## Include output and graphics in your discussions in the appropriate places, do not attach output separately.

### statistics

##### Description

Include output and graphics in your discussions in the appropriate places, do not attach output separately. Do not display output or graphics without comments. You will be graded on the thoroughness of your analysis and your discussions.

All data sets can be found in asta.

1. Let St represent the monthly sales data in sales (n = 150), and let Lt be the leading indicator in lead.

.      (a)  Fit an ARIMA model to St, the monthly sales data. Discuss your model fitting in a step-by-step fashion, presenting your: (A) initial examination of the data, (B) transformations and differencing orders, if necessary, (C) initial identification of the dependence orders, (D) parameter estimation, (E) residual diagnostics and model choice.

.      (b)  Use the CCF and lag plots between ∇St and ∇Lt to argue that a regression of ∇St on ∇Lt−3 is reasonable. [Note: In lag2.plot(), the first named series is the one that gets lagged.]

.      (c)  Fit the regression model ∇St = β0 + β1∇Lt−3 + xt, where xt is an ARMA process (explain how you decided on your model for xt). Discuss your results.

.       If you have to work with various transformations of series in x and y, you have to align the data: x = ts(rnorm(100), start= 2001, freq=4)  y = ts(rnorm(100), start= 2002, freq=4) dog = ts.intersect( lag(x,-1), diff(y,2) ) xnew = dog[,1] # dog has 2 columns, the first is lag(x,-1) ... ynew = dog[,2] # ... and the second column is diff(y,2) plot(dog) # now you can manipulate xnew and ynew simultaneously

2. Fit an ARIMA model of your choice to the unemployment data, UnempRate following (A) – (E) of Problem 1a. Then use the estimated model to forecast the next 12 months.

3. Investigate whether the quarterly growth rate of US GDP (gdp) exhibits GARCH behavior. If so, fit an appropriate model to the growth rate.

4. Fit a state space model to the Johnson & Johnson earnings in jj. Plot the data with (a) the smoothers, and (b) the filters, superimposed each with error bounds (two separate graphs). Compare the results of (a) and (b). In addition, what does the estimated value of φ tell you about the growth rate in the earnings?

5. x Crude oil prices in dollars per barrel are in oil. Fit an ARIMA(p,d,q) model to the growth rate following (A) – (E) of Problem 1a.

y Investigate whether the growth rate of the weekly oil price from the previous problem exhibits GARCH behavior. If so, fit an appropriate model to the growth rate.