Walex

An analyst is investigating the hypothesis that the beta of a fund is equal to one. The analyst takes 60 monthly returns for the fund and regresses them against the Wilshire 5000. The test statistic is 1.97 and the p-value is 0.05. Which of the following is CORRECT?

A) If beta is equal to 1, the likelihood that the absolute value of the test statistic would be greater than or equal to 1.97 is 5%.

B) If beta is equal to 1, the likelihood that the absolute value of the test statistic is equal to 1.97 is less than or equal to 5%.

C) The proportion of occurrences when the absolute value of the test statistic will be higher when beta is equal to 1 than when beta is not equal to 1 is less than or equal to 5%.

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A statistical test computes the likelihood of a test statistic being higher than a certain value assuming the null hypothesis is true.

Walex

Seventy-two monthly stock returns for a fund between 1997 and 2002 are regressed against the market return, measured by the Wilshire 5000, and two dummy variables. The fund changed managers on January 2, 2000. Dummy variable one is equal to 1 if the return is from a month between 2000 and 2002. Dummy variable number two is equal to 1 if the return is from the second half of the year. There are 36 observations when dummy variable one equals 0, half of which are when dummy variable two also equals 0. The following are the estimated coefficient values and standard errors of the coefficients.

Coefficient	Value	Standard error
Market	1.43000	0.319000
Dummy 1	0.00162	0.000675
Dummy 2	−0.00132	0.000733

What is the p-value for a test of the hypothesis that the new manager outperformed the old manager?

A) Between 0.01 and 0.05.

B) Lower than 0.01.

C) Between 0.05 and 0.10.

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Dummy variable one measures the effect on performance of the change in managers. The t-statistic is equal to 0.00162 / 0.000675 = 2.400, which is higher than the t-value (with 72 - 3 - 1 = 68 degrees of freedom) of approximately 2.39 for a p-value of between 0.01 and 0.005 for a 1 tailed test.

Walex

David Black wants to test whether the estimated beta in a market model is equal to one. He collected a sample of 60 monthly returns on a stock and estimated the regression of the stock’s returns against those of the market. The estimated beta was 1.1, and the standard error of the coefficient is equal to 0.4. What should Black conclude regarding the beta if he uses a 5% level of significance? The null hypothesis that beta is:

A) equal to one is rejected.

B) not equal to one cannot be rejected.

C) equal to one cannot be rejected.

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The calculated t-statistic is t = (1.1 − 1.0) / 0.4 = 0.25. The critical t-value for (60 − 2) = 58 degrees of freedom is approximately 2.0. Therefore, the null hypothesis that beta is equal to one cannot be rejected.

Walex

You have been asked to forecast the level of operating profit for a proposed new branch of a tire store. This forecast is one component in forecasting operating profit for the entire company for the next fiscal year. You decide to conduct multiple regression analysis using "branch store operating profit" as the dependent variable and three independent variables. The three independent variables are "population within 5 miles of the branch," "operating hours per week," and "square footage of the facility." You used data on the company's existing 23 branches to develop the model (n=23).

Regression of Operating Profit on Population, Operating Hours, and Square Footage
Dependent Variable	Operating Profit (Y)
Independent Variables	Coefficient Estimate	t-value
Intercept	103,886	2.740
Population within 5 miles (X1)	4.372	2.133
Operating hours per week (X2)	214.856	0.258
Square footage of facility (X3)	56.767	2.643
R2	0.983
Adjusted R2	0.980
F-Statistic	360.404
Standard error of the model	19,181

	Correlation Matrix
	Y	X1	X2	X3
Y	1.00
X1	0.99	1.00
X2	0.69	0.67	1.00
X3	0.99	0.99	.71	1.00

Degrees of Freedom	.20	.10	.05	.02	.01
3	1.638	2.353	3.182	4.541	5.841
19	1.328	1.729	2.093	2.539	2.861
23	1.319	1.714	2.069	2.50	2.807

You want to evaluate the statistical significance of the slope coefficient of an independent variable used in this regression model. For 95% confidence, you should compare the t-statistic to the critical value from a t-table using:

A) 24 degrees of freedom and 0.05 level of significance for a one-tailed test.

B) 19 degrees of freedom and 0.05 level of significance for a one-tailed test.

C) 19 degrees of freedom and 0.05 level of significance for a two-tailed test.

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The degrees of freedom are [n − k − 1]. Here, n is the number of observations in the regression (23) and k is the number of independent variables (3). df = [23 − 3 − 1] = 19 Because the null hypothesis is that the slope coefficient is equal to zero, this is a two-tailed test.

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The probability of finding a value of t for variable X1 that is as large or larger than |2.133| when the null hypothesis is true is:

A) between 5% and 10%.

B) between 1% and 2%.

C) between 2% and 5%.

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The degree of freedom is	= (n − k − 1)
	= (23 − 3 − 1)
	= 19

In the table above, for 19 degrees of freedom, the value 2.133 would lie between a 2% chance (alpha of 0.02) or 2.539 and a 5% chance (alpha of 0.05) or 2.093.

Walex

Seventy-two monthly stock returns for a fund between 1997 and 2002 are regressed against the market return, measured by the Wilshire 5000, and two dummy variables. The fund changed managers on January 2, 2000. Dummy variable one is equal to 1 if the return is from a month between 2000 and 2002. Dummy variable number two is equal to 1 if the return is from the second half of the year. There are 36 observations when dummy variable one equals 0, half of which are when dummy variable two also equals zero. The following are the estimated coefficient values and standard errors of the coefficients.

Coefficient	Value	Standard error
Market	1.43000	0.319000
Dummy 1	0.00162	0.000675
Dummy 2	−0.00132	0.000733

What is the p-value for a test of the hypothesis that the beta of the fund is greater than 1?

A) Between 0.05 and 0.10.

B) Between 0.01 and 0.05.

C) Lower than 0.01.

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The beta is measured by the coefficient of the market variable. The test is whether the beta is greater than 1, not zero, so the t-statistic is equal to (1.43 − 1) / 0.319 = 1.348, which is in between the t-values (with 72 − 3 − 1 = 68 degrees of freedom) of 1.29 for a p-value of 0.10 and 1.67 for a p-value of 0.05.

Walex

Consider the following estimated regression equation, with standard errors of the coefficients as indicated:

Salesi = 10.0 + 1.25 R&Di + 1.0 ADVi – 2.0 COMPi + 8.0 CAPi
where the standard error for R&D is 0.45, the standard error for ADV is 2.2, the standard error for COMP 0.63, and the standard error for CAP is 2.5.

The equation was estimated over 40 companies. Using a 5% level of significance, what are the hypotheses and the calculated test statistic to test whether the slope on R&D is different from 1.0?

A) H0: bR&D ≠ 1 versus Ha: bR&D = 1; t = 2.778.

B) H0: bR&D = 1 versus Ha: bR&D≠ 1; t = 0.556.

C) H0: bR&D = 1 versus Ha: bR&D≠1; t = 2.778.

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The test for “is different from 1.0” requires the use of the “1” in the hypotheses and requires 1 to be specified as the hypothesized value in the test statistic. The calculated t-statistic = (1.25-1)/.45 = 0.556

Walex

A dependent variable is regressed against three independent variables across 25 observations. The regression sum of squares is 119.25, and the total sum of squares is 294.45. The following are the estimated coefficient values and standard errors of the coefficients.

Coefficient	Value	Standard error
1	2.43	1.4200
2	3.21	1.5500
3	0.18	0.0818

For which of the coefficients can the hypothesis that they are equal to zero be rejected at the 0.05 level of significance?

A) 3 only.

B) 2 and 3 only.

C) 1 and 2 only.

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The values of the t-statistics for the three coefficients are equal to the coefficients divided by the standard errors, which are 2.43 / 1.42 = 1.711, 3.21 / 1.55 = 2.070, and 0.18 / 0.0818 = 2.200. The statistic has 25 − 3 − 1 = 21 degrees of freedom. The critical value for a p-value of 0.025 (because this is a two-sided test) is 2.080, which means only coefficient 3 is significant.

Walex

63 monthly stock returns for a fund between 1997 and 2002 are regressed against the market return, measured by the Wilshire 5000, and two dummy variables. The fund changed managers on January 2, 2000. Dummy variable one is equal to 1 if the return is from a month between 2000 and 2002. Dummy variable number two is equal to 1 if the return is from the second half of the year. There are 36 observations when dummy variable one equals 0, half of which are when dummy variable two also equals 0. The following are the estimated coefficient values and standard errors of the coefficients.

Coefficient	Value	Standard error
Market	1.43000	0.319000
Dummy 1	0.00162	0.000675
Dummy 2	0.00132	0.000733

What is the p-value for a test of the hypothesis that performance in the second half of the year is different than performance in the first half of the year?

A) Between 0.05 and 0.10.

B) Between 0.01 and 0.05.

C) Lower than 0.01.

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The difference between performance in the second and first half of the year is measured by dummy variable 2. The t-statistic is equal to 0.00132 / 0.000733 = 1.800, which is between the t-values (with 63 − 3 − 1 = 59 degrees of freedom) of 1.671 for a p-value of 0.10, and 2.00 for a p-value of 0.05 (note that the test is a two-sided test).

Walex

Kathy Williams, CFA, and Nigel Faber, CFA, have been managing a hedge fund over the past 18 months. The fund’s objective is to eliminate all systematic risk while earning a portfolio return greater than the return on Treasury Bills. Williams and Faber want to test whether they have achieved this objective. Using monthly data, they find that the average monthly return for the fund was 0.417%, and the average return on Treasury Bills was 0.384%. They perform the following regression (Equation I):

(fund return)t = b0 + b1 (T-bill return) t + b2 (S&P 500 return) t + b3 (global index return) t + et

The correlation matrix for the independent variables appears below:

	S&P 500	Global Index
T-bill	0.163	0.141
S&P 500		0.484

In performing the regression, they obtain the following results for Equation I:

Variable	Coefficient	Standard Error
Intercept	0.232	0.098
T-bill return	0.508	0.256
S&P 500 Return	−0.0161	0.032
Global index return	0.0037	0.034

R2 = 22.44%
adj. R2 = 5.81%
standard error of forecast = 0.0734 (percent)

In addition to the regular summary statistics, Williams computes the correlation coefficient for the residuals, i.e., correlation of the last 17 residuals on the lag of those residuals. The value of the correlation coefficient is 0.605.
Williams argues that the equation may suffer from multicollinearity and reruns the regression omitting the return on the global index. This time, the regression (Equation II) is:

(fund return) t = b0 + b1 (T-bill return) t + b2 (S&P 500 return) t +et

The results for Equation II are:

Variable	Coefficient	Standard Error
Intercept	0.232	0.095
T-bill return	0.510	0.246
S&P 500 return	−0.015	0.028

R2 = 22.37%
adj. R2 = 12.02%
standard error of forecast = 0.0710 (percent)

The correlation of the residuals on their lagged values for this regression 0.558.
Finally, Williams reruns the regression omitting the return on the S&P 500 as well. This time, the regression (Equation III) is:

(fund return) t = b0 + b1 (T-bill return) t +et

The results for Equation III are:

Variable	Coefficient	Standard Error
Intercept	0.229	0.093
T-bill return	0.4887	0.2374

R2 = 20.94%
adj. R2 = 16.00%
standard error of forecast = 0.0693 (percent)

The correlation of the residuals on their lagged values for this regression 0.604. In the regression using Equation I, which of the following hypotheses can be rejected at a 5% level of significance in a two-tailed test? (The corresponding independent variable is indicated after each null hypothesis.)

A) H0: b2 = 0 (S&P 500)

B) H0: b1 = 0 (T-bill)

C) H0: b0 = 0 (intercept)

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The critical t-value for 18 − 3 − 1 = 14 degrees of freedom in a two-tailed test at a 5% significance level is 2.145. Although the t-statistic for T-bill is close at 0.508 / 0.256 = 1.98, it does not exceed the critical value. Only the intercept’s coefficient has a significant t-statistic for the indicated test: t = 0.232 / 0.098 = 2.37. (Study Session 3, LOS 12.b)

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In the regression using Equation II, which of the following hypothesis or hypotheses can be rejected at a 5% level of significance in a two-tailed test? (The corresponding independent variable is indicated after each null hypothesis.)

A) H0: b0 = 0 (intercept) and b1 = 0 (T-bill) only.

B) H0: b0 = 0 (intercept) only.

C) H0: b1 = 0 (T-bill) and H0: b2 = 0 (S&P 500) only.

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The critical t-value for 18 − 2 − 1 = 15 degrees of freedom in a two-tailed test at a 5% significance level is 2.131. The t-statistics on the intercept, T-bill and S&P 500 coefficients are 2.442, 2.073, −0.536, respectively. Therefore, only the coefficient on the intercept is significant. (Study Session 3, LOS 12.b)

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With respect to multicollinearity and Williams’ removal of the global index variable when running regression Equation II, Williams had:

A) reason to be suspicious, but she took the wrong step to cure the problem.

B) no reason to be suspicious, but took a correct step to improve the analysis.

C) reason to be suspicious and took the correct step to cure the problem.

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Investigating multicollinearity is justified for two reasons. First, the S&P 500 and the global index have a significant degree of correlation. Second, neither of the market index variables are significant in the first specification. The correct step is to remove one of the variables, as Williams did, to see if the remaining variable becomes significant. (Study Session 3, LOS 12.j)

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At a 5% level of significance, which of the equations suffers from serial correlation?

A) Equation I only.

B) Equations I, II, and III.

C) Equations I and III only.

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Using the correlations of the residuals, the DW statistics are 2 × (1 − 0.605) = 0.79, 0.88, and 0.79 for Equations I, II, III, respectively. The critical values for the DW test are 0.93 for Equation I, 1.05 for Equation II, and 1.16 for Equation III. Note that in the calculation of the DW statistics with the correlation coefficient of the residuals, we have made the simplifying assumption that the sample size is large enough to use the DW = 2(1 − r) method. (Study Session 3, LOS 12.i)

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Which of the following problems, multicollinearity and/or serial correlation, can bias the estimates of the slope coefficients?

A) Serial correlation, but not multicollinearity.

B) Both multicollinearity and serial correlation.

C) Multicollinearity, but not serial correlation.

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Multicollinearity can bias the coefficients because the shared movement of the independent variables. Serial correlation biases the standard errors of the slope coefficients. (Study Session 3, LOS 12.j)

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If we expect that next month the T-bill rate will equal its average over the last 18 months, using Equation III, calculate the 95% confidence interval for the expected fund return.

A) 0.296 to 0.538.

B) 0.270 to 0.564.

C) 0.259 to 0.598.

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The forecast is 0.417 = 0.229 + 0.4887 × (0.384). The 95% confidence interval is Y ± (tc × sf) and tc for 16 degrees of freedom for a 2 tailed test = 2.120. The 95% confidence interval = 0.417 ± (2.120)(.0693) = 0.270 to 0.564. (Study Session 3, LOS 12.c)

Walex

Consider the following estimated regression equation, with standard errors of the coefficients as indicated:

Salesi = 10.0 + 1.25 R&Di + 1.0 ADVi − 2.0 COMPi + 8.0 CAPi
where the standard error for R&D is 0.45, the standard error for ADV is 2.2, the standard error for COMP 0.63, and the standard error for CAP is 2.5.

Sales are in millions of dollars. An analyst is given the following predictions on the independent variables: R&D = 5, ADV = 4, COMP = 10, and CAP = 40.

The predicted level of sales is closest to:

A) $310.25 million.

B) $320.25 million.

C) $360.25 million.

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Predicted sales	= $10 + 1.25 (5) + 1.0 (4) −2.0 (10) + 8 (40)
	= 10 + 6.25 + 4 − 20 + 320 = $320.25