StAR {StReg}

Student's t Autoregression (StAR)

Description

Maximum likelihood estimation of Student's t autoregression model is the purpose of this function. It can be used to estimate the linear autoregressive function (conditional mean) and the quadratic autoskedastic function (conditional variance). Users can specify the model with deterministic variables such as trends and dummies in matrix form.

Usage

StAR(Data,Trend=1,lag=1,v=1,maxiter=1000,meth="BFGS",hes="FALSE",init="na")

Arguments

Data

A data vector with one column. Cannot be empty.

Trend

A matrix with columns representing deterministic variables like trends and dummies. If 1 (default), model with only constant intercept is estimated. If 0, the model is estimated without an intercept term.

lag

A positive integer (default value is 1) as lag length.

v

A scalar (default value is 1) greater than or equal to 1. Degrees of freedom parameter.

maxiter

Maximum number of iteration. Must be an integer bigger than 10.

meth

One of the optimization method from optim function (default value is BFGS). See details of optim function.

hes

Logical (default value is FALSE). If TRUE produces estimated hessian matrix and the standard errors of estimates.

init

If na (default), initial values for optimization are generated from a uniform distribution. A vector of initial values can also be used (not recommended). The length of the init vector must be equal to the number of parameters of the joint distribution.

Details

For the functional form of the autoregressive function and the autoskedastic function, see Spanos (1994) and Poudyal (2012).

Value

beta

coefficients of the autoregressive function including the coefficients of trends in matrix form with standard errors and p-values. If some of the standard errors are NA's, the StAR() function has to be run again.

var.coef

coefficients of the autoregressive function, standard errors and p-values if hes=TRUE.

like

maximum log likelihood value.

sigma

contemporary variance covariance matrix.

cvar

(v/(v+lag*l-2))*sigma*cvar is the fitted value of the autoskedastic function where l is the rank of Data

trend

estimated trend in the variables.

res

non-standardized residuals

fitted

fitted values of the autoregressive function.

init

estimates of the joint distribution parameters. It can be used as new initial value init in StVAR() to improve optimization further.

hes

estimated hessian matrix.

S

variance covariance matrix of the joint distribution.

R.squared

R-squared of the model.

F.stat

F-statistics of the model.

ad

Anderson-Darling test for Student's t distribution.

Author(s)

Niraj Poudyal [e-mail]

References

Poudyal, N. and Spanos, A. (2022), Model Validation and DSGE Modeling. Econometrics, 10 (2), 17.

Spanos, A. (1994), On Modeling Heteroskedasticity: the Student's t and Elliptical Linear Regression Models. Econometric Theory, 10: 286-315.

Examples

## StAR Model#####
## Random number seed
set.seed(4093)

## Creating trend variable.
t <- seq(1,50,1)

# Generating data on y. 
y <-  0.004 + 0.0045*t - 0.09*t^2 + 50*rt(50,df=5)
T <- length(y)
# The trend matrix
Trend <- cbind(1,poly(t,2,raw=TRUE))

# Estimating the model
star <- StAR(y,lag=1,Trend=Trend,v=5,maxiter=2000)

# Generate arbitrary dates
dates <- seq(as.Date("2014/1/1"), as.Date("2016/1/1"), "weeks")

## Plotting the variable y, its estimated trend and the fitted value. 

oldpar <- par(mfcol=c(3,1))
matplot(dates[2:T],cbind(y[2:T],star$fitted,star$trend),xlab="Months",type='l',
lty=c(1,2,3),lwd=c(1,1,3),col=c("black","blue","black"),ylab=" ",xaxt="n")
axis.Date(1, at = seq(as.Date("2014/1/1"), as.Date("2016/1/1"), "months"),labels=TRUE)
legend("bottomleft",legend=c("data","trend","fitted values"),lty=c(1,2,3),lwd=c(1,1,3),
col=c("black","blue","black"),cex=.85)
hist(star$res,main="",xlab="") ## Histogram of y
matplot(dates[2:T],cbind(star$cvar),xlab="Months",type='l',lty=2,lwd=1,
ylab="fitted variance",xaxt="n")
axis.Date(1,at=seq(as.Date("2014/1/1"),as.Date("2016/1/1"),"months"),labels=TRUE)
par(oldpar)