Transcript Econ 399 Chapter7b

7.4 DV’s and Groups
Often it is desirous to know if two different
groups follow the same or different regression
functions
-One way to test this is to use DV’s to allow for
ALL different intercepts and slopes for the two
groups, and test if all DV terms=0:
y   0  1 x1   2 x2  ...   k xk
  0 DV  1 DVx1   2 DVx2  ...   k DVxk  u
H0 : v  0
v 0,1, 2,.... k
H a : H 0 is not true
7.4 DV’s and Groups
As we’ve already seen in our discussion of F
tests, the significance or insignificance of one
variable is quite separate from the joint
significance of a set of variables
-Therefore an F test must be done using the
restricted model:
y   0  1 x1   2 x2  ...   k xk  u
However, in the case of many x variables, this
results in a large number of interaction variables,
and may be difficult
7.4 DV’s and Groups
Alternately, one can run the regression in
question (without DV’s) for the first and second
group, and record SSR1 and SSR2
-a full regression is then run with all observations
included to find SSRp
-a test F statistic is then formed as:
[ SSRP  ( SSR1  SSR2 )] /( k  1)
F
( SSR1  SSR2 ) /[ n  2(k  1)]
Which is usually called the CHOW STATISTIC and
is only valid under homoskedasticity
7.4 DV’s and Groups
This F value is compared to F* from our tables
with k+1, n-2(k+1) degrees of freedom
-Note that no valid R2 form of this equation exists
-the null hypothesis as listed allows for no
difference between groups
-if it is not rejected, the two groups test
statistically identical (at a certain α)
-if one wants to allow for an INTERCEPT
difference in the two groups, the full regression is
run with a single DV to distinguish the groups
7.5 Binary Dependent Variables
-Thus far we have only considered
QUANTITATIVE values for our dependent
variable, but we can also have dependent
variables analyzing a QUALITATIVE event
-ie: failing or passing the Midterm
-in the simplest case, y has 2 outcomes
-ie: MT=1 if passed, =0 otherwise
MT   0  1Study   2 Intel  3 Bribe  u
7.5 Binary Dependent Variables
-Here our y value reflect the probability of
“success”, that is,
P(y  1 | X)  E(y | X)   0  1 x1  ...   k xk
-This is also called the RESPONSE PROBABILITY
-Since probabilities must sum to one, we also
have that
P(y  0 | X)  1  P(y  1 | X)
-the regression with a binary dependent variable
is also called the LINEAR PROBABILITY MODEL
(LPM)
7.5 Binary Dependent Variables
-in the LPM, Bj can no longer be interpretted as
the change in y given a one unit change in x.
Instead,
P(y  1 | X)   j x j
(7.28)
-Our estimated regression becomes:
yˆ  ˆ0  ˆ1 x1  ˆ2 x2  ...  ˆk xk
-Where yhat is our predicted probability of
“success” and Bjhat predicts the change in the
probability of success due to a one unit
increase in xj
7.5 Binary Example
-Assume that our above example regressed as:
Mˆ T  0.38  0.01Study  0.005Intel  0.04 Bribe
-This reflects some LIMITATIONS of the LPM:
1) If bribe (expressed in tens of thousands of
dollars)=100 ($1 million), then MThat>4
(400% chance of passing). Ie: estimated
probabilities can be negative or over 1.
2) This assumes that the probability increase of
the first hour of studying (1%) is the same as
the probability increase of the 49th hour (1%).
7.5 LPM Fixing
-One way around this is to redefine predicted
~
values:
y  1 if ŷ  0.5
 0 if ŷ  0.5
-One advantage of this redefinition is we can
obtain a new goodness-of-fit measure as
PERCENT CORRECTLY PREDICTED
-as now true and predicted values are both
either zero and 1
-the number of matches over the number of
observations is our goodness of fit
7.5 LPM and Heteroskedasticity
-Because y is binary, the LPM does violate one
Gauss-Markov assumption:
Var( y | x)  p( X )[1  p X ]
(7.30)
-where p(X) is short for the probability of success
-therefore, heteroskedasticity exists and MLR.5 is
violated
-therefore t and F statistics are not valid until het
is corrected as discussed in chapter 8
7.5 DV Party
-It is also possible to include Dummy Variables as
both the dependent variable and as independent
variables, for example,
MT   0  1Study   2 Intel  3CheetSheet  u
-Where CheetSheet=1 if a cheat sheet is
prepared, =0 otherwise
-in this case the estimated coefficient of the
independent DV gives the increase in probability
of success if not the base case
7.6 Policy Analysis and Program Evaluation
-Policy Analysis and Program Evaluation is
generally done using the regression and
hypothesis test:
yˆ  ˆ0  ˆ1 x1  ˆ2 x2  ...  ˆDV DV
H 0 :  DV  0
H a :  DV  0
-Where the DV represents the group possibly
needing a program or participation in the
program
-If H0 is not rejected, the program is not needed
or is not effective
7.6 Evaluation and Analysis Difficulties
This evaluation and analysis process has two
inherent difficulties:
1) Omitted Variables
-if a variable correlated with the DV is
omitted, its estimated coefficient and test is
invalid
-due to past group discrimination, groups are
often correlated with factors such as income,
education, etc.
2) Self-Selection Problem
-often participation in a program is not
random
7.6 Omitted Example
Consider the following equation:
MT  0  1Study   2 Intel  3ClassSection  u
-Where we are testing whether midterm
achievement is a function of which class
section one is in
-However, the choice of class can depend when
you eat lunch (among other factors), which
can affect Midterm achievement
-Therefore by not including eating (which is
correlated to our DV), our estimate is biased
7.6 Sample Selection Problems
Program evaluation MUST assume that
participation in the program or group (and
thus inclusion in the control group) is random
-however, people often CHOOSE inclusion or noninclusion (ie: people choose to study, people
choose to speed)
-since these decisions are influences by parts of
the error term, our OLS estimation is biased:
E (u | DV )  E (u | DV )
Ie: one doesn’t study due to their drug (Survivor)
addiction, something that they may not report
and thus cannot be included in the regression