Reading 12: Multiple Regression and Issues in Regression Analy 2011, determine, interpret, describe

土豆妮

Session 3: Quantitative Methods for Valuation
Reading 12: Multiple Regression and Issues in Regression Analysis

LOS a: Formulate a multiple regression equation to describe the relation between a dependent variable and several independent variables, determine the statistical significance of each independent variable, and interpret the estimated coefficients and their p-values.

 

 

Which of the following statements regarding the results of a regression analysis is least accurate? The:

A) 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.

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

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

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

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土豆妮

Consider the following estimated regression equation, with calculated t-statistics of the estimates as indicated:

AUTO_t = 10.0 + 1.25 PI_t + 1.0 TEEN_t – 2.0 INS_t

with a PI calculated t-statstic of 0.45, a TEEN calculated t-statstic of 2.2, and an INS calculated t-statstic of 0.63.

The equation was estimated over 40 companies. Using a 5% level of significance, which of the independent variables significantly different from zero?

A) PI and INS only.

B) PI only.

C) TEEN only.

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The critical t-values for 40-3-1 = 36 degrees of freedom and a 5% level of significance are ± 2.028. Therefore, only TEEN is statistically significant.

土豆妮

Jacob Warner, CFA, is evaluating a regression analysis recently published in a trade journal that hypothesizes that the annual performance of the S& 500 stock index can be explained by movements in the Federal Funds rate and the U.S. Producer Price Index (PPI). Which of the following statements regarding his analysis is most accurate?

A) If the p-value of a variable is less than the significance level, the null hypothesis can be rejected.

B) If the p-value of a variable is less than the significance level, the null hypothesis cannot be rejected.

C) If the t-value of a variable is less than the significance level, the null hypothesis cannot be rejected.

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The p-value is the smallest level of significance for which the null hypothesis can be rejected. Therefore, for any given variable, if the p-value of a variable is less than the significance level, the null hypothesis can be rejected and the variable is considered to be statistically significant.

土豆妮

Which of the following statements most accurately interprets the following regression results at the given significance level?

Variable		p-value
Intercept		0.0201
X1		0.0284
X2		0.0310
X3		0.0143

A) The variables X1 and X2 are statistically significantly different from zero at the 2% significance level.

B) The variable X2 is statistically significantly different from zero at the 3% significance level.

C) The variable X3 is statistically significantly different from zero at the 2% significance level.

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The p-value is the smallest level of significance for which the null hypothesis can be rejected. An independent variable is significant if the p-value is less than the stated significance level. In this example, X3 is the variable that has a p-value less than the stated significance level.

土豆妮

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) Intercept term.

C) p-value.

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

土豆妮

The standard error of estimate for Smith’s regression is closest to:

A) 0.53

B) 0.16

C) 0.56

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The formula for the Standard Error of the Estimate (SEE) is:

The SEE equals the standard deviation of the regression residuals. A low SEE implies a high R². (Study Session 3, LOS 12.f)

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Is Smith correct or incorrect regarding Concerns 1 and 2?

A) Only correct on one concern and incorrect on the other.

B) Correct on both Concerns.

C) Incorrect on both Concerns.

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Smith’s Concern 1 is incorrect. Heteroskedasticity is a violation of a regression assumption, and refers to regression error variance that is not constant over all observations in the regression. Conditional heteroskedasticity is a case in which the error variance is related to the magnitudes of the independent variables (the error variance is “conditional” on the independent variables). The consequence of conditional heteroskedasticity is that the standard errors will be too low, which, in turn, causes the t-statistics to be too high. Smith’s Concern 2 also is not correct. Multicollinearity refers to independent variables that are correlated with each other. Multicollinearity causes standard errors for the regression coefficients to be too high, which, in turn, causes the t-statistics to be too low. However, contrary to Smith’s concern, multicollinearity has no effect on the F-statistic. (Study Session 3, LOS 12.i)

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The most recent change in foreclosure share was +1 percent. Smith decides to base her analysis on the data and methods provided in Exhibits 4 and 5, and determines that the two-step ahead forecast for the change in foreclosure share (in percent) is 0.125, and that the mean reverting value for the change in foreclosure share (in percent) is 0.071. Is Smith correct?

A) Smith is correct on the two-step ahead forecast for change in foreclosure share only.

B) Smith is correct on the mean-reverting level for forecast of change in foreclosure share only.

C) Smith is correct on both the forecast and the mean reverting level.

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Forecasts are derived by substituting the appropriate value for the period t-1 lagged value.

So, the one-step ahead forecast equals 0.30%. The two-step ahead (%) forecast is derived by substituting 0.30 into the equation.

ΔForeclosure Share_t+1 = 0.05 + 0.25(0.30) = 0.125

Therefore, the two-step ahead forecast equals 0.125%.

(Study Session 3, LOS 13.d)

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Assume for this question that Smith finds that the foreclosure share series has a unit root. Under these conditions, she can most reliably regress foreclosure share against the change in interest rates (ΔINT) if:

A) ΔINT does not have unit root.

B) ΔINT has unit root and is not cointegrated with foreclosure share.

C) ΔINT has unit root and is cointegrated with foreclosure share.

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The error terms in the regressions for choices A, B, and C will be nonstationary. Therefore, some of the regression assumptions will be violated and the regression results are unreliable. If, however, both series are nonstationary (which will happen if each has unit root), but cointegrated, then the error term will be covariance stationary and the regression results are reliable. (Study Session 3, LOS 13.k)

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土豆妮

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:

Coefficient Estimates
		Standard Error
Predictor	Coefficient	of the Coefficient
Intercept	78	13.710
?1 GDP	30.22	12.120
?2$ Fuel	?412.39	183.981

Analysis of Variance Table (ANOVA)
Source	Degrees of Freedom	Sum of Squares	Mean Square
Regression		291.30	145.65
Error	27	132.12
Total	29	423.42

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.

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

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

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

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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.

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

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

C) only GDP changes explain changes in sales.

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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.

土豆妮

In a recent analysis of salaries (in $1,000) of financial analysts, a regression of salaries on education, experience, and gender is run. Gender equals one for men and zero for women. The regression results from a sample of 230 financial analysts are presented below, with t-statistics in parenthesis.

Salaries = 34.98 + 1.2 Education + 0.5 Experience + 6.3 Gender

                (29.11)          (8.93)                (2.98)                (1.58)

What is the expected salary (in $1,000) of a woman with 16 years of education and 10 years of experience?

A) 54.98.

B) 59.18.

C) 65.48.

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34.98 + 1.2(16) + 0.5(10) = 59.18

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Holding everything else constant, do men get paid more than women? Use a 5% level of significance. No, since the t-value:

A) does not exceed the critical value of 1.96.

B) exceeds the critical value of 1.96.

C) does not exceed the critical value of 1.65.

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H₀: b_gender ≤ 0
H_a: b_gender > 0

t-value of 1.58 < 1.65 (critical value)

土豆妮

 

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):

SALES = 0.000  +  0.004 POP + 1.031 INCOME + 2.002 ADV
(0.113)	(0.005)	(0.337)	(2.312)

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.

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Predicted sales for next year are:

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