Fixed Effects Model

In this repository, I will show some basic applications of fixed effects (FE) models to a real world data (World Development Indicators) and some post-estimation hypothesis testing in Stata. The focus will be on the use of three Stata commands:

• xtreg with the fe option;
• areg with the absorb( ) option;
• reghdfe with the absorb( ) option.

Before the start, I want to show my appreciation to Professor Jack Porter, Professor Mikkel Sølvsten, and Professor Bruce Hansen; without their lectures, I couldn't have a good understanding of fixed effects models and other related topics.

Basics of FE Models

With probability one, the fixed effects model is the most popular model in panel data methods. The simplest form of this model is $$Y_{it} = X_{it}'\beta + u_i + \varepsilon_{it}$$ where

• $Y_{it}$ is the dependent variable.
• $X_{it}$ is a $k \times 1$ vector of regressors.
• $\beta$ is the coefficient of interest.
• $u_i$ is a time-invariant unobserved missing variable, such as an individual's birth year. $u_i$ is expected to be correlated with the regressors $X_{it}$.
• $\varepsilon_{it}$ is an idiosyncratic error term. Due to the presence of $u_i$, it's impossible to identify $\beta$ under a simple regular assumption such as $E(X_{it} \varepsilon_{it}) = 0$. This is the reason why FE models were born. By applying the within transformation to the equation above, we can eliminate $u_i$ and then construct an unbiased estimator for $\beta$ by generalized least squares (GLS).

The key difference between FE models and random effects (RE) models is

• In a FE model, we don't impose any assumptions on $u_i$.
• In a RE model, we make three assumptions on $u_i$:
  • $E(u_i|\mathbf{X}_i) = 0$;
  • $E(u_i^2|\mathbf{X}_i) = \sigma_u^2$;
  • $E(u_i\varepsilon_{it}|\mathbf{X}_i)=0$.

See Chapter 17 in Bruce Hansen's Econometrics for details.

Data

The data I use are from the freely available World Development Indicators (WDI) provided by The World Bank. My data selection is:

• Country: All 217 countries. Note that the WDI also constains data for regions (such as East Asia & Pacific); if you include them in your dataset by mistake, then you have to drop them manually or with the help of some statistical softwares.
• Series: Four variables, including
  • GDP per capita (constant 2015 US$)
  • Exports of goods and services (constant 2015 US$)
  • Imports of goods and services (constant 2015 US$)
  • Labor force, total
• Time: Years from 2000 to 2021.

Note: The default format of WDI data file is not (directly) suitable for panel data analysis. I recommend to customize the layout of data format on the WDI webpage before downloading the data; otherwise, you may spend some time reshaping the dataset to a panel. A good guideline for downloading WDI is in slide 21 of Professor Oscar Torres-Reyna's "Finding Data" (here).

In the dataset cleaning and construction process,

1. Observations are dropped if information on any of the four country-level variables is missing.
2. After step 1, if a country has only one observation, then it is dropped as well.
3. I generate a new variable, named trade, by summation of export and import.
4. Finally, gdppc, labor, and trade are log transformed.

The complete coding can be found here.

Summary Statistics

After setting a panel, we can use the xtsum command to see the means, standard deviations, minima, maxima, and number of observations for the data. This command differs from the famous sum command because it by default decomposes the standard deviations, minima, and maxima into between and within components. A between is a time-average, individual spesific mean, and a within is the variable $X_{it}$ minus its time average plus its globle mean. See its documentation for details.

In addition, I create a scatter plot for seeing the heterogeneity of GDP per capita across years. As it shows, the mean value does not show an evident change in the 22-year window.

Finally, I compare the distributions of the levels and the natural logarithms of three country-level variables. From the figure, we can easily understand why log transformation is so important in some cases.

Three Kingdoms! xtreg, areg, and reghdfe!

There are four important differences among these three commands:

• Vanriance-covariance matrix estimation is different; in particular, the degree-of-freedom adjustments are different.
• $R^2$'s are calculated in different ways.
• The options for adding fixed effects are different.
• Computation speeds are different. As Sergio Correia (the author of the reghdfe package) claimed, even with only one level of fixed effects, reghdfe is faster than xtreg and areg. I first tried the reghdfe package in 2021, when I was working on my graduation dissertation for Master degree. My comment is: Good!

The complete coding for running FE models can be found here.