For this project, you should work with the monthly time series of the assigned commodity prices, spanning from January of 1995 to December of 2015. The project consists of a set of time series estimation and forecasting exercises, pre-sented in a list of assignments below. You should address all these items. You should submit a hard copy of the report, and also send me an R file, containing all the coding for your project.

1. Calculate real prices from the obtained nominal prices. using U.S. producer price index as a deflator, and transform these real prices to the natural logarithm. From here forward, work with these transformed time series of commodity prices, denoted by p_t.

2. Plot the time series.

3. Plot the autocorrelogram for up to 24 lap.

4. Based on your observations from 2-3, discuss any apparent features of the time series.

5. Divide the series into the in-sample estimation environment (observations up to and including Dec 2009) and the out-of-sample forecasting environment (observations beginning from Jan 2010 and onwards).

6. Using in-sample observations to obtain parameter estimates, calculate and plot point forecasts (for the duration equivalent to the out-of-sample window size) and the associated 95% interval forecasts using the following models:

(a) random walk:
(b) linear (deterministic) trend model:
(c) linear (deterministic) trend model with seasonal component;
(d) autoregressive models of orders 1 through 6.

7. Based on your observations from 6, discuss any (additional) characteristics of the time series, particularly in terms of their apparent dynamics.

8. Using in-sample observations (again), estimate vector autoregressive model of orders 1 through 6, where the vector of variables is given by x_t = (p_t, r_t), and r_t where r_t, is natural logarithm of the real crude oil prices.

9. Test for in-sample Granger causality between the crude oil prices and the commodity prices in the considered vector autoregressive models.

10. Within the rolling forecasting environment, generate one-step-ahead fore-casts throughout the out-of-sample set using the following models:

(a) random walk;

(b) linear (deterministic) trend model with seasonal component;

(c) autoregressive model of order p, where the optimal p is selected based on AIC (or BIC);

(d) vector autoregressive model of order p. where the optimal pis the same as above.

11. For all the aforementioned models:

(a) obtain one-step ahead forecast errors, and calculate and report the out-of-sample root mean square forecast error (FINISFE) measures:

(b) test the hypotheses of (i) error unbiasedness; (ii) error efficiency; and (iii) no autocorrelation.

12. Combine forecasts from 10 using the equal-weights scheme, and calculate and report the associated out-of-sample RIVSFE measure.

13. Based on your observations from 10 and 11:

(a) discuss which of the considered models is preferred in terms of accu¬rate forecasting of the commodity price series.

(b) is there evidence of out-of-sample Granger causality between the crude oil prices and the commodity prices?