Seasonal ARMA forecasting Seasonal ARMA forecasting and and

Fitting the bivariate data to Fitting the bivariate data to GARCHGARCH

John DOEJohn DOE

OutlineOutlinePart 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 (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

2. 2. Fitting the data Fitting the data to Seasonal ARIMA modelto Seasonal ARIMA model

SARIMAfitting

Since the number of cases are strictly positive

and non stationary in the variance, the log was taken

SARIMAfitting

Then the number of cases was seasonally

and lag 1 differenced

SARIMAfitting

SARIMAfitting

For Baltimore For New York City

Model AIC Model AIC

(0,1,28)x(4,1,0)12 0.6668533 (0,1,28)x(5,1,0)12 -1.089954

(2,1,28)x(4,1,0)12 0.6555881 (2,1,28)x(5,1,0)12 -1.015811

(14,1,28)x(4,1,0)12 0.6725279 (11,1,28)x(5,1,0)12

-1.024259

For Baltimore, was selected,

12)0,1,5()28,1,0(

ti

iit

i

ii

i

ii aBBALBBBB

28241212 1)ln(11)1)(1(

12)0,1,4()28,1,2(

For New York City, was selected,

ti

iit

i

ii aBNYCBBB

2851212 1)ln(1)1)(1(

Parameter estimates for BaltimoreSARIMAfitting

Estimate Estimate Estimate

AR1 -0.0251 MA11 -0.0703 MA23 0.1741

AR2 -0.5102 MA12 -0.3713 MA24 -0.4022

MA1 -0.1634 MA13 -0.0059 MA25 0.2684

MA2 0.5935 MA14 -0.4141 MA26 -0.1641

MA3 -0.2383 MA15 0.1019 MA27 0.1697

MA4 -0.0606 MA16 -0.1736 MA28 0.2311

MA5 -0.1774 MA17 0.0952 SAR1 -0.5997

MA6 -0.0807 MA18 -0.0489 SAR2 -0.1742

MA7 -0.3268 MA19 0.2081 SAR3 -0.2425

MA8 -0.051 MA20 0.0440 SAR4 -0.2760

MA9 -0.2102 MA21 0.1740

MA10 0.0755 MA22 0.0204

Parameter estimates for New York CitySARIMAfitting

Estimate Estimate 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

The diagnostic plots of the fitted model SARIMAfitting

PredictionsData and predictions for Baltimore

PredictionsData and predictions for New York City

2. Fitting the bivariate data 2. Fitting the bivariate data to GARCH modelto GARCH model

GARCHfitting

GARCHfitting

1. We consider the OLS estimation for the model

ttt NYCBal 10

• Baltimore and New York City are geographically

close to each other.

• Measles is the infectious diseases

tt NYClBa 06941.04826.174ˆ

GARCHfitting

2. We can compute OLS residuals and fit the residuals to AR(p) model.

ttt BallBa ˆ̂ AR(12) was selected.

GARCHfitting

3. Get the residuals, , of AR(12) and calculate the portmanteau statistics, ,on the squared series. Use the following

formulas.

tn̂)(kQ tn̂

k

i

ti

in

nnnkQ

1

22 )ˆ(ˆ)2()(

n

t t

it

in

t t

tin

nnn

1

222

22

1

22

2

)ˆˆ(

)ˆˆ)(ˆˆ(ˆˆ

n

ttnn 1

22 ˆ1̂

,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

GARCHfitting

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.

)(kQk p-value

1 66.77152 3.330669e-16

2 109.5179 0

3 121.1315 0

4 122.6261 0

5 123.5836 0

6 124.9370 0

7 130.0145 0

8 131.3887 0

9 146.4859 0

10 147.6449 0

)(kQ

GARCHfitting

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

AIC.

Parameter estimatesGARCHfitting

11222

2110

2 ˆˆˆ ttttt aannn

Coefficient Value St.E

8.3439 0.3087

0.7903 0.1731

0.0464 0.0949

-0.5694 0.1687

1.3597 0.2417

0.0464 0.1731

0̂

1̂

2̂

111ˆˆˆ

22 ˆˆ

1̂

GARCHfitting

So I would suggest the following model.

GARCH(1,2).

ttt NYCBal 10

ttt en

22

21

21

2 0464.03597.15694.03439.8ˆ tttt nn

ttttt n 12122211