1、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 C
2、ross 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,
3、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
4、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 ex
5、pectationMeaning= 内生性 PRF (Population Regression Function): sth. fixed but unknown,OLS,Minimize uu sample regression function (SRF) The point is always on the OLS regression line.,拟合值与残差,PRF:,OLS,Coefficient of determinationthe fraction of the sample variation in y that is explained by x. the square
6、 of the sample correlation coefficient between and Low R-squareds,Units of Measurement,If one of the dependent variables is multiplied by the constant cwhich means each value in the sample is multiplied by cthen the OLS intercept and slope estimates are also multiplied by c. If one of the independen
7、t 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 d
8、ependent 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)-b1log(c2),Bi-Logarithmic A a Constant elasticity Change of units of measurement
9、P45, error b0*b0+log(c1)-b1log(c2) 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 mea
10、n (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
11、unobservables 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 pre
12、cisely estimate The more spread out is the sample of xi -s, the easier it is to find the relationship betweenE(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,Erro
13、rs (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 :,R
14、egression 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 f
15、or 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 related on average to educ and exper. Multiple linear regression model:,:the c
16、eteris 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 th
17、at 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 Parib
18、us Effects,Step of OLS: (1) :the OLS residuals from a multiple regressionof x1 on (2) :the OLS estimator from a simple regression of y onmeasures the effect of x1 on y after x2, xk have been partialled or netted out. Two special cases in which the simple regression of y on x1 will produce the same O
19、LS 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 whet
20、her 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
21、 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 paramete
22、rs 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 independent variables Theorem (Unbiasedness) Under the four assumptions above, we have:,rank (X)=K,Notice 1: Zero conditional mean,Exo
23、genous 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
24、(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:
25、,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 var
26、iable 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
27、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 only exception to this is when x1 and
28、 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 cor
29、related; they just cannot be perfectly correlated. Ceteris Paribus effect If we did not allow for any correlation among the independent variables, then multiple regression would not be very useful for econometric analysis. Significance,Cases of Perfect Collinearity,When can independent variables be
30、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 o
31、f 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 number, obtained from a particular sample, which usually is not equal to the population parameter. When we say that OLS is unbiased und
32、er 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 a
33、ffect the unbiasedness of the OLS estimators. including irrelevant variables can have undesirable 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 unobservables affecting y i
34、s more spread out. Gauss-Markov assumptions (for cross-sectional regression): 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 mor
35、e explanatory variablesnot 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 o
36、ther independent variables: High (but not perfect) correlation between two or 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 corr
37、elation 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 oth
38、er 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 a
39、nd 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 increase Therefor, the multicollinearity induced by adding x2 becomes less important as the sample si
40、ze 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 EconometricsA Modern Approach, Chap 2-3.,
