What is the conditional mean:

For every AR(1) model below:

a. Do a three-period ahead forecasting using the given initial values and statistics.  Write a 95% confidence interval for each forecast.

b. Do a long-run (unconditional) forecasting and write a 95% confidence interval.

a)  y_t = 1.6 + .75y_t-1 + e_t,                     y_t = 2,                          s² = 1.21        

b)  y _t = 2.5 + .3y _t-1 + e_t,                      y_t = 10,                                    s² = 6.25        

c)  y_t = 1.2 - .2y_t-1 + e_t,                        y_t = 1.5,                       s² = .49          

d)  Dy_t = 2.5 - .8Dy_t-1 + e_t,                  y_t = 6,  y_t-1 = 5,            s² = 3.69        

For every AR(2) model below:

a. Do a three-period ahead forecasting using the given initial values and statistics.  Write a 95% confidence interval for each forecast.

b. Do a long-run (unconditional) forecasting and write a 95% confidence interval.

a)  y_t =  6 + .7y_t-1 + .12y_t-2+ e_t,                        y_o = 5, y₁ = 6,              s² = 1.21        

b)  y _t = 2.5 + .3y _t-1 - .28y_t-2 + e_t,                     y_o = 1, y₁ = 2,              s² = 6.25        

c)  y_t = 1.2 - .2y_t-1 - .35y_t-2 + e_t,                        y_o = 1.5, y₁ = 2,           s² = .49          

d)  y_t = 2.5 - .07y_t-1 + .06y_t-2 + e_t,                    y_o = 6,    y₁ = 5,            s² = 3.69        

For the ARCH model, Y_t = 8.5 + .6Y_t-1 ,     Y_t = 10,     e_t = .5

                                               (3.2)  (2.8)

                                                e²_t = 1.2 + .2e²_t-1

1. What is the conditional mean of Y at times t+1, t+2, t+3?
2. What is the conditional variance of Y at times t+1, t+2, t+3?
3. What is the unconditional (long-run) mean of Y?
4. What is the unconditional (long-run) variance of Y?
5. Write 95% confidence interval for the long-run forecast of Y.
6. Write 95% confidence interval for forecasts of Y at time t+1, t+2, and t+3.

For the following ARCH model, Y_t = 4.5  + 0.4Y_t-1, Y_t = 5,     e_t = .3

                                                                   (3.2)  (2.8)

                                                                     e²_t = 2.6 + .8e²_t-1

1. What is the conditional mean of Y at times t+1, t+2, t+3?
2. What is the conditional variance of Y at times t+1, t+2, t+3?
3. What is the unconditional (long-run) mean of Y?
4. What is the unconditional (long-run) variance of Y?
5. Write 95% confidence interval for the long-run forecast of Y.
6. Write 95% confidence interval for forecasts of Y at time t+1, t+2, and t+3.

1.  Use The ARIMA file in the course site in BB. For each variables in the ARIMA file: 

a) Test for the stationarity of the variable.

b)  Do the correlogram of the variable and decide the order of the ARIMA(p, d, q).

c)  Run the best ARIMA model.

d) Do three period ex-ante dynamic forecasting and write 95% confidence intervals.

e) Do three period ex-post static forecasting and write 95% confidence intervals.

f) Do long-run forecasting of each variable and write 95% confidence interval.