# Problem 1.
> da=read.table("m-mortg.txt")
> dim(da)  % Find the dimension of the data matrix
[1] 370   4
> da[1,]    % View the first row of the data
    V1 V2 V3   V4
1 1976  6  1 8.85
> mort=log(da[,4])
> acf(mort)
> acf(diff(mort))
> m1=arima(mort,order=c(0,1,3))
> m1
arima(x = mort, order = c(0, 1, 3))

Coefficients:
         ma1      ma2      ma3
      0.5269  -0.0491  -0.1362
s.e.  0.0512   0.0576   0.0509

sigma^2 estimated as 0.0006853:  log likelihood = 820.43,  aic = -1632.85
> tsdiag(m1)
> Box.test(m1$residuals,lag=12,type='Ljung')

        Box-Ljung test

data:  m1$residuals 
X-squared = 9.652, df = 12, p-value = 0.6465

> 1-pchisq(9.652,9)   % Take into account the number of fitted coefficients
[1] 0.3793851

> predict(m1,4)
$pred
Time Series:
Start = 371 
End = 374 
Frequency = 1 
[1] 1.804799 1.805716 1.808650 1.808650

$se
Time Series:
Start = 371 
End = 374 
Frequency = 1 
[1] 0.02617852 0.04778130 0.06147972 0.07080488

#Problem 2
> da=read.table("m-gs2.txt")
> dim(da)
> da[1,]
> gs2=log(da[,4])
> 
> plot(gs2,mort)
> m2=lm(mort~gs2)
> summary(m2)

lm(formula = mort ~ gs2)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.24264 -0.10086 -0.01074  0.09251  0.31197 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.23309    0.02422   50.91   <2e-16 ***
gs2          0.52823    0.01279   41.30   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Residual standard error: 0.1239 on 368 degrees of freedom
Multiple R-Squared: 0.8225,     Adjusted R-squared: 0.8221 
F-statistic:  1706 on 1 and 368 DF,  p-value: < 2.2e-16 

> acf(m2$residuals,lag=12)
> y=diff(mort)
> x=diff(gs2)
> m2=lm(y~x)
> summary(m2)

lm(formula = y ~ x)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.087185 -0.012493 -0.001316  0.010620  0.127094 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.0005758  0.0010942  -0.526    0.599    
x            0.3445959  0.0177031  19.465   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Residual standard error: 0.02101 on 367 degrees of freedom
Multiple R-Squared: 0.508,      Adjusted R-squared: 0.5066 
F-statistic: 378.9 on 1 and 367 DF,  p-value: < 2.2e-16 

> acf(m2$residuals,lag=12)
> m3=arima(y,order=c(0,0,3),xreg=x)
> m3
arima(x = y, order = c(0, 0, 3), xreg = x)

Coefficients:
         ma1      ma2      ma3  intercept       x
      0.3251  -0.0722  -0.1174    -0.0006  0.3161
s.e.  0.0532   0.0541   0.0498     0.0012  0.0189

sigma^2 estimated as 0.0003914:  log likelihood = 923.88,  aic = -1835.76

> tsdiag(m3,gof.lag=12)
> Box.test(m3$residuals,lag=12,type='Ljung')
        Box-Ljung test
data:  m3$residuals 
X-squared = 7.9935, df = 12, p-value = 0.7856

> 1-pchisq(7.9935,8)
[1] 0.4341053

# Problem 3
> da=read.table("m-dec1-8006.txt")
> dim(da)
> da[1,]

> jan=rep(c(1,rep(0,11)),27)
> dec1=da[,2]
> acf(dec1,lag=36)
> m4=arima(dec1,order=c(0,0,1),xreg=jan)
> m4
arima(x = dec1, order = c(0, 0, 1), xreg = jan)

Coefficients:
         ma1  intercept     jan
      0.2483     0.0077  0.1224
s.e.  0.0519     0.0045  0.0121

sigma^2 estimated as 0.003975:  log likelihood = 435.74,  aic = -863.48
> tsdiag(m4,gof.lag=24)
> Box.test(m4$residuals,lag=24,type='Ljung')
        Box-Ljung test
data:  m4$residuals 
X-squared = 24.3446, df = 24, p-value = 0.442

> 1-pchisq(24.34,22)
[1] 0.329663

#Problem 4
> m5=arima(dec1,order=c(0,0,1),seasonal=list(order=c(1,0,1),period=12))
> m5
arima(x = dec1, order = c(0, 0, 1), seasonal = list(order = c(1, 0, 1), period = 12))

Coefficients:
         ma1    sar1     sma1  intercept
      0.2409  0.9995  -0.9830     0.0179
s.e.  0.0517  0.0014   0.0241     0.0131

sigma^2 estimated as 0.00409:  log likelihood = 420.42,  aic = -830.83
> tsdiag(m5,gof.lag=24)
> Box.test(m5$residuals,lag=24,type='Ljung')
        Box-Ljung test
data:  m5$residuals 
X-squared = 23.9223, df = 24, p-value = 0.466

> 1-pchisq(23.92,21)
[1] 0.2969509

#Problem 5
> da=read.table("q-aa-earn.txt")
> dim(da)
> aa=da[,4]
> plot(aa,type='l')
> acf(aa)
> pacf(aa)
> m6=arima(aa,order=c(1,0,0)) % Lag-1 PACF is clearly significant
> m6
arima(x = aa, order = c(1, 0, 0))

Coefficients:
         ar1  intercept
      0.8004     0.2908
s.e.  0.0776     0.0713

sigma^2 estimated as 0.01390:  log likelihood = 43.34,  aic = -80.68

> acf(m6$residuals)  % Shows a seasonal lag
> pacf(m6$residuals)  % Also shows a seasonal lag

% First, entertain an seasonal MA model.
> m7=arima(aa,order=c(1,0,0),seasonal=list(order=c(0,0,1),period=4))
> m7
arima(x = aa, order = c(1, 0, 0), seasonal = list(order = c(0, 0, 1), period = 4))

Coefficients:
         ar1    sma1  intercept
      0.7573  0.5574     0.3015
s.e.  0.0850  0.1142     0.0804

sigma^2 estimated as 0.01089:  log likelihood = 49.97,  aic = -91.94
> tsdiag(m7,gof.lag=12)
> Box.test(m7$residuals,lag=12,type='Ljung')
        Box-Ljung test
data:  m7$residuals 
X-squared = 3.56, df = 12, p-value = 0.9901

> predict(m7,4)
$pred
Time Series:
Start = 62 
End = 65 
Frequency = 1 
[1] 0.5452940 0.6766207 0.7408668 0.5678639

$se
 ...
[1] 0.1043482 0.1308912 0.1439203 0.1508851

% Next, entertain a seasonal AR model.
> m8=arima(aa,order=c(1,0,0),seasonal=list(order=c(1,0,0),period=4))
> m8
arima(x = aa, order = c(1, 0, 0), seasonal = list(order = c(1, 0, 0), period = 4))

Coefficients:
         ar1    sar1  intercept
      0.7206  0.6992     0.3204
s.e.  0.0893  0.1346     0.1311

sigma^2 estimated as 0.01005:  log likelihood = 51.82,  aic = -95.64
> tsdiag(m8,gof.lag=12)
> Box.test(m8$residuals,lag=12,type='Ljung')
        Box-Ljung test
data:  m8$residuals 
X-squared = 6.8164, df = 12, p-value = 0.8695

> predict(m8,4)
$pred
Time Series:
Start = 62 
End = 65 
Frequency = 1 
[1] 0.5521602 0.7379300 0.8352861 0.6088761

$se
 ...
[1] 0.1002715 0.1235951 0.1341167 0.1392674