Problem A

Let $Z_t$ be a sequence of independent normal random variables, each with mean 0 and variance $\sigma^2$, and let $a$, $b$, and $c$ be constants. Which, if any, of the following processes are stationary? for each stationary process specify the mean, autocovariance, and autcorrelations functions.

$X_t = a+ bZ_t + cZ_{t-2}$

$X_t = a+ bZ_0$

Problem B

Use the following time series:

airpass{itsmr}: Number of international airline passengers, 1949 to 1960.

lake{itsmr}: Level of Lake Huron, 1875 to 1972.

And perform the following:

Plot and examine the time series and its ACF.

Estimate the Trend and/or Season being considered using Moving Average Smoothing and Regression, if applicable

Plot both trend estimates and the original series in one figure.

Test the hypothesis whether or not the the residuals data are IID.

State your conclusions whether or not further analysis should be pursued. Why?

Use the differencing operator to eliminate the Trend/Season components and redo steps 4 and 5.

{r} # Load packages library(itsmr) library(tidyverse) library(tsibble) library(feasts)