Question

Fit an appropriate seasonal ARIMA model to the log-transformed Johnson and Johnson earnings series (jj) of Example 1.1. Use the estimated model to forecast the next 4 quarters.

**using R**

**using library "astsa" **

**using data "jj"

*** the data used in example 1.1 is the same data meant to be used for this question, but for this question we are supposed to log transform it ***

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to access data:

install.packages("astsa")

data("so2")

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Answer 1

UR_DATA <- read_csv("C:/Users/swapnil.soni01/Desktop/UR_DATA.csv")
attach(UR_DATA)

initial examination of the data,   

UR=as.ts(UR)
plot(UR)

#NO SEASONALITY; no trend

#Stationarity test
adf.test(UR)

transformations, if necessary, &
initial identification of the dependence orders and degree of differencing,
#now let's difference series & check
UR_d1=diff(UR)
adf.test(UR_d1)
#now series is stationary

parameter estimation,

#ACF & PACF plot
acf(UR_d1) #MA order=1
pacf(UR_d1) #AR order=2

#estimate ARIMA model
model=arima(UR,order=c(2,1,1))
Call:
arima(x = UR, order = c(2, 1, 1))

Coefficients:
ar1 ar2 ma1
-0.7999 -0.4548 0.1290
s.e. 0.3290 0.2110 0.3552

sigma^2 estimated as 0.05501: log likelihood = 0.46, aic = 7.08

residual diagnostics and model choice.
plot(model$residuals)
#test for autocorrelation
Box.test(model$residuals)
Box-Pierce test

data: model$residuals
X-squared = 0.06046, df = 1, p-value = 0.8058
yes residuals are free from autocorrelation (so, white noise)

(F) Use the estimated model to forecast the next 12 months.
forecast(model)
plot(forecast(model))

Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
27 3.492477 3.191907 3.793047 3.032795 3.952159
28 3.479451 3.163029 3.795874 2.995525 3.963377
29 3.474190 3.134513 3.813867 2.954699 3.993682
30 3.484323 3.092380 3.876266 2.884898 4.083748
31 3.478610 3.066688 3.890533 2.848629 4.108591
32 3.478572 3.040522 3.916622 2.808633 4.148511
33 3.481201 3.013985 3.948416 2.766656 4.195745
34 3.479115 2.990920 3.967311 2.732485 4.225745
35 3.479588 2.968223 3.990952 2.697524 4.261652
36 3.480158 2.946336 4.013981 2.663747 4.296570

Note: We used Indian unemployment data from WB source:

year	UR
1991	3.996
1992	3.901
1993	4.06
1994	3.7
1995	3.974
1996	3.951
1997	4.386
1998	4.119
1999	4.215
2000	4.31
2001	3.775
2002	4.316
2003	3.929
2004	3.889
2005	4.4
2006	4.331
2007	3.724
2008	4.154
2009	3.906
2010	3.55
2011	3.537
2012	3.623
2013	3.574
2014	3.53
2015	3.49
2016	3.458