JoeyDVivre

Given: Y = 2.83 + 1.5X
What is the predicted value of the dependent variable when the value of an independent variable equals 2?

A) 2.83

B) -0.55

C) 5.83

---

Y [/td][td]= 2.83 + (1.5)(2)
	= 2.83 + 3
	= 5.83

JoeyDVivre

A simple linear regression is run to quantify the relationship between the return on the common stocks of medium sized companies (Mid Caps) and the return on the S&P 500 Index, using the monthly return on Mid Cap stocks as the dependent variable and the monthly return on the S&P 500 as the independent variable. The results of the regression are shown below:

	Coefficient	Standard Error of Coefficient	t-Value
Intercept	1.71	2.950	0.58
S&P 500	1.52	0.130	11.69
R2 = 0.599

Use the regression statistics presented above and assume this historical relationship still holds in the future period. If the expected return on the S&P 500 over the next period were 11%, the expected return on Mid Cap stocks over the next period would be:

A) 20.3%.

B) 33.8%.

C) 18.4%.

---

Y = intercept + slope(X)

JoeyDVivre

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

   
Coefficient
Standard Error
t-statistic
P-value

Intercept
5.65
1.27
4.44
0.01

Age
0.53
?
1.33
0.21

Years of Education
2.32
0.41
?
0.01

   

Anova
df
SS
MS
F

Regression
?
215.10
?
?

Error
?
115.10
?


Total
?
?




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

A) 0.40; 5.66.

B) 0.53; 2.96.

C) 0.32; 1.65.





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

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



--------------------------------------------------------------------------------
The mean square regression (MSR) is:
A) 6.72.

B) 107.55.

C) 102.10.





--------------------------------------------------------------------------------
df for Regression = k = 2

MSR = RSS / df = 215.10 / 2 = 107.55



--------------------------------------------------------------------------------
The mean square error (MSE) is:
A) 3.60.

B) 3.58.

C) 7.11.





--------------------------------------------------------------------------------

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

MSE = SSE / df = 115.10 / 32 = 3.60



--------------------------------------------------------------------------------
What is the R2 for the regression?
A) 76%.  

B) 62%.  

C) 65%.  





--------------------------------------------------------------------------------

SST = RSS + SSE

= 215.10 + 115.10

= 330.20

R2= RSS / SST = 215.10 / 330.20 = 0.65



--------------------------------------------------------------------------------
What is the predicted income of a 40-year-old person with 16 years of education?
A) $62,120.  

B) $74,890.

C) $63,970.  





--------------------------------------------------------------------------------

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

            = 5.65 + 0.53 (40) + 2.32 (16)

            = 63.97 or $63,970



--------------------------------------------------------------------------------
What is the F-value?
A) 14.36.  

B) 1.88.

C) 29.88.





--------------------------------------------------------------------------------
F = MSR / MSE = 107.55 / 3.60 = 29.88

JoeyDVivre

Consider the following analysis of variance (ANOVA) table:

Source	Sum of squares	Degrees of freedom	Mean square
Regression	550	1	550.000
Error	750	38	19.834
Total	1,300	39

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

A) 0.965.

B) 27.730.

C) 0.423.

---

F = Mean Square of Regression / Mean Square of Error = 550 / 19.834 = 27.730.

JoeyDVivre

Consider the following analysis of variance (ANOVA) table:

Source	Sum of squares	Degrees of freedom	Mean square
Regression	200	1	200
Error	400	40	10
Total	600	41

The R2 and the F-statistic are, respectively:

A) R2 = 33% and F = 2.0.

B) R2 = 33% and F = 20.0.

C) R2 = 50% and F = 2.0.

---

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

JoeyDVivre

Consider the following analysis of variance (ANOVA) table:

Source	Sum of squares	Degrees of freedom	Mean square
Regression	500	1	500
Error	750	50	15
Total	1,250	51

The R2 and the F-statistic are, respectively:

A) R2 = 0.67 and F = 0.971.

B) R2 = 0.40 and F = 33.333.

C) R2 = 0.40 and F = 0.971.

---

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

JoeyDVivre

Consider the following analysis of variance (ANOVA) table:

Source	Sum of squares	Degrees of freedom	Mean square
Regression	556	1	556
Error	679	50	13.5
Total	1,235	51

The R2 for this regression is:

A) 0.45.

B) 0.55.

C) 0.82.

---

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

JoeyDVivre

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

Analysis of Variance Table (ANOVA)
Source	Degrees of freedom (df)	Sum of Squares	Mean Square (SS/df)	F-statistic
Regression	5	18,500	3,700
Error	94	600.66	6.39
Total	99	19,100.66

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

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

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

---

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

JoeyDVivre

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

A) 0.55.

B) 0.30.

C) 0.99.

---

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

JoeyDVivre

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

Regression Statistics
R²	??
Standard Error	??
Observations	20
ANOVA
	df	SS	MS	F	Significance F
Regression	1	23,516	23,516	?	?
Residual	18	?	7
Total	19	23,644
Regression Equation
		Coefficients	Std. Error	t-statistic	P-value
Intercept		5.2900	1.6150	?	?
Slope		0.8700	0.0152	?	?

What is the value of the correlation coefficient?

A) −0.9973.

B) 0.8700.

C) 0.9973.

---

R2 is the correlation coefficient squared, taking into account whether the relationship is positive or negative. Since the value of the slope is positive, the TIM fund and the index are positively related. R2 is calculated by taking the (RSS / SST) = 0.99459. (0.99459)1/2 = 0.9973. (Study Session 3, LOS 11.i)

---

What is the sum of squared errors (SSE)?

A) 128.

B) 23,644.

C) 23,515.

---

SSE = SST − RSS = 23,644 − 23,516 = 128. (Study Session 3, LOS 11.i)

---

What is the value of R2?

A) 0.9946.

B) 0.0055.

C) 0.9471.

---

R2 = RSS / SST = 23,516 / 23,644 = 0.9946. (Study Session 3, LOS 11.i)

---

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

1%

5%

A) Yes No

B) No No

C) Yes Yes

---

The test statistic is t = b / std error of b = 5.29 / 1.615 = 3.2755.
Critical t-values are ± 2.101 for the degrees of freedom = n − k − 1 = 18 for alpha = 0.05. For alpha = 0.01, critical t-values are ± 2.878. At both levels (two-tailed tests) we can reject H0 that b = 0. (Study Session 3, LOS 11.g)

---

What is the value of the F-statistic?

A) 3,359.

B) 0.0003.

C) 0.9945.

---

F = mean square regression / mean square error = 23,516 / 7 = 3,359. (Study Session 3, LOS 11.i)

---

Heteroskedasticity can be defined as:

A) independent variables that are correlated with each other.

B) error terms that are dependent.

C) nonconstant variance of the error terms.

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Heteroskedasticity occurs when the variance of the residuals is not the same across all observations in the sample. Autocorrelation refers to dependent error terms. (Study Session 3, LOS 12.i)