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).

Usage

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.

Value

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

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

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. doi:10.1016/j.ijforecast.2023.10.003

See also

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

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
#> [1] 0.5736559
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)