# 1.3 Autocovariances and Autocorrelations (Solution to Homework)

# Ex 1.16 generate a time series Y_t = b0+b1*t+eps_t, for t, t = 1,..., 100, where b0 and b1 
# are not equal to 0 and the eps_t are independent normal random variables with mean mu and variance sig_1^2
# if t <= 69 but variance sig_2^2 if t >= 70. Plot the exponentially filtered variables Y*_t for different 
# values of the smoothing parameter alpha and compare the results.

b0 <- 2; b1 <- 0.05; mu <- 0; sig1 <- 1; sig2 <- 2
t <- 1:100
Y <- b0 + b1*t + rnorm(100,mu,c(rep(sig1,69),rep(sig2,100-69)))
plot(Y,type="l")
a1 <- 0.3; a2 <- 0.1
Ys1 <- Ys2 <- Y
for(i in t[-1])
{
    Ys1[i] <- a1*Y[i] + (1-a1)*Ys1[i-1]
    Ys2[i] <- a2*Y[i] + (1-a2)*Ys2[i-1]
}
lines(Ys1,col="blue")
lines(Ys2,col="red")

# Ex 1.20: Consider the percentages to annual bancruptcies among all US companies between 
# 1867 and 1932. Compute and plot the empirical autocovariance function and the empirical
# autocorrelation function
bankrupt <- read.table("bankrupt.txt")
bankrupt <- ts(bankrupt[,2],start=1867)
plot(bankrupt)
acf(bankrupt)
acf(bankrupt,type="covariance")


# Ex 1.21: Plot the correlogram for dierent values of n.
t <- 1:100
acf(t)
n <- 100
k <- 0:20
r <- 1 - 3*k/n + 2*k*(k^2-1)/(n*(n^2-1))
points(k,r,col="red")