# Problem 1.
> da=read.table("m-dec19.txt",header=T)
> d1=da[,2]
> d9=da[,3]
> out1=acf(d1,lag=24)
> print(out1$acf,digit=1)
 [1,]  1.000
 [2,]  0.225
 [3,] -0.002
 [4,] -0.076
 [5,] -0.030
 [6,] -0.022
 [7,] -0.051
 [8,] -0.050
 [9,] -0.085
[10,] -0.079
[11,] -0.004
[12,]  0.076
[13,]  0.268
[14,]  0.018
[15,] -0.052
[16,] -0.092
[17,] -0.024
[18,] -0.042
[19,] -0.091
[20,] -0.080
[21,] -0.051
[22,] -0.058
[23,] -0.049
[24,]  0.053
[25,]  0.247
> out2=pacf(d1,lag=24)
> print(out2$acf,digit=1)
 [1,]  0.225
 [2,] -0.055
 [3,] -0.067
 [4,]  0.003
 [5,] -0.020
 [6,] -0.050
 [7,] -0.032
 [8,] -0.075
 [9,] -0.056
[10,]  0.016
[11,]  0.060
[12,]  0.240
[13,] -0.106
[14,] -0.029
[15,] -0.057
[16,] -0.004
[17,] -0.047
[18,] -0.070
[19,] -0.035
[20,] -0.001
[21,] -0.048
[22,] -0.061
[23,]  0.037
[24,]  0.168
> Box.test(d1,lag=12,type='Ljung')
        Box-Ljung test
data:  d1 
X-squared = 69.6524, df = 12, p-value = 3.72e-10
> tt=out1$acf[13]*sqrt(length(d1))
> tt
[1] 5.637147

#Problem 2.
> out1=acf(d9,lag=12)
> print(out1$acf,digit=1)
 [1,]  1.000
 [2,]  0.137
 [3,] -0.062
 [4,] -0.033
 [5,] -0.059
 [6,] -0.009
 [7,]  0.003
 [8,] -0.011
 [9,] -0.078
[10,] -0.027
[11,]  0.043
[12,] -0.036
[13,]  0.005
> Box.test(d9,lag=12,type='Ljung')
        Box-Ljung test
data:  d9 
X-squared = 16.8115, df = 12, p-value = 0.1568

#Problem 3.
> da=read.table("m-cpileng.txt")
> cpi=da[,4]
> ct=diff(log(cpi))*100
> out=acf(ct,lag=12)
> print(out$acf,digit=2)
 [1,] 1.00
 [2,] 0.58
 [3,] 0.60
 [4,] 0.56
 [5,] 0.50
 [6,] 0.57
 [7,] 0.55
 [8,] 0.53
 [9,] 0.56
[10,] 0.55
[11,] 0.56
[12,] 0.50
[13,] 0.50
> out=pacf(ct,lag=12)
> print(out$acf,digit=1)
 [1,]  0.58
 [2,]  0.39
 [3,]  0.21
 [4,]  0.07
 [5,]  0.22
 [6,]  0.16
 [7,]  0.07
 [8,]  0.12
 [9,]  0.11
[10,]  0.12
[11,] -0.05
[12,]  0.01
> Box.test(ct,lag=12,type='Ljung')
        Box-Ljung test
data:  ct 
X-squared = 2186.646, df = 12, p-value < 2.2e-16
> out=acf(diff(ct),lag=12)
> print(out$acf,digit=1)
 [1,]  1.000
 [2,] -0.516
 [3,]  0.062
 [4,]  0.022
 [5,] -0.147
 [6,]  0.096
 [7,]  0.008
 [8,] -0.056
 [9,]  0.042
[10,] -0.033
[11,]  0.100
[12,] -0.086
[13,]  0.042
> m1=arima(ct,order=c(1,0,5))
> m1
arima(x = ct, order = c(1, 0, 5))
Coefficients:
         ar1      ma1     ma2      ma3      ma4     ma5  intercept
      0.9782  -0.8160  0.0652  -0.0997  -0.1062  0.1913     0.3247
s.e.  0.0094   0.0409  0.0511   0.0518   0.0478  0.0416     0.0755

sigma^2 estimated as 0.03388:  log likelihood = 163.6,  aic = -311.21
#Problem 4.
> da=read.table("q-gnprate.txt")
> gnp=da[,1]
> m2=arima(gnp,order=c(3,0,0))
> m2
arima(x = gnp, order = c(3, 0, 0))
Coefficients:
         ar1     ar2      ar3  intercept
      0.4172  0.2003  -0.1648     0.0168
s.e.  0.0636  0.0679   0.0642     0.0011

sigma^2 estimated as 9.313e-05:  log likelihood = 769.83,  aic = -1529.67
> p1=c(1,-m2$coef[1:3])
> roots=polyroot(p1)
> roots
[1]  1.542932+0.928342i -1.870974+0.000000i  1.542932-0.928342i
> Mod(roots)
[1] 1.800683 1.870974 1.800683
> kk=2*pi/acos(1.543/1.801)
> kk
[1] 11.59539
> mf=predict(m2,4)
> names(mf)
[1] "pred" "se"  
> print(mf$pred,digit=2)
Time Series:
Start = 240 
End = 243 
Frequency = 1 
[1] 0.014 0.016 0.017 0.017
> print(mf$se,digit=2)
Time Series:
Start = 240 
End = 243 
Frequency = 1 
[1] 0.0097 0.0105 0.0111 0.0111

#Problem 5.
> m9=arima(d9,order=c(0,0,1))
> m9
arima(x = d9, order = c(0, 0, 1))

Coefficients:
         ma1  intercept
      0.1593     0.0109
s.e.  0.0499     0.0029

sigma^2 estimated as 0.002700:  log likelihood = 683.03,  aic = -1360.07
> tsdiag(m9)
> Box.test(m9$residuals,lag=12,type='Ljung')
        Box-Ljung test
data:  m9$residuals 
X-squared = 8.2257, df = 12, p-value = 0.7672

> mf=predict(m9,4)
> print(mf$pred,digit=2)
Time Series:
Start = 445 
End = 448 
Frequency = 1 
[1] 0.009 0.011 0.011 0.011
> print(mf$se,digit=2)
Time Series:
Start = 445 
End = 448 
Frequency = 1 
[1] 0.052 0.053 0.053 0.053