AUTOt = 10.0 + 1.25 PIt + 1.0 TEENt – 2.0 INSt
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.

ARARi = b0 + b1Sizei + b2Indexi + ei
Where:

ARARi	=	the average annual risk-adjusted percent returns for the fund i over the 1998-2002 time period.
Sizei	=	the natural logarithm of the average assets under management for fund i.
Indexi	=	the percentage of assets in fund i that were managed to an indexing strategy.

Partial Results from Regression ARAR on Size and Extent of Indexing
	Coefficients	Standard Error	t-Statistic
Intercept	???	0.55	−5.2
Size	0.6	0.18	???
Index	1.1	???	2.1

ARARi = b0 + b1Sizei + b2Indexi + ei
Where:

ARARi	=	the average annual risk-adjusted percent returns for the fund i over the 1998-2002 time period.
Sizei	=	the natural logarithm of the average assets under management for fund i.
Indexi	=	the percentage of assets in fund i that were managed to an indexing strategy.

Partial Results from Regression ARAR on Size and Extent of Indexing
	Coefficients	Standard Error	t-Statistic
Intercept	???	0.55	−5.2
Size	0.6	0.18	???
Index	1.1	???	2.1

Expected Sales = 12.6 + 1.6 (Marketing Expenditures) + 1.2 (# of Salespeople)

• Point 1: The SEE is the standard deviation of the differences between the estimated values for the independent variables and the actual observations for the independent variable.
• Point 2: Any violation of the basic assumptions of a multiple regression model is going to affect the SEE.
• Point 3: If there is a strong relationship between the variables and the SSE is small, the individual estimation errors will also be small.

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.

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.  

Coefficient Estimates
Parameter	Coefficient	Standard Error of Coefficient
Intercept	9.50	3.40
EMP	-4.50	1.25
GDP	4.20	0.76
INV	-0.30	0.16

• Regression sum of squares (RSS) = 126.00
• Sum of squared errors (SSE) = 267.00
• Durbin-Watson statistic (DW) = 1.34

Abbreviated Table of the Student’s t-distribution (One-Tailed Probabilities)
df	p = 0.10	p = 0.05	p = 0.025	p = 0.01	p = 0.005
3	1.638	2.353	3.182	4.541	5.841
10	1.372	1.812	2.228	2.764	3.169
50	1.299	1.676	2.009	2.403	2.678
100	1.290	1.660	1.984	2.364	2.626
120	1.289	1.658	1.980	2.358	2.617
200	1.286	1.653	1.972	2.345	2.601

Critical Values for Durbin-Watson Statistic (α = 0.05)
	K=1		K=2		K=3		K=4		K=5
n	dl	du	dl	du	dl	du	dl	du	dl	du
20	1.20	1.41	1.10	1.54	1.00	1.68	0.90	1.83	0.79	1.99
50	1.50	1.59	1.46	1.63	1.42	1.67	1.38	1.72	1.34	1.77
>100	1.65	1.69	1.63	1.72	1.61	1.74	1.59	1.76	1.57	1.78

H0: bGDP = 3.50
Ha: bGDP ≠ 3.50

• Employment growth = 2.0%
• GDP growth = 1.0%
• Private investment growth = -1.0%

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.

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% confidence level is 2.120.
Which of the independent variables is statistically different from zero at the 95% confidence level?

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?

	Coefficient	Standard Error	t-statistics
Intercept	66,500	59,292	1.12
House Size	74.30	21.11	3.52
Number of Bedrooms	10306	3230	3.19

R2 = 0.56; F = 40.73

sales = b0 + (b1 × CPI) + (b2 × IP) + (b3 × GDP) + εt
Where:
sales = $ change in sales (in 000’s)
CPI = change in the consumer price index
IP = change in industrial production (millions)
GDP = change in GDP (millions)
All changes in variables are in percentage terms.

Assumption 1:	There is a linear relationship between the dependent and independent variables.
Assumption 2:	The independent variables are not random, and there is no correlation between any two of the independent variables.
Assumption 3:	The residual term is normally distributed with an expected value of zero.
Assumption 4:	The residuals are serially correlated.
Assumption 5:	The variance of the residuals is constant.

Hypothesis 1:	The coefficient on GDP is negative.
Hypothesis 2:	The intercept term is equal to –4.
Hypothesis 3:	A 2.6% increase in the CPI will result in an increase in sales of more than 12.0%.
Hypothesis 4:	A 1% increase in industrial production will result in a 1% decrease in sales.

Figure 1: Partial table of the Student’s t-distribution (One-tailed probabilities)

df	p = 0.10	p = 0.05	p = 0.025	p = 0.01	p = 0.005
170	1.287	1.654	1.974	2.348	2.605
176	1.286	1.654	1.974	2.348	2.604
180	1.286	1.653	1.973	2.347	2.603

Figure 2: Partial F-Table critical values for right-hand tail area equal to 0.05

	df1 = 1	df1 = 3	df1 = 5
df2 = 170	3.90	2.66	2.27
df2 = 176	3.89	2.66	2.27
df2 = 180	3.89	2.65	2.26

Figure 3: Partial F-Table critical values for right-hand tail area equal to 0.025

	df1 = 1	df1 = 3	df1 = 5
df2 = 170	5.11	3.19	2.64
df2 = 176	5.11	3.19	2.64
df2 = 180	5.11	3.19	2.64

foreclosure share = b0 + b1(ΔINT) + b2(STIM) + b3(CRISIS) + ε

where:	<P td > [td=1,1,700]<P td > [/td]
Foreclosure share	=	the percentage of all outstanding mortgages foreclosed upon during the quarter
ΔINT	=	the quarterly change in the 1-year Treasury bill rate (e.g., ΔINT = 2 for a two percentage point increase in interest rates)
STIM	=	1 for quarters in which a Federal fiscal stimulus package was in place
CRISIS	=	1 for quarters in which the median house price is one standard deviation below its 5-year moving average

Exhibit 2: Foreclosure Share Regression Results

Exhibit 3: Foreclosure Share Regression Equation ANOVA Table

Exhibit 5. Autocorrelation Analysis

Exhibit 6: Critical Values for Student’s t-Distribution

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

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

Table 1: Regression Output

Variable	Coefficient	Standard Error of the Coefficient	t-statistic	p-value
Intercept	0.043	0.01159	3.71	< 0.001
Ln(No. of Analysts)	−0.027	0.00466	−5.80	< 0.001
Ln(Market Value)	0.006	0.00271	2.21	0.028

Table 2: ANOVA

	Degrees of Freedom	Sum of Squares	Mean Square
Regression	2	0.103	0.051
Residual	194	0.559	0.003
Total	196	0.662

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

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

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

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.

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

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

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

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

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)

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

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)

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

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)

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:

AUTOt = 10.0 + 1.25 PIt + 1.0 TEENt – 2.0 INSt

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. The predicted value of AUTO if PI is 4, TEEN is 0.30, and INS = 0.6 is closest to:

---

Predicted AUTO [/td][td]= 10 + 1.25 (4) + 1.0 (0.30) – 2.0 (0.6)
	= 10 + 5 + 0.3 – 1.2
	= 14.10

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作者: Walex    时间: 2012-3-26 17:14

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

AUTOt = 10.0 + 1.25 PIt + 1.0 TEENt – 2.0 INSt

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. The predicted value of AUTO if PI is 4, TEEN is 0.30, and INS = 0.6 is closest to:

A) 14.90.

B) 17.50.

C) 14.10.

---

Predicted AUTO [/td][td]= 10 + 1.25 (4) + 1.0 (0.30) – 2.0 (0.6)
	= 10 + 5 + 0.3 – 1.2
	= 14.10

---

作者: Walex    时间: 2012-3-26 17:14

Wanda Brunner, CFA, is trying to calculate a 95% confidence interval (df = 40) for a regression equation based on the following information:

	Coefficient	Standard Error
Intercept	-10.60%	1.357
DR	0.52	0.023
CS	0.32	0.025

What are the lower and upper bounds for variable DR?

A) 0.481 to 0.559.

B) 0.474 to 0.566.

C) 0.488 to 0.552.

---

The critical t-value is 2.02 at the 95% confidence level (two tailed test). The estimated slope coefficient is 0.52 and the standard error is 0.023. The 95% confidence interval is 0.52 ± (2.02)(0.023) = 0.52 ± (0.046) = 0.474 to 0.566.

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作者: Walex    时间: 2012-3-26 17:15

Wanda Brunner, CFA, is trying to calculate a 98% confidence interval (df = 40) for a regression equation based on the following information:

	Coefficient	Standard Error
Intercept	-10.60%	1.357
DR	0.52	0.023
CS	0.32	0.025

Which of the following are closest to the lower and upper bounds for variable CS?

A) 0.267 to 0.374.

B) 0.274 to 0.367.

C) 0.260 to 0.381.

---

The critical t-value is 2.42 at the 98% confidence level (two tailed test). The estimated slope coefficient is 0.32 and the standard error is 0.025. The 98% confidence interval is 0.32 ± (2.42)(0.025) = 0.32 ± (0.061) = 0.260 to 0.381.

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作者: Walex    时间: 2012-3-26 17:16

An analyst is interested in forecasting the rate of employment growth and instability for 254 metropolitan areas around the United States. The analyst’s main purpose for these forecasts is to estimate the demand for commercial real estate in each metro area. The independent variables in the analysis represent the percentage of employment in each industry group.

Regression of Employment Growth Rates and Employment Instability on Industry Mix Variables for 254 U.S. Metro Areas
	Model 1		Model 2
Dependent Variable	Employment Growth Rate		Relative Employment Instability
Independent Variables	Coefficient Estimate	t-value	Coefficient Estimate	t-value
Intercept	–2.3913	–0.713	3.4626	0.623
% Construction Employment	0.2219	4.491	0.1715	2.096
% Manufacturing Employment	0.0136	0.393	0.0037	0.064
% Wholesale Trade Employment	–0.0092	–0.171	0.0244	0.275
% Retail Trade Employment	–0.0012	–0.031	–0.0365	–0.578
% Financial Services Employment	0.0605	1.271	–0.0344	–0.437
% Other Services Employment	0.1037	2.792	0.0208	0.338
R²	0.289		0.047
Adjusted R²	0.272		0.024
F-Statistic	16.791		2.040
Standard error of estimate	0.546		0.345

Based on the data given, which independent variables have both a statistically and an economically significant impact (at the 5% level) on metropolitan employment growth rates?

A) "% Manufacturing Employment," "% Financial Services Employment," "% Wholesale Trade Employment," and "% Retail Trade" only.

B) "% Construction Employment" and "% Other Services Employment" only.

C) "% Wholesale Trade Employment" and "% Retail Trade" only.

---

The percentage of construction employment and the percentage of other services employment have a statistically significant impact on employment growth rates in U.S. metro areas. The t-statistics are 4.491 and 2.792, respectively, and the critical t is 1.96 (95% confidence and 247 degrees of freedom). In terms of economic significance, construction and other services appear to be significant. In other words, as construction employment rises 1%, the employment growth rate rises 0.2219%. The coefficients of all other variables are too close to zero to ascertain any economic significance, and their t-statistics are too low to conclude that they are statistically significant. Therefore, there are only two independent variables that are both statistically and economically significant: "% of construction employment" and "% of other services employment".
Some may argue, however, that financial services employment is also economically significant even though it is not statistically significant because of the magnitude of the coefficient. Economic significance can occur without statistical significance if there are statistical problems. For instance, the multicollinearity makes it harder to say that a variable is statistically significant. (Study Session 3, LOS 12.m)

---

The coefficient standard error for the independent variable “% Construction Employment” under the relative employment instability model is closest to:

A) 0.3595.

B) 0.0818.

C) 2.2675.

---

The t-statistic is computed by t-statistic = slope coefficient / coefficient standard error. Therefore, the coefficient standard error =
= slope coefficient/the t-statistic = 0.1715/2.096 = 0.0818. (Study Session 3, LOS 12.a)

---

Which of the following best describes how to interpret the R2 for the employment growth rate model? Changes in the value of the:

A) independent variables cause 28.9% of the variability of the employment growth rate.

B) independent variables explain 28.9% of the variability of the employment growth rate.

C) employment growth rate explain 28.9% of the variability of the independent variables.

---

The R2 indicates the percent variability of the dependent variable that is explained by the variability of the independent variables. In the employment growth rate model, the variability of the independent variables explains 28.9% of the variability of employment growth. Regression analysis does not establish a causal relationship. (Study Session 3, LOS 12.f)

---

Using the following forecasts for Cedar Rapids, Iowa, the forecasted employment growth rate for that city is closest to:

Construction employment	10%
Manufacturing	30%
Wholesale trade	5%
Retail trade	20%
Financial services	15%
Other services	20%

A) 3.15%.

B) 5.54%.

C) 3.22%.

---

The forecast uses the intercept and coefficient estimates for the model. The forecast is:
= −2.3913 + (0.2219)(10) + (0.0136)(30) + (−0.0092)(5) + (−0.0012)(20) + (0.0605)(15) + (0.1037)(20) = 3.15%. (Study Session 3, LOS 12.c)

---

The 95% confidence interval for the coefficient estimate for “% Construction Employment” from the relative employment instability model is closest to:

A) 0.0897 to 0.2533.

B) –0.0740 to 0.4170.

C) 0.0111 to 0.3319.

---

With a sample size of 254, and 254 − 6 − 1 = 247 degrees of freedom, the critical value for a two-tail 95% t-statistic is very close to the two-tail 95% statistic of 1.96. Using this critical value, the formula for the 95% confidence interval for the jth coefficient estimate is:95% confidence interval = . But first we need to figure out the coefficient standard error:

Hence, the confidence interval is 0.1715 ± 1.96(0.08182).
With 95% probability, the coefficient will range from 0.0111 to 0.3319, 95% CI = {0.0111 < b1 < 0.3319}. (Study Session 3, LOS 11.f)

---

One possible problem that could jeopardize the validity of the employment growth rate model is multicollinearity. Which of the following would suggest the existence of multicollinearity?

A) The variance of the observations has increased over time.

B) There is high positive correlation between “% Manufacturing Employment,” “% Service Sector Employment,” and “% Construction Employment.”

C) The employment growth rate displays a steady upward trend during the period covered by the data.

---

The problem of multicollinearity involves the existence of high correlation between two or more independent variables. Clearly, as service employment rises, construction employment must rise to facilitate the growth in these sectors. Alternatively, as manufacturing employment rises, the service sector must grow to serve the broader manufacturing sector.

• The variance of observations suggests the possible existence of heteroskedasticity.
• A steady upward trend displayed by the employment growth rate suggests the possible existence of autocorrelation.

(Study Session 3, LOS 12.j)

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作者: bapswarrior    时间: 2012-3-27 10:13

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 linear relationship.

B) homoskedasticity.

C) a normal distribution.

---

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

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作者: bapswarrior    时间: 2012-3-27 10:14

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

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

B) The error term is normally distributed.

C) The independent variables are not random.

---

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

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作者: bapswarrior    时间: 2012-3-27 10:14

Assume that in a particular multiple regression model, it is determined that the error terms are uncorrelated with each other. Which of the following statements is most accurate?

A) Unconditional heteroskedasticity present in this model should not pose a problem, but can be corrected by using robust standard errors.

B) This model is in accordance with the basic assumptions of multiple regression analysis because the errors are not serially correlated.

C) Serial correlation may be present in this multiple regression model, and can be confirmed only through a Durbin-Watson test.

---

One of the basic assumptions of multiple regression analysis is that the error terms are not correlated with each other. In other words, the error terms are not serially correlated. Multicollinearity and heteroskedasticity are problems in multiple regression that are not related to the correlation of the error terms.

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作者: bapswarrior    时间: 2012-3-27 10:15

An analyst runs a regression of monthly value-stock returns on five independent variables over 48 months. The total sum of squares is 430, and the sum of squared errors is 170. Test the null hypothesis at the 2.5% and 5% significance level that all five of the independent variables are equal to zero.

A) Rejected at 2.5% significance and 5% significance.

B) Rejected at 5% significance only.

C) Not rejected at 2.5% or 5.0% significance.

---

The F-statistic is equal to the ratio of the mean squared regression (MSR) to the mean squared error (MSE).
RSS = SST – SSE = 430 – 170 = 260
MSR = 260 / 5 = 52
MSE = 170 / (48 – 5 – 1) = 4.05
F = 52 / 4.05 = 12.84
The critical F-value for 5 and 42 degrees of freedom at a 5% significance level is approximately 2.44. The critical F-value for 5 and 42 degrees of freedom at a 2.5% significance level is approximately 2.89. Therefore, we can reject the null hypothesis at either level of significance and conclude that at least one of the five independent variables explains a significant portion of the variation of the dependent variable.

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作者: bapswarrior    时间: 2012-3-27 10:15

Consider the following analysis of variance table:

Source	Sum of Squares	Df	Mean Square
Regression	20	1	20
Error	80	20	4
Total	100	21

The F-statistic for a test of the overall significance of the model is closest to:

A) 0.20

B) 5.00

C) 0.05

---

The F-statistic is equal to the ratio of the mean squared regression to the mean squared error.
F = MSR / MSE = 20 / 4 = 5.

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作者: bapswarrior    时间: 2012-3-27 10:16

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

What is the p-value for the test of the hypothesis that all three of the coefficients are equal to zero?

A) Between 0.025 and 0.05.

B) Between 0.05 and 0.10.

C) lower than 0.025.

---

This test requires an F-statistic, which is equal to the ratio of the mean regression sum of squares to the mean squared error.
The mean regression sum of squares is the regression sum of squares divided by the number of independent variables, which is 119.25 / 3 = 39.75.
The residual sum of squares is the difference between the total sum of squares and the regression sum of squares, which is 294.45 − 119.25 = 175.20. The denominator degrees of freedom is the number of observations minus the number of independent variables, minus 1, which is 25 − 3 − 1 = 21. The mean squared error is the residual sum of squares divided by the denominator degrees of freedom, which is 175.20 / 21 = 8.34.
The F-statistic is 39.75 / 8.34 = 4.76, which is higher than the F-value (with 3 numerator degrees of freedom and 21 denominator degrees of freedom) of 3.07 at the 5% level of significance and higher than the F-value of 3.82 at the 2.5% level of significance. The conclusion is that the p-value must be lower than 0.025.
Remember the p-value is the probability that lies above the computed test statistic for upper tail tests or below the computed test statistic for lower tail tests.

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作者: bapswarrior    时间: 2012-3-27 10:17

Consider the following analysis of variance (ANOVA) table:

Source	Sum of squares	Degrees of freedom	Mean square
Regression	20	1	20
Error	80	40	2
Total	100	41

The F-statistic for the test of the fit of the model is closest to:

A) 0.10.

B) 10.00.

C) 0.25.

---

The F-statistic is equal to the ratio of the mean squared regression to the mean squared error.
F = MSR/MSE = 20 / 2 = 10.

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作者: bapswarrior    时间: 2012-3-27 10:18

Which of the following statements about the F-statistic is least accurate?

A) Rejecting the null hypothesis means that only one of the independent variables is statistically significant.

B) F = MSR/MSE.

C) dfnumerator = k and dfdenominator = n − k − 1.

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An F-test assesses how well the set of independent variables, as a group, explains the variation in the dependent variable. That is, the F-statistic is used to test whether at least one of the independent variables explains a significant portion of the variation of the dependent variable.

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作者: bapswarrior    时间: 2012-3-27 10:19

Toni Williams, CFA, has determined that commercial electric generator sales in the Midwest U.S. for Self-Start Company is a function of several factors in each area: the cost of heating oil, the temperature, snowfall, and housing starts. Using data for the most currently available year, she runs a cross-sectional regression where she regresses the deviation of sales from the historical average in each area on the deviation of each explanatory variable from the historical average of that variable for that location. She feels this is the most appropriate method since each geographic area will have different average values for the inputs, and the model can explain how current conditions explain how generator sales are higher or lower from the historical average in each area. In summary, she regresses current sales for each area minus its respective historical average on the following variables for each area.

• The difference between the retail price of heating oil and its historical average.
• The mean number of degrees the temperature is below normal in Chicago.
• The amount of snowfall above the average.
• The percentage of housing starts above the average.

Williams used a sample of 26 observations obtained from 26 metropolitan areas in the Midwest U.S. The results are in the tables below. The dependent variable is in sales of generators in millions of dollars.

Coefficient Estimates Table
Variable	Estimated Coefficient	Standard Error of the Coefficient
Intercept	5.00	1.850
$ Heating Oil	2.00	0.827
Low Temperature	3.00	1.200
Snowfall	10.00	4.833
Housing Starts	5.00	2.333

Analysis of Variance Table (ANOVA)
Source	Degrees of Freedom	Sum of Squares	Mean Square
Regression	4	335.20	83.80
Error	21	606.40	28.88
Total	25	941.60

One of her goals is to forecast the sales of the Chicago metropolitan area next year. For that area and for the upcoming year, Williams obtains the following projections: heating oil prices will be $0.10 above average, the temperature in Chicago will be 5 degrees below normal, snowfall will be 3 inches above average, and housing starts will be 3% below average.
In addition to making forecasts and testing the significance of the estimated coefficients, she plans to perform diagnostic tests to verify the validity of the model’s results.According to the model and the data for the Chicago metropolitan area, the forecast of generator sales is:

A) $55 million above average.

B) $35.2 million above the average.

C) $65 million above the average.

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The model uses a multiple regression equation to predict sales by multiplying the estimated coefficient by the observed value to get:

[5 + (2 × 0.10) + (3 × 5) + (10 × 3) + (5 × (−3))] × $1,000,000 = $35.2 million.

(Study Session 3, LOS 12.c)

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

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

C) at least one of the independent variables has 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) = (83.80 / 28.88) = 2.9017. From the F distribution table (4 df numerator, 21 df denominator) the critical F value is 2.84. Because 2.9017 is greater than 2.84, Williams rejects the null hypothesis and concludes that at least one of the independent variables has explanatory power. (Study Session 3, LOS 12.e)

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With respect to testing the validity of the model’s results, Williams may wish to perform:

A) both a Durbin-Watson test and a Breusch-Pagan test.

B) a Durbin-Watson test, but not a Breusch-Pagan test.

C) a Breusch-Pagan test, but not a Durbin-Watson test.

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Since this is not an autoregression, a test for serial correlation is appropriate so the Durbin-Watson test would be used. The Breusch-Pagan test for heteroskedasticity would be a good idea. (Study Session 3, LOS 12.i)

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Williams decides to use two-tailed tests on the individual variables, at a 5% level of significance, to determine whether electric generator sales are explained by each of them individually. Williams concludes that:

A) all of the variables explain sales.

B) all of the variables except snowfall explain sales.

C) all of the variables except snowfall and housing starts explain sales.

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The calculated t–statistics are:

Heating Oil: (2.00 / 0.827) = 2.4184 Low Temperature: (3.00 / 1.200) = 2.5000 Snowfall: (10.00 / 4.833) = 2.0691 Housing Starts: (5.00 / 2.333) = 2.1432

All of these values are outside the t–critical value (at (26 − 4 − 1) = 21 degrees of freedom) of 2.080, except the change in snowfall. So Williams should reject the null hypothesis for the other variables and conclude that they explain sales, but fail to reject the null hypothesis with respect to snowfall and conclude that increases or decreases in snowfall do not explain sales. (Study Session 3, LOS 12.b)

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When Williams ran the model, the computer said the R2 is 0.233. She examines the other output and concludes that this is the:

A) neither the unadjusted nor adjusted R2 value, nor the coefficient of correlation.

B) adjusted R2 value.

C) unadjusted R2 value.

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This can be answered by recognizing that the unadjusted R-square is (335.2 / 941.6) = 0.356. Thus, the reported value must be the adjusted R2. To verify this we see that the adjusted R-squared is: 1− ((26 − 1) / (26 − 4 − 1)) × (1 − 0.356) = 0.233. Note that whenever there is more than one independent variable, the adjusted R2 will always be less than R2. (Study Session 3, LOS 12.f)

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In preparing and using this model, Williams has least likely relied on which of the following assumptions?

A) There is a linear relationship between the independent variables.

B) A linear relationship exists between the dependent and independent variables.

C) The disturbance or error term is normally distributed.

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Multiple regression models assume that there is no linear relationship between two or more of the independent variables. The other answer choices are both assumptions of multiple regression. (Study Session 3, LOS 12.d)

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作者: bapswarrior    时间: 2012-3-27 10:21

Manuel Mercado, CFA has performed the following two regressions on sales data for a given industry. He wants to forecast sales for each quarter of the upcoming year.

Model ONE
Regression Statistics
Multiple R	0.941828
R2	0.887039
Adjusted R2	0.863258
Standard Error	2.543272
Observations	24

Durbin-Watson test statistic = 0.7856

ANOVA
	df	SS	MS	F	Significance F
Regression	4	965.0619	241.2655	37.30006	9.49E−09
Residual	19	122.8964	6.4682
Total	23	1087.9583

	Coefficients	Standard Error	t-Statistic
Intercept	31.40833	1.4866	21.12763
Q1	−3.77798	1.485952	−2.54246
Q2	−2.46310	1.476204	−1.66853
Q3	−0.14821	1.470324	−0.10080
TREND	0.851786	0.075335	11.20848

Model TWO
Regression Statistics
Multiple R	0.941796
R2	0.886979
Adjusted R2	0.870026
Standard Error	2.479538
Observations	24

Durbin-Watson test statistic = 0.7860

	df	SS	MS	F	Significance F
Regression	3	964.9962	321.6654	52.3194	1.19E−09
Residual	20	122.9622	6.14811
Total	23	1087.9584

	Coefficients	Standard Error	t-Statistic
Intercept	31.32888	1.228865	25.49416
Q1	−3.70288	1.253493	−2.95405
Q2	−2.38839	1.244727	−1.91881
TREND	0.85218	0.073991	11.51732

The dependent variable is the level of sales for each quarter, in $ millions, which began with the first quarter of the first year. Q1, Q2, and Q3 are seasonal dummy variables representing each quarter of the year. For the first four observations the dummy variables are as follows: Q11,0,0,0), Q20,1,0,0), Q30,0,1,0). The TREND is a series that begins with one and increases by one each period to end with 24. For all tests, Mercado will use a 5% level of significance. Tests of coefficients will be two-tailed, and all others are one-tailed.Which model would be a better choice for making a forecast?

A) Model TWO because serial correlation is not a problem.

B) Model ONE because it has a higher R2.

C) Model TWO because it has a higher adjusted R2.

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Model TWO has a higher adjusted R2 and thus would produce the more reliable estimates. As is always the case when a variable is removed, R2 for Model TWO is lower. The increase in adjusted R2 indicates that the removed variable, Q3, has very little explanatory power, and removing it should improve the accuracy of the estimates. With respect to the references to autocorrelation, we can compare the Durbin-Watson statistics to the critical values on a Durbin-Watson table. Since the critical DW statistics for Model ONE and TWO respectively are 1.01 (>0.7856) and 1.10 (>0.7860), serial correlation is a problem for both equations. (Study Session 3, LOS 12.f)

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Using Model 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.

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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)

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Which of the coefficients that appear in both models 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.

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

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If it is determined that conditional heteroskedasticity is present in model one, which of the following inferences are most accurate?

A) Regression coefficients will be biased but standard errors will be unbiased.

B) Both the regression coefficients and the standard errors will be biased.

C) Regression coefficients will be unbiased but standard errors will be biased.

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Presence of conditional heteroskedasticity will not affect the consistency of regression coefficients but will bias the standard errors leading to incorrect application of t-tests for statistical significance of regression parameters. (Study Session 3, LOS 12.i)

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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.

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The fourth quarter serves as the base quarter, and for the fourth quarter, Q1 = Q2 = Q3 = 0. Had the model included a Q4 as specified, we could not have had an intercept. In that case, for Model 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 the Model ONE. In a model that included Q1, Q2, Q3, and Q4 but no intercept, for example:

Q1 = 31.40833 + (−3.77798) = 27.63035

Such a model would produce the same estimated values for the dependent variable. (Study Session 3, LOS 12.h)

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If Mercado determines that Model 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, but by less than $1,000,000.

C) grow by more than $1,000,000.

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The specification of Model 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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作者: bapswarrior    时间: 2012-3-27 10:22

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

A) The adjusted-R2 is greater than the R2 in multiple regression.

B) The adjusted-R2 not appropriate to use in simple regression.

C) It is possible for the adjusted-R2 to decline as more variables are added to the multiple regression.

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

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作者: bapswarrior    时间: 2012-3-27 10:24

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.

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

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作者: bapswarrior    时间: 2012-3-27 10:24

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 will be higher, but the adjusted R2 and F-statistic could be higher or lower.

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

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

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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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作者: bapswarrior    时间: 2012-3-27 10:25

An analyst regresses the return of a S&P 500 index fund against the S&P 500, and also regresses the return of an active manager against the S&P 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.

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The index fund regression should provide a higher R2 than the active manager regression. R2 is the sum of squares regression divided by the total sum of squares.

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作者: bapswarrior    时间: 2012-3-27 10:25

May Jones estimated a regression that produced the following analysis of variance (ANOVA) table:

Source	Sum of squares	Degrees of freedom	Mean square
Regression	20	1	20
Error	80	40	2
Total	100	41

The values of R2 and the F-statistic for the fit of the model are:

A) R2 = 0.25 and F = 0.909.

B) R2 = 0.20 and F = 10.

C) R2 = 0.25 and F = 10.

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R2 = RSS / SST = 20 / 100 = 0.20
The F-statistic is equal to the ratio of the mean squared regression to the mean squared error.F = 20 / 2 = 10

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作者: bapswarrior    时间: 2012-3-27 10:26

Wilson estimated a regression that produced the following analysis of variance (ANOVA) table:

Source	Sum of squares	Degrees of freedom	Mean square
Regression	100	1	100.0
Error	300	40	7.5
Total	400	41

The values of R2 and the F-statistic for the fit of the model are:

A) R2 = 0.25 and F = 13.333.

B) R2 = 0.20 and F = 13.333.

C) R2 = 0.25 and F = 0.930.

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R2 = RSS / SST = 100 / 400 = 0.25
The F-statistic is equal to the ratio of the mean squared regression to the mean squared error.
F = 100 / 7.5 = 13.333

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作者: bapswarrior    时间: 2012-3-27 10:27

Which of the following statements regarding the analysis of variance (ANOVA) table is least accurate? The:

A) F-statistic is the ratio of the mean square regression to the mean square error.

B) standard error of the estimate is the square root of the mean square error.

C) F-statistic cannot be computed with the data offered in the ANOVA table.

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The F-statistic can be calculated using an ANOVA table. The F-statistic is MSR/MSE.

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作者: bapswarrior    时间: 2012-3-27 10:29

The F-statistic is the ratio of the mean square regression to the mean square error. The mean squares are provided directly in the analysis of variance (ANOVA) table. Which of the following statements regarding the ANOVA table for a regression is CORRECT?

A) If the F-statistic is less than its critical value, we can reject the null hypothesis that all coefficients are equal to zero.

B) R2 = SSRegression / SSTotal.

C) R2 = SSError / SSTotal.

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The coefficient of determination is the proportion of the total variation of the dependent variable that is explained by the independent variables.

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作者: bapswarrior    时间: 2012-3-27 10:32

An analyst is trying to determine whether stock market returns are related to size and the market-to-book ratio, through the use of multiple regression. However, the analyst uses returns of portfolios of stocks instead of individual stocks in the regression. Which of the following is a valid reason why the analyst uses portfolios? The use of portfolios:

A) will increase the power of the test by giving the test statistic more degrees of freedom.

B) reduces the standard deviation of the residual, which will increase the power of the test.

C) will remove the existence of multicollinearity from the data, reducing the likelihood of type II error.

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The use of portfolios reduces the standard deviation of the returns, which reduces the standard deviation of the residuals.

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作者: bapswarrior    时间: 2012-3-27 10:34

Lynn Carter, CFA, is an analyst in the research department for Smith Brothers in New York. She follows several industries, as well as the top companies in each industry. She provides research materials for both the equity traders for Smith Brothers as well as their retail customers. She routinely performs regression analysis on those companies that she follows to identify any emerging trends that could affect investment decisions.
Due to recent layoffs at the company, there has been some consolidation in the research department. Two research analysts have been laid off, and their workload will now be distributed among the remaining four analysts. In addition to her current workload, Carter will now be responsible for providing research on the airline industry. Pinnacle Airlines, a leader in the industry, represents a large holding in Smith Brothers’ portfolio. Looking back over past research on Pinnacle, Carter recognizes that the company historically has been a strong performer in what is considered to be a very competitive industry. The stock price over the last 52-week period has outperformed that of other industry leaders, although Pinnacle’s net income has remained flat. Carter wonders if the stock price of Pinnacle has become overvalued relative to its peer group in the market, and wants to determine if the timing is right for Smith Brothers to decrease its position in Pinnacle.  
Carter decides to run a regression analysis, using the monthly returns of Pinnacle stock and airlines industry.

Analysis of Variance Table (ANOVA)
Source	df (Degrees of Freedom)	SS (Sum of Squares)	Mean Square (SS/df)
Regression	1	3,257 (RSS)	3,257 (MSR)
Error	8	298 (SSE)	37.25 (MSE)
Total	9	3,555 (SS Total)

Which of the following are least likely to be major assumptions regarding linear regression?

A) The independent variable is correlated with the residuals.

B) A linear relationship exists between the dependent and independent variables.

C) The variance of the residual term is constant.

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Although the linear regression model is fairly insensitive to minor deviations from any of these assumptions, the independent variable is typically uncorrelated with the residuals. (Study Session 3, LOS 11.d)

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Carter wants to test the strength of the relationship between the two variables. She calculates a correlation coefficient of 0.72. This means that the two variables:

A) are perfectly correlated.

B) have no linear relationship.

C) have a positive linear relationship.

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If the correlation coefficient (r) is greater that 0 and less than 1, then the two variables are said to be positively correlated. (Study Session 3, LOS 11.a)

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Based upon the information presented in the ANOVA table, what is the standard error of the estimate?

A) 6.10.

B) 57.07.

C) 37.25.

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The standard error of the estimate (SEE) measures the “fit” of the regression line, and the smaller the standard error, the better the fit. The SSE can be calculated as √(MSE) = √(SSE / (n − 2) = √(298 / 8) = 6.10. (Study Session 3, LOS 12.g)

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Based upon the information presented in the ANOVA table, what is the coefficient of determination?

A) 0.916, indicating the variability of company returns explains about 91.6% of the variability of industry returns.

B) 0.084, indicating that the variability of industry returns explains about 8.4% of the variability of company returns.

C) 0.916, indicating that the variability of industry returns explains about 91.6% of the variability of company returns.

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The coefficient of determination (R2) is the percentage of the total variation in the dependent variable explained by the independent variable.
The R2 = (RSS / SS) Total = (3,257 / 3,555) = 0.916. This means that the variation of independent variable (the airline industry) explains 91.6% of the variations in the dependent variable (Pinnacle stock). (Study Session 3, LOS 12.g)

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Based upon her analysis, Carter has derived the following regression equation: Ŷ = 1.75 + 3.25X1. The predicted value of the Y variable equals 50.50, if the:

A) predicted value of the independent variable equals 15.

B) predicted value of the dependent variable equals 15.

C) coefficient of the determination equals 15.

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Note that the easiest way to answer this question is to plug numbers into the equation.
The predicted value for Y = 1.75 + 3.25(15) = 50.50.
The variable X1 represents the independent variable. (Study Session 3, LOS 13.a)

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Carter realizes that although regression analysis is a useful tool when analyzing investments, there are limitations. Carter made a list of points describing limitations that Smith Brothers equity traders should be aware of when applying her research to their investment decisions.

• Point 1: Data derived from regression analysis may be homoskedastic.
• Point 2: Data from regression relationships tends to exhibit parameter instability.
• Point 3: Results of regression analysis may exhibit autocorrelation.
• Point 4: The variance of the error term changes over time.

When reviewing Carter’s list, one of the Smith Brothers’ equity traders points out that not all of the points describe regression analysis limitations. Which of Carter’s points most accurately describes the limitations to regression analysis?

A) Points 2, 3, and 4.

B) Points 1, 2, and 3.

C) Points 1, 3, and 4.

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One of the basis assumptions of regression analysis is that the variance of the error terms is constant, or homoskedastic. Any violation of this assumption is called heteroskedasticity. Therefore, Point 1 is incorrect, but Point 4 is correct. Points 2 and 3 also describe limitations of regression analysis. (Study Session 3, LOS 11.j)

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作者: bapswarrior    时间: 2012-3-27 10:35

The management of a large restaurant chain believes that revenue growth is dependent upon the month of the year. Using a standard 12 month calendar, how many dummy variables must be used in a regression model that will test whether revenue growth differs by month?

A) 13.

B) 11.

C) 12.

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The appropriate number of dummy variables is one less than the number of categories because the intercept captures the effect of the other effect. With 12 categories (months) the appropriate number of dummy variables is 11 = 12 – 1. If the number of dummy variables equals the number of categories, it is possible to state any one of the independent dummy variables in terms of the others. This is a violation of the assumption of the multiple linear regression model that none of the independent variables are linearly related

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作者: bapswarrior    时间: 2012-3-27 10:37

A fund has changed managers twice during the past 10 years. An analyst wishes to measure whether either of the changes in managers has had an impact on performance. The analyst wishes to simultaneously measure the impact of risk on the fund’s return. R is the return on the fund, and M is the return on a market index. Which of the following regression equations can appropriately measure the desired impacts?

A) R = a + bM + c1D1 + c2D2 + c3D3 + ε, where D1 = 1 if the return is from the first manager, and D2 = 1 if the return is from the second manager, and D3 = 1 is the return is from the third manager.

B) The desired impact cannot be measured.

C) R = a + bM + c1D1 + c2D2 + ε, where D1 = 1 if the return is from the first manager, and D2 = 1 if the return is from the third manager.

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The effect needs to be measured by two distinct dummy variables. The use of three variables will cause collinearity, and the use of one dummy variable will not appropriately specify the manager impact.

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作者: bapswarrior    时间: 2012-3-27 10:37

Jill Wentraub is an analyst with the retail industry. She is modeling a company’s sales over time and has noticed a quarterly seasonal pattern. If she includes dummy variables to represent the seasonality component of the sales she must use:

A) three dummy variables.

B) one dummy variables.

C) four dummy variables.

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Three. Always use one less dummy variable than the number of possibilities. For a seasonality that varies by quarters in the year, three dummy variables are needed.

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作者: bapswarrior    时间: 2012-3-27 10:38

Consider the following model of earnings (EPS) regressed against dummy variables for the quarters:

EPSt = α + β1Q1t + β2Q2t + β3Q3t
where:
EPSt is a quarterly observation of earnings per share
Q1t takes on a value of 1 if period t is the second quarter, 0 otherwise
Q2t takes on a value of 1 if period t is the third quarter, 0 otherwise
Q3t takes on a value of 1 if period t is the fourth quarter, 0 otherwise

Which of the following statements regarding this model is most accurate? The:

A) EPS for the first quarter is represented by the residual.

B) significance of the coefficients cannot be interpreted in the case of dummy variables.

C) coefficient on each dummy tells us about the difference in earnings per share between the respective quarter and the one left out (first quarter in this case).

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The coefficients on the dummy variables indicate the difference in EPS for a given quarter, relative to the first quarter.

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作者: bapswarrior    时间: 2012-3-27 10:39

An analyst wishes to test whether the stock returns of two portfolio managers provide different average returns. The analyst believes that the portfolio managers’ returns are related to other factors as well. Which of the following can provide a suitable test?

A) Paired-comparisons.

B) Difference of means.

C) Dummy variable regression.

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The difference of means and paired-comparisons tests will not account for the other factors

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作者: bapswarrior    时间: 2012-3-27 10:40

An analyst is trying to determine whether fund return performance is persistent. The analyst divides funds into three groups based on whether their return performance was in the top third (group 1), middle third (group 2), or bottom third (group 3) during the previous year. The manager then creates the following equation: R = a + b1D1 + b2D2 + b3D3 + ε, where R is return premium on the fund (the return minus the return on the S&P 500 benchmark) and Di is equal to 1 if the fund is in group i. Assuming no other information, this equation will suffer from:

A) heteroskedasticity.

B) serial correlation.

C) multicollinearity.

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When we use dummy variables, we have to use one less than the states of the world. In this case, there are three states (groups) possible. We should have used only two dummy variables. Multicollinearity is a problem in this case. Specifically, a linear combination of independent variables is perfectly correlated. X1 + X2 + X3 = 1.
There are too many dummy variables specified, so the equation will suffer from multicollinearity

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作者: bapswarrior    时间: 2012-3-27 10:40

Suppose the analyst wants to add a dummy variable for whether a person has an undergraduate college degree and a graduate degree. What is the CORRECT representation if a person has both degrees?

Undergraduate Degree Dummy Variable

Graduate Degree Dummy Variable

A) 1 1

B) 0 0

C) 0 1

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Assigning a zero to both categories is appropriate for someone with neither degree. Assigning one to the undergraduate category and zero to the graduate category is appropriate for someone with only an undergraduate degree. Assigning zero to the undergraduate category and one to the graduate category is appropriate for someone with only a graduate degree. Assigning a one to both categories is correct since it reflects the possession of both degrees.

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作者: bapswarrior    时间: 2012-3-27 10:41

The amount of the State of Florida’s total revenue that is allocated to the education budget is believed to be dependent upon the total revenue for the year and the political party that controls the state legislature. Which of the following regression models is most appropriate for capturing the effect of the political party on the education budget? Assume Yt is the amount of the education budget for Florida in year t, X is Florida’s total revenue in year t, and Dt = {1 if the legislature has a Democratic majority in year t, 0 otherwise}.

A) Yt = b1Dt + b2Xt + et.

B) Yt = b0 + b1Dt + b2Xt + et.

C) Yt = b0 + b1Dt + et.

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In this application, b0, b1, and b2 are estimated by regressing Yt against a constant, Dt, and Xt.The estimated relationships for the two parties are:

Non-Democrats: Ŷ = b0 + b2Xt
Democrats: Ŷ = (b0 + b1) + b2Xt

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作者: bapswarrior    时间: 2012-3-27 10:43

Raul Gloucester, CFA, is analyzing the returns of a fund that his company offers. He tests the fund’s sensitivity to a small capitalization index and a large capitalization index, as well as to whether the January effect plays a role in the fund’s performance. He uses two years of monthly returns data, and runs a regression of the fund’s return on the indexes and a January-effect qualitative variable. The “January” variable is 1 for the month of January and zero for all other months. The results of the regression are shown in the tables below.

Regression Statistics
Multiple R	0.817088
R2	0.667632
Adjusted R2	0.617777
Standard Error	1.655891
Observations	24

ANOVA
	df	SS	MS
Regression	3	110.1568	36.71895
Residual	20	54.8395	2.741975
Total	23	164.9963

	Coefficients	Standard Error	t-Statistic
Intercept	-0.23821	0.388717	-0.61282
January	2.560552	1.232634	2.077301
Small Cap Index	0.231349	0.123007	1.880778
Large Cap Index	0.951515	0.254528	3.738359

Gloucester will perform an F-test for the equation. He also plans to test for serial correlation and conditional and unconditional heteroskedasticity.
Jason Brown, CFA, is interested in Gloucester’s results. He speculates that they are economically significant in that excess returns could be earned by shorting the large capitalization and the small capitalization indexes in the month of January and using the proceeds to buy the fund. The percent of the variation in the fund’s return that is explained by the regression is:

A) 66.76%.

B) 81.71%.

C) 61.78%.

---

The R2 tells us how much of the change in the dependent variable is explained by the changes in the independent variables in the regression: 0.667632.

---

In a two-tailed test at a five percent level of significance, the coefficients that are significant are:

A) the large cap index only.

B) the January effect and the small capitalization index only.

C) the January effect and the large capitalization index only.

---

For a two-tailed test with 20 = 24 – 3 – 1 degrees of freedom and a five percent level of significance, the critical t-statistic is 2.086. Only the coefficient for the large capitalization index has a t-statistic larger than this.

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Which of the following best summarizes the results of an F-test (5 percent significance) for the regression? The F-statistic is:

A) 13.39 and the critical value is 3.10.

B) 9.05 and the critical value is 3.86.

C) 13.39 and the critical value is 3.86.

---

The F-statistic is the ratio of the Mean Square of the Regression divided by the Mean Square Error (Residual): 13.39 = 36.718946 / 2.74197510. The F-statistic has 3 and 20 degrees of freedom, so the critical value, at a 5 percent level of significance = 3.10.

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The best test for unconditional heteroskedasticity is:

A) the Durbin-Watson test only.

B) neither the Durbin-Watson test nor the Breusch-Pagan test.

C) the Breusch-Pagan test only.

---

The Durbin-Watson test is for serial correlation. The Breusch-Pagan test is for conditional heteroskedasticity; it tests to see if the size of the independent variables influences the size of the residuals. Although tests for unconditional heteroskedasticity exist, they are not part of the CFA curriculum, and unconditional heteroskedasticity is generally considered less serious than conditional heteroskedasticity.

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In the month of January, if both the small and large capitalization index have a zero return, we would expect the fund to have a return equal to:

A) 2.799.

B) 2.322.

C) 2.561.

---

The forecast of the return of the fund would be the intercept plus the coefficient on the January effect: 2.322 = -0.238214 + 2.560552.

---

Assuming (for this question only) that the F-test was significant but that the t-tests of the independent variables were insignificant, this would most likely suggest:

A) serial correlation.

B) conditional heteroskedasticity.

C) multicollinearity.

---

When the F-test and the t-tests conflict, multicollinearity is indicated.

---

作者: bapswarrior    时间: 2012-3-27 10:46

John Rains, CFA, is a professor of finance at a large university located in the Eastern United States. He is actively involved with his local chapter of the Society of Financial Analysts. Recently, he was asked to teach one session of a Society-sponsored CFA review course, specifically teaching the class addressing the topic of quantitative analysis. Based upon his familiarity with the CFA exam, he decides that the first part of the session should be a review of the basic elements of quantitative analysis, such as hypothesis testing, regression and multiple regression analysis. He would like to devote the second half of the review session to the practical application of the topics he covered in the first half.
Rains decides to construct a sample regression analysis case study for his students in order to demonstrate a “real-life” application of the concepts. He begins by compiling financial information on a fictitious company called Big Rig, Inc. According to the case study, Big Rig is the primary producer of the equipment used in the exploration for and drilling of new oil and gas wells in the United States. Rains has based the information in the problem on an actual equity holding in his personal portfolio, but has simplified the data for the purposes of the review course.
Rains constructs a basic regression model for Big Rig in order to estimate its profitability (in millions), using two independent variables: the number of new wells drilled in the U.S. (WLS) and the number of new competitors (COMP) entering the market:

Profits = b0 + b1WLS – b2COMP + ε

Based on the model, the estimated regression equation is:

Profits = 22.5 + 0.98(WLS) − 0.35(COMP)

Using the past 5 years of quarterly data, he calculated the following regression estimates for Big Rig, Inc:

	Coefficient	Standard Error
Intercept	22.5	2.465
WLS	0.98	0.683
COMP	0.35	0.186

Using the information presented, the t-statistic for the number of new competitors (COMP) coefficient is:

A) 1.435.

B) 1.882.

C) 9.128.

---

To test whether a coefficient is statistically significant, the null hypothesis is that the slope coefficient is zero. The t-statistic for the COMP coefficient is calculated as follows:
(0.35 – 0.0) / 0.186 = 1.882
(Study Session 3, LOS 11.g)

---

Rains asks his students to test the null hypothesis that states for every new well drilled, profits will be increased by the given multiple of the coefficient, all other factors remaining constant. The appropriate hypotheses for this two-tailed test can best be stated as:

A) H0: b1 ≤ 0.98 versus Ha: b1 > 0.98.

B) H0: b1 = 0.35 versus Ha: b1 ≠ 0.35.

C) H0: b1 = 0.98 versus Ha: b1 ≠ 0.98.

---

The coefficient given in the above table for the number of new wells drilled (WLS) is 0.98. The hypothesis should test to see whether the coefficient is indeed equal to 0.98 or is equal to some other value. Note that hypotheses with the “greater than” or “less than” symbol are used with one-tailed tests. (Study Session 3, LOS 11.g)

---

Continuing with the analysis of Big Rig, Rains asks his students to calculate the mean squared error(MSE). Assume that the sum of squared errors (SSE) for the regression model is 359.

A) 21.118.

B) 17.956.

C) 18.896.

---

The MSE is calculated as SSE / (n – k – 1). Recall that there are twenty observations and two independent variables. Therefore, the SEE in this instance = 359 / (20 – 2 − 1) = 21.118. (Study Session 3, LOS 11.i)

---

Rains now wants to test the students’ knowledge of the use of the F-test and the interpretation of the F-statistic. Which of the following statements regarding the F-test and the F-statistic is the most correct?

A) The F-test is usually formulated as a two-tailed test.

B) The F-statistic is used to test whether at least one independent variable in a set of independent variables explains a significant portion of the variation of the dependent variable.

C) The F-statistic is almost always formulated to test each independent variable separately, in order to identify which variable is the most statistically significant.

---

An F-test assesses how well a set of independent variables, as a group, explains the variation in the dependent variable. It tests all independent variables as a group, and is always a one-tailed test. The decision rule is to reject the null hypothesis if the calculated F-value is greater than the critical F-value. (Study Session 3, LOS 11.i)

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One of the main assumptions of a multiple regression model is that the variance of the residuals is constant across all observations in the sample. A violation of the assumption is known as:

A) robust standard errors.

B) heteroskedasticity.

C) positive serial correlation.

---

Heteroskedasticity is present when the variance of the residuals is not the same across all observations in the sample, and there are sub-samples that are more spread out than the rest of the sample. (Study Session 3, LOS 12.i)

---

Rains reminds his students that a common condition that can distort the results of a regression analysis is referred to as serial correlation. The presence of serial correlation can be detected through the use of:

A) the Breusch-Pagen test.

B) the Hansen method.

C) the Durbin-Watson statistic.

---

The Durbin-Watson test (DW ≈ 2(1 − r)) can detect serial correlation. Another commonly used method is to visually inspect a scatter plot of residuals over time. The Hansen method does not detect serial correlation, but can be used to remedy the situation. Note that the Breusch-Pagen test is used to detect heteroskedasticity. (Study Session 3, LOS 12.i)

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作者: bapswarrior    时间: 2012-3-27 10:48

An analyst is trying to estimate the beta for a fund. The analyst estimates a regression equation in which the fund returns are the dependent variable and the Wilshire 5000 is the independent variable, using monthly data over the past five years. The analyst finds that the correlation between the square of the residuals of the regression and the Wilshire 5000 is 0.2. Which of the following is most accurate, assuming a 0.05 level of significance? There is:

A) evidence of serial correlation but not conditional heteroskedasticity in the regression equation.

B) evidence of conditional heteroskedasticity but not serial correlation in the regression equation.

C) no evidence that there is conditional heteroskedasticity or serial correlation in the regression equation.

---

The test for conditional heteroskedasticity involves regressing the square of the residuals on the independent variables of the regression and creating a test statistic that is n × R2, where n is the number of observations and R2 is from the squared-residual regression. The test statistic is distributed with a chi-squared distribution with the number of degrees of freedom equal to the number of independent variables. For a single variable, the R2 will be equal to the square of the correlation; so in this case, the test statistic is 60 × 0.22 = 2.4, which is less than the chi-squared value (with one degree of freedom) of 3.84 for a p-value of 0.05. There is no indication about serial correlation.

---

作者: bapswarrior    时间: 2012-3-27 10:49

Which of the following is least likely a method used to detect heteroskedasticity?

A) Durbin-Watson test.

B) Test of the variances.

C) Breusch-Pagan test.

---

The Durbin-Watson test is used to detect serial correlation. The Breusch-Pagan test is used to detect heteroskedasticity.

---

作者: bapswarrior    时间: 2012-3-27 10:50

Consider the following graph of residuals and the regression line from a time-series regression:

These residuals exhibit the regression problem of:<

A) autocorrelation.

B) homoskedasticity.

C) heteroskedasticity.

---

The residuals appear to be from two different distributions over time. In the earlier periods, the model fits rather well compared to the later periods.

---

作者: bapswarrior    时间: 2012-3-27 10:51

Which of the following statements regarding heteroskedasticity is least accurate?

A) Multicollinearity is a potential problem only in multiple regressions, not simple regressions.

B) The presence of heteroskedastic error terms results in a variance of the residuals that is too large.

C) Heteroskedasticity only occurs in cross-sectional regressions.

---

If there are shifting regimes in a time-series (e.g., change in regulation, economic environment), it is possible to have heteroskedasticity in a time-series.

---

作者: bapswarrior    时间: 2012-3-27 10:52

Which of the following statements regarding heteroskedasticity is least accurate?

A) The assumption of linear regression is that the residuals are heteroskedastic.

B) Heteroskedasticity results in an estimated variance that is too large and, therefore, affects statistical inference.

C) Heteroskedasticity may occur in cross-section or time-series analyses.

---

The assumption of regression is that the residuals are homoskedastic (i.e., the residuals are drawn from the same distribution).

---

作者: bapswarrior    时间: 2012-3-27 10:53

Which of the following conditions will least likely affect the statistical inference about regression parameters by itself?

A) Multicollinearity.

B) Conditional heteroskedasticity.

C) Unconditional heteroskedasticity.

---

Unconditional heteroskedasticity does not impact the statistical inference concerning the parameters.

---

作者: bapswarrior    时间: 2012-3-27 10:54

George Smith, an analyst with Great Lakes Investments, has created a comprehensive report on the pharmaceutical industry at the request of his boss. The Great Lakes portfolio currently has a significant exposure to the pharmaceuticals industry through its large equity position in the top two pharmaceutical manufacturers. His boss requested that Smith determine a way to accurately forecast pharmaceutical sales in order for Great Lakes to identify further investment opportunities in the industry as well as to minimize their exposure to downturns in the market. Smith realized that there are many factors that could possibly have an impact on sales, and he must identify a method that can quantify their effect. Smith used a multiple regression analysis with five independent variables to predict industry sales. His goal is to not only identify relationships that are statistically significant, but economically significant as well. The assumptions of his model are fairly standard: a linear relationship exists between the dependent and independent variables, the independent variables are not random, and the expected value of the error term is zero.
Smith is confident with the results presented in his report. He has already done some hypothesis testing for statistical significance, including calculating a t-statistic and conducting a two-tailed test where the null hypothesis is that the regression coefficient is equal to zero versus the alternative that it is not. He feels that he has done a thorough job on the report and is ready to answer any questions posed by his boss.
However, Smith’s boss, John Sutter, is concerned that in his analysis, Smith has ignored several potential problems with the regression model that may affect his conclusions. He knows that when any of the basic assumptions of a regression model are violated, any results drawn for the model are questionable. He asks Smith to go back and carefully examine the effects of heteroskedasticity, multicollinearity, and serial correlation on his model. In specific, he wants Smith to make suggestions regarding how to detect these errors and to correct problems that he encounters. Suppose that there is evidence that the residual terms in the regression are positively correlated. The most likely effect on the statistical inferences drawn from the regressions results is for Smith to commit a:

A) Type I error by incorrectly rejecting the null hypotheses that the regression parameters are equal to zero.

B) Type II error by incorrectly failing to reject the null hypothesis that the regression parameters are equal to zero.

C) Type I error by incorrectly failing to reject the null hypothesis that the regression parameters are equal to zero.

---

One problem with positive autocorrelation (also known as positive serial correlation) is that the standard errors of the parameter estimates will be too small and the t-statistics too large. This may lead Smith to incorrectly reject the null hypothesis that the parameters are equal to zero. In other words, Smith will incorrectly conclude that the parameters are statistically significant when in fact they are not. This is an example of a Type I error: incorrectly rejecting the null hypothesis when it should not be rejected. (Study Session 3, LOS 12.i)

---

Sutter has detected the presence of conditional heteroskedasticity in Smith’s report. This is evidence that:

A) two or more of the independent variables are highly correlated with each other.

B) the variance of the error term is correlated with the values of the independent variables.

C) the error terms are correlated with each other.

---

Conditional heteroskedasticity exists when the variance of the error term is correlated with the values of the independent variables.
Multicollinearity, on the other hand, occurs when two or more of the independent variables are highly correlated with each other. Serial correlation exists when the error terms are correlated with each other. (Study Session 3, LOS 12.i)

---

Suppose there is evidence that the variance of the error term is correlated with the values of the independent variables. The most likely effect on the statistical inferences Smith can make from the regressions results is to commit a:

A) Type II error by incorrectly failing to reject the null hypothesis that the regression parameters are equal to zero.

B) Type I error by incorrectly rejecting the null hypotheses that the regression parameters are equal to zero.

C) Type I error by incorrectly failing to reject the null hypothesis that the regression parameters are equal to zero.

---

One problem with heteroskedasticity is that the standard errors of the parameter estimates will be too small and the t-statistics too large. This will lead Smith to incorrectly reject the null hypothesis that the parameters are equal to zero. In other words, Smith will incorrectly conclude that the parameters are statistically significant when in fact they are not. This is an example of a Type I error: incorrectly rejecting the null hypothesis when it should not be rejected. (Study Session 3, LOS 12.i)

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Which of the following is most likely to indicate that two or more of the independent variables, or linear combinations of independent variables, may be highly correlated with each other? Unless otherwise noted, significant and insignificant mean significantly different from zero and not significantly different from zero, respectively.

A) The R2 is low, the F-statistic is insignificant and the Durbin-Watson statistic is significant.

B) The R2 is high, the F-statistic is significant and the t-statistics on the individual slope coefficients are insignificant.

C) The R2 is high, the F-statistic is significant and the t-statistics on the individual slope coefficients are significant.

---

Multicollinearity occurs when two or more of the independent variables, or linear combinations of independent variables, may be highly correlated with each other. In a classic effect of multicollinearity, the R2 is high and the F-statistic is significant, but the t-statistics on the individual slope coefficients are insignificant. (Study Session 3, LOS 12.j)

---

Suppose there is evidence that two or more of the independent variables, or linear combinations of independent variables, may be highly correlated with each other. The most likely effect on the statistical inferences Smith can make from the regression results is to commit a:

A) Type I error by incorrectly rejecting the null hypothesis that the regression parameters are equal to zero.

B) Type I error by incorrectly failing to reject the null hypothesis that the regression parameters are equal to zero.

C) Type II error by incorrectly failing to reject the null hypothesis that the regression parameters are equal to zero.

---

One problem with multicollinearity is that the standard errors of the parameter estimates will be too large and the t-statistics too small. This will lead Smith to incorrectly fail to reject the null hypothesis that the parameters are statistically insignificant. In other words, Smith will incorrectly conclude that the parameters are not statistically significant when in fact they are. This is an example of a Type II error: incorrectly failing to reject the null hypothesis when it should be rejected. (Study Session 3, LOS 12.j)

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Using the Durbin-Watson test statistic, Smith rejects the null hypothesis suggested by the test. This is evidence that:

A) the error terms are correlated with each other.

B) two or more of the independent variables are highly correlated with each other.

C) the error term is normally distributed.

---

Serial correlation (also called autocorrelation) exists when the error terms are correlated with each other.
Multicollinearity, on the other hand, occurs when two or more of the independent variables are highly correlated with each other. One assumption of multiple regression is that the error term is normally distributed. (Study Session 3, LOS 12.i)

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作者: bapswarrior    时间: 2012-3-27 11:00

An analyst is estimating whether a fund’s excess return for a quarter is related to interest rates and last quarter’s excess return. The regression equation is found to have unconditional heteroskedasticity and serial correlation. Which of the following is most accurate? Parameter estimates will be:

A) inaccurate and statistical inference about the parameters will not be valid.

B) accurate but statistical inference about the parameters will not be valid.

C) inaccurate but statistical inference about the parameters will be valid.

---

One of the independent variables is a lagged value of the dependent variable. This means that serial correlation will cause an inaccurate parameter estimate. Serial correlation always impacts the statistical inference about the parameters. Unconditional heteroskedasticity never impacts statistical inference or parameter accuracy.

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作者: bapswarrior    时间: 2012-3-27 11:01

During the course of a multiple regression analysis, an analyst has observed several items that she believes may render incorrect conclusions. For example, the coefficient standard errors are too small, although the estimated coefficients are accurate. She believes that these small standard error terms will result in the computed t-statistics being too big, resulting in too many Type I errors. The analyst has most likely observed which of the following assumption violations in her regression analysis?

A) Multicollinearity.

B) Homoskedasticity.

C) Positive serial correlation.

---

Positive serial correlation is the condition where a positive regression error in one time period increases the likelihood of having a positive regression error in the next time period. The residual terms are correlated with one another, leading to coefficient error terms that are too small.

---

作者: bapswarrior    时间: 2012-3-27 11:01

Which of the following is least likely a method of detecting serial correlations?

A) The Durbin-Watson test.

B) A scatter plot of the residuals over time.

C) The Breusch-Pagan test.

---

The Breusch-Pagan test is a test of the heteroskedasticity and not of serial correlation

---

作者: kasinkei    时间: 2012-3-27 11:15

Which of the following statements regarding serial correlation that might be encountered in regression analysis is least accurate?

A) Negative serial correlation causes a failure to reject the null hypothesis when it is actually false.

B) Serial correlation occurs least often with time series data.

C) Positive serial correlation typically has the same effect as heteroskedasticity.

---

Serial correlation, which is sometimes referred to as autocorrelation, occurs when the residual terms are correlated with one another, and is most frequently encountered with time series data.

---

作者: kasinkei    时间: 2012-3-27 11:16

Alex Wade, CFA, is analyzing the result of a regression analysis comparing the performance of gold stocks versus a broad equity market index. Wade believes that serial correlation may be present, and in order to prove his theory, should use which of the following methods to detect its presence?

A) The Durbin-Watson statistic.

B) The Breusch-Pagan test.

C) The Hansen method.

---

The Durbin-Watson statistic is the most commonly used method for the detection of serial correlation, although residual plots can also be utilized. For a large sample size, DW ≈ 2(1-r), where r is the correlation coefficient between residuals from one period and those from a previous period. The DW statistic is then compared to a table of DW statistics that gives upper and lower critical values for various sample sizes, levels of significance and numbers of degrees of freedom to detect the presence or absence of serial correlation.

---

作者: kasinkei    时间: 2012-3-27 11:17

An analyst is estimating whether a fund’s excess return for a month is dependent on interest rates and whether the S&P 500 has increased or decreased during the month. The analyst collects 90 monthly return premia (the return on the fund minus the return on the S&P 500 benchmark), 90 monthly interest rates, and 90 monthly S&P 500 index returns from July 1999 to December 2006. After estimating the regression equation, the analyst finds that the correlation between the regressions residuals from one period and the residuals from the previous period is 0.145. Which of the following is most accurate at a 0.05 level of significance, based solely on the information provided? The analyst:

A) cannot conclude that the regression exhibits either serial correlation or heteroskedasticity.

B) can conclude that the regression exhibits serial correlation, but cannot conclude that the regression exhibits heteroskedasticity.

C) can conclude that the regression exhibits heteroskedasticity, but cannot conclude that the regression exhibits serial correlation.

---

The Durbin-Watson statistic tests for serial correlation. For large samples, the Durbin-Watson statistic is equal to two multiplied by the difference between one and the sample correlation between the regressions residuals from one period and the residuals from the previous period, which is 2 × (1 − 0.145) = 1.71, which is higher than the upper Durbin-Watson value (with 2 variables and 90 observations) of 1.70. That means the hypothesis of no serial correlation cannot be rejected. There is no information on whether the regression exhibits heteroskedasticity.

---

作者: kasinkei    时间: 2012-3-27 11:17

An analyst is estimating whether a fund’s excess return for a month is dependent on interest rates and whether the S&P 500 has increased or decreased during the month. The analyst collects 90 monthly return premia (the return on the fund minus the return on the S&P 500 benchmark), 90 monthly interest rates, and 90 monthly S&P 500 index returns from July 1999 to December 2006. After estimating the regression equation, the analyst finds that the correlation between the regressions residuals from one period and the residuals from the previous period is 0.199. Which of the following is most accurate at a 0.05 level of significance, based solely on the information provided? The analyst:

A) can conclude that the regression exhibits serial correlation, but cannot conclude that the regression exhibits multicollinearity.

B) cannot conclude that the regression exhibits either serial correlation or multicollinearity.

C) can conclude that the regression exhibits multicollinearity, but cannot conclude that the regression exhibits serial correlation.

---

The Durbin-Watson statistic tests for serial correlation. For large samples, the Durbin-Watson statistic is approximately equal to two multiplied by the difference between one and the sample correlation between the regressions residuals from one period and the residuals from the previous period, which is 2 × (1 − 0.199) = 1.602, which is less than the lower Durbin-Watson value (with 2 variables and 90 observations) of 1.61. That means the hypothesis of no serial correlation is rejected. There is no information on whether the regression exhibits multicollinearity.

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作者: kasinkei    时间: 2012-3-27 11:18

Which of the following is least accurate regarding the Durbin-Watson (DW) test statistic?

A) If the residuals have negative serial correlation, the DW statistic will be greater than 2.

B) In tests of serial correlation using the DW statistic, there is a rejection region, a region over which the test can fail to reject the null, and an inconclusive region.

C) If the residuals have positive serial correlation, the DW statistic will be greater than 2.

---

A value of 2 indicates no correlation, a value greater than 2 indicates negative correlation, and a value less than 2 indicates a positive correlation. There is a range of values in which the DW test is inconclusive.

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作者: kasinkei    时间: 2012-3-27 11:21

Miles Mason, CFA, works for ABC Capital, a large money management company based in New York. Mason has several years of experience as a financial analyst, but is currently working in the marketing department developing materials to be used by ABC’s sales team for both existing and prospective clients. ABC Capital’s client base consists primarily of large net worth individuals and Fortune 500 companies. ABC invests its clients’ money in both publicly traded mutual funds as well as its own investment funds that are managed in-house. Five years ago, roughly half of its assets under management were invested in the publicly traded mutual funds, with the remaining half in the funds managed by ABC’s investment team. Currently, approximately 75% of ABC’s assets under management are invested in publicly traded funds, with the remaining 25% being distributed among ABC’s private funds. The managing partners at ABC would like to shift more of its client’s assets away from publicly-traded funds into ABC’s proprietary funds, ultimately returning to a 50/50 split of assets between publicly traded funds and ABC funds. There are three key reasons for this shift in the firm’s asset base. First, ABC’s in-house funds have outperformed other funds consistently for the past five years. Second, ABC can offer its clients a reduced fee structure on funds managed in-house relative to other publicly traded funds. Lastly, ABC has recently hired a top fund manager away from a competing investment company and would like to increase his assets under management.ABC Capital’s upper management requested that current clients be surveyed in order to determine the cause of the shift of assets away from ABC funds. Results of the survey indicated that clients feel there is a lack of information regarding ABC’s funds. Clients would like to see extensive information about ABC’s past performance, as well as a sensitivity analysis showing how the funds will perform in varying market scenarios. Mason is part of a team that has been charged by upper management to create a marketing program to present to both current and potential clients of ABC. He needs to be able to demonstrate a history of strong performance for the ABC funds, and, while not promising any measure of future performance, project possible return scenarios. He decides to conduct a regression analysis on all of ABC’s in-house funds. He is going to use 12 independent economic variables in order to predict each particular fund’s return. Mason is very aware of the many factors that could minimize the effectiveness of his regression model, and if any are present, he knows he must determine if any corrective actions are necessary. Mason is using a sample size of 121 monthly returns. In order to conduct an F-test, what would be the degrees of freedom used (dfnumerator; dfdenominator)?

A) 108; 12.

B) 11; 120.

C) 12; 108.

---

Degrees of freedom for the F-statistic is k for the numerator and n − k − 1 for the denominator.

k = 12n − k − 1 = 121 − 12 − 1 = 108

(Study Session 3, LOS 12.e)

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In regard to multiple regression analysis, which of the following statements is most accurate?

A) Adjusted R2 always decreases as independent variables increase.

B) Adjusted R2 is less than R2.

C) R2 is less than adjusted R2.

---

Whenever there is more than one independent variable, adjusted R2 is less than R2. Adding a new independent variable will increase R2, but may either increase or decrease adjusted R2.

R2 adjusted = 1 − [((n − 1) / (n − k − 1)) × (1 − R2)]Where:
n = number of observations
K = number of independent variables
R2 = unadjusted R2

(Study Session 3, LOS 12.f)

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Which of the following tests is used to detect autocorrelation?

A) Durbin-Watson.

B) Residual Plot.

C) Breusch-Pagan.

---

Durbin-Watson is used to detect autocorrelation. Breusch-Pagan and the residual plot are methods to detect heteroskedasticity. (Study Session 3, LOS 12.i)

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One of the most popular ways to correct heteroskedasticity is to:

A) adjust the standard errors.

B) use robust standard errors.

C) improve the specification of the model.

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Using generalized least squares and calculating robust standard errors are possible remedies for heteroskedasticity. Improving specifications remedies serial correlation. The standard error cannot be adjusted, only the coefficient of the standard errors. (Study Session 3, LOS 12.i)

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Which of the following statements regarding the Durbin-Watson statistic is most accurate? The Durbin-Watson statistic:

A) can only be used to detect positive serial correlation.

B) is approximately equal to 1 if the error terms are not serially correlated.

C) only uses error terms in its computations.

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The formula for the Durbin-Watson statistic uses error terms in its calculation. The Durbin-Watson statistic is approximately equal to 2 if there is no serial correlation. A Durbin-Watson statistic less than 2 indicates positive serial correlation, while a Durbin-Watson statistic greater then 2 indicates negative serial correlation. (Study Session 3, LOS 12.i)

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If a regression equation shows that no individual t-tests are significant, but the F-statistic is significant, the regression probably exhibits:

A) multicollinearity.

B) heteroskedasticity.

C) serial correlation.

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Common indicators of multicollinearity include: high correlation (>0.7) between independent variables, no individual t-tests are significant but the F-statistic is, and signs on the coefficients that are opposite of what is expected. (Study Session 3, LOS 12.j)

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作者: kasinkei    时间: 2012-3-27 11:22

An analyst is testing to see whether a dependent variable is related to three independent variables. He finds that two of the independent variables are correlated with each other, but that the correlation is spurious. Which of the following is most accurate? There is:

A) no evidence of multicollinearity and serial correlation.

B) evidence of multicollinearity but not serial correlation.

C) evidence of multicollinearity and serial correlation.

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Just because the correlation is spurious, does not mean the problem of multicollinearity will go away. However, there is no evidence of serial correlation.

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作者: kasinkei    时间: 2012-3-27 11:22

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) y2.

C) 3.

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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.

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作者: kasinkei    时间: 2012-3-27 13:12

Which of the following is a potential remedy for multicollinearity?

A) Omit one or more of the collinear variables.

B) Take first differences of the dependent variable.

C) Add dummy variables to the regression.

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The first differencing is not a remedy for the collinearity, nor is the inclusion of dummy variables. The best potential remedy is to attempt to eliminate highly correlated variables.

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作者: kasinkei    时间: 2012-3-27 13:14

Which of the following statements regarding multicollinearity is least accurate?

A) Multicollinearity may be present in any regression model.

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

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

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Multicollinearity is not an issue in simple linear regression.

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作者: kasinkei    时间: 2012-3-27 13:19

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) conditional heteroskedasticity.

B) serial correlation.

C) multicollinearity.

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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.

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作者: kasinkei    时间: 2012-3-27 13:20

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?

	Age	Education	Experience	Income
Age	1.00
Education	0.50	1.00
Experience	0.95	0.55	1.00
Income	0.60	0.65	0.89	1.00

A) Experience may be a redundant variable.

B) Age should be excluded from the regression.

C) Education may be unnecessary.

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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.

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作者: kasinkei    时间: 2012-3-27 13:21

When two or more of the independent variables in a multiple regression are correlated with each other, the condition is called:

A) serial correlation.

B) multicollinearity.

C) conditional heteroskedasticity.

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Multicollinearity refers to the condition when two or more of the independent variables, or linear combinations of the independent variables, in a multiple regression are highly correlated with each other. This condition distorts the standard error of estimate and the coefficient standard errors, leading to problems when conducting t-tests for statistical significance of parameters.

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作者: kasinkei    时间: 2012-3-27 13:21

When utilizing a proxy for one or more independent variables in a multiple regression model, which of the following errors is most likely to occur?

A) Multicollinearity.

B) Heteroskedasticity.

C) Model misspecification.

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By using a proxy for an independent variable in a multiple regression analysis, there is some degree of error in the measurement of the variable.

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作者: kasinkei    时间: 2012-3-27 13:22

When constructing a regression model to predict portfolio returns, an analyst runs a regression for the past five year period. After examining the results, she determines that an increase in interest rates two years ago had a significant impact on portfolio results for the time of the increase until the present. By performing a regression over two separate time periods, the analyst would be attempting to prevent which type of misspecification?

A) Incorrectly pooling data.

B) Using a lagged dependent variable as an independent variable.

C) Forecasting the past.

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The relationship between returns and the dependent variables can change over time, so it is critical that the data be pooled correctly. Running the regression for multiple sub-periods (in this case two) rather than one time period can produce more accurate results.

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作者: kasinkei    时间: 2012-3-27 13:23

Which of the following is least likely to result in misspecification of a regression model?

A) Transforming a variable.

B) Using a lagged dependent variable as an independent variable.

C) Measuring independent variables with errors.

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A basic assumption of regression is that the dependent variable is linearly related to each of the independent variables. Frequently, they are not linearly related and the independent variable must be transformed or the model is misspecified. Therefore, transforming an independent variable is a potential solution to a misspecification. Methods used to transform independent variables include squaring the variable or taking the square root.

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作者: kasinkei    时间: 2012-3-27 13:23

An analyst is building a regression model which returns a qualitative dependant variable based on a probability distribution. This is least likely a:

A) probit model.

B) discriminant model.

C) logit model.

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A probit model is a qualitative dependant variable which is based on a normal distribution. A logit model is a qualitative dependant variable which is based on the logistic distribution. A discriminant model returns a qualitative dependant variable based on a linear relationship that can be used for ranking or classification into discrete states.

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作者: kasinkei    时间: 2012-3-27 13:24

Which of the following questions is least likely answered by using a qualitative dependent variable?

A) Based on the following company-specific financial ratios, will company ABC enter bankruptcy?

B) Based on the following subsidiary and competition variables, will company XYZ divest itself of a subsidiary?

C) Based on the following executive-specific and company-specific variables, how many shares will be acquired through the exercise of executive stock options?

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The number of shares can be a broad range of values and is, therefore, not considered a qualitative dependent variable.

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作者: kasinkei    时间: 2012-3-27 13:24

Which of the following is NOT a model that has a qualitative dependent variable?

A) Logit.

B) Event study.

C) Discriminant analysis.

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An event study is the estimation of the abnormal returns--generally associated with an informational event—that take on quantitative values.

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作者: kasinkei    时间: 2012-3-27 13:25

A high-yield bond analyst is trying to develop an equation using financial ratios to estimate the probability of a company defaulting on its bonds. Since the analyst is using data over different economic time periods, there is concern about whether the variance is constant over time. A technique that can be used to develop this equation is:

A) multiple linear regression adjusting for heteroskedasticity.

B) logit modeling.

C) dummy variable regression.

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The only one of the possible answers that estimates a probability of a discrete outcome is logit modeling.

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作者: kasinkei    时间: 2012-3-27 13:25

What is the main difference between probit models and typical dummy variable models?

A) There is no difference--a probit model is simply a special case of a dummy variable regression.

B) A dummy variable represents a qualitative independent variable, while a probit model is used for estimating the probability of a qualitative dependent variable.

C) Dummy variable regressions attempt to create an equation to classify items into one of two categories, while probit models estimate a probability.

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Dummy variables are used to represent a qualitative independent variable. Probit models are used to estimate the probability of occurrence for a qualitative dependent variable.

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作者: kasinkei    时间: 2012-3-27 13:26

An analyst has run several regressions hoping to predict stock returns, and wants to translate this into an economic interpretation for his clients.

Return = 3.0 + 2.0Beta – 0.0001MarketCap (in billions) + ε

A correct interpretation of the regression most likely includes:

A) a billion dollar increase in market capitalization will drive returns down by 0.01%.

B) a stock with zero beta and zero market capitalization will return precisely 3.0%.

C) prediction errors are always on the positive side.

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The coefficient of MarketCap is 0.01%, indicating that larger companies have slightly smaller returns. Note that a company with no market capitalization would not be expected to have a return at all. Error terms are typically assumed to be normally distributed with a mean of zero

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作者: kasinkei    时间: 2012-3-27 13:27

Mary Steen estimated that if she purchased shares of companies who announced restructuring plans at the announcement and held them for five days, she would earn returns in excess of those expected from the market model of 0.9%. These returns are statistically significantly different from zero. The model was estimated without transactions costs, and in reality these would approximate 1% if the strategy were effected. This is an example of:

A) statistical significance, but not economic significance.

B) statistical and economic significance.

C) a market inefficiency.

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The abnormal returns are not sufficient to cover transactions costs, so there is no economic significance to this trading strategy. This is not an example of market inefficiency because excess returns are not available after covering transactions costs.

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