Average Multi-Horizon Superior Predictive Ability Test

Implements the average multi-horizon SPA test of Quaedvlieg (2021), which tests whether a weighted average of horizon-wise expected loss differentials favors the benchmark, allowing inferior performance at some horizons to be compensated by superior performance at others. Bootstrap follows Algorithm 1 (moving-block bootstrap, block length L).

Usage

aspa_mh_test(loss_diff, weights, L, B = 999L, level = 0.05)

Arguments

loss_diff: A T x H matrix of loss differentials, defined as L_bench - L_comp at each horizon. Positive values indicate the benchmark has higher loss (i.e. is worse) than the competitor.
weights: Numeric vector of length H of horizon weights. They need not sum to 1; uniform weights rep(1/H, H) reproduce the unweighted average SPA of the original paper.
L: Integer block length for the moving-block bootstrap.
B: Integer number of bootstrap replications. Default 999.
level: Significance level. Default 0.05.

Value

A list with class "aspa_mh_test" containing statistic, pvalue, reject, level, L, B, T, H, weights, d_bar.

Details

The aSPA statistic is $$t^{aSPA} = \sqrt{T}\, \bar d^w / \sqrt{\hat\omega^{QS}(\bar d^w)},$$ where $d^w_t = \sum_h w_h d_{h,t}$ and $\hat\omega^{QS}$ is the Quadratic-Spectral HAC long-run variance (Andrews, 1991). Critical values are obtained via a moving-block bootstrap that recenters the weighted series under the null. The p-value is upper-tail (mean(t < t_b)); rejection requires the weighted-average differential to be large and positive (the benchmark loses on the weighted average).

This is a port of the Matlab reference implementation accompanying Quaedvlieg (2021); the helpers .mbb_indices, .mbb_variance, and .qs_lrvar translate Get_MBB_ID.m, MBB_Variance.m, and QS.m respectively.

References

Quaedvlieg, R. (2021). Multi-Horizon Forecast Comparison. Journal of Business & Economic Statistics, 39(1), 40-53.

Examples

set.seed(2)
ld <- matrix(rnorm(200 * 4, mean = 0.2), 200, 4)
aspa_mh_test(ld, weights = rep(0.25, 4), L = 3, B = 199)
#> 
#> ╭────────────────────────────────────────────────────╮
#> │           Average Multi-Horizon SPA Test           │
#> │                 (Quaedvlieg, 2021)                 │
#> ├────────────────────────────────────────────────────┤
#> │ H0: Benchmark has aSPA (weighted avg superior)     │
#> │ H1: Benchmark worse on weighted average            │
#> ├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
#> │ Test Results:                                      │
#> │  aSPA statistic: 7.7808                            │
#> │  P-value (MBB): 0.0000                             │
#> │  Decision: Rejected ***                            │
#> ├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
#> │ Details:                                           │
#> │  Observations (T): 200                             │
#> │  Horizons (H): 4                                   │
#> │  Block length (L): 3                               │
#> │  Bootstrap replications: 199                       │
#> │  Significance level: 0.0500                        │
#> ╰────────────────────────────────────────────────────╯
#>