Assignment_3_S07_Solution

# Assignment_3_S07_Solution - 1 The original form can be...

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1. The original form can be written as: Zt - a*Zt-2 = et + b*et-2 (1-a*B^2)Zt = (1+ b*B^2) et-2 If Zt is weakly stationary, all the roots of (1-a*B^2) = 0 should be outside the unit circle, assume a is not 0, B= a 1 or - a 1 if a> 0 and a 1 i or - a 1 i if a<0 So a 1 > 1 and i a 1 >1, which gives ) 1 , 0 ( ) 0 , 1 ( a Since Zt=a*Zt-2 +et + b*et-2 Zt+2=a*Zt + et+2 + b*et = a*Zt + b*et 2-step-ahead predictor: t t t e b Z a Z ˆ ˆ ˆ * * 2 + = + where and are estimated coefficients of a and b. * a * b 2. a) > mydata<-data\$stock[1:370] > data<-read.table("C:/companyGdata.txt",header=T) > acf(mydata)

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0 5 10 15 20 25 0.0 0.4 0.8 Lag ACF Series mydata Comment: This acf plot shows long range dependence, which decays hyperbolically. All lags are significant up to lag 25. > pacf(mydata) 5 1 01 52 02 5 -0.2 0.2 0.6 1.0 Lag Partial ACF mydata
Comment: The data are correlated. Few lags are significant up to lag 13. b) > lgdata=log(mydata) > ts.plot(diff(lgdata)) (take log transformation and difference once) Time diff(lgdata) 0 100 200 300 -0.2 0.0 0.1 0.2 0.3 > acf(diff(lgdata) )

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0 5 10 15 20 25 0.0 0.4 0.8 Lag ACF Series diff(lgdata) Comment: All the lags are inside the border, the data is not correlated. > pacf(diff(lgdata)) 5 1 01 52 02 5 -0.10 0.00 0.05 0.10 Lag Partial ACF Series diff(lgdata)
Comment: All the lags are inside the border except lag 23, which is simply a sample error. The data is not correlated. c) > ar.yw(diff(lgdata),order.max=10) Call: ar.yw.default(x = diff(lgdata), order.max = 10) Coefficients: 1 -0.0899 Order selected 1 sigma^2 estimated as 0.003201

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Assignment_3_S07_Solution - 1 The original form can be...

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