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Answer Key toAsst 11(40 points)

- # 10, Page 157
- The Durbin-Watson statistic does not tell you anything when applied to a cross-section regression. Data in a cross-section regression are often in alphabetical order, but that order has nothing to do with the underlying characteristics of the data. You could change the order of the data in a cross-section, and you would get a different value for the Durbin-Watson statistic.

#11, Page 157

- The results of the regression are:
- Dependent Variable is DVDEXP
- Variable Coefficient Standard Error t-Statistic p-Value
- C 81.46 23.47 3.47 0.00
- INCOME 0.061 0.0098 6.27 0.00
- PRICE -3.12 0.88 -3.54 0.00
- RAINFALL 7.52 2.37 3.17 0.00
- Observations: 24
- R2 = 0.79
- Adjusted R2 = 0.75
- Residual Sum of Squares = 5038.14
- F-statistic = 24.49
- Durbin-Watson statistic: 2.32
- The Durbin-Watson statistic comes out to 2.32. Since it is greater than 2, a test for negative autocorrelation should be conducted. From the Durbin-Watson statistic table, when k=3 and N=24, for a 5% error level, dU=1.10 and dl=1.66. Since we need to check for negative (rather than positive) autocorrelation, these values must be subtracted from 4.
- 4-1.66=2.34
- 4-1.10=2.90
- Since the Durbin-Watson statistic from our results, 2.32, is lower than 2.34, we do not reject the null hypothesis of no autocorrelation (with a 5% error level). There is not enough evidence of autocorrelation to justify taking any corrective action, but it is close though.

#12, Page 157

- The results are:
- Dependent Variable is DVDEXP
- Variable Coefficient Standard Error t-Statistic p-Value
- C 71.18 24.99 2.85 0.01
- INCOME 0.062 0.0088 7.05 0.00
- PRICE -2.78 0.98 -2.82 0.01
- RAINFALL 8.29 2.30 3.60 0.00
- AR(1) -0.22 0.23 -0.95 0.36
- Observations: 23
- R2 = 0.78
- Adjusted R2 = 0.73
- Residual Sum of Squares = 4597.67
- F-statistic = 15.82
- Durbin-Watson statistic: 1.67
- Comparing these results to those of the last question, they are not exactly the same, but similar. Note that the AR(1) term is statistically insignificant. This is consistent with the fact that we did not find autocorrelation in the previous question.

#13, Page 157

- Results for Microsoft Revenue Model Including T
- Dependent Variable is REVENUE
- Variable Coefficient Standard Error T-Statistic P-Value
- Constant -49.99 88.32 -0.57 0.57
- MARKETING 8.07 0.73 10.97 0.00
- SECONDQ 10.84 80.77 0.13 0.89
- THIRDQ 124.24 84.21 1.48 0.15
- FOURTHQ 75.97 82.01 0.93 0.36
- T -25.81 8.06 -3.20 0.00
- Observations: 55
- R2 = 0.96
- Adjusted R2 = 0.96
- Residual Sum of Squares = 2,220,798
- F-statistic = 247.41
- Durbin-Watson statistic: 0.68

b. Yes, it is statistically significant at even a 1% error level. It only adds to our understanding slightly. It tells us that Microsoft’s real revenues have been falling over time when MARKETING and the quarter dummy variables are accounted for. It would really add to our understanding if we could figure out why this is happening, so the T variable doesn’t really tell us that much.

c. Using a 5% error level, dU=1.77, dL=1.34. The Durbin-Watson statistic from our results, 0.68, is below dL=1.34. The null hypothesis of no autocorrelation is rejected at a 5% error level; there is evidence of autocorrelation. This is the same result that we got originally (Table 7.B) when we did not include T in the model.

Chapter 8: Homoskedasticity(up to Page 173)

- In this graph, there are 5 different observations on each X (each representing a different value of Y)
- Note that the errors on each observation of X have a mean of zero and the same variance across all values of Xs

Heteroskedasticity

- Suppose X is income and Y is consumption and we have cross sectional data.
- Note that at low levels of income there is not much variation in consumption but at higher levels if income there is more variation in consumption.
- The reason is that families that have little income spend a large portion (maybe all) of their income. But families that have a large amount of income vary greatly on how much of their income the spend.
- That is, the error on each observation of X comes from a distribution with a mean of zero but a different variance
- In this case the variance of error increases as the level of income increases.
- Heteroskedacity is problem that is more common in cross sectional data sets.

Consequences Heteroskedasticity

- Unbiased estimates (if the functional form is correct and there are no missing variables) but wrong standard errors
- OLS tend to underestimate the standard errors
- You should know how this affects the results of the t-test of significance.

A casual way to look for heteroskedasticity

- Run the regression
- Look at the graph of residuals
- Is there a sign of heteroskedasticity?
- If it looks like the residuals are not evenly distributed around a mean of zero, then yes.
- But we need a more formal test
- Note: We will not cover the Park Test

The White Test

- Set the null and alternative hypotheses:
- – Ho: homoskedasticity
- – Ha: heteroskedasticity
- Estimate the original regression
- Use the squared residuals as a dependent variable in a second equation that includes Xs, X2s and product of each pair of Xs
- Find nR2
- Find Critcal chi-squared on page 324 (df = number of of independent variables in the second equation)
- If nR2 > critical chi-squaredreject Ho

EViews does the estimations automatically

- Estimate the original regression as usual
- On the regression output click on “View”
- Then on “Residual Test”
- The choose White Heteroskedastcity (cross terms included)
- On the top of your output, you will see observations times R2

Asst 12Due: Sunday, December 2 before 10 PM(20 points)

Questions #13 and 14 on Page 181

Note

- We will have classes during the week of December 3
- Our first class is on Tuesday, December 4 at 8 AM in Thomas 223.
- Don’t miss it.

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