3. The three models AR(2), MA(1), and ARMA(2,1) are fitted to the following time series:
Year Quarter 1 Quarter 2 Quarter 3 Quarter XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX110
The results using R are as follows: AR(2):
an
ar2 intercept
0.2618 st. error XXXXXXXXXXo-2 = 25.02
XXXXXXXXXXlog likelihood = -36.4
XXXXXXXXXX
ARMA(2,1):
ma1
MA(1): intercept
0.1899 st. error XXXXXXXXXXo-2 = 25.66
XXXXXXXXXXlog likelihood = -36.51
an ar2 ma1 intercept
st. error o-2 = 21.31
XXXXXXXXXX
XXXXXXXXXXlog likelihood = -36.02
XXXXXXXXXX XXXXXXXXXX
The models are ranked using Akaike Information Criterion (AIC): AIC = —2x log-likelihood+2 x number of free parameters. Determine the order from the best to worst model. Give full explanation on how you arrived to your answer. Show calculations.
A. AR(2), MA(1), ARMA (2,1) B. AR(2), ARMA (2,1), MA(1) C. MA(1), AR(2), ARMA (2,1) D. MA(1), ARMA (2,1), AR(2) E. ARMA (2,1), AR(2), MA(1).
4. In modeling the weekly sales of a certain commodity over the past six months, the time series model Xt — cbiXt_i = Zt + 91Zt_1 was thought to be appropriate. Suppose the model was fitted and the autocorrelations of the residuals were:
k XXXXXXXXXX13w (k XXXXXXXXXX XXXXXXXXXXst. dev /3/-v (k XXXXXXXXXX XXXXXXXXXX
Is the assumed model really appropriate? If not, how would you modify the model? Explain.