Tests the null hypothesis of equal mean squared forecast error between the (restricted) benchmark and the (unrestricted) alternative nested model, using the MSE-F statistic of McCracken (2007). Under the null, the alternative does not reduce MSFE relative to the benchmark.
Value
A list with class "mse_f_test" containing:
- statistic
The MSE-F test statistic.
- msfe1
Mean squared forecast error of the benchmark.
- msfe2
Mean squared forecast error of the alternative.
- n
Number of out-of-sample observations.
- h
Forecast horizon.
Details
The MSE-F statistic is: $$\text{MSE-F} = (T - h + 1) \cdot \frac{\text{MSFE}_1 - \text{MSFE}_2}{\text{MSFE}_2}$$ where \(T\) is the number of out-of-sample observations and \(h\) is the forecast horizon.
Critical values are non-standard and depend on the number of extra regressors in the alternative model (\(k_2\)) and on \(\pi = P/R\) (out-of-sample size relative to estimation window). See McCracken (2007, Table 1) for asymptotic critical values.
References
McCracken, M.W. (2007). Asymptotics for out of sample tests of Granger causality. Journal of Econometrics, 140(2), 719-752.
Clark, T.E. and McCracken, M.W. (2001). Tests of Equal Forecast Accuracy and Encompassing for Nested Models. Journal of Econometrics, 105(1), 85-110.
Examples
set.seed(42)
e1 <- rnorm(100)
e2 <- rnorm(100, sd = 0.9)
mse_f_test(e1, e2)
#>
#> ╭────────────────────────────────────────────────────╮
#> │ MSE-F Test for Equal Forecast Accuracy │
#> │ (McCracken, 2007) │
#> ├────────────────────────────────────────────────────┤
#> │ H0: MSFE benchmark <= MSFE alternative │
#> │ H1: Alternative reduces MSFE │
#> ├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
#> │ Test Results: │
#> │ MSE-F statistic: 62.3868 │
#> │ MSFE benchmark: 1.0746 │
#> │ MSFE alternative: 0.6618 │
#> ├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
#> │ Details: │
#> │ Observations (n): 100 │
#> │ Forecast horizon (h): 1 │
#> │ Note: Critical values are non-standard. │
#> │ See McCracken (2007, Table 1). │
#> ╰────────────────────────────────────────────────────╯
#>