Estimate a heterogeneous log-ARCH-X model

The function larch() estimates a heterogeneous log-ARCH-X model, which is a generalisation of the dynamic log-variance model in Pretis, Reade and Sucarrat (2018). Internally, estimation is undertaken by a call to larchEstfun. The log-variance specification can contain log-ARCH terms, log-HARCH terms, asymmetry terms ('leverage'), the log of volatility proxies made up of past returns and other covariates ('X'), for example Realised Volatility (RV), volume or the range.

Usage

larch(e, vc=TRUE, arch = NULL, harch = NULL, asym = NULL, asymind = NULL,
  log.ewma = NULL, vxreg = NULL, zero.adj = NULL, 
  vcov.type = c("robust", "hac"), qstat.options = NULL,
  normality.JarqueB = FALSE, tol = 1e-07, singular.ok = TRUE,  plot = NULL)

Arguments

e: 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
vc: logical. TRUE includes an intercept in the log-variance specification. Currently, vc cannot be set to any other value than 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
harch: either NULL (default) or an integer vector, say, c(5,10). The (log of) heterogeneous ARCH terms (Muller et al. 1997) to include
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
asymind: either NULL (default or an integer vector. The indicator asymmetry terms to include
log.ewma: either NULL (default) or a vector of the lengths of the volatility proxies, see leqwma. The terms serve as (log of) volatility proxies similar to RVs in the HAR-model of Corsi (2009). Here, the log.ewma terms are made up of past e's
vxreg: either NULL (default) or a numeric vector or matrix, say, a zoo object. If both e and vxreg are zoo objects, then their samples are chosen to match
zero.adj: NULL (default) or a strictly positive numeric scalar. If NULL, the zeros in the squared residuals are replaced by the 10 percent quantile of the non-zero squared residuals. If zero.adj is a strictly positive numeric scalar, then this value is used to replace the zeros of the squared e's
vcov.type: character. "robust" (default) or "hac" (partial matching is allowed). If "robust", the robust variance-covariance matrix of the White (1980) type is used. If "hac", 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 of the standardised residuals, whereas the second value sets the lag-order of the ARCH diagnostic test of the standardised residuals. 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
tol: numeric value. The tolerance (the default is 1e-07) for detecting linear dependencies in the columns of the regressors (see ols and qr). Only used if LAPACK is FALSE (default)
singular.ok: logical. If TRUE (default), the regressors are checked for singularity, and the ones causing it are automatically removed. If FALSE, then the function returns an error
plot: NULL (default) or logical. If TRUE, the fitted values and the residuals are plotted. If NULL, then the value set by options determines whether a plot is produced or not

Details

No details for the moment

Value

A list of class 'larch'

References

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

F. Corsi (2009): 'A Simple Approximate Long-Memory Model of Realized Volatility', Journal of Financial Econometrics 7, pp. 174-196

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

U. Muller, M. Dacorogna, R. Dave, R. Olsen, O. Pictet and J. von Weizsacker (1997): 'Volatilities of different time resolutions - analyzing the dynamics of market components'. Journal of Empirical Finance 4, pp. 213-239

F. Pretis, J. Reade and G. 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

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

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

Author

Genaro Sucarrat: https://www.sucarrat.net/

Examples

##Simulate some data:
set.seed(123)
e <- rnorm(40)
x <- matrix(rnorm(40*2), 40, 2)

##estimate a log-variance specification with a log-ARCH(4)
##structure:
larch(e, arch=1:4)
#> 
#> Date: Mon Apr  7 09:08:29 2025 
#> Dependent var.: e 
#> Variance-Covariance: Robust (default) 
#> No. of observations: 36 
#> Sample: 5 to 40 
#> 
#> Log-variance equation:
#> 
#>             coef std.error  t-stat p-value  
#> vconst -0.887676  0.327234 -2.7127 0.04214 *
#> arch1   0.095588  0.119125  0.8024 0.45874  
#> arch2  -0.203163  0.123729 -1.6420 0.16151  
#> arch3   0.051273  0.117283  0.4372 0.68022  
#> arch4  -0.362734  0.129071 -2.8103 0.03753 *
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics and fit:
#> 
#>                    Chi-sq df p-value
#> Ljung-Box AR(1)   0.37831  1  0.5385
#> Ljung-Box ARCH(5) 3.09101  5  0.6860
#>                         
#> Log-lik.(n=36): -42.9794

##estimate a log-variance specification with a log-ARCH(4)
##structure, a log-HARCH(5) term and a first-order asymmetry/leverage
##term:
larch(e, arch=1:4, harch=5, asym=1)
#> 
#> Date: Mon Apr  7 09:08:29 2025 
#> Dependent var.: e 
#> Variance-Covariance: Robust (default) 
#> No. of observations: 35 
#> Sample: 6 to 40 
#> 
#> Log-variance equation:
#> 
#>              coef  std.error  t-stat p-value  
#> vconst -0.9798902  0.2823343 -3.4707 0.01040 *
#> arch1   0.0048135  0.1473334  0.0327 0.97485  
#> arch2  -0.1643365  0.1243636 -1.3214 0.22791  
#> arch3  -0.0620451  0.1267956 -0.4893 0.63957  
#> arch4  -0.3875460  0.1369073 -2.8307 0.02538 *
#> harch5 -0.1215850  0.1446098 -0.8408 0.42825  
#> asym1  -0.0805486  0.2242473 -0.3592 0.73004  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics and fit:
#> 
#>                   Chi-sq df p-value
#> Ljung-Box AR(1)   0.2082  1  0.6482
#> Ljung-Box ARCH(5) 5.1941  5  0.3927
#>                         
#> Log-lik.(n=35): -43.0298

##estimate a log-variance specification with a log-ARCH(4)
##structure, an asymmetry/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:
larch(e, arch=1:4, asym=1, log.ewma=list(length=10), vxreg=log(x^2))
#> 
#> Date: Mon Apr  7 09:08:29 2025 
#> Dependent var.: e 
#> Variance-Covariance: Robust (default) 
#> No. of observations: 30 
#> Sample: 11 to 40 
#> 
#> Log-variance equation:
#> 
#>                   coef std.error  t-stat p-value  
#> vconst       -0.891993  0.451570 -1.9753 0.07966 .
#> arch1        -0.195556  0.212075 -0.9221 0.38053  
#> arch2        -0.370430  0.203381 -1.8214 0.10188  
#> arch3        -0.115784  0.150598 -0.7688 0.46169  
#> arch4        -0.393912  0.235987 -1.6692 0.12942  
#> asym1        -0.114630  0.217382 -0.5273 0.61072  
#> logEqWMA(10)  4.263016  2.054292  2.0752 0.06779 .
#> vxreg1        0.072216  0.112665  0.6410 0.53751  
#> vxreg2       -0.105502  0.119036 -0.8863 0.39852  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostics and fit:
#> 
#>                     Chi-sq df p-value
#> Ljung-Box AR(1)   0.028824  1  0.8652
#> Ljung-Box ARCH(5) 2.708187  5  0.7449
#>                         
#> Log-lik.(n=30): -36.3623