Cross-sectional sequential reconciliation-combination

This function applies a sequential method that first reconciles the base forecasts from each expert to satisfy the linear constraints, and then combines the reconciled forecasts obtained so far. cssrc may be applied only in 'balanced' cases (e.g., \(n_j = n\) \(\forall j\), see Girolimetto and Di Fonzo, 2024)

Usage

cssrc(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).

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, mimeo.

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: csscr()

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 <- cssrc(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 <- cssrc(base = base, cons_mat = C, comb = "wls", res = res, fc = "sa") # same results

# WARNING!
reco_v <- cssrc(base = base, agg_mat = A, comb = "wls", res = res, fc = "var")
round(C %*% t(reco_v), 3) # Incoherent forecasts
#>         h-1   h-2
#> [1,] -0.113 0.071