Consider the annual real GDP of 14 countries from 1960 to 2011 for 52 observations. The data are obtained from FRED of the Federal Reserve Bank of St. Louis, in 2005 millions of U.S. dollars, and in the file a-rgdp-14 . txt. The first row of the file contains names of the 14 countries. Let \(z_{t}\) be the annual real GDP growth rates, in percentages, of the 14 countries. The \(z_{t}\) series has 51 observations. Divide the data into two parts with the second part consisting of the last 6 observations for forecasting purpose. The first part has 45 observations and is used in modeling. In this problem, we focus on 1-step ahead forecasting for United States and South Korea. Do the followings separately for United States and South Korea.

(a) Find a univariate time series model to produce 1-step ahead forecasts and compute the mean square error of forecasts in the forecasting subsample.

(b) Apply diffusion index method with \(\boldsymbol{x}_{t}=\left(\boldsymbol{z}_{t-1}^{\prime}, \boldsymbol{z}_{t-2}^{\prime}\right)^{\prime}\) and \(m=1,3,5\) to produce forecasts and compute the mean square error of forecasts.

(c) Let \(v_{t}\) be the series of simple average of the 14 GDP growth rates. Here, \(v_{t}\) may represent the global GDP growth rates. Use a bivariate time series model (country and \(v_{t}\) ) to produce 1 -step ahead forecasts. Also, compute the mean square error of forecasts in the forecasting subsample.