Estimate an AR-X model with log-ARCH-X errors

Estimation by OLS, two-step OLS if a variance specification is specified: In the first the mean specification (AR-X) is estimated, whereas in the second step the log-variance specification (log-ARCH-X) is estimated.

The AR-X mean specification can contain an intercept, AR-terms, lagged moving averages of the regressand and other conditioning covariates ('X'). The log-variance specification can contain log-ARCH terms, asymmetry or 'leverage' terms, log(EqWMA) where EqWMA is a lagged equally weighted moving average of past squared residuals (a volatility proxy) and other conditioning covariates ('X').

Usage

arx(y, mc=TRUE, ar=NULL, ewma=NULL, mxreg=NULL, vc=FALSE,
  arch=NULL, asym=NULL, log.ewma=NULL, vxreg=NULL, zero.adj=0.1,
  vc.adj=TRUE, vcov.type=c("ordinary", "white", "newey-west"),
  qstat.options=NULL, normality.JarqueB=FALSE, user.estimator=NULL,
  user.diagnostics=NULL, tol=1e-07, LAPACK=FALSE, singular.ok=TRUE,
  plot=NULL)

Arguments

y: numeric vector, time-series or zoo object. Missing values in the beginning and at the end of the series is allowed, as they are removed with the na.trim command
mc: logical. TRUE (default) includes an intercept in the mean specification, whereas FALSE does not
ar: either NULL (default) or an integer vector, say, c(2,4) or 1:4. The AR-lags to include in the mean specification. If NULL, then no lags are included
ewma: either NULL (default) or a list with arguments sent to the eqwma function. In the latter case a lagged moving average of y is included as a regressor
mxreg: either NULL (default) or a numeric vector or matrix, say, a zoo object, of conditioning variables. Note that, if both y and mxreg are zoo objects, then their samples are chosen to match
vc: logical. TRUE includes an intercept in the log-variance specification, whereas FALSE (default) does not. If the log-variance specification contains any other item but the log-variance intercept, then vc is set to TRUE
arch: either NULL (default) or an integer vector, say, c(1,3) or 2:5. The log-ARCH lags to include in the log-variance specification
asym: either NULL (default) or an integer vector, say, c(1) or 1:3. The asymmetry (i.e. 'leverage') terms to include in the log-variance specification
log.ewma: either NULL (default) or a vector of the lengths of the volatility proxies, see leqwma
vxreg: either NULL (default) or a numeric vector or matrix, say, a zoo object, of conditioning variables. If both y and mxreg are zoo objects, then their samples are chosen to match
zero.adj: numeric value between 0 and 1. The quantile adjustment for zero values. The default 0.1 means the zero residuals are replaced by the 10 percent quantile of the absolute residuals before taking the logarithm
vc.adj: logical. If TRUE (default), then the log-variance intercept is adjusted by the estimate of E[ln(z^2)], where z is the standardised error. This adjustment is needed for the conditional scale to be equal to the conditional standard deviation. If FALSE, then the log-variance intercept is not adjusted
vcov.type: character vector, "ordinary" (default), "white" or "newey-west". If "ordinary", then the ordinary variance-covariance matrix is used for inference. If "white", then the White (1980) heteroscedasticity-robust matrix is used. If "newey-west", then the Newey and West (1987) heteroscedasticity and autocorrelation-robust matrix is used
qstat.options: NULL (default) or an integer vector of length two, say, c(1,1). The first value sets the lag-order of the AR diagnostic test, whereas the second value sets the lag-order of the ARCH diagnostic test. If NULL, then the two values of the vector are set automatically
normality.JarqueB: FALSE (default) or TRUE. If TRUE, then the results of the Jarque and Bera (1980) test for non-normality in the residuals are included in the estimation results.
user.estimator: NULL (default) or a list with one entry, name, containing the name of the user-defined estimator. Additional items, if any, are passed on as arguments to the estimator in question
user.diagnostics: NULL (default) or a list with two entries, name and pval, see the user.fun argument in diagnostics
tol: numeric value (default = 1e-07). The tolerance for detecting linear dependencies in the columns of the regressors (see qr function). Only used if LAPACK is FALSE (default) and user.estimator is NULL.
LAPACK: logical. If TRUE, then use LAPACK. If FALSE (default), then use LINPACK (see qr function). Only used if user.estimator is NULL.
singular.ok: logical. If TRUE, then the regressors are checked for singularity, and the ones causing it are automatically removed.
plot: NULL or logical. If TRUE, then the fitted values and the residuals are plotted. If NULL (default), then the value set by options determines whether a plot is produced or not.

Details

For an overview of the AR-X model with log-ARCH-X errors, see Pretis, Reade and Sucarrat (2018): doi:10.18637/jss.v086.i03 .

The arguments user.estimator and user.diagnostics enables the specification of user-defined estimators and user-defined diagnostics. To this end, 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 'arx'

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

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/

Halbert White (1980): 'A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity', Econometrica 48, pp. 817-838.

Whitney K. Newey and Kenned D. West (1987): 'A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix', Econometrica 55, pp. 703-708.

Author

Jonas Kurle:		https://www.jonaskurle.com/
Moritz Schwarz:		https://www.inet.ox.ac.uk/people/moritz-schwarz/
Genaro Sucarrat:		http://www.sucarrat.net/

Examples

##Simulate from an AR(1):
set.seed(123)
y <- arima.sim(list(ar=0.4), 70)

##estimate an AR(2) with intercept:
arx(y, mc=TRUE, ar=1:2)
#> 
#> Date: Sat Jul 27 15:29:46 2024 
#> Dependent var.: y 
#> Method: Ordinary Least Squares (OLS) 
#> Variance-Covariance: Ordinary 
#> No. of observations (mean eq.): 68 
#> Sample: 3 to 70 
#> 
#> Mean equation:
#> 
#>             coef std.error  t-stat p-value  
#> mconst  0.013715  0.115112  0.1191 0.90553  
#> ar1     0.323324  0.125262  2.5812 0.01211 *
#> ar2    -0.040814  0.124257 -0.3285 0.74361  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics and fit:
#> 
#>                      Chi-sq df p-value
#> Ljung-Box AR(3)   3.8157196  3  0.2821
#> Ljung-Box ARCH(1) 0.0087708  1  0.9254
#>                           
#> SE of regression   0.94884
#> R-squared          0.09647
#> Log-lik.(n=68)   -91.41661

##Simulate four independent Gaussian regressors:
xregs <- matrix(rnorm(4*70), 70, 4)

##estimate an AR(2) with intercept and four conditioning
##regressors in the mean:
arx(y, ar=1:2, mxreg=xregs)
#> 
#> Date: Sat Jul 27 15:29:46 2024 
#> Dependent var.: y 
#> Method: Ordinary Least Squares (OLS) 
#> Variance-Covariance: Ordinary 
#> No. of observations (mean eq.): 68 
#> Sample: 3 to 70 
#> 
#> Mean equation:
#> 
#>             coef std.error  t-stat p-value  
#> mconst -0.020030  0.117600 -0.1703 0.86532  
#> ar1     0.314052  0.130041  2.4150 0.01875 *
#> ar2    -0.058036  0.129614 -0.4478 0.65591  
#> mxreg1 -0.045191  0.126042 -0.3585 0.72118  
#> mxreg2  0.108048  0.126116  0.8567 0.39494  
#> mxreg3  0.159350  0.133675  1.1921 0.23785  
#> mxreg4  0.145276  0.114408  1.2698 0.20898  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics and fit:
#> 
#>                    Chi-sq df p-value
#> Ljung-Box AR(3)   2.92995  3  0.4026
#> Ljung-Box ARCH(1) 0.17116  1  0.6791
#>                           
#> SE of regression   0.95304
#> R-squared          0.14454
#> Log-lik.(n=68)   -89.71711

##estimate a log-variance specification with a log-ARCH(4)
##structure:
arx(y, mc=FALSE, arch=1:4)
#> 
#> Date: Sat Jul 27 15:29:46 2024 
#> Dependent var.: y 
#> Method: Ordinary Least Squares (OLS) 
#> Sample: 1 to 70 
#> 
#> Log-variance equation:
#> 
#>             coef std.error  t-stat p-value
#> vconst -0.334531  0.471100  0.5042  0.4776
#> arch1   0.040073  0.128641  0.3115  0.7565
#> arch2  -0.076960  0.133758 -0.5754  0.5672
#> arch3  -0.124580  0.133945 -0.9301  0.3560
#> arch4  -0.058274  0.134204 -0.4342  0.6657
#> 
#> Diagnostics and fit:
#> 
#>                   Chi-sq df p-value  
#> Ljung-Box AR(1)   5.8181  1 0.01586 *
#> Ljung-Box ARCH(5) 3.6677  5 0.59818  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>                           
#> SE of regression   0.97479
#> R-squared         -0.00002
#> Log-lik.(n=66)   -91.89688

##estimate a log-variance specification with a log-ARCH(4)
##structure and an asymmetry/leverage term:
arx(y, mc=FALSE, arch=1:4, asym=1)
#> 
#> Date: Sat Jul 27 15:29:46 2024 
#> Dependent var.: y 
#> Method: Ordinary Least Squares (OLS) 
#> Sample: 1 to 70 
#> 
#> Log-variance equation:
#> 
#>             coef std.error  t-stat p-value  
#> vconst -0.123582  0.468464  0.0696 0.79193  
#> arch1   0.246437  0.174802  1.4098 0.16376  
#> arch2  -0.071564  0.131724 -0.5433 0.58895  
#> arch3  -0.130442  0.131916 -0.9888 0.32672  
#> arch4  -0.018810  0.134119 -0.1402 0.88893  
#> asym1  -0.378539  0.221001 -1.7128 0.09191 .
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics and fit:
#> 
#>                   Chi-sq df p-value  
#> Ljung-Box AR(1)   5.2578  1 0.02185 *
#> Ljung-Box ARCH(5) 1.4461  5 0.91921  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>                           
#> SE of regression   0.97479
#> R-squared         -0.00002
#> Log-lik.(n=66)   -95.13333

##estimate a log-variance specification with a log-ARCH(4)
##structure, an asymmetry or leverage term, a 10-period log(EWMA) as
##volatility proxy, and the log of the squareds of the conditioning
##regressors in the log-variance specification:
arx(y, mc=FALSE,
  arch=1:4, asym=1, log.ewma=list(length=10), vxreg=log(xregs^2))
#> 
#> Date: Sat Jul 27 15:29:46 2024 
#> Dependent var.: y 
#> Method: Ordinary Least Squares (OLS) 
#> Sample: 1 to 70 
#> 
#> Log-variance equation:
#> 
#>                   coef std.error  t-stat p-value
#> vconst        0.487403  0.749811  0.4225  0.5157
#> arch1         0.267782  0.202055  1.3253  0.1912
#> arch2        -0.037977  0.146493 -0.2592  0.7965
#> arch3        -0.083006  0.147989 -0.5609  0.5774
#> arch4         0.057241  0.157793  0.3628  0.7183
#> asym1        -0.363850  0.249324 -1.4593  0.1509
#> logEqWMA(10) -1.111548  1.102564 -1.0081  0.3183
#> vxreg1       -0.122789  0.184280 -0.6663  0.5083
#> vxreg2        0.062479  0.189227  0.3302  0.7427
#> vxreg3        0.185292  0.167040  1.1093  0.2727
#> vxreg4        0.069712  0.148582  0.4692  0.6410
#> 
#> Diagnostics and fit:
#> 
#>                   Chi-sq df p-value  
#> Ljung-Box AR(1)   4.8085  1 0.02832 *
#> Ljung-Box ARCH(5) 4.1348  5 0.53018  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>                           
#> SE of regression   0.97479
#> R-squared         -0.00002
#> Log-lik.(n=60)   -90.89802

##estimate an AR(2) with intercept and four conditioning regressors
##in the mean, and a log-variance specification with a log-ARCH(4)
##structure, an asymmetry or leverage term, a 10-period log(EWMA) as
##volatility proxy, and the log of the squareds of the conditioning
##regressors in the log-variance specification:
arx(y, ar=1:2, mxreg=xregs,
  arch=1:4, asym=1, log.ewma=list(length=10), vxreg=log(xregs^2))
#> 
#> Date: Sat Jul 27 15:29:46 2024 
#> Dependent var.: y 
#> Method: Ordinary Least Squares (OLS) 
#> Variance-Covariance: Ordinary 
#> No. of observations (mean eq.): 68 
#> Sample: 3 to 70 
#> 
#> Mean equation:
#> 
#>             coef std.error  t-stat p-value  
#> mconst -0.020030  0.117600 -0.1703 0.86532  
#> ar1     0.314052  0.130041  2.4150 0.01875 *
#> ar2    -0.058036  0.129614 -0.4478 0.65591  
#> mxreg1 -0.045191  0.126042 -0.3585 0.72118  
#> mxreg2  0.108048  0.126116  0.8567 0.39494  
#> mxreg3  0.159350  0.133675  1.1921 0.23785  
#> mxreg4  0.145276  0.114408  1.2698 0.20898  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Log-variance equation:
#> 
#>                    coef  std.error  t-stat p-value  
#> vconst       -0.6418794  0.6313424  1.0337 0.30930  
#> arch1         0.0964675  0.1671815  0.5770 0.56668  
#> arch2         0.0100427  0.1557870  0.0645 0.94887  
#> arch3        -0.0358162  0.1435422 -0.2495 0.80405  
#> arch4        -0.0870830  0.1454666 -0.5986 0.55228  
#> asym1        -0.4784833  0.2605120 -1.8367 0.07258 .
#> logEqWMA(10) -0.3209869  0.9299380 -0.3452 0.73151  
#> vxreg1       -0.1216583  0.1350725 -0.9007 0.37235  
#> vxreg2        0.0616220  0.1342431  0.4590 0.64833  
#> vxreg3       -0.0017352  0.1265075 -0.0137 0.98911  
#> vxreg4       -0.0216166  0.1155589 -0.1871 0.85242  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics and fit:
#> 
#>                   Chi-sq df p-value
#> Ljung-Box AR(3)   3.7660  3  0.2879
#> Ljung-Box ARCH(5) 1.1414  5  0.9504
#>                           
#> SE of regression   0.95304
#> R-squared          0.14454
#> Log-lik.(n=58)   -75.68205