End-of-sample instability test on a Gaussian state-space filter

Tests whether the last m observations of a series are consistent with a linear-Gaussian state-space model fitted on the rest, in the regime where m is small (even m = 1) and smaller than the parameter count – where ordinary structural-break tests (Chow, sup-Wald) are undefined because the post-break parameters cannot be estimated. The statistic is the sum of the last m squared standardised one-step innovations from the filter; a break inflates it.

Usage

gmm_eos_test(
  prior,
  dynamics,
  measurement,
  y,
  m = 1L,
  method = c("chisq", "andrews"),
  alpha = 0.05
)

Arguments

prior: A single-component gmm giving the state prior (the test is defined for a fitted linear-Gaussian model; multi-component priors are not yet supported).
dynamics: A list with A (the state-transition matrix), Q (the process-noise covariance) and an optional offset b, or a function function(t) returning such a list, exactly as for gmm_filter(). Gaussian-sum (mixture) process noise is rejected: the calibrations below are defined for Gaussian innovations only.
measurement: A list with C (the observation matrix), R (the observation-noise covariance) and an optional offset d, or a function function(t) returning such a list, exactly as for gmm_filter(). Gaussian-sum measurement noise is rejected for the same reason.
y: A numeric vector or an n x d matrix of observations.
m: Integer; the number of end-of-sample observations to test, 1 <= m < nrow(y). The tiny-m regime (m = 1, 2, 3) is the point of the test.
method: Either "chisq" (parametric) or "andrews" (subsampling calibration after Andrews' P-test, asymptotic in n).
alpha: The nominal level used to set reject.

Value

An object of class gmm_eos_test: a list with the statistic, the p_value, the logical reject, the method, m, alpha, and the in-sample block statistics used for the subsampling calibration.

Details

Two calibrations are offered. method = "chisq" refers the statistic to a chi-square distribution on m * ncol(y) degrees of freedom, which is exact when the standardised innovations are Gaussian. method = "andrews" is a subsampling calibration after the P-test of Andrews (2003): the statistic's rank among the \(n - 2m + 1\) overlapping m-blocks of the earlier innovations gives the p-value \((1 + k) / (n - 2m + 2)\), where \(k\) is the number of blocks at least as large as the statistic. It differs from Andrews' test in two ways: the p-value counts the tested block itself, and the block statistics are computed at the supplied model instead of at estimates that leave out each block. It does not assume Gaussian innovations, so it suits heavy-tailed observation noise, but its validity is asymptotic: n large with m fixed, and innovations that are stationary and ergodic before the tested block. Its size equals the nominal level only when the end block is exchangeable with the earlier blocks, and in short series it can over-reject. The model is supplied exactly as for gmm_filter().

Two finite-sample cautions. The chi-square calibration is exact when the model is given; with parameters estimated on a short series it over-rejects (the end-of-sample vignette reports its size by series length). The subsampling calibration can also over-reject in short series, so neither method holds its nominal size when the model is estimated on little data. The subsampling p-value has a floor of 1 / (n - 2m + 2), so it can reject at level 0.05 only when n > 2m + 18.

References

Andrews, D. W. K. (2003). End-of-Sample Instability Tests. Econometrica, 71(6), 1661–1694.

Examples

prior <- gmm(weights = 1, means = list(0), covariances = list(matrix(10)))
dyn   <- list(A = matrix(1), Q = matrix(0.04))
meas  <- list(C = matrix(1), R = matrix(1))
set.seed(1)
y <- c(rnorm(119), 6)                # a stable series with a jump at the end
gmm_eos_test(prior, dyn, meas, y, m = 1L, method = "andrews")
#> $statistic
#> [1] 29.10382
#> 
#> $p_value
#> [1] 0.008333333
#> 
#> $reject
#> [1] TRUE
#> 
#> $method
#> [1] "andrews"
#> 
#> $m
#> [1] 1
#> 
#> $alpha
#> [1] 0.05
#> 
#> $df
#> [1] 1
#> 
#> $in_sample_blocks
#>   [1] 3.554750e-02 2.911332e-01 2.622507e-01 2.935641e+00 2.740067e-02
#>   [6] 7.895244e-01 2.170989e-01 3.560326e-01 1.118222e-01 2.736059e-01
#>  [11] 1.474811e+00 5.579593e-04 8.706621e-01 4.854571e+00 1.485184e+00
#>  [16] 3.742538e-03 5.791188e-04 7.207274e-01 3.412181e-01 7.431643e-02
#>  [21] 2.677158e-01 8.993352e-02 1.558138e-01 4.800755e+00 3.212741e-01
#>  [26] 2.176074e-02 4.452547e-02 1.857312e+00 4.746573e-02 4.000792e-01
#>  [31] 1.875224e+00 4.043256e-02 7.793913e-02 2.920177e-02 1.788863e+00
#>  [36] 5.051060e-02 2.733394e-02 2.813809e-02 1.407948e+00 4.448343e-01
#>  [41] 8.603388e-02 1.027993e-01 3.569358e-01 1.312872e-01 6.894557e-01
#>  [46] 4.858162e-01 1.594944e-01 4.797690e-01 5.285386e-02 5.052012e-01
#>  [51] 2.102961e-02 6.326433e-01 4.458853e-02 1.340566e+00 1.878720e+00
#>  [56] 2.617749e+00 6.391465e-01 1.606209e+00 1.785440e-01 8.511620e-02
#>  [61] 4.230079e+00 2.750905e-01 5.297272e-02 1.683986e-01 1.069329e+00
#>  [66] 1.162422e-05 3.265654e+00 2.189644e+00 5.903947e-04 3.412866e+00
#>  [71] 5.214728e-04 1.191428e+00 9.073974e-02 1.325579e+00 1.518131e+00
#>  [76] 1.514492e-01 1.198558e-01 1.407395e-02 2.671228e-02 2.180238e-01
#>  [81] 1.321657e-01 8.941230e-03 1.602585e+00 1.982918e+00 5.822228e-01
#>  [86] 1.511109e-01 9.587286e-01 1.896176e-01 6.426711e-02 1.310650e-02
#>  [91] 4.082348e-01 1.125279e+00 6.820047e-01 6.754606e-02 1.030703e+00
#>  [96] 9.832867e-03 3.034342e+00 6.243341e-01 1.529166e+00 1.108467e-01
#> [101] 1.645748e-01 7.144235e-02 4.141931e-01 1.938593e-01 1.404771e-01
#> [106] 3.552470e+00 3.515232e-01 4.364804e-01 4.234652e-03 1.507889e+00
#> [111] 1.192417e+00 5.428482e-01 1.233330e+00 9.516459e-01 1.582138e-01
#> [116] 2.436178e-01 1.144764e-01 5.765224e-02 2.531887e-01
#> 
#> attr(,"class")
#> [1] "gmm_eos_test"