14
Introduction Chapter to panel data
Overview
Increasingly, researchers are now using panel data where possible in preference to cross-sectional data. One
major reason is that dynamics may be explored with panel data in a way that is seldom possible with crosssectional data. Another is that panel data offer the possibility of a solution to the pervasive problem of omitted
variable bias. A further reason is that panel data sets often contain very large numbers of observations and the
quality of the data is high. This chapter describes fixed effects regressions and random effects regression,
alternative techniques that exploit the structure of panel data.
Learning outcomes
After working through the corresponding chapter in the text, studying the corresponding slideshows, and doing
the starred exercises in the text and the additional exercises in this guide, you should be able to:
explain the differences between panel data, cross-sectional data, and time series data
explain what the benefits that can be obtained using panel data
explain the differences between OLS pooled regressions, fixed effects regressions, and random effects
regressions
explain the potential advantages of the fixed effects model over pooled OLS
explain the differences between the within-groups, first differences, and least squares dummy variables
variants of the fixed effects model
explain the assumptions required for the use of the random effects model
explain the advantages of the random effects model over the fixed effects model when the assumptions are
valid
explain how to use a DurbinWuHausman test to determine whether the random effects model may be used
instead of the fixed effects model
Additional exercises
A14.1 The NLSY2000 data set contains the following data for a sample of 2,427 males and 2,392 females for the
years 19802000: years of work experience, EXP, years of schooling, S, and age, AGE. A researcher
investigating the impact of schooling on willingness to work regresses EXP on S, including potential
work experience, PWE, as a control. PWE was defined as
PWE = AGE S 5
The following regressions were performed for males and females separately:
(1) an ordinary least squares (OLS) regression pooling the observations
(3) a within-groups fixed effects regression
(3) a random effects regression
The results of these regressions are shown in the table. Standard errors are given in parentheses.
April 2011
2
Males
OLS
Females
FE
RE
OLS
FE
RE
S
0.78
(0.01)
0.65
(0.01)
0.72
(0.01)
0.89
(0.01)
0.71
(0.02)
0.85
(0.01)
PWE
0.83
(0.003)
0.94
(0.001)
0.94
(0.001)
0.74
(0.004)
0.88
(0.002)
0.87
(0.002)
10.16
(0.09)
dropped
10.56
(0.14)
11.11
(0.12)
dropped
12.39
(0.19)
constant
R2
0.79
n
24,057
24,057
24,057
DWH 2(2)
0.71
18,758
10.76
18,758
18,758
1.43
Explain why the researcher included PWE as a control.
Evaluate the results of the DurbinWuHausman tests
For males and females separately, explain the differences in the coefficients of S in the OLS and FE
regressions.
For males and females separately, explain the differences in the coefficients of PWE in the OLS and
FE regressions.
A14.2 Using the NLSY2000 data set, a researcher fits OLS and fixed effects regressions of the logarithm of
hourly wages on schooling, years of work experience, EXP, ASVABC score, and dummies MALE,
ETHBLACK, and ETHHISP for being male, black, or hispanic. Schooling was split into years of high
school, SH, and years of college, SC. The results are shown in the table, with standard errors placed in
parentheses.
OLS
FE
RE
SH
0.026
(0.002)
0.005
(0.007)
0.016
(0.004)
SC
0.063
(0.001)
0.073
(0.004)
0.067
(0.002)
EXP
0.033
(0.0004)
0.032
(0.0003)
ASVABC
0.012
(0.0003)
0.011
(0.001)
MALE
0.193
(0.004)
0.197
(0.009)
ETHBLACK
0.040
(0.007)
0.030
(0.015)
ETHHISP
0.047
(0.008)
0.033
(0.018)
constant
5.639
(0.028)
5.751
(0.051)
0.0367
R2
DWH (3)
2
April 2011
0.033
(0003)
9.31
3
If an individual reported being in high school or college, the observation for that individual for that year
was deleted from the sample. As a consequence, the observations for most individuals in the sample
begin when the formal education of that individual has been completed. However a small minority of
individuals, having apparently completed their formal education and having taken employment,
subsequently resumed their formal education, either to complete high school with a general educational
development (GED) degree equivalent to the high school diploma, or to complete one or more years of
college.
Discuss the differences in the estimates of the coefficient of SH.
Discuss the differences in the estimates of the coefficient of SC.
A14.3 A researcher has data on G, the average annual rate of growth of GDP 20012005, and S, the average
years of schooling of the workforce in 2005, for 28 European Union countries. She believes that G
depends on S and on E, the level of entrepreneurship in the country, and a disturbance term u:
G = 1 + 2S + 3E + u
(1)
u may be assumed to satisfy the usual regression model assumptions. Unfortunately the researcher does not
have data on E.
Explain intuitively and mathematically the consequences of performing a simple regression of G on S.
For this purpose S and E may be treated as nonstochastic variables.
The researcher does some more research and obtains data on G*, the average annual rate of growth of GDP
19962000, and S*, the average years of schooling of the workforce in 2000, for the same countries. She
thinks that she can deal with the unobservable variable problem by regressing G, the change in G, on S,
the change in S, where
G = G G*
S = S S*
assuming that E would be much the same for each country in the two periods. She fits the equation
G = 1 + 2S + w
(2)
where w is a disturbance term that satisfies the usual regression model assumptions.
Compare the properties of the estimators of the coefficient of S in (1) and of the coefficient of S in (2).
Explain why in principle you would expect the estimate of 1 in (2) not to be significant. Suppose that
nevertheless the researcher finds that the coefficient is significant. Give two possible explanations.
Random effects regressions have potential advantages over fixed effect regressions.
Could the researcher have used a random effects regression in the present case?
A14.4 A researcher has the following data for 3,763 respondents in the United States National Longitudinal
Survey of Youth 1979 : hourly earnings in dollars in 1994 and 2000, years of schooling as recorded in
1994 and 2000, and years of work experience as recorded in 1994 and 2000. The respondents were aged
1421 in 1979, so they were aged 2936 in 1994 and 3542 in 2000. 371 of the respondents had
increased their formal schooling between 1994 and 2000, 210 by one year, 101 by two years, 47 by three
years, and 13 by more than three years, mostly at college level in non-degree courses. The researcher
performs the following regressions:
(1) the logarithm of hourly earnings in 1994 on schooling and work experience in 1994
(2) the logarithm of hourly earnings in 2000 on schooling and work experience in 2000
(3) the change in the logarithm of hourly earnings from 1994 to 2000 on the changes in schooling and
work experience in that interval.
April 2011
4
The results are shown in columns (1) (3) in the table (t statistics in parentheses), and are presented at a
seminar.
Dependent
variable
Schooling
Experience
Cognitive
ability score
Male
Black
Hispanic
Change in
schooling
Change in
experience
constant
R2
n
April 2011
(1)
log earnings
1994
(2)
log earnings
2000
0.114
(30.16)
0.052
(18.81)
0.116
(28.99)
0.038
(14.59)
0.214
(12.03)
0.149
(5.23)
0.039
(1.11)
0.229
(11.77)
0.199
(6.44)
0.053
(1.38)
4.899
(74.59)
0.265
3,763
5.023
(65.02)
0.243
3,763
(3)
Change in log
earnings
19942000
0.090
(5.00)
0.024
(2.75)
0.102
(2.13)
0.007
3,763
(4)
log earnings
2000
0.108
(24.53)
0.037
(14.10)
0.004
(4.79)
0.230
(11.88)
0.167
(5.29)
0.071
(1.84)
4.966
(63.69)
0.248
3,763
(5)
Change in log
earnings
19942000
0.006
(0.16)
0.003
(0.15)
0.389
(3.05)
0.0002
371
The researcher is unable to explain why the coefficient of the change in schooling in regression (3) is
so much lower than the schooling coefficients in (1) and (2). Someone says that it is because he has
left out relevant variables such as cognitive ability, region of residence, etc, and the coefficients in (1)
and (2) are therefore biased. Someone else says that cannot be the explanation because these
variables are also omitted from regression (3). Explain what would be your view.
He runs regressions (1) and (2) again, adding a measure of cognitive ability. The results for the 2000
regression are shown in column (4). The results for 1994 were very similar. Discuss possible reasons
for the fact that the estimate of the schooling coefficient differs from those in (2) and (3).
Someone says that the researcher should not have included a constant in regression (3). Explain why
she made this remark and assess whether it is valid.
Someone else at the seminar says that the reason for the relatively low coefficient of schooling in
regression (3) is that it mostly represented non-degree schooling. Hence one would not expect to
find the same relationship between schooling and earnings as for the regular pre-employment
schooling of young people. Explain in general verbal terms what investigation the researcher should
undertake in response to this suggestion.
Another person suggests that the small minority of individuals who went back to school or college in
their thirties might have characteristics different from those of the individuals who did not, and that
this could account for a different coefficient. Explain in general verbal terms what investigation the
researcher should undertake in response to this suggestion.
Finally, another person says that it might be a good idea to look at the relationship between earnings
and schooling for the subsample who went back to school or college, restricting the analysis to these
371 individuals. The researcher responds by running the regression for that group alone. The result is
shown in column (5) in the table. The researcher also plots a scatter diagram, reproduced below,
showing the change in the logarithm of earnings and the change in schooling. For those with one extra
year of schooling, the mean change in log earnings was 0.40. For those with two extra years, 0.37.
For those with three extra years, 0.47. What conclusions might be drawn from the regression results?
5
4
3
change in log earnings
2
1
0
1
0
2
3
4
5
-1
-2
-3
-4
change in schooling
Answer to the starred exercise in the text
14.9
(This exercise should have had a star.)
The NLSY2000 data set contains the following data for a sample of 2,427 males and 2,392 females for the
years 19802000: weight in pounds, years of schooling, age, marital status in the form of a dummy
variable MARRIED defined to be 1 if the respondent was married, 0 if single, and height in inches.
Hypothesizing that weight is influenced by schooling, age, marital status, and height, the following
regressions were performed for males and females separately:
(1) an ordinary least squares (OLS) regression pooling the observations
(2) a within-groups fixed effects regression
(3) a random effects regression
The results of these regressions are shown in the table. Standard errors are given in parentheses.
Males
Females
OLS
FE
RE
OLS
FE
RE
Years of
schooling
0.98
(0.09)
0.02
(0.23)
0.45
(0.16)
1.95
(0.12)
0.60
(0.27)
1.25
(0.18)
Age
1.61
(0.04)
1.64
(0.02)
1.65
(0.02)
2.03
(0.05)
1.66
(0.03)
1.72
(0.03)
Married
3.70
(0.48)
2.92
(0.33)
3.00
(0.32)
8.27
(0.59)
3.08
(0.46)
1.98
(0.44)
Height
5.07
(0.08)
dropped
4.95
(0.18)
3.48
(0.10)
dropped
3.38
(0.21)
209.52
(5.39)
dropped
209.81
(12.88)
105.90
(6.62)
dropped
107.61
(13.43)
constant
R2
n
DWH 2(3)
April 2011
0.27
17,299
17,299
17,299
7.22
0.17
13,160
13,160
13,160
92.94
6
Explain why height is excluded from the FE regression.
Evaluate, for males and females separately, whether the fixed effects or random effects model
should be preferred.
For males and females separately, compare the estimates of the coefficients in the OLS and FE
models and attempt to explain the differences.
Explain in principle how one might test whether individual-specific fixed effects jointly have
significant explanatory power, if the number of individuals is small. Explain why the test is not
practical in this case.
Answer: Height is constant over observations. Hence, for each individual,
HEIGHTit HEIGHTi 0
for all t, where HEIGHTi is the mean height for individual i for the observations for that individual.
Hence height has to be dropped from the regression model.
The critical value of chi-squared, with three degrees of freedom, is 7.82 at the 5 percent level and
16.27 at the 0.1 percent level. Hence there is a possibility that the random effects model may be
appropriate for males, but it is definitely not appropriate for females.
Males
The OLS regression suggests that schooling has a small (one pound less per year of schooling) but
highly significant negative effect on weight. The fixed effects regression eliminates the effect,
indicating that an unobserved effect is responsible: males with unobserved qualities that have a positive
effect on educational attainment, controlling for other measured variables, have lower weight as a
consequence of the same unobserved qualities. We cannot compare estimates of the effect of height
since it is dropped from the FE regression. The effect of age is the same in the two regressions. There
is a small but highly significant positive effect of being married, the OLS estimate possibly being
inflated by an unobserved effect.
Females
The main, and very striking, difference is in the marriage coefficient. The OLS regression suggests that
marriage reduces weight by eight pounds, a remarkable amount. The FE regression suggests the
opposite, that marriage leads to an increase in weight that is similar to that for males. The clear
implication is that women who weigh less are relatively successful in the marriage market, but once
they are married they put on weight .
For schooling the story is much the same as for males, except that the OLS coefficient is much
larger and the coefficient remains significant at the 5 percent level in the FE regression. The effect of
age appears to be exaggerated in the OLS regression, for reasons that are not obvious.
One might test whether individual-specific fixed effects jointly have significant explanatory power
by performing a LSDV regression, eliminating the intercept in the model and adding a dummy variable
for each individual. One would compare RSS for this regression with that for the regression without the
dummy variables, using a standard F test. In the present case it is not a practical proposition because
there are more than 17,000 males and 13,000 females.
Answers to the additional exercises
A14.1
Explain why the researcher included PWE as a control.
Clearly work actual experience is positively influenced by PWE. Omitting it would cause the
coefficient of S to be biased downwards since PWE and S are negatively correlated.
April 2011
7
Evaluate the results of the DurbinWuHausman tests
With two degrees of freedom, the critical value of chi-squared is 5.99 at the 5 percent level and 9.21 at
the 1 percent level. Thus the random effects model is rejected for males but seemingly not for
females.
For males and females separately, explain the differences in the coefficients of S in the OLS and FE
regressions.
For both sexes the OLS estimate is greater than the FE estimate. One possible reason is that some
unobserved characteristics, for example drive, are positively correlated with both acquiring schooling,
and seeking and gaining employment.
For males and females separately, explain the differences in the coefficients of PWE in the OLS and
FE regressions.
Since S and PWE are negatively correlated, these same unobserved characteristics would cause the
OLS estimate of the coefficient of PWE to be biased downwards.
A14.2 First, note that the DWH statistic is significant at the 5 percent level (critical value 7.82) but not at the 1
percent level (critical value 11.35).
The coefficients of SH and SC in the OLS regression is an estimate of the impact of variations in years
of high school and years of college among all the individuals in the sample. Most individuals in fact
completed high school and so had SH = 12. However, a small minority did not and this variation made
possible the estimation of the SH coefficient. The majority of the remainder did not complete any years
of college and therefore had SC = 0, but a substantial minority did have a partial or complete college
education, some even pursuing postgraduate studies, and this variation made possible the estimation of
the SC coefficient.
Most individuals completed their formal education before entering employment. For them,
SH it SH i for all t and hence SH it SH i 0 for all t. As a consequence, the observations for such
individuals provide no variation in the SH variable. Likewise they provide no variation in the SC
variable. If all observations pertained to such individuals, schooling would be washed out in the FE
regression along with other unchanging characteristics such as sex, ethnicity, and ASVABC score. The
schooling coefficients in the FE regression therefore relate to those individuals who returned to formal
education after a break in which they found employment.
The fact that these individuals account for a relatively small proportion of the observations in the data
set has an adverse effect on the precision of the FE estimates of the coefficients of SH and SC. This is
reflected in standard errors that are much larger than those obtained in the OLS pooled regression.
Discuss the differences in the estimates of the coefficient of SH.
Most of the variation in SH in the FE regressions come from individuals earning the GED degree.
This degree provides an opportunity for high school drop-outs to make good their shortfall by taking
courses and passing the examinations required for this diploma. These course may be civilian or
military adult education classes, but very often they are programmes offered to those in jail. In
principle the GED should be equivalent to the high school diploma, but there is some evidence that
standards are sometimes lower. The results in the table appear to corroborate this view. The OLS
regression indicates that a year of high school raises earnings by 2.6 percent, with the coefficient
being highly significant, whereas the FE coefficient indicates that the effect is only 0.5 percent and not
significant.
Discuss the differences in the estimates of the coefficient of SC.
Some of the variation in SC in the FE regressions comes from individuals entering employment for a
year or two after finishing high school and then going to college, resuming their formal education.
April 2011
8
However most comes from individuals returning college for a year or two after having been
employment for a number of years. A typical example is a high school graduate who has settled down
in an occupation and who has then decided to upgrade his or her professional skills by taking a twoyear associate of arts degree. Similarly one encounters college graduates who upgrade to masters
level after having worked for some time. One would expect such students to be especially well
motivatedthey are often undertaking studies that a relevant to an established career, and they are
often bearing high opportunity costs from loss of earnings while studyingand accordingly one might
expect the payoff in terms of increased earnings to be relatively high. This seems to be borne out in a
comparison of the OLS and FE estimates of the coefficient of SC, though the difference is not
dramatic.
On the surface, this exercise appeared to be about how one might use FE to eliminate the bias in OLS
pooled regression caused by unobserved effects. Has the analysis been successful in is respect?
Absolutely not. In particular, the apparent conclusion that high school education has virtually no effect
on earnings should not be taken at face value. The reason is that the issue of biases attributable to
unobserved effects has been overtaken by the much more important issue of the difference in the
interpretation of the SH and SC coefficients discussed in the exercise. This illustrates a basic point in
econometrics: understanding the context of the data if often just as important as being proficient at
technical analysis.
A14.3
Explain intuitively and mathematically the consequences of performing a simple regression of G on S.
For this purpose S and E may be treated as nonstochastic variables.
If one fits the regression
G b1 b2 S ,
then
S S G G
S S
S S S
b2
i
i
2
i
i
2 3
1
2
i
3 Ei u i 1 2 S 3 E u
S S E E S S u u
S S
S S
i
S
i
S
2
i
i
i
2
2
i
i
Taking expectations, and making use of the invitation to treat S and E as nonstochastic,
E b2 2 3
S S E E S S E u u
S S
S S
S S E E
S S
i
i
i
i
2 3
i
2
i
2
i
i
2
i
Hence the estimator is biased unless S and E happen to be uncorrelated in the sample.
consequence, the standard errors will be invalid.
Compare the properties of the estimators of the coefficient of S in (1) and of the coefficient of S in (2).
Given (1), the differenced model should have been
G = 2S + w
where w = u u*.
April 2011
As a
9
The estimator of the coefficient of S in (2) should be unbiased, while that of S in (1) will be subject
to omitted variable bias. However:
it is possible that the bias in (1) may be small. This would be the case if E were a relatively
unimportant determinant of G or if its correlation with S were low.
it is possible that the variance in S is smaller than that of S. This would be the case if S were
changing slowly in each country, or if the rate of change of S were similar in each country.
Thus there may be a trade-off between bias and variance and it is possible that the estimator of 2 using
specification (1) could actually be superior according to some criterion such as the mean square error. It
should be noted that the inclusion of 1 in (2) will make the estimation of 2 even less efficient.
Explain why in principle you would expect the estimate of 1 in (2) not to be significant. Suppose that
nevertheless the researcher finds that the coefficient is significant. Give two possible explanations.
If specification (1) is correct, there should be no intercept in (2) and for this reason the estimate of the
intercept should not be significantly different from zero. If it is significant, this could have occurred as a
matter of Type I error. Alternatively, it might indicate a shift in the relationship between the two time
periods. Suppose that (1) should have included a dummy variable set equal to 0 in the first time period
and 1 in the second. d1 would then be an estimate of its coefficient.
Could the researcher have used a random effects regression in the present case?
Random effects requires the sample to be drawn randomly from a population and for unobserved
effects to be uncorrelated with the regressors. The first condition is not satisfied here, so random
effects would be inappropriate
A14.4
The researcher is unable to explain why the coefficient of the change in schooling in regression (3) is
so much lower than the schooling coefficients in (1) and (2). Someone says that it is because he has
left out relevant variables such as cognitive ability, region of residence, etc, and the coefficients in (1)
and (2) are therefore biased. Someone else says that cannot be the explanation because these
variables are also omitted from regression (3). Explain what would be your view.
Suppose that the true model is
LGEARN 1 2 S 3 EXP 4 ASVABC 5 MALE
6 ETHBLACK 7 ETHHISP 8 X 8 u
where X8 is some further fixed characteristic of the respondent. ASVABC and X8 are absent from
regressions (1) and (2) and so those regressions will be subject to omitted variable bias. In particular,
since ASVABC is likely to be positively correlated with S, and to have a positive coefficient, its
omission will tend to bias the coefficient of S upwards.
However, if the specification is valid for both 1994 and 2000 and unchanged, one can eliminate
the omitted variable bias by taking first differences as in regression (3):
LGEARN 2 S 3 EXP u
By fitting this specification one should obtain unbiased estimates of the coefficients of schooling and
experience, and the former should therefore be smaller than in (1) and (2). Note that all the fixed
characteristics have been washed out. The suggestion that ASVABC should have been included in (3)
is therefore incorrect.
Note that (3) should not have included an intercept. This is discussed in later in the question.
April 2011
He runs regressions (1) and (2) again, adding a measure of cognitive ability. The results for the 2000
regression are shown in column (4). The results for 1994 were very similar. Discuss possible reasons
for the fact that the estimate of the schooling coefficient differs from those in (2) and (3).
10
The estimate of the coefficient of S differs from that in (2) because the omitted variable bias
attributable to the omission of ASVABC in that specification has now been corrected. However it is
still biased if X8 (representing other omitted characteristics) is a determinant of earnings and is
correlated with S. This partial rectification of the omitted variable problem accounts for the fact that
the coefficient of S in (4) lies between those in (2) and (3).
Someone says that the researcher should not have included a constant in regression (3). Explain why
she made this remark and assess whether it is valid.
Given the specification in (1) and (2), there should have been no intercept in the first differences
specification (3). One would therefore expect the estimate of the intercept to be somewhere near zero
in the sense of not being significantly different from it. Nevertheless, it is significantly different at the
5 percent level. However, suppose that the relationship shifted between 1994 and 2000, and that the
shift could be represented by a dummy variable D equal to zero in 1994 and 1 in 2000, with
coefficient . Then (3) should have an intercept . Its estimate, 0.102, suggests that earnings grew by
10 percent from 1994 to 2000, holding other factors constant. This seems entirely reasonable, perhaps
even a little low.
Alternatively, the apparently significant t statistic might have arisen as a matter of Type I error.
Someone else at the seminar says that the reason for the relatively low coefficient of schooling in
regression (3) is that it mostly represented non-degree schooling. Hence one would not expect to
find the same relationship between schooling and earnings as for the regular pre-employment
schooling of young people. Explain in general verbal terms what investigation the researcher should
undertake in response to this suggestion.
Divide S into two variables, schooling as of 1994 and extra schooling as of 2000, with separate
coefficients. Then use a standard F test (or t test) of a restriction to test whether the coefficients are
significantly different.
Another person suggests that the small minority of individuals who went back to school or college in
their thirties might have characteristics different from those of the individuals who did not, and that
this could account for a different coefficient. Explain in general verbal terms what investigation the
researcher should undertake in response to this suggestion.
The issue is sample selection bias and an appropriate procedure would be that proposed by Heckman.
One would use probit analysis with an appropriate set of determinants to model the decision to return
to school between 1994 and 2000, and a regression model to explain variations in the logarithm of
earnings of those respondents who do return to school, linking the two models by allowing their
disturbance terms to be correlated. One would test whether the estimate of this correlation is
significantly different from zero.
Finally, another person says that it might be a good idea to look at the relationship between earnings
and schooling for the subsample who went back to school or college, restricting the analysis to these
371 individuals. The researcher responds by running the regression for that group alone. The result
is shown in column (5) in the table. The researcher also plots a scatter diagram, reproduced below,
showing the change in the logarithm of earnings and the change in schooling. For those with one
extra year of schooling, the mean change in log earnings was 0.40. For those with two extra years,
0.37. For those with three extra years, 0.47. What conclusions might be drawn from the regression
results?
The schooling coefficient is effectively zero! [These are real data, incidentally.] The scatter diagram
shows why. Irrespective of whether the respondent had one, two, or three years of extra schooling,
the gain is about the same, on average. (These are the only categories with large numbers of
observations, given the information at the beginning of the question, confirmed by the scatter
diagram.) So the results indicate that the fact of going back to school, rather than the duration of the
April 2011
11
schooling, is the relevant determinant of the change in earnings. The intercept indicates that this
subsample on average increased their earnings between 1994 and 2000 by 38.9 percent. (As a first
approximation. The actual proportion would be better estimated as e0.389 1 = 0.476.) This figure is
confirmed by the diagram, and it would appear to be much greater than the effect of regular schooling.
One explanation could be sample selection bias, as already discussed. A more likely possibility is that
the respondents were presented with opportunities to increase their earnings substantially if they
undertook certain types of formal course, and they took advantage of these opportunities.
April 2011