Tests for weak white noise

Carry out tests for weak white noise under GARCH, GARCH-type, and stochastic volatility null hypotheses.

Usage

acfGarchTest(acr, x, nlags, interval = 0.95)

acfWnTest(acr, x, nlags, interval = 0.95, ...)

Arguments

acr: autocorrelations.
x: time series.
nlags: how many lags to use.
interval: If not NULL, compute also confidence intervals with the specified coverage probability.
...: additional arguments for the computation of the variance matrix under the null hypothesis, passed on to nvarOfAcfKP.

Details

Unlike the autocorrelation IID test, the time series is needed here to estimate the covariance matrix of the autocorrelations under the null hypothesis.

acfGarchTest performs a test for uncorrelatedness of a time series. The null hypothesis is that the time series is GARCH, see Francq and Zakoian (2010) .

acfWnTest performs a test for uncorrelatedness of a time series under a weaker null hypothesis. The null hypothesis is that the time series is GARCH-type or from a stochasitc volatily model, see Kokoszka and Politis (2011) .

See the references for details and precise specification of the hypotheses.

The format of the return value is the same as for acfIidTest.

Value

a list with components "test" and "ci"

References

Francq C, Zakoian J (2010). GARCH models: structure, statistical inference and financial applications. John Wiley & Sons. ISBN 978-0-470-68391-0.

Kokoszka PS, Politis DN (2011). “Nonlinearity of ARCH and stochastic volatility models and Bartlett's formula.” Probability and Mathematical Statistics, 31(1), 47--59.

Author

Georgi N. Boshnakov

Examples

## see also the examples for \code{\link{whiteNoiseTest}}
set.seed(1234)
n <- 5000
x <- sarima:::rgarch1p1(n, alpha = 0.3, beta = 0.55, omega = 1, n.skip = 100)
x.acf <- autocorrelations(x)
x.pacf <- partialAutocorrelations(x)

acfGarchTest(x.acf, x = x, nlags = c(5,10,20))
#> $test
#>       h         Q      pval
#> [1,]  5  2.867048 0.7204741
#> [2,] 10  9.064633 0.5259808
#> [3,] 20 24.420187 0.2245192
#> 
#> $ci
#>               int           
#>  [1,] -0.05923087 0.05923087
#>  [2,] -0.05326386 0.05326386
#>  [3,] -0.05066191 0.05066191
#>  [4,] -0.04522221 0.04522221
#>  [5,] -0.04507543 0.04507543
#>  [6,] -0.04189048 0.04189048
#>  [7,] -0.04210249 0.04210249
#>  [8,] -0.03890700 0.03890700
#>  [9,] -0.03517673 0.03517673
#> [10,] -0.03572033 0.03572033
#> [11,] -0.03155470 0.03155470
#> [12,] -0.03222239 0.03222239
#> [13,] -0.03105026 0.03105026
#> [14,] -0.03010171 0.03010171
#> [15,] -0.02875639 0.02875639
#> [16,] -0.02686352 0.02686352
#> [17,] -0.02775162 0.02775162
#> [18,] -0.02700132 0.02700132
#> [19,] -0.02673210 0.02673210
#> [20,] -0.02709457 0.02709457
#> 
acfGarchTest(x.pacf, x = x, nlags = c(5,10,20))
#> $test
#>       h         Q      pval
#> [1,]  5  2.884453 0.7177943
#> [2,] 10  9.183271 0.5148034
#> [3,] 20 27.271537 0.1277959
#> 
#> $ci
#>               int           
#>  [1,] -0.05923087 0.05923087
#>  [2,] -0.05326386 0.05326386
#>  [3,] -0.05066191 0.05066191
#>  [4,] -0.04522221 0.04522221
#>  [5,] -0.04507543 0.04507543
#>  [6,] -0.04189048 0.04189048
#>  [7,] -0.04210249 0.04210249
#>  [8,] -0.03890700 0.03890700
#>  [9,] -0.03517673 0.03517673
#> [10,] -0.03572033 0.03572033
#> [11,] -0.03155470 0.03155470
#> [12,] -0.03222239 0.03222239
#> [13,] -0.03105026 0.03105026
#> [14,] -0.03010171 0.03010171
#> [15,] -0.02875639 0.02875639
#> [16,] -0.02686352 0.02686352
#> [17,] -0.02775162 0.02775162
#> [18,] -0.02700132 0.02700132
#> [19,] -0.02673210 0.02673210
#> [20,] -0.02709457 0.02709457
#> 

# do not compute CI's:
acfGarchTest(x.pacf, x = x, nlags = c(5,10,20), interval = NULL)
#> $test
#>       h         Q      pval
#> [1,]  5  2.884453 0.7177943
#> [2,] 10  9.183271 0.5148034
#> [3,] 20 27.271537 0.1277959
#> 

## plot methods call acfGarchTest() suitably if 'x' is given:
plot(x.acf, data = x)

plot(x.pacf, data = x)


## use 90% limits:
plot(x.acf, data = x, interval = 0.90)


acfWnTest(x.acf, x = x, nlags = c(5,10,20))
#> $test
#>       h         Q      pval
#> [1,]  5  2.695352 0.7468293
#> [2,] 10  8.157398 0.6134657
#> [3,] 20 20.691112 0.4155097
#> 
#> $ci
#>               int           
#>  [1,] -0.05923087 0.05923087
#>  [2,] -0.05326386 0.05326386
#>  [3,] -0.05066191 0.05066191
#>  [4,] -0.04522221 0.04522221
#>  [5,] -0.04507543 0.04507543
#>  [6,] -0.04189048 0.04189048
#>  [7,] -0.04210249 0.04210249
#>  [8,] -0.03890700 0.03890700
#>  [9,] -0.03517673 0.03517673
#> [10,] -0.03572033 0.03572033
#> [11,] -0.03155470 0.03155470
#> [12,] -0.03222239 0.03222239
#> [13,] -0.03105026 0.03105026
#> [14,] -0.03010171 0.03010171
#> [15,] -0.02875639 0.02875639
#> [16,] -0.02686352 0.02686352
#> [17,] -0.02775162 0.02775162
#> [18,] -0.02700132 0.02700132
#> [19,] -0.02673210 0.02673210
#> [20,] -0.02709457 0.02709457
#> 
nvarOfAcfKP(x, maxlag = 20)
#>  [1] 4.5663595 3.6926586 3.3406961 2.6618118 2.6445613 2.2840442 2.3072224
#>  [8] 1.9702857 1.6105891 1.6607521 1.2959906 1.3514172 1.2548862 1.1793864
#> [15] 1.0763228 0.9392902 1.0024221 0.9489510 0.9301220 0.9555168
whiteNoiseTest(x.acf, h0 = "arch-type", x = x, nlags = c(5,10,20))
#> $test
#>       h         Q      pval
#> [1,]  5  2.695352 0.7468293
#> [2,] 10  8.157398 0.6134657
#> [3,] 20 20.691112 0.4155097
#> 
#> $ci
#>               int           
#>  [1,] -0.05923087 0.05923087
#>  [2,] -0.05326386 0.05326386
#>  [3,] -0.05066191 0.05066191
#>  [4,] -0.04522221 0.04522221
#>  [5,] -0.04507543 0.04507543
#>  [6,] -0.04189048 0.04189048
#>  [7,] -0.04210249 0.04210249
#>  [8,] -0.03890700 0.03890700
#>  [9,] -0.03517673 0.03517673
#> [10,] -0.03572033 0.03572033
#> [11,] -0.03155470 0.03155470
#> [12,] -0.03222239 0.03222239
#> [13,] -0.03105026 0.03105026
#> [14,] -0.03010171 0.03010171
#> [15,] -0.02875639 0.02875639
#> [16,] -0.02686352 0.02686352
#> [17,] -0.02775162 0.02775162
#> [18,] -0.02700132 0.02700132
#> [19,] -0.02673210 0.02673210
#> [20,] -0.02709457 0.02709457
#>