General-to-Specific (GETS) Modelling of an AR-X model (the mean specification) with log-ARCH-X errors (the log-variance specification).

The starting model, an object of the 'arx' class, is referred to as the General Unrestricted Model (GUM). The getsm function undertakes multi-path GETS modelling of the mean specification, whereas getsv does the same for the log-variance specification. The diagnostic tests are undertaken on the standardised residuals, and the keep option enables regressors to be excluded from possible removal.

Usage

##GETS-modelling of mean specification:
getsm(object, t.pval=0.05, wald.pval=t.pval, vcov.type=NULL, 
    do.pet=TRUE, ar.LjungB=list(lag=NULL, pval=0.025), 
    arch.LjungB=list(lag=NULL, pval=0.025), normality.JarqueB=NULL, 
    user.diagnostics=NULL, info.method=c("sc","aic","aicc", "hq"),
    gof.function=NULL, gof.method=NULL, keep=NULL, include.gum=FALSE,
    include.1cut=TRUE, include.empty=FALSE, max.paths=NULL, tol=1e-07,
    turbo=FALSE, print.searchinfo=TRUE, plot=NULL, alarm=FALSE)

##GETS modelling of log-variance specification:
getsv(object, t.pval=0.05, wald.pval=t.pval,
    do.pet=TRUE, ar.LjungB=list(lag=NULL, pval=0.025),
    arch.LjungB=list(lag=NULL, pval=0.025), normality.JarqueB=NULL,
    user.diagnostics=NULL, info.method=c("sc","aic","aicc","hq"),
    gof.function=NULL, gof.method=NULL, keep=c(1), include.gum=FALSE,
    include.1cut=TRUE, include.empty=FALSE, max.paths=NULL, tol=1e-07,
    turbo=FALSE, print.searchinfo=TRUE, plot=NULL, alarm=FALSE)

Arguments

object

an object of class 'arx'

t.pval

numeric value between 0 and 1. The significance level used for the two-sided regressor significance t-tests

wald.pval

numeric value between 0 and 1. The significance level used for the Parsimonious Encompassing Tests (PETs). By default, it is the same as t.pval

vcov.type

the type of variance-covariance matrix used. If NULL (default), then the type used in the estimation of the 'arx' object is used. This can be overridden by either "ordinary" (i.e. the ordinary variance-covariance matrix) or "white" (i.e. the White (1980) heteroscedasticity robust variance-covariance matrix)

do.pet

logical. If TRUE (default), then a Parsimonious Encompassing Test (PET) against the GUM is undertaken at each regressor removal for the joint significance of all the deleted regressors along the current path. If FALSE, then a PET is not undertaken at each regressor removal

ar.LjungB

a list with named items lag and pval, a two-element numeric vector where the first element contains the lag and the second the p-value, or NULL. In the first case, lag contains the order of the Ljung and Box (1979) test for serial correlation in the standardised residuals, and pval contains the significance level. If lag=NULL (default), then the order used is that of the estimated 'arx' object. If ar.Ljungb=NULL, then the standardised residuals are not checked for serial correlation

arch.LjungB

a list with named items lag and pval, a two-element numeric vector where the first element contains the lag and the second the p-value, or NULL. In the first case, lag contains the order of the Ljung and Box (1979) test for serial correlation in the squared standardised residuals, and pval contains the significance level. If lag=NULL (default), then the order used is that of the estimated 'arx' object. If arch.Ljungb=NULL, then the standardised residuals are not checked for ARCH

normality.JarqueB

a value between 0 and 1, or NULL. In the former case, the Jarque and Bera (1980) test for non-normality is conducted using a significance level equal to the numeric value. If NULL, then no test for non-normality is undertaken

user.diagnostics

NULL or a list with two entries, name and pval, see the user.fun argument in diagnostics

info.method

character string, "sc" (default), "aic" or "hq", which determines the information criterion to be used when selecting among terminal models. The abbreviations are short for the Schwarz or Bayesian information criterion (sc), the Akaike information criterion (aic) and the Hannan-Quinn (hq) information criterion

gof.function

NULL (default) or a list, see getsFun. If NULL, then infocrit is used

gof.method

NULL (default) or a character, see getsFun. If NULL and gof.function is also NULL, then the best goodness-of-fit is characterised by a minimum value

keep

the regressors to be excluded from removal in the specification search. Note that keep=c(1) is obligatory when using getsv. This excludes the log-variance intercept from removal. The regressor numbering is contained in the reg.no column of the GUM

include.gum

logical. If TRUE, then the GUM (i.e. the starting model) is included among the terminal models. If FALSE (default), then the GUM is not included

include.1cut

logical. If TRUE, then the 1-cut model is added to the list of terminal models. If FALSE (default), then the 1-cut is not added, unless it is a terminal model in one of the paths

include.empty

logical. If TRUE, then an empty model is included among the terminal models, if it passes the diagnostic tests, even if it is not equal to one of the terminals. If FALSE (default), then the empty model is not included (unless it is one of the terminals)

max.paths

NULL (default) or an integer greater than 0. If NULL, then there is no limit to the number of paths. If an integer (e.g. 1), then this integer constitutes the maximum number of paths searched (e.g. a single path)

tol

numeric value. The tolerance for detecting linear dependencies in the columns of the variance-covariance matrix when computing the Wald-statistic used in the Parsimonious Encompassing Tests (PETs), see the qr.solve function

turbo

logical. If TRUE, then (parts of) paths are not searched twice (or more) unnecessarily, thus yielding a significant potential for speed-gain. However, the checking of whether the search has arrived at a point it has already been comes with a slight computational overhead. Accordingly, if turbo=TRUE, then the total search time might in fact be higher than if turbo=FALSE. This happens if estimation is very fast, say, less than quarter of a second. Hence the default is FALSE

print.searchinfo

logical. If TRUE (default), then a print is returned whenever simiplification along a new path is started

plot

NULL or logical. If TRUE, then the fitted values and the residuals of the final model are plotted after model selection. If FALSE, then they are not. If NULL (default), then the value set by options determines whether a plot is produced or not

alarm

logical. If TRUE, then a sound or beep is emitted (in order to alert the user) when the model selection ends

Details

For an overview, see Pretis, Reade and Sucarrat (2018): doi:10.18637/jss.v086.i03 .

The arguments user.diagnostics and gof.function enable the specification of user-defined diagnostics and a user-defined goodness-of-fit function. For the former, see the documentation of diagnostics. For the latter, the principles of the same arguments in getsFun are followed, see its documentation under "Details", and Sucarrat (2020): https://journal.r-project.org/archive/2021/RJ-2021-024/.

Value

A list of class 'gets'

References

C. Jarque and A. Bera (1980): 'Efficient Tests for Normality, Homoscedasticity and Serial Independence'. Economics Letters 6, pp. 255-259. doi:10.1016/0165-1765(80)90024-5

G. Ljung and G. Box (1979): 'On a Measure of Lack of Fit in Time Series Models'. Biometrika 66, pp. 265-270

Felix Pretis, James Reade and Genaro Sucarrat (2018): 'Automated General-to-Specific (GETS) Regression Modeling and Indicator Saturation for Outliers and Structural Breaks'. Journal of Statistical Software 86, Number 3, pp. 1-44. doi:10.18637/jss.v086.i03

Genaro Sucarrat (2020): 'User-Specified General-to-Specific and Indicator Saturation Methods'. The R Journal 12:2, pages 388-401. https://journal.r-project.org/archive/2021/RJ-2021-024/

Author

Genaro Sucarrat, http://www.sucarrat.net/

See also

Extraction functions: coef.gets, fitted.gets, paths, plot.gets, print.gets,
residuals.gets, summary.gets, terminals, vcov.gets

Related functions: arx, eqwma, leqwma, zoo, getsFun, qr.solve

Examples

##Simulate from an AR(1):
set.seed(123)
y <- arima.sim(list(ar=0.4), 80)

##Simulate four independent Gaussian regressors:
xregs <- matrix(rnorm(2*80), 80, 2)

##estimate an AR(2) with intercept and four conditioning
##regressors in the mean, and a log-ARCH(3) with log(xregs^2) as
##regressors in the log-variance:
gum01 <- arx(y, mc=TRUE, ar=1:2, mxreg=xregs, arch=1:3,
  vxreg=log(xregs^2))

##GETS model selection of the mean:
meanmod01 <- getsm(gum01)
#> 
#> GUM mean equation:
#> 
#>        reg.no. keep      coef std.error  t-stat  p-value   
#> mconst       1    0  0.052513  0.101817  0.5158 0.607583   
#> ar1          2    0  0.312720  0.115433  2.7091 0.008401 **
#> ar2          3    0 -0.062448  0.118204 -0.5283 0.598886   
#> mxreg1       4    0  0.166409  0.101452  1.6403 0.105250   
#> mxreg2       5    0 -0.013699  0.111233 -0.1232 0.902323   
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> GUM log-variance equation:
#> 
#>             coef std.error  t-stat p-value  
#> vconst  0.406367  0.414376  0.9617 0.32675  
#> arch1   0.192973  0.117722  1.6392 0.10572  
#> arch2   0.245569  0.125909  1.9504 0.05519 .
#> arch3  -0.076098  0.118621 -0.6415 0.52331  
#> vxreg1 -0.196355  0.127679 -1.5379 0.12865  
#> vxreg2  0.158193  0.131686  1.2013 0.23374  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics:
#> 
#>                   Chi-sq df p-value
#> Ljung-Box AR(3)   3.9061  3 0.27179
#> Ljung-Box ARCH(4) 4.8830  4 0.29952
#> 
#> 4 path(s) to search
#> Searching: 
#> 1 
#> 2 
#> 3 
#> 4 
#> 
#>   Path 1: 1 5 3 4 
#>   Path 2: 3 5 1 4 
#>   Path 3: 4 5 3 1 
#>   Path 4: 5 1 3 4 
#> 
#> Terminal models:
#> 
#>                 info(sc)      logl  n k
#> spec 1 (1-cut): 2.776498 -106.1051 78 1
#> 
#> Retained regressors (final model):
#> 
#>   ar1 

##GETS model selection of the log-variance:
varmod01 <- getsv(gum01)
#> GUM log-variance equation:
#> 
#>        reg.no. keep      coef std.error  t-stat p-value  
#> vconst       1    1  0.406367  0.414376  0.9617 0.32675  
#> arch1        2    0  0.192973  0.117722  1.6392 0.10572  
#> arch2        3    0  0.245569  0.125909  1.9504 0.05519 .
#> arch3        4    0 -0.076098  0.118621 -0.6415 0.52331  
#> vxreg1       5    0 -0.196355  0.127679 -1.5379 0.12865  
#> vxreg2       6    0  0.158193  0.131686  1.2013 0.23374  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics:
#> 
#>                   Chi-sq df p-value
#> Ljung-Box AR(3)   3.9061  3 0.27179
#> Ljung-Box ARCH(4) 4.8830  4 0.29952
#> 
#> 5 path(s) to search
#> Searching: 
#> 1 
#> 2 
#> 3 
#> 4 
#> 5 
#> 
#>   Path 1: 2 4 6 5 3 
#>   Path 2: 3 4 6 5 
#>   Path 3: 4 6 5 3 
#>   Path 4: 5 4 6 3 
#>   Path 5: 6 4 5 3 
#> 
#> Terminal models:
#> 
#>                 info(sc)      logl  n k
#> spec 1 (1-cut): 2.632361 -96.55478 75 1
#> spec 2:         2.748094 -98.73603 75 2
#> 
#> Retained regressors (final model):
#> 
#>   vconst 

##GETS model selection of the mean with the mean intercept
##excluded from removal:
meanmod02 <- getsm(gum01, keep=1)
#> 
#> GUM mean equation:
#> 
#>        reg.no. keep      coef std.error  t-stat  p-value   
#> mconst       1    1  0.052513  0.101817  0.5158 0.607583   
#> ar1          2    0  0.312720  0.115433  2.7091 0.008401 **
#> ar2          3    0 -0.062448  0.118204 -0.5283 0.598886   
#> mxreg1       4    0  0.166409  0.101452  1.6403 0.105250   
#> mxreg2       5    0 -0.013699  0.111233 -0.1232 0.902323   
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> GUM log-variance equation:
#> 
#>             coef std.error  t-stat p-value  
#> vconst  0.406367  0.414376  0.9617 0.32675  
#> arch1   0.192973  0.117722  1.6392 0.10572  
#> arch2   0.245569  0.125909  1.9504 0.05519 .
#> arch3  -0.076098  0.118621 -0.6415 0.52331  
#> vxreg1 -0.196355  0.127679 -1.5379 0.12865  
#> vxreg2  0.158193  0.131686  1.2013 0.23374  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics:
#> 
#>                   Chi-sq df p-value
#> Ljung-Box AR(3)   3.9061  3 0.27179
#> Ljung-Box ARCH(4) 4.8830  4 0.29952
#> 
#> 3 path(s) to search
#> Searching: 
#> 1 
#> 2 
#> 3 
#> 
#>   Path 1: 3 5 4 
#>   Path 2: 4 5 3 
#>   Path 3: 5 3 4 
#> 
#> Terminal models:
#> 
#>                 info(sc)      logl  n k
#> spec 1 (1-cut): 2.656706 -99.25482 78 2
#> 
#> Retained regressors (final model):
#> 
#>   mconst   ar1 

##GETS model selection of the mean with non-default
#serial-correlation diagnostics settings:
meanmod03 <- getsm(gum01, ar.LjungB=list(pval=0.05))
#> 
#> GUM mean equation:
#> 
#>        reg.no. keep      coef std.error  t-stat  p-value   
#> mconst       1    0  0.052513  0.101817  0.5158 0.607583   
#> ar1          2    0  0.312720  0.115433  2.7091 0.008401 **
#> ar2          3    0 -0.062448  0.118204 -0.5283 0.598886   
#> mxreg1       4    0  0.166409  0.101452  1.6403 0.105250   
#> mxreg2       5    0 -0.013699  0.111233 -0.1232 0.902323   
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> GUM log-variance equation:
#> 
#>             coef std.error  t-stat p-value  
#> vconst  0.406367  0.414376  0.9617 0.32675  
#> arch1   0.192973  0.117722  1.6392 0.10572  
#> arch2   0.245569  0.125909  1.9504 0.05519 .
#> arch3  -0.076098  0.118621 -0.6415 0.52331  
#> vxreg1 -0.196355  0.127679 -1.5379 0.12865  
#> vxreg2  0.158193  0.131686  1.2013 0.23374  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics:
#> 
#>                   Chi-sq df p-value
#> Ljung-Box AR(3)   3.9061  3 0.27179
#> Ljung-Box ARCH(4) 4.8830  4 0.29952
#> 
#> 4 path(s) to search
#> Searching: 
#> 1 
#> 2 
#> 3 
#> 4 
#> 
#>   Path 1: 1 5 3 4 
#>   Path 2: 3 5 1 4 
#>   Path 3: 4 5 3 1 
#>   Path 4: 5 1 3 4 
#> 
#> Terminal models:
#> 
#>                 info(sc)      logl  n k
#> spec 1 (1-cut): 2.776498 -106.1051 78 1
#> 
#> Retained regressors (final model):
#> 
#>   ar1 

##GETS model selection of the mean with very liberal
##(20 percent) significance levels:
meanmod04 <- getsm(gum01, t.pval=0.2)
#> 
#> GUM mean equation:
#> 
#>        reg.no. keep      coef std.error  t-stat  p-value   
#> mconst       1    0  0.052513  0.101817  0.5158 0.607583   
#> ar1          2    0  0.312720  0.115433  2.7091 0.008401 **
#> ar2          3    0 -0.062448  0.118204 -0.5283 0.598886   
#> mxreg1       4    0  0.166409  0.101452  1.6403 0.105250   
#> mxreg2       5    0 -0.013699  0.111233 -0.1232 0.902323   
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> GUM log-variance equation:
#> 
#>             coef std.error  t-stat p-value  
#> vconst  0.406367  0.414376  0.9617 0.32675  
#> arch1   0.192973  0.117722  1.6392 0.10572  
#> arch2   0.245569  0.125909  1.9504 0.05519 .
#> arch3  -0.076098  0.118621 -0.6415 0.52331  
#> vxreg1 -0.196355  0.127679 -1.5379 0.12865  
#> vxreg2  0.158193  0.131686  1.2013 0.23374  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics:
#> 
#>                   Chi-sq df p-value
#> Ljung-Box AR(3)   3.9061  3 0.27179
#> Ljung-Box ARCH(4) 4.8830  4 0.29952
#> 
#> 3 path(s) to search
#> Searching: 
#> 1 
#> 2 
#> 3 
#> 
#>   Path 1: 1 5 3 
#>   Path 2: 3 5 1 
#>   Path 3: 5 1 3 
#> 
#> Terminal models:
#> 
#>                 info(sc)     logl  n k
#> spec 1 (1-cut): 2.767504 -103.576 78 2
#> 
#> Retained regressors (final model):
#> 
#>   ar1   mxreg1 

##GETS model selection of log-variance with all the
##log-ARCH terms excluded from removal:
varmod03 <- getsv(gum01, keep=2:4)
#> Warning: Regressor 1 included into 'keep'
#> GUM log-variance equation:
#> 
#>        reg.no. keep      coef std.error  t-stat p-value  
#> vconst       1    1  0.406367  0.414376  0.9617 0.32675  
#> arch1        2    1  0.192973  0.117722  1.6392 0.10572  
#> arch2        3    1  0.245569  0.125909  1.9504 0.05519 .
#> arch3        4    1 -0.076098  0.118621 -0.6415 0.52331  
#> vxreg1       5    0 -0.196355  0.127679 -1.5379 0.12865  
#> vxreg2       6    0  0.158193  0.131686  1.2013 0.23374  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics:
#> 
#>                   Chi-sq df p-value
#> Ljung-Box AR(3)   3.9061  3 0.27179
#> Ljung-Box ARCH(4) 4.8830  4 0.29952
#> 
#> 2 path(s) to search
#> Searching: 
#> 1 
#> 2 
#> 
#>   Path 1: 5 6 
#>   Path 2: 6 5 
#> 
#> Terminal models:
#> 
#>                 info(sc)      logl  n k
#> spec 1 (1-cut): 2.811774 -96.80654 75 4
#> 
#> Retained regressors (final model):
#> 
#>   vconst   arch1   arch2   arch3