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Seasonal ARMA forecasting and Fitting the bivariate data to GARCH

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

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Seasonal ARMA forecasting and Fitting the bivariate data to GARCH

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

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

  7. SARIMA fitting For Baltimore, was selected, For New York City, was selected,

  8. SARIMA fitting Parameter estimates for Baltimore

  9. SARIMA fitting Parameter estimates for New York City

  10. SARIMA fitting The diagnostic plots of the fitted model

  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.

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

  19. GARCH fitting Parameter estimates

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

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