ARMA(2,1) PACF and ACF interpretation

Question

I have two main questions. 1. Why does my best fit not match the actual? THe best fit based on AIC and BIC is AR1 the model is ARMA(2,1)

2. Why does my ACF and PACF for ARMA(2,1) look like it should be ARMA(1,1) based off the image of the chart? (There is one clear lag then drops off towards 0 for both ACF and PACF)

Here are my formulas

set.seed(170)
x <- arima.sim(model = list(ar = c(0.2, -0.1), ma = 0.1), n = 230)
ts.plot(x)
acf2(x)

Here is the coding used for to find AIC and BIC for question 1.

AR1_fit <- sarima(x, p=1, d=0, q=0)
AR1_fit

AR2_fit <- sarima(x, p=2, d=0, q=0)
AR2_fit

MA1_fit <- sarima(x, p=0, d=0, q=1)
MA1_fit

MA2_fit <- sarima(x, p=0, d=0, q=2)
MA2_fit

ARMA11_fit <- sarima(x, p=1, d=0, q=1)
ARMA11_fit

ARMA12_fit <- sarima(x, p=1, d=0, q=2)
ARMA12_fit

ARMA21_fit <- sarima(x, p=2, d=0, q=1)
ARMA21_fit

ARMA22_fit <- sarima(x, p=2, d=0, q=2)
ARMA22_fit

Answer 1

The simple reason is the random component. You fitted an ARMA(2,1) model but due to the random variable in every step, it is possible that this random factor ensure that the ARMA(2,1) model looks like an ARMA(1,1) model. This can happen and in another seed the AIC and BIC might select an ARIMA (1,2) as the best model fit and even the acf and pacf could look like the model is an ARMA(1,2). This is just due to the error term. Try to repeat your code 1000 different times with different seeds and you will see that in most cases the AIC and BIC will choose an ARMA(2,1) and the acf and pacf look like ARMA(2,1) too. Hope it answers both questions.