Confidence and acceptance intervals in package sarima

Compute confidence and acceptance intervals for sample autocorrelations under assumptions chosen by the user.

Usage

# S4 method for SampleAutocorrelations
confint(object, parm, level = 0.95, se = FALSE, maxlag, ..., assuming)

Arguments

object

an object containing sample autocorrelations (sacfs).

parm

which parameters to include, as for stats::confint.

level

coverage level, such as 0.95.

se

If TRUE return also standard errors.

assuming

under what assumptions to do the computations? Currently can be "iid", "garch", a fitted model, or a theoretical model, see Details.

maxlag

maximal lag to include

...

further arguments for se.

Details

For lags not fixed by the assumed model the computed intervals are confidence intervals.

The autocorrelations postulated by the null model (argument assuming) are usually fixed for some lags. For such lags it doesn't make sense to talk about confidence intervals. We use the term acceptance interval in this case since the sacfs for such lags fall in the corresponding intervals with high probability if the null model is correct.

If assuming is "iid" (strong white noise), then all autocorrelations in the null model are fixed (to zero) and the limits of the resulting acceptance intervals are ethose from the familiar plots produced by base-R's function acf.

If assuming is a fitted MA(q) model, e.g. obtained from arima(), then for lags \(1,\ldots,q\) we get confidence intervals, while for lags greater than \(q\) the intervals are acceptance intervals.

The autocorrelations of ARMA models with non-trivial autoregressive part may also have structural patterns of zeroes (for example some seasonal models), leading to acceptance intervals for such lags.

If assuming specifies a theoretical (non-fitted) model, then the autocorrelation function of the null model is completely fixed and we get acceptance intervals for all lags.

The return value is a matrix with one row for each requested lag, containg the lag, lower bound, upper bound, estimate for acf(lag) and the value of acf(lag) under H0 (if fixed) and NA if not fixed under H0. The null model is stored as attribute "assuming".

Note: When assuming = "garch" it is currently necessary to submit the time series from which the autocorrelations were computed as argument x.

Value

a matrix as described in section ‘Details’;

if se = TRUE, a column giving the standard errors of the sample autocorrelations is appended.

See also

se

vignette("white_noise_tests", package = "sarima")

vignette("garch_tests_example", package = "sarima")

Examples

set.seed(1234)
v1 <- arima.sim(n = 100, list(ma = c(0.8, 0.1), sd = 1))
v1.acf <- autocorrelations(v1, maxlag = 10)

confint(v1.acf, parm = 1:4, assuming = "iid")
#>       Lag      2.5 %    97.5 %  Estimate H0_fixed
#> Lag_1   1 -0.1959964 0.1959964 0.6194318        0
#> Lag_2   2 -0.1959964 0.1959964 0.1960955        0
#> Lag_3   3 -0.1959964 0.1959964 0.1071118        0
#> Lag_4   4 -0.1959964 0.1959964 0.1348527        0
#> attr(,"assuming")
#> [1] "iid"
confint(v1.acf,  assuming = "iid", maxlag = 4) # same
#>       Lag      2.5 %    97.5 %  Estimate H0_fixed
#> Lag_1   1 -0.1959964 0.1959964 0.6194318        0
#> Lag_2   2 -0.1959964 0.1959964 0.1960955        0
#> Lag_3   3 -0.1959964 0.1959964 0.1071118        0
#> Lag_4   4 -0.1959964 0.1959964 0.1348527        0
#> attr(,"assuming")
#> [1] "iid"

## a fitted MA(2) - rho_1, rho_2 not fixed, the rest fixed
ma2fitted <- arima(v1, order = c(0,0,2), include.mean=FALSE)
confint(v1.acf, assuming = ma2fitted, maxlag = 4)
#>       Lag       2.5 %    97.5 %  Estimate H0_fixed
#> Lag_1   1  0.48912998 0.7497336 0.6194318       NA
#> Lag_2   2 -0.04684288 0.4390339 0.1960955       NA
#> Lag_3   3 -0.24926724 0.2492672 0.1071118        0
#> Lag_4   4 -0.24926724 0.2492672 0.1348527        0
#> attr(,"assuming")
#> 
#> Call:
#> arima(x = v1, order = c(0, 0, 2), include.mean = FALSE)
#> 
#> Coefficients:
#>          ma1     ma2
#>       0.9902  0.1126
#> s.e.  0.0884  0.0910
#> 
#> sigma^2 estimated as 0.9474:  log likelihood = -139.99,  aic = 285.99

## a theoretical MA(2) model, all acfs fixed under H0
ma2 <- MaModel(ma = c(0.8, 0.1), sigma2 = 1)
confint(v1.acf, assuming = ma2, maxlag = 4)
#>       Lag      2.5 %    97.5 %  Estimate H0_fixed
#> Lag_1   1  0.7463301 1.0136699 0.6194318     0.88
#> Lag_2   2 -0.1394966 0.3394966 0.1960955     0.10
#> Lag_3   3 -0.2460700 0.2460700 0.1071118     0.00
#> Lag_4   4 -0.2460700 0.2460700 0.1348527     0.00
#> attr(,"assuming")
#> An object of class "MaModel"
#> mean:  0 
#> sigmaSq:  1 
#> 
#> slot "ar":
#> An object of class "BJFilter"
#> order:  0 
#> Coefficients: 
#> numeric(0)
#> 
#> slot "ma":
#> An object of class "SPFilter"
#> order:  2 
#> Coefficients: 
#> [1] 0.8 0.1

# a weak white noise null
confint(v1.acf, assuming = "garch", maxlag = 4, x = v1)
#>       Lag      2.5 %    97.5 %  Estimate H0_fixed
#> Lag_1   1 -0.2414976 0.2414976 0.6194318        0
#> Lag_2   2 -0.1919697 0.1919697 0.1960955        0
#> Lag_3   3 -0.1882971 0.1882971 0.1071118        0
#> Lag_4   4 -0.1626543 0.1626543 0.1348527        0
#> attr(,"assuming")
#> [1] "garch"