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Seasonal ARMA forecasting and Fitting the bivariate data to GARCH John DOE
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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
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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
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2. Fitting the data to Seasonal ARIMA model SARIMA fitting
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Since the number of cases are strictly positive and non stationary in the variance, the log was taken SARIMA fitting
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Then the number of cases was seasonally and lag 1 differenced SARIMA fitting
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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,
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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
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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
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The diagnostic plots of the fitted model SARIMA fitting
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Predictions Data and predictions for Baltimore
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Predictions Data and predictions for New York City
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2. Fitting the bivariate data to GARCH model GARCH fitting
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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
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GARCH fitting 2. We can compute OLS residuals and fit the residuals to AR(p) model. AR(12) was selected.
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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
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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
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GARCH fitting 2. Fit the identified ARMA(2,1) model on the squared residuals, which has the smallest AIC.
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Parameter estimates GARCH fitting CoefficientValueSt.E 8.34390.3087 0.79030.1731 0.04640.0949 -0.56940.1687 1.35970.2417 0.04640.1731
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GARCH fitting So I would suggest the following model. GARCH(1,2).
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