Are the number of bedrooms and minimum lease
significant predictors of monthly rent in the
multiple regression model
we estimated in class?
Hal Snarr
Westminster College
Salt Lake City, UT
Professor
Economics
Fall 2015 semester
December 2, 2015
Descriptive Statistics
mean
std dev
min
max
count
correlation with rent
number of bedrooms
1.56
0.56
0
3
101
0.37
minimum lease
8.86
4.06
0
12
101
0.12
monthly rent
952.21
303.00
388
1780
101
1
Simple Regression 1
I estimated the following simple regression model
y = b 0 + b 9· x9 + e
where
y = monthly rent
x9 = number of bedrooms
b 9 = the variable’s coefficient that cannot be known, but can be estimated
e = the errors that cannot be known, but can be estimated
The coefficient significance test for number of bedrooms:
H0: b 9 = 0
Ha: b 9 < > 0
Simple Regression 1
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.37
0.14
0.13
282.92
101
ANOVA
Regression
Residual
Total
Intercept
number of bedrooms
df
1
99
100
SS
1256904.26
7924250.37
9181154.63
MS
1256904.26
80042.93
F
15.70
Coefficients
636.35
201.91
Standard Error
84.53
50.95
t Stat
7.53
3.96
P-value
0.00
0.00
Simple Regression 2
I estimated the following simple regression model
y = b 0 + b 5· x5 + e
where
y = monthly rent
x5 = minimum lease
b 5 = the variable’s coefficient that cannot be known, but can be estimated
e = the errors that cannot be known, but can be estimated
The coefficient significance test for minimum lease:
H0: b 5 = 0
Ha: b 5 < > 0
Simple Regression 2
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.12
0.01
0.00
302.31
101
ANOVA
df
Regression
Residual
Total
Intercept
minimum lease
1
99
100
SS
133391.71
9047762.92
9181154.63
Coefficients
872.53
8.99
Standard Error
72.48
7.44
MS
133391.71
91391.54
t Stat
F
1.46
P-value
12.04
1.21
0.00
0.23
Multiple Regression 1
coefficient significance t tests
I estimated the following multiple regression model
y = b 0 + b 9· x9 + b 5· x 5 + e
where
y = monthly rent
x9 = number of bedrooms
b 9 = number of bedroom’s coefficient that cannot be known, but can be estimated
x5 = minimum lease
b 5 = minimum lease’s coefficient that cannot be known, but can be estimated
e = the errors that cannot be known, but can be estimated
The coefficient significance test for number of bedrooms:
H0: b 9 = 0
Ha: b 9 < > 0
The coefficient significance test for minimum lease:
H0: b 5 = 0
Ha: b 5 < > 0
Multiple Regression 1
coefficient significance t tests
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.397
0.158
0.140
280.924
101
ANOVA
df
Regression
Residual
Total
Intercept
number of bedrooms
minimum lease
2
98
100
SS
1447140.508
7734014.126
9181154.634
Coefficients
533.322
206.827
10.758
Standard Error
107.000
50.692
6.929
MS
723570.254
78918.511
t Stat
F
9.169
P-value
4.984
4.080
1.553
0.000
0.000
0.124
Multiple Regression 1
model significance F test
I estimated the following multiple regression model
y = b 0 + b 9· x9 + b 5· x 5 + e
where
y = monthly rent
x9 = number of bedrooms
b 9 = number of bedroom’s coefficient that cannot be known, but can be estimated
x5 = minimum lease
b 5 = minimum lease’s coefficient that cannot be known, but can be estimated
e = the errors that cannot be known, but can be estimated
There is one model significance test:
H0: b 9 = b 5 = 0
Ha: b k < > 0
Multiple Regression 1
model significance F test
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.397
0.158
0.140
280.924
101
ANOVA
df
Regression
Residual
Total
Intercept
number of bedrooms
minimum lease
2
98
100
SS
1447140.508
7734014.126
9181154.634
Coefficients
533.322
206.827
10.758
Standard Error
107.000
50.692
6.929
MS
723570.254
78918.511
t Stat
F
9.169
P-value
4.984
4.080
1.553
0.000
0.000
0.124
Multiple Regression 2
coefficient significance t tests
In class, we estimated the following regression model
y = b 0 + b 1 · x1 + b 2 · x2 + ··· + b 36 · x36 + e
where
y = monthly rent
x = the independent variables (number of bedrooms, number of bathrooms…)
b = the parameters that cannot be known, but can be estimated
e = the errors that cannot be known, but can be estimated
Although there are 36 individual significance tests, one for each variable in our multiple
regression, I will conduct only the following significance tests for number of bedrooms and
minimum lease length.
H0: b 9 = 0
Ha: b 9 < > 0
H0: b 5 = 0
Ha: b 5 < > 0
Multiple Regression 2
coefficient significance t tests
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.92
0.85
0.77
145.51
101
ANOVA
Regression
Residual
Total
Intercept
Nearest Park
df
36
63
99
SS
7700058.00
1333874.00
9033932.00
MS
213890.00
21173.00
F
10.10
Coefficients Standard Error
263.90
155.00
t Stat
1.70
P-value
0.09
30.83
40.88
0.75
0.45
:
:
:
:
-0.13
4.37
-0.03
0.98
:
:
:
:
121.37
44.11
2.75
0.01
:
:
:
:
Is in Sugarhouse
373.99
87.73
4.26
0.00
Is west of State Street
-33.11
44.63
-0.74
0.46
-116.56
54.11
-2.15
0.04
:
minimum lease
:
number of bedrooms
:
Is South of I-80
Multiple Regression 2
model significance F test
In class, we estimated the following regression model
y = b 0 + b 1 · x1 + b 2 · x2 + ··· + b 36 · x36 + e
where
y = monthly rent
x = the independent variables (number of bedrooms, number of bathrooms…)
b = the parameters that cannot be known, but can be estimated
e = the errors that cannot be known, but can be estimated
There is one model significance test:
H0: b 1 = b 2 = … = b 36 = 0
Ha: b k < > 0
Multiple Regression 2
model significance F test
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.92
0.85
0.77
145.51
101
ANOVA
Regression
Residual
Total
Intercept
Nearest Park
df
36
63
99
SS
7700058.00
1333874.00
9033932.00
MS
213890.00
21173.00
F
10.10
Coefficients Standard Error
263.90
155.00
t Stat
1.70
P-value
0.09
30.83
40.88
0.75
0.45
:
:
:
:
-0.13
4.37
-0.03
0.98
:
:
:
:
121.37
44.11
2.75
0.01
:
:
:
:
Is in Sugarhouse
373.99
87.73
4.26
0.00
Is west of State Street
-33.11
44.63
-0.74
0.46
-116.56
54.11
-2.15
0.04
:
minimum lease
:
number of bedrooms
:
Is South of I-80
Conclusions
•
The F test in the multiple regression indicated that our final regression model was
significant, not all of the 36 coefficients are zero.
•
The adjusted R2 implies that 77% of the variation in monthly rent is explained by the
model, the 36 independent variables.
•
The coefficient of number of bedrooms decreased from a value of around 200 in our
early specifications to 121.37 in the final multiple regression specification.
•
The coefficient of minimum lease hovered around zero in all specifications.
•
Number of bedrooms is a significant predictor of rent in all specifications at the 10%
level of significance.
Thus, holding all else equal, increasing the number of
bedrooms by one raises SLC monthly rent by $121.37
•
Minimum lease is an insignificant predictor of rent in all specifications at the 10%
level of significance.
Thus, our results suggest that minimum lease does not affect SLC monthly rent.