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1、Econometrics I Fall 2011,Instructor: 冯强 Office: 博学 1223 Phone: 6449-3318 The best way to contact me is by email: ,Brief Overview of the Course,Economics suggests interesting relations, often with policy implications, but virtually never suggests quantitative magnitudes of causal effects. For example
2、: What is the price elasticity of public transportation? Say, a 1yuan reduction in price, by how much can we expect the volume changes? What is the effect of reducing class size on student achievement? What is the effect on earnings of a year of education? What is the effect on GDP (or inflation) of
3、 a 1 percentage point increase in interest rates by the Central Bank?,Other Example of Econometric Studies,What might be the effect a new regulation on the housing price in Beijing? What will be the impact of eliminating residence requirement (户口) on wage rate for college graduate (preferable UIBE s
4、tudents)? The short term or long term impact of the imposition征收 of odd奇/even偶 plates driving days for the demand for cars? What is the effect of GDP on electricity usage? What kind of problem are you interested in using econometrics to study? Can you come up with questions of this kind?,The focus o
5、f this course is the use of statistical and econometric methods to quantify causal effects Ideally, we would like an experiment: Transit prices; class size; returns to education; Central Bank But almost always we must use observational 观测的(nonexperimental) data. Observational data poses major challe
6、nges: consider estimation of returns to education Confounding混淆 effects (omitted省略 factors, such as ability) simultaneous causality 同时因果关系 The higher is the income, the more time one can afford to stay in school “correlation 相关性does not imply causation” High income of the Western world is correlated
7、 with their height, does that mean the taller is the people, the richer they are?,In this course you will:,Learn methods for estimating causal effects using observational data; Lean some basic theories behind the methods in econometrics Learn to produce (you do the analysis) and consume (evaluate th
8、e work of others) econometric applications; and Practice “producing” in your problem sets.,Causal Relations,Q：Which of the following has a causal relationship? Circumference 围and height of a tree No causal relationship, but can be used for prediction nevertheless Wage and output But higher wage can
9、lead to higher moral and higher output Weight and gas consumption of a truck But energy efficient engine uses less gas Cell phone fees and length of calls But long distance calls costs more,Is there a relationship between wage level and mobility?,The difference between experimental data and observat
10、ional data,Designed Experiment can easily test causal relationships, for example: The effect of a kind of fertilizer on tomato crops The effect of a medication on patients blood pressure There is no simple designed experiment for social science For each 1% increase in price, what is the percentage d
11、rop in transit volume? Q: How should we conduct such an experiment on the price elasticity of public transportation? Can each persons bus fair be determined randomly in Beijing, and see how the change in price affects a persons transit decision?,Types of data:,Cross-sectional data (截面数据) e.g. record
12、ings of every students weight for today Time series (时间序列) e.g. the weight record of a person over a year. Panel data (面板数据)the combination of cross-section and time series e.g. the weight records of all the students here for each day and for over a year.,The identification of data type:,Q: the data
13、 in the published 2010 Statistical Abstract of China is typically of what kind? A: Cross-sectional, because it is different entities data for the same time period Q: What kind of data is the published stock market activities? A: Time-series, for it is the realization of a variables value over time.,
14、地 区 年末人数（万） 平均劳动报酬 北 京 514 39,684 天 津 195 27,628 河 北 501 16,456 山 西 366 18,106 内蒙古 243 18,382 辽 宁 498 19,365 吉 林 266 16,393 黑龙江 497 15,894 上 海 333 37,585 江 苏 679 23,657 浙 江 611 27,570 安 徽 338 17,610 福 建 427 19,424 江 西 283 15,370 山 东 898 19,135 A) Cross Section B) Time Series C) Panel D) Not Sure,Wha
15、t kind of data is this?,What kind of data is this?,年 GDP （养殖业 制造业 其他） 1978 3645.2 1018.4 1745.2 881.6 1980 4545.6 1359.4 2192.0 994.2 1985 9016.0 2541.6 3866.6 2607.8 1986 10275.2 2763.9 4492.7 3018.6 1987 12058.6 3204.3 5251.6 3602.7 1988 15042.8 3831.0 6587.2 4624.6 1989 16992.3 4228.0 7278.0 5486
16、.3 1990 18667.8 5017.0 7717.4 5933.4 A) Cross Section B) Time Series C) Panel D) Not Sure,What kind of data is this?,年 GDP 养殖业 制造业 其他 1978 3645.2 1018.4 1745.2 881.6 1980 4545.6 1359.4 2192.0 994.2 1985 9016.0 2541.6 3866.6 2607.8 1986 10275.2 2763.9 4492.7 3018.6 1987 12058.6 3204.3 5251.6 3602.7 1
17、988 15042.8 3831.0 6587.2 4624.6 1989 16992.3 4228.0 7278.0 5486.3 1990 18667.8 5017.0 7717.4 5933.4 A) Cross Section B) Time Series C) Panel D) Not Sure,A) Cross Section B) Time Series C) Panel D) Not Sure,What kind of data is this?,Review of Probability and Statistics,Empirical problem: Class size
18、 and educational outcome Policy question: What is the effect of reducing class size by one student per class? by 8 students/class? What is the right outcome measure (“dependent variable”)? parent satisfaction student personal development future adult welfare and/or earnings performance on standardiz
19、ed tests,What do data say about the class size/test score relation?,The California Test Score Data Set All K-6 and K-8 California school districts (n = 420)地区 Variables: 5th grade test scores (Stanford-9 achievement test, combined math and reading), district average Student-teacher ratio (STR) = no.
20、 of students in the district divided by no. full-time equivalent teachers 全职教师,Initial look at the data: (You should already know how to interpret this table),This table doesnt tell us anything about the relationship between test scores and the STR.,Scatterplot of test score v. student-teacher ratio
21、,Do districts with smaller classes have higher test scores? What does this figure show?,How can we get some numerical evidence on whether districts with low STRs have higher test scores?,There are 3 related numerical measurements: Compare average test scores in districts with low STRs to those with
22、high STRs (“estimation”) Test the hypothesis that the mean test scores in the two types of districts are the same, against the alternative hypothesis that they differ (“hypothesis testing”) Estimate an interval for the difference in the mean test scores, high v. low STR districts (“confidence interv
23、al”),Initial data analysis: Compare districts with “small” (STR 20) and “large” (STR 20) class sizes:,Estimation of (= difference between group means) Test the hypothesis that = 0 Construct a confidence interval for ,1. Estimation,= 657.4 650.0 = 7.4 where and Is this a large difference in a real-wo
24、rld sense? Standard deviation across districts = 19.1 Difference between 60th and 75th percentiles of test score distribution is 667.6 659.4 = 8.2 Is this a big enough difference to be important for school reform discussions, for parents, or for a school committee?,2. Hypothesis testing,Difference-i
25、n-means test: compute the t-statistic, (remember this?) where SE( ) is the “standard error” of the subscripts s and l refer to “small” and “large” STR districts; and (etc.),Compute the difference-of-means t-statistic:,Q: Can we reject the null hypothesis that =0? A: Yes, since |t| 1.96, we can rejec
26、t (at the 5% significance level) the null hypothesis that the two means are the same.,3. Confidence interval,A 95% confidence interval for the difference between the means is, ( ) 1.96SE( ) = 7.4 1.961.83 = (3.8, 11.0) Q: Are the following two statements equivalent ? The 95% confidence interval for
27、doesnt include 0; The hypothesis that = 0 is rejected at the 5% level. A: Yes, they are.,This should all be familiar But:,What is the underlying framework that justifies all this? Estimation: Why estimate by ? Testing: What is the standard error of , really? Why reject = 0 if |t| 1.96? Confidence in
28、tervals (interval estimation): What is a confidence interval, really?,Review of Statistical Concepts,We will review the following in turn The probability framework for statistical inference Estimation Hypothesis testing Confidence Intervals,1. The probability framework for statistical inference,Here
29、 are some key concepts: Population Random variable Y Population distribution of Y “Moments” of the population distribution Conditional distributions Simple random sampling,Population The group or collection of entities of interest Here, “all possible” school districts “All possible” means “all possi
30、ble” circumstances that lead to specific values of STR, test scores We will think of populations as infinitely large; the task is to make inferences about a large population based on a sample from the population,Random variable Y Numerical summary of a random outcome Here, the numerical value of dis
31、trict average test scores (or district STR), once we choose a year/district to sample. Population distribution of Y The probabilities of different values of Y that occur in the population, for ex. PrY = 650 (when Y is discrete) or: The probabilities of sets of these values, for ex. PrY 650 (when Y i
32、s continuous).,总体分布实例： 美国男女成人身高的（正态）分布,=175 cm =7.1 cm,身高（英寸）,男性,女性,问：这两曲线里，哪个是男的，那个是女的？,问：为什么女的曲线比男的高？,Normal Distribution Example,The height of the curve at x is determined by the function:,x,If x is distributed as a normal variable, then it is designated as: x N(, ),There are an infinite number o
33、f normal curves,“Moments” of the population distribution,mean = expected value = E(Y) = Y = long-run average value of Y over repeated realizations of Y variance = var(Y)=E(Y Y)2 = = measure of the squared spread of the distribution standard deviation = = Y,Conditional distributions The distribution
34、of Y, given value(s) of some other random variable, X Ex: the distribution of test scores, given that STR 20 Moments of conditional distributions conditional mean = mean of conditional distribution = E(Y|X = x) (important notation) Example: E(Test scores|STR 20), the mean of test scores for district
35、s with small class sizes conditional variance = variance of conditional distribution = var(Y|X = x),The difference in means is the difference between the means of two conditional distributions: = E(Test scores|STR 20) E(Test scores|STR 20) Other examples of conditional means: Wages of all female wor
36、kers (Y = wages, X = gender) One-year mortality rate of those given an experimental treatment (Y = live/die; X = treated/not treated) The conditional mean is a new term for the familiar idea of the group average,Inference about means, conditional means, and differences in conditional means We would
37、like to know (test score gap; gender wage gap; effect of experimental treatment), but we dont know it. Therefore we must collect and use data that permits making statistical inferences about from either Experimental data, or, if not possible, from Observational data,Simple random sampling Choose an
38、individual (district, entity) at random from the population Randomness and data Prior to sample selection, the value of Y for is random because the individual selected is random Once the individual is selected and the value of Y is observed, then Y is just a number not random The data set is (Y1, Y2
39、, Yn), where Yi = value of Y for the ith individual (district, entity) sampled,Implications of simple random sampling,Because individuals #1 and #2 are selected at random, the value of Y1 has no information content for Y2. Thus: Y1, Y2 are independently distributed Y1 and Y2 come from the same distr
40、ibution, that is, Y1, Y2 are identically distributed That is, a consequence of simple random sampling is that Y1 and Y2 are independently and identically distributed (i.i.d.). More generally, under simple random sampling, the set of the sampled values Yi, i = 1, n, are i.i.d,2. Estimation,Sample ave
41、rage is the natural estimator of the population mean. But: What are the properties of this estimator? Why should we use Y rather than some other estimator? For example: Y1 (take simply the first observation) maybe unequal weights not simple average median(Y1, Yn) from a sample of size n,To answer th
42、ese questions we need to characterize the sampling distribution of The individuals in the sample are drawn at random. Thus the values of (Y1, Yn) are random Thus functions of (Y1, Yn), such as , are random: had a different sample been drawn, they would have taken on a different value The distributio
43、n of over different possible samples of size n is called the sampling distribution of . The mean and variance of are the mean and variance of its sampling distribution, E( ) and var( ). Related to var( ) is the idea of the covariance,The covariance between r.v.s X and Z is, cov(X,Z) = E(X X)(Z Z) =
44、XZ The covariance is a measure of the linear association between X and Z; its units are units of X times units of Z cov(X,Z) () 0: X and Z positive (negative) relation between X and Z If X and Z are independently distributed, then cov(X,Z) = 0 (but not vice versa! Why not? ) The covariance of a r.v.
45、 with itself is its variance: cov(X,X) = E(XX)(XX) = E(XX)2 =,The correlation coefficient is defined in terms of the covariance: corr(X,Z) = = rXZ Some notes on correlation coefficient: 1 corr(X,Z) 1 corr(X,Z) = 1 (-1) mean perfect positive (negative) linear association corr(X,Z) = 0 means no linear
46、 association If E(X|Z) = const (not a function of Z), then corr(X,Z) = 0 (not necessarily vice versa),The correlation coefficient measures linear association,The correlation coefficient measures linear association,Q: Corr(x,y)=0, but are x and y independent?,A: No!,The sampling distribution of,The i
47、ndividuals in the sample are drawn at random. Thus the values of (Y1, Yn) are random Thus functions of (Y1, Yn), such as , are random: had a different sample been drawn, they would have taken on a different value Since each sample mean is different, there must be a distribution of the sample mean, o
48、r sampling distribution The sampling distribution of is distribution of over different possible samples of size n.,(样本均值分布),Sampling distribution of the sample mean from a survey of siblings,The sampling distribution of,Demonstrations of the Sampling Distribution on the Web : http:/ http:/www.stat.tamu.edu/west/ph/sampledist.html Lets try both binomial and uniform distributions The mean and variance of are the mean and variance of its sampling distribution, E( ) and var( ). To compute var( ), we need the covariance (协方差),The mean and variance of