Econometrics report Class number: No number: Eglish name: Chinese name:
Contents Background and Data Analysis 2-5 and model T-test 6-8 F-test 8-10 Summary,and,suggestion 11
BACKGROUND ●The report below is about the food sales , I instance the resident population (10 000 ) , per capita income the first year , meat sales , egg sales , the fish sales . ●In order to build mathematical models to understand the relationship of each variable and its food sales , and I take statistics of Tianjin from 1994 to 2007 the demand for food Among Y X1 X2 X3 X4 X5 1 98.4500 153.2000 560.2000 6.5300 1.2300 1.8900 2 100.7000 190.0000 603.1100 9.1200 1.3000 2.0300 3 102.8000 240.3000 668.0500 8.1000 1.8000 2.7100 4 133.9500 301.1200 715.4700 10.1000 2.0900 3.0000 5 140.1300 361.0000 724.2700 10.9300 2.3900 3.2900 6 143.1100 420.0000 736.1300 11.8500 3.9000 5.2400 7 146.1500 491.7760 748.9100 12.2800 5.1300 6.8300 8 144.6000 501.0000 760.3200 13.5000 5.4700 8.3600 9 146.9400 529.2000 774.9200 15.2900 6.0900 10.0700 10 158.5500 552.7200 785.3000 18.1000 7.9700 12.5700 11 169.6800 771.7600 795.5000 19.6100 10.1800 15.1200 12 162.1400 811.8000 804.8000 17.2200 11.7900 18.2500 13 170.0900 988.4300 814.9400 18.6000 11.5400 20.5900 14 178.6900 1094.6500 828.7300 23.5300 11.6800 23.3700
E7.2 E7.3 E7.4
-------------------------------------------- (1) (2) ahe ahe -------------------------------------------- age 0.605*** 0.585*** (15.02) (16.02) female -3.664*** (-17.65) bachelor 8.083*** (38.00) _cons 1.082 -0.636 (0.93) (-0.59) （表2）Robust ci in parentheses *** p<0.01, ** p<0.05, * p<0.1 -------------------------------------------- N 7711 7711 -------------------------------------------- t statistics in parentheses * p<0.10, ** p<0.05, *** p<0.01 (表1) （1） 建立ahe 对age 的回归。截距估计值是1.082，斜率估计值是0.605。 （2） ①建立ahe 对age ，female 和bachelor 的回归。Age 对收入的效应的估计值是0.585。 ② age 回归系数的95%置信区间： (0.514，0.657) （3） 设H 0:βa,（2）-βa,（1）=0 H1:βa,（2）-βa （1）≠0 由表3，得SE ，SE(βa,（2）-βa,（1）)=√（0.0403）2+(0.0365)2=0.054 t=（0.605-0.585）/0.054=0.37<1.96 所以不拒绝原假设，即在5%显著水平下age 对ahe 的效应估计没有显著差异，所以（1）中的回归没有遭遇遗漏变量偏差。 （4） B ob’s predicted ahe=0.585×26-3.664×0+8.083×0-0.636=＄14.574 Alexis ’s predicted ahe=0.585×30-3.664×1+8.083×1-0.636=＄21.333 VARIABLES ahe age 0.585*** (0.514 - 0.657) female -3.664*** (-4.071 - -3.257) bachelor 8.083*** (7.666 - 8.500) Constant -0.636 (-2.759 - 1.487) Observations 7,711 R-squared 0.200
绪论 一、计量经济学概述 1、什么是计量经济学 Ｒ.Frish(挪威）1926年提出：Ｅcnometrics 定义：经济学、数学及统计学的三者结合 三园图： 依据经济理论、数据资料为基础，运用数学、统计学和计算机技术，以建立经济计量模型为主要手段，定量分析带有随机性特征的经济变量之间关系的规律，验证或发展经济理论、评价经济政策及预测经济活动的一门应用经济学科。 例：前提假设条件：消费主要取决于收入、并随着收入增长呈线性增长、边际消费递减等，则可设定消费C及Y具有下述理论计量经济模型：
u Y C ++=βα 其中：100<<<βα、，u 为随机扰动项（表示：除收入外其它因素对消费的影响） 利用数据资料n i Y C i i ,...,2,1),,(= 并进一步作计量经济学假设：假设模型满足经典（古典）条件， 则可采用普通最小二乘法估计模型参数建立样本数据经验模型，比如 Y C 67.038.2+= 检验模型：t 检验、F 检验、拟合优度检验，经济理论检验、计量经济检验 应用： 2、计量经济学的特点 （1）计量性： （2）模型性: （3）随机性： （4）实证性: 3、计量经济学内容范畴 （1）经典计量经济分析模型和方法 单方程计量经济分析模型和方法（一元、多元线性回归模型和方法） 估计： OLS （普通最小二乘法）、ML （极大似然法）、
GMM（广义矩法）、 BAYES法 检验：t检验、F检验、拟合优度检验 预测：点预测、区间预测 联立方程计量经济分析模型和方法 识别：结构式法、简化式法 估计： IlS（间接）、2SLS（二阶段）、3SLS（三阶段）、LIML（有限 信息极大似然）、FLML（完全信息ML）、最小方差比等 预测：简化式的多重多元线性回归 （2）非经典计量经济分析模型和方法 异方差性线性回归模型（估计：ＧＬＳ、ＷＬＳ、数学变换法；检验） 自相关性线性回归模型(估计：ＧＬＳ、广义差分变换；检验） 多重共线性线性回归模型 随机解释变量线性回归模型 非正态扰动线性回归模型 非线性回归模型 虚变量线性回归模型 误差变量线性回归模型
Chapter 9. Assessing Studies Based on Multiple Regression 9.1 Internal and External Validity Multiple regression has some key virtues: ?It provides an estimate of the effect on Y of arbitrary changes ΔX. ?It resolves the problem of omitted variable bias, if an omitted variable can be measured and included. ?It can handle nonlinear relations (effects that vary with the X’s)
Still, OLS might yield a biased estimator of the true causal effect. A Framework for Assessing Statistical Studies Internal and External Validity ?Internal validity: The statistical inferences about causal effects are valid for the population being studied.
?External validity: The statistical inferences can be generalized from the population and setting studied to other populations and settings, where the “setting” refers to the legal, policy, and physical environment and related salient features.
P ART T WO Solutions to Empirical Exercises
Chapter 3 Review of Statistics Solutions to Empirical Exercises 1. (a) Average Hourly Earnings, Nominal $’s Mean SE(Mean) 95% Confidence Interval AHE199211.63 0.064 11.50 11.75 AHE200416.77 0.098 16.58 16.96 Difference SE(Difference) 95% Confidence Interval AHE2004 AHE1992 5.14 0.117 4.91 5.37 (b) Average Hourly Earnings, Real $2004 Mean SE(Mean) 95% Confidence Interval AHE199215.66 0.086 15.49 15.82 AHE200416.77 0.098 16.58 16.96 Difference SE(Difference) 95% Confidence Interval AHE2004 AHE1992 1.11 0.130 0.85 1.37 (c) The results from part (b) adjust for changes in purchasing power. These results should be used. (d) Average Hourly Earnings in 2004 Mean SE(Mean) 95% Confidence Interval High School13.81 0.102 13.61 14.01 College20.31 0.158 20.00 20.62 Difference SE(Difference) 95% Confidence Interval College High School 6.50 0.188 6.13 6.87
Econometrics 专业词汇中英文对照（按课件顺序） Ch1-3 Causal effects：因果影响，指的是当x变化时，会引起y的变化；Elasticity：弹性；correlation (coefficient) 相关（系数），相关系数没有单位，unit free； estimation：估计；hypothesis testing：假设检验；confidence interval：置信区间；difference-in-means test：均值差异检验，即检验两个样本的均值是否相同； standard error：标准差；statistical inference：统计推断； Moments of distribution：分布的矩函数；conditional distribution (means)：条件分布（均值）；variance：方差；standard deviation：标准差（指总体方差的平方根）； standard error：标准误差，指样本方差的平方根；skewness：偏度，度量分布的对称性；kurtosis：峰度，度量厚尾性，即度量离散程度；joint distribution：联合分布；conditional expectation：条件期望（指总体）；randomness：随机性 i.i.d., independently and identically distributed：独立同分布的； sampling distribution：抽样分布，指的是当抽取不同的随机样本时，统计量的取值会有所不同，而当取遍所有的样本量为n的样本时，统计量有一个取值规律，即抽样分布，即统计量的随机性来自样本的随机性 consistent (consistency)：相合的（相合性），指当样本量趋于无穷大时，估计量依概率收敛到真实值；此外，在统计的语言中，还有一个叫模型选择的相合性，指的是能依概率选取到正确的模型 Central limit theory：中心极限定理；unbiased estimator：无偏估计量； uncertainty：不确定性；approximation：逼近；least squares estimator：最小二乘估计量；provisional decision：临时的决定，用于假设检验，指的是，我们现在下的结论是基于现在的数据的，如果数据变化，我们的结论可能会发生变化 significance level：显著性水平，一般取0.05或者0.01,0.1，是一个预先给定的数值，指的是在原假设成立的假设下，我们可能犯的错误的概率，即拒绝原假设的概率； p-value：p-值，指的是观测到比现在观测到的统计量更极端的概率，一般p-值很小的时候要拒绝原假设，因为这说明要观测到比现在观测到的统计量更极端的情况的概率很小，进而说明现在的统计量很极端。当p-值小于显著性水平时，在该显著性水平下拒绝原假设 Ch4-5 Linear regression：线性回归；ordinary least squares (OLS)：最小二乘； sample regression line：指的是由样本得到的回归方程；measure of fit：拟合程度的度量；regression R2：回归R平方，指的是Y的方差中被X所解释的部分，属于[0,1]，越接近1，说明拟合越好，其中拟合指的是回归方程对Y的解释程度 standard error of the regression (SER)：回归标准差，越小说明拟合越好 degree of freedom：自由度，指的是可以自由变化的参数个数；outliers：异常值； simple random sampling：简单随机抽样，指的是完全的随机抽样，得到的样本满足独立同分布的性质 t-statistics：t统计量；two-sided hypothesis：双边假设检验，指的是被择假设是双边的；null hypothesis：原假设，零假设；alternative hypothesis：被择假设，对立于原假设binary：0、1的分布；homoskedasticity:同方差的，可以写成homogeneity；
E4.1 E4.2 E4.3 E4.4
E4.1 VARIABLES ahe age 0.605 (0.0245) Constant 1.082 (0.688) Observations 7,711 R-squared 0.029 Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 1. ① 截距估计值estimated intercept: 1.082 ② 斜率估计值estimated slope: 0.605 回归方程：ahe= 1.082+0.605*age ③ 当工人年长 1 岁，平均每小时工资增加0.605 美元。 2. Bob: 0.605*26+1.082=16.812 （美元） Alexis: 0.605*30+1.082=19.232 （美元） 答：预测Bob 的收入为每小时16.812美元，Alexis为19.232 美元。 3. 年龄不能解释不同个体收入变化的大部分。因为R-squared 反映了因变量的 全部变化能通过回归关系被自变量充分解释的比例，而分析得R-squared 的值为0.029，解释度低，说明年龄不能解释不同个体收入变化的大部分
E4.1 (0.0449) Observations 463 R-squared 0.036 Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 ① 截距估计值： 3.998 斜率估计值： 0.133 回归方程： Course_Eval=3.998+0.133*beauty lave_esruo 0a u ty a e 1. 答：两者看上去有微弱的正相关关系 2. VARIABLES course eval beauty Constant 0.133 (0.0550) 3.998
Chapter 6. Linear Regression with Multiple Regressors 6.1 Omitted Variable Bias（遗漏变量偏差） OLS estimate of the Test Score/STR relation: n TestScore= 698.9 – 2.28×STR, R2 = .05, SER = 18.6 (10.4) (0.52) Is this a credible estimate of the causal effect on test scores of a change in the student-teacher ratio? 1
No: there are omitted confounding factors (family income; whether the students are native English speakers) that bias the OLS estimator: STR could be “picking up” the effect of these confounding factors. 2
Omitted Variable Bias The bias in the OLS estimator that occurs as a result of an omitted factor is called omitted variable bias. For omitted variable bias to occur, the omitted factor “Z” must be: 1.a determinant of Y; and 2.correlated with the regressor X. 3
WEEK 10: MACROECONOMETRICS Introduction 1.The concept of stationarity 2.Spurious regressions 3.Testing for unit roots 4.Cointegration analysis
1. S TATIONARITY Conditions for t y to be a stationary time series process i. t E y constant t ii. t Var y constant t iii. ,t t k Cov y y constant t and all k≠0 Autoregressive time series 1t t t y y - Notice no constant and t is a white noise error term. - AR(1) model – time series behaviour of t y is largely explained by its value in the previous period. - Necessary condition for stationarity 1 , if , 1 series is explosive and if 1 have a unit root.
Example 1 – Stationary AR(1) Model STATA code set obs 500 /*set number of observations*/ gen time=_n /*create time trend*/ gen y=0 if time==1 /* first observation set y=0*/ gen e=rnormal(0, 1) /*create a random number*/ replace y=(0.67*y[_n-1])+e if time~=1 /*AR(1) model =0.67*/ twoway (line y time) /*line plot*/
Chapter 8. Nonlinear Regression Functions 8.1 A General Strategy for Modeling Nonlinear Regression Functions ?Everything so far has been linear in the X’s ?The approximation that the regression function is linear might be good for some variables, but not for others.
?The multiple regression framework can be extended to handle regression functions that are nonlinear in one or more X.
The TestScore – STR relation looks approximately linear…
But the TestScore – average district income relation looks like it is nonlinear.
If a relation between Y and X is nonlinear: ?The effect on Y of a change in X depends on the value of X – that is, the marginal effect of X is not constant ?A linear regression is mis-specified – the functional form is wrong ?The estimator of the effect on Y of X is biased – it needn’t even be right on average. 遗漏高次项会带来遗漏变量偏 差。例如： () 2 012 Y X X u βββ =+++，显然X与2 X相关。
一、 Two-sample t test with equal variances Group Obs Mean Std.Err. Std.Dev. 95% Conf. Interval 1992 7,612 11.62 0.0644 5.619 11.49 11.74 2012 7,440 19.80 0.124 10.69 19.56 20.04 combined 15,052 15.66 0.0770 9.442 15.51 15.81 diff -8,183 0.139 -8.455 -7.911 diff = mean(1992) - mean(2012) t = -58.9871 Ho: diff = 0 degrees of freedom = 15050 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(T < t) = 0.0000 Pr(|T| > |t|) = 0.0000 Pr(T > t) = 1.0000 二、 Two-sample t test with equal variances Group Obs Mean Std.Err. Std.Dev. 95% Conf. Interval 1992 7,612 15.64 0.0867 7.564 15.47 15.81 2012 7,440 19.80 0.124 10.69 19.56 20.04 combined 15,052 17.69 0.0772 9.471 17.54 17.85 diff -4.164 0.151 -4.459 -3.869 diff = mean(1992) - mean(2012) t = -27.6423 Ho: diff = 0 degrees of freedom = 15050 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(T < t) = 0.0000 Pr(|T| > |t|) = 0.0000 Pr(T > t) = 1.0000
Chapter 5. Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals 5.1 Testing Hypotheses about One of the Regression Coefficients（对单一系数的假设检验） Suppose a skeptic suggests that reducing the number of students in a class has no effect on learning or, specifically, test scores. The skeptic thus asserts the hypothesis, 1
H0: β1 = 0 We wish to test this hypothesis using data – reach a tentative conclusion whether it is correct or incorrect. Null hypothesis and two-sided alternative: H0: β1 = 0 vs. H1: β1≠ 0 or, more generally, 2
H0: β1 = β1,0 vs. H1: β1≠β1,0 where β1,0 is the hypothesized value under the null（β1,0是一个具体的数）. Null hypothesis and one-sided alternative: H0: β1 = β1,0 vs. H1: β1 < β1,0 In economics, it is almost always possible to come up with stories in which an effect could “go either way,” so it is 3
E4.1 E4.2 E4.3 E4.4
VARIABLES ahe age 0.605 (0.0245) Constant 1.082 (0.688) Observations 7,711 R-squared 0.029 Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 1.①截距估计值estimated intercept:1.082 ②斜率估计值estimated slope:0.605 回归方程：ahe= 1.082+0.605*age ③当工人年长1岁，平均每小时工资增加0.605美元。 2.Bob: 0.605*26+1.082=16.812（美元） Alexis: 0.605*30+1.082=19.232（美元） 答：预测Bob的收入为每小时16.812美元，Alexis为19.232美元。 3.年龄不能解释不同个体收入变化的大部分。因为R-squared反映了因变量的 全部变化能通过回归关系被自变量充分解释的比例，而分析得R-squared的值为0.029，解释度低，说明年龄不能解释不同个体收入变化的大部分。
1. 答：两者看上去有微弱的正相关关系 2. VARIABLES course_eval beauty 0.133 (0.0550) Constant 3.998 (0.0449) Observations 463 R-squared 0.036 Robust standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 ①截距估计值：3.998 斜率估计值：0.133 回归方程：Course_Eval=3.998+0.133*beauty
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Chapter 2. Review of Probability 2.1 Random Variables and Probability Distributions 概率Probability： 在大量重复实验下，事件发生的频率趋向的某个稳定值。例如，记事件“下雨”为A，其发生的概率为P()A。
条件概率Conditional Probability ： 例：已知明天会出太阳，下雨的概率有多大？ 记事件“出太阳”为B 。则在出太阳的前提条件下降雨的“条件概率”（conditional probability ）为， P() P()P()A B A B B ∩≡ 其中，“∩”表示事件的交集（intersection ），故P()A B ∩为“太阳雨”的概率，参见图2.1。条件概率是计量经济学的重要概念之一。
图2.1、条件概率示意图
独立事件Independence ： 如果条件概率等于无条件概率，即P()P()A B A =，即B 是否发生不影响A 的发生，则称,A B 为相互独立的随机事件。此 时，P() P()P()P()A B A B A B ∩≡=，故 P()P()P()A B A B ∩= 也可以将此式作为独立事件的定义。
全概公式 如果事件组{}12,,,(2)n B B B n ≥ 两两互不相容， ()0(1,,) i P B i n >?= ，且12n B B B ∪∪∪ 为必然事件（即在 12,,,n B B B 中必然有某个i B 发生，“∪”表示事件的并集， union ），则对任何事件A 都有（无论A 与{}12,,,n B B B 是否有任何关系）， 1P()P()P()n i i i A B A B ==∑ 全概公式把世界分成了n 个可能的情形，再把每种情况下的条件概率“加权平均”而汇总成无条件概率（权重