Which model fits your data better?
- Exercise #6 except part (d), Chapter 4, Diebold (Elements of Forecasting 4e). Use the dataset HW1Q1.wf1 (HW1Q1.dat for others). 2. Exercise #7, Chapter 4, Diebold (Elements of Forecasting 4e). Use the dataset HW1Q2.wf1 (HW1Q2.dat for others). 3. Consider a forecasting model with a trend yt = β0 + β1t + εt , (1) where t is the t index for t = 1, 2, • • • , T. The ordinary least squares estimator of β0 and β1 are given as βˆ 1 = PT t=1 (yt − y¯) (t − t) PT t=1 (t − t) 2 , βˆ 0 = ¯y − βˆ 1t, ewhere y¯ = 1 T X T t=1 yt , t = 1 T X T t=1 t. (a) Suppose that instead of t, you use the lag of t, which is L(t) = t − 1. L is the “lag” operator L(xt) = xt−1. Now estimate a model yt = γ0 + γ1L(t) + εt . (2) Find the ordinary least squares estimator for γ0 and γ1 in terms of βˆ 0 and βˆ 1. (b) Suppose that the unit of yt is in billions but you mistakenly used wt = yt/1000, which is in trillions, and estimate a model wt = δ0 + δ1t + εt . (3) Find the ordinary least squares estimator for δ0 and δ1 in terms of βˆ 0 and βˆ 1. 1 In the dataset HW1Q3.wf1 (HW1Q3.dat for others), you can find real GDP of the US (in billions of chained 2009 dollars) over the period 1960Q1-2014Q4. Let yt denote real GDP of the US. (c) Generate two linear time trends t = t and L(t) = t − 1. That is, t = 1 and L(t) = 0 in 1960Q1, t = 2 and L(t) = 1 in 1960Q2, and so on. Run a regression of yt on t and L(t), respectively (estimate equation (1) and (2)), and confirm your answer in part (a). (d) Generate wt = yt/1000. That is, wt is real GDP in trillions of chained 2009 dollars. Run a regression of wt on t (estimate equation (3)) and confirm your answer in part (b). 4. Go to FRED by the Federal Reserve Bank of St. Louis (https://research.stlouisfed.org/fred2/) and download a macroeconomic or financial variable of your interest. Choose a quarterly or monthly variable with more than 100 observations. (a) Do you think it is better to take log transformations of your variable or not? Why? If yes, plot the logs of your variable against t. Otherwise, plot the variable itself. This can make a serious difference. (b) Let yt denote your data (log of your data if you decide to take logs). Fit the following three models yt = β0 + β1t + εt , yt = β0 + β1t + β2 (t) 2 + εt , yt = β0 + β1t + β2 (t) 2 + β3 (t) 3 + εt , to your data. Which model fits your data better? Explain. (c) Compute 1, 5, 10, 50-year ahead point and interval forecasts based on the model that fits the data best in part (b). Does your 50-year ahead forecast make sense? Explain. 5. From the FRED database, choose any two variables that you believe to be truly unrelated but which both have a clear trend with time. Regress one variable on the other and explain your result. Would you use either variable to help forecast the other? Why or why not?