Projection matrix for optimal combination cross-temporal reconciliation

This function computes the projection or the mapping matrix \(\mathbf{M}\) and \(\mathbf{G}\), respectively, such that \(\widetilde{\mathbf{y}} = \mathbf{M}\widehat{\mathbf{y}} = \mathbf{S}_{ct}\mathbf{G}\widehat{\mathbf{y}}\), where \(\widetilde{\mathbf{y}}\) is the vector of the reconciled forecasts, \(\widehat{\mathbf{y}}\) is the vector of the base forecasts, \(\mathbf{S}_{ct}\) is the cross-temporal structural matrix, and \(\mathbf{M} = \mathbf{S}_{ct}\mathbf{G}\). For further information regarding on the structure of these matrices, refer to Girolimetto et al. (2023).

Usage

ctprojmat(agg_mat, cons_mat, agg_order, comb = "ols", res = NULL,
          mat = "M", tew = "sum", ...)

Arguments

agg_mat

A (\(n_a \times n_b\)) numeric matrix representing the cross-sectional aggregation matrix. It maps the \(n_b\) bottom-level (free) variables into the \(n_a\) upper (constrained) variables.

cons_mat

A (\(n_a \times n\)) numeric matrix representing the cross-sectional zero constraints. It spans the null space for the reconciled forecasts.

agg_order

Highest available sampling frequency per seasonal cycle (max. order of temporal aggregation, \(m\)), or a vector representing a subset of \(p\) factors of \(m\).

comb

A string specifying the reconciliation method. For a complete list, see ctcov.

res

A (\(n \times N(k^\ast+m)\)) optional numeric matrix containing the in-sample residuals at all the temporal frequencies ordered from the lowest frequency to the highest frequency (columns) for each variable (rows). This matrix is used to compute some covariance matrices.

mat

A string specifying which matrix to return: "M" (default) for \(\mathbf{M}\) and "G" for \(\mathbf{G}\).

tew

A string specifying the type of temporal aggregation. Options include: "sum" (simple summation, default), "avg" (average), "first" (first value of the period), and "last" (last value of the period).

...

Arguments passed on to ctcov

mse

If TRUE (default) the residuals used to compute the covariance matrix are not mean-corrected.

shrink_fun

Shrinkage function of the covariance matrix, shrink_estim (default).

Value

The projection matrix \(\mathbf{M}\) (mat = "M") or the mapping matrix \(\mathbf{G}\) (mat = "G").

References

Girolimetto, D., Athanasopoulos, G., Di Fonzo, T. and Hyndman, R.J. (2024), Cross-temporal probabilistic forecast reconciliation: Methodological and practical issues. International Journal of Forecasting, 40, 3, 1134-1151. doi:10.1016/j.ijforecast.2023.10.003

See also

Utilities: FoReco2matrix(), aggts(), balance_hierarchy(), commat(), csprojmat(), cstools(), cttools(), df2aggmat(), lcmat(), recoinfo(), res2matrix(), shrink_estim(), teprojmat(), tetools(), unbalance_hierarchy()

Examples

# Cross-temporal framework (Z = X + Y, annual-quarterly)
A <- t(c(1,1)) # Aggregation matrix for Z = X + Y
Mct <- ctprojmat(agg_mat = A, agg_order = 4, comb = "ols")
Gct <- ctprojmat(agg_mat = A, agg_order = 4, comb = "ols", mat = "G")