BoxCox.Arima: Box-Cox Analysis for "Arima" Objects

Description

Implements Box-Cox analysis for "Arima" class objects, the output from arima, a R built-in function. Variance change in time series is an important topic. In some cases using a Box-Cox transformation will provide a much simpler analysis than the much more complex ARMA-GARCH approach. See US Tobacco series example given below for an example.

Usage

"BoxCox"(object, interval = c(-1, 1), type = "BoxCox", InitLambda = "none", ...)

Arguments

object

output from arima, a R built-in function

interval

interval to be searched for the optimal transformation

type

ignored unless, InitLambda!="none". Type of transformation, default is "BoxCox". Otherwise a simple power transformation.

InitLambda

default "none". Otherwise a numerical value giving the transformation parameter.

...

optional arguments passed to optimize

Value

No value returned. Graphical output is produced as side-effect. The plot shows relative likelihood function as well as the MLE and a confidence interval.

Details

If no transformation is used on the data, then the original data is used. But if a transformation has already been used, we need to inverse transform the data to recover the untransformed data.

For $lambda!=0$, the Box-Cox transformation is of x is $(x^lambda-1)/lambda$. If the minimum data value is <= 0,="" a="" small="" positive="" constant,="" equal="" to="" the="" negative="" of="" minimum="" plus="" 0.25,="" is="" added="" all="" data="" values.<="" p="">

The log of the Jacobian is $ (lambda-1)*sum(log(z[(D+1):n])$, where $lambda$ is the transformation, n=length(z), z is the vector of data and D = d + ds*s, where d is the degree of regular differencing, ds is the degree of seasonal differencing and s is the seasonal period. The correct expression for the loglikelihood function was first given in Hipel and McLeod (1977, eqn. 10). Using the wrong expression for the Jacobian has a disasterous effect in many situations. For example with the international airline passenger time series, the MLE for lambda would be about 1.958 instead of close to zero.

If the minimum data value is <= 0,="" a="" small="" positive="" constant,="" equal="" to="" the="" negative="" of="" minimum="" plus="" 0.25,="" is="" added="" all="" data="" values.<="" p="">

References

Hipel, K.W. and McLeod, A.I. (1977). Advances in Box-Jenkins Modelling. Part 1, Model Construction. Water Resources Research 13, 567-575.

Examples

Run this code

## Not run:  #not run to save time!
# #Tobacco Production
#  plot(USTobacco)
#  USTobacco.arima<-arima(USTobacco,order=c(0,1,1))
#  BoxCox(USTobacco.arima)
# #
#  air.arima<-arima(AirPassengers, c(0,1,1), seasonal=list(order=c(0,1,1), period=12))
#  BoxCox(air.arima)
# #
# #In this example, we fit a model to the square-root of the sunspots and
# #back transform in BoxCox.
# sqrtsun.arima<-arima(sqrt(sunspot.year),c(2,0,0))
# BoxCox(sqrtsun.arima, InitLambda=0.5, type="power")
# #
# #Back transform with AirPassengers
# Garima<-arima(log(AirPassengers), c(0,1,1), seasonal=list(order=c(0,1,1),period=12))
# BoxCox(Garima, InitLambda=0)
# ## End(Not run)

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