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Seasonal ARMA forecasting and Fitting the bivariate data to GARCH John DOE.

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Presentation on theme: "Seasonal ARMA forecasting and Fitting the bivariate data to GARCH John DOE."— Presentation transcript:

1 Seasonal ARMA forecasting and Fitting the bivariate data to GARCH John DOE

2 Outline Part I : Data description for the project Part II : Fitting the data to Seasonal ARIMA model and Forecasting Part III: Fitting the bivariate data to GARCH model

3 1. Data description MEASLBAL.DAT (http://www.robihyndman.com/TSDL/epi/measlbal.dat) Monthly reported number of cases of measles, Baltimore, Jan. 1939 to June 1972. MEASLNYC,DAT (http://www.robihyndman.com/TSDL/epi/measlnyc.dat) Monthly reported number of cases of measles, New York city, 1928-1972. Jan. 1939 to June 1972

4 2. Fitting the data to Seasonal ARIMA model SARIMA fitting

5 Since the number of cases are strictly positive and non stationary in the variance, the log was taken SARIMA fitting

6 Then the number of cases was seasonally and lag 1 differenced SARIMA fitting

7 SARIMA fitting For BaltimoreFor New York City ModelAICModelAIC (0,1,28)x(4,1,0)120.6668533(0,1,28)x(5,1,0)12-1.089954 (2,1,28)x(4,1,0)120.6555881(2,1,28)x(5,1,0)12-1.015811 (14,1,28)x(4,1,0)120.6725279(11,1,28)x(5,1,0)12-1.024259 For Baltimore, was selected, For New York City, was selected,

8 Parameter estimates for Baltimore SARIMA fitting Estimate AR1-0.0251MA11-0.0703MA230.1741 AR2-0.5102MA12-0.3713MA24-0.4022 MA1-0.1634MA13-0.0059MA250.2684 MA20.5935MA14-0.4141MA26-0.1641 MA3-0.2383MA150.1019MA270.1697 MA4-0.0606MA16-0.1736MA280.2311 MA5-0.1774MA170.0952SAR1-0.5997 MA6-0.0807MA18-0.0489SAR2-0.1742 MA7-0.3268MA190.2081SAR3-0.2425 MA8-0.051MA200.0440SAR4-0.2760 MA9-0.2102MA210.1740 MA100.0755MA220.0204

9 Parameter estimates for New York City SARIMA fitting Estimate MA1 0.1696 MA13 -0.1589 MA25 0.0705 MA2 0.0064 MA14 -0.1221 MA26 0.1183 MA3 -0.0679 MA15 -0.2073 MA27 0.0697 MA4 -0.1088 MA16 -0.0864 MA28 0.0766 MA5 -0.0949 MA17 0.0432 SAR1 -0.8291 MA6 -0.1407 MA18 0.1078 SAR2 -0.3674 MA7 -0.1385 MA19 0.0245 SAR3 -0.4394 MA8 -0.0638 MA20 0.1434 SAR4 -0.4480 MA9 -0.1631 MA21 0.0076 SAR5 -0.2535 MA10 -0.1373 MA22 0.0679 MA11 -0.0722 MA23 0.1556 MA12 -0.2022 MA24 -0.1542

10 The diagnostic plots of the fitted model SARIMA fitting

11 Predictions Data and predictions for Baltimore

12 Predictions Data and predictions for New York City

13 2. Fitting the bivariate data to GARCH model GARCH fitting

14 GARCH fitting 1. We consider the OLS estimation for the model Baltimore and New York City are geographically close to each other. Measles is the infectious diseases

15 GARCH fitting 2. We can compute OLS residuals and fit the residuals to AR(p) model. AR(12) was selected.

16 GARCH fitting 3. Get the residuals,, of AR(12) and calculate the portmanteau statistics,,on the squared series. Use the following formulas.,where Q<-function(k){n<-length(nhat) lohat<-c(rep(0,k)) Q<-c(rep(0,k)) for(i in 1:k){ fir<-(nhat^2-sig.sq) term<-fir[1:(n-i)]*fir[(1+i):n] lohat[i]<-sum(term)/sum((nhat^2-sig.sq)^2)} for(i in 1:k){ Q[i]<-lohat[i]^2/(n-i)} Qk<-n*(n+2)*sum(Q) pvalue<-(1-pchisq(Qk,k)) list(term=term,lohat=lohat,Qk=Qk,pvalue=pvalue)} R-code

17 GARCH fitting We know that the significance of the statistic Occurring only for a small value of k indicates an ARCH model, and a persistent significance for a large value of k implies a GARCH model. Since we could see the latter pattern, I would suggest GARCH modeling. kp-value 166.771523.330669e-16 2109.51790 3121.13150 4122.62610 5123.58360 6124.93700 7130.01450 8131.38870 9146.48590 10147.64490

18 GARCH fitting 2. Fit the identified ARMA(2,1) model on the squared residuals, which has the smallest AIC.

19 Parameter estimates GARCH fitting CoefficientValueSt.E 8.34390.3087 0.79030.1731 0.04640.0949 -0.56940.1687 1.35970.2417 0.04640.1731

20 GARCH fitting So I would suggest the following model. GARCH(1,2).


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