StLM {StReg}

Student's t Linear Regression Model (StLM)

Description

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

Usage

StLM(y,X,Trend=1,v=1,maxiter=1000,meth="BFGS",hes="FALSE",init="na")

Arguments

y

A vector representing dependent variable. Cannot be empty.

X

A data matrix whose columns represent exogenous variables. 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.

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 regression function and the skedastic function, see Spanos (1994) and Poudyal (2012).

Value

beta

coefficients of the regression 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 StLM() function has to be run again.

var.coef

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

like

maximum log likelihood value.

sigma

contemporary variance covariance matrix.

cvar

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

trend

estimated trend in the variable y.

res

non-standardized residuals

fitted

fitted values of the regression function.

init

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

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].p

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

## stlm Model#####
## Random number seed
set.seed(7504)

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

# Generating data on y, x and z. 
x <-  rt(50,df=5) ; set.seed(7505)
z <- 5*rt(50,df=5) ; set.seed(7506)

y <-  4 + 0.045*t + 2*x + 5*z + rt(50,25,df=5)

# The trend matrix
Trend <- cbind(1,poly(t,1,raw=FALSE))
colnames(Trend) <- c("const","t")

# Estimating the model
stlrm <- StLM(y,cbind(x,z),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[1:length(y)],cbind(y[1:length(y)],stlrm$fit,stlrm$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(stlrm$res,main="",xlab="") ## Histogram of y
matplot(dates[1:length(y)],cbind(stlrm$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)