Ch-12 Correlation and Regression Analysis

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第八章相关与回归分析Correlation and Regression Analysis

n xt yt xt yt
83142 73 321
2 2 [8713 (73) ][8 14111 (321) ]
50
40
30
0.886
20
10
0 0 2 4 6 8 10 12 14
树干的直径, x
r = 0.886 → 表明 x 和 y 具有高度线性相关关系。
Chap 08-12
2

假定3：误差项之间不存在序列相关关系，其协方差为零；假定4：自变量是给定的变量，与随机误差项线性无关；

假定5：随机误差项服从正态分布；
Chap 08-22
最小二乘估计

在根据样本数据确定样本回归方程时，总是希望 y 的估计值尽可能地接近其实际观测值，即残差 et 的总量越小越好。由于 et 有正有负，简单的代数和会相互抵消，因此为了数学上便于处理，我们采用残差平方和作为衡量总偏差的尺度。所谓最小二乘法，就是根据这一思路，通过使残差平方和最小来估计回归系数的方法。
Excel 输出结果
Excel 相关分析的输出结果工具 / 数据分析 / 相关系数
树的高度树的高度树干的直径 1 0.886231 树干的直径 1
树的高度与树干的直径的相关系数
Chap 08-13
相关系数的特点

r的取值在-1与1之间；当ｒ＝０时，Ｘ与Ｙ的样本观测值之间没有线性关系；在大多数情况下，０＜｜ｒ｜＜１，即Ｘ与Ｙ的样本观测值之间存在着一定的线性关系，当ｒ＞０时，Ｘ与Ｙ为正相关，当ｒ＜０时，Ｘ与Ｙ为负相关。如果｜ｒ｜＝１，则表明Ｘ与Ｙ完全线性相关，当ｒ＝１时，称为完全正相关，而ｒ＝－１时，称为完全负相关。ｒ是对变量之间线性相关关系的度量。ｒ＝０只是表明两个变量之间不存在线性关系，但它并不意味着Ｘ与Ｙ之间不存在其他类型的关系。

国外记忆和认知课件ch12

Cognition 7e, Margaret Matlin Chapter 12
The Propositional Calculus
Cognition 7e, Margaret Matlin
Chapter 12
Deductive Reasoning
An Overview of Conditional Reasoning
Chapter 12
Deductive Reasoning
An Overview of Conditional Reasoning
the propositional calculus antecedent consequent
Cognition 7e, Margaret Matlin
Chapter 12
The Confirmation Bias
The Standard Wason Selection Task
confirmation bias—people would rather try to confirm a hypothesis than try to disprove it
Variations on the Wason Selection Task
Confirmation Bias
Variations on the Wason Selection Task
(continued)
Griggs and Cox (1982)—drinking age example
If a person drinks an alcoholic drink, then they must be over the age of 21 years old.
Cognition 7e, Margaret Matlin

计量经济学ch12

y 1 0 +1x et
3
1 2
12
y1
1 2
12
0 1
1 2
12
x1
1 2
12
u1
y1 1 2 1 2 0 +1x uˆ1
4
3 和 4 一并用于OLS回归即可得到参数的BLUE估计。

H0 : 0；H1 : 0。
Copyright © 2007 Thomson Asia Pte. Ltd. All rights res1e1rved.
D.W检验步骤:
（1）计算DW值
（2）给定，由n和k的大小查DW分布表，得临界值dL和dU （3）比较、判断
若 0<DW<dL
e
Copyright © 2007 Thomson Asia Pte. 3
12.1 含序列相关误差时OLS的性质(续)
考虑简单回归模型中OLS斜率估计量的方差：
y 0 1xt ut , 假定x 0
1的OLS估计量可写为：ˆ1 1 SSTx-1
n t 1
xt
ut
n
n
n
n
uˆt uˆt1 2
uˆt2 uˆt21 2 uˆtuˆt1
DW t2 n
t2
t2 n
t2
uˆt2
uˆt
t 1
t 1

n
uˆtuˆt
1

21
t2 n

uˆt
21 ˆ

t 1
Copyright © 2007 Thomson Asia Pte. Ltd. All rights rese8rved.

第9讲相关分析与回归分析教案

r

2 xy

x y
__
__
(x- x)(y- y)
__
__
(x- x)2 (y- y)2
n xy x y n x2 ( x)2 n y2 ( y)2
【专栏】在相关分析中，定性分析或经济理论分析重要吗?
相关系数(Correlation Coefficient)
第一、二节－相关分析概述－相关关系测定
变量间关系
血压 ~ 年龄
动物死亡率 ~ 毒物剂量
两个随机变量之间的关系
体重~身高
肺活量~体重
都是描述两个随机变量之间的关系。
相关: 血压和年龄关联的程度如何? 动物死亡率与毒物剂量关联的程度如何?
回归: 人群中,平均而言, 血压如何随年龄变化? 毒性实验中, 动物死亡率如何随剂量变化?
Y 和 X之间的函数关系
指数函数
对数函数
正弦函数
对应于给定的 X值, 相应的Y 值是确定的.
但现在给定一个X值，Y可能是不确定的。（是上页）
一三是正线性相关，3图相关性好。 2,4是负先关性，4负的绝对值比2好一些。
分散
相关性好就是绝对值好些
集中上面都是线性相关
x不论怎么变，Y都
是在一个范围走动，下前三没有相关性烧饼，给定一个X，
na b x y a x b x2 xy
求方程组：
b

n xy x y n x2 ( x)2
b求出来都代入到a公式里面
a y b x 其实就是（y bx）
n
n
例 - 某省1991~1998年人口资料如表所示，要求拟合时点的趋势方程，并

回归分析

Regression Analysis 回归分析
y

x
5
Regression Analysis
变量间的关系
（函数关系）
函数关系的例子
回归分析
某种商品的销售额 (y) 与销售量 (x) 之间的关系可表示为 y = p x (p 为单价) 圆的面积(S)与半径之间的关系可表示为S = r2
样本相关系数的定义公式是：
r
( X X )(Y Y ) ( X X ) (Y Y )
t t 2 t t
2
上式中， X 和 Y 分别是Ｘ和Ｙ的样本平均数。样本相关系数是根据样本观测值计算的，抽取的样本不同，其具体的数值也会有所差异。容易证明，样本相关系数是总体相关系数的一致估计量。
r的取值相关程度
|r|＜0.3 不线性相关
0.3≤|r|<0.5 0.5≤|r|<0.8
|r|≥0.8
低度线性相中度线性相高度线性关关相关
23
Regression Analysis 回归分析
•
3.如果|ｒ|=1，则表明Ｘ与Ｙ完全线性相关，当ｒ=1时，称为完全正相关，而ｒ=-1时，称为完全负相关。
相关分析（Correlation Analysis）是用于度量两个
数值变量间的关联程度
3
Regression Analysis 回归分析
一、函数关系与相关关系
1.函数关系
当一个或几个变量取一定的值时，另一个变量有确定值与之相对应，我们称这种关系为确定性的函数关系。
4
（函数关系）
（1）是一一对应的确定关系（2）设有两个变量 x 和 y ，变量 y 随变量 x 一起变化，并完全依赖于 x ，当变量 x 取某个数值时， y 依确定的关系取相应的值，则称 y 是 x 的函数，记为 y = f (x)，其中 x 称为自变量，y 称为因变量（3）各观测点落在一条线上

几种发表性偏倚评估方法介绍

。
③Ro
sen
tha
l
N
法
fs
是分
别
计
算量值是否为“0”的统计量 U, 根据 U 值进行分
段 ,删除部分无统计意义的研究 ,并计算相应的安全系数 ,计算公式为〔5〕:
N fs =
∑U 1164
2
-
k
其中 U 为每一个独立研究效应值是否为“0”检验统计
量 , k为已收集的独立研究的个数。
可能更贴切。
鉴于 Rosenthal N fs存在的种种问题 , O rw in在 1983 年在 Rosenthal N fs的基础上进行了完善 ,在本文我们称做 O rw in’s N fs法。O rw in’s N fs法主要解决以下两个问题〔6〕: ①O rw in’s N fs法让研究者确定的是总的效应量变为某一特定值 (而不是“0”)最少需要多少个未发表
在此 ,介绍一种称之为“trim and fill”的方法〔7〕。 The trim and fill法实际上是一种迭代算法 ,先从漏斗图的阳性面 ( the positive side) (漏斗图中研究文献多的一边 )去处一些极小样本量的研究 ,重新计算总的效应量 ,然后将去除的这些原始研究逐步加入到公式中重新计算 ,如此反复 ,直到漏斗图围绕重新计算的总的效应量呈左右对称。 trim and fill法属于一种非参法 ,计算过程也较为复杂 ,但其作用是较为明显的 ,一些常用的统计软件 ,如 STATA、Comp rehensive M eta A2 nalysis软件都能执行相应的操作。利用 trim and fill 法还能大致估计出未发表文献的数量。
概念定义为 :当 M eta分析结果有统计学意义时 ,为排

衍生品市场ch12

Chapter 12: Financial Risk Management: Techniques and Applications
Risk management is akin to a dialysis machine. It it doesn’t work, you might have a noble obituary, but you’re dead. Ben Golub Derivatives Strategy, October, 2000, p. 26
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 15: 4
Why Practice Risk Management? (continued)

The Benefits of Risk Management What are the benefits of risk management, in light of the Modigliani-Miller principle that corporate financial decisions provide no value because shareholders can execute these transactions themselves? Firms can practice risk management more effectively. There may tax advantages from the progressive tax system. Risk management reduces bankruptcy costs. Managers are trying to reduce their own risk.

第八章相关与回归分析Correlation and Regression Analysis

变量之间的函数关系和相关关系在一定条件下可以相互转化。客观现象的函数关系可以用数学分析的方法去研究，而研究客观现
象的相关关系必须借助于统计学中的相关与回归分析方法。
Chap 08-4
相关关系的类型
从相关关系涉及的变量数量看：单相关和复相关一个变量对另一变量的相关关系，称为单相关；一个变量对两个以上变量的相关关系时，称为复相关；从变量相关关系的表现形式看：线性相关和非线性相关从变量相关关系变化的方向看：正相关和负相关从变量相关的程度看：完全相关〔函数关系〕、不完全相
或:
r
n xtyt xt yt
[n ( xt2)( xt)2]n [( yt2)( yt)2]
Chap 08-7
2 简单线性相关与回归分析
2.1 简单线性相关系数及检验 2.2 总体回归函数与样本回归函数 2.3 回归系数的估计 2.4 简单线性回归模型的检验 2.5 简单线性回归模型预测
Chap 08-8
相关系数
总体相关系数〔 population correlation coefficient〕 ρ 是反映两变量之间线性相关程度的一种特征值，表现为一个常数。
关、不相关
Chap 08-5
相关分析与回归分析
而样本回归函数中的和是随机变量，其具体数值随所抽取的样本观测值不同而变动。
是当 x 等于 0 时 y 的平均估计值 S越小说明实际观测点与所拟合的样本回归线的离差程度越小，即样本回归线具有较强的代表性，反之，S越大说明实际观测点与所拟合的样本回归线的离差程度越大，即回归线的代表性越差。
Chap 08-1
本节学习目标
通过本节的学习，你应该能够:
理解和掌握相关分析和回归分析的原理估计一元线性回归模型，并对模型进行检验利用计算机软件估计多元线性回归模型，并对模型进行

大学生自我控制量表的修订

大学生自我控制量表的修订谭树华，郭永玉(华中师范大学心理学院，湖北武汉430079)【摘要】目的：修订自我控制量表(SCS)，考察其心理测量学指标。

方法：对799名武汉市大学生进行测查，对量表进行验证性因素分析和信、效度检验。

结果：验证性因素分析的结果显示，SCS的五因素结构拟合较好。

SCS的内部一致性信度为0.862，重测信度为0.850。

以被试的平均学分绩、人际关系满意感、生活满意感、心理健康水平为效标，与SCS的相关分别为0.146；0.280；0.163；0.317。

结论：SCS符合心理测量学的要求；可作为测量我国大学生自我控制能力的工具。

【关键词】自我控制量表；信度；效度中图分类号:R395.1文献标识码:A文章编号:1005-3611(2008)05-0468-03Revision of Self-Control Scale for Chinese College StudentsTAN Shu-hua，GUO Yong-yuSchool of Psychology，Central China Normal University，Wuhan430079，China【Abstract】Objective：To revise the Self-Control Scale(SCS).Methods：Data were collected from a sample of799col-lege students of Wuhan and analyzed by Confirmatory Factor Analysis,reliability test and validity test.Results：The re-sults of Confirmatory Factor Analysis showed that the revised SCS was five-factor construct and had good construct validi-ties.The Cronbach’sαcoefficient of the SCS scale was0.862,and the reliability coefficient of the test-retest stability co-efficient was0.850.The correlation between total score of SCS and that of Grade Point Average(GPA)was0.146;the cor-relation between total score of SCS and that of Interpersonal Satisfaction Scale(ISS)was0.280;the correlation between total score of SCS and that of Life Satisfaction Scale(LSS)was0.163,and the correlation between total score of SCS and that of General Health Questionnaire(CHQ)was0.317.Conclusion：The revised scale of SCS has good psychometric quality and can be used in Chinese college students.【Key words】Self-control Scale；Reliability；Validity人们最健康、最幸福的时候就是自我和环境完全匹配的时候，只是日常生活中个体和环境很难完全匹配，但匹配的程度可以通过调整自我来得到最大化地提升[1]。

CorrelationandCausationReview-1

Review -1•Two types of correlational study–When same items have values on two scorevariables, correlate the scores on one with the scores on the other•Measure degree of correlation in terms of Pearsoncoefficient r•Predict value on one variable from that on theother using the regression line: y=a x+b–When one nominal variable divides a population intotwo or more sub-populations, compare the twopopulations on another (score) variable in terms oftheir central tendencies•If the means are different, predict the value on thescore variable depending on the value of thenominal variableReview -2•In both types of correlational studies, one commonly makes inferences from a sample to an actual (total)population–Does what is found in the sample apply to the actualpopulation?–Addressed in terms of statistical significance•Is the result in the sample one that would beunlikely to happen by chance if there weren’t acorrelation or a difference in the actual population?•The p value specifies the likelihood of the result inthe sample happening by chance (in drawing thesample)–p < .05 indicates there is less than 5% chanceof the result happening by chanceReview -3•In testing a claim about differences in the means of two sub-populations, one tests the null-hypothesis–There is no difference in the means•The strategy is to try to reject the null hypothesis in terms of the results in the sample–If the differences in means in the sample arestatistically significant (at a choosen level), one infers that the null hypothesis is false•Therefore, the means differ in the real populations –If the differences in means in the sample are notstatistically signfiicant (at the choosen level), onecannot reject the null hypothesis•Whatever differences there might be, they will nothave been detected.Review -4•Two types of errors–Type 1 error: concluding that there is a differencebetween the two groups in the population when there is no difference–Type 2 error: concluding that there is no (detectable)difference between the two groups in the populationwhen there is a difference•To reduce Type 1 error: demand a higher p-value before accepting that there really is a difference•To reduce Type 2 error: use a larger sample size –Which is more likely to produce a statisticallysignificant difference if there really is a difference inthe two groupsThe Logic of CorrelationalResearch•To confirm or falsify a correlational claim based on a sample, we use modus tollens. The first premise in each case, though, is different•Confirming a correlational claim:If there is no difference between means in thepopulation, then there will not be a statisticallysignificant difference in my sampleThere is a statistically significant difference in meansin my sampleˆThere is a difference between means in thepopulation•We pick the level of significance in the first premise according to how great a risk of error we can acceptAvoid Confirming the Consequence Array•What about this argument?If there is a difference between means in the XXXpopulation, then there will be a statistically significantdifference in my sampleThere is a statistically significant difference in meansin my sampleˆThere is a difference between means in thepopulation•It is INVALIDThe Logic of CorrelationalResearch -2•Falsifying a correlational claimIf there is a (detectable) difference between means inthe population, then there will be a statisticallysignificant difference in my sampleThere is no statistically significant difference in meansin my sampleˆThere is no (detectable) difference between meansin the population•The truth of the first premise depends upon using a largeenough sampleQuest for finding causes•When something happens, we ask “Why?” We want to know what caused the event–Why are we interested in causes?•Knowing the causes frequently providesunderstanding•Knowing causes empowers us to intervene•These two tend to go together–Why do these barrels produce better beer?»Learning the reason is more hops providesunderstanding»And a procedure for making better beer–How does HIV cause AIDS?»Knowing about protease inhibitors explains»And tells us a good place to interveneWhat is a cause?•The roots of appeal of causation lie in ourdoing something to produce and effect–We want to move a rock, so we push it–We want to see a friend so we walk to herapartment–We want to stay warm so we put on a jacket •Independent of our own action, a cause is something which brings about or increases the likelihood of an effect–The cause of the explosion was the spark fromthe generatorCorrelation and Causation • A major reason people are interested in correlationsis that they might be indicative of causation •Correlations per se only allow you to predict–The correlation of unprotected sex with having a baby nine months later allows you to predict thatif you engage in unprotected sex, you are morelikely to have a baby nine months later •Causation tells you how to change the effect–Knowing that unprotected sex causes (increases the likelihood of) having a baby nine months laterallows you to take action to have or not have ababyCorrelations Point to Causation •Statistical relations between variables that exceed what is statistically expected are typically due to causalrelations–Although not necessary direct causal relations•Examples:–Consumption of red wine and reduced heart attacks –Book that have a green cover and books that do notselling many copies–Good study habits and good gradesCorrelation Symmetrical;Causation Asymmetrical•Being run into in a traffic accidentmight be a cause for the big dentin your car•Having a big dent in your car iscorrelated with having a car accident, but it is not thecause of having a car accident•Causation is directional, correlation is symmetrical –So when correlation points to causation, we still need to establish the directionProblem of Directionality •Does watching violence on TV result in aggressive behavior?•Or do the factors that generate aggressive behavior cause children to watch more violence on TVCausal Loops•Sometimes X causes Y and then Y causes more X –The causation here is still directional, but works inboth directions•Back pain may be the cause of a person limping –but walking with a limp may cause further back painSnoring and Obesity•There is a positive correlation between obesity and snoring•Does obesity cause (increased) snoring?–Yes—via fat buildup in the back of the throat•But fat build up also causes sleep apnea–Sleeper stops breathing momentarily and wakes up •As a result of sleep apnea, sufferer is tired and avoids physical activity–Thereby getting more obeseObesity SnoringSleepApneaRelating Correlation and Causation •Establishing correlation does not establish causation –But it is a big part of the project!•If X causes Y, then one expects a correlation between X and Y–The greater the value of X (if X is a score variable), the greater the value of Y–Individuals exhibiting X (if X is a nominal variable) will have greater values of YIndependent/Dependent Variables •Independent variable–The variable that is thought to be the cause–The variable that is altered/manipulated in anexperiment–The treatment in a clinical trial•Dependent variable–The variable that is thought to be the effect–The variable that one is trying to predict/explain–The outcome•The dependent variable depends on the independent variableMeasured versus Manipulated•The strongest tests of causation claims involve manipulation of variables ÆExperiments•In some contexts, a researcher does not or cannot manipulate the independent variable–Immoral to assign people to categories such ashaving unprotected sex–Cannot assign people to categories such as beingfemale•If we are nonetheless considering causes in such a case, we refer to a measured independent variable •When it is possible to manipulate the independent variable (conduct an experiment), we speak of amanipulated independent variableMeasures and Data•Often causal relations are specified in general terms:–Violence on TV causes violent behavior in school •The variables used to operationalize such variables are sometimes referred to as measures. The specific values on these variables are data–“The number of gun firings on a given TV show is agood measure of violence on the show. We haverelated data on gun firings to data on two measures of aggressive behavior by those watching the show.”•The measure: Violence operationalized as # of gunfirings•Data on # of gun firingsCorrelation without direct causation •Sometimes one variable isdirectly related causally toanother•But sometimes thecausation is via someother linkCorrelations without directcausation•Ice cream sales and the number of shark attacks on swimmers are correlated•SAT scores and college grades are correlated•Skirt lengths and stock prices are highly correlated (as stock prices go up, skirt lengths get shorter).•The number of cavities in elementary school children and vocabulary size have a strong positive correlationWhen causationsuspected•Driving red cars is positively correlated with having traffic accidents•Why? Several possible causal scenarios–accident-prone drivers prefer red–people become more aggressive when driving red cars–more dangerous cars tend to be painted red (sports cars)–the color red is harder to see and is more likely to be involved in a 2-car accidentExtraneous Variables•Given the number of possible variables to consider, in any given sample some will be correlated with thedependent variable of interest•If these are not the variables we are focusing on, we term them extraneous•But–What we term extraneous may in fact be the causallyrelevant variable–So, care must be taken to rule out any causal linkbetween these extraneous variables and thedependent variableLimits of correlation•Fluoride in water is correlated with lower rate of tooth decay•But why?–Fluoride reduces cavities–People in cities with fluoride enjoy better diets–People in cities with fluoride practice better dental hygiene–People in cities with fluoride have better genetics–Water in cities with fluoride contains other minerals (calcium) that help prevent tooth decay•These additional variables are extraneous from the point of view of the first hypothesis, but they might be the true causesTelling Causal Stories Can be Fun •Correlation: Amount of ice cream sold correlates with increaseddeaths by drowning:“Increases in nuclear power generator accidents (Chernobyl, Three Mile Island...) have resulted in greenhouse gas increases, ozonelayer reduction, average world temperature rise and increases in the fraction of heavy water in rain. Concerns about nuclear catastrophe have resulted in increases in eating disorders, especially amongthose with a genetic predisposition to obesity. Heavy water in rainhas resulted in an increase in the specific gravity of cream produced by cows, while the increasing world temperature has resulted in anincreasing attendance at beach resorts, coupled with increasedconsumption of ice cream. The increased weight of fat worried people whose centre of gravity has been lowered by a rising consumption of heavy ice cream has caused an increased number of deaths bydrowning.” Dr. Paul Gardner, Monash University, AustraliaTelling Causal Stories can be Fun -2•Correlation: Number of fire trucks and amount of firedamage:“While this could be another case of intentionally startingfires in effort to attract the fire people, this seems highlyunlikely. Firefighter salaries are modest. The only logicalexplanation is that the community just feels so darn safeknowing that there are more fire trucks around, that theysimply are not as careful and concerned with fire safety.They feel so confident that a truck would rescue them in aninstant, before a fire could spread very far, so they are justcareless. With this inappropriate assumption and subsequent increase in fires, the firefighters are even less able to arriveat a scene on time. Thus, more damage occurs.” KatieBrandt, Purdue University IndianapolisBeyond causal story telling Array•If a causal relation exists between two variables, then if we can directly manipulate values on one (the independent variable), we should affect values on the other (the dependent variable)•An experiment is precisely an attempt to demonstrate causal relations by manipulating the independent variable and measuring the affect on the dependent variable.The Logic of Causal Research•To confirm or falsify a causal claim based on a correlation, we use modus tollens. The first premise in each case,though, is different•Confirming a causal claim:If X is not a cause of Y [and there is no alternativeplausible hypothesis], then there will not be astatistically significant difference in Y when X is presentThere is a statistically significant difference in Y when X is present [and there is no alternative plausiblehypothesis]ˆX is a cause of Y•Whether the first premise is true depends critically on how we set up the test of the causal hypothesis—whether we make it very unlikely that anything else could produce a difference in YAvoiding Confirming theConsequence•Don’t use this argument: XXXIf X is a cause of Y, then there will be a statisticallysignificant difference in Y when X is presentThere is a statistically significant difference in in Ywhen X is presentˆX is the cause of YIt is invalid!The Logic of Causal Research -2•Falsifying a causal claimIf X were the cause of Y [and auxiliary assumptionsare true and the experimental set up is adequate],then there would be a statistically significantdifference in Y when X is presentThere is no statistically significant difference in Ywhen X is present [and auxiliary assumptions are true and the experimental set up is adequate]ˆX is not the cause of Y•The truth of the first premise depends critically on how we set up the test of the causal claim。

计量经济学导论CH12习题答案

CHAPTER 12TEACHING NOTESMost of this chapter deals with serial correlation, but it also explicitly considers heteroskedasticity in time series regressions. The first section allows a review of what assumptions were needed to obtain both finite sample and asymptotic results. Just as with heteroskedasticity, serial correlation itself does not invalidate R-squared. In fact, if the data are stationary and weakly dependent, R-squared and adjusted R-squared consistently estimate the population R-squared (which is well-defined under stationarity).Equation (12.4) is useful for explaining why the usual OLS standard errors are not generally valid with AR(1) serial correlation. It also provides a good starting point for discussing serial correlation-robust standard errors in Section 12.5. The subsection on serial correlation with lagged dependent variables is included to debunk the myth that OLS is always inconsistent with lagged dependent variables and serial correlation. I do not teach it to undergraduates, but I do to master’s students.Section 12.2 is somewhat untraditional in that it begins with an asymptotic t test for AR(1) serial correlation (under strict exogeneity of the regressors). It may seem heretical not to give the Durbin-Watson statistic its usual prominence, but I do believe the DW test is less useful than the t test. With nonstrictly exogenous regressors I cover only the regression form of Durbin’s test, as the h statistic is asymptotically equivalent and not always computable.Section 12.3, on GLS and FGLS estimation, is fairly standard, although I try to show how comparing OLS estimates and FGLS estimates is not so straightforward. Unfortunately, at the beginning level (and even beyond), it is difficult to choose a course of action when they are very different.I do not usually cover Section 12.5 in a first-semester course, but, because some econometrics packages routinely compute fully robust standard errors, students can be pointed to Section 12.5 if they need to learn something about what the corrections do. I do cover Section 12.5 for a master’s level course in applied econometrics (after the first-semester course).I also do not cover Section 12.6 in class; again, this is more to serve as a reference for more advanced students, particularly those with interests in finance. One important point is that ARCH is heteroskedasticity and not serial correlation, something that is confusing in many texts. If a model contains no serial correlation, the usual heteroskedasticity-robust statistics are valid. I have a brief subsection on correcting for a known form of heteroskedasticity and AR(1) errors in models with strictly exogenous regressors.100101SOLUTIONS TO PROBLEMS12.1 We can reason this from equation (12.4) because the usual OLS standard error is anestimate of σthe AR(1) parameter, ρ, tends to be positive in time series regression models. Further, the independent variables tend to be positive correlated, so (x t - x )(x t +j - x ) – which is what generally appears in (12.4) when the {x t } do not have zero sample average – tends to be positive for most t and j . With multiple explanatory variables the formulas are more complicated but have similar features.If ρ < 0, or if the {x t } is negatively autocorrelated, the second term in the last line of (12.4)could be negative, in which case the true standard deviation of 1ˆβis actually less than σ12.2 This statement implies that we are still using OLS to estimate the βj . But we are not using OLS; we are using feasible GLS (without or with the equation for the first time period). In other words, neither the Cochrane-Orcutt nor the Prais-Winsten estimators are the OLS estimators (and they usually differ from each other).12.3 (i) Because U.S. presidential elections occur only every four years, it seems reasonable to think the unobserved shocks – that is, elements in u t – in one election have pretty much dissipated four years later. This would imply that {u t } is roughly serially uncorrelated.(ii) The t statistic for H 0: ρ = 0 is -.068/.240 ≈ -.28, which is very small. Further, theestimate ˆρ= -.068 is small in a practical sense, too. There is no reason to worry about serial correlation in this example.(iii) Because the test based on ˆt ρ is only justified asymptotically, we would generally be concerned about using the usual critical values with n = 20 in the original regression. But any kind of adjustment, either to obtain valid standard errors for OLS as in Section 12.5 or a feasible GLS procedure as in Section 12.3, relies on large sample sizes, too. (Remember, FGLS is not even unbiased, whereas OLS is under TS.1 through TS.3.) Most importantly, the estimate of ρ ispractically small, too. With ˆρso close to zero, FGLS or adjusting the standard errors would yield similar results to OLS with the usual standard errors.12.4 This is false, and a source of confusion in several textbooks. (ARCH is often discussed as a way in which the errors can be serially correlated.) As we discussed in Example 12.9, the errors in the equation return t = β0 + β1return t-1 + u t are serially uncorrelated, but there is strong evidence of ARCH; see equation (12.51).12.5 (i) There is substantial serial correlation in the errors of the equation, and the OLS standarderrors almost certainly underestimate the true standard deviation in ˆEZβ. This makes the usual confidence interval for βEZ and t statistics invalid.102(ii) We can use the method in Section 12.5 to obtain an approximately valid standard error.[See equation (12.43).] While we might use g = 2 in equation (12.42), with monthly data we might want to try a somewhat longer lag, maybe even up to g = 12.12.6 With the strong heteroskedasticity in the errors it is not too surprising that the robuststandard error for 1ˆβ differs from the OLS standard error by a substantial amount: the robust standard error is almost 82% larger. Naturally, this reduces the t statistic. The robust t statistic is .059/.069≈ .86, which is even less significant than before. Therefore, we conclude that, once heteroskedasticity is accounted for, there is very little evidence that return t-1 is useful for predicting return t .SOLUTIONS TO COMPUTER EXERCISES12.7 Regressing ˆt uon 1ˆt u -, using the 69 available observations, gives ˆρ≈ .292 and se(ˆρ) ≈ .118. The t statistic is about 2.47, and so there is significant evidence of positive AR(1) serial correlation in the errors (even though the variables have been differenced). This means we should view the standard errors reported in equation (11.27) with some suspicion.12.8 (i) After estimating the FDL model by OLS, we obtain the residuals and run the regressionˆt uon 1ˆt u -, using 272 observations. We get ˆρ≈ .503 and ˆt ρ≈ 9.60, which is very strong evidence of positive AR(1) correlation.(ii) When we estimate the model by iterated C-O, the LRP is estimated to be about 1.110.(iii) We use the same trick as in Problem 11.5, except now we estimate the equation by iterated C-O. In particular, writegprice t = α0 + θ0gwage t + δ1(gwage t -1 – gwage t ) + δ2(gwage t-2 – gwage t )+ + δ12(gwage t -12 – gwage t ) + u t ,Where θ0 is the LRP and {u t } is assumed to follow an AR(1) process. Estimating this equationby C-O gives 0ˆθ≈ 1.110 and se(0ˆθ)≈ .191. The t statistic for testing H 0: θ0 = 1 is (1.110 – 1)/.191≈ .58, which is not close to being significant at the 5% level. So the LRP is not statistically different from one.12.9 (i) The test for AR(1) serial correlation gives (with 35 observations) ˆρ≈ –.110, se(ˆρ)≈ .175. The t statistic is well below one in absolute value, so there is no evidence of serial correlation in the accelerator model. If we view the test of serial correlation as a test of dynamic misspecification, it reveals no dynamic misspecification in the accelerator model.(ii) It is worth emphasizing that, if there is little evidence of AR(1) serial correlation, there is no need to use feasible GLS (Cochrane-Orcutt or Prais-Winsten).10312.10 (i) After obtaining the residuals ˆt ufrom equation (11.16) and then estimating (12.48), we can compute the fitted values ˆth = 4.66 – 1.104 return t for each t . This is easily done in a single command using most software packages. It turns out that 12 of 689 fitted values are negative. Among other things, this means we cannot directly apply weighted least squares using the heteroskedasticity function in (12.48).(ii) When we add 21t return - to the equation we get2ˆi u= 3.26 - .789 return t-1 + .297 21t return - + residual t (0.44) (.196) (.036)n = 689, R 2 = .130.So the conditional variance is a quadratic in return t -1, in this case a U-shape that bottoms out at .789/[2(.297)] ≈ 1.33. Now, there are no fitted values less than zero.(iii) Given our finding in part (ii) we can use WLS with the ˆth obtained from the quadratic heteroskedasticity function. When we apply WLS to equation (12.47) we obtain 0ˆβ≈ .155 (se ≈ .078) and 1ˆβ≈ .039 (se ≈ .046). So the coefficient on return t-1, once weighted least squares has been used, is even less significant (t statistic ≈ .85) than when we used OLS.(iv) To obtain the WLS using an ARCH variance function we first estimate the equation in(12.51) and obtain the fitted values, ˆth . The WLS estimates are now 0ˆβ≈ .159 (se ≈ .076) and 1ˆβ≈ .024 (se ≈ .047). The coefficient and t statistic are even smaller. Therefore, once we account for heteroskedasticity via one of the WLS methods, there is virtually no evidence that E(return t |return t -1) depends linearly on return t -1.12.11 (i) Using the data only through 1992 givesdemwins= .441 - .473 partyWH + .479 incum + .059 partyWH ⋅gnews (.107) (.354) (.205) (.036)- .024 partyWH ⋅inf (.028)n = 20, R 2 = .437, 2R = .287.The largest t statistic is on incum , which is estimated to have a large effect on the probability of winning. But we must be careful here. incum is equal to 1 if a Democratic incumbent is running and –1 if a Republican incumbent is running. Similarly, partyWH is equal to 1 if a Democrat is currently in the White House and –1 if a Republican is currently in the White House. So, for an incumbent Democrat running, we must add the coefficients on partyWH and incum together, and this nets out to about zero.104The economic variables are less statistically significant than in equation (10.23). The gnews interaction has a t statistic of about 1.64, which is significant at the 10% level against a one-sided alternative. (Since the dependent variable is binary, this is a case where we must appeal to asymptotics. Unfortunately, we have only 20 observations.) The inflation variable has the expected sign but is not statistically significant.(ii) There are two fitted values less than zero, and two fitted values greater than one.(iii) Out of the 10 elections with demwins = 1, 8 of these are correctly predicted. Out of the 10 elections with demwins = 0, 7 are correctly predicted. So 15 out of 20 elections through 1992 are correctly predicted. (But, remember, we used data from these years to obtain the estimated equation.)(iv) The explanatory variables are partyWH = 1, incum = 1, gnews = 3, and inf = 3.019. Therefore, for 1996,demwins= .441 - .473 + .479 + .059(3) - .024(3.019) ≈ .552.Because this is above .5, we would have predicted that Clinton would win the 1996 election, as he did.(v) The regression of ˆt uon 1ˆt u - produces ˆρ ≈ -.164 with heteroskedasticity-robust standard error of about .195. (Because the LPM contains heteroskedasticity, testing for AR(1) serial correlation in an LPM generally requires a heteroskedasticity-robust test.) Therefore, there is little evidence of serial correlation in the errors. (And, if anything, it is negative.)(vi) The heteroskedasticity-robust standard errors are given in [⋅] below the usual standard errors:demwins= .441 - .473 partyWH + .479 incum + .059 partyWH ⋅gnews (.107) (.354) (.205) (.036)[.086] [.301] [.185] [.030]– .024 partyWH ⋅inf(.028) [.019]n = 20, R 2 = .437, 2R = .287.In fact, all heteroskedasticity-robust standard errors are less than the usual OLS standard errors, making each variable more significant. For example, the t statistic on partyWH ⋅gnews becomes about 1.97, which is notably above 1.64. But we must remember that the standard errors in the LPM have only asymptotic justification. With only 20 observations it is not clear we should prefer the heteroskedasticity-robust standard errors to the usual ones.10512.12 (i) The regression ˆt uon 1ˆt u - (with 35 observations) gives ˆρ≈ -.089 and se(ˆρ)≈ .178; there is no evidence of AR(1) serial correlation in this equation, even though it is a static model in the growth rates.(ii) We regress gc t on gc t-1 and obtain the residuals ˆt u. Then, we regress 2ˆt u on gc t -1 and 21t gc -(using 35 observations), the F statistic (with 2 and 32 df ) is about 1.08. The p -value isabout .352, and so there is little evidence of heteroskedasticity in the AR(1) model for gc t . This means that we need not modify our test of the PIH by correcting somehow for heteroskedasticity.12.13 (i) The iterated Prais-Winsten estimates are given below. The estimate of ρ is, to three decimal places, .293, which is the same as the estimate used in the final iteration of Cochrane-Orcutt:log()chn imp = -37.08 + 2.94 log(chempi ) + 1.05 log(gas ) + 1.13 log(rtwex )(22.78) (.63) (.98) (.51)- .016 befile6 - .033 affile6 - .577 afdec6(.319) (.322) (.342)n = 131, R 2 = .202(ii) Not surprisingly, the C-O and P-W estimates are quite similar. To three decimal places,they use the same value of ˆρ(to four decimal places it is .2934 for C-O and .2932 for P-W). The only practical difference is that P-W uses the equation for t = 1. With n = 131, we hope this makes little difference.12.14 (i) This is the model that was estimated in part (vi) of Computer Exercise 10.17. Aftergetting the OLS residuals, ˆt u , we run the regression 1垐 on ,2,...,108.t t u u t -= (Included anintercept, but that is unimportant.) The coefficient on 1ˆt u- is ˆρ=.281 (se = .094). Thus, there is evidence of some positive serial correlation in the errors (t ≈ 2.99). I strong case can be made that all explanatory variables are strictly exogenous. Certainly there is no concern about the time trend, the seasonal dummy variables, or wkends , as these are determined by the calendar. It is seems safe to assume that unexplained changes in prcfat today do not cause future changes in the state-wide unemployment rate. Also, over this period, the policy changes were permanent once they occurred, so strict exogeneity seems reasonable for spdlaw and beltlaw . (Given legislative lags, it seems unlikely that the dates the policies went into effect had anything to do with recent, unexplained changes in prcfat .(ii) Remember, we are still estimating the βj by OLS, but we are computing different standard errors that have some robustness to serial correlation. Using Stata 7.0, I get垐.0671, se().0267spdlaw spdlaw ββ== and 垐.0295, se().0331beltlaw beltlaw ββ=-=. The t statistic forspdlaw has fallen to about 2.5, but it is still significant. Now, the t statistic on beltlaw is less than one in absolute value, so there is little evidence that beltlaw had an effect on prcfat .(iii) For brevity, I do not report the time trend and monthly dummies. The final estimate of ρρ=is ˆ.289:prcf at= 1.009 + … + .00062 wkends-.0132 unem(.102) (.00500) (.0055)+ .0641 spdlaw -.0248 beltlaw(.0268) (.0301)n = 108, R2 = .641There are no drastic changes. Both policy variable coefficients get closer to zero, and the standard errors are bigger than the incorrect OLS standard errors [and, coincidentally, pretty close to the Newey-West standard errors for OLS from part (ii)]. So the basic conclusion is the same: the increase in the speed limit appeared to increase prcfat, but the seat belt law, while it is estimated to decrease prcfat, does not have a statistically significant effect.12.15 (i) Here are the OLS regression results:avgprc=-.073 -.0040 t- .0101 mon- .0088 tues + .0376 wed + .0906 thurs log()(.115) (.0014) (.1294) (.1273) (.1257) (.1257)n = 97, R2 = .086The test for joint significance of the day-of-the-week dummies is F = .23, which gives p-value = .92. So there is no evidence that the average price of fish varies systematically within a week.(ii) The equation isavgprc=-.920 -.0012 t- .0182 mon- .0085 tues + .0500 wed + .1225 thurs log()(.190) (.0014) (.1141) (.1121) (.1117) (.1110)+ .0909 wave2 + .0474 wave3(.0218) (.0208)n = 97, R2 = .310Each of the wave variables is statistically significant, with wave2 being the most important. Rough seas (as measured by high waves) would reduce the supply of fish (shift the supply curve back), and this would result in a price increase. One might argue that bad weather reduces the demand for fish at a market, too, but that would reduce price. If there are demand effects captured by the wave variables, they are being swamped by the supply effects.106107 (iii) The time trend coefficient becomes much smaller and statistically insignificant. We can use the omitted variable bias table from Chapter 3, Table 3.2 (page 92) to determine what is probably going on. Without wave2 and wave3, the coefficient on t seems to have a downward bias. Since we know the coefficients on wave2 and wave3 are positive, this means the wave variables are negatively correlated with t . In other words, the seas were rougher, on average, at the beginning of the sample period. (You can confirm this by regressing wave2 on t and wave3 on t .)(iv) The time trend and daily dummies are clearly strictly exogenous, as they are just functions of time and the calendar. Further, the height of the waves is not influenced by past unexpected changes in log(avgprc ).(v) We simply regress the OLS residuals on one lag, getting ˆ垐.618,se().081,7.63.t ρρρ===Therefore, there is strong evidence of positive serial correlation.(vi) The Newey-West standard errors are 23垐se().0234 and se().0195.wave wave ββ== Given thesignificant amount of AR(1) serial correlation in part (v), it is somewhat surprising that these standard errors are not much larger compared with the usual, incorrect standard errors. In fact,the Newey-West standard error for 3ˆwave βis actually smaller than the OLS standard error.(vii) The Prais-Winsten estimates arelog()avgprc = -.658 - .0007 t + .0099 mon + .0025 tues + .0624 wed +.1174 thurs (.239) (.0029) (.0652) (.0744) (.0746) (.0621)+ .0497 wave2 + .0323 wave3(.0174) (.0174)n = 97, R 2 = .135The coefficient on wave2 drops by a nontrivial amount, but it still has a t statistic of almost 3. The coefficient on wave3 drops by a relatively smaller amount, but its t statistic (1.86) is borderline significant. The final estimate of ρ is about .687.。

【大学课件】相关分析Correlation Analysis

虛無假設H0：兩變項X與Y不相關（相關係數為0, ＝0）對立假設H1：兩變項X與Y相關（相關係數不為0, 0）
當雙尾的機率p小於設定的顯著水準（如0.05或0.01）時，則否定虛無假設，即相關係數不為零（兩變項
以籃球得分為例。一個籃球隊獲勝場次與
每場的平均得分有關連嗎？

從散佈圖中可看出，它們具有線性關聯。我們再從 1994、1995 NBA 球季分析資料得知， Pearson 的相關係數 (0.581) 在 0.01 水準時是有意義的。於是可能猜想，每季所贏得的場次愈多，則對手的得分愈少。這些變數為負相關 (0.401)，而相關在 0.05 水準時最顯著。

Spearman’s Rho（）等級相關係數（順序變項） Kendall‘s tau-b （）等級相關係數（concordant和諧）

相關係數範圍的值在 1 (一百分比負關聯) 到 +1 (一百分比正關聯) 之間。其中，數值 0表示沒有任何線性關係。在解析結果時，請不要因為顯著的相關，而逕下任何跟因果相關的結論。

Spearman’s
Rhห้องสมุดไป่ตู้（）等級相關係數
相關顯著性訊號

相關係數在 .05 水準顯著時，會以一個星號標示，而在 .01水準顯著時，會以兩個星號標示。
等級觀察值

轉換＞等級觀察值
等級變項之相關係數為Spearman相關係數
相關 RANK of RANK of MIDTERM FINAL RANK of MIDTE RM Pearso n 相關 1.000 .825** 顯著性 (雙尾) . .003 叉積平方和 82.000 67.250 共變異數 9.111 7.472 個數 10 10 RANK of FINAL Pearso n 相關 .825** 1.000 顯著性 (雙尾) .003 . 叉積平方和 67.250 81.000 共變異數 7.472 9.000 個數 10 10 **. 在顯著水準為0.01時 (雙尾)，相關顯著。

沈阳理工大学徐静霞版统计学 (12)第7章相关与回归分析

相关系数的经验解释
1. 2. 3. 4. |r|0.8时，可视为两个变量之间高度相关 0.5|r|<0.8时，可视为中度相关 0.3|r|<0.5时，视为低度相关 |r|<0.3时，说明两个变量之间的相关程度极弱，可视为不相关 5. 上述解释必须建立在对相关系数的显著性进行检验的基础之上
7.1.3.1 定性分析 7.1.3.2 相关表
7.1.3.3相关图
散点图
(scatter diagram)

非线性相关
完全正线性相关
完全负线性相关

负线性相关
相关关系
(correlation)
1. 一个变量的取值不能由另一个变量唯一确定； 2. 当变量 x 取某个值时，变量 y y 的取值对应着一个分布； 3. 各观测点分布在直线周围； 4. 数学形式 : 式中的为随机误差项 , 反映自变量以外随机因素的影响。

y f (x)

作为一门统计学的入门读物，本教材以经典的一元线性回归分析为主来介绍回归分析的基本思想，希望能起到一个抛砖引玉的作用。
回归分析研究什么？
研究某些实际问题时往往涉及到多个变量。在这些变量中，有一个变量是研究中特别关注的，称为因变量，而其他变量则看成是影响这一变量的因素，称为自变量。假定因变量与自变量之间有某种关系，并把这种关系用适当的数学模型表达出来，那么，就可以利用这一模型根据给定的自变量来预测因变量，这就是回归要解决的问题。在回归分析中，只涉及一个自变量时称为一元回归，涉及多个自变量时则称为多元回归。如果因变量与自变量之间是线性关系，则称为线性回归(linear regression)；如果因变量与自变量之间是非线性关系则称为非线性回归 (nonlinear regression)

correlation- analysis and validation process

correlation- analysis and validation process1. 引言1.1 概述在各个领域中，相关性分析和验证过程是非常重要的研究方法。

相关性分析帮助我们了解和评估两个或多个变量之间的关系，而验证过程则用于检验并确认这些关系的有效性和可靠性。

无论是在科学研究、市场调查还是工程设计中，正确地进行相关性分析和验证过程对于推动知识的发展以及决策的制定具有重要意义。

1.2 文章结构本文将按照以下结构组织讨论相关性分析和验证过程的内容：第一部分是引言部分，提供文章背景以及本文的目标和意义。

第二部分是正文部分，将介绍有关相关性分析和验证过程的一般概念，并探讨其理论基础和应用领域。

第三部分是关于相关性分析流程的详细讨论。

将阐述数据收集与准备的重要性，并提供选择合适的相关性分析方法的指导原则。

我们还将深入探讨如何解释和评估相应结果。

第四部分是有关验证过程的内容。

将说明如何选择样本并进行数据处理，同时定义合适的验证指标与设计实施验证实验。

最后，我们将分析和总结验证结果。

最后一部分是结论部分，总结本文的发现和观点，并对相关性分析和验证方法进行评价和展望。

1.3 目的本文旨在提供读者对相关性分析和验证过程的充分理解，并为实际应用中的研究人员和决策者提供有用的指导原则。

通过详细介绍相关性分析流程和验证过程的步骤与方法，希望能帮助读者准确地进行数据分析、评估变量之间的关系，并有效地验证相关研究结果。

2. 正文在本文中，将介绍以"correlation-analysis and validation process"为主题的相关性分析和验证过程。

我们将探讨相关性分析的基本概念和步骤，并介绍验证过程的关键要素。

相关性分析是一种统计方法，用于测量两个或多个变量之间的关系。

通过了解变量之间的相关性，我们可以确定它们是否具有相互依赖或影响，并可能了解它们之间的强度和方向。

在进行相关性分析之前，首先需要进行数据收集和准备。

关于实验数据回归的书 -回复

关于实验数据回归的书-回复以下是一些关于实验数据回归的书籍推荐：1. "Applied Linear Regression" by Michael H. Kutner, Christopher J. Nachtsheim, and John Neter: 这本书详细介绍了线性回归的理论和应用，包括实验设计和数据分析。

2. "Regression Analysis by Example" by Samprit Chatterjee and Ali S. Hadi: 这本书通过实例解释了回归分析的基本概念和方法，包括数据准备、模型选择和评估等。

3. "An Introduction to Statistical Learning with Applications in R" by Gareth James, Daniela Witten, Trevor Hastie, and Robert Tibshirani: 这本书涵盖了多种统计学习方法，包括回归分析，并提供了R语言实现的例子。

4. "The Elements of Statistical Learning: Data Mining, Inference, and Prediction" by Trevor Hastie, Robert Tibshirani, and Jerome Friedman: 这本书深入探讨了统计学习的各种方法，包括回归分析，并强调了在实际数据集上的应用。

5. "Experimental Design and Analysis for Engineers" by Gary O.Evans: 这本书专门针对工程师和科研人员，介绍了实验设计和数据分析的方法，包括回归分析。

以上书籍都是关于实验数据回归的经典著作，适合不同背景和需求的读者参考学习。

Ch-12 Correlation and Regression Analysis

第八章相关与回归分析Correlation and Regression Analysis

国外记忆和认知课件ch12

计量经济学ch12

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第9讲相关分析与回归分析教案

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衍生品市场ch12

第八章相关与回归分析Correlation and Regression Analysis

大学生自我控制量表的修订

CorrelationandCausationReview-1

相关与回归分析课件

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计量经济学导论CH12习题答案

【大学课件】相关分析Correlation Analysis

沈阳理工大学徐静霞版统计学 (12)第7章相关与回归分析

correlation- analysis and validation process

关于实验数据回归的书 -回复

Ch-12 Correlation and Regression Analysis

第八章相关与回归分析Correlation and Regression Analysis

国外记忆和认知课件ch12

计量经济学ch12

相关分析(Correlate)

第9讲 相关分析与回归分析 教案

回归分析

几种发表性偏倚评估方法介绍

衍生品市场ch12

第八章相关与回归分析Correlation and Regression Analysis

大学生自我控制量表的修订

CorrelationandCausationReview-1

相关与回归分析课件

相关分析与回归分析

计量经济学导论CH12习题答案

【大学课件】相关分析Correlation Analysis

沈阳理工大学徐静霞版统计学 (12)第7章 相关与回归分析

correlation- analysis and validation process

关于实验数据回归的书 -回复

第9讲相关分析与回归分析教案

沈阳理工大学徐静霞版统计学 (12)第7章相关与回归分析