Cross-sectional sequential combination-reconciliation

This function performs a two-step process designed to first combine forecasts from multiple models or experts and then apply reconciliation techniques to ensure coherence.

Usage

csscr(base, fc = "sa", comb = "ols", res = NULL, mse = TRUE, shrink = TRUE,
      nnw = FALSE, factorized = FALSE, ...)

Arguments

base

A list of \(p\) numeric (\(h \times n\)) matrix or multivariate time series (mts class) containing the base forecasts to be reconciled; \(h\) is the forecast horizon, \(n\) is the total number of time series (\(n = n_u + n_b\)) and \(p\) is the total number of experts.

fc

A string specifying the combination method:

• "sa" - (default) simple average (equal weights).

• "var" - (uses res) weights derived from the inverse of forecasts variances/MSE as proposed by Bates and Granger (1969).

• "cov" - (uses res) weights derived using the whole forecast error covariance matrix, as proposed by Newbold and Granger (1974).

comb

A string specifying the reconciliation method: "ols", "wls", "shr", "sam" (see FoReco). If comb = "none", no reconciliation is performed and the combined forecasts are directly returned.

res

A list of \(p\) numeric (\(N \times n\)) matrix containing the in-sample residuals. This input is used to compute some covariance matrices.

mse

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

shrink

If TRUE (default), the covariance matrix for fc = "cov" is shrunk.

nnw

If TRUE for fc = "cov", the weights are constrained to be non-negative (Conflitti et al., 2015). The default is FALSE.

factorized

Value to be passed to the quadprog::solve.QP, only when nnw = TRUE.

...

Arguments passed on to FoReco::csrec (e.g., agg_mat or cons_mat).

Value

A (\(h \times n\)) numeric matrix of cross-sectional combined and reconciled forecasts.

References

Bates, J. and Granger, C. W. (1969), The combination of forecasts, Operations Research Quarterly, 20, 451–468. doi:10.1057/jors.1969.103 .

Conflitti, C., De Mol, C., and Giannone, D. (2015), Optimal combination of survey forecasts. International Journal of Forecasting, 31(4), 1096–1103. doi:10.1016/j.ijforecast.2015.03.009 .

Girolimetto, D. and Di Fonzo, T. (2024), Coherent forecast combination for linearly constrained multiple time series. doi:10.48550/arXiv.2412.03429 .

Newbold, P. and Granger, C. W. (1974), Experience with forecasting univariate time series and the combination of forecasts, Journal of the Royal Statistical Society, A, 137, 131–146. doi:10.2307/2344546

See also

Sequential coherent combination: cssrc()

Examples

set.seed(123)
# (2 x 3) base forecasts matrix (simulated), expert 1
base1 <- matrix(rnorm(6, mean = c(20, 10, 10)), 2, byrow = TRUE)
# (10 x 3) in-sample residuals matrix (simulated), expert 1
res1 <- t(matrix(rnorm(n = 30), nrow = 3))

# (2 x 3) base forecasts matrix (simulated), expert 2
base2 <- matrix(rnorm(6, mean = c(20, 10, 10)), 2, byrow = TRUE)
# (10 x 3) in-sample residuals matrix (simulated), expert 2
res2 <- t(matrix(rnorm(n = 30), nrow = 3))

# Base forecasts' and residuals' lists
base <- list(base1, base2)
res <- list(res1, res2)

# Aggregation matrix for Z = X + Y
A <- t(c(1,1))
reco <- csscr(base = base, agg_mat = A, comb = "wls", res = res, fc = "sa")

# Zero constraints matrix for Z - X - Y = 0
C <- t(c(1, -1, -1))
reco <- csscr(base = base, cons_mat = C, comb = "wls", res = res, fc = "sa") # same results

# Incoherent combined forecasts
fc_comb <- csscr(base = base, comb = "none", fc = "sa")
round(C %*% t(fc_comb), 3) # Incoherent forecasts
#>         h-1    h-2
#> [1,] -0.484 -0.626