API documentation

@kwdef struct Forecast <: PosteriorSample

It contains a result of the scenario analysis, the conditional prediction for yields, factors = [PCs macros], and term premiums.

• yields
• factors
• TP: term premium forecasts

@kwdef struct Hyperparameter

• p::Int
• q::Matrix
• nu0
• Omega0::Vector
• mean_phi_const::Vector = zeros(length(Omega0)): It is a prior mean of a constant term in our VAR.

@kwdef struct LatentSpace <: PosteriorSample

When the model goes to the JSZ latent factor space, the statistical parameters in struct Parameter are also transformed. This struct contains the transformed parameters. Specifically, the transformation is latents[t,:] = T0P_ + inv(T1X)*PCs[t,:].

In the latent factor space, the transition equation is data[t,:] = KPXF + GPXFXF*vec(data[t-1:-1:t-p,:]') + MvNormal(O,OmegaXFXF), where data = [latent macros].

• latents::Matrix
• kappaQ
• kQ_infty
• KPXF::Vector
• GPXFXF::Matrix
• OmegaXFXF::Matrix

@kwdef struct Parameter <: PosteriorSample

It contains statistical parameters of the model that are sampled from function posterior_sampler.

• kappaQ
• kQ_infty::Float64
• phi::Matrix{Float64}
• varFF::Vector{Float64}
• SigmaO::Vector{Float64}
• gamma::Vector{Float64}

abstract type PosteriorSample

It is a super-set of structs Parameter, ReducedForm, LatentSpace, YieldCurve, TermPremium, Forecast.

@kwdef struct ReducedForm <: PosteriorSample

It contains statistical parameters in terms of the reduced form VAR(p) in P-dynamics. lambdaP and LambdaPF are parameters in the market prices of risks equation, and they only contain the first dQ non-zero equations.

• kappaQ
• kQ_infty
• KPF
• GPFF
• OmegaFF::Matrix
• SigmaO::Vector
• lambdaP
• LambdaPF
• mpr::Matrix(market prices of risks, T, dP)

@kwdef struct Scenario

It contains scenarios to be conditioned in the scenario analysis. When y = [yields; macros] is a observed vector in our measurement equation, Scenario.combinations*y = Scenario.values constitutes the scenario at a specific time. Vector{Scenario} is used to describe a time-series of scenarios.

combinations and values should be Matrix and Vector. If values is a scalar, combinations would be a matrix with one raw vector and values should be one-dimensional vector, for example [values].

• combinations::Matrix
• values::Vector

@kwdef struct TermPremium <: PosteriorSample

The yields are decomposed into the term premium(TP) and the expectation hypothesis component(EH). Each components have constant terms(const_TP and const_EH) and time-varying components(timevarying_TP and timevarying_EH). factorloading_EH and factorloading_TP are coefficients of the pricing factors for the time varying components. Each column of the outputs indicates the results for each maturity.

We do not store time-varying components in TermPremium, and the time-varying components are the separate outputs in function term_premium.

• TP
• EH
• factorloading_TP
• factorloading_EH
• const_TP
• const_EH

@kwdef struct YieldCurve <: PosteriorSample

It contains a fitted yield curve. yields[t,:] = intercept + slope*latents[t,:] holds.

• latents::Matrix: latent pricing factors in LatentSpace
• yields
• intercept
• slope

getindex(x::PosteriorSample, c::Symbol)

For struct <: PosteriorSample, struct[:name] calls objects in struct.

getindex(x::Vector{<:PosteriorSample}, c::Symbol)

For struct <: PosteriorSample, struct[:name] calls objects in struct. Output[i] = $i'$th posterior sample

mean(x::Vector{<:PosteriorSample})

Output[:variable name] returns the corresponding posterior mean.

median(x::Vector{<:PosteriorSample})

Output[:variable name] returns the corresponding posterior median.

quantile(x::Vector{<:PosteriorSample}, q)

Output[:variable name] returns a quantile of the corresponding posterior distribution.

std(x::Vector{<:PosteriorSample})

Output[:variable name] returns the corresponding posterior standard deviation.

var(x::Vector{<:PosteriorSample})

Output[:variable name] returns the corresponding posterior variance.

AR_res_var(TS::Vector, p)

It derives an MLE error variance estimate of an AR(p) model

Input

• univariate time series TS and the lag p

output(2)

residual variance estimate, AR(p) coefficients

GQ_XX(; kappaQ)

kappaQ governs a conditional mean of the Q-dynamics of X, and its slope matrix has a restricted form. This function shows that restricted form.

Output

• slope matrix of the Q-conditional mean of X

LDL(X)

This function generate a matrix decomposition, called LDLt. X = L*D*L', where L is a lower triangular matrix and D is a diagonal. How to conduct it can be found at Wikipedia.

Input

• Decomposed Object, X

Output(2)

L, D

• Decomposed result is X = L*D*L'

PCA(yields, p, proxies=[]; rescaling=false, dQ=[])

It derives the principal components from yields.

Input

• yields[p+1:end, :] is used to construct the affine transformation, and then all yields[:,:] are transformed into the principal components.
• Since signs of PCs is not identified, we use proxies to identify the signs. We flip PCs to make cor(proxies[:, i]. PCs[:,i]) > 0. If proxies is not given, we use the following proxies as a default: [yields[:, end] yields[:, end] - yields[:, 1] 2yields[:, Int(floor(size(yields, 2) / 3))] - yields[:, 1] - yields[:, end]].
• size(proxies) = (size(yields[p+1:end, :], 1), dQ)
• If rescaling == true, all PCs and OCs are normalized to have an average std of yields.

Output(4)

PCs, OCs, Wₚ, Wₒ, mean_PCs

• PCs, OCs: first dQ and the remaining principal components
• Wₚ, Wₒ: the rotation matrix for PCs and OCs, respectively
• mean_PCs: the mean of PCs before demeaned.
• PCs are demeaned.

calibrate_mean_phi_const(mean_kQ_infty, std_kQ_infty, nu0, yields, macros, tau_n, p; mean_phi_const_PCs=[], medium_tau=collect(24:3:48), iteration=1000, data_scale=1200, kappaQ_prior_pr=[], τ=[])

The purpose of the function is to calibrate a prior mean of the first dQ constant terms in our VAR. Adjust your prior setting based on the prior samples in outputs.

Input

• mean_phi_const_PCs is your prior mean of the first dQ constants. Our default option set it as a zero vector.
• iteration is the number of prior samples.
• τ::scalar is a maturity for calculating the constant part in the term premium.
  • If τ is empty, the function does not sampling the prior distribution of the constant part in the term premium.

Output(2)

prior_λₚ, prior_TP

• samples from the prior distribution of λₚ
• prior samples of constant part in the τ-month term premium

conditional_forecasts(S::Vector, τ, horizon, saved_params, yields, macros, tau_n; mean_macros::Vector=[], data_scale=1200)

Input

scenarios, a result of the posterior sampler, and data

• S[t] = conditioned scenario at time size(yields, 1)+t.
  • If we need an unconditional prediction, S = [].
  • If you are conditionaing a scenario, I assume S = Vector{Scenario}.
• τ is a vector. The term premium of τ[i]-bond is forecasted for each i.
  • If τ is set to [], the term premium is not forecasted.
• horizon: maximum length of the predicted path. It should not be small than length(S).
• saved_params: the first output of function posterior_sampler.
• mean_macros::Vector: If you demeaned macro variables, you can input the mean of the macro variables. Then, the output will be generated in terms of the un-demeaned macro variables.

Output

• Vector{Forecast}(, iteration)
• t'th rows in predicted yields, predicted factors, and predicted TP are the corresponding predicted value at time size(yields, 1)+t.
• Mathematically, it is a posterior samples from future observation|past observation,scenario.

dcurvature_dτ(τ; kappaQ)

This function calculate the first derivative of the curvature factor loading w.r.t. the maturity.

Input

• kappaQ: The decay parameter
• τ: The maturity that the derivative is calculated

Output

• the first derivative of the curvature factor loading w.r.t. the maturity

dimQ()

It returns the dimension of Q-dynamics under the standard ATSM.

erase_nonstationary_param(saved_params; threshold=1)

It filters out posterior samples that implies an unit root VAR system. Only stationary posterior samples remain.

Input

• saved_params is the first output of function posterior_sampler.
• Posterior samples with eigenvalues of the P-system greater than threshold are removed.

Output(2):

stationary samples, acceptance rate(%)

• The second output indicates how many posterior samples remain.

fitted_YieldCurve(τ0, saved_latent_params::Vector{LatentSpace}; data_scale=1200)

It generates a fitted yield curve.

Input

• τ0 is a set of maturities of interest. τ0 does not need to be the same as the one used for the estimation.
• saved_latent_params is a transformed posterior sample using function latentspace.

Output

• Vector{YieldCurve}(,# of iteration)
• yields and latents contain initial observations.

generative(T, dP, tau_n, p, noise::Float64; kappaQ, kQ_infty, KPXF, GPXFXF, OmegaXFXF, data_scale=1200)

This function generate a simulation data given parameters. Note that all parameters are the things in the latent factor state space (that is, parameters in struct LatentSpace). There is some differences in notations because it is hard to express mathcal letters in VScode. So, mathcal{F} in my paper is expressed in F in the VScode. And, "F" in my paper is expressed as XF.

Input:

• noise = variance of the measurement errors

Output(3)

yields, latents, macros

• yields = Matrix{Float64}(obs,T,length(tau_n))
• latents = Matrix{Float64}(obs,T,dimQ())
• macros = Matrix{Float64}(obs,T,dP - dimQ())

hessian(f, x, index=[])

It calculates the Hessian matrix of a scalar function f at x. If index is not empty, it calculates the Hessian matrix of the function with respect to the selected variables.

ineff_factor(saved_params)

It returns inefficiency factors of each parameter.

Input

• Vector{Parameter} from posterior_sampler

Output

• Estimated inefficiency factors are in Tuple(kappaQ, kQ_infty, gamma, SigmaO, varFF, phi). For example, if you want to load an inefficiency factor of phi, you can use Output.phi.
• If fix_const_PC1==true in your optimized struct Hyperparameter, Output.phi[1,1] can be weird. So you should ignore it.

isstationary(GPFF; threshold)

It checks whether a reduced VAR matrix has unit roots. If there is at least one unit root, return is false.

Input

• GPFF should not include intercepts. Also, GPFF is dP by dP*p matrix that the coefficient at lag 1 comes first, and the lag p slope matrix comes last.
• Posterior samples with eigenvalues of the P-system greater than threshold are removed. Typically, threshold is set to 1.

Output

• boolean

latentspace(saved_params, yields, tau_n; data_scale=1200)

This function translates the principal components state space into the latent factor state space.

Input

• data_scale::scalar: In typical affine term structure model, theoretical yields are in decimal and not annualized. But, for convenience(public data usually contains annualized percentage yields) and numerical stability, we sometimes want to scale up yields, so want to use (data_scale*theoretical yields) as variable yields. In this case, you can use data_scale option. For example, we can set data_scale = 1200 and use annualized percentage monthly yields as yields.

Output

• Vector{LatentSpace}(, iteration)
• latent factors contain initial observations.

log_marginal(PCs, macros, rho, tuned::Hyperparameter, tau_n, Wₚ; psi=[], psi_const=[], medium_tau, kappaQ_prior_pr, fix_const_PC1)

This file calculates a value of our marginal likelihood. Only the transition equation is used to calculate it.

Input

• tuned is a point where the marginal likelihood is evaluated.
• psi_const and psi are multiplied with prior variances of coefficients of the intercept and lagged regressors in the orthogonalized transition equation. They are used for imposing zero prior variances. A empty default value means that you do not use this function. [psi_const psi][i,j] is corresponds to phi[i,j].

Output

• the log marginal likelihood of the VAR system.

loglik_mea(yields, tau_n; kappaQ, kQ_infty, phi, varFF, SigmaO, data_scale)

This function generate a log likelihood of the measurement equation.

Output

• the measurement equation part of the log likelihood

loglik_tran(PCs, macros; phi, varFF)

It calculate log likelihood of the transition equation.

Output

• log likelihood of the transition equation.

phi_2_phi₀_C(; phi)

It divide phi into the lagged regressor part and the contemporaneous regerssor part.

Output(3)

phi0, C = C0 + I, C0

• phi0: coefficients for the lagged regressors
• C: coefficients for the dependent variables when all contemporaneous variables are in the LHS of the orthogonalized equations. Therefore, the diagonals of C is ones. Note that since the contemporaneous variables get negative signs when they are at the RHS, the signs of C do not change whether they are at the RHS or LHS.

phi_varFF_2_OmegaFF(; phi, varFF)

It construct OmegaFF from statistical parameters.

Output

• OmegaFF

posterior_sampler(yields, macros, tau_n, rho, iteration, tuned::Hyperparameter; medium_tau=collect(24:3:48), init_param=[], psi=[], psi_const=[], gamma_bar=[], kappaQ_prior_pr=[], mean_kQ_infty=0, std_kQ_infty=0.1, fix_const_PC1=false, data_scale=1200)

This is a posterior distribution sampler.

Input

• iteration: # of posterior samples
• tuned: optimized hyperparameters used during estimation
• init_param: starting point of the sampler. It should be a type of Parameter.
• psi_const and psi are multiplied with prior variances of coefficients of the intercept and lagged regressors in the orthogonalized transition equation. They are used for imposing zero prior variances. A empty default value means that you do not use this function. [psi_const psi][i,j] is corresponds to phi[i,j]. The entries of psi and psi_const should be nearly zero(e.g., 1e-10), not exactly zero.

Output(2)

Vector{Parameter}(posterior, iteration), acceptance rate of the MH algorithm

prior_kappaQ(medium_tau, pr)

The function derive the maximizer decay parameter kappaQ that maximize the curvature factor loading at each candidate medium-term maturity. And then, it impose a discrete prior distribution on the maximizers with a prior probability vector pr.

Input

• medium_tau::Vector(candidate medium maturities, # of candidates)
• pr::Vector(probability, # of candidates)

Output

• discrete prior distribution that has a support of the maximizers kappaQ

reducedform(saved_params, yields, macros, tau_n; data_scale=1200)

It converts posterior samples in terms of the reduced form VAR parameters.

Input

• saved_params is the first output of function posterior_sampler.

Output

• Posterior samples in terms of struct ReducedForm

scenario_analysis(S::Vector, τ, horizon, saved_params, yields, macros, tau_n; mean_macros::Vector=[], data_scale=1200)

Input

scenarios, a result of the posterior sampler, and data

• S[t] = conditioned scenario at time size(yields, 1)+t.
  • Set S = [] if you need an unconditional prediction.
  • If you are conditionaing a scenario, I assume S = Vector{Scenario}.
• τ is a vector of maturities that term premiums of interest has.
• horizon: maximum length of the predicted path. It should not be small than length(S).
• saved_params: the first output of function posterior_sampler.
• mean_macros::Vector: If you demeaned macro variables, you can input the mean of the macro variables. Then, the output will be generated in terms of the un-demeaned macro variables.

Output

• Vector{Forecast}(, iteration)
• t'th rows in predicted yields, predicted factors, and predicted TP are the corresponding predicted value at time size(yields, 1)+t.
• Mathematically, it is a posterior distribution of E[future obs|past obs, scenario, parameters].

term_premium(tau_interest, tau_n, saved_params, yields, macros; data_scale=1200)

This function generates posterior samples of the term premiums.

Input

• maturity of interest tau_interest for Calculating TP
• saved_params from function posterior_sampler

Output(3)

saved_TP, saved_tv_TP, saved_tv_EH

• saved_TP::Vector{TermPremium}(, iteration)
• saved_tv_TP::Vector{Array}(, iteration)
• saved_tv_EH::Vector{Array}(, iteration)
• Both the term premiums and expectation hypothesis components are decomposed into the time-invariant part and time-varying part. For the maturity tau_interest[i] and j-th posterior sample, the time-varying parts are saved in saved_tv_TP[j][:, :, i] and saved_tv_EH[j][:, :, i]. The time-varying parts driven by the k-th pricing factor is stored in saved_tv_TP[j][:, k, i] and saved_tv_EH[j][:, k, i].

tuning_hyperparameter(yields, macros, tau_n, rho; populationsize=50, maxiter=10_000, medium_tau=collect(24:3:48), upper_q=[1 1; 1 1; 10 10; 100 100], mean_kQ_infty=0, std_kQ_infty=0.1, upper_nu0=[], mean_phi_const=[], fix_const_PC1=false, upper_p=18, mean_phi_const_PC1=[], data_scale=1200, kappaQ_prior_pr=[], init_nu0=[], is_pure_EH=false, psi_common=[], psi_const=[])

It optimizes our hyperparameters by maximizing the marginal likelhood of the transition equation. Our optimizer is a differential evolutionary algorithm that utilizes bimodal movements in the eigen-space(Wang, Li, Huang, and Li, 2014) and the trivial geography(Spector and Klein, 2006).

Input

• When we compare marginal likelihoods between models, the data for the dependent variable should be the same across the models. To achieve that, we set a period of dependent variable based on upper_p. For example, if upper_p = 3, yields[4:end,:] and macros[4:end,:] are the data for our dependent variable. yields[1:3,:] and macros[1:3,:] are used for setting initial observations for all lags.
• populationsize and maxiter are options for the optimizer.
  • populationsize: the number of candidate solutions in each generation
  • maxtier: the maximum number of iterations
• The lower bounds for q and nu0 are 0 and dP+2.
• The upper bounds for q, nu0 and VAR lag can be set by upper_q, upper_nu0, upper_p.
  • Our default option for upper_nu0 is the time-series length of the data.
• If you use our default option for mean_phi_const,
  1. mean_phi_const[dQ+1:end] is a zero vector.
  2. mean_phi_const[1:dQ] is calibrated to make a prior mean of λₚ a zero vector.
  3. After step 2, mean_phi_const[1] is replaced with mean_phi_const_PC1 if it is not empty.
• mean_phi_const = Matrix(your prior, dP, upper_p)
• mean_phi_const[:,i] is a prior mean for the VAR(i) constant. Therefore mean_phi_const is a matrix only in this function. In other functions, mean_phi_const is a vector for the orthogonalized VAR system with your selected lag.
• When fix_const_PC1==true, the first element in a constant term in our orthogonalized VAR is fixed to its prior mean during the posterior sampling.
• data_scale::scalar: In typical affine term structure model, theoretical yields are in decimal and not annualized. But, for convenience(public data usually contains annualized percentage yields) and numerical stability, we sometimes want to scale up yields, so want to use (data_scale*theoretical yields) as variable yields. In this case, you can use data_scale option. For example, we can set data_scale = 1200 and use annualized percentage monthly yields as yields.
• is_pure_EH::Bool: When mean_phi_const=[], is_pure_EH=false sets mean_phi_const to zero vectors. Otherwise, mean_phi_const is set to imply the pure expectation hypothesis under mean_phi_const=[].
• psi_const and psi = kron(ones(1, lag length), psi_common) are multiplied with prior variances of coefficients of the intercept and lagged regressors in the orthogonalized transition equation. They are used for imposing zero prior variances. A empty default value means that you do not use this function. [psi_const psi][i,j] is corresponds to phi[i,j]. The entries of psi_common and psi_const should be nearly zero(e.g., 1e-10), not exactly zero.

Output(2)

optimized Hyperparameter, optimization result

• Be careful that we minimized the negative log marginal likelihood, so the second output is about the minimization problem.

Aₓ(aτ_, tau_n)

Input

• aτ_ is an output of function aτ.

Output

• Aₓ

Aₚ(Aₓ_, Bₓ_, T0P_, Wₒ)

Input

• Aₓ_, Bₓ_, and T0P_ are outputs of function Aₓ, Bₓ, and T0P, respectively.

Output

• Aₚ

Bₓ(bτ_, tau_n)

Input

• bτ_ is an output of function bτ.

Output

• Bₓ

Bₚ(Bₓ_, T1X_, Wₒ)

Input

• Bₓ_ and T1X_ are outputs of function Bₓ and T1X, respectively.

Output

• Bₚ

Kphi(i, V, Xphi, dP)

Minnesota(l, i, j; q, nu0, Omega0, dQ=[])

It return unscaled prior variance of the Minnesota prior.

Input

• lag l, dependent variable i, regressor j in the VAR(p)
• q[:,1] and q[:,2] are [own, cross, lag, intercept] shrikages for the first dQ and remaining dP-dQ equations, respectively.
• nu0(d.f.), Omega0(scale): Inverse-Wishart prior for the error-covariance matrix of VAR(p).

Output

• Minnesota part in the prior variance

NIG_NIG(y, X, β₀, B₀, α₀, δ₀)

Normal-InverseGamma-Normal-InverseGamma update

• prior: β|σ² ~ MvNormal(β₀,σ²B₀), σ² ~ InverseGamma(α₀,δ₀)
• likelihood: y|β,σ² = Xβ + MvNormal(zeros(T,1),σ²I(T))

Output(2)

β, σ²

• posterior sample

PCs_2_latents(yields, tau_n; kappaQ, kQ_infty, KPF, GPFF, OmegaFF, data_scale)

Notation XF is for the latent factor space and notation F is for the PC state space.

Input

• data_scale::scalar: In typical affine term structure model, theoretical yields are in decimal and not annualized. But, for convenience(public data usually contains annualized percentage yields) and numerical stability, we sometimes want to scale up yields, so want to use (data_scale*theoretical yields) as variable yields. In this case, you can use data_scale option. For example, we can set data_scale = 1200 and use annualized percentage monthly yields as yields.

Output(6)

latent, kappaQ, kQ_infty, KPXF, GPXFXF, OmegaXFXF

• latent factors contain initial observations.

S(i; Omega0)

S_hat(i, m, V, yphi, Xphi, dP; Omega0)

T0P(T1X_, Aₓ_, Wₚ, c)

Input

• T1X_ and Aₓ_ are outputs of function T1X and Aₓ, respectively. c is a sample mean of PCs.

Output

• T0P

T1X(Bₓ_, Wₚ)

Input

• Bₓ_ if an output of function Bₓ.

Output

• T1X

_conditional_forecasts(S, τ, horizon, yields, macros, tau_n; kappaQ, kQ_infty, phi, varFF, SigmaO, mean_macros, data_scale)

_scenario_analysis(S, τ, horizon, yields, macros, tau_n; kappaQ, kQ_infty, phi, varFF, SigmaO, mean_macros, data_scale)

_scenario_analysis_unconditional(τ, horizon, yields, macros, tau_n; kappaQ, kQ_infty, phi, varFF, SigmaO, mean_macros, data_scale)

_termPremium(τ, PCs, macros, bτ_, T0P_, T1X_; kappaQ, kQ_infty, KPF, GPFF, ΩPP, data_scale)

This function calculates a term premium for maturity τ.

Input

• data_scale::scalar = In typical affine term structure model, theoretical yields are in decimal and not annualized. But, for convenience(public data usually contains annualized percentage yields) and numerical stability, we sometimes want to scale up yields, so want to use (data_scale*theoretical yields) as variable yields. In this case, you can use data_scale option. For example, we can set data_scale = 1200 and use annualized percentage monthly yields as yields.

Output(4)

TP, timevarying_TP, const_TP, jensen

• TP: term premium of maturity τ
• timevarying_TP: contributions of each [PCs macros] on TP at each time $t$ (row: time, col: variable)
• const_TP: Constant part of TP
• jensen: Jensen's Ineqaulity part in TP
• Output excludes the time period for the initial observations.

_unconditional_forecasts(τ, horizon, yields, macros, tau_n; kappaQ, kQ_infty, phi, varFF, SigmaO, mean_macros, data_scale)

aτ(N, bτ_, tau_n, Wₚ; kQ_infty, ΩPP, data_scale)
aτ(N, bτ_; kQ_infty, ΩXX, data_scale)

The function has two methods(multiple dispatch).

Input

• When Wₚ ∈ arguments: It calculates aτ using ΩPP.
• Otherwise: It calculates aτ using ΩXX = OmegaXFXF[1:dQ, 1:dQ], so parameters are in the latent factor space. So, we do not need Wₚ.
• bτ_ is an output of function bτ.
• data_scale::scalar: In typical affine term structure model, theoretical yields are in decimal and not annualized. But, for convenience(public data usually contains annualized percentage yields) and numerical stability, we sometimes want to scale up yields, so want to use (data_scale*theoretical yields) as variable yields. In this case, you can use data_scale option. For example, we can set data_scale = 1200 and use annualized percentage monthly yields as yields.

Output

• Vector(Float64)(aτ,N)
• For i'th maturity, Output[i] is the corresponding aτ.

bτ(N; kappaQ, dQ)

It solves the difference equation for bτ.

Output

• for maturity i, bτ[:, i] is a vector of factor loadings.

jensens_inequality(τ, bτ_, T1X_; ΩPP, data_scale)

This function evaluate the Jensen's Ineqaulity term. All term is invariant with respect to the data_scale, except for this Jensen's inequality term. So, we need to scale down the term by data_scale.

Output

• Jensen's Ineqaulity term for aτ of maturity τ.

loglik_mea2(yields, tau_n; kappaQ, kQ_infty, phi, varFF, SigmaO, data_scale)

This function is the same as loglik_mea but it requires ΩPP as an input.

longvar(v)

It calculates the long-run variance of v using the quadratic spectral window with selection of bandwidth of Andrews(1991). We use the AR(1) approximation.

Input

• Time-series Vector v

Output

• Estimated 2πh(0) of v, where h(x) is the spectral density of v at x.

mle_error_covariance(yields, macros, tau_n, p)

It calculates the MLE estimates of the error covariance matrix of the VAR(p) model.

phi_hat(i, m, V, yphi, Xphi, dP)

phi_varFF_2_ΩPP(; phi, varFF, dQ=[])

It construct ΩPP from statistical parameters.

Output

• ΩPP

post_SigmaO(yields, tau_n; kappaQ, kQ_infty, ΩPP, gamma, p, data_scale)

Posterior sampler for the measurement errors

Output

• Vector{Dist}(IG, N-dQ)

post_gamma(; gamma_bar, SigmaO)

Posterior sampler for the population measurement error

Output

• Vector{Dist}(Gamma,length(SigmaO))

post_kQ_infty(mean_kQ_infty, std_kQ_infty, yields, tau_n; kappaQ, phi, varFF, SigmaO, data_scale)

Output

• Full conditional posterior distribution

post_kappaQ(yields, prior_kappaQ_, tau_n; kQ_infty, phi, varFF, SigmaO, data_scale)

Input

• prior_kappaQ_ is a output of function prior_kappaQ.

Output

• Full conditional posterior distribution

post_kappaQ2(yields, prior_kappaQ_, tau_n; kappaQ, kQ_infty, phi, varFF, SigmaO, data_scale, x_mode, inv_x_hess)

It conducts the Metropolis-Hastings algorithm for the reparameterized kappaQ under the unrestricted JSZ form. x_mode and inv_x_hess constitute the mean and variance of the Normal proposal distribution.

• Reparameterization: kappaQ[1] = x[1] kappaQ[2] = x[1] + x[2] kappaQ[3] = x[1] + x[2] + x[3]
• Jacobian: [1 0 0 1 1 0 1 1 1]
• The determinant = 1

post_phi_varFF(yields, macros, mean_phi_const, rho, prior_kappaQ_, tau_n; phi, psi, psi_const, varFF, q, nu0, Omega0, kappaQ, kQ_infty, SigmaO, fix_const_PC1, data_scale)

Full-conditional posterior sampler for phi and varFF

Input

• prior_kappaQ_ is a output of function prior_kappaQ.
• When fix_const_PC1==true, the first element in a constant term in our orthogonalized VAR is fixed to its prior mean during the posterior sampling.

Output(3)

phi, varFF, isaccept=Vector{Bool}(undef, dQ)

• It gives a posterior sample.

prior_C(; Omega0::Vector)

We translate the Inverse-Wishart prior to a series of the Normal-Inverse-Gamma (NIG) prior distributions. If the dimension is dₚ, there are dₚ NIG prior distributions. This function generates Normal priors.

Output:

• unscaled prior of C in the LDLt decomposition, OmegaFF = inv(C)*diagm(varFF)*inv(C)'

Important note

prior variance for C[i,:] = varFF[i]*variance of output[i,:]

prior_gamma(yields, p)

There is a hierarchcal structure in the measurement equation. The prior means of the measurement errors are gamma[i] and each gamma[i] follows Gamma(1,gamma_bar) distribution. This function decides gamma_bar empirically. OLS is used to estimate the measurement equation and then a variance of residuals is calculated for each maturities. An inverse of the average residual variances is set to gamma_bar.

Output

• hyperparameter gamma_bar

prior_phi0(mean_phi_const, rho::Vector, prior_kappaQ_, tau_n, Wₚ; psi_const, psi, q, nu0, Omega0, fix_const_PC1)

This part derives the prior distribution for coefficients of the lagged regressors in the orthogonalized VAR.

Input

• prior_kappaQ_ is a output of function prior_kappaQ.
• When fix_const_PC1==true, the first element in a constant term in our orthogonalized VAR is fixed to its prior mean during the posterior sampling.

Output

• Normal prior distributions on the slope coefficient of lagged variables and intercepts in the orthogonalized equation.
• Output[:,1] for intercepts, Output[:,1+1:1+dP] for the first lag, Output[:,1+dP+1:1+2*dP] for the second lag, and so on.

Important note

prior variance for phi[i,:] = varFF[i]*var(output[i,:])

prior_varFF(; nu0, Omega0::Vector)

We translate the Inverse-Wishart prior to a series of the Normal-Inverse-Gamma (NIG) prior distributions. If the dimension is dₚ, there are dₚ NIG prior distributions. This function generates Inverse-Gamma priors.

Output:

• prior of varFF in the LDLt decomposition,OmegaFF = inv(C)*diagm(varFF)*inv(C)'
• Each element in the output follows Inverse-Gamma priors.

yphi_Xphi(PCs, macros, p)

This function generate the dependent variable and the corresponding regressors in the orthogonalized transition equation.

Output(4)

yphi, Xphi = [ones(T - p) Xphi_lag Xphi_contemporaneous], [ones(T - p) Xphi_lag], Xphi_contemporaneous

• yphi and Xphi is a full matrix. For i'th equation, the dependent variable is yphi[:,i] and the regressor is Xphi.
• Xphi is same to all orthogonalized transtion equations. The orthogonalized equations are different in terms of contemporaneous regressors. Therefore, the corresponding regressors in Xphi should be excluded. The form of parameter phi do that task by setting the coefficients of the excluded regressors to zeros. In particular, for last dP by dP block in phi, the diagonals and the upper diagonal elements should be zero.

ν(i, dP; nu0)