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1、第一章1.Econometrics（计量经济学）：the social science in which the tools of economic theory, mathematics, and statistical inference are applied to the analysis of economic phenomena. the result of a certain outlook on the role of economics, consists of the application of mathematical statistics to economic da
2、ta to lend empirical support to the models constructed by mathematical economics and to obtain numerical results. 2.Econometric analysis proceeds along the following lines计量经济学分析步骤1)Creating a statement of theory or hypothesis. 建立一个理论假说2)Collecting data.收集数据3)Specifying the mathematical model of the
3、ory. 设定数学模型4)Specifying the statistical, or econometric, model of theory. 设立统计或经济计量模型5)Estimating the parameters of the chosen econometric model.估计经济计量模型参数6)Checking for model adequacy : Model specification testing. 核查模型的适用性：模型设定检验7)Testing the hypothesis derived from the model. 检验自模型的假设8)Using the
4、model for prediction or forecasting.利用模型进行预测Step2：收集数据? Three types of data三类可用于分析的数据1)Time series(时间序列数据 ):Collected over a period of time, are collected at regular intervals.按时间跨度收集得到2)Cross-sectional截面数据 :Collected over a period of time, are collected at regular intervals.按时间跨度收集得到3)Pooled data合并
5、数据（上两种的结合）Step3：设定数学模型1.plot scatter diagram or scattergram 2.write the mathematical model Step4：设立统计或经济计量模型? CLFPR is dependent variable应变量? CUNR is independent or explanatory variable独立或解释变量（自变量）? We give a catchall variable U to stand for all these neglected factors ? In linear regression analysi
6、s our primary objective is to explain the behavior of the dependent variable in relation to the behavior of one or more other variables, allowing for the data that the relationship between them is inexact. 线性回归分析的主要目标就是解释一个变量（应变量）与其他一个或多个变量（自变量）只见的行为关系，当然这种关系并非完全正确Step5：估计经济计量模型参数? In short, the est
7、imated regression line gives the relationship between average CLFPR and CUNR 简言之，估计的回归直线给出了平均应变量和自变量之间的关系? That is, on average, how the dependent variable responds to a unit change in the 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 1 页，共 26 页independent variable. 单位因变量的变化引起的自变量平均变化量的多少。Step6：核查模型
8、的适用性：模型设定检验The purpose of developing an econometric model is not to capture total reality, but just its salient features. Step7：检验自模型的假设Why do we perform hypothesis testing? We want to find our whether the estimated model makes economic sense and whether the results obtains conform with the underlyi
9、ng economic theory. 第二章1.The meaning of regression （回归）Regression analysis is concerned with the study of the relationship between one variable called the dependent or explained variable, and one or more other variables called independent or explanatory variables. 2.Objectives of regression 1)Estima
10、te the mean, or average, and the dependent values given the independent values 2)Test hypotheses about the nature of the dependence -hypotheses suggested by the underlying economic theory 3)Predict or forecast the mean value of the dependent variable given the values of the independents 4)One or mor
11、e of the preceding objectives combined 3.Population Regression Line （PRL）In short, the PRL tells us how the mean, or average, value of Y is related to each value of X in the whole population 4.The dependence of Y on X, technically called the regression of Y on X. 5.How do we explain it? A students S
12、.A.T. score, say, the ith individual, corresponding to a specific family income can be expressed as the sum of two components 1)The component can be called the systematic, or deterministic, component. 2)May be called the nonsystematic or random component 6.What is the nature of U(stochastic error) t
13、erm ？1)The error term may represent the influence of those variables that are not explicitly included in the model. 误差项代表了未纳入模型变量的影响2)Some intrinsic randomness in the math score is bound to occur that can not be explained even we include all relevant variables. 即使模型包括了决定性数学分数的所有变量，内在随机性也不可避免，这是做任何努力
14、都无法解释的。3)U may also represent errors of measurement. U 还代表了度量误差4)The principle of Ockham s razor - the description be kept as simple as possible until proved inadequate - would suggest that we keep our regression model as simple as possible. “奥卡姆剃刀原则”，描述应该尽可能简单，只要不遗漏重要信息。这表明回归模型应尽可能简单。7.How do we es
15、timate the PRF （population regression function ）? Unfortunately, in practice, We rarely have the entire population in our disposal, 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 2 页，共 26 页often we have only a sample from this population. 8.Granted that the SRF is only an approximation of PRF. Can w
16、e find a method or a procedure that will make this approximation as close as possible? SRF 仅仅是PRF的近似，那么能不能找到一种方法使这种近似尽可能接近真实呢？9.Special meaning of “linear”1)Linearity in the variables 变量线性The conditional mean value of the dependent variable is a linear function of the independent variables 2)Lineari
17、ty in the Parameters参数线性The conditional mean of the dependent variable is a linear function of the parameters, the B s; it may or may not be linear in the variables. 第三章1.Unless we are willing to assume how the stochastic U terms are generated, we will not be able to tell how good an SRF is as an es
18、timate of the true PRF. 只有假定了随机误差的生成过程，才能判定SRF对 PRF 拟合的是好是坏。2.Classical Linear Regression Model 1)Assumption 1: The regression model is linear in the parameters. It may or may not be linear in the variables. 回归模型是参数线性的， 但不一定是变量线性的。2)Assumption 2: The explanatory variables X is uncorrelated with the
19、disturbance term U. X s are nonstochastic, U is stochastic. 解释变量 X 与扰动误差项 u 不相关 . X 是非随机的， U 是随机的。3)Assumption 3: Given the value of Xi, the expected, or mean value of the disturbance term U is zero. 给定 Xi，扰动项的期望或均值为零。Disturbance U represent all those factors that are not specifically introduced in
20、the model干扰项 U 代表了所有未纳入模型的影响因素。4)Assumption 4:The variance of each Ui is constant, or homoscedastic. U 的方差为常数，或同方差。Homoscedasticity （同方差） : a.This assumption simply means that the conditional distribution of each Y population corresponding to the given value of X has the same variance. 该假定表明，与给定的 X
21、相对应的每个 Y 的条件分布具有同方差。b.The individual Y values are spread around their mean values with the same variance. 即每个 Y 值以相同的方差分布在其均值周围。5)Assumption 5:There is no correlation between two error terms, this is the assumption of no-autocorrelation. 无自相关假定，即两个误差项之间不相关。6)Assumption 6:The regression model is corr
22、ectly specified.回归模型是正确假定的。 There is no specification bias or specification error in the model. 实证分析的模型不存在设定偏差或设定误差。This assumption can be explained informally as follows. An econometric investigation begins with the specification of the econometric model underlying the phenomenon of interest. 3.Var
23、iances and Standard errors of OLS estimators 普通最小二乘估计量的方差与标准误:One immediate result of the assumptions introduced is that they enable us to 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 3 页，共 26 页estimate the variances and standard errors of the OLS estimators given in Eq.(2.16) and (2.17). 4.We sho
24、uld know: Variances of the estimators Standard errors of the estimators 5.What is the value of The homoscedastic is estimated from formula 6.Standard Error of the Regression (SER) 回归标准误Is simply the standard deviation of the Y values about the estimated regression line. Y 值偏离估计回归的标准差。7.Summary of ma
25、th S.A.T.score function 1) Interpretation The standard deviation, or standard error, is 0.000245, is a measure of variability of b2 from sample to sample. If we can say that our computed b2 lies within a certain number of standard deviation units from the true B2, we can state with some confidence h
26、ow good the computed SRF is as an estimator of the true PRF. 2）Sampling Distribution 抽样分布Once we determine the sampling distribution of our two estimators, the task of hypothesis testing becomes straightforward. 一旦确定了两个估计量的抽样分布，那么假设检验就是举手之劳的事情。8.Why do we use OLS ? The properties of OLS estimators T
27、he method of OLS is used popularly not only because it is easy to use but also because it has some strong theoretical properties. OLS法得到广泛使用， 不仅是因为它简单易行，还因为它具有很强的理论性质。9.Gauss-Markov theorem 高斯-马尔科夫定理Given the assumptions of the classical linear regression model (CLRM), the OLS estimators have minimu
28、m variance in the class of linear estimators.The OLS estimators are BLUE (best linear unbiased estimators) 满足古典线性模型的基本假定，则在所有线性据计量中，OLS 估计两具有最小方差性，即OLS 是最优线性无偏估计量（ BLUE）10.BLUE property 最优线性无偏估计量的性质1)B1 and B2 are linear estimators. B1 和 B2 是线性估计量2)They are unbiased , that is E(b1)=B1, E(b2)=B2. B1
29、和 B2 是无偏估计两3)The OLS estimator of the error variance is unbiased. 误差方差的 OLS 估计量是无偏的4)b1 and b2 are efficient estimators.B1和 B2 是有效估计量Var(b1) is less than the variance of any other linear unbiased estimator of B1 Var(b2) is less than the variance of any other linear unbiased estimator of B2 11.Monte
30、Carlo simulation 蒙特卡洛模拟Do the experiment at lab Do it by Excell. =NORMINV(RAND(),0,2) Do it by matlab.= NORMINV(uniform(),MU,SIGMA) 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 4 页，共 26 页Do it by Stata. =invnorm(uniform() 12.Central Limit Theorem s 中心极限定理If there is a large number of independent a
31、nd identically distributed (iid) random variables, then, with a few exceptions , the distribution of their sum tends to be a normal distribution as the number of such variables increases indefinitely. 随着变量个数的无限增加，独立同分布随机变量近似服从正态分布13.Recall U, the error term represents the influence of all those forc
32、es that affect Y but are not specifically included in the regression model because there are so many of them and the individual effect of any one such force on Y may be too minor. 误差项代表了未纳入回归模型的其他所有因素的影响。因为在这些影响中， 每种因素对 Y 的影响都很微弱If all these forces are random, if we let U represent the sum of all th
33、ese forces, then by invoking the CLT, we can assume that the error term U follows the normal distribution.如果所有这些影响因素都是随机的，用U 代表所有这些影响因素之和，那么根据中心极限定理，可以假定误差项服从正态分布。14.Another property of normal distribution 另一个正态分布的性质Any linear function of a normally distributed variable is itself normally distribute
34、d. 正态变量的性质函数仍服从正态分布。15.Hypothesis testing 假设检验Having known the distribution of OLS estimators b1 and b2, we can proceed the topic of hypothesis testing. 16.Null hypothesis 零假设“zero” null hypothesis is deliberately chosen to find out whether Y is related to X al all, which is also called straw man hy
35、pothesis. 之所以选择这样一个假设是为了确定 Y 是否与 X 有关，也称为稻草人假设。17.We need some formal testing procedure to reject or receive the null hypothesis and make the skeptical guys shut up. 需要正规的检验过程拒绝或接受零假设18. If our null hypothesis is B2=0 and the computed b2=0.0013, we can find out the probability of obtaining such a va
36、lue from the Z, the standard normal distribution. 如果零假设为 B2=0，计算得到 b2=0.0013，那么根据标准正态分布Z，能够求得获此 b2值的概率 If the probability is very small, we can reject the null hypothesis. 如果这个概率非常小，则拒绝零假设。If the probability is larger, say , greater than 10 percent, we may not reject the null hypothesis. 如果这概率比较大，比如
37、大于10%，就不拒绝零假设。19.We don t know the 2 We must know the true 2, but we can estimate it by using ?220.What will happen if we replace by its estimator -hat 22222222,()ninbBtxor moregenerallybBtse b:精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 5 页，共 26 页21.Let us assume that , the level of significance
38、 or the probability of committing a type I error, is fixed at 5 percent. 假定，显著水平成犯第一类错误的概率为5%。22.red area = rejection region for 2-sided test 23.Loop and ball a.This is a 95% confidence interval for B2 给出了 B2 的一个 95%的置信区间。b.in repeated applications 95 out of 100 such intervals will include the true
39、B2重复上述过程，100 个这样的区间中将有95 个包括真实的B2。c.Such a confidence interval is known as the region of acceptance (of H0) and the area outside the confidence interval is known as the rejection region (of H0)用假设检验的语言把这样的置信区间称为（H0 的）接受区域，把置信区间以外的区间成为（H0 的）拒绝区域24.回归系数的假设检验目的：简单线性回归中，检验X 对 Y 是否真有显著影响基本概念回顾 : 临界值与概率、大
40、概率事件与小概率事件相对于显著性水平的临界值为 : t（单侧）或2t（双侧）计算的统计量为:*t25.Conclusions Since this interval does not include the null-hypothesized value of 0.因为这个区间没有包括零假(1-a) t 0 f(t) -tc tc a/2 a/2 统计量 t 2t2t*t0 （ 大 概 率 事件）（ 小 概 率事件）1精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 6 页，共 26 页设值 0。 We can reject the null hyp
41、othesis that annual family income is not related to math S.A.T. Scores.所以拒绝假设：家庭年收入对数学SAT 没有影响。 Put positively, income does have a relationship to math S.A.T. scores. 换言之，收入确实与数学SAT 有关系。26.A cautionary note Although the statement given is true, we cannot say that the probability is 95 percent that t
42、he particular interval includes B2, for this interval is not a random interval, it is fixed, therefore, the probability is either 1 ore 0 that the interval includes B2.虽然式子3.26 为真，但不能说某个特定区间式 3.27 包括真实B2 的概率为95%，因为与式子3.26 不同，式3.27 是固定的，而不是一根随机区间， 所以区间3.27 包括 B2 的概率为1 或 0.We can only say that if we c
43、onstruct 100 intervals like this interval, 95 out of 100 such intervals will include the true B2.我们只能说， 如果建立 100 个像式 3.27 这样的区间，则有 95 个区间包括真实的B2.We can not guarantee that this particular interval will necessarily includes B2.并不能保证某个区间一定有B2. 27.The test of significance approach to hypothesis testing
44、假设检验的显著性检验方法Hypothesis testing is that of a test statistic and the sampling distribution of the test statistic under the null hypothesis, H0. 假设检验方法涉及两个重要的概念检验统计量和零假设下检验统计量的抽样分布。 The decision to accept or reject H0 is made on the basis of the value of the test statistic obtained from the sample data
45、.根据从样本数据求得的检验统计量的值决定接受或拒绝零假设。28.T test We can use the t value computed here ad the test statistic, which follows the t distribution with (n-2) d.f. 可以计算出t 值作为检验统计量，它服从自由度为（n-2）的 t 分布。29.Instead of arbitrarily choosing the value , we can find the p value (the exact level of significance) and reject t
46、he null hypothesis if the computed P value is sufficiently low.为了避免选择显著水平的随意性，通常求出p 值（精确的显著水平） ，如果计算的p 值充分小，则拒绝零假设。30.Conclusions In the case of two-sided t test 双边检验情况中If the computed |t|, the absolute value of t, exceeds the critical t value at the chosen level of significance, we can reject the n
47、ull hypothesis.如果计算得到的 |t|值超过临界t 值，则拒绝零假设。31.P value The P value of that t statistic of 5.4354 is about 0.0006. t统计量（ 5.4354）的 p 值（概率值）约为 0.0006。The smaller the p value, the more confident we are when reject the null hypothesis.p值越小， 在拒绝零假设的时候就越有自信。Thus if we were to reject the null hypothesis that
48、the true slope coefficient is zero at this P value, we would be wrong in six out of ten thousand occasions. 如果在这个p 值水平之上拒绝零假设：真实的斜率系数为0，则犯错误的机会有万分之六。32.How can we computed t We first compute the t value as if the null hypothesis were that B2=0, we still get the t 0.0013 05.43540.000245t首先计算在零假设B2=0
49、下的 t 值 Since this value exceeds any of the critical values shown in the preceding table, following the rules laid down. t 值大与上表给出的任何临界值， 附录 D 表 D-2 列出的规则， We can reject the hypothesis that annual family income has no relationship to math S.A.T. Scores. 拒绝零假设：家庭年收入对数学SAT 没有影响。33.How good is the fitte
50、d regression line: the coefficient of determination r2 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 7 页，共 26 页On the basis of t test both the estimated intercept and slope coefficients are statistically significant (i.e. significantly different from zero) suggests that the SRF seems to “fit ” the