perARMA (version 1.7)

parmaf: PARMA coefficients estimation

Description

Procedure parmaf enables the estimation of parameters of the chosen representation of PARMA(p,q) model. For general PARMA we use non-linear optimization methods to obtain minimum of negative logarithm of likelihood function using loglikef procedure. Intitial values of parameters are computed using Yule-Walker equations.

Usage

parmaf(x, T_t, p, q, af, bf, ...)

Value

list of values:

a

is matrix of Fourier coefficients determining phi.

b

is matrix of Fourier corfficients determining theta.

negloglik

minimum value of negative logarithm of likehood function.

aicval

value of AIC criterion.

fpeval

value of FPE criterion.

bicval

value of BIC criterion.

resids

series of residuals.

Arguments

x

input time series.

T_t

period length of PC-T structure.

p

maximum PAR order, which is a number of columns in af.

q

maximum PMA order, which is a number of columns in bf diminished by 1.

af

\(T \times p\) logical values matrix pointing to active frequency components for phi.

bf

\(T \times (q+1)\) logical matrix pointing to active frequency components for theta.

...

Other arguments:
a0 starting value for a, where a is Fourier representation of phi (use phi=ab2phth(a) to recover phi); if a0 is not defined Yule Walker method is used to estimate it;
b0 starting values for b, where b is Fourier representation of theta (use del=ab2phth(b[,1]) to recover del and use theta = ab2phth(b[,2:q+1]) to recover theta); if b0 is not defined Yule Walker method is used to estimate it;
stype numeric parameter connected with covariance matrix computation, so far should be equal to 0 to use procedure R_w_ma (see R_w_ma description). In the future also other values of stype will be available for full covariance matrix computation.

Author

Harry Hurd

Details

In order to obtain maximum likelihood estimates of model parameters a and b we use a numerical optimization method to minimalize value of y (as negative value of logarithm of loglikelihood function returned by loglikef) over parameter values. Internally, parameters a and b are converted to phi and theta as needed via function ab2phth. For the present we use optim function with defined method="BFGS" (see code for more details).

References

Box, G. E. P., Jenkins, G. M., Reinsel, G. (1994), Time Series Analysis, 3rd Ed., Prentice-Hall, Englewood Cliffs, NJ.

Brockwell, P. J., Davis, R. A., (1991), Time Series: Theory and Methods, 2nd Ed., Springer: New York.

Jones, R., Brelsford, W., (1967), Time series with periodic structure, Biometrika 54, 403-408.

Vecchia, A., (1985), Maximum Likelihood Estimation for Periodic Autoregressive Moving Average Models, Technometrics, v. 27, pp.375-384.

See Also

loglikef, perYW, R_w_ma,

Examples

######## simulation of periodic series
T=12
nlen=480
p=2
 a=matrix(0,T,p)
q=1
 b=matrix(0,T,q)
a[1,1]=.8                 
a[2,1]=.3                
                                                 
a[1,2]=-.9               
phia<-ab2phth(a) 
phi0=phia$phi            
phi0=as.matrix(phi0)       
      
b[1,1]=-.7	          
b[2,1]=-.6                  
thetab<-ab2phth(b)       
theta0=thetab$phi  
theta0=as.matrix(theta0) 
del0=matrix(1,T,1)        
makeparma_out<-makeparma(nlen,phi0,theta0,del0)                      
y=makeparma_out$y

## Do not run 
## It could take more than one minute

# \donttest{
############ fitting of PARMA(0,1) model
p=0
q=1
af=matrix(0,T,p)
bf=matrix(0,T,q+1)
bf[1,1]=1
bf[1:3,2]=1

parmaf(y,T,p,q,af,bf)

########### fitting of PARMA(2,0) model
p=2
q=0
af=matrix(0,T,p)
bf=matrix(0,T,q+1)
af[1:3,1]=1       
af[1:3,2]=1
bf[1,1]=1
parmaf(y,T,p,q,af,bf)
############ fitting of PARMA(2,1) model
p=2
q=1
af=matrix(0,T,p)
bf=matrix(0,T,q+1)
af[1:3,1]=1       
af[1:3,2]=1
bf[1,1]=1
bf[1:3,2]=1
parmaf(y,T,p,q,af,bf)
# }