bapswarrior

An analyst is trying to determine whether fund return performance is persistent. The analyst divides funds into three groups based on whether their return performance was in the top third (group 1), middle third (group 2), or bottom third (group 3) during the previous year. The manager then creates the following equation: R = a + b1D1 + b2D2 + b3D3 + ε, where R is return premium on the fund (the return minus the return on the S&P 500 benchmark) and Di is equal to 1 if the fund is in group i. Assuming no other information, this equation will suffer from:

A) heteroskedasticity.

B) serial correlation.

C) multicollinearity.

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When we use dummy variables, we have to use one less than the states of the world. In this case, there are three states (groups) possible. We should have used only two dummy variables. Multicollinearity is a problem in this case. Specifically, a linear combination of independent variables is perfectly correlated. X1 + X2 + X3 = 1.
There are too many dummy variables specified, so the equation will suffer from multicollinearity

bapswarrior

Suppose the analyst wants to add a dummy variable for whether a person has an undergraduate college degree and a graduate degree. What is the CORRECT representation if a person has both degrees?

Undergraduate Degree Dummy Variable

Graduate Degree Dummy Variable

A) 1 1

B) 0 0

C) 0 1

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Assigning a zero to both categories is appropriate for someone with neither degree. Assigning one to the undergraduate category and zero to the graduate category is appropriate for someone with only an undergraduate degree. Assigning zero to the undergraduate category and one to the graduate category is appropriate for someone with only a graduate degree. Assigning a one to both categories is correct since it reflects the possession of both degrees.

bapswarrior

The amount of the State of Florida’s total revenue that is allocated to the education budget is believed to be dependent upon the total revenue for the year and the political party that controls the state legislature. Which of the following regression models is most appropriate for capturing the effect of the political party on the education budget? Assume Yt is the amount of the education budget for Florida in year t, X is Florida’s total revenue in year t, and Dt = {1 if the legislature has a Democratic majority in year t, 0 otherwise}.

A) Yt = b1Dt + b2Xt + et.

B) Yt = b0 + b1Dt + b2Xt + et.

C) Yt = b0 + b1Dt + et.

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In this application, b0, b1, and b2 are estimated by regressing Yt against a constant, Dt, and Xt.The estimated relationships for the two parties are:

Non-Democrats: Ŷ = b0 + b2Xt
Democrats: Ŷ = (b0 + b1) + b2Xt

bapswarrior

Raul Gloucester, CFA, is analyzing the returns of a fund that his company offers. He tests the fund’s sensitivity to a small capitalization index and a large capitalization index, as well as to whether the January effect plays a role in the fund’s performance. He uses two years of monthly returns data, and runs a regression of the fund’s return on the indexes and a January-effect qualitative variable. The “January” variable is 1 for the month of January and zero for all other months. The results of the regression are shown in the tables below.

Regression Statistics
Multiple R	0.817088
R2	0.667632
Adjusted R2	0.617777
Standard Error	1.655891
Observations	24

ANOVA
	df	SS	MS
Regression	3	110.1568	36.71895
Residual	20	54.8395	2.741975
Total	23	164.9963

	Coefficients	Standard Error	t-Statistic
Intercept	-0.23821	0.388717	-0.61282
January	2.560552	1.232634	2.077301
Small Cap Index	0.231349	0.123007	1.880778
Large Cap Index	0.951515	0.254528	3.738359

Gloucester will perform an F-test for the equation. He also plans to test for serial correlation and conditional and unconditional heteroskedasticity.
Jason Brown, CFA, is interested in Gloucester’s results. He speculates that they are economically significant in that excess returns could be earned by shorting the large capitalization and the small capitalization indexes in the month of January and using the proceeds to buy the fund. The percent of the variation in the fund’s return that is explained by the regression is:

A) 66.76%.

B) 81.71%.

C) 61.78%.

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The R2 tells us how much of the change in the dependent variable is explained by the changes in the independent variables in the regression: 0.667632.

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In a two-tailed test at a five percent level of significance, the coefficients that are significant are:

A) the large cap index only.

B) the January effect and the small capitalization index only.

C) the January effect and the large capitalization index only.

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For a two-tailed test with 20 = 24 – 3 – 1 degrees of freedom and a five percent level of significance, the critical t-statistic is 2.086. Only the coefficient for the large capitalization index has a t-statistic larger than this.

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Which of the following best summarizes the results of an F-test (5 percent significance) for the regression? The F-statistic is:

A) 13.39 and the critical value is 3.10.

B) 9.05 and the critical value is 3.86.

C) 13.39 and the critical value is 3.86.

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The F-statistic is the ratio of the Mean Square of the Regression divided by the Mean Square Error (Residual): 13.39 = 36.718946 / 2.74197510. The F-statistic has 3 and 20 degrees of freedom, so the critical value, at a 5 percent level of significance = 3.10.

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The best test for unconditional heteroskedasticity is:

A) the Durbin-Watson test only.

B) neither the Durbin-Watson test nor the Breusch-Pagan test.

C) the Breusch-Pagan test only.

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The Durbin-Watson test is for serial correlation. The Breusch-Pagan test is for conditional heteroskedasticity; it tests to see if the size of the independent variables influences the size of the residuals. Although tests for unconditional heteroskedasticity exist, they are not part of the CFA curriculum, and unconditional heteroskedasticity is generally considered less serious than conditional heteroskedasticity.

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In the month of January, if both the small and large capitalization index have a zero return, we would expect the fund to have a return equal to:

A) 2.799.

B) 2.322.

C) 2.561.

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The forecast of the return of the fund would be the intercept plus the coefficient on the January effect: 2.322 = -0.238214 + 2.560552.

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Assuming (for this question only) that the F-test was significant but that the t-tests of the independent variables were insignificant, this would most likely suggest:

A) serial correlation.

B) conditional heteroskedasticity.

C) multicollinearity.

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When the F-test and the t-tests conflict, multicollinearity is indicated.

bapswarrior

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:

A) 1.435.

B) 1.882.

C) 9.128.

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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)

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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 = 0.35 versus Ha: b1 ≠ 0.35.

C) H0: b1 = 0.98 versus Ha: b1 ≠ 0.98.

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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)

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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.

A) 21.118.

B) 17.956.

C) 18.896.

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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)

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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 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.

C) The F-statistic is almost always formulated to test each independent variable separately, in order to identify which variable is the most statistically significant.

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An F-test assesses how well a set of independent 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)

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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) heteroskedasticity.

C) positive serial correlation.

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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)

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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 Hansen method.

C) the Durbin-Watson statistic.

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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)

bapswarrior

An analyst is trying to estimate the beta for a fund. The analyst estimates a regression equation in which the fund returns are the dependent variable and the Wilshire 5000 is the independent variable, using monthly data over the past five years. The analyst finds that the correlation between the square of the residuals of the regression and the Wilshire 5000 is 0.2. Which of the following is most accurate, assuming a 0.05 level of significance? There is:

A) evidence of serial correlation but not conditional heteroskedasticity in the regression equation.

B) evidence of conditional heteroskedasticity but not serial correlation in the regression equation.

C) no evidence that there is conditional heteroskedasticity or serial correlation in the regression equation.

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The test for conditional heteroskedasticity involves regressing the square of the residuals on the independent variables of the regression and creating a test statistic that is n × R2, where n is the number of observations and R2 is from the squared-residual regression. The test statistic is distributed with a chi-squared distribution with the number of degrees of freedom equal to the number of independent variables. For a single variable, the R2 will be equal to the square of the correlation; so in this case, the test statistic is 60 × 0.22 = 2.4, which is less than the chi-squared value (with one degree of freedom) of 3.84 for a p-value of 0.05. There is no indication about serial correlation.

bapswarrior

Which of the following is least likely a method used to detect heteroskedasticity?

A) Durbin-Watson test.

B) Test of the variances.

C) Breusch-Pagan test.

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The Durbin-Watson test is used to detect serial correlation. The Breusch-Pagan test is used to detect heteroskedasticity.

bapswarrior

Consider the following graph of residuals and the regression line from a time-series regression:

These residuals exhibit the regression problem of:<

A) autocorrelation.

B) homoskedasticity.

C) heteroskedasticity.

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The residuals appear to be from two different distributions over time. In the earlier periods, the model fits rather well compared to the later periods.

bapswarrior

Which of the following statements regarding heteroskedasticity is least accurate?

A) Multicollinearity is a potential problem only in multiple regressions, not simple regressions.

B) The presence of heteroskedastic error terms results in a variance of the residuals that is too large.

C) Heteroskedasticity only occurs in cross-sectional regressions.

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If there are shifting regimes in a time-series (e.g., change in regulation, economic environment), it is possible to have heteroskedasticity in a time-series.

bapswarrior

Which of the following statements regarding heteroskedasticity is least accurate?

A) The assumption of linear regression is that the residuals are heteroskedastic.

B) Heteroskedasticity results in an estimated variance that is too large and, therefore, affects statistical inference.

C) Heteroskedasticity may occur in cross-section or time-series analyses.

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The assumption of regression is that the residuals are homoskedastic (i.e., the residuals are drawn from the same distribution).