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, and 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.

• For ordinary least squares reconciliation:

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

• For weighted least squares reconciliation:

  • "str" - structural variances.

  • "csstr" - cross-sectional structural variances.

  • "testr" - temporal structural variances.

  • "wlsh" - hierarchy variances (uses res).

  • "wlsv" - series variances (uses res).

• For generalized least squares (uses res) reconciliation:

  • "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.

• Others (no for reconciliation):

  • "bu" - bottom-up 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 or validation errors 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 errors 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. (2023a), 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(), ctmvn(), ctrec(), ctsmp(), 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
cov2 <- ctcov("str", agg_mat = A, agg_order = 4)               # STR
cov3 <- ctcov("csstr", agg_mat = A, agg_order = 4)             # CSSTR
cov4 <- ctcov("testr", n = 3, agg_order = 4)                   # TESTR
cov5 <- ctcov("wlsv", agg_order = 4, res = res)                # WLSv
cov6 <- ctcov("wlsh", agg_order = 4, res = res)                # WLSh
cov7 <- ctcov("shr", agg_order = 4, res = res)                 # SHR
cov8 <- ctcov("sam", agg_order = 4, res = res)                 # SAM
cov9 <- ctcov("acov", agg_order = 4, res = res)                # ACOV
cov10 <- ctcov("Sshr", agg_order = 4, res = res)               # Sshr
cov11 <- ctcov("Ssam", agg_order = 4, res = res)               # Ssam
cov12 <- ctcov("hshr", agg_order = 4, res = res)               # Hshr
cov13 <- ctcov("hsam", agg_order = 4, res = res)               # Hsam
cov14 <- ctcov("hbshr", agg_mat = A, agg_order = 4, res = res) # HBshr
cov15 <- ctcov("hbsam", agg_mat = A, agg_order = 4, res = res) # HBsam
cov16 <- ctcov("bshr", agg_mat = A, agg_order = 4, res = res)  # Bshr
cov17 <- ctcov("bsam", agg_mat = A, agg_order = 4, res = res)  # Bsam
cov18 <- ctcov("bdshr", agg_order = 4, res = res)              # BDshr
cov19 <- ctcov("bdsam", agg_order = 4, res = res)              # BDsam

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