Time series model update

lectures/continuous_time/covid_sde.md

@@ -604,5 +604,4 @@ In this case, there are significant differences between the early and late death
 
 This bleak simulation has assumed that no individuals has long-term immunity and that there will be no medical advancements on that time horizon - both of which are unlikely to be true.
 
-Nevertheless, it suggests that the timing of lifting lockdown has a more profound impact after 18 months if we allow stochastic shocks imperfect immunity.
-
+Nevertheless, it suggests that the timing of lifting lockdown has a more profound impact after 18 months if we allow stochastic shocks imperfect immunity.

lectures/continuous_time/seir_model.md

@@ -517,4 +517,3 @@ Despite its richness, the model above is fully deterministic.  The policy $\bar{
 One way that randomness can lead to aggregate fluctuations is the granularity that comes through the discreteness of individuals.  This topic, the connection between SDEs and the Langevin equations typically used in the approximation of chemical reactions in well-mixed media is explored in further lectures on continuous time Markov chains.
 
 Instead, in the {doc}`next lecture <covid_sde>`, we will concentrate on randomness that comes from aggregate changes in behavior or policy.
-

lectures/time_series_models/additive_functionals.md

@@ -266,7 +266,7 @@ function AMF_LSS_VAR(A, B, D, F = nothing; ν = nothing)
     # construct BIG state space representation
     lss = construct_ss(A, B, D, F, ν, nx, nk, nm)
 
-    return (A = A, B = B, D = D, F = F, ν = ν, nx = nx, nk = nk, nm = nm, lss = lss)
+    return (;A, B, D, F, ν, nx, nk, nm, lss)
 end
 
 AMF_LSS_VAR(A, B, D) =
@@ -335,14 +335,14 @@ function multiplicative_decomp(A, B, D, F, ν, nx)
 end
 
 function loglikelihood_path(amf, x, y)
-    @unpack A, B, D, F = amf
+    (;A, B, D, F) = amf
     k, T = size(y)
     FF = F * F'
     FFinv = inv(FF)
     temp = y[:, 2:end]-y[:, 1:end-1] - D*x[:, 1:end-1]
     obs =  temp .* FFinv .* temp
     obssum = cumsum(obs)
-    scalar = (logdet(FF) + k * log(2π)) * (1:T)
+    scalar = (logdet(FF) + k * log(2pi)) * (1:T)
 
     return -(obssum + scalar) / 2
 end
@@ -356,7 +356,7 @@ end
 function plot_additive(amf, T; npaths = 25, show_trend = true)
 
     # pull out right sizes so we know how to increment
-    @unpack nx, nk, nm = amf
+    (;nx, nk, nm) = amf
 
     # allocate space (nm is the number of additive functionals - we want npaths for each)
     mpath = zeros(nm*npaths, T)
@@ -428,7 +428,7 @@ end
 
 function plot_multiplicative(amf, T, npaths = 25, show_trend = true)
     # pull out right sizes so we know how to increment
-    @unpack nx, nk, nm = amf
+    (;nx, nk, nm) = amf
     # matrices for the multiplicative decomposition
     H, g, ν_tilde = multiplicative_decomp(A, B, D, F, ν, nx)
 
@@ -508,7 +508,7 @@ end
 function plot_martingales(amf, T, npaths = 25)
 
     # pull out right sizes so we know how to increment
-    @unpack A, B, D, F, ν, nx, nk, nm = amf
+    (;A, B, D, F, ν, nx, nk, nm) = amf
     # matrices for the multiplicative decomposition
     H, g, ν_tilde = multiplicative_decomp(A, B, D, F, ν, nx)
 
@@ -571,7 +571,7 @@ function plot_given_paths(T, ypath, mpath, spath, tpath, mbounds, sbounds;
     trange = 1:T
 
     # allocate transpose
-    mpathᵀ = Matrix(mpath')
+    mpath_T = Matrix(mpath')
 
     # create figure
     plots=plot(layout = (2, 2), size = (800, 800))
@@ -589,7 +589,7 @@ function plot_given_paths(T, ypath, mpath, spath, tpath, mbounds, sbounds;
 
     # plot martingale component
     plot!(plots[2], trange, mpath[1, :], color = :magenta, label = "")
-    plot!(plots[2], trange, mpathᵀ, alpha = 0.45, color = :magenta, label = "")
+    plot!(plots[2], trange, mpath_T, alpha = 0.45, color = :magenta, label = "")
     ub = mbounds[2, :]
     lb = mbounds[1, :]
     plot!(plots[2], ub, fillrange = lb, alpha = 0.25, color = :magenta, label = "")
@@ -645,18 +645,18 @@ Random.seed!(42);
 ```
 
 ```{code-cell} julia
-ϕ_1, ϕ_2, ϕ_3, ϕ_4 = 0.5, -0.2, 0, 0.5
-σ = 0.01
+phi_1, phi_2, phi_3, phi_4 = 0.5, -0.2, 0, 0.5
+sigma = 0.01
 ν = 0.01 # growth rate
 
 ## A matrix should be n x n
-A = [ϕ_1 ϕ_2 ϕ_3 ϕ_4;
+A = [phi_1 phi_2 phi_3 phi_4;
        1   0   0   0;
        0   1   0   0;
        0   0   1   0]
 
 # B matrix should be n x k
-B = [σ, 0, 0, 0]
+B = [sigma, 0, 0, 0]
 
 D = [1 0 0 0] * A
 F = [1, 0, 0, 0] ⋅ vec(B)

lectures/time_series_models/arma.md

@@ -237,10 +237,10 @@ plt_1=plot()
 plt_2=plot()
 plots = [plt_1, plt_2]
 
-for (i, ϕ) in enumerate((0.8, -0.8))
+for (i, phi) in enumerate((0.8, -0.8))
     times = 0:16
-    acov = [ϕ.^k ./ (1 - ϕ.^2) for k in times]
-    label = L"autocovariance, $\phi = %$ϕ$"
+    acov = [phi.^k ./ (1 - phi.^2) for k in times]
+    label = L"autocovariance, $\phi = %$phi$"
     plot!(plots[i], times, acov, color=:blue, lw=2, marker=:circle, markersize=3,
           alpha=0.6, label=label)
     plot!(plots[i], legend=:topright, xlabel="time", xlim=(0,15))
@@ -254,9 +254,9 @@ plot(plots[1], plots[2], layout=(2,1), size=(700,500))
 tags: [remove-cell]
 ---
 @testset begin
-  ϕ = 0.8
+  phi = 0.8
   times = 0:16
-  acov = [ϕ.^k ./ (1 - ϕ.^2) for k in times]
+  acov = [phi.^k ./ (1 - phi.^2) for k in times]
   @test acov[4] ≈ 1.422222222222223 # Can't access acov directly because of scoping.
   @test acov[1] ≈ 2.7777777777777786
 end
@@ -480,19 +480,19 @@ It is a nice exercise to verify that {eq}`ma1_sd_ed` and {eq}`ar1_sd_ed` are ind
 Plotting {eq}`ar1_sd_ed` reveals the shape of the spectral density for the AR(1) model when $\phi$ takes the values 0.8 and -0.8 respectively
 
 ```{code-cell} julia
-ar1_sd(ϕ, ω) = 1 ./ (1 .- 2 * ϕ * cos.(ω) .+ ϕ.^2)
+ar1_sd(phi, omega) = 1 ./ (1 .- 2 * phi * cos.(omega) .+ phi.^2)
 
-ω_s = range(0, π, length = 180)
+omega_s = range(0, pi, length = 180)
 
 plt_1=plot()
 plt_2=plot()
 plots=[plt_1, plt_2]
 
-for (i, ϕ) in enumerate((0.8, -0.8))
-    sd = ar1_sd(ϕ, ω_s)
-    label = L"spectral density, $\phi = %$ϕ$"
-    plot!(plots[i], ω_s, sd, color=:blue, alpha=0.6, lw=2, label=label)
-    plot!(plots[i], legend=:top, xlabel="frequency", xlim=(0,π))
+for (i, phi) in enumerate((0.8, -0.8))
+    sd = ar1_sd(phi, omega_s)
+    label = L"spectral density, $\phi = %$phi$"
+    plot!(plots[i], omega_s, sd, color=:blue, alpha=0.6, lw=2, label=label)
+    plot!(plots[i], legend=:top, xlabel="frequency", xlim=(0,pi))
 end
 plot(plots[1], plots[2], layout=(2,1), size=(700,500))
 ```
@@ -502,9 +502,9 @@ plot(plots[1], plots[2], layout=(2,1), size=(700,500))
 tags: [remove-cell]
 ---
 @testset begin
-  @test ar1_sd(0.8, ω_s)[18] ≈ 9.034248169239635
-  @test ar1_sd(-0.8, ω_s)[18] ≈ 0.3155260821833043
-  @test ω_s[1] == 0.0 && ω_s[end] ≈ π && length(ω_s) == 180 # Grid invariant.
+  @test ar1_sd(0.8, omega_s)[18] ≈ 9.034248169239635
+  @test ar1_sd(-0.8, omega_s)[18] ≈ 0.3155260821833043
+  @test omega_s[1] == 0.0 && omega_s[end] ≈ pi && length(omega_s) == 180 # Grid invariant.
 end
 ```
 
@@ -536,20 +536,20 @@ products on the right-hand side of {eq}`sumpr` is large.
 These ideas are illustrated in the next figure, which has $k$ on the horizontal axis
 
 ```{code-cell} julia
-ϕ = -0.8
+phi = -0.8
 times = 0:16
-y1 = [ϕ.^k ./ (1 - ϕ.^2) for k in times]
-y2 = [cos.(π * k) for k in times]
+y1 = [phi.^k ./ (1 - phi.^2) for k in times]
+y2 = [cos.(pi * k) for k in times]
 y3 = [a * b for (a, b) in zip(y1, y2)]
 
-# Autocovariance when ϕ = -0.8
+# Autocovariance when phi = -0.8
 plt_1 = plot(times, y1, color=:blue, lw=2, marker=:circle, markersize=3,
              alpha=0.6, label=L"\gamma(k)")
 plot!(plt_1, seriestype=:hline, [0], linestyle=:dash, alpha=0.5,
       lw=2, label="")
 plot!(plt_1, legend=:topright, xlim=(0,15), yticks=[-2, 0, 2])
 
-# Cycles at frequence π
+# Cycles at frequence pi
 plt_2 = plot(times, y2, color=:blue, lw=2, marker=:circle, markersize=3,
              alpha=0.6, label=L"cos(\pi k)")
 plot!(plt_2, seriestype=:hline, [0], linestyle=:dash, alpha=0.5,
@@ -583,20 +583,20 @@ not matched, the sequence $\gamma(k) \cos(\omega k)$ contains
 both positive and negative terms, and hence the sum of these terms is much smaller
 
 ```{code-cell} julia
-ϕ = -0.8
+phi = -0.8
 times = 0:16
-y1 = [ϕ.^k ./ (1 - ϕ.^2) for k in times]
-y2 = [cos.(π * k/3) for k in times]
+y1 = [phi.^k ./ (1 - phi.^2) for k in times]
+y2 = [cos.(pi * k/3) for k in times]
 y3 = [a * b for (a, b) in zip(y1, y2)]
 
-# Autocovariance when ϕ = -0.8
+# Autocovariance when phi = -0.8
 plt_1 = plot(times, y1, color=:blue, lw=2, marker=:circle, markersize=3,
              alpha=0.6, label=L"\gamma(k)")
 plot!(plt_1, seriestype=:hline, [0], linestyle=:dash, alpha=0.5,
       lw=2, label="")
 plot!(plt_1, legend=:topright, xlim=(0,15), yticks=[-2, 0, 2])
 
-# Cycles at frequence π
+# Cycles at frequence pi
 plt_2 = plot(times, y2, color=:blue, lw=2, marker=:circle, markersize=3,
              alpha=0.6, label=L"cos(\pi k/3)")
 plot!(plt_2, seriestype=:hline, [0], linestyle=:dash, alpha=0.5,
@@ -755,7 +755,7 @@ using QuantEcon, Random
 function plot_spectral_density(arma, plt)
     (w, spect) = spectral_density(arma, two_pi=false)
     plot!(plt, w, spect, lw=2, alpha=0.7,label="")
-    plot!(plt, title="Spectral density", xlim=(0,π),
+    plot!(plt, title="Spectral density", xlim=(0,pi),
           xlabel="frequency", ylabel="spectrum", yscale=:log)
     return plt
 end
@@ -839,9 +839,9 @@ We'll use the model $X_t = 0.5 X_{t-1} + \epsilon_t - 0.8 \epsilon_{t-2}$
 
 ```{code-cell} julia
 Random.seed!(42) # For reproducible results.
-ϕ = 0.5;
-θ = [0, -0.8];
-arma = ARMA(ϕ, θ, 1.0)
+phi = 0.5;
+theta = [0, -0.8];
+arma = ARMA(phi, theta, 1.0)
 quad_plot(arma)
 ```
 
@@ -864,7 +864,7 @@ end
 The call
 
 ```{code-block} julia
-arma = ARMA(ϕ, θ, σ)
+arma = ARMA(phi, theta, sigma)
 ```
 
 creates an instance `arma` that represents the ARMA($p, q$) model
@@ -874,15 +874,15 @@ X_t = \phi_1 X_{t-1} + ... + \phi_p X_{t-p} +
     \epsilon_t + \theta_1 \epsilon_{t-1} + ... + \theta_q \epsilon_{t-q}
 $$
 
-If `ϕ` and `θ` are arrays or sequences, then the interpretation will
+If `phi` and `theta` are arrays or sequences, then the interpretation will
 be
 
-* `ϕ` holds the vector of parameters $(\phi_1, \phi_2,..., \phi_p)$
-* `θ` holds the vector of parameters $(\theta_1, \theta_2,..., \theta_q)$
+* `phi` holds the vector of parameters $(\phi_1, \phi_2,..., \phi_p)$
+* `theta` holds the vector of parameters $(\theta_1, \theta_2,..., \theta_q)$
 
-The parameter `σ` is always a scalar, the standard deviation of the white noise.
+The parameter `sigma` is always a scalar, the standard deviation of the white noise.
 
-We also permit `ϕ` and `θ` to be scalars, in which case the model will be interpreted as
+We also permit `phi` and `theta` to be scalars, in which case the model will be interpreted as
 
 $$
 X_t = \phi X_{t-1} + \epsilon_t + \theta \epsilon_{t-1}