HW#6

Question 1

Figure 9.32 shows the ACFs for 36 random numbers, 360 random numbers and 1,000 random numbers.

1. Explain the differences among these figures. Do they all indicate that the data are white noise?

2. Why are the critical values at different distances from the mean of zero? Why are the autocorrelations different in each figure when they each refer to white noise?

Question 2

A classic example of a non-stationary series are stock prices. Plot the daily closing prices for Amazon stock (contained in gafa_stock), along with the ACF and PACF. Explain how each plot shows that the series is non-stationary and should be differenced.

Question 3

For the following series, find an appropriate Box-Cox transformation and order of differencing in order to obtain stationary data. If any type of difference were taken, please write down the differences you chose above using backshift operator notation.

1. Turkish GDP from global_economy.

2. Accommodation takings in the state of Tasmania from aus_accommodation.

3. Monthly sales from souvenirs.

Question 4

Simulate and plot some data from simple ARIMA models. a. Use the following R code to generate data from an AR(1) model with \(\phi_{1} = 0.6\) and \(\sigma^2=1\). The process starts with \(y_1=0\).

ar1 <- function(phi, n = 100) {
  y <- numeric(n)
  e <- rnorm(n)
  for (i in 2:n) {
    y[i] <- phi * y[i - 1] + e[i]
  }
  tsibble(idx = seq_len(n), y = y, index = idx)
}
ar1(0.5)

## # A tsibble: 100 x 2 [1]
##      idx       y
##    <int>   <dbl>
##  1     1  0     
##  2     2  1.00  
##  3     3  2.08  
##  4     4 -0.550 
##  5     5 -0.496 
##  6     6 -0.870 
##  7     7  1.01  
##  8     8 -0.285 
##  9     9 -0.0253
## 10    10 -2.04  
## # … with 90 more rows

2. Produce a time plot for the series. How does the plot change as you change \(\phi_1\)?

3. Write your own code to generate data from an MA(1) model with \(\theta_{1} = 0.6\) and \(\sigma^2=1\).

4. Produce a time plot for the series. How does the plot change as you change \(\theta_1\)?

5. Generate data from an ARMA(1,1) model with \(\phi_{1} = 0.6\), \(\theta_{1} = 0.6\) and \(\sigma^2=1\).

6. Generate data from an AR(2) model with \(\phi_{1} =-0.8\), \(\phi_{2} = 0.3\) and \(\sigma^2=1\). (Note that these parameters will give a non-stationary series.)

7. Graph the latter two series and compare them.

Question 5

Consider aus_airpassengers, the total number of passengers (in millions) from Australian air carriers for the period 1970-2011.

1. Use ARIMA() to find an appropriate ARIMA model. What model was selected. Check that the residuals look like white noise. Plot forecasts for the next 10 periods.

2. Write the model in terms of the backshift operator.

3. Plot forecasts from an ARIMA(0,1,0) model with drift and compare these to part a.

4. Plot forecasts from an ARIMA(2,1,2) model with drift and compare these to part b. Remove the constant and see what happens.

5. Plot forecasts from an ARIMA(0,2,1) model with a constant. What happens?

Question 6

For the United States GDP series (from global_economy):

1. If necessary, find a suitable Box-Cox transformation for the data;

2. fit a suitable ARIMA model to the transformed data using ARIMA();

3. try some other plausible models by experimenting with the orders chosen;

4. choose what you think is the best model and check the residual diagnostics;

5. produce forecasts of your fitted model. Do the forecasts look reasonable?

6. compare the results with what you would obtain using ETS() (with no transformation).

Question 7

Consider aus_arrivals, the quarterly number of international visitors to Australia from several countries for the period 1981 Q1 – 2012 Q3. a. Select one country and describe the time plot.

2. Use differencing to obtain stationary data.

3. What can you learn from the ACF graph of the differenced data?

4. What can you learn from the PACF graph of the differenced data?

5. What model do these graphs suggest?

6. Does ARIMA() give the same model that you chose? If not, which model do you think is better?

7. Write the model in terms of the backshift operator, then without using the backshift operator.

Question 8

1. Select a time series from Quandl. Then copy its short URL and import the data. Make sure you select a data set that is Free.

#Quandl::Quandl("????")
# Copy the short URL into the ????

2. Plot graphs of the data, and try to identify an appropriate ARIMA model.

3. Do residual diagnostic checking of your ARIMA model. Are the residuals white noise?

4. Use your chosen ARIMA model to forecast the next four years.

5. Now try to identify an appropriate ETS model.

6. Do residual diagnostic checking of your ETS model. Are the residuals white noise?

7. Use your chosen ETS model to forecast the next four years.

8. Which of the two models do you prefer?