Econ 388 R. Butler 2014 revisions Lecture 1
I. Introduction: names, prayers, responsibility for learning, grading and syllabus
II. Simple (Bivariate) Regression: test behavioral relationships or make predictions.
A. Linear Relationships: Some Background
Linear relationship: Y = a + bX
with data variables: Y,X and parameters:
a=intercept (the point on the Y-axis where the line crosses)
b=slope (of the straight line = rise/run)
B. FHSS salaries and teaching experience: Is there a relationship between salary and
teaching experience. For the following data we can check this by looking at a plot of the
data or the simple correlation between Y(annual salary) and X (experience):
FHSS faculty (see the lec4corr.ppt file):
Annual Salary(2)
Experience (in years)
male
45,000
2
1
60,600
7
0
70,000
10
1
85,000
18
1
50,800
6
0
64,000
8
1
62,500
8
0
87,000
15
1
92,000
25
0
89,500
22
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(We ignore the dummy variable for male for the moment, but male=1 for males and
male=0 for female faculty).
Probabilistic linear relationships:
yi   0   1 x i  i (this is the ‘population’ regression—our theoretical model)
where  = error term
When we estimate this linear relationship, we get specific estimates for the slope (  1 ),
intercept (  0 ), and error term (the estimated error term is called the residual ̂ i ):


yi   0   1 x i  ˆ i (this is the ‘sample’ regression—estimated with data)
Once we pick values for the intercept and slope parameters (respectively, those will be
ˆ0 , ˆ1 ), we have a line from which we can test hypothesis (about the population
regression) or make predictions.
*What does the hypothesis that experience leads to higher salaries indicate about
what we expect to find for the slope and intercept values?
*What does the hypothesis that new professors (with no teaching experience)
make $40,000 a year indicate about the expected slope and intercept values?
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E. Ordinary least squares criterion for choosing the values of ˆ0 , ˆ1 : minimize the
following sum of squares (these are called the “residual sum of squares,” “error sum of
squares,” or “unexplained sum of squares” since there are the deviations in the dependent
variable that are unexplained by the regression model):
n
min
 ( y - ŷ ) ,
2
i
i
where ŷi = ˆ 0 + ˆ 1 x i = value of y predicted by the regression line
i =1
This yields the following estimators (see Wooldridge, chapter 2):
yˆi  ˆ0  ˆ1 xi
 4483  (2128.7)(12)
 $70, 427.00 ( predicted salary when exp  12 years)
n
estimators: ˆ1 =
 (x - x)(y
i
i
- y)
i =1
;
n
 (x - x )
2
ˆ0 = y - ˆ 1 x
i
i =1
𝑦̂𝑖 = 𝛽̂0 + 𝛽̂1 𝑥𝑖
=44883 + 2128.7 (12)
=$70,427 (which is the predicted starting salary when experience=12 years)
y  70, 640
x  12.10
Estimate: ˆ (45000  70640)(2  12.1)  (60, 600  70, 640)(7  12.1)  etc.
1 
(2  12.10)2  (7  12.1) 2  etc.
Estimators are formulas; estimates are application of the formulas to specific samples.
Using these formulas we find that ̂ 0 =44883 and ˆ1 =2128.7. What do these estimates
tell us about the structure of wages at FHSS?
What is the predicted wage for someone with 12 years of experience? What is the
predicted wage for someone with 13 years of experience? What is the difference in these
two predicted wages?
STATA code (get into the do-editor of Stata, type this, save it as file (*.do), and execute
it using the “do” option on the toolbar to the upper right on top of the toolbar)
# delimit ;
input salary experience male;
45000
2
1;
60600
7
0;
70000
10
1;
85000
18
1;
50800
6
0;
64000
8
1;
62500
8
0;
87000
15
1;
92000
25
0;
2
89500
end;
regress salary experience;
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1;
and to get the predicted salary at 12 years we can do a trick with a (0,-1) variable:
# delimit ;
input salary experience pred_trick;
45000 2 0;
60600 7 0;
70000 10 0;
85000 18 0;
50800 6 0;
64000 8 0;
62500 8 0;
87000 15 0;
92000 25 0;
89500 22 0;
0
12 -1;
end;
regress salary experience pred_trick;
III. Multivariate Regression
This just means we have two or more independent variables (i.e., two or more slope
regressors). To run this regression in STATA, we just need to change the last line of the
program above to the following:
regress salary experience male;
and we would get the following sample regression function (for our 10 observations from
FHSS faculty):
salary  44, 256  2,106 exp erience  4,836 male
Notice that the intercept and slope coefficient for experience didn’t change much. The
2106 coefficient now means “holding gender constant, each additional year of experience
(either for males or females) increases salary by $2,106.” What does the coefficient for
the male variable indicate? What is the expected salary of a female faculty member with
12 years of experience?
Note that the 𝛽̂ type variables are estimators that depend on y, and since y is a random
variable, so are the 𝛽̂ type variables. Random variables are usefully characterized by
their mean, their standard deviation (square root of the estimated variance), and their
shape.
IV. Multivariate Regression Application: Difference-in-Difference to measure regime
change (like new light rail system on local economy, or new laws on the incomes or
health or employment status of the affected individuals)
Econometrics is often used to make causal inferences: When a state builds a new light
rail system (LR), it is often argued that this improves the quality of life of the local
residents. There are many dimensions to quality of life, but let’s suppose that we are
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interested in the wages of those in the metropolitan area where the light rail is installed in
the short run (before immigration possibly changes the size of the labor market). We
could address this by looking at wages before and after this change, but the danger is that
their salaries may have risen or fallen for other reasons not related to the change in the
law. That is, we need to know what would have happened to them in the absence of the
law change—sometimes, this is called the counterfactual.
A difference-in-difference approach to estimating the impact of LR on wages picks a
“reasonable comparison” group (in this case wages of those in a similar labor market who
did not adopt a LR system, and examines the before and after of this group (with no LR
system) to the before and after results of those living in the metro area where LR (light
rail train system) was built. The treatment group is workers living in a metro area with
LR; the control group, workers living in a metro area without LR, during the same
periods. Difference in difference estimation exploits temporal variation between the
treatment and control groups, by adjusting the wage incomes after minus incomes before
for the treatment group (call this the “a-b” change in income). The adjustment is to
assume that control group is just like the treatment group, except that they didn’t get the
treatment. So the control group’s “incomes after” minus “incomes before” (this change
in income is denoted as “c-d”) represents what would have happened to the treatment
group, if only they had not received the treatment. If the control group is suitable in just
this way, the basic strategy is to use the difference in difference, or “(a-b) – (c-d)” as the
net effect of LR on , accounting for other trends in wages (in the “c-d” difference) that
would have occurred in the absence of the treatment.
For concreteness, assume that a mythical place called SLC put in a LR system in 1999,
and another mythical place with an analogous urban population, call it Boise, didn’t put
in a LR system. Here are the average wages before and after for the two cities
Real Wages for Two Metro Areas
Year City SLC (LR-treatment grp)
1998
$30,000
2000
$34,000
Boise(no LR-control grp)
$25,000
$27,500
The first difference, $4,000, is the increase in wage income in SLC after LR. However,
note that even in places without LR, urban wages were increasing: wages rose $2,500 in
Boise). So the real effect of the LR in SLC (assuming the Boise data is providing the
right counterfactual information) is $1,500, as indicated by the following difference in
difference calculation:
Diff-in-diff = ($34,000-$30,000) – ($27,500-$25,000)=$1,500.
Since these differences are not the result of a randomized experiment, it is very often
useful to include the difference in difference estimation technology into a regression
framework so that we can additionally control for other factors that might affect the
outcomes (in this case, workers’ age, marital status, and educational attainment). To
see how the difference in difference is estimated with regression coefficients, consider
workers’ wages as determined by being in the treatment group (a metro area with LR),
after versus before the law was passed (another dummy variable that takes the value of
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1 after the LR is operational and zero before), the interaction of treatment group and the
LR, and other factors (demographic variables such as those indicated above). Then this
model is for individual i at year t (  ,  ,  ,  are parmenters, the variables with
subscripts are the data variables):
wagei ,t  X i ,t   treatmenti   after _ LRt   (treatmenti * after _ LR)i ,t    it
Hence the expected wage (with E ( it )  0 ) for the treatment group before LR is (that
is, before the LR is operational, setting the appropriate dummy variable terms to zero
and one) :
wagei ,t  X T i ,t   
and the expected wage for the treatment group after the LR is operational is
wagei ,t  X T i ,t       
So the first difference for the treatment group is just
( X T i ,t        )  ( X T i ,t    )     .
For the control group, a similar calculation indicates that their before and after wage
difference equals
( X C i ,t    )  X C i , t    ,
So that the difference in difference regression equivalent (controlling for other
characteristics, X) is just the coefficient  , the coefficient of “treatment*after LR”
interaction term.
SAS APPENDIX
SAS code (get into the SAS editor, type this, save it as a file (*.sas), and execute it by
clicking the “run” option—little guy running—on the toolbar to the upper right)
data one;
input salary experience male;
cards; *you can use ‘datalines’ instead of ‘cards’ here;
45000
2
1
60600
7
0
70000
10
1
85000
18
1
5
50800
64000
62500
87000
92000
89500
6
8
8
15
25
22
0
1
0
1
0
1
run;
proc reg;
model salary=experience male;
run;
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