Coefficient Interpretation

1.1 Level-Level
Suppose the dependent variable and independent variable of interest are both in level form. For
example, suppose we want to know the e ect of an additional year of schooling on wages and from
available data we t the following equation
W
i
= +
1
Ed
i
+
2
Age
i
+
3
Exp
i
+ 
(1)
In this case, the coecient on the variable of interest can be interpreted as the marginal e ect. The
i
marginal e ect is how the dependent variable changes when the independent variable changes by
an additional unit holding all other variables in the equation constant (i.e. partial derivative) or
Therefore,
j
@W
@Ed
i
i
=
1
can be interpreted as the change in wages from a one unit increase (or state change
if dummy variable) of X
j
holding all other independent variables constant.
Example: Suppose the tted equation for (1) is
^
W
i
= 3:5 + 0:75Ed
i
+ 0:25Age
i
+ 0:30Exp
i
+ 
i
Based on the data used in this regression, an additional year of education corresponds to an increase
in hourly wages of $0.75. Similarly, an additional year of experience is associated with a $0.30 per
hour wage increase.
1.2 Log-Log
Consider interpreting coecients from a regression where the dependent and independent variable
of interest are in log form. The coecients can no longer be interpreted as marginal e ects. Suppose
economic theory suggests estimation of our wage equation with the dependent variable in log form
and inclusion of community volunteer hours per week (Comm) also in log form. The equation of
interest is now
log(W
i
) = +
1
Ed
i
+
2
Age
i
+
3
Exp
i
+
4
log(Comm
i
) + 
(2)
We would like to interpret the coecient on the community volunteer variable (
). To better understand
the interpretation, consider taking the di erential of (2) holding all independent variables
constant except Comm
since d[log(X)] =
1
X
i
.
d[log(W
1
W
i
i
)] = d[log(Comm
dW
i
=
1
Comm
i
i
)]
dComm
4
i

d(X). This nal equation can be rearranged such that,
100dW
W
i
100dComm
Comm
i
i
1
i
=
4
4
i
4
where the left hand side is the (partial) elasticity of W with respect to Comm. Elasticity is the
ratio of the percent change in one variable to the percent change in another variable. The coecient
in a regression is a partial elasticity since all other variables in the equation are held constant.
Therefore,
can be interpreted as the percent change in hourly wages from a one percent increase
in community volunteer hours per week holding education, age and experience constant.
4
Example Suppose that the tted equation for (2) is
^
log(W
i
) = 3:26 + 0:24Ed
i
+ 0:08Age
i
+ 0:16Exp
i
+ 1:2 log(Comm
)
Based on these regression results, a one percent increase in community volunteer hours per week is
associated with a 1.2% increase in hourly wages.
1.3 Log-Level
In equation (2), education, age and experience are in level terms while the dependent variable
(wage) is in log terms. We would like to interpret the coecients on these variables. First, consider
education. Take the di erential holding all other independent variables constant.
Multiply both sides by 100 and rearrange,
100  dW
Therefore, 100 
1
W
i
100 
d[log W
i
1
dW
W
i
i
] = dEd
i
= dEd
= 100  dEd
=
100dW
W
i
dEd
i
i
i
i
i

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1
1
1
=
%W
unit Ed
can be interpreted as the percentage change in W
i
i
i
i
for a unit increase in Ed
,
holding all other independent variables constant. Similar derivations can derive the interpretation
for the coecients on age and experience.
Example: Consider the tted equation for (2)
^
log(W
i
) = 3:26 + 0:24Ed
i
+ 0:08Age
i
+ 0:16Exp
i
+ 1:2 log(Comm
)
Therefore, holding all other independent variables constant, an additional year of schooling is
associated with a 24% increase in hourly wages. Similarly, an additional year of experience is
associated with a 16% increase in hourly wages.
1.4 Level-Log
Consider a regression where the dependent variable is in level terms and the independent variable
of interest is in log terms. For example, consider the following equation
W
i
= +
1
Ed
i
+
2
Age
i
+
3
Exp
i
+
4
log(Comm
i
) + 
(3)
Recall from the section on level-level regressions that the coecients on education, age and experi-
i
ence can be interpreted as marginal e ects. We would like to interpret the coecient on community
2
i
i
volunteer hours (
). Again, take the di erential on both sides, holding all independent variables
constant except community volunteer hours:
4
Divide both sides by 100 and rearrange,

Therefore,

4
100
4
dW
dW
100
=
i
i
= d[log(Comm
=
1
Comm
dW
i
100dComm
Comm
i
i
i
i
)]
dComm
4
i

4
=
unit W
%Comm
i
i
can be interpreted as the increase in hourly wages from a one percent increase in
community volunteer hours per week.
Example: Suppose that the tted equation for (3) is
^
W
i
= 3 + 0:67Ed
i
+ 0:28Age
i
+ 0:34Exp
i
+ 13:2 log(Comm
)
Therefore, holding education, age and experience constant, a one percent increase in community
volunteer hours per week is associated with a $0.132 increase in hourly wages.
3
i