###----------------------------------------------------------------------------
### MA(1),  MA(2)
###----------------------------------------------------------------------------

T <- 1000

## Generate the epsilons
epsilon <- rnorm(T, sd = 0.1)

epsilon_t<- epsilon[1:(T-1)]
epsilon_t1 <- epsilon[2:T]


## Generate Y
y_t <- 40 + epsilon_t + 0.8*epsilon_t1


## Plot the time series
par(mfrow = c(1, 3))
plot(y_t, type = "b")

## Plot the sample ACF
acf(y_t)

## Plot the sample PACF
pacf(y_t)


## Question: Generate MA(3) model and check the ACF patterns.
## y_t  =  40 + epsilon_t + 0.7*epsilon_t1 - 0.28*epsilon_t2


###----------------------------------------------------------------------------
### AR(1)
###----------------------------------------------------------------------------

T <- 100

## Generate the epsilons
epsilon <- rnorm(T)

y <- 0
for(t in 2:T)
    {
      y[t] <-  8 + 0.8 * y[t-1]+ epsilon[t]
    }

## Plot the time series
par(mfrow = c(1, 2))
plot(y, type = "l")

## Plot the sample ACF
acf(y)



###----------------------------------------------------------------------------
### AR(2)
###----------------------------------------------------------------------------

T <- 1000

## Generate the epsilons
epsilon <- rnorm(T, sd = 0.5)
y <- NULL
y[1] <- 0
y[2] <- 1
y[3] <- 3
for(t in 4:T)
    {
      y[t] <-  4 + 0.4 * y[t-1] +0.5 * y[t-2] - 0.7 * y[t-3] + epsilon[t]
    }

## Plot the time series
par(mfrow = c(1, 3))
plot(y, type = "l")

## Plot the sample ACF
acf(y)

## Plot the sample PACF
pacf(y)


###----------------------------------------------------------------------------
### Estimate ARIMA model (Bank Loan example)
###----------------------------------------------------------------------------

loan <- read.table("TimeSeriesTable5.5.csv", header = TRUE)
y <- loan$Applications

## Plot the time series and ACF,  PACF

par(mfrow = c(2, 2))

plot(loan, type = "b")
acf(y)
pacf(y)


## We try to fit an AR(2) model. WHY?
AR2 <- arima(y, order = c(1, 0, 1))
AR2

## The residual plot
epsilon <- as.numeric(AR2$residuals)


par(mfrow = c(3, 2))

plot(loan, type = "b")
plot(epsilon, type = "l")
hist(epsilon)
qqnorm(epsilon)
qqline(epsilon)

acf(epsilon)
pacf(epsilon)