**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.

- Y
_{t} = 21.5 + .87Y_{t-1} + e _{t}
- Y
_{t} = 1.75 + .87Y_{t-1} +.32Y_{t-2 + } e _{t}
- Y
_{t} = 1.75 + .87DY_{t-1} +.32DY_{t-2 + } e _{t} - .87e_{t-1}
- Y
_{t} = 1.75 - .87Y_{t-1} +.32Y_{t-2 + } e _{t}
- Y
_{t} = 1.22 +_{ }e _{t} - .87e_{t-1}

- Y
_{t} = .25 + 1.32Y_{t-1} +_{ } e _{t} + .67e_{t-1} - 12e_{t-1}
- DY
_{t} = 1.05 + .80DY_{t-1} + e _{t} - .87e_{t-1}
- DY
_{t} = .35 + .87Y_{t-1} + e _{t}
- DY
_{t} = 1.05 + .97DY_{t-1} +.02DY_{t-2 + } e _{t} - .67e_{t-1}
- 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.