Forecasting

We have two kinds of forecasts.

1. Baseline Forecasts
2. Scenario Forecasts (Scenario Analysis)

All of two forecasts are conditional forecasts, because they are based on information in data. The difference is that the scenario forecast assumes additional scenarios that describe future paths of some variables.

Baseline forecasts and scenario forecasts can be represented either as the posterior distribution of predicted objects or as the posterior distribution of conditional expectations of predicted objects. To summarize:

1. Posterior Distribution of Predicted Objects
   • In other words, Distribution of future objects conditional on past observations and the scenario
   • Function: conditional_forecasts
2. Posterior Distribution of Conditional Expectations of Predicted Objects
   • In other words, Posterior Distribution of "E[future object|past obs, scenario, parameters]"
   • Function: scenario_analysis

In this summary, for baseline forecasts, the scenario is the empty set.

The first one is the full Bayesian treatment, so it is mathematically strict. However, it can be difficult to derive meaningful implications from the prediction because of its wide prediction intervals. The second one consider only parameter uncertainty, so it underestimates the prediction uncertainty. However, it is appropriate when users make decisions based on the expected path of future variables. We recommend the second version (scenario_analysis).

The required inputs and the type of the output are the same between conditional_forecasts and scenario_analysis. That is,

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

and

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

projections::Vector{Forecast} contains the results of the forecasting. τ is a vector, and the term premium of τ[i]-bond is forecasted for each i. If τ is set to [], the term premium is not forecasted. horizon is the forecasting horizon. horizon should not be smaller than length(S). saved_params::Vector{Parameter} is the output of posterior_sampler.

Users can use the same yields, tau_n and macros they employed when executing posterior_sampler. If one wishes to compute conditional forecasts using observations up to a certain point, they can simply use yields and macros from the initial period up to that point. However, parameter uncertainty is incorporated independently of yields and macros through saved_params.

If you use demeaned macro data, option mean_macros is useful. If the sample mean of macro data is specified as the input value for mean_macros, projections contains conditional forecasts of non-demeaned macro variables. The sample mean of macro data can be calculated as follows.

mean_macros = mean(raw_macros_data, dims=1)[1, :]

S determines whether we are computing a baseline forecast or a scenario forecast. How S is set will be described in the following sections.

Baseline Forecasts

Do

S = []

It sets a scenario to an empty set, so the package calculate baseline forecasts.

Scenario Forecasts

S should be Vector{Scenario}. S can be initialized by

S = Vector{Scenario}(undef, len)

len is the length of S. For example, if the scenario is assumed for the next 5 time periods, len=5.

S[i] represents the scenario for future variables at time T+i, where T refers to the time of the last observation in macros and yields. The type of S[i] is Scenario, and struct Scenario has two fields: combinations::Matrix and values::Vector. The fields in S[i] are implicitly defined by

S[i].combination*[yields[T+i,:]; macros[T+i, :]] == S[i].values

[yields[T+i,:]; macros[T+i, :]] is a predicted variable that is not observed. Scenario forecasts are calculated assuming that the above equation holds at time T+i, based on S[i] set by users. The number of rows in S[i].combination and the length of S[i].values are the same, and this length represents the number of scenarios assumed at time T+i.

Setting the two fields of S[i] is straightforward. Suppose that the content of the scenarios at time T+i is

combs*[yields[T+i,:]; macros[T+i, :]] == vals

Then, users can assign the content to S[i] by executing

S[i] = Scenario(combinations=combs, values=vals)