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", agg_mat = NULL, agg_order = NULL, tew = "sum",
      res = NULL, n = NULL, 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.

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 \(m\).

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).

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.

n

Cross-sectional number of variables.

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)