Selecting an ARIMA(1,0,q) Model by BIC with Fixed AR Order

# Automate selection of BIC-minimizing ARIMA(1,0,X) model


# Automate selection of BIC-minimizing ARIMA(1,0,X) model












I want to estimate an ARIMA(1,0,X) model. The MA(X) in the model is selected to minimize BIC. I have the following code employing the function `auto.arima` from "forecast" package in R:

```
 auto.arima(logret$appl, d=0, max.p=1, max.q=10,
 max.order=NA, max.d=0,start.p=1, start.q=0,
 stationary=FALSE, seasonal=FALSE,
 ic=c("bic"), stepwise=TRUE, trace=TRUE,
 allowdrift=TRUE, allowmean=TRUE, lambda=NULL, parallel=FALSE, num.cores=NULL)
```

The following is the result of the above code

```
 ARIMA(1,0,0) with non-zero mean : -6592.886
 ARIMA(0,0,0) with non-zero mean : -6597.561
 ARIMA(1,0,0) with non-zero mean : -6592.886
 ARIMA(0,0,1) with non-zero mean : -6592.48
 ARIMA(0,0,0) with zero mean     : -6604.679
 ARIMA(1,0,1) with non-zero mean : -6586.55

Best model: ARIMA(0,0,0) with non-zero mean 

Series: logreturns$REL.CP 
ARIMA(0,0,0) with non-zero mean 

Coefficients:
      intercept
          0e+00
s.e.      5e-04

sigma^2 estimated as 0.0002818:  log likelihood=3305.9
AIC=-6607.81   AICc=-6607.8   BIC=-6597.56
```

My problem requires to keep `AR(1)` and `d=0` in all the models where I test MA lags. Is there a way to fix AR lags? What I really want is MA(1) coefficient out of this arima(1,0, X) model. Thanks

## Answer by Richard Hardy (score 1, accepted)

https://quant.stackexchange.com/a/25564

You can do it manually. Let `x` be the data series. The code below considers all moving-average lag orders between `0` and `max.q` and prints out the BIC-minimizing lag order and the corresponding estimated model:

```
m=list() # I will save estimated ARIMA(1,0,q) models here
BIC=c()  # I will save the corresponding BIC values here
max.q=10 # the maximum MA order you want to consider
n=length(x)
for(q in 0:max.q){
 m[[q+1]]=arima(x,order=c(1,0,q),method="ML")
 BIC[q+1]=AIC(m[[q+1]],k=log(n))
}
print(paste("BIC-optimal MA order is",which.min(BIC)-1)) # info message
print(m[[which.min(BIC)]]) # print the estimated BIC-optimal model
```

Edit:

To respond to a request in the comments, I am including a function that returns the estimated MA1 coefficient:

```
MA1fromARIMA10q=function(x,max.q=10,...){
 # x is a data vector (a time series)
 # max.q is the maximum MA order to be considered
 m=list() # I will save estimated ARIMA(1,0,q) models here
 BIC=c()  # I will save the corresponding BIC values here
 n=length(x)
 for(q in 0:max.q){
  m[[q+1]]=arima(x,order=c(1,0,q),...)
  BIC[q+1]=AIC(m[[q+1]],k=log(n))
 }
 q=which.min(BIC)-1
 if(q>0) MA1=coef(m[[q+1]])[2] else(MA1=NA) # if MA order is 0, assign NA
 return(MA1)
}
```