Covariances of sample autocorrelations

Compute covariances of autocorrelations.

Usage

nvcovOfAcf(model, maxlag)

nvcovOfAcfBD(acf, ma, maxlag)

acfOfSquaredArmaModel(model, maxlag)

Arguments

model: a model, see Details.
maxlag: a positive integer number, the maximal lag.
acf: autocorrelations.
ma: a positive integer number, the order of the MA(q) model. The default is the maximal lag available in acf.

Details

nvcovOfAcf computes the unscaled asymptotic autocovariances of sample autocorrelations of ARMA models, under the classical assumptions when the Bartlett's formulas are valid. It works directly with the parameters of the model and uses Boshnakov (1996). Argument model can be any specification of ARMA models for which autocorrelations() will work, e.g. a list with components "ar", "ma", and "sigma2".

nvcovOfAcfBD computes the same quantities but uses the formula given by Brockwell & Davis (1991) (eq. (7.2.6.), p. 222), which is based on the autocorrelations of the model. Argument acf contains the autocorrelations.

For nvcovOfAcfBD, argument ma asks to treat the provided acf as that of MA(ma). Only the values for lags up to ma are used and the rest are set to zero, since the autocorrelations of MA(ma) models are zero for lags greater than ma. To force the use of all autocorrelations provided in acf, set ma to the maximal lag available in acf or omit ma, since this is its default.

acfOfSquaredArmaModel(model, maxlag) is a convenience function which computes the autocovariances of the "squared" model, see Boshnakov (1996).

Value

an (maxlag,maxlag)-matrix

References

Boshnakov GN (1996). “Bartlett's formulae---closed forms and recurrent equations.” Ann. Inst. Statist. Math., 48(1), 49--59. ISSN 0020-3157, doi:10.1007/BF00049288 .

Brockwell PJ, Davis RA (1991). Time series: theory and methods. 2nd ed.. Springer Series in Statistics. Berlin etc.: Springer-Verlag..

Author

Georgi N. Boshnakov

Note

The name of nvcovOfAcf stands for “n times the variance-covariance matrix”, so it needs to be divided by n to get the asymptotic variances and covariances.

Examples

## MA(2)
ma2 <- list(ma = c(0.8, 0.1), sigma2 = 1)
nv <- nvcovOfAcf(ma2, maxlag = 4)
d <- diag(nvcovOfAcf(ma2, maxlag = 7))
cbind(ma2 = 1.96 * sqrt(d) / sqrt(200), iid = 1.96/sqrt(200))
#>             ma2       iid
#> [1,] 0.09452061 0.1385929
#> [2,] 0.16935276 0.1385929
#> [3,] 0.17400093 0.1385929
#> [4,] 0.17400093 0.1385929
#> [5,] 0.17400093 0.1385929
#> [6,] 0.17400093 0.1385929
#> [7,] 0.17400093 0.1385929

acr <- autocorrelations(list(ma = c(0.8, 0.1)), maxlag = 7)
nvBD <- nvcovOfAcfBD(acr, 2, maxlag = 4)
all.equal(nv, nvBD) # TRUE
#> [1] TRUE