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Milky Way, Inc. is a large manufacturer of children’s toys and games based in the United States. Their products have high name brand recognition, and have been sold in retail outlets throughout the United States for nearly fifty years. The founding management team was bought out by a group of investors five years ago. The new management team, led by Russell Stepp, decided that Milky Way should try to expand its sales into the Western European market, which had never been tapped by the former owners. Under Stepp’s leadership, additional personnel are hired in the Research and Development department, and a new marketing plan specific to the European market is implemented. Being a new player in the European market, Stepp knows that it will take several years for Milky Way to establish its brand name in the marketplace, and is willing to make the expenditures now in exchange for increased future profitability.
Now, five years after entering the European market, Stepp is reviewing the results of his plan. Sales in Europe have slowly but steadily increased over since Milky Way’s entrance into the market, but profitability seems to have leveled out. Stepp decides to hire a consultant, Ann Hays, CFA, to review and evaluate their European strategy. One of Hays’ first tasks on the job is to perform a regression analysis on Milky Way’s European sales. She is seeking to determine whether the additional expenditures on research and development and marketing for the European market should be continued in the future.
Hays begins by establishing a relationship between the European sales of Milky Way (in millions of dollars) and the two independent variables, the number of dollars (in millions) spent on research and development (R&D) and marketing (MKTG). Based upon five years of monthly data, Hays constructs the following estimated regression equation:
Estimated Sales = 54.82 + 5.97 (MKTG) + 1.45 (R&D)
Additionally, Hays calculates the following regression estimates:
| Coefficient | Standard Error |
Intercept | 54.82 | 3.165 |
MKTG | 5.97 | 1.825 |
R&D | 1.45 | 0.987 |
Hays begins the analysis by determining if both of the independent variables are statistically significant. To test whether a coefficient is statistically significant means to test whether it is statistically significantly different from: A)
| the upper tail critical value. |
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The magnitude of the coefficient reveals nothing about the importance of the independent variable in explaining the dependent variable. Therefore, it must be determined if each independent variable is statistically significant. The null hypothesis is that the slope coefficient for each independent variable equals zero. (Study Session 3, LOS 11.a)
The t-statistic for the marketing variable is calculated to be:
The t-statistic for the marketing coefficient is calculated as follows: (5.97– 0.0) / 1.825 = 3.271. (Study Session 3, LOS 11.g)
Hays formulates a test structure where the decision rule is to reject the null hypothesis if the calculated test statistic is either larger than the upper tail critical value or lower than the lower tail critical value. At a 5% significance level with 57 degrees of freedom, assume that the two-tailed critical t-values are tc = ±2.004. Based on this information, Hays makes the following conclusions:- Point 1: The intercept term is statistically significant.
- Point 2: Both independent variables contribute to explaining states for Milky Way, Inc.
- Point 3: If an F-test were being used, the null hypothesis would be rejected.
Which of Hays’ conclusions are CORRECT?
Hays’ Point 1 is correct. The t-statistic for the intercept term is (54.82 – 0) / 3.165 = 17.32, which is greater than the critical value of 2.004, so we can conclude that the intercept term is statistically significant.
Hays’ Point 2 is incorrect. The t-statistic for the R&D term is (1.45 – 0) / 0.987 = 1.469, which is not greater than the critical value of 2.004. This means that only MKTG can be said to contribute to explaining sales for Milky Way, Inc.
Hays’ Point 3 is correct. An F-test tests whether at least one of the independent variables is significantly different from zero, where the null hypothesis is that all none of the independent variables are significant. Since we know that MKTG is a significant variable (t-statistic of 3.271), we can reject the hypothesis that none of the variables are significant. (Study Session 3, LOS 11.i)
Hays is aware that part, but not all, of the total variation in expected sales can be explained by the regression equation. Which of the following statements correctly reflects this relationship? | | C)
| SST = RSS + SSE + MSE. |
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RSS (Regression sum of squares) is the portion of the total variation in Y that is explained by the regression equation. The SSE (Sum of squared errors), is the portion of the total variation in Y that is not explained by the regression. The SST is the total variation of Y around its average value. Therefore, SST = RSS + SSE. These sums of squares will always be calculated for you on the exam, so focus on understanding the interpretation of each. (Study Session 3, LOS 11.i)
Hays decides to test the overall effectiveness of the both independent variables in explaining sales for Milky Way. Assuming that the total sum of squares is 389.14, the sum of squared errors is 146.85 and the mean squared error is 2.576, calculate and interpret the R2. A)
| The R2 equals 0.242, indicating that the two independent variables account for 24.2% of the variation in monthly sales. |
| B)
| The R2 equals 0.623, indicating that the two independent variables account for 37.7% of the variation in monthly sales. |
| C)
| The R2 equals 0.623, indicating that the two independent variables account for 62.3% of the variation in monthly sales. |
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The R2 is calculated as (SST – SSE) / SST. In this example, R2 equals (389.14–146.85) / 389.14 = .623 or 62.3%. This indicates that the two independent variables together explain 62.3% of the variation in monthly sales. The value for mean squared error is not used in this calculation. (Study Session 3, LOS 11.i)
Stepp is concerned about the validity of Hays’ regression analysis and asks Hays if he can test for the presence of heteroskedasticity. Hays complies with Stepp’s request, and detects the presence of unconditional heteroskedasticity. Which of the following statements regarding heteroskedasticity is most correct? A)
| Heteroskedasticity can be detected either by examining scatter plots of the residual or by using the Durbin-Watson test. |
| B)
| Unconditional heteroskedasticity does create significant problems for statistical inference. |
| C)
| Unconditional heteroskedasticity usually causes no major problems with the regression. |
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Unconditional heteroskedasticity occurs when the heteroskedasticity is not related to the level of the independent variables. This means that it does not systematically increase or decrease with changes in the independent variable(s). Note that heteroskedasticity occurs when the variance of the residuals is different across all observations in the sample and can be detected either by examining scatter plots or using a Breusch-Pagen test. (Study Session 3, LOS 12.i) |
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发表于 2012-3-26 15:22
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