Seasonal ARMA forecasting and Fitting the bivariate data to GARCH John DOE.
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Transcript of Seasonal ARMA forecasting and Fitting the bivariate data to GARCH John DOE.
![Page 1: Seasonal ARMA forecasting and Fitting the bivariate data to GARCH John DOE.](https://reader035.fdocuments.in/reader035/viewer/2022062423/5697c0051a28abf838cc5159/html5/thumbnails/1.jpg)
Seasonal ARMA forecasting Seasonal ARMA forecasting and and
Fitting the bivariate data to Fitting the bivariate data to GARCHGARCH
John DOEJohn DOE
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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
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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. 2. Fitting the data Fitting the data to Seasonal ARIMA modelto Seasonal ARIMA model
SARIMAfitting
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Since the number of cases are strictly positive
and non stationary in the variance, the log was taken
SARIMAfitting
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Then the number of cases was seasonally
and lag 1 differenced
SARIMAfitting
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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(
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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
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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
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The diagnostic plots of the fitted model SARIMAfitting
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PredictionsData and predictions for Baltimore
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PredictionsData and predictions for New York City
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2. Fitting the bivariate data 2. Fitting the bivariate data to GARCH modelto GARCH model
GARCHfitting
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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
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GARCHfitting
2. We can compute OLS residuals and fit the residuals to AR(p) model.
ttt BallBa ˆ̂ AR(12) was selected.
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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
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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
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GARCHfitting
2. Fit the identified ARMA(2,1) model on the squared residuals , which has the smallest
AIC.
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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
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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