1-      a-  What are the five components of a time series?

b- Briefly explain how you would estimate each component.

c- What does deterministc trend mean?  How do you detrend a variable that has determininstic trend?

d- What does stochastic trend mean?  How do you detrend a variable that has stochastic trend?

e- What would be the properties of the estimated coefficients if you use improper method of detrending?

2- You have estimated the forecasting model for price of a product as AR(1)

                        Pt  =  2.2 + .7P_t-1 ,            e ~ N(0, .49)               

                                 (.90)  (.08)

The last observation of the price series is $3.00.

a)      Do a three-period forecasting for t+1, t+2, and t+3.

b)      Write 95% confidence range for each forecast in (b).

c)      Do a long-run forecast for P.

d)     Write a 95% confidence interval for the long-run forecast of P.

3- Show that any AR(1) process can be written as MA(oo) and any MA(1) process can be written as an AR(oo) process.

4.  Write the order of the following ARIMA models.  For each model explain whether the specification is a correct specification or not.  For each model explain whether the model is stationary or not.

1. Y_t = 21.5 + .87Y_t-1 + e _t
2. Y_t = 1.75 + .87Y_t-1 +.32Y_{t-2 +}  e _t 
3. Y_t = 1.75 + .87DY_t-1 +.32DY_{t-2 +}  e _t - .87e_t-1
4. Y_t = 1.75 - .87Y_t-1 +.32Y_{t-2 +}  e _t  
5. Y_t = 1.22 +  e _t - .87e_t-1  


6. Y_t = .25 + 1.32Y_t-1 +  e _t + .67e_t-1 - 12e_t-1
7. DY_t = 1.05 + .80DY_t-1 + e _t - .87e_t-1  
8. DY_t = .35 + .87Y_t-1 + e _t 
9. DY_t = 1.05 + .97DY_t-1 +.02DY_{t-2 +}  e _t - .67e_t-1
10. Y_t =  .75 + e _t - .87e_t-1 + 1.6 e_t-1

5.  a. What is auto-correlation function (ACF), explain?  

b.  What is partial auto-correlation function (PACF), explain?

c.  Write the best model that will explain each of the following ACF - PACF schedules.