LinearHypothesisTest

Perform Linear Hypothesis (Linear Restriction) Tests. Currently allows to test hypothesis of the following form for GLM and FixedEffectModels models:

• $ \beta_1 = \beta_2$
• $ \beta_2 + 3 \times \beta_2 =5$
• $ 0.5 \times \beta_1 + \beta_2 + \beta_3 = \beta_4$
• $ 0.5 \times \beta_1 + \beta_2 + \beta_3 = 0$
• $ (0.5 \times \beta_1 + \beta_2 + \beta_3)/3 = 0$
• $ (0.5 \times \beta_1 + \beta_2 - \beta_3) \times 1.5 = 0$

The function also allows for checking multiple linear restrictions at once:

• $ \beta_1 = \beta_2, \beta_3=0$

The LinearHypothesisTests function either computes a finite-sample F statistic or an asymptotic Chi-squared statistic for carrying out a Wald-test-based comparison between a model and a linearly restricted model.

Package Installation

The package is NOT registered with the General Registry but you can install the package either by using the Pkg REPL mode (press ] to enter):

pkg> add http://github.com/eohne/LinearHypothesisTest.jl

or by using Pkg functions

julia> using Pkg; Pkg.add("http://github.com/eohne/LinearHypothesisTest.jl")

Rules of defining hypothesis:

1. The first element in the hypothesis should not be negative. Instead of "-coef1 + coef2 =0" write "coef2 - coef1 = 0". Instead of " -coef1 -coef2 =0" do the equivalent test of "coef1 + coef2 =0"

2. Use fractions instead of divisions except if the entire left-hand side is divided. Instead of "coef1 /2 + coef2 = 0" write "0.5 * coef1 + coef2=0". If you want to test "1/3 * coef1 + coef2 =0" you can do this "$(1/3) * coef1 + coef2 =0". Dividing the entire lhs is ok (e.g. "(coef1 + coef2)/2 =0")

Example

The below regression etc are just toy examples to show how the functions work.

Load some required packages and data:

using GLM, FixedEffectModels, CSV, DataFrames
using LinearHypothesisTest
data = CSV.File("test/lalonde.csv") |> DataFrame;

m_lm =  lm(@formula(treat ~ age + educ + race + married + nodegree ),data);
m_fe = reg(data,@formula(treat ~ age + educ + race + married + nodegree  + fe(re74)),Vcov.cluster(:Column1));

Example Linear Model (GLM) m_lm

julia> m_lm
StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}, Matrix{Float64}}

treat ~ 1 + age + educ + race + married + nodegree

Coefficients:
─────────────────────────────────────────────────────────────────────────────────
                    Coef.  Std. Error       t  Pr(>|t|)    Lower 95%    Upper 95%
─────────────────────────────────────────────────────────────────────────────────
(Intercept)    0.371995    0.126153      2.95    0.0033   0.124247     0.619744
age            0.00135367  0.00164096    0.82    0.4097  -0.00186898   0.00457632
educ           0.0183177   0.00818698    2.24    0.0256   0.00223943   0.0343959
race: hispan  -0.449351    0.0497604    -9.03    <1e-17  -0.547075    -0.351628
race: white   -0.539054    0.0334271   -16.13    <1e-48  -0.604701    -0.473407
married       -0.109849    0.0336998    -3.26    0.0012  -0.176031    -0.0436663
nodegree       0.103103    0.0439286     2.35    0.0192   0.0168322    0.189373
─────────────────────────────────────────────────────────────────────────────────

Let's test if the coefficent of age + 2*educ = married:

julia> LinearHypothesisTests(m_lm, "age + 2*educ = married")

F-Test
==============================
Number of Restrictions: 1
Value:                  3.534
5.0% Crit Value:        3.857
P-Value:                6.06%
St Error:       [0.038]
==============================

Let's test multiple restrictions:

julia> LinearHypothesisTests(m_lm, ["age =educ","married =0 "])

F-Test
==============================
Number of Restrictions: 2
Value:                  8.724
5.0% Crit Value:        3.011
P-Value:                0.02%
St Error:       [0.008, 0.034]
==============================

Example Fixed Effects (FixedEffectModels.jl) m_fe:

julia> m_fe
                                 FixedEffectModel
==================================================================================
Number of obs:                       266  Converged:                          true
dof (model):                           6  dof (residuals):                     265
R²:                                0.359  R² adjusted:                       0.320
F-statistic:                     19.0944  P-value:                           0.000
R² within:                         0.294  Iterations:                            1
==================================================================================
                 Estimate  Std. Error    t-stat  Pr(>|t|)    Lower 95%   Upper 95%
──────────────────────────────────────────────────────────────────────────────────
age            0.00445118  0.00324399   1.37213    0.1712  -0.00193609   0.0108385
educ           0.0220794   0.0137164    1.60971    0.1087  -0.00492753   0.0490864
race: hispan  -0.423725    0.100488    -4.21668    <1e-04  -0.621581    -0.225869
race: white   -0.531621    0.0599516   -8.86749    <1e-15  -0.649663    -0.413578
married       -0.130539    0.0730434   -1.78715    0.0751  -0.274359     0.0132798
nodegree       0.0991155   0.0807286    1.22776    0.2206  -0.0598355    0.258066
==================================================================================

julia> LinearHypothesisTests(m_fe, "age + educ + race: hispan = married")

F-Test
==============================
Number of Restrictions: 1
Value:                  3.611
5.0% Crit Value:        3.877
P-Value:                5.85%
St Error:       [0.14]
==============================