ctcov function

Cross-temporal covariance matrix approximation

This function provides an approximation of the cross-temporal base forecasts errors covariance matrix using different reconciliation methods (Di Fonzo and Girolimetto, 2023, Girolimetto et al., 2023).

ctcov(comb = "ols", n = NULL, agg_mat = NULL, agg_order = NULL, res = NULL,
      tew = "sum", mse = TRUE, shrink_fun = shrink_estim, ...)

Arguments

• comb: A string specifying the reconciliation method.

  • Ordinary least squares:

    • "ols" (default) - identity error covariance.

  • Weighted least squares:

    • "str" - structural variances.
    • "csstr" - cross-sectional structural variances.
    • "testr" - temporal structural variances.
    • "wlsh" - hierarchy variances (uses res).
    • "wlsv" - series variances (uses res).

  • Generalized least squares (uses res):

    • "acov" - series auto-covariance.
    • "bdshr"/"bdsam" - shrunk/sample block diagonal cross-sectional covariance.
    • "Sshr"/"Ssam" - series shrunk/sample covariance.
    • "shr"/"sam" - shrunk/sample covariance.
    • "hbshr"/"hbsam" - shrunk/sample high frequency bottom time series covariance.
    • "bshr"/"bsam" - shrunk/sample bottom time series covariance.
    • "hshr"/"hsam" - shrunk/sample high frequency covariance.

• n: Cross-sectional number of variables.

• 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.

• 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 $m$.

• 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.

• 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).

• 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).

• ...: Not used.

Returns

A ($n(k^\ast+m) \times n(k^\ast+m)$) symmetric matrix.

Examples

set.seed(123)
# Aggregation matrix for Z = X + Y
A <- t(c(1,1))
# (3 x 70) in-sample residuals matrix (simulated),
# agg_order = 4 (annual-quarterly)
res <- rbind(rnorm(70), rnorm(70), rnorm(70))

cov1 <- ctcov("ols", n = 3, agg_order = 4)                     # OLS methods
cov2 <- ctcov("str", agg_mat = A, agg_order = 4)               # STR methods
cov3 <- ctcov("csstr", agg_mat = A, agg_order = 4)             # CSSTR methods
cov4 <- ctcov("testr", n = 3, agg_order = 4)                   # TESTR methods
cov5 <- ctcov("wlsv", agg_order = 4, res = res)                # WLSv methods
cov6 <- ctcov("wlsh", agg_order = 4, res = res)                # WLSh methods
cov7 <- ctcov("shr", agg_order = 4, res = res)                 # SHR methods
cov8 <- ctcov("sam", agg_order = 4, res = res)                 # SAM methods
cov9 <- ctcov("acov", agg_order = 4, res = res)                # ACOV methods
cov10 <- ctcov("Sshr", agg_order = 4, res = res)               # Sshr methods
cov11 <- ctcov("Ssam", agg_order = 4, res = res)               # Ssam methods
cov12 <- ctcov("hshr", agg_order = 4, res = res)               # Hshr methods
cov13 <- ctcov("hsam", agg_order = 4, res = res)               # Hsam methods
cov14 <- ctcov("hbshr", agg_mat = A, agg_order = 4, res = res) # HBshr methods
cov15 <- ctcov("hbsam", agg_mat = A, agg_order = 4, res = res) # HBsam methods
cov16 <- ctcov("bshr", agg_mat = A, agg_order = 4, res = res)  # Bshr methods
cov17 <- ctcov("bsam", agg_mat = A, agg_order = 4, res = res)  # Bsam methods
cov18 <- ctcov("bdshr", agg_order = 4, res = res)              # BDshr methods
cov19 <- ctcov("bdsam", agg_order = 4, res = res)              # BDsam methods

# Custom covariance matrix
ctcov.ols2 <- function(comb, x) diag(x)
cov20 <- ctcov(comb = "ols2", x = 21) # == ctcov("ols", n = 3, agg_order = 4)

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. tools:::Rd_expr_doi("10.1016/j.ijforecast.2021.08.004")

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

Cross-temporal framework: ctboot(), ctbu(), ctlcc(), ctmo(), ctrec(), cttd(), cttools(), iterec(), tcsrec()