sm: Estimate an AR-X Model with Log-ARCH-X Errors

Description

Estimation is by OLS in two stages. In the first the AR-X mean specification is estimated, whereas in the second stage the residuals from the first are used to fit a log-ARCH-X model to the log-variance. The natural logarithm of the squared residuals constitutes the regressand in the second step.

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

sm(y, mc=NULL, ar=NULL, ewma=NULL, mx=NULL, arch=NULL, asym=NULL, log.ewma=NULL, vx=NULL, p=2, zero.adj=0.1, vc.adj=TRUE, varcov.mat=c("ordinary", "white"), qstat.options=NULL, tol=1e-07, LAPACK=FALSE, verbose=TRUE, smpl=NULL)

Arguments

numeric vector, time-series or zoo object. Note that missing values in the beginning or at the end of the series is allowed, as they are removed with the na.trim command from the zoo package

logical, TRUE or FALSE (default). TRUE includes intercept in the specification, FALSE does not

integer vector, say, c(2,4) or 1:4. The AR-lags to include in the specification

ewma

list of arguments sent to the leqwma function

numeric matrix, time-series or zoo object of conditioning covariates. Note that missing values in the beginning or at the end of the series is allowed, as they are removed with the na.trim command from the zoo package

arch

integer vector, say, c(1,3) or 2:5. The ARCH-lags to include in the log-volatility specification

asym

integer vector, say, c(1) or 1:3. The asymmetry or leverage terms to include in the log-volatility specification

log.ewma

NULL (default) or a list. If NULL then log(EWMA) is not included as volatility proxy. If a list, then log(EWMA) is included as a volatility proxy.

numeric value greater than zero. The power of the log-volatility specification.

zero.adj

numeric value between 0 and 1. The quantile adjustment for zero values. The default 0.1 means that the zero residuals are replaced by means of the 10 percent quantile of the absolute residuals before taking the logarithm

vc.adj

logical, TRUE (default) or FALSE. If true then the log-volatility constant is adjusted by means of the estimate of E[log(z^2)]. This adjustment is needed for the standardised residuals to have unit variance. If FALSE then the log-volatility constant is not adjusted

varcov.mat

character vector, "ordinary" or "white". If "ordinary" then the ordinary variance-covariance matrix is used for inference. Otherwise the White (1980) heteroscedasticity robust matrix is used

qstat.options

NULL or an integer vector of length two, say, c(2,5). The first value sets the order of the AR diagnostic test, whereas the second value sets the order of the ARCH diagnostic test. NULL (default) sets the vector to c(1,1)

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

LAPACK

logical, TRUE or FALSE (default). If true use LAPACK otherwise use LINPACK (see qr() function)

verbose

logical, TRUE (default) or FALSE. FALSE returns less output and is therefore faster

smpl

Either NULL (default; the whole sample is used for estimation) or a two-element vector of dates with the start and end dates of the sample to be used in estimation. For example, smpl=c("2001-01-01", "2009-12-31")

Value

call: the function call
mean.fit: zoo-object with the fitted values of the mean specification
resids: zoo-object with the residuals of the mean specification
volatility.fit: zoo-object with the fitted values of the volatility (sigma^p) specification
resids.ustar: zoo-object with the residuals of the AR-representation of the log-volatility specification
resids.std: zoo-object with the standardised residuals
Elogzp: estimate of E(log(z^p))
mean.results: data frame with the estimation results of the mean specification
volatility.results: data frame with the estimation results of the log-volatility specification
diagnostics: data frame with selected diagnostics of the standardised residuals

Details

See Sucarrat and Escribano (2012)

References

Genaro Sucarrat and Alvaro Escribano (2012): 'Automated Financial Model Selection: General-to-Specific Modelling of the Mean and Volatility Specifications', Oxford Bulletin of Economics and Statistics 74, Issue no. 5 (October), pp. 716-735

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

Examples

Run this code

#Generate AR(1) model and independent normal regressors:
set.seed(123)
y <- arima.sim(list(ar=0.4), 200)
xregs <- matrix(rnorm(4*200), 200, 4)

#estimate AR(2) with intercept:
sm(y, mc=TRUE, ar=1:2)

#estimate AR(2) with intercept and four conditioning regressors
#in the mean:
sm(y, mc=TRUE, ar=1:2, mx=xregs)

#estimate a log-volatility specification with a log-ARCH(4)
#structure:
sm(y, arch=1:4)

#estimate a log-volatility specification with a log-ARCH(4)
#structure and an asymmetry or leverage term:
sm(y, arch=1:4, asym=1)

#estimate a log-volatility specification with a log-ARCH(4)
#structure, an asymmetry or leverage term, a 30-period log(EWMA) as
#volatility proxy, and the squareds of the conditioning regressors
#in the log-volatility specification:
sm(y, arch=1:4, asym=1, log.ewma=list(length=30), vx=log(xregs^2))

#estimate AR(2) with intercept and four conditioning regressors
#in the mean, and a log-volatility specification with a log-ARCH(4)
#structure, an asymmetry or leverage term, a 30-period log(EWMA) as
#volatility proxy, and the squareds of the conditioning regressors
#in the log-volatility specification:
sm(y, mc=TRUE, ar=1:2, mx=xregs, arch=1:4, asym=1,
  log.ewma=list(length=30), vx=log(xregs^2))

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