Diagnostic checks for mixture autoregressive models

Carry out diagnostic checks and tests on fitted mixAR models.

Usage

# S3 method for MixAR
tsdiag(object, gof.lag = NULL, y, ask = interactive(), ..., 
       plot = interactive(), std.resid = FALSE)

mixAR_diag(model, y, ...)

Arguments

model,object

the model on which to perform the checks, an object from class MixAR. model can also be the output list from fit_mixAR.

gof.lag

Goodness of fit lag(s) for the Ljung-Box tests. Vector containing one or more positive integers. max(gof.lag) is the maximal lag in the acf and pacf plots.

how many lags to compute for acf and pacf? The default is as that of lag.max for acf.

y

a time series, currently a numeric vector.

ask

if TRUE, ask (using a menu) which plot to present. Otherwise just plot the selected plots. ask is ignored if only one plot is selected with argument plot.

plot

if TRUE, the default, produce diagnostic plots. If FALSE don't produce plots. Otherwise, a numeric vector of integers defining a subset of plots to consider, see Details.

std.resid

if TRUE standardise the ordinary residuals using the conditional standard deviations. NOTE: the default is currently FALSE but it may soon be changed to TRUE.

...

for mixAR_diag, passed on to tsdiag.

Details

It is recommended to use tsdiag. mixAR_diag is essentially deprecated and is still here for compatibility with old code. Moreover, the tsdiag method is more flexible. The only advantage of mixAR_diag is that it accepts also a list for argument model but this is equivalent to calling tsdiag with object = model$model.

The function calculates several types of residuals, provides diagnostic plots for each of them, and returns numerical results. The following choices are currently available:

ACF/PACF of residuals,
ACF/PACF of U_residuals,
ACF/PACF of tau_residuals,
ACF/Histogram of tau_residuals.

In interactive sessions the user is presented with a menu to select plot(s) from the available ones. The choice can be restricted to a subset of them by giving argument plot a vector of integers. This is most useful to select a particular plot, with somethinng like plot = 2 in the call to tsdiag. plot is used as an index vector, so plot = -1 would remove the first item listed above from the offered alternatives.

Transformations on the data are performed, as described in Smith (1985).

Four types of residuals are computed:

ordinary residuals

difference (possibly scaled) between observed values and point predictions.

U_residuals/PIT residuals

probability integral transform of the data using the CDF of the conditional distributions implied by the fitted model. For a good model these should resemble an IID sequence uniformly distributed on (0,1).

V_residuals

set of transformed U_residuals with the quantile function of the standard normal distribution (qnorm). For a good model these should resemble an IID sequence from N(0,1).

tau_residuals

These residuals are calculated as the component specific residual e_tk divided by its corresponding scale sigma_k, according to under which component y_t has largest density. Under correct model specification, these should be jointly Normal. Shapiro-Wilk test is performed on this set of residual to assess the hypothesis.

is applied to V- and tau-residuals.

Value

returns invisibly a list with class "tsdiagMixAR", currently containing the following components:

residuals: ordinary residuals,
U_residuals: see Details,
V_residuals: see Details,
tau_residuals: see Details,
BIC: the value of the BIC criterion, a number.

Each component, except BIC, is a list containing the residuals in component value, Ljung-Box test in "Ljung-Box" and possibly other tests suitable for the corresponding type of residuals.

References

Smith JQ (1985). “Diagnostic checks of non-standard time series models.” Journal of Forecasting, 4(3), 283-291. doi:10.1002/for.3980040305 , https://onlinelibrary.wiley.com/doi/pdf/10.1002/for.3980040305, https://onlinelibrary.wiley.com/doi/abs/10.1002/for.3980040305.

Wong CS, Li WK (2000). “On a mixture autoregressive model.” J. R. Stat. Soc., Ser. B, Stat. Methodol. , 62(1), 95-115.

Author

Davide Ravagli and Georgi N. Boshnakov

Note

This function should be used for diagnostic checking of MixARGaussian objects only.

Examples

model1 <- new("MixARGaussian", prob = c(0.5, 0.5), scale = c(1, 2),
              arcoef = list(-0.5, 1.1))
set.seed(123)
y <- mixAR_sim(model1, 400, c(0,0,0), nskip = 100) 

fit1 <- fit_mixAR(y, model1)
d <- tsdiag(fit1$model, c(10, 20, 50), y)
d
#> 
#>  -------------------------------------------------- 
#> Tests on the ordinary residuals
#>  -------------------------------------------------- 
#> $`Ljung-Box`
#>        df   p.value
#> Lag_10 10 0.1363658
#> Lag_20 20 0.1693712
#> Lag_50 50 0.4213514
#> 
#> 
#>  -------------------------------------------------- 
#> Tests on the U_residuals
#>  -------------------------------------------------- 
#> $`Ljung-Box`
#>        df    p.value
#> Lag_10 10 0.43658682
#> Lag_20 20 0.29918541
#> Lag_50 50 0.03460944
#> 
#> $KS
#> 
#> 	Asymptotic one-sample Kolmogorov-Smirnov test
#> 
#> data:  cdf
#> D = 0.028344, p-value = 0.9057
#> alternative hypothesis: two-sided
#> 
#> 
#> 
#>  -------------------------------------------------- 
#> Tests on the V_residuals
#>  -------------------------------------------------- 
#> $`Ljung-Box`
#>        df    p.value
#> Lag_10 10 0.68338359
#> Lag_20 20 0.47238093
#> Lag_50 50 0.06870134
#> 
#> $`Shapiro-Wilk`
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  v
#> W = 0.99784, p-value = 0.8928
#> 
#> 
#> 
#>  -------------------------------------------------- 
#> Tests on the tau_residuals
#>  -------------------------------------------------- 
#> $`Ljung-Box`
#>        df   p.value
#> Lag_10 10 0.8708327
#> Lag_20 20 0.9790725
#> Lag_50 50 0.6138649
#> 
#> $`Shapiro-Wilk`
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  err2
#> W = 0.99563, p-value = 0.3328
#> 
#> 
## This will put each plot in a separate file (mydiag01.pdf, ..., mydiag04.pdf)
## pdf("mydiag%02d.pdf", onefile = FALSE)
## d <- tsdiag(fit1$model, c(10, 20, 50), y,  ask = FALSE)
## dev.off()