1. Analysis of Cross Section and Panel Data
Yan Zhang
School of Economics, Fudan University
CCER, Fudan University

2. Introductory Econometrics A Modern Approach
Yan Zhang
School of Economics, Fudan University
CCER, Fudan University

3. Analysis of Cross Section and Panel Data
Part Regression Analysis on Cross Sectional Data

4. Chap 4. Multiple Regression Analysis：Inference
E., Var., BLUE
the full sample distribution of the OLS estimators
Assumption 6 (Normality):
Assumption Assumption 3 and 5
(zero conditional mean; homogenous)
古典假设Classical Linear Model (CLM) Assumptions: Assumption 1－6
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

5. 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 yi)
The Population Assumptions of the CLM:
Assumption Given x, the distribution of y is normal
Clearly wrong e.g. narr86; prate How?
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

6. The Normality of the OLS estimators
Normality of u normal sampling distributions of the OLS estimators:
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
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

7. 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 yi are not from a normal distribution, the OLS estimators 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
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

8. 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, x1 is correlated with u but the other independent v.-s are uncorrelated with u, all of the OLS estimators are generally inconsistent.
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

9. 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(yx) 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 
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

10. 4.2 TESTING HYPOTHESES: A Single Population Parameter
The t Test:
Null Hypothesis:
t Statistic of :
Significance Level a:
the prob. of rejecting H0 when it is true
Confidence Level: 1－a

11. Testing Against One-Sided Alternatives
Alternative Hypothesis:
Critical Value c and Rejection Rule:
If , reject H0 and accept H1 at the significant level a
Only need know: degree of freedom; significant level
If df>120，the critical value of normal distribution

12. Testing Against Two-Sided Alternatives
Alternative Hypothesis:
Critical Value c and Rejection Rule:
If , reject H0 and accept H1 at the significant level
c is the 97.5th，(1-a/2)th percentile in the t distribution with n  k  1 degrees of freedom.

13. 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 H0.
once the p-value has been computed, a classical test can be carried out at any desired level.
one-sided p-value: just divide the two-sided p-value by 2.

14. Some Guidelines Language: “在5%的水平下不能拒绝零假设”
“we fail to reject H0 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 模型和函数性质（遗漏变量；内生性；异方差）

15. 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:

16. Example 4.2: Students Performance and School Size
The Estimated Equation:
Hypothesis Tests：
df:
The t statistic on enroll:
we fail to reject H0 in favor of H1 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?

17. 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)

18. 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

19. 4.4 Testing Hypothesis about a single linear combination of the parameters
E.g.
Null Hypothesis:
Meaning: Under H0, 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:

20. Testing Hypothesis about a single linear combination of the parameters
E.g.
Note: a/2

21. 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:

22. The R-Squared Form of the F Statistic
P Value:
The F Statistic for Overall Significance of a Regression
Testing General Linear Restrictions
SST

23. 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.

24. 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

25. Example 4.10: Salary-Pension Tradeoff of Teachers

26. Chap 6. Multiple Regression Analysis：Further Issues
Analysis of Cross Section and Panel Data Part Regression Analysis on Cross Sectional Data
Chap 6. Multiple Regression Analysis：Further Issues

27. 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R2，不改变检验结果
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

28. 6.1 Effects of Data Scaling on OLS Statistics
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

29. 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

30. 6.2 More on Functional Form
Logarithmic Functional Forms
Models with Quadratics
Models with Interaction Terms
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

31. 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.
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

32. 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—such as the unemployment rate, the participation rate in a pension plan, the percentage of students passing a standardized exam, the arrest rate on reported crimes—can appear in either original or logarithmic form, although there is a tendency to use them in level forms.
a percentage change and a percentage point change
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

33. 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 R2 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)
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

34. 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
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

35. 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.
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

36. 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

37. 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
×(2.59)=
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

38. 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 R2, not , that appears in the F statistic
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

39. 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.
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

40. 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
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

41. 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
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

42. 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.
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

43. Predicting y When log( y) Is the Dependent Variable
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

44. The Goodness-of-fit Measure in the y Model and the log( y) Model
it is not legitimate to compare R2 from models where y is the dependent v. in one case and log(y) is the dependent v. in the other.
Solution：
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

45. Chap 7. Multiple Regression Analysis with Qualitative Information:
Analysis of Cross Section and Panel Data Part Regression Analysis on Cross Sectional Data
Chap 7. Multiple Regression Analysis with Qualitative Information:
Binary (or Dummy) Variables

46. 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.
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

47. Interpreting Coefficients on Dummy Explanatory V. When the Dependent V
Interpreting Coefficients on Dummy Explanatory V. When the Dependent V. Is log(y)
Percentage Interpretation
Example：
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

48. 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 xj 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
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

49. 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.
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

50. 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
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

51. 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:
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

52. Example 7.6 (7.1,7.5) The Determination of log Hourly Wage:
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

53. 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.
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

54. 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
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

55. 7.4.2 Testing for Differences in Regression Functions Across Groups
H0: 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 R2 form of the test if separate regressions have been estimated for each group; the R2 form of the test can be used only if interactions have been included to create the unrestricted model.
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.
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

56. 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
Another problem that often arises in policy and program evaluation is that individuals (or firms or cities) choose whether or not to participate in certain behaviors or programs.

57. 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).
OLS has the smallest variance among unbiased estimators; we no longer have to restrict our comparison to estimators that are linear in the yi.

58. References
Jeffrey M. Wooldridge, Introductory Econometrics——A Modern Approach, Chap 4－7.