硕士计量101IntroductiontoEconom

Econometrics I Fall 2011Instructor: 冯强Office: 博学 1223 Phone: 6449-3318The best way to contact me is by email: Brief Overview of the Course n Economics suggests interesting relations, often with policy implications, but virtually never suggests quantitative magnitudes of causal effects.n For example:What is the price elasticity of public transportation?n 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 a 1 percentage point increase in interest rates by the Central Bank? Other Example of Econometric Studiesn What might be the effect a new regulation on the housing price in Beijing?n What will be the impact of eliminating residence requirement (户口) on wage rate for college graduate (preferable UIBE students)?n 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?n What is the effect of GDP on electricity usage?n What kind of problem are you interested in using econometrics to study? n Can you come up with questions of this kind? n The focus of this course is the use of statistical and econometric methods to quantify causal effectsn Ideally, we would like an experiment:n Transit prices; class size; returns to education; Central Bank n But almost always we must use observational 观测的(nonexperimental) data.n Observational data poses major challenges: 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 n The higher is the income, the more time one can afford to stay in school “correlation 相 关 性 does not imply causation”n High income of the Western world is correlated with their height, does that mean the taller is the people, the richer they are? In this course you will: n Learn methods for estimating causal effects using observational data;n Lean some basic theories behind the methods in econometricsn Learn to produce (you do the analysis) and consume (evaluate the work of others) econometric applications; andn Practice “producing” in your problem sets. Causal Relationsn Q：Which of the following has a causal relationship?Circumference 围and height of a treen No causal relationship, but can be used for prediction neverthelessWage and outputn But higher wage can lead to higher moral and higher outputWeight and gas consumption of a truck n But energy efficient engine uses less gasCell phone fees and length of callsn But long distance calls costs more Is there a relationship between wage level and mobility? The difference between experimental data and observational datan Designed Experiment can easily test causal relationships, for example:The effect of a kind of fertilizer on tomato cropsThe effect of a medication on patients blood pressuren There is no simple designed experiment for social scienceFor each 1% increase in price, what is the percentage drop 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:n Cross-sectional data (截面数据) e.g. recordings of every students weight for todayn Time series (时间序列)e.g. the weight record of a person over a year.n Panel data (面板数据)the combination of cross-section and time seriese.g. the weight records of all the students here for each day and for over a year. The identification of data type:n Q: the data 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 periodn 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. 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 SureWhat 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 1745.2 881.61980 4545.6 1359.4 2192.0 994.21985 9016.0 2541.6 3866.6 2607.81986 10275.2 2763.9 4492.7 3018.61987 12058.6 3204.3 5251.6 3602.71988 15042.8 3831.0 6587.2 4624.61989 16992.3 4228.0 7278.0 5486.31990 18667.8 5017.0 7717.4 5933.4A) 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 1745.2 881.61980 4545.6 1359.4 2192.0 994.21985 9016.0 2541.6 3866.6 2607.81986 10275.2 2763.9 4492.7 3018.61987 12058.6 3204.3 5251.6 3602.71988 15042.8 3831.0 6587.2 4624.61989 16992.3 4228.0 7278.0 5486.31990 18667.8 5017.0 7717.4 5933.4 A) Cross Section B) Time Series C) Panel D) Not Sure 收 入张三 李 四 。 王 麻 子2000年23145 43251 652342001年25389 46239 67341。2009年30125 52395 70128A) Cross Section B) Time Series C) Panel D) Not SureWhat kind of data is this? Review of Probability and Statistics n Empirical problem: Class size and educational outcomen Policy question: What is the effect of reducing class size by n one student per class? n by 8 students/class?n What is the right outcome measure (“dependent variable”)?parent satisfactionstudent personal developmentfuture adult welfare and/or earnings performance on standardized tests What do data say about the class size/test score relation?The California Test Score Data Setn All K-6 and K-8 California school districts (n = 420)地 区n Variables:5th grade test scores (Stanford-9 achievement test, combined math and reading), district averageStudent-teacher ratio (STR) = no. 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) n This table doesnt tell us anything about the relationship between test scores and the STR. Scatterplot of test score v. student-teacher ratio n Do districts with smaller classes have higher test scores? n 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:1. Compare average test scores in districts with low STRs to those with high STRs (“estimation”)2. 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”)3. Estimate an interval for the difference in the mean test scores, high v. low STR districts (“confidence interval”) Initial data analysis: Compare districts with “small” (STR 1.96, we can reject (at the 5% significance level) the null hypothesis that the two means are the same. 3. Confidence interval n A 95% confidence interval for the difference between the means is,n ( ) 1.96SE( )= 7.4 1.961.83 = (3.8, 11.0)n Q: Are the following two statements equivalent ?The 95% confidence interval for doesnt include 0;The hypothesis that = 0 is rejected at the 5% level.n A: Yes, they are. s lY Y s lY Y This should all be familiarBut: n What is the underlying framework that justifies all this?n Estimation: Why estimate by ?n Testing: What is the standard error of , really? Why reject = 0 if |t| 1.96?n Confidence intervals (interval estimation): What is a confidence interval, really?s lY Y s lY Y Review of Statistical ConceptsWe will review the following in turn1. The probability framework for statistical inference2. Estimation3. Hypothesis testing4. Confidence Intervals 1. The probability framework for statistical inferencen Here are some key concepts: Population Random variable Y Population distribution of Y “Moments” of the population distribution Conditional distributions Simple random sampling n Population The group or collection of entities of interest Here, “all possible” school districts “All possible” means “all possible” 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 n Random variable YNumerical summary of a random outcomeHere, the numerical value of district average test scores (or district STR), once we choose a year/district to sample. n Population distribution of YThe 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 is 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 ExampleThe height of the curve at x is determined by the function: 2121( ) 2 xf x e xIf x is distributed as a normal variable, then it is designated as:x N(, )nThere are an infinite number of normal curves “Moments” of the population distribution n mean = expected value = E(Y) = Y = long-run average value of Y over repeated realizations of Yn variance = var(Y)=E(Y Y)2 = = measure of the squared spread of the distributionn standard deviation = = Y2Y variance n Conditional distributionsThe distribution of Y, given value(s) of some other random variable, XEx: the distribution of test scores, given that STR 20n Moments of conditional distributionsconditional mean = mean of conditional distribution = E(Y|X = x) (important notation) n Example: E(Test scores|STR 20), the mean of test scores for districts with small class sizes conditional variance = variance of conditional distribution = var(Y|X = x) n The difference in means is the difference between the means of two conditional distributions:n = E(Test scores|STR () 0: X and Z positive (negative) relation between X and Zn If X and Z are independently distributed, then cov(X,Z) = 0 (but not vice versa! Why not? )n The covariance of a r.v. with itself is its variance:n cov(X,X) = E(X X)(XX) = E(XX)2 = 2X n The correlation coefficient is defined in terms of the covariance:corr(X,Z) = = rXZn Some notes on correlation coefficient:1 corr(X,Z) 1corr(X,Z) = 1 (-1) mean perfect positive (negative) linear associationcorr(X,Z) = 0 means no linear associationIf E(X|Z) = const (not a function of Z), then corr(X,Z) = 0 (not necessarily vice versa) cov( , )var( )var( ) XZX ZX ZX Z The correlation coefficient measures linear association The correlation coefficient measures linear associationQ: Corr(x,y)=0, but are x and y independent?A: No! The sampling distribution ofn The individuals in the sample are drawn at random.n Thus the values of (Y1, Yn) are randomn Thus functions of (Y1, Yn), such as , are random: had a different sample been drawn, they would have taken on a different valuen Since each sample mean is different, there must be a distribution of the sample mean, or sampling distribution n The sampling distribution of is distribution of over different possible samples of size n. YY(样本均值分布)Y Y Sampling distribution of the sample mean from a survey of siblings The sampling distribution ofn Demonstrations of the Sampling Distribution on the Web :http:/ Lets try both binomial and uniform distributions n The mean and variance of are the mean and variance of its sampling distribution, E( ) and var( ).n To compute var( ), we need the covariance (协方差) YYY YY n The mean and variance of the sampling distribution of n mean: n Variance: Y( )E Y 11( ) n iiE Yn 11 ( ) n ii E Yn 11 n Y Yin2var( ) ( ) Y E Y E Y2 YE Y 211 ( ) n i YiE Yn2 1 11 ( )( ) n n i Y j Yi jE Y Yn n So, var( )Y 2 1 11 ( )( ) n n i Y j Yi j E Y Yn2 1 11 cov( , ) n n i ji j Y Yn2 21 1 1,1 1var( ) cov( , ) n n ni i ji i j j iY Y Yn n22 11 0 n Yin 2 Yn 2 1 11 ( )( ) n n i Y j Yi jE Y Yn Why?Y i and Yj are independent n Summary: E( ) = Y and var( ) = .n Implications: is an unbiased estimator of Y (that is, E( ) = Y)var( ) is inversely proportional to nspread of sampling distribution is proportional to 1/in this sense, the sampling uncertainty arising from using to make inferences about Y is proportional to 1/ Y Y 2YnY Y nnYY n For small sample sizes, the distribution of is complicated. n BUT: when n is large, it is not!(1) As n increases, the distribution of becomes more tightly centered around Y: the sampling uncertainty decreases as n increases (recall that var( ) = /n)n An estimator is consistent if the probability that its falls within an interval of the true population value tends to one as the sample size increases.Y 2Y Y The Law of Large Numbers:n If (Y1,Yn) are i.i.d. and , then is a consistent estimator of Y, that is, Pr| Y| 1 as n n which can be written as: Y (“ converges in probability to Y”)n (Proof: as n , var( ) = 0, which implies that Pr| Y| 1.) 2Y YYpY Y Y 2YnY (2) Central limit theorem (CLT)n If (Y1,Yn) are i.i.d. and 0 , then when n is large the distribution of is well approximated by a normal distribution:n is approximately distributed N(Y, ) (“normal distribution with mean Y and variance /n”)n is approximately distributed N(0,1) (standard normal)n That is, “standardized” is approximately distributed as N(0,1) n The approximation gets better as n increases 2YYY 2Yn 2Y( )/ Y Yn Y ( ) /var( ) YYY E Y YY nY Summary:n for (Y1,Yn) i.i.d. with 0 Y,0 (1-sided, )H0: E(Y) = Y,0 vs. H1: E(Y) Y,0 (1-sided, )H 0: E(Y) = Y,0 vs. H1: E(Y) Y,0 (2-sided) n p-value = probability of drawing a statistic (e.g. ) at least as adverse to the null as the value actually computed with your data, assuming that the null hypothesis is true. n The significance level of a test is a pre-specified probability of incorrectly rejecting the null, when the null is true.n Calculating the p-value based on :p-value = , Where is the value of actually observed (nonrandom) n In other words, p-value is the probability that the sampling distribution of the sample mean deviates from the hypo. pop. mean is greater then the observed deviation Y0 ,0 ,0Pr | | | |actH Y YY Y YactY Y ,0| | YY ,0| |act YY n p-value = , n To compute the p-value, you need the distribution of . n If n is large, we can use the large-n normal approximation: p-value , probability under left+right N(0,1) tails 0 ,0 ,0Pr | | | |actH Y YY Y Y 0 ,0 ,0Pr | | | | actH Y YY Y0 ,0 ,0Pr | | | |/ / actY YH Y YY Yn n Let denote the std. dev. of the distribution of : Y Y n In practice, Y is unknown it too must be estimatedn Estimator of the variance of Y: n Fact: If (Y1,Yn) are i.i.d. and E(Y4) , then n Why does the law of large numbers apply? n Because is a sample average.n Technical note: we assume E(Y4) because here the average is not of Yi, but of its square.2 211 ( )1 nY iis Y Yn 2 2pY Ys2Ys 4. Confidence intervalsn A 95% confidence interval for Y meansA. an interval that contains the true value of Y in 95% of repeated samples.B. The probability that the true value of Y falls within the confidence interval is 0.95.n What is random here? A. the confidence interval B. the population parameter, Y,n The interval will differ from one sample to the next; Y, is not random, we just dont know it. Homeworkn Appendix B, B.3, B.10n Appendix C, C.1, C.4