Statistical Inference

Inference for Parameters

You can obtain posterior samples of the term structure model parameters using reducedform.

reduced_params = reducedform(saved_params, yields, macros, tau_n; data_scale=1200, pca_loadings=[])

yields is a T by N matrix, T is the length of the sample period and N is the number of bond maturities in the data. tau_n is an N-Vector that contains maturities in the data. For example, if there are two maturities, 3 and 24 months, in the monthly term structure model, tau_n=[3; 24]. macros is a T by dP-dQ matrix in which each column represents an individual macroeconomic variable.

Each entry in reduced_params::Vector{ReducedForm} is a joint posterior sample of the parameters.

Yield Curve Interpolation

First, transform the parameter space from the principal component space to the latent factor space. This is done using latentspace. Then, use fitted_yieldcurve to obtain fitted yields. Specifically,

saved_latent_params = latentspace(saved_params, yields, tau_n; data_scale=1200, pca_loadings=[])
fitted_yields = fitted_yieldcurve(tau_vec, saved_latent_params::Vector{LatentSpace}; data_scale=1200)

tau_vec is a vector containing the maturities for which you want to calculate fitted yields through interpolation. fitted_yields::Vector{YieldCurve} contains the interpolation results.

Term Premiums

term_premium calculates the term premium of the bonds. tau_interest contains the maturities of interest and should be a Vector (at least a one-dimensional vector), in strictly increasing order without duplicates.

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

yields and macros are the data used to estimate saved_params. If the yield curves to decompose are the same as yields, leave both decomp_yields and decomp_macros as []. To decompose another yield curve dataset while keeping the estimated model fixed, supply decomp_yields and, if the model includes macro variables, decomp_macros. For example:

saved_TP, saved_tv_TP, saved_tv_EH = term_premium(tau_interest, tau_n, saved_params, yields, macros;
    data_scale=1200, decomp_yields=new_yields, decomp_macros=new_macros)

decomp_yields may have a different number of observations and a different observation frequency, but must use the same yield units/scaling and the same maturity columns in the same order as yields and tau_n. The PCA rotation, ordering, signs, and centering are determined from the estimation sample yields; the new yields are projected through that fixed transformation. The estimated model frequency, maturity units, data_scale, and P- and Q-dynamics remain unchanged. External decomposition data are supported only for P-dynamics with p=1; outputs exclude the first observation of that sample.

When supplying decomp_yields, also supply decomp_macros if the model includes macro variables; otherwise leave decomp_macros empty. Its rows must match decomp_yields, and its columns must match the number and order of variables in macros. It cannot be supplied without decomp_yields.

saved_TP::Vector{TermPremium} contains the results of the term premium calculations. Both the term premiums and expectation hypothesis components are decomposed into time-invariant and time-varying parts. For the maturity tau_interest[i], the time-varying parts are saved in saved_tv_TP[:, :, i] and saved_tv_EH[:, :, i]. The time-varying parts driven by the j-th pricing factor are stored in saved_tv_TP[:, j, i] and saved_tv_EH[:, j, i].