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Reading 12: Multiple Regression and Issues in Regression Analy

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

LOS f: distinguish between and interpret the R2 and adjusted R2 in multiple regression.

 

 

Which of the following statements regarding the R2 is least accurate?

A)
The adjusted-R2 not appropriate to use in simple regression.
B)
The adjusted-R2 is greater than the R2 in multiple regression.
C)
It is possible for the adjusted-R2 to decline as more variables are added to the multiple regression.


 

The adjusted-R2 will always be less than R2in multiple regression.

Which of the following statements regarding the R2 is least accurate?

A)
The R2 of a regression will be greater than or equal to the adjusted-R2 for the same regression.
B)
The F-statistic for the test of the fit of the model is the ratio of the mean squared regression to the mean squared error.
C)
The R2 is the ratio of the unexplained variation to the explained variation of the dependent variable.


The R2 is the ratio of the explained variation to the total variation.

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An analyst is estimating a regression equation with three independent variables, and calculates the R2, the adjusted R2, and the F-statistic. The analyst then decides to add a fourth variable to the equation. Which of the following is most accurate?

A)
The R2 and F-statistic will be higher, but the adjusted R2 could be higher or lower.
B)
The R2 will be higher, but the adjusted R2 and F-statistic could be higher or lower.
C)
The adjusted R2 will be higher, but the R2 and F-statistic could be higher or lower.


The R2 will always increase as the number of variables increase. The adjusted R2 specifically adjusts for the number of variables, and might not increase as the number of variables rise. As the number of variables increases, the regression sum of squares will rise and the residual T sum of squares will fall—this will tend to make the F-statistic larger. However, the number degrees of freedom will also rise, and the denominator degrees of freedom will fall, which will tend to make the F-statistic smaller. Consequently, like the adjusted R2, the F-statistic could be higher or lower.

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An analyst regresses the return of a S& 500 index fund against the S& 500, and also regresses the return of an active manager against the S& 500. The analyst uses the last five years of data in both regressions. Without making any other assumptions, which of the following is most accurate? The index fund:

A)
regression should have higher sum of squares regression as a ratio to the total sum of squares.
B)
should have a higher coefficient on the independent variable.
C)
should have a lower coefficient of determination.


The index fund regression should provide a higher R2, which is the sum of squares regression divided by the total sum of squares.

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Using Equation ONE, what is the sales forecast for the second quarter of the next year?

A)
$51.09 million.
B)
$56.02 million.
C)
$46.31 million.


The estimate for the second quarter of the following year would be (in millions):

31.4083 + (?2.4631) + (24 + 2) × 0.851786 = 51.091666. (Study Session 3, LOS 12.c)


Which of the coefficients that appear in both equations are not significant at the 5% level in a two-tailed test?

A)
The coefficients on Q1 and Q2 only.
B)
The coefficient on Q2 only.
C)
The intercept only.


The absolute value of the critical T-statistics for Equation ONE and TWO are 2.093 and 2.086, respectively. Since, the t-statistics for Q2 in Equations ONE and TWO are ?1.6685 and ?1.9188, respectively, these fall below the critical values for both equations. (Study Session 3, LOS 12.a)


Conditional heteroskedasticity is a problem for:

A)
neither equation.
B)
Equation ONE but not Equation TWO.
C)
both Equations ONE and TWO.


Mercado would use the Breusch-Pagan test for heteroskedasticity. Mercado regressed the squared residuals from each equation on the respective independent variables. The R2 values were 0.008801 and 0.006313 respectively. The test-statistic for the Breusch-Pagan test is n × (R2) which is distributed as a Chi-squared statistic with degrees of freedom equal to the number of independent variables. Assuming a 5% level of significance, the respective critical values are 7.815 and 9.488. The respective test statistic values are 24 × 0.008801 = 0.2112 and 24 × 0.006313 = 0.1515. Both computed test statistics are much less than their respective critical values; thus, Mercado would conclude that conditional heteroskedasticity is not a problem. (Study Session 3, LOS 12.i)


Mercado probably did not include a fourth dummy variable Q4, which would have had 0, 0, 0, 1 as its first four observations because:

A)
it would have lowered the explanatory power of the equation.
B)
the intercept is essentially the dummy for the fourth quarter.
C)
it would not have been significant.


The fourth quarter serves as the base quarter, and for the fourth quarter, Q1 = Q2 = Q3 = 0. Had the equation included a Q4 as specified, we could not have had an intercept. In that case, for Equation ONE for example, the estimate of Q4 would have been 31.40833. The dummies for the other quarters would be the 31.40833 plus the estimated dummies from Equation ONE. In an equation that included Q1, Q2, Q3, and Q4 but no intercept, for example:

Q1 = 31.40833 + (?3.77798) = 27.63035
Such an equation would produce the same estimated values for the dependent variable. (Study Session 3, LOS 12.h)


If Mercado determines that Equation TWO is the appropriate specification, then he is essentially saying that for each year, value of sales from quarter three to four is expected to:

A)
remain approximately the same.
B)
grow by more than $1,000,000.
C)
grow, but by less than $1,000,000.


The specification of Equation TWO essentially assumes there is no difference attributed to the change of the season from the third to fourth quarter. However, the time trend is significant. The trend effect for moving from one season to the next is the coefficient on TREND times $1,000,000 which is $852,182 for Equation TWO. (Study Session 3, LOS 13.a)


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