hist(rnorm(2000))
#white noise
WN<-ts(rnorm(1400), frequency=4, start=c(1980,1))
ts.plot(WN,ylab="White noise")

MA_1<-WN+0.9*lag(WN,1)
ts.plot(MA_1,ylab="Moving average")
acf(MA_1,lag.max = 15)

MA_2<-WN+0.9*lag(WN)+0.6*lag(WN,2)
ts.plot(MA_2,ylab="Moving average")
acf(MA_2,lag.max = 15)

acf(WN,lag.max=20)

AR_1<-0*WN
AR_1[1]<-WN[1]
for(i in 2:length(AR_1))
AR_1[i]<-0.99*AR_1[i-1]+WN[i]

ts.plot(AR_1)

AR_1<-arima.sim(model=list(ar=c(.9)),n=(50))#simule tout seul


op <-par(no.readonly=TRUE)
layout(matrix(c(1,1,2,3),2,2,byrow=TRUE) )

ts.plot(AR_1,ylab="Autoregressive")
acf(AR_1, main='Autocorrelations')
pacf(AR_1, main='Parital autocorrelations')
par(op)

#un ARMA(2,1)
ARMA_12<-arima.sim(model=list(ar=c(.8),ma=c(.7,.6)),n=1000)#simule tout seul

op <-par(no.readonly=TRUE)
layout(matrix(c(1,1,2,3),2,2,byrow=TRUE) )

ts.plot(ARMA_12,ylab="Autoregressive")
acf(ARMA_12, main='Autocorrelations')
pacf(ARMA_12, main='Parital autocorrelations')
par(op)

tsdisplay(ARMA_12, lag.max=10)

library(forecast)
bestmodel<-auto.arima(ARMA_12, ic="bic")

Box.test(bestmodel$residuals, type= "Ljung-Box")
acf(bestmodel$residuals)

par(mfrow=c(1,1))
pacf(residuals(bestmodel))


#un ARMA(2,1)
ARMA_12<-arima.sim(model=list(ar=c(.8),ma=c(.7,.6)),n=1000)#simule tout seul
ts.plot(ARMA_12)

bestmodel<-auto.arima(ARMA_12,d=0,seasonal = F,ic="bic")

forecast.Arima(bestmodel)

par(mfrow=c(1,1))

plot(forecast.Arima(bestmodel,h=15),xlim=c(970,1015))


-----------------------
#Random walk
shock=ts(rnorm(140), frequency=4, start=c(1970,1))
RW<-diffinv(shock)
ts.plot(RW,ylab="Random walk")

RW<-diffinv(shock+.2)
ts.plot(RW,ylab="Random walk + trend")

#install.packages("fpp")
library(fpp)
consumption<-diffinv(usconsumption[,1])/sqrt(usgdp)

ts.plot((consumption))
acf(consumption, 20)
pacf(consumption, 20)

acf(diff(consumption), 20)
pacf(diff(consumption), 20)
ts.plot(diff(consumption))

income<-diffinv(usconsumption[,2])/sqrt(usgdp)
ts.plot(income,consumption,col = c("red","blue"))
legend("topleft", legend = c("income", "consumption"), 
       fill = c("red","blue"))
ts.plot(diff(income))
acf(diff(income), 20)
pacf(diff(income), 20)

install.packages("urca")
library(urca)
df=ur.df(income, type="trend")
summary(df)
#first value of test-stat  (-2.7) is closer to 0 than levels (-3.13):
#accept null hypothese -> cointegration.
#check the trend (third value and levels line): 10.9 is larger than 8.4:
#very significant, there is a trend. 
#The same for the constant. We accept the full model.

df=ur.df(consumption, type="trend",lags=1)
summary(df)
#first value of test-stat  (-1.97) is closer to 0 than 10% level (-3.13):
#accept null hypothese -> cointegration.
#check the trend (third value and levels line): 6.6 is larger than 6.5:
#significant, there is a trend. 
#No constant. We accept the trend model without constant.

coint=ca.jo(usconsumption)
summary(coint)
#test for cointegration between variables.
#compare column test with the levels:
#for r<=1 test is larger than 1% level so there is a very significant cointegration relation

res= lm(consumption~time(consumption))
summary(res)

ts.plot(consumption-1.4537*income)
usconsumption