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 MEASLBAL.DAT ( Monthly reported number of cases of measles, Baltimore, Jan to June MEASLNYC,DAT ( Monthly reported number of cases of measles, New York city, Jan to June 1972

2. Fitting the data to Seasonal ARIMA model SARIMA fitting

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

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

SARIMA fitting For BaltimoreFor New York City ModelAICModelAIC (0,1,28)x(4,1,0) (0,1,28)x(5,1,0) (2,1,28)x(4,1,0) (2,1,28)x(5,1,0) (14,1,28)x(4,1,0) (11,1,28)x(5,1,0) For Baltimore, was selected, For New York City, was selected,

Parameter estimates for Baltimore SARIMA fitting Estimate AR MA MA AR MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA SAR MA MA SAR MA MA SAR MA MA SAR MA MA MA MA

Parameter estimates for New York City SARIMA fitting Estimate MA MA MA MA MA MA MA MA MA MA MA MA MA MA SAR MA MA SAR MA MA SAR MA MA SAR MA MA SAR MA MA MA MA MA MA

The diagnostic plots of the fitted model SARIMA fitting

Predictions Data and predictions for Baltimore

Predictions Data and predictions for New York City

2. Fitting the bivariate data to GARCH model GARCH fitting

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

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

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

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 e

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

Parameter estimates GARCH fitting CoefficientValueSt.E

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