To answer your questions, you basically need to know how the residuals i.e. $e_t$ are calculated in an arma model. Because then $\hat{X_{t}}=X_{t}-e_{t}$. Let's first generate a fake data ($X_t$) from arima(.5,.6) and fit the arma model (without mean):
library(forecast)
n=1000
ts_AR <- arima.sim(n = n, list(ar = 0.5,ma=0.6))
f=arima(ts_AR,order=c(1,0,1),include.mean=FALSE)
summary(f)
Series: ts_AR
ARIMA(1,0,1) with zero mean
Coefficients:
ar1 ma1
0.4879 0.5595
s.e. 0.0335 0.0317
sigma^2 estimated as 1.014: log likelihood=-1426.7
AIC=2859.4 AICc=2859.42 BIC=2874.12
Training set error measures:
ME RMSE MAE MPE MAPE MASE
Training set 0.02102758 1.00722 0.8057205 40.05802 160.1078 0.6313145
Now I create the residuals as follows: $e_1=0$ (since there is no residual at 1) and for $t=2,...,n$ we have: $e_t=X_t-Ar*X_{t-1}-Ma*e_{t-1}$, where $Ar$ and $Ma$ are the estimated auto-regressive and moving average part in above fitted model. Here is the code:
e = rep(1,n)
e[1] = 0 ##since there is no residual at 1, e1 = 0
for (t in (2 : n)){
e[t] = ts_AR[t]-coef(f)[1]*ts_AR[t-1]-coef(f)[2]*e[t-1]
}
Once you find the residuals $e_{t}$, the fitted values are just $\hat{X_{t}}=X_{t}-e_{t}$. So in the following, I compared the first 10 fitted values obtained from R and the ones I can calculate from $e_{t}$ I created above (i.e. manually).
cbind(fitted.from.package=fitted(f)[1:10],fitted.calculated.manually=ts_AR[1:10]-e[1:10])
fitted.from.package fitted.calculated.manually
[1,] -0.4193068 -1.1653515
[2,] -0.8395447 -0.5685977
[3,] -0.4386956 -0.6051324
[4,] 0.3594109 0.4403898
[5,] 2.9358336 2.9013738
[6,] 1.3489537 1.3682191
[7,] 0.5329436 0.5219576
[8,] 1.0221220 1.0283511
[9,] 0.6083310 0.6048668
[10,] -0.5371484 -0.5352324
As you see there are close but not exactly the same. The reason is that when I created the residuals I set $e_{1}=0$. There are other choices though. For example based on the help file to arima, the residuals and their variance found by a Kalman filter and therefore their calculation of $e_t$ will be slightly different from me. But as time goes on they are converging.
Now for the Ar(1) model. I fitted the model (without mean) and directly show you how to calculate the fitted values using the coefficients. This time I didn't calculate the residuals. Note that I reported the first 10 fitted values removing the first one (as again it would be different depending on how you define it). As you can see, they are completely the same.
f=arima(ts_AR,order=c(1,0,0),include.mean=FALSE)
cbind(fitted.from.package=fitted(f)[2:10],fitted.calculated.manually=coef(f)*ts_AR[1:9])
fitted.from.package fitted.calculated.manually
[1,] -0.8356307 -0.8356307
[2,] -0.6320580 -0.6320580
[3,] 0.0696877 0.0696877
[4,] 2.1549019 2.1549019
[5,] 2.0480074 2.0480074
[6,] 0.8814094 0.8814094
[7,] 0.9039184 0.9039184
[8,] 0.8079823 0.8079823
[9,] -0.1347165 -0.1347165
