(b.i) Plot y_{t}. Do any time intervals, of the order 100 years, exist where one can observe behavior comparable to that observed in the global temperature records in Figure 1.2? (This phenomenon is discussed in the Climate Change Report in the section on A PALEOCLIMATIC PERSPECTIVE; page 8 of the report.)Also, in part (d): The formula for γ_{u} (h): if |h|≥ 1 should be if |h|> 1.
(b.ii) Examine the ACF of y_{t} and comment.
# AICc - see Section 2.2 > log(gnpgr.ma$sigma2)+(222+3)/(222-3-2) # MA(2) [1] -8.287855 > log(gnpgr.ar$sigma2)+(222+2)/(222-2-2) # AR(1) [1] -8.284898 # SIC or BIC - see Section 2.2 > log(gnpgr.ma$sigma2)+(3*log(222)/222) # MA(2) [1] -9.251712 > log(gnpgr.ar$sigma2)+(2*log(222)/222) # AR(1) [1] -9.263748
> sarima(prod,1,1,1,2,1,1,12) # ARIMA(1,1,1)×(2,1,1)_{12} ar1 ma1 sar1 sar2 sma1 0.5753 -0.2709 -0.2153 -0.2800 -0.4968 s.e. 0.1120 0.1300 0.0784 0.0619 0.0712 sigma^2 estimated as 1.351: log likelihood = -568.22, aic = 1148.43 $AIC $AICc $BIC [1] 1.327435 [1] 1.333430 [1] 0.3801087 > sarima(prod,1,1,1,0,1,3,12) # ARIMA(1,1,1)×(0,1,3)_{12} ar1 ma1 sma1 sma2 sma3 0.5706 -0.2608 -0.7432 -0.1397 0.2782 s.e. 0.1119 0.1295 0.0535 0.0647 0.0526 sigma^2 estimated as 1.314: log likelihood = -564.24, aic = 1140.48 $AIC $AICc $BIC [1] 1.299957 [1] 1.305952 [1] 0.3526299
n^{-1} |
n Σ t=1 |
n Σ s=1 |
γ(s − t) ... |
Call: garch(formula.mean = gnpr ~ar(1), formula.var= ~garch(1, 0)) Mean Equation: gnpr ~ ar(1) Conditional Variance Equation: ~ garch(1, 0) Conditional Distribution: gaussian -------------------------------------------------------------- Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) C 0.0052876 8.295e-004 6.374 5.400e-010 AR(1) 0.3637468 7.902e-002 4.603 3.535e-006 A 0.0000725 7.035e-006 10.305 0.000e+000 ARCH(1) 0.2012735 7.037e-002 2.860 2.321e-003 -------------------------------------------------------------- AIC(4) = -1436.53 BIC(4) = -1422.92 Normality Test: -------------------------------------------------------------- Jarque-Bera P-value Shapiro-Wilk P-value 8.425 0.01481 0.9826 0.4778 Ljung-Box test for standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 13.33 0.3452 12 Ljung-Box test for squared standardized residuals: -------------------------------------------------------------- Statistic P-value Chi^2-d.f. 27.03 0.007653 12 Lagrange multiplier test: -------------------------------------------------------------- Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 Lag 6 Lag 7 Lag 8 2.09 0.2293 1.04 -0.2954 0.7298 -0.8147 1.55 2.008 Lag 9 Lag 10 Lag 11 Lag 12 C -1.705 1.097 2.392 -0.6218 -0.7981 TR^2 P-value F-stat P-value 25.43 0.01289 2.631 0.03724
Figure 6.8 Interest rate for three-month treasury bills (dotted line--circles) and quarterly inflation rate (line--squares) in the Consumer Price Index, 1953:1 to 1965:2.In the example, the series should be y_{t} = quarterly inflation rate and z_{t} = quarterly interest rate. The model in the example remains the same; that is, y_{t} = α + β_{t} z_{t} + v_{t} is the observation equation. The results of the analysis are correct for the correctly named series.