Econ 302.
Econometrics
Bilkent University
Department of Economics
Taskin
Homework Exercise 3
Pooled and Panel data exercise
Part I:
1. Fixed Effect Model:
Suppose that the panel data estimation with group fixed effects model is represented by the following
equation:
yit = ati + bit xit + uit
or in matrix form the equation is;
é
ê
ê
ê
ê
ê
ê
ê
êë
y1 ù é I 0 . . . 0 ùé a1 ù é X1 ù é
ú ê
ú ê
ú ê
úê
y2 ú ê 0 I . . . 0 úê a 2 ú ê X2 ú ê
ú
ú ê
úê . ú ê
. ú=ê
+
ê
ú
ê
úB + ê
ê
ú
. ú ê
.
ê
ú
ê
ú ê
ú
ú
ê
ú
ê
ú ê
.
ê
ú .
ú ê 0 0 0 0 0 I úê
ú ê
ú ê
yn ûú ë
ûêë a n úû êë Xn ûú êë
u1 ù
ú
u2 ú
ú
. ú
. ú
. ú
ú
un úû
where there are n group and T observations on each group.
The equation can also be written as:
y = éë d1 d2 . . . dn
é a ù
ú+ u
X ùûê
êë b úû
a)What are the coefficients that will be estimated in the above system and how are the interpretation?
b) Suppose that the model is formulated with an overall constant term and n-1 dummy variables (by
dropping last dummy variable). Examine the effect this will have on the coefficients of the set of dummy
variables and the OLS estimates of the slopes?
2. Random Effects model:
2. The random effects model:
The regression equation is:
yit = ai + bit xit + uit
Instead of treating
a i as fixed we assume that ai = a + ei for i =1, 2....n .
e
Where i is the random error with mean zero and variance
equation will be:
s e2 . If the substitute for , a i the regression
yit = a + bit xit + uit + ei
and the new error is wi = ei + uit where i =1, 2....n
The assumptions of the model are:
uit ~ N(0, s u2 ) and ei ~ N(0, s e2 ) Furthermore, E(eiuit ) = 0, E(ei e j ) = 0 (i ¹ j), and
E(uit uis ) = E(uit u jt ) = E(uit u js ) = 0
Drive the form for the Var(w) matrix. This will give you the special form of GLS correction relevant in the
Random Effects model.
(hint: write out the error vector for the NxT observations)
Part 2:
Computer Exercise:
Topic: Implementation of seat belt rules are seen as a policy rule designed to decrease the fatality rates in
traffic crashes in all countries. The following questions and data will give you an econometric analysis of the
impact of the change in the seat belt rules on the severity of traffic accidents. The relationship between
fatality and seat belt usage is modelled in a panel data framework with data collected across states and time
period 1983-1997, for US. The description of the data can be found in the file: Seat-belts_description.pdf.
(Stock and Watson) The work file designed for pooled estimation via EVIEWS is in seatbelts2.xls.
1.
In the following model:
fatalityratei = b0 + b1sb_usei + b2 speed65i + b3speed70i + b4ba08i + b5dage21i + b6 ln(incomei )+ b7agei + ui
ESTIMATIONS:
a) Compute summary statistics of the variables in the given data, and plot the dependent
variable across different groups.
b) Pool data across different groups and estimate a ‘pooled estimation’, ie. estimate with
common intercept and slope coefficients. Examine the significance of the variables.
c) Estimate the equation using fixed effect model. Formulate to include group (state) fixed
effects. Examine the significance of the variables.
d) Estimate the equation using time fixed effects model. Examine the significance of the
variables.
e) Estimate the equation using both cross section and time fixed effects model. Examine the
significance of the variables.
CORRECTIONS
f)
Continuing from part (c ); estimate the model with group weights (correction for different
error variance across groups,
Var(uit ) = s i2 where i =1...n ). Compare the coefficient
estimates and their significance.
g) Continuing from part (d )Estimate the model with correction for different error variance
across time, ie Var(uit ) = s t where
2
t =1983...1997 ). Compare the coefficient estimates
and their significance.
h) Estimate the model using White’s Heteroscedasticity consistent standard errors, with the
assumption that error variances changes across groups (states). Evaluate the estimation
results.
i) Estimate the model using White’s Heteroscedasticity consistent standard errors, with the
assumption that error variances changes over time (years). Evaluate the estimation results.
(not reported)
j) Estimate the model using Random Effects estimation model, with fixed effects for states
and white’s heteroschedasticity consistent standard errors.
ANALYZE THE FOLLOWING STATEMENTS ACCORDING TO THE ESTIMATION
RESULTS:
k) Does the estimated coefficient suggest that increased seat belts reduce fatalities? How does
this conclusion change across different estimations when you add fixed effects? Why?
l) Do the results change when you add time fixed effects?
m) Explain which regression is most reliable? Explain why?
n) Using the results in (h) discuss the size of the coefficient on sb_use. How many lives would
be saved if seat belts use increase from 50% to 80%
Report the results of these estimations into the following table, where each cell reports the
coefficient value and relevant t_stat.
Dependent variable: Number of fatalities per million of traffic miles
Regressor
(1)
(2)
(3)
(4)
SB_USE
(5)
(6)
(7)
example:
XXX*
(t-stat)
SPEED65
SPEED70
BA08
DAGE
LN(INCOME
AGE
Method of
estimation
STATE
EFFECTS
TIME EFFECTS
WHITE’S
ST. ERROR
GSL
CORRECTION
(GROUP)
GSL
CORRECTION
(YEAR)
no
yes
no
yes
no
yes
yes
no
no
no
no
yes
no
no
no
yes
no
no
yes
no
yes
no
no
no
yes
no
no
no
no
no
no
no
yes
no
no
RANDOM
EFFECTS
MODEL
no
no
no
no
no
no
Restrictions
Tests
State effects=0
Time effects=0
Haussman Test
# of lives saved
due to seat belts
usage
No Speed Limit
*, **, *** indicates that the coefficient is significant at 1%, 5% and 10% respectively.
yes