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.

Reason Why You Need to Run `reducedform` in Addition to `posterior_sampler`

We estimate the $\mathbb{P}$-VAR by transforming it into a recursive VAR form. Therefore, Parameter, the output of posterior_sampler, contains parameters in the recursive VAR. In contrast, ReducedForm, the output of reducedform, contains parameters in the original reduced-form $\mathbb{P}$-VAR.

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).

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

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].