Iterative cross-temporal reconciliation

This function performs the iterative procedure described in Di Fonzo and Girolimetto (2023), which produces cross-temporally reconciled forecasts by alternating forecast reconciliation along one single dimension (either cross-sectional or temporal) at each iteration step.

Usage

iterec(base, cslist, telist, res = NULL, itmax = 100, tol = 1e-5,
       type = "tcs", norm = "inf", verbose = TRUE)

Arguments

base: A (\(n \times h(k^\ast+m)\)) numeric matrix containing the base forecasts to be reconciled; \(n\) is the total number of variables, \(m\) is the max. order of temporal aggregation, \(k^\ast\) is the sum of (a subset of) (\(p-1\)) factors of \(m\), excluding \(m\), and \(h\) is the forecast horizon for the lowest frequency time series. The row identifies a time series, and the forecasts in each row are ordered from the lowest frequency (most temporally aggregated) to the highest frequency.
cslist: A list of elements for the cross-sectional reconciliation. See csrec for a complete list (excluded base and res).
telist: A list of elements for the temporal reconciliation. See terec for a complete list (excluded base and res).
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.
itmax: Max number of iteration (100, default).
tol: Convergence tolerance (1e-5, default).
type: A string specifying the uni-dimensional reconciliation order: temporal and then cross-sectional ("tcs") or cross-sectional and then temporal ("cst").
norm: Norm used to calculate the temporal and the cross-sectional incoherence: infinity norm ("inf", default), one norm ("one"), and 2-norm ("two").
verbose: If TRUE, reconciliation information are printed.

Value

A (\(n \times h(k^\ast+m)\)) numeric matrix of cross-temporal reconciled forecasts.

References

Di Fonzo, T. and Girolimetto, D. (2023), Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives, International Journal of Forecasting, 39, 1, 39-57. doi:10.1016/j.ijforecast.2021.08.004

Examples

set.seed(123)
# (3 x 7) base forecasts matrix (simulated), Z = X + Y and m = 4
base <- rbind(rnorm(7, rep(c(20, 10, 5), c(1, 2, 4))),
              rnorm(7, rep(c(10, 5, 2.5), c(1, 2, 4))),
              rnorm(7, rep(c(10, 5, 2.5), c(1, 2, 4))))
# (3 x 70) in-sample residuals matrix (simulated)
res <- rbind(rnorm(70), rnorm(70), rnorm(70))

A <- t(c(1,1)) # Aggregation matrix for Z = X + Y
m <- 4 # from quarterly to annual temporal aggregation

rite <- iterec(base = base,
               cslist = list(agg_mat = A, comb = "shr"),
               telist = list(agg_order = m, comb = "wlsv"),
               res = res)
#> ── Iterative heuristic cross-temporal forecast reconciliation ──────────────────
#> Legend: i = iteration; s = step. Norm = "inf".
#> 
#>   i.s |        Temporal | Cross-sectional |
#>     0 |            3.36 |            1.79 |
#>   1.1 |            0.00 |            1.91 |
#>   1.2 |        2.02e-01 |            0.00 |
#>   2.1 |            0.00 |        2.42e-02 |
#>   2.2 |        2.56e-03 |            0.00 |
#>   3.1 |            0.00 |        3.18e-04 |
#>   3.2 |        3.37e-05 |            0.00 |
#>   4.1 |            0.00 |        4.50e-06 |
#>   4.2 |        4.62e-07 |            0.00 |
#> 
#> ✔ Convergence achieved at iteration 4.
#> ────────────────────────────────────────────────────────────────────────────────