ctprojmat function

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).

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).

Returns

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

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")

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. tools:::Rd_expr_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()