John Rains, CFA, is a professor of finance at a large university located in the Eastern United States. He is actively involved with his local chapter of the Society of Financial Analysts. Recently, he was asked to teach one session of a Society-sponsored CFA review course, specifically teaching the class addressing the topic of quantitative analysis. Based upon his familiarity with the CFA exam, he decides that the first part of the session should be a review of the basic elements of quantitative analysis, such as hypothesis testing, regression and multiple regression analysis. He would like to devote the second half of the review session to the practical application of the topics he covered in the first half.
Rains decides to construct a sample regression analysis case study for his students in order to demonstrate a “real-life” application of the concepts. He begins by compiling financial information on a fictitious company called Big Rig, Inc. According to the case study, Big Rig is the primary producer of the equipment used in the exploration for and drilling of new oil and gas wells in the United States. Rains has based the information in the problem on an actual equity holding in his personal portfolio, but has simplified the data for the purposes of the review course.
Rains constructs a basic regression model for Big Rig in order to estimate its profitability (in millions), using two independent variables: the number of new wells drilled in the U.S. (WLS) and the number of new competitors (COMP) entering the market:
Profits = b0 + b1WLS – b2COMP + ε
Based on the model, the estimated regression equation is:
Profits = 22.5 + 0.98(WLS) ? 0.35(COMP)
Using the past 5 years of quarterly data, he calculated the following regression estimates for Big Rig, Inc:
|
|
Coefficient |
Standard Error |
|
Intercept |
22.5 |
2.465 |
|
WLS |
0.98 |
0.683 |
|
COMP |
0.35 |
0.186 |
Using the information presented, the t-statistic for the number of new competitors (COMP) coefficient is:
To test whether a coefficient is statistically significant, the null hypothesis is that the slope coefficient is zero. The t-statistic for the COMP coefficient is calculated as follows:
(0.35 – 0.0) / 0.186 = 1.882
(Study Session 3, LOS 11.g)
Rains asks his students to test the null hypothesis that states for every new well drilled, profits will be increased by the given multiple of the coefficient, all other factors remaining constant. The appropriate hypotheses for this two-tailed test can best be stated as:
|
A) |
H0: b1 ≤ 0.98 versus Ha: b1 > 0.98. | |
|
B) |
H0: b1 = 0.98 versus Ha: b1 ≠ 0.98. | |
|
C) |
H0: b1 = 0.35 versus Ha: b1 ≠ 0.35. | |
The coefficient given in the above table for the number of new wells drilled (WLS) is 0.98. The hypothesis should test to see whether the coefficient is indeed equal to 0.98 or is equal to some other value. Note that hypotheses with the “greater than” or “less than” symbol are used with one-tailed tests. (Study Session 3, LOS 11.g)
Continuing with the analysis of Big Rig, Rains asks his students to calculate the mean squared error(MSE). Assume that the sum of squared errors (SSE) for the regression model is 359.
The MSE is calculated as SSE / (n – k – 1). Recall that there are twenty observations and two independent variables. Therefore, the SEE in this instance = 359 / (20 – 2 ? 1) = 21.118. (Study Session 3, LOS 11.i)
Rains now wants to test the students’ knowledge of the use of the F-test and the interpretation of the F-statistic. Which of the following statements regarding the F-test and the F-statistic is the most correct?
|
A) |
The F-test is usually formulated as a two-tailed test. | |
|
B) |
The F-statistic is almost always formulated to test each independent variable separately, in order to identify which variable is the most statistically significant. | |
|
C) |
The F-statistic is used to test whether at least one independent variable in a set of independent variables explains a significant portion of the variation of the dependent variable. | |
An F-test assesses how well a set of impendent variables, as a group, explains the variation in the dependent variable. It tests all independent variables as a group, and is always a one-tailed test. The decision rule is to reject the null hypothesis if the calculated F-value is greater than the critical F-value. (Study Session 3, LOS 11.i)
One of the main assumptions of a multiple regression model is that the variance of the residuals is constant across all observations in the sample. A violation of the assumption is known as:
|
A) |
robust standard errors. | |
|
B) |
positive serial correlation. | |
|
|
Heteroskedasticity is present when the variance of the residuals is not the same across all observations in the sample, and there are sub-samples that are more spread out than the rest of the sample. (Study Session 3, LOS 12.i)
Rains reminds his students that a common condition that can distort the results of a regression analysis is referred to as serial correlation. The presence of serial correlation can be detected through the use of:
|
A) |
the Breusch-Pagen test. | |
|
B) |
the Durbin-Watson statistic. | |
|
|
The Durbin-Watson test (DW ≈ 2(1 ? r)) can detect serial correlation. Another commonly used method is to visually inspect a scatter plot of residuals over time. The Hansen method does not detect serial correlation, but can be used to remedy the situation. Note that the Breusch-Pagen test is used to detect heteroskedasticity. (Study Session 3, LOS 12.i)
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发表于 2011-3-3 14:29
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500 has increased or decreased during the month. The analyst collects 90 monthly return premia (the return on the fund minus the return on the S&
