Analysis of Cross Section
and Panel Data
Yan Zhang
School of Economics, Fudan University
CCER, Fudan University
Introductory Econometrics
A Modern Approach
Yan Zhang
School of Economics, Fudan University
CCER, Fudan University
Analysis of Cross Section and Panel Data
Part 1. Regression Analysis on
Cross Sectional Data
Chap 8. Heteroskedasticity

Heteroskedasticity

Robust statistic
 Heteroskedasticity-robust t statistic
Heteroskedasticity-robust s.e. (White-Huber-Eicker s.e.)
 Heteroskedasticity-robust F statistic
 Heteroskedasticity-robust LM statistic
异方差检验方法：B-P方法；White方法；
 异方差处理：GLS
 FGLS

8.1 Consequences of
Heteroskedasticity for OLS
Heteroskedasticity
 Not Change:

 Unbiasedness
 Consistency

Change:
 Biased variance of OLS estimator,
 Invalid t, F, LM statistic
 OLS No longer BLUE
 OLS no longer asymptotically efficient

Solutions:
 Modify the OLS test statistics
 More efficient estimator
8.2 Heteroskedasticity-Robust
Inferences after OLS Estimation
 Heteroskedasticity-Robust
procedures
how to adjust standard errors, t, F, and LM
statistics so that they are valid in the presence of
heteroskedasticity of unknown form.
 Robust
statistic
Heteroskedasticity-robust t statistic
Heteroskedasticity-robust s.e. (White-Huber-
Eicker s.e.)
Heteroskedasticity-robust F statistic
Heteroskedasticity-robust LM statistic (E.g 5.3, 8.3)
Example 8.1 (7.6, 7.1, 7.5) The
Determination of log Hourly Wage:

Stata Command
 use WAGE1
 generate marrmale=married*(1-female)
 generate marrfem=married*female
 generate singfem=(1-married)*female
 reg lwage marrmale marrfem singfem educ exper expersq
tenure tenursq
 test tenure tenuresq
 reg lwage marrmale marrfem singfem educ exper expersq
tenure tenursq, robust
 test tenure tenuresq (Heteroskedasticity-robust F statistic)
Example 8.1 (7.6, 7.1, 7.5) The
Determination of log Hourly Wage:
 Heteroskedasticity-Robust
 The
Standard Error
same coef., R-squared and Adjusted Rsquared (Unbias)
 Different s.e., t-statistic, p-value, CI, F-statistic
Example 8.1 (7.6): Notice

Dummy Variables:
 Same “marriage premium”; (0,1); (1,1); (1,0); (0,0)
 Different “marriage premium”; (1,0,0); (0,1,0); (0,0,1);
(0,0,0)
 Adding Interaction Term

Inference:
 we can use this equation to obtain the estimated difference
between any two groups.
 Unfortunately, we cannot use it for testing whether the
estimated difference between single and married women is
statistically significant. to choose one of these groups to be
the base group and to reestimate the equation.
Example 8.2: The Determination of GPA
 Stata
Command
use GPA3
describe
reg cumgpa sat hsperc tothrs female black
white if spring==1
test black white
regress cumgpa sat hsperc tothrs female
black white if spring>0, robust
test black white (Heteroskedasticity-robust F statistic)
The Determination of GPA:补充
Chap 7.4.3——Chow Statistic

Stata Command
 gen fmsat=female*sat
 gen fmhsperc=female* hsperc
 gen fmtothrs=female*tothrs
 label variable fmsat “=female*sat”
 label variable fmhsperc "=female*hsperc"
 label variable fmtothrs"=female*tothrs"
 reg cumgpa female sat fmsat hsperc fmhsperc tothrs
fmtothrs if spring==1
 test female fmsat fmhsperc fmtothrs
 test fmsat fmhsperc fmtothrs
 reg cumgpa female sat hsperc tothrs if spring==1
8.3 Testing for Heteroskedasticity
Heteroskedasticity-Robust s.e.——不需知道是否存在
异方差
 Testing for Heteroskedasticity

 The Breusch-Pagan Test (B-P Test)
 The White Test

Basic Methods
 BP Test
 White Test
8.3.1 The Breusch-Pagan Test
(BP Test) for Heteroskedasticity
 The
Breusch-Pagan Test (B-P Test)
 Basic Methods
Heteroskedasticity
BP Test
Heteroskedasticity
White Test
8.3.2 The White Test
for Heteroskedasticity

The White Test (B-P Test)
 adds the squares and cross products of all of the
independent variables to equation (8.14).

The procedure of White Test
Notice: Problems with
Heteroskedasticity Tests

Can we always take a rejection using one of the
heteroskedasticity tests as evidence of
heteroskedasticity?
 appropriate provided we maintain Assumptions MLR.1
through MLR.4.
 But, if MLR.3 is violated—in particular, if the functional
form of E(y x) is misspecified—then a test for
heteroskedastcity can reject H0, even if Var(y/x) is constant.
If we omit one or more quadratic terms in a regression
model or use the level model when we should use the log,
a test for heteroskedasticity can be significant.
8.4 Weighted Least Square
Estimations (WLS)
Methods:
Var(u/x)=σ2 Ω
Ω-1 =P’P
Y=XB+u
B^=(X’ Ω-1 X) -1(X’ Ω-1 Y)
 WLS

 Weight: The efficient procedure, GLS, weights each
squared residual by the inverse of the conditional variance
of ui given xi
 Examples of GLS
 The R-squares of OLS and WLS are not comparable
Example 8.6: Family Saving Function
Marginal Propensity to Save:
 STATA Command:

 help weights; use saving
 reg sav inc; reg sav inc size educ age black;
the var. of the error
is proportional to
the level of income
This means that, as
income increases,
the variability in
savings increases.
 test size educ age black
 reg sav inc [pw=1/inc]; reg sav inc size educ age black
[pw=1/inc]
 test size educ age black

Notice:
 not comparable R-squares
 compare the coef.——either is good
 adding demographic control variables——individually and
jointly insignificant
Notice: The Weights
Unknown
 One case where the weights needed for WLS arise
naturally from an underlying econometric model.

 Individual level data——data across some group （firm-
level） or geographic region
 A similar weighting arises when we are using per capita
data at the city, county, state, or country level. If the
individual-level equation satisfies the Gauss-Markov
assumptions, then the error in the per capita equation has a
variance proportional to one over the size of the population.
Therefore, weighted least squares with weights equal to the
population is appropriate.
8.4.2 Feasible GLS: The Heteroskedasticity F. Must Be Estimated
FGLS Estimator: Using the estimator, , instead of
hi in the GLS transformation yields an estimator
(model the function h and use the data to estimate the
unknown parameters)
 One FGLS:


Procedure:
Notice: The Properties of FGLS
The FGLS estimator is neither unbiased, nor BLUE
 The FGLS estimator is still consistent, and
asymptotically more efficient than OLS
 for large sample sizes, FGLS is an attractive
alternative to OLS when there is evidence of
heteroskedasticity that inflates the standard errors of
the OLS estimates.
 The FGLS estimator measures the marginal impact of
each xj on y,
 The F statistic with WLS: same weights in both
restricted and unrestricted models

Compare with the OLS and WLS
Estimators

the OLS and WLS estimates can be substantially different.
 Not a big problem in the e.g.——all the coefficients maintain the same
signs, and the biggest changes are on variables that were statistically
insignificant when the equation was estimated by OLS.
 The OLS and WLS estimates will always differ due to sampling error.
The issue is whether their difference is enough to change important
conclusions.
 If OLS and WLS produce statistically significant estimates that differ in
sign, or the difference in magnitudes of the estimates is practically large,
we should be suspicious. Typically, this indicates that one of the other
Gauss-Markov assumptions is false, particularly the zero conditional
mean assumption on the error (MLR.3).
Hausman Test
Example 8.7: Demand for Cigarettes
 Homework
 OLS
 BP test
 FGLS
 Interpretation
Chap 9. More on Specification and
Data Problem
 Functional
Form Misspecification
Heteroskedasticity
assumption 3, zero conditional mean
correlation between the error, u, and one or
more of the explanatory variables.
Endogenous Explanatory Variable
 Specific
problem and Solutions
Omitting v.
Proxy Variable
measurement error
 Data
Problem
9.1 Functional Form Misspecification


表现：多元回归模型没有正确的解释因变量和观测到的解释变量之间的
关系
E.g.:
 Explanatory variables:
 Log-wage: the return to working experience, exper2
 Biased estimator of all coef.s
 Log-wage: the return to education, female*educ
explanation
 Explained Variable
 Log-wage
wage
 Unobservable key variable

检验：the F test for joint exclusion restrictions.
 增加一个显著变量的平方项，进行联合显著性检验
 缺点：无法确定函数形式误设的确切原因；使用大量自由度
 一般情形下，对数形式和平方项
RESET as a General Test for
Functional Form Misspecification

RESET: Regression Specification Error Test (回归设定误差检验)
Idea：RESET adds polynomials in the OLS fitted values to
equation (9.2) to detect general kinds of functional form
misspecification.

Drawbacks:

 it provides no real direction on how to proceed if the model is rejected.
 Just a Functional Form Test, Misguide on omitted v. and
heteroskedasticity
 RESET has no power for detecting omitted v. whenever they have
expectations that are linear in the included independent v. in the
model
 if the functional form is properly specified, RESET has no power
for detecting heteroskedasticity.
Test against Non-nested Alternatives


Construct a Comprehensive Model

Davidson-Mackinnon Test

Problems with non-nested test
 a clear winner need not emerge (reject or accept
simultaneously)
 rejecting one does not mean the other is right
 difficult when the non-nested models have different
dependent variables
9.2 Using Proxy Variables for
Unobserved Explanatory Variables

Unobserved omitted v.
Proxy v.
 Loosely speaking, a proxy variable is something that is
related to the unobserved variable that we would like to
control for in our analysis.
Ability & IQ

Plug-in solution to the omitted variables problem
 When does the plug-in solution give consistent estimators?
The error u is uncorrelated with x1, x2, and x3*, in
addition, u is uncorrelated with x3.
The error v3 is uncorrelated with x1, x2, and x3.
 Influence: consistent estimator of
Proxy Variables: cases of Bias

the average level of ability not only changes with IQ,
but also with educ and exper.

Biased estimator of
 Upward bias of proxy variable IQ
Proxy Variables: Lagged Dependent V.
account for historical factors that cause current
differences in the dependent variable that are
difficult to account for in other ways.
 E.g.: Crime rate & expenditure on law enforcement

 the main reason for putting crime-1 in the equation is that
cities with high historical crime rates may spend more on
crime prevention.
 Hardly perfect, but better

Other way
 differentials
9.3 Properties of OLS under
Measurement Error

Measurement Error: use an imprecise measure of an
economic v. in a regression model
 Marginal tax rate (average)

Differences between proxy variables and measurement error
 Different conceptually:
 In the proxy variable case, looking for a v. that is somehow
associated with the unobserved v.
 In the measurement error case, the v. that we do not observe has a
well-defined, quantitative meaning, but our recorded measures of
it may contain error.
 Different primary interests:
 In the proxy variable case, we are usually concerned with the
effects of the other independent v.
 In the measurement error case, the mis-measured independent v.

只有当可搜集到数据的变量与影响个人决策的变量不同时，
测量误差才成为问题
9.3.1 Measurement Error in the
Dependent Variable




The bottom line is that measurement error in the dependent v.
can cause biases in OLS if it is systematically related to one or
more of the explanatory v.-s. If the measurement error is just
a random reporting error that is independent of the
explanatory v.-s, then OLS is perfectly appropriate.
Measurement Error (the difference between the observed
value and the actual value)
The usual assumption is that the measurement error in y is
statistically independent of each explanatory v. If this is true,
then the OLS estimators from (9.19) are unbiased and
consistent. Further, the usual OLS inference procedures (t, F,
and LM statistics) are valid. (larger var. of OLS estimators)
Multiplicative measurement error:
9.3.2 Measurement Error in the
Independent Variables




Much more important problem
Maintained assumption: u is uncorrelated with x1* and x1.
If:
Then OLS has all of its nice properties.
If:
(Classical Error-in-variables, CEV)
Then biased and inconsistent estimator;
 Attenuation Bias（衰减偏误）：on average (or in large samples), the
estimated OLS effect will be attenuated.
 If the variance of x1* is large, relative to the variance in the
measurement error, then the inconsistency in OLS will be small.
多个变量时，一般的一个变量的
ME会导致所有估计量有偏、不一致

9.4 Missing Data, Nonrandom Samples
and Outlying Observations

Missing Data
 Reduce the sample size
 Missing at random
 某些样本缺失数据的概率更大

非随机抽样
Nonrandom Samples
 Certain types of nonrandom sampling do not cause bias or
inconsistency in OLS.
 sample selection based on the independent variables：Exogenous
sample selection.
 sample selection based on the dependent variable：Endogenous
sample selection.
Biased and Inconsistent
Outlying Observations

Outlying Observations (Influential Observations)
 Loosely speaking, an observation is an outlier if dropping it from a
regression analysis makes the OLS estimates change by a practically
“large” amount.
 Entering mistakes; sampling from a small population if one or several
members of the population are very different in some relevant aspect
from the rest of the population.


OLS results should probably be reported with and without
outlying observations in cases where one or several data
points substantially change the results.
Certain functional forms are less sensitive to outlying
observations.
 logarithmic transformation significantly narrows the range of the data
and also yields functional forms that can explain a broader range of
data.
Outlying Observations：LAD

least absolute deviations (LAD):
 The LAD estimator minimizes the sum of the absolute deviation of the
residuals, rather than the sum of squared residuals.
 Compared with OLS, LAD gives less weight to large residuals. Thus, it
is less influenced by changes in a small number of observations.

Drawbacks：
 there are no formulas for the estimators
 LAD consistently estimates the parameters in the population
regression function (the conditional mean), only when the distribution
of the error term u is symmetric.
 if the error u is normally distributed, LAD is less efficient
(asymptotically) than OLS.

Robust Regression:
Example 9.1 (1.1, 3.5, 5.3, 7.12, 8.3)
Economic Model of Crime
 Economics
of Crime (Gary Becker, 1968)
 Economic Model: choice of labor supply
 Econometric
 Functional
Model:
Form Misspecification?
 Proxy Variable?
Example 9.1 (1.1, 3.5, 5.3, 7.12, 8.3)
Economic Model of Crime
 Data:
CRIME1.RAW contains data on arrests
during the year 1986 and other information
on 2,725 men born in either 1960 or 1961 in
California. Each man in the sample was
arrested at least once prior to 1986.
 Functional Form Misspecification?
Quadratic terms?
 Proxy Variable?
E.g. 9.4 (CRIME2.RAW)
Example 9.1 (1.1, 3.5, 5.3, 7.12, 8.3)
Economic Model of Crime

Regression and Results
 reg narr86 pcnv ptime86 qemp86 avgsen(+)
 问题：判刑时间越长，增加犯罪活动？(e.g. 3.5; avgsen的系数不
显著)
 reg narr86 pcnv ptime86 qemp86 avgsen tottime
 test avgsen tottime (F, LM)
 二值因变量线性概率模型LPM（e.g. 7.12）
 gen arr86=narr86; replace arr86=1 if arr86>0
 reg arr86 pcnv avgsen tottime ptime86 qemp86 (解释)
 Quadratic terms （显著的项增加其平方项，看其显著性）
 RESET
 predict yhat; predict resid, resid; (drop)

Proxy Variable?
 E.g. 9.4 (CRIME2.RAW)；作业9.2; 9.4; 9.3(习题9.7);
References
Jeffrey
M. Wooldridge, Introductory
Econometrics——A Modern
Approach, Chap 4－7.