Math

...

alternative hypothesis: stationary

Warning message:

In adf.test(x1, alternative = "stationary") :

p-value smaller than printed p-value

From the analysis above, since p-value = 0.01 0, the difference is stationary

(c)

[pic 2][pic 3]

From the original plot seems not to be stationary for mean is not constant. However, after taking difference, it seems to be stationary with constant mean. Analysis from the plot is proved by formal test in (a) and (b)

(d) SACF and SPACF for original data:

[pic 4][pic 5]

From SACF => it is exp shape => AR(1) model. From PACF when lag >=2, SPACF=0 which also suggest this is AR(1) model

SACF and SPACF for difference:

[pic 6][pic 7]

From SACF and SPACF, residual (difference) is obviously white noise which will also suggest AR(1) model.

e.

Candidate model: AR(1) or ARIMA(0,1,0)

AR(1) or ARIMA(1,0,0):

> fit

Call:

arima(x = x, order = c(1, 0, 0))

Coefficients:

ar1 intercept

1 55.7266

s.e. 0 NaN

sigma^2 estimated as 0.7362: log likelihood = -66.33, aic = 138.65

ARIMA(0,1,0):

> fit

Call:

arima(x = x, order = c(0, 1, 0))

sigma^2 estimated as 0.7506: log likelihood = -65.05, aic = 132.1

Diagnose:

[pic 8]

From plot above residual of original data should be white noise and independent.

AIC of ARIMA(0,1,0)

f.

From e:

The original data suggest: AR(1): Yt = Yt-1 +55.7266

Coefficients:

ar1 intercept

1 55.7266

s.e.