariant feature across time🡪this means that each group are i.i.d. Your sample is randomly drawn from a larger population.	there might be some features that are shared within a group
When you have omitted variables bias?	removing unobserved heterogeneity BETWEEN different groups in your data. -->cross-sectional variation	When you face situations where observations WITHIN each group are not i.i.d. (independently and identically distributed).
Rule of thumb	Cross-sectional data	Time series or longitudinal data

For more information, please check: https://alex.miller.im/posts/when-to-use-fixed-effects-vs-clustered-standard-errors-panel-data/?fbclid=IwAR0wGdcJFNJ7bbfoiH6IhOMFNzz9Sdhh0T81xlEUR6baaMeY79DjdwoAhq4

Random effect	Predicted estimates (random effect)
	Random intercept	Random slope	Random intercept and random slope
Description	Each group has the same slope, but different starting points. Since they have different starting points, each group has different predict value (y) even with the same value of x.	Each group has different slopes, but have the same starting points. Since they have different slopes, each group has different predict value (y) even with the same value of constant (alpha).	Each group has different slopes, but have different starting points
Example	US’s GDP in 2010 is higher than Zimbabwe’s GDP in 2010. We know from the model that GDP’s effect on political trust is 0.05. (y=alpha+0.05x). Assuming both US and Zimbabwe have the same value of GDP in 2011, we get the predicted random intercept for US (130+0.05*20=131)and Zimbabwe (30+0.05*20=31)	US and UK have the same value of GDP, but the effect of GDP(x) on political trust(y) is different since they have different electoral system. Hence, 1 unit increase of GDP in US causes 0.3 percentage of decreasing political trust. But, 1 unit increase of GDP in UK causes 0.7 percentage of increasing political trust.

For more information, please check: http://www.bristol.ac.uk/cmm/learning/videos/random-slopes.html

fixed effect固定效果: 指在控制其他變數之下，重視比較個體"之間"行為的差異性。它的對應模random effects隨機效果: 指在控制其他變數之下，重視整個母體的特性，因此從中隨機選取足以代表母體的樣本並推論母體狀況。

我們通常可以透過 Hausman 檢定來justify應該要選fixed effect model或是random effect model。因為fixed effect會吃掉比較多的degree of freedom(因為我們放了很多虛擬變數)，當fixed effect和random effect所產生的係數沒有極大差異時，我們會選擇random effect，這表示omitted variables bias並不是大問題。但當這兩個模型的係數有極大差異時，我們會選擇fixed effect model。

Reference:

Clark, T. S., & Linzer, D. A. (2015). Should I use fixed or random effects?. Political science research and methods, 3(2), 399-408.

Esarey, J., & Menger, A. (2019). Practical and effective approaches to dealing with clustered data. Political Science Research and Methods, 7(3), 541-559.