AIM 2: Define and interpret the partial slope coefficient.

1、Which of the following statements regarding the results of a regression analysis is FALSE? The:

A) slope coefficient in a multiple regression is the value of the dependent variable for a given value of the independent variable.

B) slope coefficient in a multiple regression is the change in the dependent variable for a one-unit change in the independent variable, holding all other variables constant.

C) intercept is the value that the dependent variable takes on if all the independent variables had a value of zero.

D) slope coefficients in the multiple regression are referred to as partial betas.

The correct answer is A

The slope coefficient is the change in the dependent variable for a one-unit change in the independent variable.

2、When interpreting the results of a multiple regression analysis, which of the following terms represents the value of the dependent variable when the independent variables are all equal to zero?

A) Slope coefficient.

B) p-value.

C) t-value.

D) Intercept term.

The correct answer is D

The intercept term is the value of the dependent variable when the independent variables are set to zero.

AIM 3: List the assumptions of the multiple linear regression model.

1、Which of the following statements least accurately describes one of the fundamental multiple regression assumptions?

A) The error term is normally distributed.

B) The variance of the error terms is not constant (i.e., the errors are heteroskedastic).

C) The independent variables are not random.

D) There is no exact linear relationship between any two or more independent variables.

The correct answer is B

The variance of the error term IS assumed to be constant, resulting in errors that are homoskedastic.

2、One of the underlying assumptions of a multiple regression is that the variance of the residuals is constant for various levels of the independent variables. This quality is referred to as:

A) a normal distribution.

B) homoskedasticity.

C) a linear relationship.

D) serial correlation.

The correct answer is B

Homoskedasticity refers to the basic assumption of a multiple regression model that the variance of the error terms is constant.

AIM 4: Explain the concept of multicollinearity and implications it has on modeling.

1、Which of the following statements regarding multicollinearity is FALSE?

A) Multicollinearity may be a problem even if the Multicollinearity is not perfect.

B) Multicollinearity makes it difficult to determine the contribution to explanation of the dependent variable of an individual explanatory variable.

C) Multicollinearity may be present in any regression model.

D) If the t-statistics for the individual independent variables are insignificant, yet he F-statistic is significant, this indicates the presence of Multicollinearity.

The correct answer is C

Multicollinearity is not an issue in simple regression.

2、A variable is regressed against three other variables, x, y, and z. Which of the following would NOT be an indication of multicollinearity? X is closely related to:

A) 3y + 2z.

B) 3.

C) y2.

D) 9y, and x is closely related to 4z.

The correct answer is C

If x is related to y2, the relationship between x and y is not linear, so multicollinearity does not exist. If x is equal to a constant (3), it will be correlated with the intercept term.

3、An analyst runs a regression of portfolio returns on three independent variables.  These independent variables are price-to-sales (P/S), price-to-cash flow (P/CF), and price-to-book (P/B).  The analyst discovers that the p-values for each independent variable are relatively high.  However, the F-test has a very small p-value.  The analyst is puzzled and tries to figure out how the F-test can be statistically significant when the individual independent variables are not significant.  What violation of regression analysis has occurred?

A) serial correlation.

B) conditional heteroskedasticity.

C) multicollinearity.

D) unconditional heteroskedasticity.

The correct answer is C

An indication of multicollinearity is when the independent variables individually are not statistically significant but the F-test suggests that the variables as a whole do an excellent job of explaining the variation in the dependent variable.

4、An analyst further studies the independent variables of a study she recently completed. The correlation matrix shown below is the result. Which statement best reflects possible problems with a multivariate regression?

A)    Age should be excluded from the regression.

B)    Education may be unnecessary.

C)   Income is not needed.

D)   Experience may be a redundant variable.

The correct answer is D

The correlation coefficient of experience with age and income, respectively, is close to +1.00. This indicates a problem of multicollinearity and should be addressed by excluding experience as an independent variable.

AIM 6: Define, calculate and interpret the variance and standard errors in a multilinear regression.

In the multivariable regression model: Y = B0 + B1 × Xli + B2 × X2i + εi, the formula for the standard errors of the estimated coefficients includes the variance of εi, which is represented by: σ2. Since the term is:

A) not known with certainty, the expression σ2 is replaced with: .

B) not known with certainty, the standard errors of the coefficients cannot be estimated.

C) not known with certainty, the expression σ2 is replaced with: .

D) known with certainty, the standard errors of the coefficients are known with certainty.

The correct answer is C

The expression:  is an estimate of σ2 when there are two independent variables and n-3 degrees of freedom.

AIM 8: Interpret regression results.

1、Consider the following analysis of variance (ANOVA) table:

The R² for this regression is:

A)    0.82.

B)    0.55.

C)   0.67.

D)   0.45.

The correct answer is D

R2 = RSS/SST = 556/1,235 = 0.45.

2、A regression equation with 4 independent variables is estimated using 20 data points. The R2 is 0.46. An analyst is testing to see whether all of the coefficients are equal to zero. The p-value for the test is:

A) lower than 0.025.

B) between 0.05 and 0.10.

C) greater than 0.10.

D) between 0.025 and 0.05.

The correct answer is D

To solve this problem, one can assume any value for the total sum of squares. In this case, assume its 1. The regression sum of squares is R2 multiplied by the total sum of squares, which is 0.46. The residual sum of squares is the difference between the total sum of squares and the regression sum of squares, which is 1 ? 0.46 = 0.54. The numerator degrees of freedom is equal to the number of independent variables, which is 4, and the mean regression sum of squares is the regression sum of squares divided by the numerator degrees of freedom, which is 0.46 / 4 = 0.115. The denominator degrees of freedom is the number of observations minus the number of independent variables, minus 1, which is 20 ? 4 ? 1 = 15. The mean squared error is the residual sum of squares divided by the denominator degrees of freedom, which is 0.54 / 15 = 0.036. The F-statistic is the ratio of the mean regression sum of squares to the mean squared error, which is 0.115 / 0.036 = 3.19, which is in between the F-values (with four numerator degrees of freedom and 15 denominator degrees of freedom) of 3.06 for a p-value of 0.05 (calculated using the F-table at 5%) and 3.80 for a p-value of 0.025 (calculated using the F-table at 2.5%).

3、A dependent variable is regressed against a single independent variable across 100 observations. The mean squared error is 2.807, and the mean regression sum of squares is 117.9. What is the correlation coefficient between the two variables?

A) 0.55.

B) 0.30.

C) 0.99.

D) 0.65.

The correct answer is A

The correlation coefficient is the square root of the R2, which can be found by dividing the regression sum of squares by the total sum of squares. The regression sum of squares is the mean regression sum of squares multiplied by the number of independent variables, which is 1, so the regression sum of squares is equal to 117.9. The residual sum of squares is the mean squared error multiplied by the denominator degrees of freedom, which is the number of observations minus the number of independent variables, minus 1, which is equal to 100 ? 1 ? 1 = 98. The residual sum of squares is then 2.807 × 98 = 275.1. The total sum of squares is the sum of the regression sum of squares and the residual sum of squares, which is 117.9 + 275.1 = 393.0. The R2 = 117.9 / 393.0 = 0.3, so the correlation is the square root of 0.3 = 0.55.

4、Erica Basenj, CFA, has been given an assignment by her boss. She has been requested to review the following regression output to answer questions about the relationship between the monthly returns of the Toffee Investment Management (TIM) High Yield Bond Fund and the returns of the index (independent variable).

What is the value of the correlation coefficient?

A)    ?0.9973.

B)    0.8700.

C)   0.9973.

D)   ?0.8700.

The correct answer is C

R2 is the correlation coefficient squared, taking into account whether the relationship is positive or negative. Since the value of the slope is positive, the TIM fund and the index are positively related. R2 is calculated by taking the (RSS / SST) = 0.99459. (0.99459)1/2 = 0.9973.

What is the sum of squared errors (SSE)?

A) 23,515.

B) 23,644.

C) 3,283.

D) 128.

The correct answer is D

SSE = SST ? RSS = 23,644 ? 23,516 = 128

What is the value of R2?

A) 0.0055.

B) 5.2900.

C) 0.9946. 

D) 0.9471.

The correct answer is D

SSE = SST ? RSS = 23,644 ? 23,516 = 128

What is the value of R2?

A) 0.0055.

B) 5.2900.

C) 0.9946. 

D) 0.9471.

The correct answer is C

R2 = RSS / SST = 23,516 / 23,644 = 0.9946.

Is the intercept term statistically significant at the 5% level of significance and the 1% level of significance, respectively?

1%   5%

A)               Yes    No

B)               No     No

C)               Yes    Yes

D)               No     Yes

The correct answer is C

The test statistic is t = b / std error of b = 5.29 / 1.615 = 3.2755.

Critical t-values are ± 2.101 for the degrees of freedom = n ? k ? 1 = 18 for alpha = 0.05. For alpha = 0.01, critical t-values are ± 2.878. At both levels (two-tailed tests) we can reject H0 that b = 0.

What is the value of the F-statistic?

A) 0.0003. 

B) 0.9945. 

C) 182. 

D) 3,359. 

The correct answer is D

F = mean square regression / mean square error = 23,516 / 7 = 3,359.

Heteroskedasticity can be defined as:

A) error terms that are dependent. 

B) a regression that changes over time. 

C) nonconstant variance of the error terms. 

D) independent variables that are correlated with each other. 

The correct answer is C

Heteroskedasticity occurs when the variance of the residuals is not the same across all observations in the sample. Autocorrelation refers to dependent error terms. Nonstationarity is when the regression changes over time.

5、A study of a sample of incomes (in thousands of dollars) of 35 individuals shows that income is related to age and years of education. The following table shows the regression results:

The standard error for the coefficient of age and t-statistic for years of education are:

A)    0.40; 5.66.

B)    0.53; 2.96.

C)   0.20; 1.96.

D)   0.32; 1.65.

The correct answer is A

standard error for the coefficient of age = coefficient / t-value = 0.53 / 1.33 = 0.40

t-statistic for the coefficient of education = coefficient / standard error = 2.32 / 0.41 = 5.66

Mean square regression (MSR) and mean square error (MSE) are:

A) 102.10; 7.11.

B) 7.38; 3.42.

C) 107.55; 3.60.

D) 6.72; 3.58.

The correct answer is C

df for Regression = k = 2

df for Error = n – k – 1 = 35 – 2 – 1 = 32

MSR = RSS / df = 215.10 / 2 = 107.55

MSE = SSE / df = 115.10 / 32 = 3.60

What is the R2 for the regression?

A) 62%. 

B) 76%. 

C) 65%. 

D) 82%. 

The correct answer is C

SST = RSS + SSE

= 215.10 + 115.10 = 330.20

R2 = RSS / SST = 215.10 / 330.20 = 0.65

What is the predicted income of a 40-year-old person with 16 years of education?

A) $63,970. 

B) $62,120. 

C) $52,780. 

D) $74,890.

The correct answer is A

income  = 5.65 + 0.53 (age) + 2.32 (education)

        = 5.65 + 0.53 (40) + 2.32 (16)= 63.97 or $63,970

What is the F-value?

A) 14.36. 

B) 2.16.

C) 29.88.

D) 1.88.

The correct answer is C

F = MSR / MSE = 107.55 / 3.60 = 29.88

6、Which statement is most accurate? Assume a 5 percent level of significance. The F-statistic is:

A)    0.0017 and the regression is said to be statistically significant.

B)    0.0017 and the regression is said to be statistically insignificant.

C)   579.03 and the regression is said to be statistically insignificant.

D)   579.03 and the regression is said to be statistically significant.

The correct answer is D

F =3,700/6.39 = 579.03 which is significant since the critical F value is between 2.29 and 2.37. The critical F value is found by using a 5% level of significance F-table and looking up the value that corresponds with 5 = k = the number of independent variables in the numerator and 100 _ 5 _ 1 = 94 df in the denominator resulting in a critical value between 2.29 and 2.37.

7、Consider the following analysis of variance (ANOVA) table:

The R² and the F-statistic are, respectively:

A)    R² = 0.40 and F = 33.333.

B)    R² = 0.67 and F = 0.971.

C)   R² = 0.40 and F = 0.971.

D)   R² = 0.67 and F = 33.333.

The correct answer is A

R2 = 500 / 1,250 = 0.40. The F-statistic is 500 / 15 = 33.33.

8、Consider the following analysis of variance (ANOVA) table:

The R² and the F-statistic are, respectively:

A)    R² = 50% and F = 2.0.

B)    R² = 33% and F = 20.0.

C)   R² = 50% and F = 0.952.

D)   R² = 33% and F = 2.0.

The correct answer is B

R2 = 200 / 600 = 0.333. The F-statistic is 200 / 10 = 20.

9、Cynthia Jones is Director of Marketing at Vancouver Industries, a large producer of athletic apparel and accessories. Approximately three years ago, Vancouver experienced increased competition in the marketplace, and consequently sales for that year declined nearly 20 percent. At that time, Jones proposed a new marketing campaign for the company, aimed at positioning Vancouver’s product lines toward a younger target audience. Although the new marketing effort was significantly more costly than previous marketing campaigns, Jones assured her superiors that the resulting increase in sales would more than justify the additional expense. Jones was given approval to proceed with the implementation of her plan.

Three years later, in preparation for an upcoming shareholders meeting, the CEO of Vancouver has asked Jones for an evaluation of the marketing campaign. Sales have increased since the inception of the new marketing campaign nearly three years ago, but the CEO is questioning whether the increase is due to the marketing expenditures or can be attributed to other factors. Jones is examining the following data on the firm's aggregate revenue and marketing expenditure over the last 10 quarters. Jones plans to demonstrate the effectiveness of marketing in boosting sales revenue. She chooses to utilize a simple linear regression model. The output is as follows:

Slope coefficient = 34.74 Standard error of slope coefficient = 9.916629313 Standard error of intercept = 92.2840128

For the questions below, assume that the regression model was estimated using a large number of observations.

What is the upper limit of the 95 percent confidence interval for the slope coefficient?

A)    62.84.

B)    54.18.

C)   46.05.

D)   36.70.

The correct answer is B

Upper Limit = coefficient + (1.96 x standard error of the coefficient) = 34.74 + (1.96 x 9.917) = 54.18

What is the lower limit of the 95 percent confidence interval for the slope coefficient?

A) 32.78.

B) 15.30.

C) 12.24.

D) 19.89.

The correct answer is B

Lower Limit = Coefficient - (1.96 x Standard Error of the coefficient)

 = 34.74 - (1.96 x 9.917)

 = 34.74 - 19.4373 = 15.30

Which of the following is the CORRECT value of the correlation coefficient between aggregate revenue and advertising expenditure?

A) 0.9500.

B) 0.3947.

C) 0.7780.

D) 0.6053.

The correct answer is C

The R2 = (SST - SSE)/SST = RSS/SST = (20,922.5 - 8,257.374) / 20,922.5 = 0.6053.

The correlation coefficient is the square root of the R2 in a simple linear regression which is the square root of 0.6053 = 0.7780.

Which of the following reports the CORRECT value and interpretation of the R2 for this regression? The R2 is:

A) 0.3947 indicating that the variability of advertising expenditure explains about 39.47% percent of the variability of aggregate revenue.

B) 0.6053 indicating that the variability of advertising expenditure explains about 60.53% of the variability in aggregate revenue.

C) 0.6053 indicating that the variability of aggregate revenue explains about 60.53% of the variability in advertising expenditure.

D) 0.3947 indicating that the variability of aggregate revenue explains about 39.47% percent of the variability of advertising expenditure

The correct answer is B

The R2 = (SST - SSE)/SST = (20,922.5 - 8,257.374) / 20,922.5 = 0.6053.

The interpretation of this R2 is that 60.53% of the variation in aggregate revenue (Y) is explained by the variation in advertising expenditure (X).

What is the y-intercept term, b0?

A) 47.6712. 

B) 34.7400.

C) 92.2840.

D) 512.3600.

The correct answer is A

The mean of the aggregate revenue (Y) is 3,645/10 = 364.50 and of the advertising expenditure (X) is 91.2/10 = 9.12. The y-intercept, b0 = MeanY – Slope * MeanX = 364.50 – 34.74 * 9.12 = 47.6712.

What is the calculated F-statistic?

A) 12.2700.

B) 92.2840.

C) 1,032.1717.

D) 0.1250.

The correct answer is A

The computed value of the F-Statistic = MSR/MSE = 12,665.12576 / 1,032.17178 = 12.27, where MSR and MSE are from the ANOVA table.

10、Given the results from a multiple regression function: 24 ? 2 × X1i + 4 × X2i, which of the following are known to be true? Based upon the equation:

?            if the observed dependent variable equals 30 and X1i = 1, then X2i = 2.

?            if both independent variables equal zero, the fitted value for the dependent variable is 24.

?            there is a negative relationship between the dependent variable and X1i.

?            given X1i = 2 and X2i = 1 the fitted value for the dependent variable is 24.

A) II and III only.

B) I and III only.

C) II, III, and IV only.

D) I, II, III, and IV.

The correct answer is C

A researcher cannot make inferences about the independent variables from observed values of the dependent variable.

11、John Conner, FRM and Chris Bond, FRM are planning to compute the coefficient of determination from a regression analysis. Conner asserts the following formula is appropriate for a two-variable regression, but it will have to be adjusted for degrees of freedom when additional variables are added to the equation: .

Bond asserts that the following formula is appropriate, and it does not have to be adjusted for degrees of freedom: .

What, if any, error have the analysts made in their assertions?

A) There are no errors in the assertions.

B) Bond’s formula has an error, but Bond is correct in that it does not have to be adjusted for degrees of freedom.

C) Conner’s formula has an error, but Conner is correct in that it must be adjusted for degrees of freedom.

D) Conner’s formula is correct, but Conner is not correct in asserting that it must be adjusted for degrees of freedom.

The correct answer is D

The formulas are both correct, and it is not necessary to adjust for degrees of freedom

12、Werner Baltz, CFA, has regressed 30 years of data to forecast future sales for National Motor Company based on the percent change in gross domestic product (GDP) and the change in price of a U.S. gallon of fuel at retail. The results are presented below. Note: results must be multiplied by $1,000,000:

In 2002, if GDP rises 2.2% and the price of fuels falls $0.15, Baltz’s model will predict Company sales in 2002 to be (in $ millions) closest to:

A)    $128.

B)    $82.

C)   $206.

D)   $254.

The correct answer is D

Sales will be closest to $78 + ($30.22 × 2.2) + [(?412.39) × (?$0.15)] = $206.34 million.

13、Baltz proceeds to test the hypothesis that none of the independent variables has significant explanatory power. He concludes that, at a 5% level of significance:

A) at least one of the independent variables has explanatory power, because the calculated F-statistic exceeds its critical value. 

B) all of the independent variables have explanatory power, because the calculated F-statistic exceeds its critical value. 

C) none of the independent variables has explanatory power, because the calculated F-statistic does not exceed its critical value. 

D) at least one of the independent variables has explanatory power, because the calculated F-statistic does not exceed its critical value. 

The correct answer is A

From the ANOVA table, the calculated F-statistic is (mean square regression / mean square error) = 145.65 / 4.89 = 29.7853. From the F distribution table (2 df numerator, 27 df denominator) the F-critical value may be interpolated to be 3.36. Because 29.7853 is greater than 3.36, Baltz rejects the null hypothesis and concludes that at least one of the independent variables has explanatory power.

14、Baltz then tests the individual variables, at a 5% level of significance, to determine whether sales are explained by individual changes in GDP and fuel prices. Baltz concludes that:

A) neither GDP nor fuel price changes explain changes in sales.

B) only GDP changes explain changes in sales.

C) both GDP and fuel price changes explain changes in sales.

D) only fuel price changes explain changes in sales.

The correct answer is C

From the ANOVA table, the calculated t-statistics are (30.22 / 12.12) = 2.49 for GDP and (?412.39 / 183.981) = ?2.24 for fuel prices. These values are both outside the t-critical value at 27 degrees of freedom of ±2.052. Therefore, Baltz is able to reject the null hypothesis that these coefficients are equal to zero, and concludes that each variable is important in explaining sales.

15、Consider the following regression equation:

Salesi = 20.5 + 1.5 R&Di + 2.5 ADVi – 3.0 COMPi

where Sales is dollar sales in millions, R&D is research and development expenditures in millions, ADV is dollar amount spent on advertising in millions, and COMP is the number of competitors in the industry.

Which of the following is NOT a correct interpretation of this regression information?

A) If a company spends $1 more on R&D (holding everything else constant), sales are expected to increase by $1.5 million.

B) If R&D and advertising expenditures are $1 million each and there are 5 competitors, expected sales are $9.5 million.

C) One more competitor will mean $3 million less in sales (holding everything else constant).

D) Increasing advertising dollars by $1 million (holding everything else constant), will result in $2.5 million additional sales.

The correct answer is A

If a company spends $1 million more on R&D (holding everything else constant), sales are expected to increase by $1.5 million. Always be aware of the units of measure for the different variables.

16、Consider the following regression equation:

Salesi = 10.0 + 1.25 R&Di + 1.0 ADVi – 2.0 COMPi + 8.0 CAPi

where Sales is dollar sales in millions, R&D is research and development expenditures in millions, ADV is dollar amount spent on advertising in millions, COMP is the number of competitors in the industry, and CAP is the capital expenditures for the period in millions of dollars.

Which of the following is NOT a correct interpretation of this regression information?

A) If a company spends $1 million more on capital expenditures (holding everything else constant), Sales are expected to increase by $8.0 million.

B) One more competitor will mean $2 million less in Sales (holding everything else constant).

C) If R&D and advertising expenditures are $1 million each, there are 5 competitors, and capital expenditures are $2 million, expected Sales are $8.25 million.

D) Increasing advertising dollars by $1 million (holding everything else constant), will result in $1 million additional Sales.

The correct answer is C

Predicted sales = $10 + 1.25 + 1 – 10 + 16 = $18.25 million.

17、Henry Hilton, CFA, is undertaking an analysis of the bicycle industry. He hypothesizes that bicycle sales (SALES) are a function of three factors: the population under 20 (POP), the level of disposable income (INCOME), and the number of dollars spent on advertising (ADV). All data are measured in millions of units. Hilton gathers data for the last 20 years. Which of the follow regression equations correctly represents Hilton’s hypothesis?

A) SALES = α x β1 POP x β2 INCOME x β3 ADV x ε.

B) INCOME = α + β1 POP + β2 SALES + β3 ADV + ε.

C) SALES = α + β1 POP + β2 INCOME + β3 ADV + ε.

D) INCOME = α + β1 POP + β2 ADV + ε.

The correct answer is C

SALES is the dependent variable. POP, INCOME, and ADV should be the independent variables (on the right hand side) of the equation (in any order). Regression equations are additive.

18、Henry Hilton, CFA, is undertaking an analysis of the bicycle industry.  He hypothesizes that bicycle sales (SALES) are a function of three factors: the population under 20 (POP), the level of disposable income (INCOME), and the number of dollars spent on advertising (ADV).  All data are measured in millions of units.  Hilton gathers data for the last 20 years and estimates the following equation (standard errors in parentheses): 

The critical t-statistic for a 95 percent confidence level is 2.120.  Which of the independent variables is statistically different from zero at the 95 percent confidence level?

A)    INCOME and ADV.

B)    POP and ADV.

C)   INCOME only.

D)   ADV only.

The correct answer is C

The calculated test statistic is coefficient/standard error. Hence, the t-stats are 0.8 for POP, 3.059 for INCOME, and 0.866 for ADV. Since the t-stat for INCOME is the only one greater than the critical t-value of 2.120, only INCOME is significantly different from zero.

19、Henry Hilton, CFA, is undertaking an analysis of the bicycle industry.  He hypothesizes that bicycle sales (SALES) are a function of three factors: the population under 20 (POP), the level of disposable income (INCOME), and the number of dollars spent on advertising (ADV).  All data are measured in millions of units.  Hilton gathers data for the last 20 years and estimates the following equation (standard errors in parentheses):

For next year, Hilton estimates the following parameters: (1) the population under 20 will be 120 million, (2) disposable income will be $300,000,000, and (3) advertising expenditures will be $100,000,000.  Based on these estimates and the regression equation, what are predicted sales for the industry for next year?

A)    $557,143,000.

B)    $509,980,000.

C)   $656,991,000.

D)   $669,471,000.

The correct answer is B

Predicted sales for next year are:

SALES = α + 0.004 (120) + 1.031 (300) + 2.002 (100) = 509,980,000.

20、Consider a study of 100 university endowment funds that was conducted to determine if the funds’ annual risk-adjusted returns could be explained by the size of the fund and the percentage of fund assets that are managed to an indexing strategy. The equation used to model this relationship is:

ARAR_i = b₀ + b₁Size_i + b₂Index_i + e_i

Where:

The table below contains a portion of the regression results from the study.

Which of the following is the most accurate interpretation of the slope coefficient for size? ARAR:

A)    will change by 1.0% when the natural logarithm of assets under management changes by 0.6, holding index constant.

B)    will change by 0.6% when the natural logarithm of assets under management changes by 1.0, holding index constant.

C)   and index will change by 1.1% when the natural logarithm of assets under management changes by 1.0.

D)   will change by 1.1% when the natural logarithm of assets under management changes by 1.0 and index changes by 0.6.

The correct answer is B

A slope coefficient in a multiple linear regression model measures how much the dependent variable changes for a one-unit change in the independent variable, holding all other independent variables constant. In this case, the independent variable size (= ln average assets under management) has a slope coefficient of 0.6, indicating that the dependent variable ARAR will change by 0.6% return for a one-unit change in size, assuming nothing else changes. Pay attention to the units on the dependent variable.

Which of the following is the estimated standard error of the regression coefficient for index?

A) 1.91.

B) 2.31.

C) 0.52.

D) 1.00.

The correct answer is C

The t-statistic for testing the null hypothesis H0:  βi = 0 is t = (bi ? 0) / σi, where βi is the population parameter for independent variable i, bi is the estimated coefficient, and σi is the coefficient standard error.  Using the information provided, the estimated coefficient standard error can be computed as bIndex / t = σIndex = 1.1 / 2.1 = 0.5238.

Which of the following is the t-statistic for size?

A) 0.30.

B) ?0.12.

C) 0.70.

D) 3.33.

The correct answer is D

The t-statistic for testing the null hypothesis H0:  βi = 0 is t = (bi ? 0) / σi, where βi is the population parameter for independent variable i, bi is the estimated coefficient, and σi is the coefficient standard error.  Using the information provided, the t-statistic for size can be computed as t = bSize / σSize = 0.6 / 0.18 = 3.3333.

Which of the following is the estimated intercept for the regression?

A) ?0.11.

B) ?9.45.

C) ?2.86.

D) ?4.65.

The correct answer is C

The t-statistic for testing the null hypothesis H0:  βi = 0 is t = (bi ? 0) / σi, where βi is the population parameter for independent variable i, bi is the estimated parameter, and σi is the parameter’s standard error.  Using the information provided, the estimated intercept can be computed as b0 = t × σ0 = ?5.2 × 0.55 = ?2.86.

Which of the following statements is most accurate regarding the significance of the regression parameters at a 5% level of significance?

A) All of the parameter estimates are significantly different than zero at the 5% level of significance.

B) The parameter estimates for the intercept are significantly different than zero. The slope coefficients for index and size are not significant.

C) The parameter estimates for the intercept and the independent variable size are significantly different than zero. The coefficient for index is not significant.

D) None of the regression parameters are significantly different than zero at the 5% level of significance.

The correct answer is A

At 5% significance and 97 degrees of freedom (100 ? 3), the critical t-value is slightly greater than, but very close to, 1.984.  The t-statistic for the intercept and index are provided as ?5.2 and 2.1, respectively, and the t-statistic for size is computed as 0.6 / 0.18 = 3.33.  The absolute value of the all of the regression intercepts is greater than tcritical = 1.984.  Thus, it can be concluded that all of the parameter estimates are significantly different than zero at the 5% level of significance.

21、Consider the following estimated regression equation, with the standard errors of the slope coefficients as noted:

Salesi = 10.0 + 1.25 R&Di + 1.0 ADVi – 2.0 COMPi + 8.0 CAPi

where the standard error for the estimated coefficient on R&D is 0.45, the standard error for the estimated coefficient on ADV is 2.2 , the standard error for the estimated coefficient on COMP is 0.63, and the standard error for the estimated coefficient on CAP is 2.5.

The equation was estimated over 40 companies. Using a 5 percent level of significance, which of the estimated coefficients are significantly different from zero?

A) R&D, ADV, COMP, and CAP.

B) R&D, COMP, and CAP only.

C) COMP and CAP only.

D) ADV and CAP only.

The correct answer is B

The critical t-values for 40-4-1 = 35 degrees of freedom and a 5 percent level of significance are ± 2.03.

22、 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:

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.

B)    slope coefficient.

C)   zero.

D)   the lower tail critical value.

The correct answer is C

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.

The t-statistic for the marketing variable is calculated to be:

A) 3.271. 

B) 1.886. 

C) 1.469. 

D) 17.321. 

The correct answer is A

The t-statistic for the marketing coefficient is calculated as follows: (5.97– 0.0) / 1.825 = 3.271

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 percent 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: Both independent variables have slope coefficients that are significantly different from zero at the 5% significance level.

?            Point 4: If an F-test were being used, the null hypothesis would be rejected.

Which of Hays’ conclusions are CORRECT?

A) Points 1 and 2. 

B) Points 1 and 4. 

C) Points 2 and 3. 

D) Points 2 and 4. 

The correct answer is B

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’ Points 2 and 3 are 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 the R&D term does not have a slope coefficient that is greater than 0 at the 5 percent significant level. It also means that only MKTG can be said to contribute to explaining sales for Milky Way, Inc.

Hays’ Point 4 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.