ARIMA giving forecasts with higher RMSE than AR

Question

I am trying to argue that ARIMA models are better than AR models i.e since AR is a subset of ARIMA, the best ARIMA model will not be worse than the best AR model, but may be better. I have used an AR(6) model, and then used auto.arima() in R which has told me that an ARIMA(1,0,2) model is optimal using AICc. I have used both of these to do a rolling window forecast, but am getting an RMSE of 3.901 for AR(6) and 4.503 for ARIMA(1,0,2). My code for the forecasting is below (I know it is not very advanced but I'm a beginner and this is the best way I could find - it matches my results by hand):

#find moving averages and residual errors
ma=rep(NA,14976)
for (i in 3:14976){
  ma[i] = mean(ds[(i-2):(i-1)])
}
frame <- ds-ma

#fit model
model <- arima(ds[1:14676],order=c(1,0,2),include.mean=TRUE,method="ML")

`%+=%` = function(e1,e2){eval.parent(substitute(e1 <- e1 + e2))}
training_data <- data[1:14676]
test_data <- data[14677:14976]
window <- 1
window1 <- 2
coef <- model$coef
history <- training_data[(length(training_data)-window+1):14676]
predictions <- list()
for (i in (1:length(test_data))){
  length <- length(history)
  lag <- array()
  for (d in ((length-window+1):length)){
    lag[d-i+1] <- history[d]}
  yhat <- coef[length(coef)]-1
  for (t in (1:window)){
    yhat %+=% (coef[t]*lag[window-t+1])}
  if (window1 != 0){
    for (j in ((window+1):(window+window1))){
      yhat %+=% (coef[j]*frame[14676+i-j+1])}
  }
  obs <- test_data[i]
  predictions <- append(predictions,yhat)
  history <- append(history,obs)
  print(predictions)
}

The graph that comes out for the ARIMA(1,0,2) forecast (compared to the actual values in the test set) looks better, but is quite raised. It seems like the intercept needs to be lower, which does give a better RMSE, but arima() gave the intercept it did so I haven't changed it.