Chapter 4: BASIC ESTIMATION TECHNIQUES
Multiple Choice
4-1 For the equation Y = a + bX, the objective of regression analysis is to
a. estimate the parameters a and b.
b. estimate the variables Y and X.
c. fit a straight line through the data scatter in such a way that the sum of the squared errors
is minimized.
d. both a and c
4-2 In a linear regression equation of the form Y = a + bX, the slope parameter b shows
a. X / Y.
b. Y / X.
c. Y / b.
d. X / b.
4-3 In a linear regression equation of the form Y = a + bX, the intercept parameter a shows
a. the value of X when Y is zero.
b. the value of Y when X is zero.
c. the amount that Y changes when X changes by one unit.
d. the amount that X changes when Y changes by one unit.
4-4 In a regression equation, the ______ captures the effects of factors that might influence the
dependent variable but aren’t used as explanatory variables.
a. intercept
b. slope parameter
c. R-square
d. random error term
4-5 The sample regression line
a. shows the actual (or true) relation between the dependent and independent variables.
b. is used to estimate the population regression line.
c. connects the data points in a sample.
d. is estimated by the population regression line.
e. maximizes the sum of the squared differences between the data points in a sample and the
sample regression line.
4-6 Which of the following is an example of a time-series data set?
a. amount of labor employed in each factory in the U.S. in 2010.
b. amount of labor employed yearly in a specific factory from 1990 through 2010.
c. average amount of labor employed at specific times of the day at a specific factory in
2010.
d. All of the above are time-series data sets.
4–7 The method of least squares
a. can be used to estimate the explanatory variables in a linear regression equation.
b. can be used to estimate the slope parameters of a linear equation.
c. minimizes the distance between the population regression line and the sample regression
line.
d. all of the above
4-8 In a linear regression equation Y = a + bX, the fitted or predicted value of Y is
a. the value of Y obtained by substituting specific values of X into the sample regression
equation.
b. the value of X associated with a particular value of Y.
c. the value of X that the regression equation predicts.
d. the values of the parameters predicted by the estimators.
e. the value of Y associated with a particular value of X in the sample.
4-9 A parameter estimate is said to be statistically significant if there is sufficient evidence that the
a. sample regression equals the population regression.
b. parameter estimated from the sample equals the true value of the parameter.
c. value of the t-ratio equals the critical value.
d. true value of the parameter does not equal zero.
4-10 An estimator is unbiased if it produces
a. a parameter from the sample that equals the true parameter.
b. estimates of a parameter that are close to the true parameter.
c. estimates of a parameter that are statistically significant.
d. estimates of a parameter that are on average equal to the true parameter.
e. both b and c
4-11 The critical value of t is the value that a t-statistic must exceed in order to
a. reject the hypothesis that the true value of a parameter equals zero.
b. accept the hypothesis that the estimated value of parameter equals the true value.
c. reject the hypothesis that the estimated value of the parameter equals the true value.
d. reject the hypothesis that the estimated value of the parameter exceeds the true value.
4-12 To test whether the overall regression equation is statistically significant one uses
a. the t-statistic.
b. the R2-statistic.
c. the F-statistic.
d. the standard error statistic.
4-13 In the regression model
Y=a+bX +cZ
, a test of the hypothesis that parameter c equals zero is
a. an F-test.
b. an R2-test.
c. a zero-statistic.
d. a t-test.
e. a Z-test.
4-14 If an analyst believes that more than one explanatory variable explains the variation in the
dependent variable, what model should be used?
a. a simple linear regression model
b. a multiple regression model
c. a nonlinear regression model
d. a log-linear model
4-15 The linear regression equation, Y = a + bX, was estimated. The following computer printout was
obtained:
DEPENDENT VARIABLE:
Y
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
18
0.3066
7.076
0.0171
VARIABLE
PARAMETER
ESTIMATE
STANDARD
ERROR
T−RATIO
INTERCEPT
15.48
5.09
3.04
X
−21.36
8.03
−2.66
Given the above information, the parameter estimate of a indicates
a. when X is zero, Y is 5.09.
b. when X is zero, Y is 15.48.
c. when Y is zero, X is –21.36.
d. when Y is zero, X is 8.03.
4-16 The linear regression equation, Y = a + bX, was estimated. The following computer printout was
obtained:
DEPENDENT VARIABLE:
Y
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
18
0.3066
7.076
0.0171
VARIABLE
PARAMETER
ESTIMATE
STANDARD
ERROR
T−RATIO
INTERCEPT
15.48
5.09
3.04
X
−21.36
8.03
−2.66
Given the above information, the parameter estimate of b indicates
a. X increases by 8.03 units when Y increases by one unit.
b. X decreases by 21.36 units when Y increases by one unit.
c. Y decreases by 2.66 units when X increases by one unit.
d. a 10-unit decrease in X results in a 213.6 unit increase in Y.
4-17 The linear regression equation, Y = a + bX, was estimated. The following computer printout was
obtained:
DEPENDENT VARIABLE:
Y
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
18
0.3066
7.076
0.0171
VARIABLE
PARAMETER
ESTIMATE
STANDARD
ERROR
T−RATIO
INTERCEPT
15.48
5.09
3.04
X
−21.36
8.03
−2.66
Given the above information, what is the critical value of t at the 1% level of significance?
a. 1.746
b. 2.120
c. 2.878
d. 2.921
Learning Objective: 04-03
4-18 The linear regression equation, Y = a + bX, was estimated. The following computer printout was
obtained:
DEPENDENT VARIABLE:
Y
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
18
0.3066
7.076
0.0171
VARIABLE
PARAMETER
ESTIMATE
STANDARD
ERROR
T−RATIO
INTERCEPT
15.48
5.09
3.04
X
−21.36
8.03
−2.66
Given the above information, which of the following statements is correct at the 1% level of
significance?
a. Both
ˆ
a
and
ˆ
b
are statistically significant.
b. Neither
ˆ
a
nor
ˆ
b
is statistically significant.
c.
ˆ
a
is statistically significant, but
ˆ
b
is not.
d.
b
is statistically significant, but
ˆ
is not.
4-19 The linear regression equation, Y = a + bX, was estimated. The following computer printout was
obtained:
DEPENDENT VARIABLE:
Y
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
18
0.3066
7.076
0.0171
VARIABLE
PARAMETER
ESTIMATE
STANDARD
ERROR
T−RATIO
INTERCEPT
15.48
5.09
3.04
X
−21.36
8.03
−2.66
Given the above information, the value of the R2 statistic indicates that
a. 0.3066% of the total variation in Y is explained by the regression equation.
b. 0.3066% of the total variation in X is explained by the regression equation.
c. 30.66% of the total variation in Y is explained by the regression equation.
d. 30.66% of the total variation in X is explained by the regression equation.
ˆ
a
4-20 The linear regression equation, Y = a + bX, was estimated. The following computer printout was
obtained:
DEPENDENT VARIABLE:
Y
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
18
0.3066
7.076
0.0171
VARIABLE
PARAMETER
ESTIMATE
STANDARD
ERROR
T−RATIO
INTERCEPT
15.48
5.09
3.04
X
−21.36
8.03
−2.66
Given the above information, the exact level of significance of
ˆ
b
is
a. 0.171 percent.
b. 1 percent.
c. 1.71 percent.
d. 2.66 percent.
e. 2.921 percent.
4-21 The linear regression equation, Y = a + bX, was estimated. The following computer printout was
obtained:
DEPENDENT VARIABLE:
Y
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
18
0.3066
7.076
0.0171
VARIABLE
PARAMETER
ESTIMATE
STANDARD
ERROR
T−RATIO
INTERCEPT
15.48
5.09
3.04
X
−21.36
8.03
−2.66
Given the above information, if X equals 20, what is the predicted value of Y?
a. 186.42
b. 165.69
c. −186.42
d. −4 11.72
4-22 A firm is experiencing theft problems at its warehouse. A consultant to the firm believes that the
dollar loss from theft each week (T) depends on the number of security guards (G) and on the
unemployment rate in the county where the warehouse is located (U measured as a percent). In
order to test this hypothesis, the consultant estimated the regression equation T = a + bG + cU and
obtained the following results:
DEPENDENT VARIABLE:
T
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
27
0.7793
42.38
0.0001
VARIABLE
PARAMETER
ESTIMATE
STANDARD
ERROR
T−RATIO
P−VALUE
INTERCEPT
5150.43
1740.72
2.96
0.0068
G
−480.92
130.66
−3.68
0.0012
U
211.0
75.0
2.81
0.0096
Based on the above information, which of the following is correct at the 1% level of significance?
a. The regression equation as a whole is statistically significant because the p-value of F is
smaller than 0.01.
b. The estimates of the parameters a, b, and c are all statistically significant because the
absolute values of their t-ratios exceed 2.797.
c. The estimates of the parameters a, b, and c are all statistically significant because the p–
values for,
ˆ
a
,
ˆ
b
and
ˆ
c
are all less than 0.01.
d. The critical value of t is 2.797.
e. all of the above
4-23 A firm is experiencing theft problems at its warehouse. A consultant to the firm believes that the
dollar loss from theft each week (T) depends on the number of security guards (G) and on the
unemployment rate in the county where the warehouse is located (U measured as a percent). In
order to test this hypothesis, the consultant estimated the regression equation T = a + bG + cU and
obtained the following results:
DEPENDENT VARIABLE:
T
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
27
0.7793
42.38
0.0001
PARAMETER
STANDARD
VARIABLE
ESTIMATE
ERROR
T−RATIO
P−VALUE
INTERCEPT
5150.43
1740.72
2.96
0.0068
G
−480.92
130.66
−3.68
0.0012
U
211.0
75.0
2.81
0.0096
Based on the above information, hiring one more guard per week will decrease the losses due to
theft at the warehouse by _________ per week.
a. $5,150
b. $211
c. $130
d. $480.92
4-24 A firm is experiencing theft problems at its warehouse. A consultant to the firm believes that the
dollar loss from theft each week (T) depends on the number of security guards (G) and on the
unemployment rate in the county where the warehouse is located (U measured as a percent). In
order to test this hypothesis, the consultant estimated the regression equation T = a + bG + cU and
obtained the following results:
DEPENDENT VARIABLE:
T
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
27
0.7793
42.38
0.0001
VARIABLE
PARAMETER
ESTIMATE
STANDARD
ERROR
T−RATIO
P−VALUE
INTERCEPT
5150.43
1740.72
2.96
0.0068
G
−480.92
130.66
−3.68
0.0012
U
211.0
75.0
2.81
0.0096
Based on the above information, if the firm hires 6 guards and the unemployment rate in the
county is 10% (U = 10), what is the predicted dollar loss to theft per week?
a. $4,375 per week
b. $5,150 per week
c. $8,300 per week
d. $9,955 per week
4-25 A firm is experiencing theft problems at its warehouse. A consultant to the firm believes that the
dollar loss from theft each week (T) depends on the number of security guards (G) and on the
unemployment rate in the county where the warehouse is located (U measured as a percent). In
order to test this hypothesis, the consultant estimated the regression equation T = a + bG + cU and
obtained the following results:
DEPENDENT VARIABLE:
T
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
27
0.7793
42.38
0.0001
VARIABLE
PARAMETER
ESTIMATE
STANDARD
ERROR
T−RATIO
P−VALUE
INTERCEPT
5150.43
1740.72
2.96
0.0068
G
−480.92
130.66
−3.68
0.0012
U
211.0
75.0
2.81
0.0096
Based on the above information, a one percent increase in the level of unemployment in the
county results in an increase in losses due to theft of __________ more losses per week.
a. $75
b. $211
c. $280
4-26 In the nonlinear function
Y=aX bZc
, the parameter c measures
a.
Y /Z.
b. the percent change in Y for a 1 percent change in Z.
c. the elasticity of Y with respect to Z.
d. both a and c
e. both b and c
4-27 Tests for statistical significance must be performed
a. because the TRUE values of the intercept and slope parameters are random variables.
b. because the ESTIMATED values of the intercept and slope parameters are not, in
general, equal to the true values of the intercept and slope parameters.
c. because the computed t-ratios are random variables and may be too large to provide
evidence that b is not equal to zero.
d. in order to determine whether or not the parameter estimates are far enough away from
zero to conclude that the true parameter values are not equal to zero.
e. both b and d
4-28 If the p-value is 10%, then the
a. level of significance is 10%.
b. level of confidence is 90%.
c. probability of a Type I error is 90%.
d. both a and b
e. null hypothesis should not be rejected if the level of significance is 5%
4-29 Suppose you are testing the statistical significance (at the 5% significance level) of a parameter
estimate from the regression equation:
Y = a + bR + cS + dW
which is estimated using a time-series sample containing monthly observations over a 30−month
time period. The critical value of the appropriate test statistic is
a. tcritical = 2.042.
b. tcritical = 2.056.
c. Fcritical = 4.22.
d. Fcritical = 7.76.
4-30 Suppose you are testing the statistical significance (at the 1% significance level) of a parameter
estimate from the regression model:
M = a + bR + cI
which is estimated using a cross−section data set on 22 firms. The critical value of the appropriate
test statistic is
a. tcritical = 2.861.
b. tcritical = −2.845.
c. tcritical = 2.845.
d. Fcritical = 5.93.
e. Fcritical = 19.44.
4-31 Refer to the following computer output from estimating the parameters of the nonlinear model
Y=aRbScTd
The computer output from the regression analysis is:
DEPENDENT VARIABLE:
LNY
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
32
0.7766
32.44
0.0001
VARIABLE
PARAMETER
ESTIMATE
STANDARD
ERROR
T−RATIO
INTERCEPT
−0.6931
0.32
−2.17
LNR
4.66
1.36
3.43
LNS
−0.44
0.24
−1.83
LNT
8.28
4.6
1.80
Based on the info above, the nonlinear relation can be transformed into the following linear
regression model:
a.
Y=ln(aRbScTd)
b.
lnY=ln(aRbScTd)
c.
lnY=ln a×ln R×lnS×lnT
d.
lnY=ln a+bln R+clnS+dlnT
4-32 Refer to the following computer output from estimating the parameters of the nonlinear model
Y=aRbScTd
The computer output from the regression analysis is:
DEPENDENT VARIABLE:
LNY
R−SQUARE
F−RATIO
P−VALUE ON F
OBSERVATIONS:
32
0.7766
32.44
0.0001
VARIABLE
PARAMETER
ESTIMATE
STANDARD
ERROR
T−RATIO
INTERCEPT
−0.6931
0.32
−2.17
LNR
4.66
1.36
3.43
LNS
−0.44
0.24
−1.83
LNT
8.28
4.6
1.80
Based on the info above, the estimated value of a is
a. −0.6931
b. 0.50