b.
f<- c(1:6)
x<-split(log(varve), f)
## Warning in split.default(log(varve), f): data length is not a multiple of
## split variable
ts.plot(x$`1`)
ts.plot(x$`2`)
ts.plot(x$`3`)
ts.plot(x$`4`)
ts.plot(x$`5`)
ts.plot(x$`6`)
ts.plot(gtemp)
acf(gtemp, lag.max = 100)
#Yes we can observe behavior comparable to that observed in the global temperature records.
d.
u<- diff(LogVarve)
t633<- c(1:633)
df33<- data.frame(u,t633)
ggplot() + geom_line(aes(x=t633, y=u), data= df33, color= "red")+ geom_point() + theme_economist() + xlab("Time")
## Don't know how to automatically pick scale for object of type ts. Defaulting to continuous
## Warning in max(vapply(evaled, length, integer(1))): no non-missing
## arguments to max; returning -Inf
t634<- c(1:634)
df34<- data.frame(varve,t634)
ggplot() + geom_line(aes(x=t634, y=varve), data= df34, color= "red")+ geom_point() + theme_economist() + xlab("Time")
## Don't know how to automatically pick scale for object of type ts. Defaulting to continuous
## Warning in max(vapply(evaled, length, integer(1))): no non-missing
## arguments to max; returning -Inf
acf(varve)
acf(u)
#Differencing produces a reasonably stationary series as can be seen from the ACF after the log/differencing transformation.
Problem 2.9
a.
fit<- lm(soi~time(soi))
summary(fit)
##
## Call:
## lm(formula = soi ~ time(soi))
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.04140 -0.24183 0.01935 0.27727 0.83866
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.70367 3.18873 4.298 2.12e-05 ***
## time(soi) -0.00692 0.00162 -4.272 2.36e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3756 on 451 degrees of freedom
## Multiple R-squared: 0.0389, Adjusted R-squared: 0.03677
## F-statistic: 18.25 on 1 and 451 DF, p-value: 2.359e-05
#Yes the coefficent for time is negative and significant therefore we can see a small negative trend in sea surface temperature.
b.
w <- periodogram(fit$residuals, plot = F)
dat <- data.frame(w$freq, w$spec)
ggplot() + geom_line(aes(x=w$freq, y=w$spec), data= dat, color= " red")+ geom_point() + theme_economist() + ylab("Spec") + xlab("Freq")
## Warning in max(vapply(evaled, length, integer(1))): no non-missing
## arguments to max; returning -Inf
Frequency1 <- 1/.083333
# 12 Months
Frequency1
## [1] 12.00005
Frequency2<- 1/.0166667
Frequency2
## [1] 59.99988
#Therefore Minor peak would indicate cycle every 5 Years.
Problem 2.11
a.
t = 1:545
df<- data.frame(oil,gas,t)
df2<- df
ggplot() + geom_line(data=df, aes(x=t,y=oil, color = "Oil")) + geom_line(data=df2, aes(x=t,y=gas, color="Gas")) + ylab("Price")+theme_economist()
## Don't know how to automatically pick scale for object of type ts. Defaulting to continuous
w = rnorm(500,0,1) # 500 N(0,1) variates
v = filter(w, sides=2, filter=rep(1/3,3)) # moving average
par(mfrow=c(2,1))
plot.ts(w, main="white noise")
plot.ts(v, ylim=c(-3,3), main="moving average")
# these plots of oil and gas do not appear to be stationary nor do they seem to resemble white noise from section 1.3 they more closely resemble the moving average plot presented.
b.
# The transformation should be applied should be transformed to the diff log because it will be easier to see magnitude of the shocks, and the white noise after transformation.
c.
DiffLogOil <- diff(log(oil))
DiffLogGas <- diff(log(gas))
df3<- data.frame(DiffLogGas, DiffLogOil )
t <- t[-545]
df3<-cbind(df3,t)
plot1 <- ggplot() + geom_line(data=df, aes(x=t, y=oil, color = "Oil")) + geom_line(data=df2, aes(x=t,y=df2$gas, color="Gas")) + theme_economist()
plot2<- ggplot()+ geom_line(data=df3, aes(x=t,y=DiffLogGas, color="DiffGas")) + geom_line(data=df3, aes(x=t,y=df3$DiffLogOil, color="DiffOil")) + theme_economist()
grid.arrange(plot2, plot1, nrow=2)
## Don't know how to automatically pick scale for object of type ts. Defaulting to continuous
## Don't know how to automatically pick scale for object of type ts. Defaulting to continuous
acf(DiffLogOil)
acf(oil)
acf(DiffLogGas)
acf(gas)
#It appears that there is a clear cyclical pattern that repeats every 24 months.
d.
ccf(DiffLogGas, DiffLogOil)
#We can see from the CCF that Oil and Gas even when transformed are very highly correlated.
e.
lag2.plot(DiffLogGas,DiffLogOil,3)
# There sems to be no relationships in the lagged plots.
#outliers definetly do exist within these plots although they are very rare.
f. (i.)
poil = diff(log(oil))
pgas = diff(log(gas))
indi = ifelse(poil < 0, 0, 1)
mess = ts.intersect(pgas, poil, poilL = lag(poil,-1), poilL2 = lag(poil,-2), poilL3 = lag(poil,-3), indi)
summary(fit <- lm(pgas~ poil + poilL+ poilL2+ poilL3 + indi, data=mess))
##
## Call:
## lm(formula = pgas ~ poil + poilL + poilL2 + poilL3 + indi, data = mess)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.18517 -0.02118 0.00040 0.02210 0.34271
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.006645 0.003472 -1.914 0.05620 .
## poil 0.682261 0.058827 11.598 < 2e-16 ***
## poilL 0.106547 0.039220 2.717 0.00681 **
## poilL2 0.044673 0.039201 1.140 0.25496
## poilL3 0.012803 0.039302 0.326 0.74474
## indi 0.012724 0.005529 2.301 0.02177 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.04173 on 535 degrees of freedom
## Multiple R-squared: 0.4579, Adjusted R-squared: 0.4529
## F-statistic: 90.4 on 5 and 535 DF, p-value: < 2.2e-16
#It would appear the price of oil has less of a positive effect which can be seen by the lags that are placed in the regression.
