Discussion 5
Po
10/27/2020
Logistics
ACF, PACF
AR(2)
\[ x_t = 0.3 x_{t-1} + 0.4 x_{t-2} + w_t \]
x1 = arima.sim(list(ar=c(0.3, -0.4)), n=100)
par(mfrow=c(2,1))
plot(0:10, ARMAacf(ar=c(0.3, -0.4), lag.max = 10),
type = "h",
xlab = "lag",
ylab = "Theoretical ACF",
main = "Theoretical ACF of AR(2) (0.3, -0.4)"
)
abline(h = 0)
acf(x1, lag.max = 10)
x1 = arima.sim(list(ar=c(0.3, -0.4)), n=100)
par(mfrow=c(2,1))
plot(1:10, ARMAacf(ar=c(0.3, -0.4), lag.max = 10, pacf=TRUE),
type = "h",
xlab = "lag",
ylab = "Theoretical PACF",
main = "Theoretical PACF of AR(2) (0.3, -0.4)"
)
abline(h = 0)
pacf(x1, lag.max = 10)
MA(2)
x2 = arima.sim(list(ma=c(0.2, 0.1)), n=100)
par(mfrow=c(2,1))
plot(0:10, ARMAacf(ma=c(0.2, 0.1), lag.max = 10),
type = "h",
xlab = "lag",
ylab = "Theoretical ACF",
main = "Theoretical ACF of MA(2) (0.2, 0.1)"
)
abline(h = 0)
acf(x2, lag.max = 10)
par(mfrow=c(2,1))
plot(1:10, ARMAacf(ma=c(0.2, 0.1), lag.max = 10, pacf=TRUE),
type = "h",
xlab = "lag",
ylab = "Theoretical PACF",
main = "Theoretical PACF of MA(2) (0.2, 0.1)"
)
abline(h = 0)
pacf(x2, lag.max = 10)
HW4
1a Show \(X_t = \sum_{j=0}^t \phi^j Z_{t-j}\).
1b/c Show \(\text{Cov}(X_{t+h}, X_t) = \phi^h \text{Var}(X_t)\).
2 Show \(\rho(1) = \frac{\theta}{1+\theta^2}\). What is \(\text{Cov}(X_{t+2} - \rho(1) X_{t+1}, X_{t} - \rho(1) X_{t+1})\)?
4 WTF? TBH the wording in this question is too fancy I don’t comprehend this neither. Let me tell you the solution directly. It is in fact \[ X_{n+2} = \phi^2_1 X_n + \phi_1 \phi_2 X_{n-1} + \phi_1 w_{n+1} + w_{n+2} \] and thus \[ E[(X_{n+2} - \phi^2_1 X_n - \phi_1 \phi_2 X_{n-1})] = 0 \quad \text{for all } k = 1, 2, \dots, n \] From all the above equations, you can guess the formula for \(X^n_{n+2}\). Please fill in the missing mathematical arguments and conclusion. Blindly copy and paste above sentences as solution will get zero mark.
Rmarkdown
Hey, most of the project submissions are very uglily written. Your answers would be easily overlooked if you just throw all raw R outputs and treat them as answers. If your answer is sitting inside raw R output and is also unfortunately missed, it will NOT credited :(.
There are some resources of RMarkdown for you to learn better organizing your work.