Hello everyone, I need to estimate a ARMA(1,1) model with Stan. My Stan model is as follows:

```
data {
int<lower=1> N;
real y[N];
}
parameters {
real mu;
real<lower = -1, upper = 1> phi;
real<lower = -1, upper = 1> theta;
real<lower=0> sigma;
}
model {
real err;
mu ~ normal(0, 10);
phi ~ normal(0, 2);
theta ~ normal(0, 2);
sigma ~ cauchy(0, 5);
err = y[1] - mu + phi * mu;
for (n in 2:N) {
err = y[n] - (mu + phi * y[n-1] + theta * err);
err ~ normal(0, sigma);
}
}
```

This model is basically same as the model in Stan documentation (https://mc-stan.org/docs/2_22/stan-users-guide/autoregressive-moving-average-models.html).

arima function from forecast package estimates this model:

X[t] - m = a[1]X[t-1] + … + a[p]X[t-p] + e[t] + b[1]e[t-1] + … + b[q]e[t-q]

Since, in my situation, p and q equal to 1, above equation induces to

X[t] - m = a[1]X[t-1] + b[1]e[t-1]

which is the same as my Stan model.

I don’t have a problem with AR(1) and MA(1) coefficients, they are similar in Bayesian and Frequentist estimations (aroma from forecast package in R). But the intercept term is differing between models.

This is ARMA(1,1) frequentist estimate:

```
Coefficients:
ar1 ma1 intercept
0.5447 -0.1733 17.3737
s.e. 0.0285 0.0335 0.0416
```

and this is the Bayesian results:

```
mean se_mean sd
mu 8.2029658 0.0162509 0.4749644
phi 0.5278543 0.0009340 0.0273189
theta -0.1524502 0.0010340 0.0312320
```

There is an interesting fact: the frequentist intercept term nearly equals two times Bayesian intercept.

I couldn’t figure it this difference. Any ideas?

I tried to change prior mean for mu but it doesn’t make any difference in result.