Analysis of Cross Section and Panel Data

Yan ZhangSchool of Economics, Fudan University

CCER, Fudan University

Introductory EconometricsA Modern Approach

Yan ZhangSchool of Economics, Fudan University

CCER, Fudan University

Analysis of Cross Section and Panel Data

Part 1. Regression Analysis on Cross Sectional Data

Chap 2. The Simple Regression Model——Practice for learning multiple Regression

Bivariate linear regression model

:the slope parameter in the relationship between y and x holding the other factors in u fixed; it is of primary interest in applied economics.

:the intercept parameter, also has its uses, although it is rarely central to an analysis.

More Discussion

:A one-unit change in x has the same effect on y, regardless of the initial value of x. Increasing returns: wage-education (f. form)

Can we draw ceteris paribus conclusions about how x affects y from a random sample of data, when we are ignoring all the other factors? Only if we make an assumption restricting how

the unobservable random variable u is related to the explanatory variable x

Classical Regression Assumptions

Feasible assumption if the intercept term is included

Linearly uncorrelated zero conditional expectation

Meaning = 内生性 PRF (Population Regression Function): sth. fixed but unknown

OLS

Minimize uu sample regressi

on function (SRF)

The point is always on the OLS regression line.

拟合值与残差

PRF:

OLS

Coefficient of determination

the fraction of the sample variation in y that is explained by x.

the square of the sample correlation coefficient between and

Low R-squareds

Units of Measurement

If one of the dependent variables is multiplied by the constant c—which means each value in the sample is multiplied by c—then the OLS intercept and slope estimates are also multiplied by c.

If one of the independent variables is divided or multiplied by some nonzero constant, c, then its OLS slope coefficient is also multiplied or divided by c respectively.

The goodness-of-fit of the model, R-squareds, should not depend on the units of measurement of our variables.

Function Form

Linear Nonlinear Logarithmic dependent variable

A Percentage change in y, semi-elasticity an increasing return to edu. Other nonlinearity: diploma effect

Bi-Logarithmic A a Constant elasticity

Change of units of measurement P45, error: b0* ＝ b0+log(c1)-b1·log(c2)

Bi-LogarithmicAaConstant elasticity

Change of units of measurement

P45, error

b0* ＝ b0+log(c1)-b1·log(c

2)Be proficient at interpreting the coef.

Unbiasedness of OLS Estimators

Statistical properties of OLS 从总体中随机抽样取出的不同样本的 OLS 估计 的分

布性质 Assumptions

Linear in parameters (f. form; advanced methods) Random sampling (time series data; nonrandom sampling) Zero conditional mean (unbiased biased; spurious cor) Sample Variation in the independent variables (colinearity)

Theorem (Unbiasedness) Under the four assumptions above, we have:

Variance of OLS Estimators

的随机抽样以 为中心，问题是 究竟距离 多远？

Assumptions Homoskedasticity:

Error variance A larger means that the distribution of the unobser

vables affecting y is more spread out. Theorem (Sampling variance of OLS estimators)

Under the five assumptions above:

Variance of y given x

Conditional mean and variance of y: Heteroskedasticity

What does depend on?

More variation in the unobservables affecting y makes it more difficult to precisely estimate

The more spread out is the sample of xi -s, the easier it is to find the relationship between

E(y x) and x As the sample size increases, so does the total

variation in the xi. Therefore, a larger sample size results in a smaller variance of the estimator

Estimating Error Variance

Errors (Disturbances) and Residuals Errors: , population Residuals: , estimated f.

Theorem (The unbiased estimator of ) Under the five assumptions above, we have:

standard error of the regression (SER): Estimating the standard deviation in y after the effect of x

has been taken out. Standard Error of :

Regression through the Origin

Regression through the Origin: Pass through E.g. income tax revenue —— income The estimator of OLS:

= only if 0 if the intercept 0, then is a biased

estimator of

Chap 3. Multiple Regression Analysis ： Estimation

Advantages of multiple regression analysis build better models for predicting the dependent variable.

E.g. generalize functional form.

Marginal propensity to consume Be more amenable to ceteris paribus analysis

Chap 3.2 Key assumption: Implication: other factors affecting wage are not relate

d on average to educ and exper. Multiple linear regression model:

:the ceteris paribus effect of xj on y

Ordinary Least Square Estimator

SPF: OLS: Minimize

F.O.C: ceteris paribus interpretations:

Holding fixed, then Thus, we have controlled for the variables

when estimating the effect of x1 on y.

Holding Other Factors Fixed

The power of multiple regression analysis is that it provides this ceteris paribus interpretation even though the data have not been collected in a ceteris paribus fashion.

it allows us to do in non-experimental environments what natural scientists are able to do in a controlled laboratory setting: keep other factors fixed.

OLS and Ceteris Paribus Effects

Step of OLS: (1) :the OLS residuals from a multiple regression

of x1 on

(2) :the OLS estimator from a simple regression

of y on measures the effect of x1 on y after x2,…, xk have b

een partialled or netted out. Two special cases in which the simple regression of y

on x1 will produce the same OLS estimate on x1 as the regression of y on x1 and x2.

Goodness-of-fit

also equal the squared correlation coef. between the actual and the fitted values of y.

R never decreases, and it usually increases when another independent variable is added to a regression.

The factor that should determine whether an explanatory variable belongs in a model is whether the explanatory variable has a nonzero partial effect on y in the population.

Regression through the origin

the properties of OLS derived earlier no longer hold for regression through the origin. the OLS residuals no longer have a zero sample

average. can actually be negative.

to calculate it as the squared correlation coefficient

if the intercept in the population model is different from zero, then the OLS estimators of the slope parameters will be biased.

The Expectation of OLS Estimator

Assumptions( 简单回归模型假定的直接推广；比较 ) Linear in parameters Random sampling Zero conditional mean No perfect co-linearity

none of the independent variables is constant; and there are no exact linear relationships among the in

dependent variables Theorem (Unbiasedness)

Under the four assumptions above, we have:

rank (X)=K

Notice 1: Zero conditional mean

Exogenous Endogenous Misspecification of function form (Chap 9)

Omitting the quadratic term The level or log of variable

Omitting important factors that correlated with any independent v. 如果被遗漏的变量与解释变量相关，则零条件方差不

成立，回归结果有偏 Measurement Error (Chap 15, IV) Simultaneously determining one or more x-s with y (Chap

16, 联立方程组 )

Omitted Variable Bias: The Simple Case

Problem ： Excluding a relevant variable or Under-specifying the model （遗漏本来应该包括在总体（真实）模型中的变量）

Omitted Variable Bias (misspecification analysis) The true population model: The underspecified OLS line: The expectation of : The Omitted variable bias:

前面 3.2节中是 x1对 x2回归

Omitted Variable Bias: Nonexistence Two cases where is unbiased:

The true population model:

is the sample covariance between x1 and x2 over the sample variance of x1

If , then 的无偏性与 x2 无关，估计时只需调整截距，将 x2 放入误差项不影响零条件均值假定

Summary of Omitted Variable Bias:

The expectation of : The Omitted variable bias:

The Size of Omitted Variable Bias

Direction Size A small bias of either sign need not be a cause for

concern. Unknown Some idea

we usually have a pretty good idea about the direction of the partial effect of x2 on y, that is, the sign of

in many cases we can make an educated guess about whether x1 and x2 are positively or negatively correlated.

E.g. (Upward/downward Bias; biased toward zero)高估！

Omitted Variable Bias: More General Cases

Suppose: x2 and x3 are uncorrelated, but that

x1 is correlated with x3. Both and will normally be biased. The o

nly exception to this is when x1 and x2 are also uncorrelated.

Difficult to obtain the direction of the bias in and

Approximation: if x1 and x2 are also uncor.

Notice 2: No Perfect Collinearity

An assumption only about x-s, nothing about the relationship between u and x-s

Assumption MLR.4 does allow the independent variables to be correlated; they just cannot be perfectly correlated. Ceteris Paribus effect If we did not allow for any correlation among the indepen

dent variables, then multiple regression would not be very useful for econometric analysis.

Significance

Cases of Perfect Collinearity

When can independent variables be perfectly collinear software—“singular” Nonlinear functions of the same variable is not an exact

linear f. Not to include the same explanatory variable measured in

different units in the same regression equation. More subtle ways

one independent variable can be expressed as an exact linear function of some or all of the other independent variables. Drop it

Key:

Notice 3: Unbiase

the meaning of unbiasedness: an estimate cannot be unbiased: an estimate is a fixed num

ber, obtained from a particular sample, which usually is not equal to the population parameter.

When we say that OLS is unbiased under Assumptions MLR.1 through MLR.4, we mean that the procedure by which the OLS estimates are obtained is unbiased when we view the procedure as being applied across all possible random samples.

Notice 4: Over-Specification

Inclusion of an irrelevant variable or over-specifying the model ： does not affect the unbiasedness of the OLS

estimators.

including irrelevant variables can have und

esirable effects on the variances of the OLS

estimators.

Variance of The OLS Estimators

Adding Assumptions Homoskedasticity:

Error variance A larger means that the distribution of the unobser

vables affecting y is more spread out. Gauss-Markov assumptions (for cross-sectional regr

ession): Assumption 1-5 Theorem (Sampling variance of OLS estimators)

Under the five assumptions above:

More about

The stastical properties of y on x=(x1, x2, …, xk)

Error variance only one way to reduce the error variance: to add more ex

planatory variables——not always possible and desirable The total sample variations in xj: SSTj

Increase the sample size

Multi-collinearity （多重共线性） The linear relationships among the independent v.

其他解释变量对 xj 的拟合优度（含截距项） If k=2 ： ： the proportion of the total variation in xj that

can be explained by the other independent variables : : : High (but not perfect) correlation between two o

r more of the in dependent variables is called multicollinearity.

Micro-numerosity: problem of small sample size

High Low SSTj one thing is clear: everything else being equal,

for estimating j, it is better to have less correlation between xj and the other x-s.

How to “solve” the multicollinearity? Increase sample size Dropping some v.? 如果删除了总体模型中

的一个变量，则会导致有偏

Notice: The influence of multicollinearity

A high degree of correlation between certain independent variables can be irrelevant as to how well we can estimate other parameters in the model.

E.g.

参见注释

Importance for economists ： controlling v.

Variances in Misspecified Models

Whether or Not to Include x2: Two Favorable Reasons

The choice of whether or not to include a particular variable in a regression model can be made by analyzing the tradeoff between bias and variance..

However, when2 0, there are two favorable reasons for including x2 in the model. any bias in does not shrink as the sample size grows; The variance of estimators both shrink to zero as n increas

e

Therefor, the multicollinearity induced by adding x2 beco

mes less important as the sample size grows. In large samples, we would prefer

Estimating : Standard Errors of the OLS Estimators

参见注释

EFFICIENCY OF OLS: THE GAUSS-MARKOV THEOREM

BLUE “Best”: smallest variance

“linear”:

“unbiased”:

定理含义：（ 1）无需寻找其他线性组合的无偏估计量；（ 2）如果 G-M假设有一个不成立，则 BLUE不成立。例如零条件均值不成立（内生性）会导致有偏；异方差不会有偏，但会使方差不再是最小。

Classical Linear Model Assumptions——Inference

本部分课程内容参考资料

Jeffrey M. Wooldridge, Introductory Econometrics——A Modern Approach, Chap 2-3.