Question

True or False: Please specify your reasons.

(i) If an independent variable in a multiple linear regression model is an exact linear combination of other independent variables, we can still calculate the least square estimators of the intercept.

(ii) For the multiple linear regression y = β0 + β1x + β2x 2 + u, β1 can be interpreted as the effect of one unit increase in x on y.

(iii) In the multiple linear regression with an intercept, (the sum of the residuals equals zero) may not hold as in the simple regression.

(iv) When adding an irrelavent regressor, R2 will decrease

Answer #1

(i) False. If there is an exact linear combination between independent variables then regression model will suffer from perfect multicollinearity and it becomes impossible to calculate the OLS estimators.

(ii)False.β1 can be interpreted as the effect of one unit
increase in X1 on Y,**keeping other factors
constant**.

(iii) False. Sum of residuals is always zero in the case of multiple regression model with an intercept.

(iv)False.R2 always increases with an addition of new variables no matter whether its relevant or irrelevant.

If you have any doubt,feel free to ask.

1.
Consider regression through the origin,
y=β1x1+β2x2+u, which of
the following statements is wrong?
a.
The degree of freedom for estimating the variance of error term
is n−2.
b.
The sum of residuals equals to 0.
c.
If the true intercept parameter doesn’t equal to 0, all slope
estimators are biased.
d.
The residual is uncorrelated with the independent variable.
2.
Which of the following statements is true of hypothesis
testing?
a.
OLS estimates maximize the sum of squared residuals....

You want to construct a multiple linear regression model. The
dependent variable is Y and independent variables are x1 and x2.
The samples and STATA outputs are provided:
Y
X1
X2
3
2
1
4
1
2
6
3
3
6
3
4
7
4
5
STATA
Y
Coef.
Std. Err.
t
P> abs. value (t)
95% confidence interval
X1
0.25
0.4677072
0.53
0.646
-1.762382 , 2.262382
X2
0.85
0.3372684
2.52
0.128
-.601149 , 2.301149
_cons
2
0.7245688
2.76
0.110...

You want to construct a multiple linear regression model. The
dependent variable is Y and independent variables are x1 and x2.
The samples and STATA outputs are provided:
Y
X1
X2
3
2
1
4
1
2
6
3
3
6
3
4
7
4
5
STATA
Y
Coef.
Std. Err.
t
P> abs. value (t)
95% confidence interval
X1
0.25
0.4677072
0.53
0.646
-1.762382 , 2.262382
X2
0.85
0.3372684
2.52
0.128
-.601149 , 2.301149
_cons
2
0.7245688
2.76
0.110...

Multiple linear regression results:
Dependent Variable: Cost
Independent Variable(s): Summated Rating
Cost = -43.111788 + 1.468875 Summated Rating
Parameter estimates:
Parameter
Estimate
Std. Err.
Alternative
DF
T-Stat
P-value
Intercept
-43.111788
10.56402
≠ 0
98
-4.0810021
<0.0001
Summated Rating
1.468875
0.17012937
≠ 0
98
8.633871
<0.0001
Analysis of variance table for multiple regression model:
Source
DF
SS
MS
F-stat
P-value
Model
1
8126.7714
8126.7714
74.543729
<0.0001
Error
98
10683.979
109.02019
Total
99
18810.75
Summary of fit:
Root MSE: 10.441273
R-squared: 0.432...

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