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