1、1Lecture OneMethodology of Econometrics2立论？结论？立论？结论？v立论:要求给出求论的路径。v结论:要求说明结论的来源。v自以为是的东西并不见得是真 v我们不是上帝！3我们的习惯是这样的吗？我们的习惯是这样的吗？v结论来自感觉（象上帝）v宏观思考（象战略家）v习惯地提出政策建议（象顾问）v得争取把一个个的大、小问题搞明白再说吧！4Mainstream Analysis ApproachesNormative AnalysisPositive Analysis(empirical analysis)5The Writer D.N.GujarativProf
2、essor of econometrics at the Military Academy at West PointvMaster of CommercevMBAvEditorial refereevAuthorvVisiting Professor6What is EconometricsvEmpirical support to the models vQuantitative analysis of actual economic phenomenavSocial science in which the tools of economic theory,mathematics,and
3、 statistical inference are applied to the analysis.vPositive help vEconomic theory _ measurements7Methodology of EconometricsvStatement of theory or hypothesisvObtaining the datavSpecification of the mathematical modelvSpecification of the econometric modelvEstimation of the parameters of the econom
4、etric modelvHypothesis testingvForecasting or predictionvUsing the model for control or policy purposes8Statement of Theory or HypothesisvPostulate(give some examples)vStatement vNote:hypothesis is not the same as an assumption 9Obtaining the DatavNature vSourcesvLimitations10Types of DataTime serie
5、s data:quantitative,qualitative (dummy variable)(SATIONARY)Cross-sectional data:(HETEROGENEITY)Pooled data:(Panel data)11Sourcesvwww.whitehouse.gov/fsbr/esbr.htmvwww.nber.org (National bureau of economic research)vwww.census.govvStats.bls.govvwww.jstor.gov12Accuracy of Data vNon-experimental in natu
6、revRound-offs and approximationsvNon-responsevSelectivity biasvAggregate levelvConfidentialityvThe results of research are only as good the quality of the data.13Specification of the mathematical modelvYi=b1+b2*Xi 0b2sample parameter-estimate-estimator distribution-population parameter-population ch
7、aracteristicsvConfirmation or refutation of economic theories on the basis of sample evidence vThe basement is statistical inference(Hypothesis testing)21Forecasting or PredictionvHypothesis or theory be confirmedvKnown or predictor variable X vPredict the future values of the dependent22Use of the
8、Model for Control or Policy PurposesvControl variable XvTarget variable YvYi=b1+b2*XivManipulate the control variable X to produce the desired level of the target variable Y23Anatomy of Classical Econometric ModelingvEconomic theoryvMathematical model of theoryvEconometric model of theoryvDatavEstim
9、ation of econometric modelvHypothesis testingvForecasting or predictionvUsing the model for control or policy purposes24第第1章章计量经济学研究的方法论计量经济学研究的方法论4第第2-3章章基本统计概念，概率分布基本统计概念，概率分布4第第4章章估计与假设估计与假设4第第5章章双变量模型的基本思想双变量模型的基本思想4第第6章章双变量模型的假设检验双变量模型的假设检验4第第7章章多元回归：估计与假设检验多元回归：估计与假设检验4第第8章章回归方程的函数形式回归方程的函数形式4
10、第第9章章虚拟变量的回归模型虚拟变量的回归模型4第第10章章多重共线性多重共线性4第第11章章异方差性异方差性4第第12章章自相关性自相关性4第第13章章模型选择：标准与检验模型选择：标准与检验4实验实验实验实验1-6625Please Give Some SuggestionsZ3-W163.COM027-62082852Thank you.26A Review of Some Statistical ConceptsLecture Two27Sample space、Sample points、EventsvPopulation is the set of all possible out
11、comes of random experiment(sample space)vSample point is the each member of this sample spacevEvent is a subset of the sample space28Probability and Random VariablesvP(A)probability(pr;p;pro)vX random variable(r v)vX the value of a random variablev0=P(A)=0,-1)(dxxfbabxaPdxxf)()(30Cumulative Distribu
12、tion Function C D F F(X)=P(X=x)(discrete)=(continuous)xdxxf)(31CDF of Discrete VRPDFCDFX times of face upf(x)(PDF)Value of X f(x)(CDF)01/16X=01/1614/16X=15/1626/16X=211/1634/16X=315/1641/16X=4132CDF11/165/1611/1615/161X234033CDF of Continuous VRPDFCDFValue of X timesf(x)PDF)Value of X timesf(x)(CDF)
13、0=X11/16X=01/161=X24/16X=15/162=X36/16X=211/163=X44/16X=315/164=X51/16X0,symmetrical S=0 or left S2,its variance is)4()2(222221)21(22kkkkkk71An ExamplevGiven k1=10 and k2=8,what is the probability of obtaining an F value (a)of 3.4 or greater;(b)of 5.8 or greater?vThese probabilities are (a)approxima
14、tely 0.05;(b)approximately 0.01.72Relationships1.If the denominator df,k2,is fairly large,the following relationship holds:2.3.Large df,the t,chi square,and F distributions approach the normal distribution,these distributions are known as the distributions related to the normal distribution.2k1Fk1KF
15、tk,1273 An ExamplevLet k1=20 and k2=120.The 5 percent critical F value for these df is 1.48 Therefore,k1F=(20)*(1.48)=29.6.vFrom the chi-square distribution for 20 df,the 5 percent critical chi-square value is about 31.41.74Lecture 3(2)Estimation and Inference75ESTIMATIONvAssume that a random variab
16、le X follows a particular probability distribution but do not know the value(s)of the parameter(s)of the distribution.vif X follows the normal distribution,we may want to know the value of its two parameters,namely,the mean and the variance.76Estimate the UnknownsvWe have a random sample of size n f
17、rom the known probability distribution;vUse the sample data to estimate the unknown probability distribution;(non-pa)vUse the sample data to estimate the unknown parameters.(pa)77Two Categories Point estimation Interval estimation.78Point EstimationvLet X be a rv with PDF f(x;),is the parameter of t
18、he distribution(for simplicity only one unknown parameter).vAssume that we know the theoretical PDF,such as the t distribution do not know the value of.we draw a random sample of size n from this known this PDF and then develop a function of the sample values),(21nxxxfL79Estimator or Estimatevprovid
19、es us an estimate of the true.is known as a statistic,or an estimator,vA particular numerical value taken by the estimator is known as an estimate.can be treated as a random variable.provides us with a rule,or formula,that tells us how we may estimate the true.80An ExamplevSample mean is an estimato
20、r of the true mean value,.If in a specific case =50,this provides an estimate of.The estimator obtained is known as a point estimator because it provides only a single(point)estimate of.Xxxxnn)(121L81Interval EstimationDef of interval estimation:vwe obtain two estimates of,by constructing two estima
21、tors1(x1,x2,xn)and2(x1,x2,xn),and say with some confidence(i.e.,probability)that the interval between 1 and 2 includes the true.vwe provide a range of possible values within which the true may lie.82Key conepts vSampling,Probability distribution,An estimator.vX is normally distributed,then the sampl
22、e mean is also normally distributed with mean=(the true mean)and variance=2/n,N (,2).v probability is 95%Xnx283Interval EstimationvMore generally,in interval estimation we construct two estimators and ,both functions of the sample X values,such thatvThe interval is known as a confidence interval of
23、size 1for,v1 being known as the confidence coefficient,v is known as the level of significance.)10(1)Pr(211284An ExamplevSuppose that the distribution of height of men in a population is normally distributed with mean=and =2.5.vA sample of 100 men drawn randomly from this population had an average h
24、eight of 67.Establish a 95%confidence interval for the mean height(=)in the population as a whole.85SolutionvAs noted,N (,2/n),which in this case becomes N (,2.52/100).v95%confidence interval as 66.5167.49nXnX96.1)(96.1XX86OLS and MLvThere are several methods of obtaining point estimators,the best k
25、nown being the method of Ordinary least-squares and the method of maximum likelihood(ML).vThe desirable statistical properties fall into two categories:small-sample,and large sample,or asymptotic.87Small-Sample PropertiesvUnbiasedness.An estimator is said to be an unbiased estimator of if the expect
26、ed value of is equal to the true;that is,vE()=0vIf this equality does not hold,then the estimator is said to be biased,and the bias is calculated as vBias()=E()0)(E88Minimum VariancevMinimum variance.1is said to be a minimum-variance estimator of if the variance of 1 is smaller than or at most equal
27、 to the variance of 2，which is any other estimator of。89EfficientvBest unbiased，or efficient，estimator.If1 and2 are two unbiased estimators of,and are two unbiased estimators of 1，and the variance of 1 is smaller than or at most equal to the variance of 2，then1 is a minimum-variance unbiased，or best
28、 unbiased，or efficient，estimator.90Linearity.vAn estimatoris said to be a linear estimator of if it is a linear function of the sample observations。Thus，the sample mean defined as)(1121nixxxnXnX91BLUEvBest linear unbiased estimator（BLUE）。If is linear，is unbiased，and has minimum variance in the class
29、 of all linear unbiased estimators of，then it is called a best linear unbiased estimator，or BLUE for short。92Hypothesis TestingvEstimation and hypothesis testing constitute the twin branches of classical statistical inference.vAssume that we have an rv X with a know PDF f(;),where is the parameter o
30、f the distribution.Having obtained a random sample of size n,we obtain the point estimator.The true is rarely known,Is the estimator “compatible”with some hypothesized value of,say,=*,?93HypothesisvIn the language of hypothesis testing=*is called the null null hypothesishypothesis and is generally d
31、enoted by Ho.The null hypothesis is tested against an alternative hypothesisalternative hypothesis,denoted by H1,which,for example,may state that*.94Simple and CompositevA simplesimple hypothesis:it specifies the value(s)of the parameter(s)of the distribution;otherwise it is called a composite compo
32、site hypothesis.vThus,if XN(,2)and we state that v Ho:=15 and=2vit is a simple hypothesis,whereas v Ho:=15 and 2vis a composite hypothesis because here the value of is not specified.95Test the Null HypothesisvTo test the null hypothesis(i.e.to test its validity),we use the sample information to obta
33、in what is known as the test statistic.The point estimator of the unknown parameter.Then we try to find out the sampling,or probability,distribution of the test statistic and use the confidence interval or test of significance approach to test the null hypothesis.96An Example The height(X)of men in
34、a population.We are told that Xi N(,2)=N(,2.52)vAverage height=67 =67 n=100vLet us assume that Ho:=*=69 H1:6997v Could the sample with =67,the test statistic,have come from the population with the mean value of 69?vwe may not reject the null hypothesis if is“sufficiently close”to*;otherwise we may r
35、eject it in favor of the alternative hypothesis.how do we decide that is“sufficiently close”to*?98Two Approaches(1)(1)Confidence interval Confidence interval(2)(2)Test of significanceTest of significance(3)Both leading to identicalidentical c o n c l u s i o n s i n a n y specific application.99Conf
36、idence Interval Confidence Interval ApproachApproachvSince Xi N(,2),we know that the test statistic is distributed as v100(1)confidence interval for based on and see whether this confidence interval includes =*?)/2,(nNXXX100vThe actual mechanics are as follows:since N(,2/n),it follows that v Pr(1.96
37、 Zi 1.96)=0.95)1,0(/NnXZi95.0)96.1/96.1Pr(nXX101 Turning to our example,we have already established a 95%confidence interval for,which is 66.51 67.49 This interval obviously does not include =69.Therefore,we can reject the null hypothesis that the true is 69 with a 95%confidence coefficient.102Regio
38、n and Critical ValuesvThe confidence interval that we have established is called the acceptance region.vThe area(s)outside the acceptance region is(are)called the region(s)of rejection of the null hypothesis.(Critical regions)vThe lower and upper limits of the acceptance region are called the critic
39、al values.103Criterion for RejectionvLanguage of hypothesis testing,if the hypothesized value falls inside the acceptance region,one may not reject the null hypothesis;votherwise one may reject it.104Two Types of ErrorvWe are likely to commit two types of errors:(1)we may reject H0 when it is,in fac
40、t,true;this is called a type error.v(2)we may not reject H0 when it is,in fact,false;this is called a type error.105v Ideally,we would like to minimize both type and type error.vBut unfortunately,for any given sample size,it is not possible to minimize both the errors simultaneously.106Level of Sign
41、ificancevIn the literature the probability of type error is designated as and is called the level of significance.vThe probability of type error is designated as.vThe probability of not committing a type error,1,is called the power of the test.107 The classical approach to hypothesis testing is to f
42、ix at level such as 0.01 or 0.05 and then try to maximize the power of the test;that is minimize.108Confidence CoefficientvConfidence coefficient(1-)is simply one minus the probability of committing a type I error.vA 95%confidence coefficient means that we are prepared to accept at the most a 5%prob
43、ability of committing a type I error.vwe do not want to reject the true hypothesis by more than 5 out of 100 times.109P valuevThe p value,or exact level of significance.Instead of preselecting at arbitrary levels,such as 1,5,or 10 percent.one can obtain the p(probability)value,or exact level of sign
44、ificance of a test statistic.vThe p value is defined as the lowest significance level at which a null hypothesis can be rejected.110 The Test of Significance Approachva test(statistic)is significant,we generally mean that we can reject the null hypothesis.vthe test statistic is regarded as significa
45、nt if the probability of our obtaining it is equal to or less than,the probability of committing a type I error.111Summarize the Steps Involved in Testing a Statistical HypothesisvStep1.State the null hypothesis Ho and the alternative hypothesis H1(e.g.Ho :=69 and H1:69).vStep2.Select the test stati
46、stic(e.g,).vStep3.Determine the probability distribution of the test statistic(e.g.,N(,2/n).112vStep4.Choose the level of significance(i.e.,the probability of committing a type I error).vStep5.Using the probability distribution of the test statistic,establish a 100(1-)%confidence interval.113vIf the
47、 value of parameter under the null hypothesis(e.g,=*=69)lies in this confidence region,the region of acceptance,do not reject the null hypothesis.But if it falls outside this interval(i.e,it falls into the region of rejection),you may reject the null hypothesis.v Keep in mind that in not rejecting o
48、r rejecting a null hypothesis you are taking a chance of being wrong percent of the time 114Examplev if=*=69,vif=0.05,the probability of obtaining a Z value of 1.96 or 1.96 is 5 percent(or 2.5 percent in each tail of the standardized normal distribution).vIn our illustrative example Z was 8.825.0/21
49、00/5.26967/*nXZi115v Z=-8 is statistically significant;that is,we reject the null hypothesis that the true*is 69.vOf course,we reached the same conclusion using the confidence interval approach to hypothesis testing.116Lecture FourTwo-variable Regression Analysis(A)Some Basic Ideas(B)Estimation(poin
50、t and interval)(C)Hypothesis Testing117Some Basic IdeasvConcept of RegressionvPopulation Regression Function（PRF）vSample Regression Function（SRF）118Two and Multiple Regression AnalysisThe more general multiple regression analysis is in many ways a logical extension of the two-variable case.So we fir
