Smooth local projections (SLP)

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Implementation of Smooth Local Projections (SLP) based on Barnichon and Brownlees (2019) - "Impulse Response Estimation by Smooth Local Projections." Original method of Local Projections is first introduced by Òscar Jordà (2005)

The code was translated from MATLAB code published from the replication file made available by C. Brownlees.

import Pkg;
Pkg.update();

using CSV, Statistics, LinearAlgebra: I;
cd(@__DIR__) #src location

include("functions.jl")

Input argument is constructed using a mutable object of following,

df = CSV.read("data.csv", DataFrame);
df = convert(Array{Float16}, df[:, 2:4]);
# Or
#df = Matrix(df[:, 2:4]);
T, k = size(df);

## Parameterization
P = 4; # Number of lags used in LP - controlled variable

# start LP at H_min=0 or 1 (H_min=1 if impose no contemporanous impact)
H_max = 20;
ind_response = 1; # Endogenous variable - response
ind_shock    = 3; # Endogenous variable related to the shock

# Packaing everything as input
r = 2; #(r-1)=order of the limit polynomial
# NOTE: (so r=2 implies the IR is shrunk towards a line )
λ = 100; 

indx     = [ind_response ind_shock H_max P];

inicjał₀ = initalz(df, indx, "reg", [r 0]);
inicjał₁ = initalz(df, indx, "smooth", [r λ]);
# Obliczanie projekcja lokalna, in order of,
#  (1) Local projection a la Jordá (2005)
#  (2) Local projection with basic parameteri tested
#  (3) Local projection with optimal parameter validated
lp₀, lp₁ = slp(inicjał₀), slp(inicjał₁);

# Cross-valudation for value of optimal value of λ → λₒ
λₒ, resvec = slpᵥ(lp₁);

# Optimal parameter
inicjał₂ = initalz(df, indx, "smooth", [r λₒ]);
lp₂ = slp(inicjał₂);
lp₂_ci = slp_ci(lp₂);

Each stripped sub/co-routine is a plot of impulse response functions, (1 - blue line) local projection using Òscar Jordà (2005); (2 - purple line) smoothed local projection using (λ = 100, for example presented below); (3 - red lines) smoothed local projection using optimal λ estimated using cross-validation method as shown in Barnichon and Brownlees (2019) (red solid line is a point estimate, dashed red lines are estimated 90% confidence set).

Following example is a replication of Barnichon and Brownlees (2019) estimation of the impulse response of Gross Domestic Product (GDP) to identified positive monetary policy shock.

# rezultat 
plot(lp₀.IR, xlabel = "Time since stimulus/impact", ylabel = "Response", label = "Jordà (2005)")
plot!(lp₁.IR, label = "SLP: λ = $(λ)", color = "purple")
plot!(zeros(length(lp₁.IR)), label = false, color = "black", line = :dot)
plot!(lp₂.IR, label = "SLP: λ optymalny = $(λₒ)", color = "red")
plot!(lp₂_ci, line = :dash, color = "red", legend = false)

# fantazyjny 
function plt_anime(ir₁, ir₂, iter)
    plot!(ir₁.IR[1:iter], label = "SLP: λ = $(λ)", color = "purple")
    plot!(ir₂.IR[1:iter], label = "SLP: λ optymalny = $(λₒ)", color = "red")
end

plot(lp₀.IR, xlabel = "Time since stimulus/impact", ylabel = "Response", label = "Jordà (2005)")
plot!(zeros(length(lp₁.IR)), label = false, color = "black", line = :dot)
plot!(lp₂_ci, line = :dash, color = "red", legend = false)
fantazyjny = @animate for iter ∈ 1:length(lp₁.IR)
    plt_anime(lp₁, lp₂, iter)
end

gif(fantazyjny, "example.gif", fps = 5)

To estimate the optimal parameter,

# Optimal parameter evaluation
# walidacja krzyżowa dla optymalny
plt = plot(resvec[:,1], resvec[:,2],
    title = "Cross-validation for optimal λ",
    xlabel = "λ",
    ylabel = "MSE",
    label = "oszacowanie", dpi=500)
param_min = minimum(resvec[:,2]) 
param_bound = minimum(resvec[:,2]) + 
  ((maximum(resvec[:,2])-minimum(resvec[:,2]))/2)
plot!((λₒ .* ones(2, 1)),
      [param_min param_bound]',
      label = "optymalny"
      )
savefig("param_optimal.png")