For Autoregressive Integrated Moving Average models (ARIMA), if the residual autocorrelation function (ACF) of both autoregressive, AR(1), and moving average, MA(1), models is the same what does this imply? Also, which model do we choose and will the forecasts be the same?
Consider the context of what autoregressive (AR) models and Moving Average (MA) models are from link:
ARIMA methodology attempts to describe the movements in a stationary time series as a function of what are called "autoregressive and moving average" parameters. These are referred to as AR parameters (autoregessive) and MA parameters (moving averages). An AR model with only 1 parameter may be written as...
$X(t) = A(1) \cdot X(t-1) + E(t)$
where $X(t)$ = time series under investigation
$A(1)$ = the autoregressive parameter of order 1
$X(t-1)$ = the time series lagged 1 period
$E(t)$ = the error term of the model
This simply means that any given value $X(t)$ can be explained by some function of its previous value, $X(t-1)$, plus some unexplainable random error, $E(t)$. If the estimated value of $A(1)$ was .30, then the current value of the series would be related to 30% of its value 1 period ago. Of course, the series could be related to more than just one past value. For example,
$X(t) = A(1) \cdot X(t-1) + A(2) \cdot X(t-2) + E(t)$
This indicates that the current value of the series is a combination of the two immediately preceding values, $X(t-1)$ and $X(t-2)$, plus some random error $E(t)$. Our model is now an autoregressive model of order 2.
Moving Average Models:
A second type of Box-Jenkins model is called a "moving average" model. Although these models look very similar to the AR model, the concept behind them is quite different. Moving average parameters relate what happens in period $t$ only to the random errors that occurred in past time periods, i.e. $E(t-1)$, $E(t-2)$, etc. rather than to $X(t-1)$, $X(t-2)$, $X(t-3)$ as in the autoregressive approaches. A moving average model with one MA term may be written as follows...
$X(t) = -B(1) * E(t-1) + E(t)$
The term $B(1)$ is called an MA of order 1. The negative sign in front of the parameter is used for convention only and is usually printed out automatically by most computer programs. The above model simply says that any given value of $X(t)$ is directly related only to the random error in the previous period, $E(t-1)$, and to the current error term, $E(t)$. As in the case of autoregressive models, the moving average models can be extended to higher order structures covering different combinations and moving average lengths.
End link quote.
Q1 ...if the residual autocorrelation function (ACF) of both autoregressive, AR(1), and moving average, MA(1), models is the same what does this imply?
A1 From the rather extensive quote above, the moving average (MA) model is dependent upon the errors of prior model estimates, and the autoregressive model (AR) depends on the magnitude of the prior datum. So, the models are quite different, and would not usually produce the same results.
Q2 What’s the best way to correct for autocorrelation: adding AR terms or adding MA terms?
A2 A partial answer to this very general question is from Link "...the problem of autocorrelated errors in a random walk model was fixed in two different ways: by adding a lagged value of the differenced series to the equation or adding a lagged value of the forecast error. Which approach is best? A rule-of-thumb for this situation...is that positive autocorrelation is usually best treated by adding an AR term to the model and negative autocorrelation is usually best treated by adding an MA term.
Caution: There are lots of different modeling assumptions that can be made so that a general read of models and much model testing is needed to determine which model should be used. Some examples of forecasting models are given as an answer on this site.
A more complete answer is that sometimes AR(1,2,3,,,n) and MA(1,2,3,,,,m) terms may be needed for a forecasting model: "The [mixture] models developed by this approach are usually called ARIMA models because they use a combination of autoregressive (AR), integration (I) - referring to the reverse process of differencing to produce the forecast, and moving average (MA) operations. An ARIMA model is usually stated as ARIMA(p,d,q). This represents the order of the autoregressive components (p), the number of differencing operators (d), and the highest order of the moving average term. For example, ARIMA(2,1,1) means that you have a second order autoregressive model with a first order moving average component whose series has been differenced once to induce stationarity."