library(readxl)
seaice <- read_excel("Documents/Data Analysis and econometrics/seaice.xlsx",skip = 1)

So I used the same data set as week but instead looked at Antartica sea ice depth rather than Artic sea ice.

library(forecast)
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
library(seasonal)
t=c(1:255)
icets<-ts(seaice$Antarctica,t,frequency=12)
plot(icets)

ice<-window(icets,end=c(t=255))
## Warning in window.default(x, ...): 'end' value not changed
library(fpp2)
## Loading required package: ggplot2
## Loading required package: fma
## Loading required package: expsmooth
library(tseries)
library(fGarch)
## Loading required package: timeDate
## Loading required package: timeSeries
## 
## Attaching package: 'timeSeries'
## The following objects are masked from 'package:seasonal':
## 
##     outlier, series
## Loading required package: fBasics
decomp=stl(icets,s.window='periodic')
autoplot(decomp)

train<-ts(icets[1:200],frequency = 12)
test<-ts(icets[201:255],frequency = 12)
ets<-ets(train,model = "ZZZ")
ets
## ETS(A,N,A) 
## 
## Call:
##  ets(y = train, model = "ZZZ") 
## 
##   Smoothing parameters:
##     alpha = 0.8572 
##     gamma = 1e-04 
## 
##   Initial states:
##     l = 8.6114 
##     s = -2.0545 2.7694 5.18 5.7245 5.2947 3.9571
##            1.7382 -0.7528 -3.5797 -6.0259 -6.7337 -5.5174
## 
##   sigma:  0.2627
## 
##      AIC     AICc      BIC 
## 540.4146 543.0233 589.8894
fcets<-forecast(ets,55)
autoplot(fcets)

accets<-accuracy(fcets,test[1:55])
accets
##                       ME      RMSE       MAE       MPE     MAPE       MASE
## Training set  0.00287898 0.2533148 0.1981419 -0.199337 3.434498 0.09509972
## Test set     -0.14446080 0.4756570 0.3973897 -5.133348 8.215553 0.19073019
##                    ACF1
## Training set 0.06909786
## Test set             NA
checkresiduals(fcets)

## 
##  Ljung-Box test
## 
## data:  Residuals from ETS(A,N,A)
## Q* = 36.903, df = 10, p-value = 5.883e-05
## 
## Model df: 14.   Total lags used: 24
myarima<-auto.arima(train)
summary(myarima)
## Series: train 
## ARIMA(1,0,1)(0,1,1)[12] 
## 
## Coefficients:
##          ar1     ma1     sma1
##       0.6601  0.1446  -0.8815
## s.e.  0.0742  0.0966   0.0829
## 
## sigma^2 estimated as 0.06378:  log likelihood=-15.82
## AIC=39.64   AICc=39.86   BIC=52.59
## 
## Training set error measures:
##                      ME      RMSE       MAE        MPE     MAPE      MASE
## Training set 0.01558477 0.2429011 0.1852697 -0.1146198 3.173879 0.5013056
##                     ACF1
## Training set 0.001921232
fcmyarima<-forecast(myarima,55)
autoplot(fcmyarima)

accmyarima<-accuracy(fcmyarima,test[1:55])
accmyarima
##                      ME      RMSE       MAE        MPE     MAPE       MASE
## Training set 0.01558477 0.2429011 0.1852697 -0.1146198 3.173879 0.08892161
## Test set     0.24148561 0.5119285 0.3936008  2.2973198 6.078179 0.18891170
##                     ACF1
## Training set 0.001921232
## Test set              NA
checkresiduals(fcmyarima)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(1,0,1)(0,1,1)[12]
## Q* = 17.519, df = 21, p-value = 0.6792
## 
## Model df: 3.   Total lags used: 24
garch1<-garch(train)
## 
##  ***** ESTIMATION WITH ANALYTICAL GRADIENT ***** 
## 
## 
##      I     INITIAL X(I)        D(I)
## 
##      1     1.870185e+01     1.000e+00
##      2     5.000000e-02     1.000e+00
##      3     5.000000e-02     1.000e+00
## 
##     IT   NF      F         RELDF    PRELDF    RELDX   STPPAR   D*STEP   NPRELDF
##      0    1  6.703e+02
##      1    2  5.332e+02  2.04e-01  2.18e+00  2.6e-02  1.5e+03  1.0e+00  1.59e+03
##      2    4  5.291e+02  7.79e-03  7.49e-03  1.3e-03  6.2e+00  5.0e-02  5.68e+00
##      3    6  5.222e+02  1.29e-02  1.28e-02  2.6e-03  1.8e+00  1.0e-01  2.94e-02
##      4    8  5.211e+02  2.21e-03  2.20e-03  5.2e-04  1.7e+01  2.0e-02  2.63e-02
##      5   10  5.190e+02  4.05e-03  4.04e-03  1.0e-03  3.1e+00  4.0e-02  3.33e-02
##      6   12  5.186e+02  7.54e-04  7.53e-04  2.1e-04  3.9e+01  8.0e-03  3.49e-02
##      7   14  5.185e+02  1.49e-04  1.49e-04  4.2e-05  2.0e+02  1.6e-03  3.85e-02
##      8   16  5.184e+02  2.95e-04  2.95e-04  8.4e-05  2.5e+01  3.2e-03  3.92e-02
##      9   18  5.183e+02  5.86e-05  5.86e-05  1.7e-05  4.8e+02  6.4e-04  3.94e-02
##     10   20  5.183e+02  1.17e-04  1.17e-04  3.3e-05  6.2e+01  1.3e-03  3.97e-02
##     11   22  5.183e+02  2.33e-05  2.33e-05  6.7e-06  1.2e+03  2.6e-04  3.97e-02
##     12   24  5.182e+02  4.66e-05  4.66e-05  1.3e-05  1.5e+02  5.1e-04  3.99e-02
##     13   26  5.182e+02  9.31e-06  9.31e-06  2.7e-06  3.0e+03  1.0e-04  3.99e-02
##     14   28  5.182e+02  1.86e-05  1.86e-05  5.4e-06  3.8e+02  2.0e-04  3.99e-02
##     15   30  5.182e+02  3.72e-06  3.72e-06  1.1e-06  7.5e+03  4.1e-05  3.99e-02
##     16   32  5.182e+02  7.44e-07  7.44e-07  2.1e-07  3.8e+04  8.2e-06  4.00e-02
##     17   34  5.182e+02  1.49e-07  1.49e-07  4.3e-08  1.9e+05  1.6e-06  4.00e-02
##     18   36  5.182e+02  2.98e-07  2.98e-07  8.6e-08  2.4e+04  3.3e-06  4.00e-02
##     19   38  5.182e+02  5.95e-08  5.95e-08  1.7e-08  4.7e+05  6.6e-07  4.00e-02
##     20   40  5.182e+02  1.19e-07  1.19e-07  3.4e-08  5.9e+04  1.3e-06  4.00e-02
##     21   42  5.182e+02  2.38e-07  2.38e-07  6.9e-08  2.9e+04  2.6e-06  4.00e-02
##     22   44  5.182e+02  4.76e-08  4.76e-08  1.4e-08  5.9e+05  5.2e-07  4.00e-02
##     23   46  5.182e+02  9.52e-09  9.52e-09  2.7e-09  2.9e+06  1.0e-07  4.00e-02
##     24   48  5.182e+02  1.90e-08  1.90e-08  5.5e-09  3.7e+05  2.1e-07  4.00e-02
##     25   50  5.182e+02  3.81e-09  3.81e-09  1.1e-09  7.4e+06  4.2e-08  4.00e-02
##     26   52  5.182e+02  7.62e-09  7.62e-09  2.2e-09  9.2e+05  8.4e-08  4.00e-02
##     27   54  5.182e+02  1.52e-08  1.52e-08  4.4e-09  4.6e+05  1.7e-07  4.00e-02
##     28   57  5.182e+02  3.05e-10  3.05e-10  8.8e-11  9.2e+07  3.4e-09  4.00e-02
##     29   59  5.182e+02  6.09e-11  6.09e-11  1.8e-11  1.2e+00  6.7e-10 -2.74e-02
##     30   61  5.182e+02  1.22e-10  1.22e-10  3.5e-11  5.8e+07  1.3e-09  4.00e-02
##     31   64  5.182e+02  2.44e-12  2.44e-12  7.0e-13  1.2e+00  2.7e-11 -2.74e-02
##     32   66  5.182e+02  4.88e-12  4.88e-12  1.4e-12  1.2e+00  5.4e-11 -2.74e-02
##     33   68  5.182e+02  9.75e-12  9.75e-12  2.8e-12  1.2e+00  1.1e-10 -2.74e-02
##     34   71  5.182e+02  1.95e-13  1.95e-13  5.6e-14  1.2e+00  2.1e-12 -2.74e-02
##     35   73  5.182e+02  3.91e-14  3.90e-14  1.1e-14  1.2e+00  4.3e-13 -2.74e-02
##     36   75  5.182e+02  7.24e-15  7.80e-15  2.2e-15  1.2e+00  8.6e-14 -2.74e-02
##     37   77  5.182e+02  1.65e-14  1.56e-14  4.5e-15  1.2e+00  1.7e-13 -2.75e-02
##     38   78  5.182e+02 -1.93e+07  3.12e-14  9.0e-15  1.2e+00  3.4e-13 -2.75e-02
## 
##  ***** FALSE CONVERGENCE *****
## 
##  FUNCTION     5.182099e+02   RELDX        8.995e-15
##  FUNC. EVALS      78         GRAD. EVALS      38
##  PRELDF       3.120e-14      NPRELDF     -2.747e-02
## 
##      I      FINAL X(I)        D(I)          G(I)
## 
##      1    1.870655e+01     1.000e+00     5.456e-01
##      2    9.780310e-01     1.000e+00     9.485e+00
##      3    3.523641e-15     1.000e+00     4.609e+01
summary(garch1)
## 
## Call:
## garch(x = train)
## 
## Model:
## GARCH(1,1)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## 0.2627 0.5491 0.9491 1.0994 1.2746 
## 
## Coefficient(s):
##     Estimate  Std. Error  t value Pr(>|t|)
## a0 1.871e+01   1.942e+01    0.963    0.335
## a1 9.780e-01   8.229e-01    1.189    0.235
## b1 3.524e-15   4.238e-01    0.000    1.000
## 
## Diagnostic Tests:
##  Jarque Bera Test
## 
## data:  Residuals
## X-squared = 20.462, df = 2, p-value = 3.603e-05
## 
## 
##  Box-Ljung test
## 
## data:  Squared.Residuals
## X-squared = 128.01, df = 1, p-value < 2.2e-16
checkresiduals(garch1)
## Warning in modeldf.default(object): Could not find appropriate degrees of
## freedom for this model.