jl_script.ipynb

Kernel: Julia 1.7.3

" Homework 1 - solution " " @author: Roberto Mendoza " " @date: 12/09/2020 "

using Random
using Distributions
using DataFrames
using Statistics 
using LinearAlgebra

Random.seed!(756)

x1 = rand(500)  # uniform distribution
x2 = rand(500) # uniform distribution
x3 = rand(500) # uniform distribution
x4 = rand(500) # uniform distribution

e = randn(500) # normal dsitribution mean = 0 y sd = 1
Y = ones(500) + 0.8.*x1 + 1.2.*x2 + 0.5.*x3 + 1.5.*x4 + e
X = hcat(ones(500),x1,x2,x3,x4)

z = randn(500)

500-element Vector{Float64}:
  1.2161218700987348
 -0.9926126764602828
  0.19191592562045984
 -0.15086947966621472
 -0.6389125810461639
 -1.3942144444104163
 -0.705889215799248
  0.37351476592213184
  1.1746314413492067
 -0.9545887894895217
  0.39410363435252194
  2.7466082381229593
 -1.0373997619198814
  ⋮
  0.13025908043652165
 -0.12163806225317268
  0.43155933378570865
 -0.2831212753900742
 -0.30073565141764164
  1.935178430836171
 -0.11737482834931355
 -1.034540049333148
  0.25990723232112944
  0.5935131458905702
  0.20677712273119342
  1.0340990198372546

function ols(X, Y , standar::Bool = true, Pvalue::Bool = true , instrumento=nothing, index=nothing)
    
    if standar && Pvalue && isnothing(instrumento) && isnothing(index)
        
        beta = inv(transpose(X) * X) * (transpose(X) * Y)
        
        
        
        y_est = X*beta
        n = size(X)[1]
        k = size(X)[2]
        df = n-k
        error_vector = Y-y_est
        sigma = sum(error_vector.^(2))/df
        Var = sigma*inv(transpose(X) * X)
        sd = Var[diagind(Var)].^(0.5)
        t_test = abs.(beta./sd)
        dist = TDist(df)
        pvalue = 2*(1 .- cdf(dist, t_test)) 
        table = DataFrame(Coeff =beta, Standar_error =sd , P_value = pvalue)
        
    elseif !isnothing(instrumento) && !isnothing(index)
        
        beta = inv(transpose(X) * X) * (transpose(X) * Y)
        
        index = index +1
        Z = X
        Z[:,index] = z
        
        beta_x = inv(transpose(Z) * Z) * (transpose(Z) * X[:,index])
        x_est = Z * beta_x
        X[:,index] = x_est
        
        beta_iv = inv(transpose(X) * X) * (transpose(X) * Y)
        
        table = DataFrame(OLS= beta,  
                       OLS_IV = beta_iv)
    end
    
    return table
end

ols (generic function with 5 methods)

 ols(X,Y)

5×3 DataFrame
 Row │ Coeff      Standar_error  P_value     
     │ Float64    Float64        Float64     
─────┼───────────────────────────────────────
   1 │ 1.39969        0.14981    0.0
   2 │ 0.0235243      0.0470551  0.617347
   3 │ 1.38447        0.165582   6.66134e-16
   4 │ 0.565652       0.161536   0.000504135
   5 │ 1.33924        0.163545   2.22045e-15

 ols(X,Y,false,false, z,1)

5×2 DataFrame
 Row │ OLS        OLS_IV    
     │ Float64    Float64   
─────┼──────────────────────
   1 │ 1.39969    1.39969
   2 │ 0.0235243  0.0235243
   3 │ 1.38447    1.38447
   4 │ 0.565652   0.565652
   5 │ 1.33924    1.33924