PS05 Q1 Rouwenhorst.jl

#PS05 Q1 Rouwenhorst; Code was adopted from Sergio and internet

using Parameters
using Random
using Distributions
using LinearAlgebra
using Statistics
using StatsBase

# Set random seed
Random.seed!(3486);

#-----------------------------------------------------------
# Define a markov process struct
    # Generate structure for markov processes 
    @with_kw struct MP
        # Model Parameters
        N::Int64 # Number of states
        grid     # Grid of discrete markov process
        Π        # Transition matrix
        PDF      # Stationary distribution
        CDF      # Stationary distribution
    end

#Q1a Simulate MC
# Define parameters
n = 10000  # Number of draws
ρ = 0.9
σ = 0.1

# Set initial value
z = zeros(n)
z[1] = 0.0

# Define distribution for error
η = Normal(0.0, σ)

# Generate AR1 using draws
for i in 2:n
    z[i] = ρ * z[i-1] + rand(η)
end

#Q1b;
#This function uses the Rouwenhorst method  
#N=5

function Rouwenhorst95_5(ρ,σ,N=5)
    # Define parameters for Rouwenhorst's approximation
        p = (1+ρ)/2
        q = p                   
            ψ = σ*sqrt((N-1)/(1-ρ^2))   # the size of the shock that corresponds to one standard deviation of the shock distribution
        s = (1-q)/(2-(p+q))     
    # Fill in transition matrix
    if N==2
        Π_z = [p 1-p ; 1-q q]
    else
        MP_aux = Rouwenhorst95_5(ρ,σ,N-1)
        o = zeros(N-1)
        Π_z = p*[MP_aux.Π o ; o' 0] + (1-p)*[o MP_aux.Π ; 0 o'] + (1-q)*[o' 0 ; MP_aux.Π o] + q*[0 o' ; o MP_aux.Π]
        # Adjust scale for double counting
        Π_z = Π_z./repeat(sum(Π_z,dims=2),1,N)
    end
    # Distribution
        PDF_z = pdf.(Binomial(N-1,1-s),(0:N-1))
        CDF_z = cumsum(PDF_z)
    # Create z grid
        z    = range(-ψ,ψ,length=N)
    # Return
        return MP(N=N,grid=z,Π=Π_z,PDF=PDF_z,CDF=CDF_z)
end


# Simulate the Markov chain
#simulates a Markov chain with Ns periods using the transition matrix Π from the MP structure. It returns a vector z_MC of simulated values of the Markov chain.

function simulation_5(Ns,MP::MP)
    # Compute conditional CDF
    Γ = cumsum(MP.Π,dims=2)
    # Allocate simulated vector
    z_ind    = zeros(Int64,Ns)
    z_MC     = zeros(Ns)
    # Starting value for simulation
    z_ind[1] = Int(ceil(length(MP.grid)/2))
    z_MC[1]  = MP.grid[z_ind[1]]
    # Simulate
    for i=2:Ns # number of simulated periods
        z_ind[i] = rand(Categorical(MP.Π[z_ind[i-1],:]))
        z_MC[i]  = MP.grid[z_ind[i]]
    end
       # Return result
    return z_MC
end

#Q1c
# Moments function for sample
function moments_5(z_MC)
    mean_MC      = mean(z_MC)
    std_MC       = std(z_MC)
    skewness_MC  = skewness(z_MC)
    kurtosis_MC  = kurtosis(z_MC)
              auto_corr_MC = cor(z_MC[1:end-1],z_MC[2:end])
              auto_corr_MC4 = zeros(4)
              for i in 2:5
                  auto_corr_MC4[i-1] = cor(z_MC[1:end-i],z_MC[1+i:end])[1]
              end
    return mean_MC, std_MC, skewness_MC, kurtosis_MC,auto_corr_MC4
end

#Report
Ns=10000
MP_T5  = Rouwenhorst95_5(ρ,σ,5)
z_MC = simulation_5(Ns, MP_T5)
mean_MC, std_MC, skewness_MC, kurtosis_MC,auto_corr_MC4 = moments_5(z_MC)


# Print the results
using DataFrames

function table_5(mean_MC, std_MC, skewness_MC, kurtosis_MC, auto_corr_MC4)
        df = DataFrame(Statistics=["Mean", "St.dev.", "Skewness", "Kurtosis", "Autocorr1", "Autocorr2", "Autocorr3", "Autocorr4"],
                    Values=[mean_MC, std_MC, skewness_MC, kurtosis_MC, auto_corr_MC4[1], auto_corr_MC4[2], auto_corr_MC4[3], auto_corr_MC4[4]])
    return df
end

df_Results_5 = table_5(mean_MC, std_MC, skewness_MC, kurtosis_MC, auto_corr_MC4)
println(df_Results_5)

using Plots
gr()

# Histograms ## There is error 
#histogram(z_MC)



#The same as above but for N=15

function Rouwenhorst95_15(ρ,σ,N=15)
    # Define parameters for Rouwenhorst's approximation
        p = (1+ρ)/2
        q = p                   
            ψ = σ*sqrt((N-1)/(1-ρ^2))   # the size of the shock that corresponds to one standard deviation of the shock distribution
        s = (1-q)/(2-(p+q))     
    # Fill in transition matrix
    if N==2
        Π_z = [p 1-p ; 1-q q]
    else
        MP_aux = Rouwenhorst95_15(ρ,σ,N-1)
        o = zeros(N-1)
        Π_z = p*[MP_aux.Π o ; o' 0] + (1-p)*[o MP_aux.Π ; 0 o'] + (1-q)*[o' 0 ; MP_aux.Π o] + q*[0 o' ; o MP_aux.Π]
        # Adjust scale for double counting
        Π_z = Π_z./repeat(sum(Π_z,dims=2),1,N)
    end
    # Distribution
        PDF_z = pdf.(Binomial(N-1,1-s),(0:N-1))
        CDF_z = cumsum(PDF_z)
    # Create z grid
        z    = range(-ψ,ψ,length=N)
    # Return
        return MP(N=N,grid=z,Π=Π_z,PDF=PDF_z,CDF=CDF_z)
end

# Simulate the Markov chain
#simulates a Markov chain with Ns periods using the transition matrix Π from the MP structure. It returns a vector z_MC of simulated values of the Markov chain.

function simulation_15(Ns,MP::MP)
    # Compute conditional CDF
    Γ = cumsum(MP.Π,dims=2)
    # Allocate simulated vector
    z_ind    = zeros(Int64,Ns)
    z_MC15     = zeros(Ns)
    # Starting value for simulation
    z_ind[1] = Int(ceil(length(MP.grid)/2))
    z_MC[1]  = MP.grid[z_ind[1]]
    # Simulate
    for i=2:Ns # number of simulated periods
        z_ind[i] = rand(Categorical(MP.Π[z_ind[i-1],:]))
        z_MC15[i]  = MP.grid[z_ind[i]]
    end
       # Return result
    return z_MC15
end

#Q1c
# Moments function for sample
function moments_15(z_MC)
    mean_MC      = mean(z_MC)
    std_MC       = std(z_MC)
    skewness_MC  = skewness(z_MC)
    kurtosis_MC  = kurtosis(z_MC)
              auto_corr_MC = cor(z_MC[1:end-1],z_MC[2:end])
              auto_corr_MC4 = zeros(4)
              for i in 2:5
                  auto_corr_MC4[i-1] = cor(z_MC[1:end-i],z_MC[1+i:end])[1]
              end
    return mean_MC, std_MC, skewness_MC, kurtosis_MC,auto_corr_MC4
end

#Report
Ns=10000
MP_T15  = Rouwenhorst95_15(ρ,σ,15)
z_MC = simulation_15(Ns, MP_T5)
mean_MC, std_MC, skewness_MC, kurtosis_MC,auto_corr_MC4 = moments_15(z_MC)


# Print the results
using DataFrames

function table_15(mean_MC, std_MC, skewness_MC, kurtosis_MC, auto_corr_MC4)
        df = DataFrame(Statistics=["Mean", "St.dev.", "Skewness", "Kurtosis", "Autocorr1", "Autocorr2", "Autocorr3", "Autocorr4"],
                    Values=[mean_MC, std_MC, skewness_MC, kurtosis_MC, auto_corr_MC4[1], auto_corr_MC4[2], auto_corr_MC4[3], auto_corr_MC4[4]])
    return df
end

df_Results_15 = table_15(mean_MC, std_MC, skewness_MC, kurtosis_MC, auto_corr_MC4)
println(df_Results_15)

#using Plots
#gr()

# Histograms ## There is error 
#histogram(z_MC)