Economics 4312 Review for Exam

1. How is the process of differencing used to identify an appropriate time series model?

2. If data is stationary, give examples of a naive model, an averaging model, an exponential smoothing model, and a Box-Jenkins model.

3. If data is not stationary, give examples of a naive model, an averaging model, and two exponential smoothing models.

4. Use the following data to fill in the appropriate columns for stationary data:

Period	Y	NF	e1	M	MAF	e2	SEF	e3
1	6						6
2	8
3	5
4	8
5	7
6	8
7

a. Fill in column 3 based upon a naive “no change” forecasting model including the forecast for period 7: Model used: NF_t+1 = Y_t

b. Fill in column 4 the ex post errors for periods 1-6 based upon the naive model. Compute the mean absolute error (deviation) for the model. Also compute the mean squared error and the mean absolute percentage error for the model.

c. Fill in column 5 for periods 3-6 a moving average of order n=3 for the data. In column 6 insert a moving average forecast for periods 4-7: Model used: MAF_t+1 = M_t

d. Compute in column 7 the ex post errors for periods 4-6 and the mean absolute error for the moving average model.

e. Compute in column 8 a single exponential smoothing forecast based upon an original forecast for period 1 equal to the first value of Y and a smoothing constant, alpha, equal to .4. Model used: SEF_t+1 = alpha x Y_t + (1 - alpha) x SEF _t

f. Compute the ex post errors for periods 2-6 in column 9. Compute the mean absolute error and compare with the other three models to determine the “best” forecasting model based upon the limited data available. What is the root mean squared error? (Note that forecasts for large samples should fall within about 1.96 root mean squared errors 95 percent of the time. This may be used to establish a “tracking system” for error terms to determine if the parameter values (alpha in this case) should be changed. Plus or minus 1.96 root mean squared errors may also be used to estimate confidence intervals for future forecasts provided they are not excessively far into the future or there is past evidence that forecasts remain within these “tracks” 95 percent of the time.)

5. Use the following nonstationary data to fill in the columns in the tables below:

Period	Y	NTF	e1	M^1	M^2	a	b	DMF	e2
1	4
2	5
3	8
4	10
5	12
6

a. Compute a naive trend forecast for periods 3-7 in column 3 based upon the following model:

NTF_t+1 = Y_t + (Y_t-Y_t-1)

b. Compute the ex post errors and mean absolute error of the model in column 4.

c. Compute a single moving average with n=2 for periods 2-5 in column 5 and a double moving average with n=2 for period 3-5 in column 6.

d. Compute the value of a = 2 x M¹-M² for periods 3-5 in column 7.

e. Compute the value of b = (2/(n-1)) x (M¹-M²) for period 3-5 in column 8

f. Compute the double moving average forecast for periods 4-7 in column 9 based upon the model:

DMF_t+T = a_(t + b_t x T

g. Compute the ex post errors and the mean absolute error of the model in column 10. Which of the two models is better based upon their errors?

6. Use the same data base to compute the following model: (Brown’s model)

Period	Y	S^1	S^2	a	b	DEF	e3
1	4	4	4
2	5
3	8
4	10
5	12
6

a. Compute the single and double smoothed series for periods 2 through 5 in columns 3 and 4 based upon initial values of 4 and a smoothing constant, alpha = .4. The following equations are used:

S¹_t+1 = alpha x Y_t+1 + (1 – alpha) x S¹_t

S² _t+1 = alpha x S¹_t+1 + (1 – alpha) x S²_t

b. Compute the values of a and b for periods 2 through 5 in columns 5 and 6 based upon the values of S¹ and S² as follows:

a_t = 2 x S¹_t - S²_t

b_t = (alpha/(1 - alpha)) x (S¹ - S²)

c. Compute the double exponential smoothed forecast values, DEF, in column 7 based upon the values of a and b as follows:

DEF_{t + T} = a_t + b_t x T

d. Compute the values of the error terms and the mean absolute error for periods 2 through 5 in column 8. How does the model perform compared with the naive and double moving average models?

7. Use the same data base to compute the following model: (Holt’s model)

Period	Y	A	T	A+T	HF	e4
1	4	4	2	6
2	5
3	8
4	10
5	12
6

a. Compute the mean component, A, for periods 2 through 5 in column 3 based upon initial values of A=4, B=2 (A+B=6) and a value of alpha=.2 using the following equation:

A_t+1 = alpha x Y_t+1 + (1 - alpha) x (A_t + T_t)

b. Compute the trend component, T, for periods 2 through 5 in column 4 based the initial value of T = 2 and a value of beta = .4 using the following equation:

T_t+1 = beta x (A_t+1 - A_t) + (1 - beta) x T_t

c. Compute the forecast values in column 6 for periods 2 through 6 based upon the following equation:

HF_t+P = A_t + T_t x P (where P is the number of periods ahead)

d. Compute the errors and the mean absolute error for periods 2 through 5 in column 7. Compare the results with the other models represented.

7. Use the data below to compute a Box-Jenkins forecast based upon estimated computer results as specified below:

Period	Y	DY	FDY	FY	e5
1	4
2	5
3	8
4	10
5	12
6

a. Sketch what the ACF for the nonstationary data would look like.

b. Compute the first difference of the data for periods 2-5 in column 2

c. Sketch what the ACF and PACF functions would look like if an AR(1) model is found.

d. Suppose the AR(1) model is estimated as follows:

DY_t = 2 + .5 x DY_t-1

Use this model to forecast the change in Y for periods 2-6 in column 4

e. Use the forecasted change to determine a forecast for periods 2-6 based upon the following equation:

Y_t+1 = Y_t + DY_t+1

f. Compute the errors terms for periods 2-5 and the mean absolute error of the model. Does the model appear to be adequate? How would you test for this result using an ACF of the error terms?

8. Sketch the ACF and PACF diagrams of the following Box-Jenkins models for stationary data:

a. AR(2)

b. MA(1)

c. ARMA(1,1)

d. AR(1)

e. MA(2)

9. If the model is adequate for forecasting, what would the ACF diagram of the error terms look like?

Review Questions for Chapter 10

1. What are some key questions that you might ask to determine forecast guidelines before initiating the forecast procedure?

2. What are three general categories that encompass the sources of forecast error?

3. What are some key external and internal forces that a forecaster needs to monitor as a source of forecast error?

4. How is the decomposition method used to identify error due to regression model specification versus measurement error? (Work exercise 5 on page 513)

5. How is a ladder chart used to track the reliability of forecast, i.e. what are the four elements of information needed and how are they used?

6. What would a modified turning point error diagram look like that (a) overestimated growth or decline, (b) underestimated growth or decline, (c) showed considerable turning point errors, (d) showed random errors?

7. What is the primary purpose of the graphical plot of forecasts that are made during sequential periods of the previous year?

8. Briefly describe how Trigg’s tracking signal is calculated and how it is used to show when a model becomes inadequate (moves from random to nonrandom error terms).