HuskyGrad2010

Assume you ran a multiple regression to gain a better understanding of the relationship between lumber sales, housing starts, and commercial construction. The regression uses lumber sales as the dependent variable with housing starts and commercial construction as the independent variables. The results of the regression are:

	Coefficient	Standard Error	t-statistics
Intercept	5.37	1.71	3.14
Housing starts	0.76	0.09	8.44
Commercial construction	1.25	0.33	3.78
The level of significance for a 95% confidence level is 1.96

Construct a 95% confidence interval for the slope coefficient for Housing Starts.

A) 0.76 ± 1.96(0.09).

B) 0.76 ± 1.96(8.44).

C) 1.25 ± 1.96(0.33).

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The confidence interval for the slope coefficient is b1 ± (tc × sb1).

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Construct a 95% confidence interval for the slope coefficient for Commercial Construction.

A) 1.25 ± 1.96(0.33).

B) 0.76 ± 1.96(0.09).

C) 1.25 ± 1.96(3.78).

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The confidence interval for the slope coefficient is b1 ± (tc × sb1).

HuskyGrad2010

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%. 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:

	Aggregate Revenue (Y)	Advertising Expenditure (X)	Y2	XY	X2
	300	7.5	90,000	2,250	56.25
	320	9.0	102,400	2,880	81.00
	310	8.5	96,100	2,635	72.25
	335	8.2	112,225	2,747	67.24
	350	9.0	122,500	3,150	81.00
	400	8.5	160,000	3,400	72.25
	430	10.0	184,900	4,300	100.00
	390	10.5	152,100	4,095	110.25
	380	9.0	144,400	3,420	81.00
	430	11.0	184,900	4,730	121.00
TOTAL	3,645	91.2	1,349,525	33,607	842.24

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

ANOVA
	Df	SS	MS
Regression	1	12,665.125760	12,665.12576
Residual	8	8,257.374238	1,032.17178
Total	9	20,922.5

Jones discusses her findings with her market research specialist, Mira Nair. Nair tells Jones that she should check her model for heteroskedasticity. She explains that in the presence of conditional heteroskedasticity, the model coefficients and t-statistics will be biased.
For the questions below, assume a t-value of 2.306.Which of the following is closest to the upper limit of the 95% confidence interval for the slope coefficient?

A) 57.61.

B) 62.84.

C) 111.72.

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Upper Limit	= coefficient + (2.306 x standard error of the coefficient)
	= 34.74 + (2.306 x 9.917) = 57.61

(Study Session 3, LOS 11.f)

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Which of the following is closest to the lower limit of the 95% confidence interval for the slope coefficient?

A) 12.24.

B) 11.87.

C) 72.84.

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Lower Limit	= Coefficient - (2.306 x Standard Error of the coefficient)
	= 34.74 - (2.306 x 9.917)
	= 34.74 - 22.87 = 11.87

(Study Session 3, LOS 11.f)

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

A) 0.9500.

B) 0.6053.

C) 0.7780.

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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. (Study Session 3, LOS 11.i)

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

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

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

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

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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). (Study Session 3, LOS 11.i)

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Is Nair’s statement about conditional heteroskedasticity CORRECT?

A) No, because coefficients will not be biased.

B) Yes, because both the coefficients and t-statistics will be biased.

C) No, because the t-statistics will not be biased.

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Conditional heteroskedasticity will result in consistent coefficient estimates but inconsistent standard errors resulting in biased t-statistics. (Study Session 3, LOS 12.i)

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What is the calculated F-statistic?

A) 0.1250.

B) 12.2700.

C) 92.2840.

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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. (Study Session 3, LOS 11.i)

HuskyGrad2010

Craig Standish, CFA, is investigating the validity of claims associated with a fund that his company offers. The company advertises the fund as having low turnover and, hence, low management fees. The fund was created two years ago with only a few uncorrelated assets. Standish randomly draws two stocks from the fund, Grey Corporation and Jars Inc., and measures the variances and covariance of their monthly returns over the past two years. The resulting variance covariance matrix is shown below. Standish will test whether it is reasonable to believe that the returns of Grey and Jars are uncorrelated. In doing the analysis, he plans to address the issue of spurious correlation and outliers.

	Grey	Jars
Grey	42.2	20.8
Jars	20.8	36.5

Standish wants to learn more about the performance of the fund. He performs a linear regression of the fund’s monthly returns over the past two years on a large capitalization index. The results are below:

ANOVA	[td=1,1,110] [td=1,1,75] [td=1,1,70] [/td]
	df	SS	MS	F
Regression	1	92.53009	92.53009	28.09117
Residual	22	72.46625	3.293921
Total	23	164.9963
	Coefficients	Standard Error	t-statistic	P-value
Intercept	0.148923	0.391669	0.380225	0.707424
Large Cap Index	1.205602	0.227467	5.30011	2.56E-05

Standish forecasts the fund’s return, based upon the prediction that the return to the large capitalization index used in the regression will be 10%. He also wants to quantify the degree of the prediction error, as well as the minimum and maximum sensitivity that the fund actually has with respect to the index.
He plans to summarize his results in a report. In the report, he will also include caveats concerning the limitations of regression analysis. He lists four limitations of regression analysis that he feels are important: relationships between variables can change over time, the decision to use a t-statistic or F-statistic for a forecast confidence interval is arbitrary, if the error terms are heteroskedastic the test statistics for the equation may not be reliable, and if the error terms are correlated with each other over time the test statistics may not be reliable. Given the variance/covariance matrix for Grey and Jars, in a one-sided hypothesis test that the returns are positively correlated H0: ρ = 0 vs. H1: ρ > 0, Standish would:

A) reject the null at the 5% but not the 1% level of significance.

B) need to gather more information before being able to reach a conclusion concerning significance.

C) reject the null at the 1% level of significance.

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First, we must compute the correlation coefficient, which is 0.53 = 20.8 / (42.2 × 36.5)0.5.
The t-statistic is: 2.93 = 0.53 × [(24 - 2) / (1 − 0.53 × 0.53)]0.5, and for df = 22 = 24 − 2, the t-statistics for the 5 and 1% level are 1.717 and 2.508 respectively. (Study Session 3, LOS 11.g)

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In performing the correlation test on Grey and Jars, Standish would most appropriately address the issue of:

A) spurious correlation but not the issue of outliers.

B) spurious correlation and the issue of outliers.

C) neither outliers nor correlation.

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Both these issues are important in performing correlation analysis. A single outlier observation can change the correlation coefficient from significant to not significant and even from negative (positive) to positive (negative). Even if the correlation coefficient is significant, the researcher would want to make sure there is a reason for a relationship and that the correlation is not caused by chance. (Study Session 3, LOS 11.b)

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If the large capitalization index has a 10% return, then the forecast of the fund’s return will be:

A) 13.5.

B) 16.1.

C) 12.2.

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The forecast is 12.209 = 0.149 + 1.206 × 10, so the answer is 12.2. (Study Session 3, LOS 11.h)

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The standard error of the estimate is:

A) 1.81.

B) 9.62.

C) 0.56.

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SEE equals the square root of the MSE, which on the ANOVA table is 72.466 / 22 = 3.294. The SEE is 1.81 = (3.294)0.5. (Study Session 3, LOS 11.i)

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A 95% confidence interval for the slope coefficient is:

A) 0.905 to 1.506.

B) 0.760 to 1.650.

C) 0.734 to 1.677.

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The 95% confidence interval is 1.2056 ± (2.074 × 0.2275). (Study Session 3, LOS 11.f)

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Of the four caveats of regression analysis listed by Standish, the least accurate is:

A) if the error terms are heteroskedastic the test statistics for the equation may not be reliable.

B) the choice to use a t-statistic or F-statistic for a forecast confidence interval is arbitrary.

C) the relationships of variables change over time.

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The t-statistic is used for constructing the confidence interval for the forecast. The F-statistic is not used for this purpose. The other possible shortfalls listed are valid. (Study Session 3, LOS 11.i)

HuskyGrad2010

What does the R2 of a simple regression of two variables measure and what calculation is used to equate the correlation coefficient to the coefficient of determination?

[td=1,1,225]R2measures:

Correlation coefficient

A) percent of variability of the independent variable that is explained by the variability of the dependent variable R2 = r2

B) percent of variability of the independent variable that is explained by the variability of the dependent variable R2 = r × 2

C) percent of variability of the dependent variable that is explained by the variability of the independent variable R2 = r2

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R2, or the Coefficient of Determination, is the square of the coefficient of correlation (r). The coefficient of correlation describes the strength of the relationship between the X and Y variables. The standard error of the residuals is the standard deviation of the dispersion about the regression line. The t-statistic measures the statistical significance of the coefficients of the regression equation. In the response: "percent of variability of the independent variable that is explained by the variability of the dependent variable," the definitions of the variables are reversed.

HuskyGrad2010

Consider the following estimated regression equation:

ROEt = 0.23 - 1.50 CEt

The standard error of the coefficient is 0.40 and the number of observations is 32. The 95% confidence interval for the slope coefficient, b1, is:

A) {0.683 < b1 < 2.317}.

B) {-2.317 < b1 < -0.683}.

C) {-2.300 < b1 < -0.700}.

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The confidence interval is -1.50 ± 2.042 (0.40), or {-2.317 < b1 < -0.683}.

HuskyGrad2010

Assume you perform two simple regressions. The first regression analysis has an R-squared of 0.80 and a beta coefficient of 0.10. The second regression analysis has an R-squared of 0.80 and a beta coefficient of 0.25. Which one of the following statements is most accurate?

A) Explained variability from both analyses is equal.

B) The influence on the dependent variable of a one-unit increase in the independent variable is the same in both analyses.

C) Results from the first analysis are more reliable than the second analysis.

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The coefficient of determination (R-squared) is the percentage of variation in the dependent variable explained by the variation in the independent variable. The R-squared (0.80) being identical between the first and second regressions means that 80% of the variability in the dependent variable is explained by variability in the independent variable for both regressions. This means that the first regression has the same explaining power as the second regression.

HuskyGrad2010

Which of the following statements about linear regression analysis is most accurate?

A) The coefficient of determination is defined as the strength of the linear relationship between two variables.

B) An assumption of linear regression is that the residuals are independently distributed.

C) When there is a strong relationship between two variables we can conclude that a change in one will cause a change in the other.

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Even when there is a strong relationship between two variables, we cannot conclude that a causal relationship exists. The coefficient of determination is defined as the percentage of total variation in the dependent variable explained by the independent variable.

HuskyGrad2010

Linear regression is based on a number of assumptions. Which of the following is least likely an assumption of linear regression?

A) Values of the independent variable are not correlated with the error term.

B) There is at least some correlation between the error terms from one observation to the next.

C) The variance of the error terms each period remains the same.

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When correlation (between the error terms from one observation to the next) exists, autocorrelation is present. As a result, residual terms are not normally distributed. This is inconsistent with linear regression.

HuskyGrad2010

The assumptions underlying linear regression include all of the following EXCEPT the:

A) disturbance term is normally distributed with an expected value of 0.

B) independent variable is linearly related to the residuals (or disturbance term).

C) disturbance term is homoskedastic and is independently distributed.

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The independent variable is uncorrelated with the residuals (or disturbance term).
The other statements are true. The disturbance term is homoskedastic because it has a constant variance. It is independently distributed because the residual for one observation is not correlated with that of another observation. Note: The opposite of homoskedastic is heteroskedastic. For the examination, memorize the assumptions underlying linear regression!

HuskyGrad2010

Which of the following is least likely an assumption of linear regression?

A) The residuals are normally distributed.

B) The variance of the residuals is constant.

C) The independent variable is correlated with the residuals.

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The assumption is that the independent variable is uncorrelated with the residuals.