Transcript Economics of the Government 政 府 经 济 学
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 4. Multiple Regression Analysis
：
Inference
E., Var., BLUE the full sample distribution of the OLS estimators
Assumption 6 (Normality): Assumption 6 Assumption 3 and 5 (zero conditional mean; homogenous)
古典假设
Classical Linear Model (CLM) Assumptions: Assumption 1
－
6
The Efficiency of the OLS estimators
The Efficiency of the OLS estimators under different assumptions:
Gauss-Markov Assumptions: BLUE, minimum variance
linear
unbiased estimators
CLM Assumptions: minimum variance unbiased estimators (among
all
, not only linear estimators in the y
i
)
The Population Assumptions of the CLM:
Assumption 6 Given x, the distribution of y is normal
Clearly wrong e.g. narr86; prate How?
The Normality of the OLS estimators
Normality of u of the OLS estimators: normal sampling distributions
Therefore, standard normal random v.:
More results:
any linear combination of the is also normally distributed, and any subset of the has a joint normal distribution. Testing Hypothesis
Whether Normality of u Can Be Assumed?
Empirical matter: transformation, log(price) CLT(
中心极限定理
) : Non-normality of the errors is not a serious problem with large sample sizes.
Even though the y
i
estimators are not from a normal distribution, the OLS are approximately normally distributed, at least in large sample sizes.
（
Chap. 5.
） 仅需
Gauss-Markov
假设；有限方差、同方差、零条件均值
Whether normality of u can be assumed?
And
Other Topics about Large Sample
Finite sample properties: unbiasedness; BLUE
Large sample properties: the asymptotic properties
Consistency
Inconsistency:
The problem does
not
go away by adding more observations to the sample.
Very hard to Derive the sign and magnitude of the inconsistency in the general k regressor case, just as to derive the bias
say, x
1
is correlated with u but the other independent v.-s are uncorrelated with u,
all
of the OLS estimators are generally inconsistent.
Other Topics about Large Sample
Large sample properties: the asymptotic properties
Asymptotic Normality
Even without the normality assumption (Assumption MLR.6), t and F statistics have approximately t and F distributions, at least in large sample sizes. (the OLS estimators are approximately normally distributed)
Key assumption: Homoskedasticity
If Var(y x) is not constant, the usual t statistics and confidence intervals are invalid no matter how large the sample size is; the central limit theorem does not bail us out when it comes to heteroskedasticity.
is a consistent estimator of
4.2 TESTING HYPOTHESES: A Single Population Parameter
The t Test:
Null Hypothesis:
t Statistic of :
Significance Level a:
the prob. of rejecting H 0 when it is true
Confidence Level: 1
－
a
Testing Against One-Sided Alternatives
Alternative Hypothesis:
Critical Value c and Rejection Rule:
If , reject H 0 and accept H 1 at the significant level a
If , reject H 0 and accept H 1 at the significant level a
Only need know: degree of freedom; significant level
If df>120
，
the critical value of normal distribution
Testing Against Two-Sided Alternatives
Alternative Hypothesis:
Critical Value c and Rejection Rule:
If , reject H 0 and accept H 1 at the significant level
c is the 97.5
th
，
(1 a/2 ) th percentile in the t distribution with n k 1 degrees of freedom.
P-Values for Test
P-values:
Given the observed value of the t statistic, what is the
smallest
significance level at which the null hypothesis would be rejected?
small p-values are evidence against the null; large p values provide little evidence against H 0 .
once the p-value has been computed, a classical test can be carried out at any desired level.
one-sided p-value: just divide by 2. the two-sided p-value
Some Guidelines
Language: “
在
5%
的水平下不能拒绝零假设” “
we fail to reject H 0 at the x% level ” Statistical significance: the size of Economic Significance: the size (and sign) of
大样本容量常常容易导致统计显著性；在样本容量逐渐扩大 时考虑使用越来越小的显著性水平，以弥补标准误的减小
some guidelines for discussing the economic and statistical significance of a variable in a multiple regression model:
统计显著性。统计显著 系数大小（注意观测单位、函数形式） 如果不显著，变量对
y
是否有预期的影响？实际中影响多大？ 增大
p
值
t
统计量很小的变量的符号：可忽略；难以解决的真正问题 型和函数性质（遗漏变量；内生性；异方差） 模
Example 4.2: Students Performance and School Size
One claim: everything else being equal, students at smaller schools fare better than those at larger schools. This hypothesis is assumed to be true even after accounting for differences in class sizes across schools.
Data: MEAP93.RAW
, 408 high schools in Michigan, 1993
Variable:
Students performance:
通过
MEAP
的
10
分制数学测验的学 生百分比 （
math10
）
School Size:
学生注册人数
(enroll)
Control factors(
可能影响学生成绩的其他因素
):
年均教师 工资
(totcomp);
千名学生教师人数
(staff) Null Hypothesis:
Example 4.2: Students Performance and School Size
The Estimated Equation:
Hypothesis Tests
：
df:
The t statistic on enroll:
we fail to reject H 0 in favor of H 1 at the 5% level.
totcomp is statistically significant even at the 1% significance level; staff is not statistically significant at the 10% level.
Function Form?
Students Performance and School Size
Function Form? Level
－
Log
Hypothesis Tests
：
The t statistic on log(enroll):
we reject in favor of at the 5% level
Interpretations: (Holding totcomp and staff fixed )
Thus, if enrollment is 10% higher at a school, is predicted to be 1.3 percentage points lower (math10 is measured as a percent).
Which One is Better? a higher R-squared for the level log model (Chap 6)
4.3 Confidence Intervals
Confidence Intervals (Interval Estimates):
Meaning:
将在
95%
的随机抽样样本中出现在置信区间
4.4 Testing Hypothesis about a single linear combination of the parameters
E.g.
Null Hypothesis: Meaning: Under H 0 , another year at a junior college and another year at a university lead to the same ceteris paribus percentage increase in wage.
t Statistic:
Another Way:
Testing Hypothesis about a single linear combination of the parameters
E.g.
Note: a/2
4.5 Testing Multiple Linear Hypothesis: The F Test
E.g.
Multiple Hypothesis Test (Joint Hypothesis Test ): F Statistic:
Unrestricted Model:
Restricted Model:
Restriction Rule:
The R-Squared Form of the
F
Statistic
F Statistic:
P Value:
The F Statistic for Overall Significance of a Regression
SST
Testing General Linear Restrictions
4.5 Reporting Regression Results
How to report multiple regression results for relatively complicated empirical projects —— Including:
the estimated OLS coefficients (interpret the estimated coefficients of the key variables; the units of measurement)
the standard errors (s.e VS t statistic)
The R-squared
The number of observations
Reporting the SSR and the standard error of the regression is sometimes a good idea, but it is not crucial.
Summarized in equation form Summarize the results in one or more tables.
Example 4.10: Salary-Pension Tradeoff of Teachers
Economic Model:
（
Standard wage model)
Econometric Model
Key variable: b/s
Null Hypothesis:
Controlling v. : enroll, staff, droprate, gadrate
Regression and Reports
Interpretations
Example 4.10: Salary-Pension Tradeoff of Teachers
Analysis of Cross Section and Panel Data
Part 1. Regression Analysis on Cross Sectional Data
Chap 6. Multiple Regression Analysis
：
Further Issues
6.1 Effects of Data Scaling on OLS Statistics
Units of Measurement Estimator; Statistic
standard errors, t statistics, F statistics, and confidence intervals.
When variables are rescaled, the coefficients, standard errors, confidence intervals, t statistics, and F statistics change in ways that preserve all measured effects and testing outcomes .
含
log
形式只改变截距，不改变斜率系数，不 改变
R 2
，不改变检验结果
6.1 Effects of Data Scaling on OLS Statistics
Beta Coefficients
E.g. test score Standardized: Z=(X-
均值
)/
标准差
(sd)
Advantages:
The importance of each v.: the scale of the regressors irrelevant, comparing the magnitudes
6.2 More on Functional Form
Logarithmic Functional Forms
Models with Quadratics
Models with Interaction Terms
6.2.1 More on Using Logarithmic Functional Forms
Approximate Accurate Small % change: not crucial Large change: mitigate or eliminate heteroskedastic or skewed Advantages of using Ln:
leads to coefficients with appealing interpretations
when y>0, models using log(y) as the dependent v. often satisfy the CLM assumptions more closely than models using the level of y
taking logs usually narrows the range of the v., in some cases by a considerable amount. This makes estimates less sensitive to outlying (or extreme) observations on the dependent or independent v.
Some Standard Rules of Thumb for Taking Logs
The Log is often taken when the v. being large integer values
a positive dollar amount, wages, salaries, firm sales, and firm market value;
population, total number of employees, and school enrollment Variables that are measured in years — such as education, experience, tenure, age, and so on — usually appear in their
original
form.
A variable that is a proportion or a percent arrest rate on reported crimes in level forms.
— — such as the unemployment rate, the participation rate in a pension plan, the percentage of students passing a standardized exam, the can appear in either original or logarithmic form, although there is a tendency to use them
a percentage change and a percentage point change
More on Taking Logs
If a variable takes on zero or negative values:
The percentage change interpretations are often closely preserved, except for changes beginning at y=0 (where the percentage change is not even defined)
One
drawback
to using a dependent v. in logarithmic form is that it is more difficult to predict the original v.
it is not legitimate to compare
R 2
from models where y is the dependent v. in one case and log(y) is the dependent v. in the other.
(see Section 6.4)
6.2.2 Models with Quadratics
Decreasing or Increasing Marginal Effects: Quadratic Functions
parabolic shape and U-shape
Turning point
If this turning point is beyond all but a small percentage of the v. in the sample, then this is not of much concern
.
Other forms:
using quadratics along with logarithms.
some care is needed in making a useful interpretation with a dependent v. in log form and an explanatory v. entering as a quadratic
a nonconstant elasticity: double-log and quadratics
other polynomial terms: a cubic and even a quartic term
6.2.3 Models with Interaction Terms
Interaction Effects:
Sometimes it is natural for the partial effect, elasticity, or semi-elasticity of the dependent v. with respect to an explanatory v. to depend on the magnitude of yet another explanatory v.
Example 6.3: Effects of Attendence
on Final Exam Performance
Dependent v.: Final Exam Performance
Standardized
outcome on a final exam (stndfnl)
Be easier to interpret a student ’ s performance relative to the rest of the class Explanatory v.-s:
Percentage of classes attended (atdrte)
prior college grade point average (priGPA), and ACT score Functional Form:
Quadratics in priGPA and ACT
class attendance might have a different effect for students who have performed differently in the past, as measured by priGPA.
an interaction between priGPA and the atdrte. Econometric Model
Example 6.3: Regression Results, Inference and Interpretations
Sample Regression Function
：
Jiont Hypothesis: What ’ s the Partial Effect of atdrte (priGPA) on stndfnl?
We must plug in interesting values of priGPA to obtain the partial effect.
The mean of priGPA :at the mean priGPA, the effect of atndrte on stndfnl is
-.0067+ .0056
×
(2.59)= .0078.
Meaning: a 10 percentage point increase in atndrte increases stndfnl by .078 standard deviations from the mean final exam score.
Statistic Significance: New Regression
replace priGPA
×
atndrte with (priGPA -2.59)
×
atndrte
.
gives the standard error of , which yields t=.0078/.0026= 3
6.3 More on Goodness-of-fit and Selection of Regressors
R-squared and Adjusted R-squared:
when a new independent v. is added to a regression, SSR and (n-k-1) both decrease, can go up or down.
increases if, and only if, the t statistic on the new v. is greater than one in absolute value. (An extension of this is that increases when a group of v.-s is added to a regression if, and only if, the F statistic for joint significance of the new v.-s is greater than unity.)
A negative indicates a very poor model fit relative to the number of degrees of freedom.
that it is R
2
, not , that appears in the F statistic
Using to Choose Between Nonnested Models
To choose a model without redundant independent v.:
Different functional form (different explanatory v.)
Limitation in using to choose between nonnested models: we cannot use it to choose between different functional forms for the
dependent
variable.
they are fitting two separate dependent variables.
Controlling for Too Many Factors in Regression Analysis
Overemphasize goodness-of-fit,:
regressed log(price) on log(assess), log(lotsize), log(sqrft), and bdrms
we should always include independent v. that affect y and are uncorrelated with all of the independent v.-s of interest.
Adding Regressors to Reduce the Error Variance
6.2 Prediction and Residual Analysis
Confidence Intervals for a Prediction from the OLS regression line.
The estimator of
The Standard Error of
Run the regression: The intercept term
Unobserved Error; Prediction Error
Residual Analysis
Residual Analysis:
whether the actual value of the dependent v. is above or below the predicted value; that is, to examine the residuals for the individual observations. This process is called residual analysis.
Predicting y When log( y) Is the Dependent Variable
The Goodness-of-fit Measure in the y Model and the log( y) Model
it is not legitimate to compare
R 2
from models where y is the dependent v. in one case and log(y) is the dependent v. in the other.
Solution
：
Analysis of Cross Section and Panel Data
Part 1. Regression Analysis on Cross Sectional Data
Chap 7. Multiple Regression Analysis with Qualitative Information: Binary (or Dummy) Variables
7.1 Dummy Explanatory Variables
Describing Dummy (Binary) Variables
赋值
0
，
1 Meaning: e.g.
:Whether there is discrimination against women? The difference in hourly wage between females and males, given the same amount of education (and the same error term u).
Intercept Shift Dummy Variable Trap:
to keep track of which group is the base (benchmark) group.
we will always include an overall intercept for the base group.
Nothing changes about the mechanics of OLS or the statistical theory when some of the
independent
v.-s are defined as dummy variables. The only difference with what we have done up until now is in the interpretation of the coefficient on the dummy variable.
Interpreting Coefficients on Dummy Explanatory
V. When the Dependent V. Is log(y) Percentage Interpretation
Example
：
7.2 The Binary Dependent Variables: The Linear Probability Model
Dependent variable y has quantitative meaning What Happens then? LPM
Interpretation of the OLS coefficients:
In the LPM, measures the change in the probability of success when x
j
changes, holding other factors fixed:
Example: the coefficient on educ means that, everything else in (7.29) held fixed, another year of education increases the probability of labor force participation by .038
Limitations of the LPM
Some Shortcomings:
predictions either less than zero or greater than one.
Linear constant marginal effect unrealistic
Smaller marginal effect of subsequent children on working probability of women
It usually works well for values of the independent variables that are near the averages in the sample.
Heteroskedasticy
Unbiasedness; incorrect statistic
Dummy Dependent and Explanatory Variables:
The coefficient measures the predicted difference in probability when the dummy v. goes from zero to one.
7.3 Using Dummy V. in Multiple Categories: Interpretation
Adding a dummy v. married
Same “ marriage premium ” for men and women;
(0,1); (1,1); (1,0); (0,0) Different “ marriage premium ”
Three dummy variables?
marrmale, marrfem, and singfem
.
(1,0,0); (0,1,0); (0,0,1); (0,0,0)
Ordinary Variable:
to define dummy variables for each value or each categories of respective information
Example 7.6 (7.1,7.5) The Determination of log Hourly Wage:
Explanatory Variables: educ, exper, tenure, marriage, gender 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 Regression: 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.
Interpretation:
Example 7.6 (7.1,7.5) The Determination of log Hourly Wage:
7.4 Interaction Effects Involving with Dummy Variables
Adding Interaction Term
The marriage premium depends on gender
the rest of the regression is necessarily identical to (7.11).
Equation (7.14) is just a different way of finding wage differentials across all gender-marital status combinations. It has no real advantages over (7.11); in fact, equation (7.11) makes it easier to test for differentials between any group and the base group of single men.
Interaction Effects: Differences in Slopes
Adding Interaction Term: Differences in Slopes
The return of education depends on gender
Hypothesis Test:
the return to education is the same for women and men.
average wages are identical for men and women who have the same levels of education:
F
test
7.4.2 Testing for Differences in Regression Functions Across Groups
H 0 : two populations or groups follow the same regression function, against the alternative that one or more of the slopes differ across the groups.
Chow Statistic:
Caution: there is no simple R separate regressions have been estimated for each group; the R
2
form of the test can be used only if interactions have been included to create the unrestricted model.
2
form of the test if One important limitation of the Chow test: regardless of the method used to implement it, is that the null hypothesis allows for no differences at all between the groups.
7.5 Policy Analysis and Program Evaluation with Dummy Variables
Policy analysis; Program evaluation
Control group; experimental (treatment) group be careful to include factors that might be systematically related to the binary independent variable of interest.
Self-Selection Problems:
The term is used generally when a binary indicator of participation might be systematically related to unobserved factors.
another way that an explanatory variable can be endogenous .
Solutions:
Data
more advanced methods
Example: the effect of the job training grants on worker productivity
Consider again the Holzer et al. (1993) study, where we are now interested in the effect of the job training grants on worker productivity (as opposed to amount of job training, example 7.3).
References
Jeffrey M. Wooldridge, Introductory Econometrics —— A Modern Approach, Chap 4
－
7.