elliptical.ts {sym.arma}

Autoregressive and Moving Average Symmetric Models

Description

Fit an SYMARMA model to a univariate time series. Fitting method: conditional maximum likelihood estimation.

Usage

elliptical.ts(Y, family="Normal", order=c(0,0,0), xreg=NULL, 
include.mean=TRUE, epsilon=0.0001, maxit=100, trace="TRUE",
index1=NULL, index2=NULL, fixed=NULL)

Arguments

Y

a univariate time series.

family

a description of the conditional distribution of each Y[t], given the set of past information. Symmetric distributions available for adjustment: Normal (Normal), Student-t (Student), Generalized Student-t (Gstudent), Exponential Power (ExpPower) (by Box & Tiao, 1973, ch 3), Logistic I (LogisticI), Logistic II (LogisticII), Generalized Logistic (Glogistic), Cauchy (Cauchy) and Contamined Normal (Cnormal). The default is to normal distribution.

order

a specification of the SYMARMA model: the three integer components (p, d, q) are the AR order, the degree of differencing, and the MA order.

xreg

optionally, a vector or matrix of external regressors, which must have the same number of rows as Y.

include.mean

should the SYMARMA model include a mean/intercept term? The default is TRUE.

epsilon

positive convergence tolerance e; the iterations converge when |fit - fit_old|/|fit| < e. Default is e=1e-04.

maxit

integer giving the maximal number of iterations. Default is 100 iterations.

trace

a logical indicating if output should be produced.

index1

The parameter to Student-t and Exponential Power distributions or the first argument to Generalized Student-t, Generalized Logistic and Contamined Normal distributions.

index2

The second argument to Generalized Student-t, Generalized Logistic (index2 = index2(index1)) and Contamined Normal distributions.

fixed

a optional numeric vector of the same length as the total number of parameters. If supplied, only NA entries in fixed will be varied.

Details

Different definitions of autoregressive and moving average models have different signs for the AR and/or MA coefficients. The dynamic component in SYMARMA model used here has

Y[t] = X[t]Beta + phi[1](Y[t-1] - X[t-1]Beta) + ... + phi[np](Y[t-np] - X[t-np]Beta) + theta[1]erro[t-1] + ... + theta[nq]erro[t-nq] + erro[t].

The estimation of the parameters that index the SYMARMA model is obtained by maximum conditional likelihood method on the first m observations, where m = max(np,nq).

The variance matrix of the estimates is found from the Hessian of the log-likelihood, and so may only be a rough guide.

Value

A list of class “Symarma” with components:

coefficients

a vector of estimated AR, MA and regression coefficients.

dispersion

the estimated dispersion parameter.

resid.raw

the ordinary residuals.

resid.stand

the standardized residuals.

fitted.values

the fitted mean values.

loglik

the maximized log-likelihood.

aic

the AIC value corresponding to the log-likelihood.

bic

the BIC value corresponding to the log-likelihood.

rmse

the Root Mean Squared Error value corresponding to the ajusted model.

iter

the number of iterations used in the fitting.

n

the number of observations in the series.

sd.coef

a vector of estimated standard deviation of the coefficients.

sd.disp

estimated standard deviation of the dispersion parameter.

family

the family object used.

X

if requested, the vector or matrix of external regressors.

Author(s)

Vinicius Quintas Souto Maior and Francisco Jose A. Cysneiros

Maintainer: Vinicius Quintas Souto Maior <[e-mail]>

References

Maior, V. Q. S. and Cysneiros, F. J. A. (2018). SYMARMA: a new dynamic model for temporal data on conditional symmetric distribution. Statitical Paper, 59, 75-97. doi: 10.1007/s00362-016-0753-z.

Wei, W. W. S. (2006). Time Series Analysis: Univariate and Multivariate Methods, 2nd edition. Pearson Addison Wesley. Section 7.2.1.

Box, M. J. and Tiao, G. C. (1973). Bayesian inference in statistical analysis. Londen: Addison-Wesley.

Examples

data(assets)
attach(assets)

# Return in the prices on Microsoft and SP500 index

N = length(msf)
.sp500 = ((sp500[2:N]-sp500[1:(N-1)])/sp500[1:(N-1)])*100
.msf = ((msf[2:N]-msf[1:(N-1)])/msf[1:(N-1)])*100

# The T-bill rates were divided by 253 to convert to a daily rate

.tbill = tbill/253

# Excess return in the prices on Microsoft and SP500 index

Y = .msf - .tbill[1:(N-1)]
X = .sp500 - .tbill[1:(N-1)]

# Period from April 4, 2002 to October 4, 2002

serie = Y[2122:2240]
aux = cbind(X[2122:2240])

# Returns best ARIMA model according to either AIC value.
# auto.arima(Y,xreg=aux,seasonal=FALSE,ic=c("aic"))

# Fit SYMARMA models

fit.1 = elliptical.ts(serie,order=c(0,0,1),xreg=aux,include.mean=FALSE,
 family="Normal")
fit.2 = elliptical.ts(serie,order=c(0,0,1),xreg=aux,include.mean=FALSE,
 family="Student", index1=4)
fit.3 = elliptical.ts(serie,order=c(3,0,1),xreg=aux,family="ExpPower",
 index1=0, fixed=c(0,0,NA,NA,NA,NA))