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 4 2010 112 104 100 96 2011 101 101 105 94 2012 106 106 108 110
The results using R are as follows: AR(2):
an
ar2 intercept
0.2618 st. error 0.3289 o-2 = 25.02
0.0868 0.3326 log likelihood = -36.4
104.227 2.372
ARMA(2,1):
ma1
MA(1): intercept
0.1899 st. error 0.2574 o-2 = 25.66
103.798 1.7373 log likelihood = -36.51
an ar2 ma1 intercept
st. error o-2 = 21.31
-0.5547 0.3329
0.4437 0.33267 log likelihood = -36.02
0.9779 104.4542 0.235 2.4584
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 1 2 3 4 5 6 7 8 13w (k) .50 -.04 .03 -.01 .01 .02 .03 -.01 st. dev /3/-v (k) .08 .10 .11 .11 .11 .11 .11 .11
Is the assumed model really appropriate? If not, how would you modify the model? Explain.