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Stability in the SSY Model





In [2]:



    


include("src/utilities.jl")
include("src/ez_model.jl")
include("src/ssy_discretized.jl")



    


















    






WARNING: replacing docs for 'compute_spec_rad :: Tuple{Array{T,2} where T}' in module 'Main'.
WARNING: replacing docs for 'AR1 :: Union{}' in module 'Main'.
WARNING: replacing docs for 'AR1 :: Tuple{}' in module 'Main'.
WARNING: replacing docs for 'ar1_to_mc :: Tuple{AR1,Integer}' in module 'Main'.
WARNING: replacing docs for 'EpsteinZin :: Union{}' in module 'Main'.
WARNING: replacing docs for 'EpsteinZin :: Tuple{Any,Any,Any}' in module 'Main'.
WARNING: replacing docs for 'EpsteinZinBY :: Tuple{}' in module 'Main'.
WARNING: replacing docs for 'EpsteinZinBKY :: Tuple{}' in module 'Main'.
WARNING: replacing docs for 'EpsteinZinSSY :: Tuple{}' in module 'Main'.










    
Out[2]:








compute_fp_ssy

















In [3]:



    


using PyPlot
plt = PyPlot



    


















    
Out[3]:








PyPlot

Spectral Radius Plot, SSY Parameters





In [6]:



    


J = 20 # grid size
R = Array{Float64}(J, J);



    











In [7]:



    


x_vals = linspace(1.25, 2.25, J)          # ψ
y_vals = linspace(0.0005, 0.008, J)        # μ`



    


















    
Out[7]:








0.0005:0.00039473684210526315:0.008

















In [8]:



    


for (i, x) in enumerate(x_vals)
    for (j, y) in enumerate(y_vals)
        ez = EpsteinZinSSY(ψ=x)
        @assert ez.θ < 0 "Detected non-negative theta value"
        K = compute_K_ssy(ez, ssy, μ=y)
        R[i, j] = compute_spec_rad(K)
    end
end



    











In [9]:



    


ez = EpsteinZinSSY() # Get a fresh copy
μ=0.0016



    


















    
Out[9]:






0.0016
















In [10]:



    


ez.ψ



    


















    
Out[10]:






1.93
















In [11]:



    


fig, ax = plt.subplots(figsize=(10, 5.7))

cs1 = ax[:contourf](x_vals, 
                    y_vals, 
                    R',
                    alpha=0.6)
                    #levels=lvs,


ctr1 = ax[:contour](x_vals, 
                    y_vals, 
                    R', 
                    levels=[1.0])

plt.clabel(ctr1, inline=1, fontsize=13)
plt.colorbar(cs1, ax=ax)

ax[:set_title]("Spectral radius")
ax[:set_xlabel]("ψ", fontsize=16)
ax[:set_ylabel]("μ", fontsize=16)


ax[:annotate]("Schorfheide, Song and Yaron ", 
         xy=(ez.ψ - 0.024, μ - 0.0001),  
         xycoords="data",
         xytext=(-225, 30),
         textcoords="offset points",
         fontsize=12,
         arrowprops=Dict("arrowstyle" => "->"))

ax[:plot]([ez.ψ], [μ],  "ko", alpha=0.6)

plt.savefig("ssy_pm.pdf")
plt.show()

Let's try varying gamma





In [12]:



    


x_vals = linspace(4.0, 12.0, J)       # γ
y_vals = linspace(0.0005, 0.006, J)      # μ



    


















    
Out[12]:








0.0005:0.00028947368421052634:0.006

















In [13]:



    


for (i, x) in enumerate(x_vals)
    for (j, y) in enumerate(y_vals)
        ez = EpsteinZinSSY(γ=x)
        @assert ez.θ < 0 "Detected non-negative theta value"
        K = compute_K_ssy(ez, ssy, μ=y)
        R[i, j] = compute_spec_rad(K)
    end
end



    











In [14]:



    


ez = EpsteinZinSSY() # Get a fresh copy
μ=0.0016



    


















    
Out[14]:






0.0016
















In [15]:



    


ez.γ



    


















    
Out[15]:






8.6
















In [16]:



    


fig, ax = plt.subplots(figsize=(10, 5.7))

#lvs = [0.0, 0.8, 1.0, 1.4, 1.8, 2.2, 4.4]
#cls = [cm.jet(i) for i in np.linspace(0.4, 1, len(lvs))]

cs1 = ax[:contourf](x_vals, 
                    y_vals, 
                    R',
                    alpha=0.6)
                    #levels=lvs,


ctr1 = ax[:contour](x_vals, 
                    y_vals, 
                    R', 
                    levels=[1.0])

plt.clabel(ctr1, inline=1, fontsize=13)
plt.colorbar(cs1, ax=ax)

ax[:set_title]("Spectral radius")
ax[:set_xlabel]("γ", fontsize=16)
ax[:set_ylabel]("μ", fontsize=16)


ax[:annotate]("Schorfheide, Song and Yaron ", 
         xy=(ez.γ - 0.084, μ - 0.0001),  
         xycoords="data",
         xytext=(-225, -30),
         textcoords="offset points",
         fontsize=12,
         arrowprops=Dict("arrowstyle" => "->"))

ax[:plot]([ez.γ], [μ],  "ko", alpha=0.6)

plt.savefig("ssy_gm.pdf")
plt.show()

And now mu and volatility





In [200]:



    


x_vals = linspace(0.0015, 0.004, J)          # sigma_bar
y_vals = linspace(0.0005, 0.005, J)         # μ



    


















    
Out[200]:








0.0005:0.00023684210526315788:0.005

















In [201]:



    


ssy = SSY_default()



    


















    
Out[201]:








SSY{Float64}(0.987, 0.232, 0.0032, 1.0, 0.994, 0.06244997998398398, 0.991, 0.0938083151964686)

















In [202]:



    


for (i, x) in enumerate(x_vals)
    for (j, y) in enumerate(y_vals)
        ez = EpsteinZinSSY()
        ssy.σ_bar = x
        @assert ez.θ < 0 "Detected non-negative theta value"
        K = compute_K_ssy(ez, ssy, μ=y)
        R[i, j] = compute_spec_rad(K)
    end
end



    











In [203]:



    


ez = EpsteinZinSSY() # Get a fresh copy
μ=0.0016
ssy = SSY_default()



    


















    
Out[203]:








SSY{Float64}(0.987, 0.232, 0.0032, 1.0, 0.994, 0.06244997998398398, 0.991, 0.0938083151964686)

















In [204]:



    


ssy.σ_bar



    


















    
Out[204]:






0.0032
















In [205]:



    


fig, ax = plt.subplots(figsize=(10, 5.7))

#lvs = [0.0, 0.8, 1.0, 1.4, 1.8, 2.2, 4.4]
#cls = [cm.jet(i) for i in np.linspace(0.4, 1, len(lvs))]

cs1 = ax[:contourf](x_vals, 
                    y_vals, 
                    R',
                    alpha=0.6)
                    #levels=lvs,


ctr1 = ax[:contour](x_vals, 
                    y_vals, 
                    R', 
                    levels=[1.0])

plt.clabel(ctr1, inline=1, fontsize=13)
plt.colorbar(cs1, ax=ax)

ax[:set_title]("Spectral radius")
ax[:set_xlabel]("σ_bar", fontsize=16)
ax[:set_ylabel]("μ", fontsize=16)

ax[:annotate]("Schorfheide, Song and Yaron ", 
         xy=(ssy.σ_bar - 0.000084, μ - 0.0001),  
         xycoords="data",
         xytext=(-175, -40),
         textcoords="offset points",
         fontsize=12,
         arrowprops=Dict("arrowstyle" => "->"))

ax[:plot]([ssy.σ_bar], [μ],  "ko", alpha=0.6)

plt.savefig("ssy_sbm.pdf")
plt.show()

rho and rho_hz





In [5]:



    


J = 20 # grid size
R = Array{Float64}(J, J);

x_vals = linspace(0.985, 0.991, J)           # ρ
y_vals = linspace(0.99, 0.996, J)           # ρ_hz

for (i, x) in enumerate(x_vals)
    for (j, y) in enumerate(y_vals)
        ssy.ρ = x
        ssy.ρ_hz = y
        @assert ez.θ < 0 "Detected non-negative theta value"
        K = compute_K_ssy(ez, ssy)
        R[i, j] = compute_spec_rad(K)
    end
end

fig, ax = plt.subplots(figsize=(10, 5.7))

#lvs = [0.0, 0.8, 1.0, 1.4, 1.8, 2.2, 4.4]
#cls = [cm.jet(i) for i in np.linspace(0.4, 1, len(lvs))]

cs1 = ax[:contourf](x_vals, 
                    y_vals, 
                    R',
                    alpha=0.6)
                    #levels=lvs,


ctr1 = ax[:contour](x_vals, 
                    y_vals, 
                    R', 
                    levels=[1.0])

plt.clabel(ctr1, inline=1, fontsize=13)
plt.colorbar(cs1, ax=ax)

ax[:set_title]("Spectral radius")
ax[:set_xlabel]("ρ", fontsize=16)
ax[:set_ylabel]("ρ_hz", fontsize=16)

plt.show()



    


















    
























In [ ]:

Content source: jstac/recursive_utility_code

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