Getting Started

Installation

To install the latest release, run:

julia> Pkg.add("Microeconometrics")

To update to the latest release, run:

julia> Pkg.update("Microeconometrics")

To install the last version, run:

julia> Pkg.add(PackageSpec(name = "Microeconometrics", rev = "master"))

To update to the last version, run:

julia> Pkg.update(PackageSpec(name = "Microeconometrics", rev = "master"))

Example I: OLS

For an introduction to the package, let's consider an example: ordinary least squares.

We first load some modules:

julia> using CSV
julia> using DataFrames
julia> using Microeconometrics

Next, we load the data:

julia> S = CSV.read("admit.csv") ; categorical!(S, :rank) ;

This sample is available here, if you wish to replicate this exercise. It comprises 400 observations and four variables (admit, gre, gpa and rank). The second command converts S[:rank] into a categorical array (a.k.a. a factor variable).

We wish to regress admit on gre, gpa, rank and an intercept. We first specify the model:

julia> M = @micromodel(response => admit, control => gre + gpa + rank + 1) ;

The syntax is due to StatsModels.jl. The @micromodel macro generalizes @formula.

We then define the correlation structure of the errors. Let's assume that observations are independent and identically distributed, so that errors are homoscedastic:

julia> C = Homoscedastic() ;

We now construct the estimation sample:

julia> D = Microdata(S, M, corr = C) ;

We can finally fit the model and visualize the results:

julia> E = fit(OLS, D)

              Estimate  St. Err.   t-stat.   p-value      C.I. (95%)  
gre             0.0004    0.0002    2.0384    0.0415    0.0000  0.0008
gpa             0.1555    0.0640    2.4317    0.0150    0.0302  0.2809
rank: 2        -0.1624    0.0677   -2.3978    0.0165   -0.2951 -0.0296
rank: 3        -0.2906    0.0702   -4.1365    0.0000   -0.4282 -0.1529
rank: 4        -0.3230    0.0793   -4.0726    0.0000   -0.4785 -0.1676
(Intercept)    -0.2589    0.2160   -1.1987    0.2306   -0.6822  0.1644

The corresponding code in Stata is:

. import delimited "admit.csv"
. regress admit gre gpa i.rank

To obtain summary statistics, we can apply the methods for regression models of StatsBase.jl:

julia> nobs(e_ols)

400

julia> r2(e_ols)

0.10040062851886422

Example II: Comparing models

To illustrate more advanced features, suppose that we want to compare specifications. As a first exercise, we wonder if we could drop the rank fixed effects.

We start with the full model. We now assume that the data are heteroscedastic. Because it is the default, we need not specify it.

julia> M₁ = @micromodel(response => admit, control => gre + gpa + rank + 1) ;
julia> D₁ = Microdata(S, M₁) ;
julia> E₁ = fit(OLS, D₁)

              Estimate  St. Err.   t-stat.   p-value      C.I. (95%)  
gre             0.0004    0.0002    2.0501    0.0404    0.0000  0.0008
gpa             0.1555    0.0653    2.3833    0.0172    0.0276  0.2834
rank: 2        -0.1624    0.0729   -2.2266    0.0260   -0.3053 -0.0194
rank: 3        -0.2906    0.0730   -3.9827    0.0001   -0.4336 -0.1476
rank: 4        -0.3230    0.0780   -4.1408    0.0000   -0.4759 -0.1701
(Intercept)    -0.2589    0.2110   -1.2268    0.2199   -0.6725  0.1547

Before we estimate the reduced model, we must first redefine the control set:

julia> M₂ = @micromodel(response => admit, :control => gre + gpa + 1) ;
julia> D₂ = Microdata(S, M₂) ;

We can now fit the reduced model:

julia> E₂ = fit(OLS, D₂)

              Estimate  St. Err.   t-stat.   p-value      C.I. (95%)  
gre             0.0005    0.0002    2.5642    0.0103    0.0001  0.0010
gpa             0.1542    0.0650    2.3737    0.0176    0.0269  0.2816
(Intercept)    -0.5279    0.2087   -2.5293    0.0114   -0.9370 -0.1188

The coefficients on gre and gpa seem to be robust. For a formal equality test, we use a Hausman test:

julia> H = hausman_test(E₁, E₂, ["gre", "gpa"]) ;
julia> tstat(H)

2-element Array{Float64,1}:
  -2.07838
  0.0749552

As it turns out, the difference between the coefficients on gre is statistically significant.

To further investigate this result, we wish to estimate separate effects by rank. The keyword subset helps us construct the appropriate Microdata:

julia> M = @micromodel(response => admit, control => gre + gpa + 1) ;

julia> I₁ = (S[:rank] .== 1) ;
julia> D₁ = Microdata(S, M, subset = I₁) ;
julia> E₁ = fit(OLS, D₁)

              Estimate  St. Err.   t-stat.   p-value      C.I. (95%)  
gre             0.0006    0.0006    1.0386    0.2990   -0.0005  0.0017
gpa             0.2508    0.1929    1.3000    0.1936   -0.1273  0.6290
(Intercept)    -0.6806    0.5662   -1.2021    0.2293   -1.7903  0.4291

julia> I₂ = (S[:rank] .== 2) ;
julia> D₂ = Microdata(S, M, subset = I₂) ;
julia> E₂ = fit(OLS, D₂)

              Estimate  St. Err.   t-stat.   p-value      C.I. (95%)  
gre             0.0004    0.0004    0.8794    0.3792   -0.0004  0.0011
gpa             0.1771    0.1028    1.7219    0.0851   -0.0245  0.3787
(Intercept)    -0.4470    0.3641   -1.2277    0.2196   -1.1607  0.2666

julia> H = chow_test(E₁, E₂, ["gre", "gpa"]) ;
julia> tstat(H)

2-element Array{Float64,1}:
  0.334261
  0.337304

We used the function chow_test because these estimates are based on different samples. The difference in the effect of gre between ranks 1 and 2 is not significant.