STATISTICS 730: APPLIED TIME SERIES ANALYSIS, Fall 2004, Professor Dickey
HOMEWORK 4: Wages and CPI (from Bureau of Labor Statistics)
DIRECTIONS: Complete the following problems below. Show all of your work to receive credit. Make sure to include output from the SAS output window and graphs from the SAS graphics window WHEN YOU NEED TO SUPPORT YOUR ANSWERS!!!
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[WAGES AND THE CPI]
A series on average hourly wages and the Consumer Price Index, CPI, will be analyzed in Homework 4 (DATA). The CPI is an index of prices used to measure the change in the cost of basic goods and services in comparison with a fixed base period. For example, the CPI for all items in 2000 was 172.2; whereas, it was 100 in the base period 1982-1984. So, it rose by 72.2%.
[1] Plot both the wage and CPI series. Based on the plot, what would you say about stationarity?
[2] Compute the ACF, PACF, and IACF functions for the wage series. What do they tell you about stationarity? Use 10 lags.
[3] Compute the ACF, PACF, and IACF functions for the first and second differenced wage series.
[4] There might be as many as two unit roots for the wage series. Let D(t) denote the first differenced series D(t)=W(t)-W(t-1) and let V(t)=D(t)-D(t-1). In PROC REG, regress V(t) on D(t-1) V(t-1) V(t-2) and V(t-3) with an intercept. Use a TEST statement to see if you can leave out V(t-3). Also, test to see if you can leave out both V(t-2) and V(t-3). Moreover, test to see if no lagged V is needed. Is the use of these standard ordinary least squares tests justified here?
Run a second regression based on the model you selected from [4] and look at the t test for D(t-1). Can the P-value from PROC REG be trusted?
Our (D-F) tables give -2.91 as a critical value of Tau_mu for series of this length. What are the null and alternative hypotheses and what is your conclusion with regard to stationarity? Here we have tested:
H0: Two unit roots
H1: One or less
[5] Verify our assumption that there are no more than 2 unit roots in PROC ARIMA. To accomplish this, use PROC ARIMA; I VAR=WAGE(1,1) STATIONARITY=(ADF=k); where k would be the number of augmenting lags V(t-1) V(t-2)... V(t-k) that you want to use. Is the t test the same as in PROC REG? How about the P-value? Here we have tested:
H0: Three unit roots
H1: Two or less
[6] If the second differences were nonstationary, we would do no further testing. However, if the second differences were stationary, we would wonder if we have overdifferenced. Perhaps there is only 1 unit root.**
Repeat the test from [4] to see if the first differenced wage series is stationary but this time use PROC ARIMA. What is your conclusion?
Give a model (AR, MA, ARMA, white noise) for the appropriately differenced (first or second differenced) series. Give some statistical support for your model choice.
[7] Wages and prices have risen over time as you can clearly see in the graphs from [1]. It may be more relevant to look at the purchasing power of the average wage earner; that is, express the wage in constant dollars. Merge the two datasets and compute the "real wage" by dividing the nominal wage by the CPI and multiplying by 100. Graph this "real wage" series.
[8] For the "real wage" in [7], how may unit roots are there? Back this up with some tests. Let's use 3 lagged differences in all the tests for consistency.
[Optional] Upon further investigation, you will see that the unit root test results are quite different if only one augmenting lag is used.
[9] For the appropriately differenced real wage series, diagnose and fit an ARMA model. Summarize results.
[10] For your model, forecast 1, 2, 3, and 4 periods ahead and give the forecast intervals.
[Optional] Plot the forecasts, intervals and historic data on the same graph.
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[APPENDIX TO HOMEWORK 4]
Turn in your COMPLETE SAS program from the enhanced editor window.
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[REFERENCES]
**Dickey, D. A. and Pantula, S.G. (1987). "Determining the Order of Differencing in AR Processes," Journal of Business and Economic Statistics 5, 455-461.
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[LINKS]
SAS Online Documentation