This homework is optional. If you are short on time I would rather you put your time into your project. If you do complete this homework and hand it in, your homework grade will be based on the best 6 out of 7 homeworks.
Derive the spectrum for an MA(1) process. Produce a plot of the spectrum showing the shape for a few values of β.
Show that if $X_t$ and $Y_t$ are independent, stationary processes with power spectral density functions $f_x(\omega)$ and $f_y(\omega)$, then $V_t = X_t + Y_t$ is also stationary with power spectral density $f_v(\omega) = f_x(\omega) + f_y(\omega)$.
The data.frame flow_df contains the average monthly river flow \(m^3/\text{sec}\) in the Mckenzie river at Mckenzie Bridge, Oregon. (I got this from http://robjhyndman.com/tsdldata/askew/askew7.dat
who quotes the source: Hipel and Mcleod (1994))
load(url("http://stat565.cwick.co.nz/data/flow_df.rda"))
qplot(time, log(flow), data = flow_df, geom = "line")
The column time contains a simple time index, the number of months since the start of the record. The column date contains a decimal representation of the date, i.e. 1911.750 is October 1911.
Estimate the spectrum of the logarithm of flow. Make sure you show evidence you experimented with the amount of smoothing, but you need only show your final plot.
Fit a smooth trend to the logarithm of flow, and estimate the spectrum of the residuals. How does this spectrum differ from the one in part 1.?
Fit a harmonic regression to the residuals from 2 using the estimated spectrum to choose the number and frequencies of the periodic components.
Examine the residuals from the harmonic regression using both the ACF/PACF and periodogram. Is there any evidence of remaining autocorrelation?