The acf is clearly declining very slowly in this case, which is consistent with their being an autoregressive part to the appropriate model. The pacf is clearly significant for lags one and two, but the question is does it them become insignificant for lags 2 and 4, indicating an AR(2) process, or does it remain significant, which would be more consistent with a mixed ARMA process? Well, given the huge size of the sample that gave rise to this acf and pacf, even a pacf value of 0.001 would still be statistically significant. Thus an ARMA process is the most likely candidate, although note that it would not be possible to tell from the acf and pacf which model from the ARMA family was more appropriate. The DGP for the data that generated this plot was y_t = 0.9 y_(t-1) - 0.3 u_(t-1) + u_t.
-Suppose you had to guess at the most likely value of a one hundred step-ahead forecast for the AR(2) model given in question 14 - what would your forecast be?
A) -0.1
B) 0.7
C) -0.27
D) 0.75

Question

Quizplus · Accepted Answer

Given the AR(2) model $y_t = 0.9y_{t-1} - 0.3y_{t-2} + u_t$, for long-term forecasts (like a hundred steps ahead), the forecast will converge to the model's mean, which can be calculated as $\mu = \frac{\phi_0}{1-\phi_1-\phi_2}$ when $\phi_0 = 0$ (no constant term is mentioned), the mean $\mu = 0$. However, the given model in the question is $y_t = 0.9 y_{t-1} - 0.3 u_{t-1} + u_t$, which seems to be a misinterpretation because it mixes AR terms with a noise term incorrectly. Assuming the correct model is an AR(2) without direct noise terms influencing the coefficients, and given no constant term, the forecast converges to 0. However, since the model provided does not directly translate to standard AR(2) form and the options do not include 0, the closest to a reasonable forecast given the misunderstanding in the model's description would be based on the long-term behavior of AR processes, which typically converge to their mean. Without a clear model specification that matches the options, we cannot directly calculate a forecast value from the provided equation. The choice is made based on the assumption that the question intends to refer to an AR(2) model's behavior over a long horizon, where forecasts tend to the model's mean or stationary state.

The Acf Is Clearly Declining Very Slowly in This Case