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(4.3a)
Expand and multiply top and bottom by n:
b2
=
nSxiyi - Sxi Syi nSxi2-(Sxi) 2
(4.3b)
Variance of b2
4.12
Given that both yi and ei have variance s2,
the variance of the estimator b2 is:
4. cov(ei,ej) = cov(yi,yj) = 0 5. xt c for every observation
6. et~N(0,s 2) <=> yt~N(b1+ b2xt,
The population parameters b1 and b2 4.4 are unknown population constants.
b2
+
nSxiEei - Sxi SEei nSxi2-(Sxi) 2
Since Eei = 0, then Eb2 = b2 .
An Unbiased Estimator
4.8
The result Eb2 = b2 means that the distribution of b2 is centered at b2.
4.6
The Expected Values of b1 and b2
The least squares formulas (estimators) in the simple regression case:
b2 =
nSxiyi - Sxi Syi nSxi22 -(Sxi) 2
b1 = y - b2x
计量经济学课件英文版 伍德里奇
Two methods to estimate
2016/9/22 Department of Statistics-Zhaoliqin 17
Some Terminology, cont.
β0 :intercept parameter β1 :slope parameter means that a one-unit increase in x changes the expected value of y by the amount β1,holding the other factors in u fixed.
Department of Statistics-Zhaoliqin 16
2016/9/22
Some Terminology, cont.
y = β0 + β1x + u, E [u|x] = 0. β0 + β1x is the systematic part of y. u is the unsystematic part of y. u is denoted the error term. Other terms for u : error shock disturbance residual (sometimes in reference to fitted error term)
6
2016/9/22
Department of Statistics-Zhaoliqin
7
Hence, we are interested in studying is the mean of wages given the years of Education that will be denoted as E [Wages|Education] Following economic theory, we assume a specific model for E[Wages|Education]
《计量经济学英文版》课件
2 ARMA模型
介绍ARMA模型的基本原 理和参数估计方法，以及 ARMA模型在经济数据分 析中的应用。
3 GARCH模型
讲解GARCH模型的原理 和估计方法，以及 GARCH模型在金融市场 波动预测中的应用。
计量经济模型的应用
金融市场分析
应用计量经济模型进行金融市 场的预测和分析。
政策评估
利用计量经济模型评估政策的 效果和影响。
面板数据模型
1
面板数据概述
介绍面板数据的特点和应用领域，以及面板数据模型的基本概念。
2
固定效应模型
讲解固定效应模型的估计和推断方法，以及固定效应模型的优缺点。
3
随机效应模型
介绍随机效应模型的估计和推断方法，以及随机效应模型与固定效应模型的比较。
时间序列模型
1 时间序列数据基本特
征
描述时间序列数据的基本 特征，如趋势、季节性和 周期性。
《计量经济学英文版》 PPT课件
本课程介绍了计量经济学的基本概念和分析方法。涵盖了经济数据与计量分 析、经典线性回归模型、面板数据模型、时间序列模型和计量经济模型的应 用。
课程概述
本节将概述本课程的内容，包括学习目标和涵盖的主要内容。通过本课程的学习，您将掌握计量经济学的基本 理论和实际应用。
经济数据与计量分析
企业ቤተ መጻሕፍቲ ባይዱ策
运用计量经济模型辅助企业的 决策制定。
总结
通过本课程的学习，您将获得计量经济学分析的基本知识和技能，能够应用计量经济模型进行实证研究，并在 实际问题中运用计量经济学的方法和工具。
• 经济数据的概念与特点 • 统计概率基础 • 经济计量分析的目的与意义
经典线性回归模型
普通最小二乘法
计量经济学导论第四版英文完整教学课件
Economics 20 - Prof. Anderson
7
The Question of Causality
Simply establishing a relationship between variables is rarely sufficient Want to the effect to be considered causal If we’ve truly controlled for enough other variables, then the estimated ceteris paribus effect can often be considered to be causal Can be difficult to establish causality
Need to use nonexperimental, or observational, data to make inferences
Important to be able to apply economic theory to real world data
Economics 20 - Prof. Anderson
3
Why study Econometrics?
An empirical analysis uses data to test a theory or to estimate a relationship
A formal economic model can be tested
Theory may be ambiguous as to the effect of some policy change – can use econometrics to evaluate the program
计量经济学(第五版)课件 ets1
8
计量经济学的三个要素
计量经济学的三个要素是经济理论、经济数据和统 计方法。对于解释经济现象来说，“没有计量的理论 ”和“没有理论的计量”都是不够的，正如计量经济 学创始人之一的弗里希所强调的那样，它们的结合是 计量经济学的发展能够取得成功的关键。
9
计量经济学是经济预测的科学
计量经济学从根上说，是对经验规律的认识以及将 这些规律推广为经济学“定律”的系统性努力，这些 “定律”被用来进行预测，即关于什么可能发生或者 什么将会发生的预测。因此，广义地说，计量经济学 可以称为经济预测的科学。
11
二．计量经济学的产生和发展 1.产生年代
计量经济学产生于上世纪三十年代。 • 1930年12月，弗里希（R. Frisch）、丁伯根 (J. Tinbergen)和费歇尔(I. Fisher)等经济学家在美国克利 夫兰成立计量经济学会。
• 1933 年 起 ， 定 期 出 版 《 计 量 经 济 学 》 (Econometrica)杂志。弗里希在该杂志发刊词中明确 提出计量经济学的范围和方法，指出计量经济学是 经济理论、数学和统计学的综合，但它又完全不同 于这三个学科中的每一个。
13
3. 学科发展环境
同时，随着科学技术的发展，各门学科相互渗透，数 学、系统论、信息论、控制论等相继进入经济研究领 域，使经济科学进一步数量化，有助于计量经济学的 发展。高速电子计算机的出现和发展，为计量经济技 术的广泛应用铺平了道路。
14
4. 发展过程
上世纪三十年代，侧重于个别商品供给与需求的计 量，基本上属于个量分析或微观分析。
自四十年代起，计量经济研究的范围扩大到整个经 济体系，其特征是宏观经济总量的计量分析，亦即总 量分析或宏观分析。
五十年代起，在计量经济学的理论和方法得到迅速 发展的同时，宏观计量经济模型在计量经济学的应用 中开始占重要地位。50年代末至60年代初是宏观计量 经济模型蓬勃发展的时期。
计量经济学的三个要素
计量经济学的三个要素是经济理论、经济数据和统 计方法。对于解释经济现象来说，“没有计量的理论 ”和“没有理论的计量”都是不够的，正如计量经济 学创始人之一的弗里希所强调的那样，它们的结合是 计量经济学的发展能够取得成功的关键。
9
计量经济学是经济预测的科学
计量经济学从根上说，是对经验规律的认识以及将 这些规律推广为经济学“定律”的系统性努力，这些 “定律”被用来进行预测，即关于什么可能发生或者 什么将会发生的预测。因此，广义地说，计量经济学 可以称为经济预测的科学。
11
二．计量经济学的产生和发展 1.产生年代
计量经济学产生于上世纪三十年代。 • 1930年12月，弗里希（R. Frisch）、丁伯根 (J. Tinbergen)和费歇尔(I. Fisher)等经济学家在美国克利 夫兰成立计量经济学会。
• 1933 年 起 ， 定 期 出 版 《 计 量 经 济 学 》 (Econometrica)杂志。弗里希在该杂志发刊词中明确 提出计量经济学的范围和方法，指出计量经济学是 经济理论、数学和统计学的综合，但它又完全不同 于这三个学科中的每一个。
13
3. 学科发展环境
同时，随着科学技术的发展，各门学科相互渗透，数 学、系统论、信息论、控制论等相继进入经济研究领 域，使经济科学进一步数量化，有助于计量经济学的 发展。高速电子计算机的出现和发展，为计量经济技 术的广泛应用铺平了道路。
14
4. 发展过程
上世纪三十年代，侧重于个别商品供给与需求的计 量，基本上属于个量分析或微观分析。
自四十年代起，计量经济研究的范围扩大到整个经 济体系，其特征是宏观经济总量的计量分析，亦即总 量分析或宏观分析。
五十年代起，在计量经济学的理论和方法得到迅速 发展的同时，宏观计量经济模型在计量经济学的应用 中开始占重要地位。50年代末至60年代初是宏观计量 经济模型蓬勃发展的时期。
计量经济学(英文PPT)Chapter 2 TWO-VARIABLE REGRESSION ANALYSIS SOME BASIC IDEAS
As the X value increases, the conditional mean value of Y increases, too. The dark circled points in Figure 2.1 show the conditional mean values of Y against the various X values.
is a linear function of X i ,say, of the type
E(Y | X i ) 1 2 X i (2.2.2)
1 and 2 are known as the regression coefficients. Equation（2.2.2）itself is known as the linear population regression function or simply linear population regression.
If we want to get the relationship between weekly family consumption expenditure (Y) and weekly family income (X).
In the hypothetical community, there is a total population of 60 families. The 60 families are divided into 10 income groups (from $80 to $260) . And assume that we get the observations given in Table 2.1.
Therefore, we can express the deviation of an individual Yi around its expected value as follows:
is a linear function of X i ,say, of the type
E(Y | X i ) 1 2 X i (2.2.2)
1 and 2 are known as the regression coefficients. Equation（2.2.2）itself is known as the linear population regression function or simply linear population regression.
If we want to get the relationship between weekly family consumption expenditure (Y) and weekly family income (X).
In the hypothetical community, there is a total population of 60 families. The 60 families are divided into 10 income groups (from $80 to $260) . And assume that we get the observations given in Table 2.1.
Therefore, we can express the deviation of an individual Yi around its expected value as follows:
计量经济学(英文PPT)Chapter 0 Introduction
consumption
is：2
,
namely,
－231.8
and
0.7194.
Thus,
the
estimated
Yˆ 231.8 0.7194 X
We see that，for the period 1980—1992, MPC≈0.72 in America, suggesting that for the sample period an increase in real income of 1 dollar led, on average, to an increase of about 72 cents in real
However, the establishment of World Econometric Society in December 29th, 1930 and the publication of its academic journal Econometrics in 1933 are generally acknowledged as a landmark of econometrics as a separate discipline.
Why a separate discipline?
Economic theory makes statements or hypotheses that are mostly qualitative in nature; It is the job of the econometrician to provide such numerical estimates.
Example: Keynesian theory of consumption
is：2
,
namely,
－231.8
and
0.7194.
Thus,
the
estimated
Yˆ 231.8 0.7194 X
We see that，for the period 1980—1992, MPC≈0.72 in America, suggesting that for the sample period an increase in real income of 1 dollar led, on average, to an increase of about 72 cents in real
However, the establishment of World Econometric Society in December 29th, 1930 and the publication of its academic journal Econometrics in 1933 are generally acknowledged as a landmark of econometrics as a separate discipline.
Why a separate discipline?
Economic theory makes statements or hypotheses that are mostly qualitative in nature; It is the job of the econometrician to provide such numerical estimates.
Example: Keynesian theory of consumption
计量经济学(英文PPT)Chapter 3 TWO-VARIABLE REGRESSION MODEL-THE PROBLEM OF ESTIMATION
Chapter 3 TWO-VARIABLE REGRESSION MODEL:
THE PROBLEM OF ESTIMATION
§3.1 THE METHOD OF ORDINARY LEAST SQUARES
The Method of ordinary least squares is attributed to Carl Friedrich Gauss, a German mathematician.
Ⅱ．The estimators which are point estimators are different from interval estimators.
Ⅲ．Once the OLS estimators are obtained from the sample data, the sample regression line can be easily obtained. The regression line thus obtained has the following properties:
Yˆi ˆ1 ˆ2 X i (Y ˆ2 X ) ˆ2 X i Y ˆ2 ( X i X )
while ∵ (Xi X ) 0
∴sum the equation above for the sample value on both sides and divide the result through by n( sum for i，then dived by n),
least-squares estimators, for they are derived from the least-
squares principle.
note：
THE PROBLEM OF ESTIMATION
§3.1 THE METHOD OF ORDINARY LEAST SQUARES
The Method of ordinary least squares is attributed to Carl Friedrich Gauss, a German mathematician.
Ⅱ．The estimators which are point estimators are different from interval estimators.
Ⅲ．Once the OLS estimators are obtained from the sample data, the sample regression line can be easily obtained. The regression line thus obtained has the following properties:
Yˆi ˆ1 ˆ2 X i (Y ˆ2 X ) ˆ2 X i Y ˆ2 ( X i X )
while ∵ (Xi X ) 0
∴sum the equation above for the sample value on both sides and divide the result through by n( sum for i，then dived by n),
least-squares estimators, for they are derived from the least-
squares principle.
note：
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8
One-Sided Alternatives (cont)
Having picked a significance level, a, we look up the (1 – a)th percentile in a t distribution with n – k – 1 df and call this c, the critical value We can reject the null hypothesis if the t statistic is greater than the critical value If the t statistic is less than the critical value then we fail to reject the null
Under the CLM assumptions, conditional on the sample values of the independent variable s
bˆ j ~ Normal b j ,Var bˆ j , so that
bˆ j b j sd bˆ j ~ Normal 0,1
7
t Test: One-Sided Alternatives
Besides our null, H0, we need an alternative hypothesis, H1, and a significance level H1 may be one-sided, or two-sided
because we have to estimate s 2by sˆ 2
Note the degrees of freedom : n k 1
5
The t Test (cont)
Knowing the sampling distribution for the standardized estimator allows us to carry out hypothesis tests Start with a null hypothesis
9
One-Sided Alternatives (cont)
yi = b0 + b1xi1 + … + bkxik + ui
H0: bj = 0
H1ห้องสมุดไป่ตู้ bj > 0
Fail to reject
H1: bj > 0 and H1: bj < 0 are one-sided H1: bj 0 is a two-sided alternative
If we want to have only a 5% probability of rejecting H0 if it is really true, then we say our significance level is 5%
"the"t statistic for bˆj :tbˆ j bˆ j se bˆ j
We will then useour t statistic along with a rejection rule to determine whether ot accept thenull hypothesis, H0
mean and variance s2: u ~ Normal(0,s2)
1
CLM Assumptions (cont)
Under CLM, OLS is not only BLUE, but is the minimum variance unbiased estimator We can summarize the population assumptions of CLM as follows
bˆj is distributed normally becauseit
is a linear combinatio n of the errors
4
The t Test
Under the CLM assumptions
bˆ j b j se bˆ j ~ tnk1
Note this is a t distributi on (vs normal)
For example, H0: bj=0
If accept null, then accept that xj has no effect on y, controlling for other x’s
6
The t Test (cont)
To performour test we first need to form
2
The homoskedastic normal distribution with a single explanatory variable
y
f(y|x)
.
.
E(y|x) = b0 + b1x
Normal distributions
x1
x2
3
Normal Sampling Distributions
Assumptions of the Classical Linear Model (CLM)
So far, we know that given the GaussMarkov assumptions, OLS is BLUE, In order to do classical hypothesis testing, we need to add another assumption (beyond the Gauss-Markov assumptions) Assume that u is independent of x1, x2,…, xk and u is normally distributed with zero
y|x ~ Normal(b0 + b1x1 +…+ bkxk, s2)
While for now we just assume normality, clear that sometimes not the case Large samples will let us drop normality
One-Sided Alternatives (cont)
Having picked a significance level, a, we look up the (1 – a)th percentile in a t distribution with n – k – 1 df and call this c, the critical value We can reject the null hypothesis if the t statistic is greater than the critical value If the t statistic is less than the critical value then we fail to reject the null
Under the CLM assumptions, conditional on the sample values of the independent variable s
bˆ j ~ Normal b j ,Var bˆ j , so that
bˆ j b j sd bˆ j ~ Normal 0,1
7
t Test: One-Sided Alternatives
Besides our null, H0, we need an alternative hypothesis, H1, and a significance level H1 may be one-sided, or two-sided
because we have to estimate s 2by sˆ 2
Note the degrees of freedom : n k 1
5
The t Test (cont)
Knowing the sampling distribution for the standardized estimator allows us to carry out hypothesis tests Start with a null hypothesis
9
One-Sided Alternatives (cont)
yi = b0 + b1xi1 + … + bkxik + ui
H0: bj = 0
H1ห้องสมุดไป่ตู้ bj > 0
Fail to reject
H1: bj > 0 and H1: bj < 0 are one-sided H1: bj 0 is a two-sided alternative
If we want to have only a 5% probability of rejecting H0 if it is really true, then we say our significance level is 5%
"the"t statistic for bˆj :tbˆ j bˆ j se bˆ j
We will then useour t statistic along with a rejection rule to determine whether ot accept thenull hypothesis, H0
mean and variance s2: u ~ Normal(0,s2)
1
CLM Assumptions (cont)
Under CLM, OLS is not only BLUE, but is the minimum variance unbiased estimator We can summarize the population assumptions of CLM as follows
bˆj is distributed normally becauseit
is a linear combinatio n of the errors
4
The t Test
Under the CLM assumptions
bˆ j b j se bˆ j ~ tnk1
Note this is a t distributi on (vs normal)
For example, H0: bj=0
If accept null, then accept that xj has no effect on y, controlling for other x’s
6
The t Test (cont)
To performour test we first need to form
2
The homoskedastic normal distribution with a single explanatory variable
y
f(y|x)
.
.
E(y|x) = b0 + b1x
Normal distributions
x1
x2
3
Normal Sampling Distributions
Assumptions of the Classical Linear Model (CLM)
So far, we know that given the GaussMarkov assumptions, OLS is BLUE, In order to do classical hypothesis testing, we need to add another assumption (beyond the Gauss-Markov assumptions) Assume that u is independent of x1, x2,…, xk and u is normally distributed with zero
y|x ~ Normal(b0 + b1x1 +…+ bkxk, s2)
While for now we just assume normality, clear that sometimes not the case Large samples will let us drop normality