FRM第二章数量分析20

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第二章数量分析

描述性统计学

A. Fundamentals of Probability概率论基本原理

1. Probabilities

1.1. Learning objectives

1.1.1. Describe an event and an event space 描绘事件及事件空间

1.1.2. Describe independent events and mutually exclusive events 独立事件及互斥事件

1.1.3. Explain the difference between independent events and conditionally independent events 独立事件及条件独立事件的区别

1.1.4. Calculate the probability of an event for a discrete probability function 计算离散概率函数

1.1.5. Define and calculate a conditional probability 计算条件概率

1.1.6. Distinguish between conditional and unconditional probabilities 区分条件概率及无条件概率

1.1.7. Explain and apple Bayes' rule 解释和应用贝叶斯法则

1.2. Basic Concepts

1.2.1. Random event: Events at may or may not occur in a random trial，A,B,C 随机事件： 事件可能发生可能不发生，大量实验中有一定的规律性

1.2.2. Random variables: Quantitative representation of random Events,X,Y,Z 随机变量：随机事件的数量表现

1.2.3. Outcome: An observed value of a random variable, x,y,z 结果：随机变量的结果取值(具体值)

1.2.4. The sample space of an experiment or random trial is the set of all possible outcomes of that experiment. It is common to refer to a sample space by the lables Ω 样本空间，随机变量所有可能的结果组成的集合

1.2.5. Event

Mutually exclusive events: Events that cannot haopen at the same time 互斥事件：事件不能同时发生

Exhaustive events: Events can not both happen at the same time but at least one of the Events must occur 对立事件：事件不能同时发生但其中一件事件必须发生

Collectively exhausive events 完备事件组 多个互斥事件，又构成样本空间

1.3. Venn Diagrams文氏图

1.3.1.

1.4. Probability of an event

1.4.1. P(A)= Number of outcomes favorable to A / Total number of outcomes A发生的次数/ 总结果发生次数

1.5. Two defining properties of probability 概率的两个定义属性

1.5.1. 0≤ P（A）≤ 1

1.5.2. if A1,A2,,,An is are collecively exhaustive Events完备事件组, then: P(A1)+P(A2)+......P(An)=1

1.6. Conditional probability条件概率: P(A | B)

1.6.1. Refers to the probability of an event that is affected by occurrence of another event 已知B事件发生的情况下，A事件发生的概率

1.6.2. P(A|B): the probability of event A, if event B happens

1.6.3. 已P(A|B) →P(B)为新的样品空间，求A发生的概率

1.7. Unconditional probability 非条件概率P(A)

1.7.1. Refers to the probability of an event regardless of the past or future occurrence of other events 仅考虑A事件发生的概率，不考虑其他

1.7.2. p(A): Also called " marginal probability边际概率"

1.8. Joint probability联合概率: P(AB)

1.8.1. The probability that both Events will occur

1.8.2. P(AB)= P(A|B) × P(B)

1.9. Independent events独立事件

1.9.1. Refer to Events that one occurrence has no influence on the occurrence of others 一个事件的发生对另一个事件发生没有影响 发生与不发生跟我关系

1.9.2. P(A|B)=P(A) or P(B|A)=P(B)

1.9.3. P(AB)=P(A|B)×P(B)= P(A)×P(B)

1.10. Dependent Events依赖事件: the above condition is not satisfied: P(AB) ≠ P(A)×P(B)

1.11. Conditional independent条件独立：A and B are conditionally independent given C if and only if。 给定C条件情况下，AB事件是独立的。

1.11.1. P(AB|C) =P(A|C)×P(B|C)

1.11.2. Example: P(BR|Y)=2/12 , P(B|Y)=6/12 , P(R|Y)=4/12 , P(BR|Y)=P(B|Y)×P(R|Y)=2/12

1.12. Multiplication rule乘法法则

1.12.1. P(AB)= P(A|B)×P(B) =P(B|A)×P(A) If A and B are mutually exclusive events互斥→ P(AB)= P(A|B)= P(B|A)= 0

1.13. Additional rules加法法则

1.13.1. P(A or B)= P(A) + P(B) - P(AB) If A and B are mutually exclusive events→ P(A or B)=P(A) +P(B)

1.14. Total Probability Formula全概率公式

1.14.1. If aB event A must result in one of the mutually exclusive Events完备事件组A1, A2, A3...An then P(B)= P( A1 )P( B | A1 )+ P( A2 )P( B | A2 )+...P( An )P( B | An) P(B)= P(A1 × B)+P(A2 × B)+....P(An × B)

1.15. Baye's Theorem贝叶斯理论(画树状图)

1.15.1. Baye's Formula: Given a set of prior probabilities for an event of interest, if you receive new information, the rule for updating your probability of the event

1.15.2.

1.15.3. ①P(Ai) is prior probability(贝叶斯框架中) 先验概率 ②P(B|Ai) is likelihood probability, the probability of occurrence of observed event B under the Ai event occurs似然概率， ③P(B|Ai) / P(B) is likelihood function似然函数 ④P(Ai|B) is posteriori probability后验概率

1.15.4. Example: An analyst determines that there us a 50% chance the economy will grow and that there is a 50% chance the economy will go into a recession. If the economy grows, there is a 60% chance that A stock will rise in a price and a 40% chance It will fall in price. If a recession occurs, there is a 15% chance A stock price will rise and an 85% chance the price will fall. Given that A stock has risen in price , what is the probability the economy has grown ?

2. Constent

2.1. Basic: Random variables / outcome /event 随机变量/ 随机事件/ 随机变量结果

2.2. Probability:

2.2.1. Conditional / unconditional / joint

2.2.2. Addition / multiplication rules

2.3. Explain and apply Baye's rule

B. Random Variables随机变量

1. Learning objectives

1.1. Describe and distinguish a probability mass function(PMF) from a cumulative distribution function(CDF) , and explain the relationship between these two. 概率质量函数，累积概率函数，定义及两者之间的关系

1.2. Understand and apply the concept of the mathematical expectation of a random variable. 了解并应用随机变量的数学期望的概念。

1.3. Describe the four common population moments.

1.4. Characterize the quantile function and quantile-based estimators. 表征分位数函数和估计分位函数

1.5. Explain the effect of the linear transformation of a random variable on the mean, Variance, standard deviation, skewness, kurtosis, median. 当随机变量经过一系列随机变量以后，均值，方差，标准差，偏度，峰度，中位数产生什么变化

2. Discrete Random Variables离散随机变量

2.1. The number of possible outcomes can be either finite or countable infinite 变量的结果是有限或者无穷可数(可以计数)的，

2.2. The probability distribution of a discrete Random variable X lists the values Xi any their probabilities pi

2.3. The first function is known as the Probability Mass Function (PMF) . This function returns the probability that a discrete Random variable takes a certain value 概率质量函数（probability mass function，简写为pmf）是离散随机变量在各特定取值上的概率

2.4. A Cumulative Distribution Function (CDF) , F(x), gives the probability that a random variable wil be less than or equal to a given value F(x)=p(X≤ x) 累积分布函数，又叫分布函数，是概率密度函数的积分，能完整描述一个实随机变量X的概率分布。

2.5. Moments矩

2.5.1. Expected Value期望值（概率×加权和） 在概率和统计学中，一个随机变量的期望值是变量的输出值乘以其机率的总和，换句话说，期望值是该变量输出值的平均数

2.5.2. Properties of Expected Value期望的性质

1. If b is a constant, E(b)=b， 若b是常数，期望值也是常数

2. If a is a constant, E(ax)=aE(X)

3. If a and b constants, then E(aX + b)=aE(X) +b

4. E(X²) ≠［E(X)］² 随机变量平方的期望值不等于期望值的平方

5. E(X +Y)=E(X)+E(Y)

6. In general, E(XY)≠E(X)E(Y); If X and Y are independent random variables 当XY相互独立时，等式成立，then E(XY)=E(X)E(Y)

2.5.3. Variance方差(二阶矩)

A measure of a dispersion-the second moment 取值与期望值的离散度

σ²=E(X-μ)²

To compute the Variance, we use following formula

σ²=E(X)²-［E(X)］²=E(X²)-μ²

The positive square root of σ²,σ is known as the Standard deviation, called volatility σ也称为标准差，也就是波动率

Measures how noisy or unpredictable that random variable is

2.5.4. Properties of Variance 方差的性质

If n is constant, then: σ²(nX)=n²σ²(X)

If n is a constant,then: σ²(X+n)= σ²(X)

If m and n are constant, then: σ²(mX + n)=m²σ²(X)

If X and Y are two independent random Variables独立随机变量,then: σ²(X ± Y)= σ²(X)+σ²(Y)

If X and Y are independent random variables and m and n are constants, then σ²(mX+nY) =m²σ²(X)+n²σ²(Y)

For computational convenience,we can get:σ²(X)= E(X²)-E(X)²,E(X²)=∑x²p(x)

2.5.5. Skewness 偏度(三阶矩) (衡量尾巴偏向左或者右)

A measure of asymmetry of a PDF- the third moment

S= E(X-μx)³/σx³

2.5.6. Symmetrical and nonsymmetrical distributions 对称性(给定均值，给定偏度）

Positively skewed (right skewed) and negatively skewed(left skewed)

2.5.7. Kurtosis峰度(四阶矩)(衡量尾巴的厚度)

A measure of tallness or flatness of a PDF- the fourth moment

K=E(X-μx)⁴ / σx⁴

For a normal distribution,the K value is 3 正态分布，K=3, 常峰

Excess kurtosis=kurtosis-3

2.6. Moments and linear Transformations

2.6.1. Let Y= a + bX ehere a and b are both constant values

The mean of Y is a+bE(X)

The Variance of Y is b²σx²

The Standard deviation of Y is | b |σx

2.6.2. If b＞0, the skewness and kurtosis of Y are identical to the skewness and kurtosis of X X偏度跟峰度与原来随机变量Y一样

2.6.3. If b＜0，then the skewness has the same magnitude but the opposite sign, The kurtosis is unaffected 偏度一样，但是符号相反，峰度不影响 ∵偏度是三阶(三次方)，峰度是四阶(四次方)

3. Continuous Random Variables连续随机变量

3.1. A continuous random variable X takes on all values in an interval of numbers 考虑X值在区间的概率

3.2. Continuous Random Variables use a Probability density function PDF概率密度函数, f(x) in place of the probability mass function

3.3. PDF can be used to compute the probability that outcomes of a continuous distribution lie within a particular range of outcomes PDF可用于计算连续分布的结果位于特定结果范围内的概率

4. Quantiles and Modes 分位函数及众数

4.1. Quanyile function 分位函数 is the inverse function of probability distribution function. Its independent variable is quantile which is between 0-1 分位函数CDF 的反函数，它的自变量是分位数，介于0-1之间

4.2. Modes众数 Is the most frequently occurring vaule of the distribution 数据集里出现最多的数

4.2.1. The distribution could have more than one mode,or no mode(bimodal 双峰, trimodal 三峰,etc.) 数据集可以有多个众数或没有众数

4.2.2. The mode of a continuous random variable corresponds to the maximum of the density function 连续随机变量的模式对应于密度函数的最大值

5. Content

5.1. Describe and distinguish pdf and CDF

5.2. The four common population moments

5.3. Characerize the quantile function and quantile-based estimators

5.4. Explain the effect of a linear transforming of a random variable on the mean, Variance, standard deviation, skewness, kurtosis, median

C. Common Univariate Random Variables一元随机变量

Learning objectives

Distinguish the key properties and identify the common occurrences of the following distributions: uniform distribution均匀分布, Bernoulli distribution伯努力分布,binomal distribution二项分布,Poisson distribution泊松分布, normal distribution正态分布, lognormal distribution对数正态分布, chi-squared distribution卡方分布, Student's t t分布, and F-distributionsF分布. 区分和识别一下分布

Describe a mixure distribution and explain the creation and characteristics of mixture distributions 描述混合分布，解释混合分布产生与特征

A. Bernoulli random variable伯努力分布(离散随机变量)

The trial produces one of two outcomes, one representing success, denoted1, the other representing failure, denoted0 又名两点分布或者0-1分布，是一个离散型概率分布。 若伯努利试验成功，则伯努利随机变量取值为1。 若伯努利试验失败，则伯努利随机变量取值为0。 记其成功概率为p，则失败概率为q = 1 − p。

P(X=1)=p, then P(X=0)=1-p

Mean=p , Variance=p(1-p)

B. Binomial random variable二项分布(离散随机变量)

Trails are all independent独立事件 A probability of success equal to p 定义：在n次独立重复的伯努利试验中，设每次试验中事件A发生的概率为p。用X表示n重伯努利试验中事件A发生的次数，则X的可能取值为0，1，…，n,且对每一个k（0≤k≤n）,事件{X=k}即为“n次试验中事件A恰好发生k次”，随机变量X的离散概率分布即为二项分布（Binomial Distribution）

①P(X)=Probability of x successes in n trials, with probability of success p on each trial ②x= number of successes in sample,(x=0,1,2,....n) ③P= Probability of "success" per trial ④q= Probability of "failure" =(1-p) ⑤n= Number of trials(sample size) 实验n次且只有两种结果，我关心其中一个结果，用二项分布

Mean= np , Variance=np(1-p)

C. Passion Distribution泊松分布(离散随机变量)

①X refers to the number of occurring ②λ refers mean occurrence of random events 单位时间平均成功次数 ③ λ=np

衡量单位时间范围内，某个事件发生次数所对应的概率。

D. Uniform Distribution均匀分布(连续随机变量)

1. Probability Density Function概率密度函数

1.1. f(x)= ① 1/b-a, a≤x≤b ② 0, otherwise

2. Cumulative Distribution Function累计分布函数

2.1. f(x)= ① 0,for x≤a, ② x-a/b-a, a＜x＜b ③ 1 ,for x≥b

3. ①期望值E(x)=a+b /2 ②方差D(x)=(b-a)²/12

E. Normal Distribution正态分布

1.

2. Properties of Normal Distribution 正态分布性质

①X~N( μ, σ²),fully describe by μ and σ²

②skewness=0, kurtosis=3 遍度=0，锋度=3

③Linear combination of normally distributed by random variable is normally distributed 随机变量为正态分布的线性组合为正态分布

3. The confidence intervals 置信区间

4. The Standard normal distribution标准正态分布（Z分布）

5. Lognormal distribution对数正态分布

BSM Model assumes that price of underlying Asset is lognormally distributed

a. InX is normal→X is lognormal

b. Y is lognormal→In Y is normal

c. Right skewed右偏

d. Bounded from below by zero从下限为零

e. It is used to Model Asset prices which never take negative values经常用来描述资産价格分布

6. Sampling Distribution 抽样分布(三种)

A. Chi-Square(χ²) Distribution

若n个相互独立的随机变量ξ₁，ξ₂，...,ξn ，均服从标准正态分布（也称独立同分布于标准正态分布），则这n个服从标准正态分布的随机变量的平方和构成一新的随机变量，其分布规律称为卡方分布（chi-square distribution）。

The Chi-squared distribution (χ²-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent strandard normal random variables K个独立标准正态分布的平方求和，

Properties of Chi-squared Distribution

a. asymmetrical, bounded below by zero 非对称，下限为0 (0,+∞)

b. Approaches the normal distribution as the degrees of freedom increase 随着自由度增加，卡方分布会更接近标准正态分布

c. Linear combination of Chi-squared distributed random variables is Chi-squared distributed 卡方分布随机变量的线性组合为卡方分布

d. 期望值E(X)=k, 方差D(X)=2k

B. t (Student's) Distribution( 用于均值假设检验)(小样本)

在概率论和统计学中，t-分布（t-distribution）用于根据小样本来估计呈正态分布且方差未知的总体的均值。如果总体方差已知（例如在样本数量足够多时），则应该用正态分布来估计总体均值。 t分布曲线形态与n（确切地说与自由度df）大小有关。与标准正态分布曲线相比，自由度df越小，t分布曲线愈平坦，曲线中间愈低，曲线双侧尾部翘得愈高；自由度df愈大，t分布曲线愈接近正态分布曲线，当自由度df=∞时，t分布曲线为标准正态分布曲线。 开创小样本统计时代

A Student's tis the distribution of

a. Where Z is a standard normal

b. W is a Chi-squared random variable

c. Z and W are independent

d. Symmetric 对称

e. 均值Mean=0, and its 方差Variance=n/(n-2) ，n 是自由度

f. Given a degree of confidence,t-distribution has a wider confidence interval than normal distribution 在给定的自由度之下，t 分布有更宽的自信区间

g. As degree of freedom increase, t-distribution has a wider confidence interval than normal distribution 自由度增加，t 分布越接近正态分布

h. As degree of freedom increase, t-distribution is becoming more peaked with thinner tails, which means smaller probabilities for extreme values 随着自由度增加，t 分布变得越来越尖，逼近于标准正态分布

C. F Distribution F 分布( 回归系数联合假设检验)

它是两个服从卡方分布的独立随机变量各除以其自由度后的比值的抽样分布，是一种非对称分布，且位置不可互换。F分布有着广泛的应用，如在方差分析、回归方程的显著性检验中都有着重要的地位。

Let χ²m and χ²n be independent variates distributed as Chi-squared with m and n degrees of freedom

Define a statistic F n,m as the Ratio of the dispersions of the two distributions

skewed to the right and also ranges between 0 and infinity 向右偏斜，范围在0 到无穷大之间

Approaches the Normal Distribution as m and n tend to infinity 当m 和n 趋于无穷大时接近正态分布

7. Exponential 指数分布(信用风险描述违约概率)

a. Probability that the time interval between two events is greater than t 两件是事件发生机率大于t 的概率

Y 表示时间间隔，

b. The CDF of an Exponential (λ) is

c. The PDF of an Exponential (λ) is

d. 解决问题：等到一件随机事件发生，需要多久？所考虑时间分布，指数分布就是泊松过程中事件间隔的分布

e. E(Y)=1/λ,D(Y)=1/λ²

f. Exponential variables are also memoryless无记忆性, P(Y≥s+t | Y≥s)

8. Beta 分布

The PDF of the x of Beta(α ，β)is Beta 取值0~1

Where B(α,β)is the Beta function

均值E(X)=α / (α+β)( 记！)

方差D(X)=αβ/ ［ (α+β+1)(α+β)² ］

9. Mixture Distribution混合分布

A Mixture Distribution is a weighted average distribution of density functions 概率密度函数的加权平均( 参数与非参数)

Typically, component distributions are parametric ，weights are based on empirical data, which is nonparametric 通常，组件分布是参数性的，权重是基于经验数据的，而经验性数据是非参数性的

Combining distributions with different μ→ change slowness 改变偏度/ bimodal 双峰

Combining distributions with different σ →change kurtosis 改变峰度

F. Content

1. 离散型Bernoulli / Binomial / Possion

2. 连续型Continuous uniform / Normal / Lognormal

3. 抽样分布x² / t / f

4. Exponential / Beta

5. Mixture

D. Multivariate Random Variables多元随机变量

1. Learning objectives(主要考计算题！)

1.1. Explain how a probability matrix矩阵can be used to express a probability mass function 解释如何用概率矩阵来表示概率质量函数

1.2. Compute the marginal and conditional distributions of a discrete bivariate random variable 计算离散二元随机变量的边际和条件分布

1.3. Compute the expectation of a bivariate discrete random variable 计算二元离散随机变量的期望值

1.4. Define covariance and explain what it measures 定义斜方差函数及计算方法

1.5. Explain the relationship between the covariance and correlation of two random Variables and how these are related to the independence of the two variables 解释两个随机变量的协方差和相关性之间的关系，以及它们与两个变量的独立性之间的关系

1.6. Explain the effects of applying linear transformations on the covariance and correlation between two random variables 说明应用线性变换对两个随机变量之间的协方差和相关性的影响

1.7. Compute the Variance of a weighted sum of two random variables 计算两个随机变量的加权和的方差

1.8. Compute the conditional expectation of a bivariate random variable 计算二元随机变量的条件期望

1.9. Describe the features of an iid sequence of random variables 描述独立同分布随机变量序列的特征

2. ① Multivariate random variables are vectors of random variables. For example, a bivariate random Variable X would be a vector with two components: X1 and X2. ① 多元随机变量是随机变量的向量。例如，双变量随机变量X 将是具有两个分量的向量：X1 和X2 。 ② Multivariate random variable are like their univariate counterparts in that they can be discrete or continuous. Both types of random variables are denoted with uppercase letters(e.g.X, Y, or Z) ② 多元随机变量就像它们的单变量对应变量一样，可以是离散的也可以是连续的。两种类型的随机变量均以大写字母表示（例如X ，Y 或Z ）

3. Discrete Random Variables

3.1. Probability Mass Function 概率质量函数

3.2. The PMF of a bivariate random variable is function that returns the probability that X=x, Put in terms of the components, this would X1=x1, X2=x2。 f x1x2(x1,x2) = Pr (X1=x1, X2=x2)

3.3. Probability matrix is a tabular representation of a PMF 概率矩阵是PMF的表格表示

3.4. The PMF describes the joint distribution of the two components［P( X1=x1, X2=x2)］ PMF描述了两个分量的联合分布

3.5. Marginal Distributions边际分布

3.5.1. The distribution of a single component of a bivariate random variable is called a marginal distribution, and it is simply a univariate random variable 双变量随机变量的单个成分的分布称为边际分布，它只是单变量随机变量

3.5.2. The marginal PMF is defined as

3.5.3. This distribution contains the probabilities of realizations of X1 and its PMF denoted by fx1(x1).

3.5.4. Returning to the previous example, marginal PMF of the recommendation when stock return is 5% is

3.6. Conditional distributions条件分布

3.6.1. The conditional probability of two events was defined as

3.6.2. The conditional distribution of X1 given X2 is defined as

3.6.3. Corresponding to X2=1 in the bivariate probability matrix, the Conditional distribution is then:

3.6.4. Conditional distributions can also be defined over a set of outcomes for one of variables 联合概率除边际概率

3.7. Conditional Expectation条件期望

3.7.1. A conditional expectation is an expectation when one random variable takes a specific value or falls into a defined range of values 条件期望是指一个随机变量取特定值或落入值的定义范围内时的期望

3.7.2. It is common in risk management to model the expected loss on a portfolio given a large negative market Return 在风险管理很普遍的情况下，在给定巨大的负市场收益的情况下，对投资组合的预期损失建模

3.7.3. Back to the example, the conditional distribution corresponding to

3.8. 例题

3.8.1. Jimmy, CFA,FRM is analyzing his portfolio consisting of Stock A and Stock B. Define X1 and X2 for the future performance of stock A and stock B. -1 is for price-decreasing scenario; 0 is for price-maintaining scenario; 1 is for price increasing scenario. The joint probability matrix is given as follows

3.8.2. ① The above left is marginal distribution for two variables ② The above right is conditional distribution of X1 given X2 equals to 1 ③ ③ The conditional expectation of X1 given X2 equals to 1 is: -1×16.7%+0×33.3%+1×50%=33.3%

4. Expectations and Moments

4.1. Expectation of a function of bivariate random variable对二元随机变量求期望，方差，斜方差

4.1.1. The expectation of a function of a bivariate random variable is defined analogously to that of a univariate random variable 对二元随机变量的函数的期望类似于对单变量随机变量的期望的定义

4.1.2. Z=g(X1,X2), The expectation is defined as

4.1.3. As in the univariate case

5. Covariance二元随机变量协方差

5.1. The covariance is a measure of dispersion that captures how the variables move together 协方差是离散度的度量，可捕获变量如何一起移动 covariance ranges -∞~+∞

6. Correlation 相关系数

6.1. Correlation Measures the co-movement(linear association)between two random variables 斜方差除各自标准差

6.1.1. Values range from +1 to -1, it has no units

6.1.2. A correlation of 0 indicates an absence of any linear relationship between the variables 相关性为0表示变量之间不存在任何线性关系

6.1.3. The bigger the absolute value of correlation coefficient, the stronger linear relationship 相关系数的绝对值越大，线性关系越强

例题：

6.1.4.

7. Covariance多元随机变量协方差

7.1. In a bivariate random variable X=(X1, X2)，there are two variances and one covariance 在二元随机变量X =（X1，X2）中，存在两个方差和一个协方差

7.2. The covariance can be expressed in terms of the correlation and the standard deviation 协方差可以用相关性和标准偏差表示

7.3. The covariance p lays are independent random vari ables and m and n are constants, then 协方差p位置是独立的随机变量，而m和n是常数，则 σ²(aX+bY)=a²σ²(X)+b²σ²(Y)

7.4. If X and Y are not independent random variables and m and n are constants,then 如果X和Y不是独立的随机变量，而m和n是常数，则 σ²(aX+by) =a²σ²(X)+b²σ²(Y)+2abCov(X,Y)

7.5.

8. covariance, Correlation, and independence

8.1. When two random variables are independent，they must have zero correlation 当两个随机变量独立，肯定不相关。 不相关≠独立

8.2. Correlation is a measure of linear dependence. If two variables have a strong linear relationship, then they have a large correlation 相关性是线性相关性的依赖性。 如果两个变量具有很强的线性关系，则它们具有较大的相关性

8.3. If the correlation between two variables is 0, then the expectation of the product of the two is the product of the expectations E(XY)=E(X)E(Y) 如果两个变量之间的相关性为0，则两个乘积的期望值为期望E（XY）= E（X）E（Y）的乘积

8.4. If the two component X1, X2 are independent, then the joint is the product of the marginals 如果两个分量X1，X2是独立的，则联合概率是各自概率相乘

9. Conditional Independence条件独立

9.1. P(AB|C)=P(A|C)×P(B|C)

9.2. Now suppose that 12.5% of the companies are in the conservative category(P(C)=12.5%) 12.5％的公司可以化为稳健型公司

10. iid Variables独立且均值方差相等且来自同一个正态分布总体(中心极限定理会用到)

10.1. Independent, identically distributed (iid) random variables are generator of an n-component multivariate random variable where each component is independent with each other 独立的，均匀分布的（iid）随机变量是n分量多元随机变量的生成器，其中每个分量彼此独立

10.2. For example, it is common to write As a generator of an n-component multivariate normal random variable where each component is independent of all other components and is distributed N(μ,σ²)

10.3. The iid assumption make it simple to compute mean and Variance of a sum of Variablesiid 假设使计算变量总和的均值和方差变得简单

10.3.1. The Expected Value of a sum of n iid random variables is

10.3.2. The Variance of a sum of n iid random variables is

10.3.3. The standard deviation of a sum of n iid random variables is

10.4. 例题

10.4.1.

11. Content

11.1. Probability matrix & Probability mass function

11.2. Marginal and conditional distribution

11.3. Expectation of a function

11.4. covariance and correlation

11.5. Conditional expection

11.6. iid sequ of random variables

推理性统计学

A. Sample Moments 样本矩

1. Learning objectives

1.1. Estimate the mean,Variance, and standard deviation using sample data 使用样本数据估算均值，方差和标准差

1.2. Explain the difference between a population moment and a sample moment 解释总体矩与样本矩之间的差异

1.3. Estimate and interpret the skewness and kurtosis of a random variable 估计并解释随机变量的偏度和峰度

1.4. Describe the bias of an estimator and explain what the bias measures 描述评估者的偏见，并解释偏见的测量结果

1.5. Explain what is mean by the statement that the mean estimator is BLUE( 最佳线性无偏估计)

1.6. Use sample data to estimate quantiles, including the median 使用样本数据估计分位数，包括中位数

1.7. Describe the consistency of an estimator and explain the usefulness of this concept 描述估计量的一致性，并解释此概念的用处

1.8. Explain how the Law of Large Numbers (LLN) and Central Limit Thoerm (CLT) apply to the sample mean 解释大数定律（LLN）和中心极限定理（CLT）如何应用于样本均值

1.9. Estimate the mean of two variables and apply the CLT 估计两个变量的平均值并应用中心期限定理

1.10. Estimate the covariance and correlation between two random variables 估计两个随机变量之间的协方差和相关性

1.11. Explain how coskewness and cokurtosis are related to skewness and kurtosis 解释斜偏度和协峰度与偏度和峰度如何相关

2. Inferential statistic推断性统计学

2.1. Interential statistic: Make forecasts, estimates, or judgements about a population from the sample actually drawn from the population (a sample) 推断统计量：根据实际从总体（样本）中抽取的样本对总体进行预测，估计或判断(以小见大) μ，σ参数，

2.2. Moreover,(sample) data can also be used to estimate higher order moments such as skewness, and kurtosis of population. Recall the first four moments are defined as 此外，（样本）数据还可用于估计高阶矩，例如偏度和人口峰度。 回想一下前四个时刻定义为

2.2.1. Skewness:

2.2.2. Kurtosis:

3. Estimating the mean

3.1. The population mean is estimated using the sample mean(i.e., average) of the data. The mean estimator is defined as 总体均值可以通过样本均值来估计，样本算数平均值

3.1.1. Note that the mean estimator is a function that transforms data into an estimate of the population mean 均值估算器是将数据转换为总体均值估算的函数

3.1.2. The mean estimator is a function of random variables, and so it is also a random variable 均值估算器是随机变量的函数，因此它也是随机变量

3.2. The expection of the mean is

3.2.1. The expected value of the mean estimator is the same as the population mean.Because the mean estimator's bias is zero,it is unbiased 均值估计器的期望值与总体均值相同。由于均值估计器的偏差为零，因此没有偏差。

3.3. Because Xi are iid,these variables are uncorrelated and so have 0 covariance.Thus: 因为Xi是iid独立同分布，所以这些变量不相关，所以协方差为0。因此：样本统计量方差

3.4. The Standard deviation of the mean estimator is 又称为标准误

3.5. The Standard deviation of the mean (or any other estimator) is known as a Standard-error 平均值（或任何其他估计量）的标准偏差称为标准误

3.5.1. 例题

The Standard Error of the sample mean is closest to σ / √n= 3% / √36 =0.5%

3.6. The Standard Error provides an indication of the accuracy of our estimate 标准误差表明了我们估算的准确性，越小越准确

4. Estimating the variance 估计方差

4.1. Given a set of n iid random variables Xi, the sample estimator of the Variance is

4.2. Unlike the mean, the sample Variance is a biased estimator 与均值不同，样本方差是有偏估计

4.3. Because the bias is known, an unbiased estimator for the Variance can be constructed as 由于偏差是已知的，因此可以将方差的无偏估计器构造为

4.3.1. This estimator is unbiased, as E［s²］=σ² 这个估计是无偏差的

4.4. The sample Standard deviation is estimated using square root of the sample Variance 样本标准偏差使用样本方差的平方根估算

4.5. The square root is a nonlinear function and so both estimators of the Standard deviation are biased 平方根是非线性函数，因此两个标准差的估计量都存在偏差

5. Estimating the skewness and the kurtosis

5.1. Recall that the skewness is a standardized version of the third central moment, and so it is unit and scale-free

5.2. skewness(X)= μ³ /σ³ μ³ is the third central moment

5.2.1. ①Skewness=0, Symmetrical distribution ②Skewness＞0, Positively skewed distribution ③Skewness＜0, Negatively skewed distribution

5.3. kurtosis is similary defined using the fourth moment standardized by the squared Variance. 平方方差标准化的第四矩来定义峰度。

5.4. kurtosis(X)= μ4 / σ⁴ , μ4 is the fourth central moment

6. Estimating the Median

6.1. The median is the value of the middle item that has been sorted into ascending or descending order 中位数是已排序为升序或降序的中间项的值

6.2. 【2.5.7.11.14 】→7 【3.7.9.10.15.20 】→9. 5

7. Best Linear Unbiased Estimator (BLUE) 最佳线性无偏

7.1. Unbiased:

7.1.1. The expected value of the estimator equals the parameter 估计量的期望值等于总体均值

7.2. Efficient (Best):

7.2.1. If the Variance of estimator is smallest among all linear unbiased estimators 如果所有线性无偏估计量中估计量的方差最小,最小最佳最有效

7.3. Consistent一致性：

7.3.1. As the sample size increases, the accuracy of the parameter estimate increase. Linearity. 随着样本数量的增加，参数估计的准确性也会提高。 线性度。

7.3.2. BLUE is a desirable property for an estimator BLUE是估计量的理想属性

8. Large sample behavior of the mean

8.1. This Law of Large Numbers (LLN) 大数定律 states that if ｛Xi｝is a sequence of iid random variables with μ≡E［Xi］,then Xi独立同分布序列， Xi期望值=μ， 当n→∞,算数平均值（a.s.)依概率收敛于μ ( ⊕ a.s.→means converges almost surely） 简单来说：当事情发生多了，事情的平均值会趋向稳定，样本数叫多时，样本均值会收敛到总体均值，

8.2. When an LLN applies to an estimator, the estimator is said to be consistent.Consistency requires that an estimator is asymptotically unbiased, and so any finite sample Bias must diminish as n increases 当一个LLN应用于一个估计量时，该估计量被认为是一致的。一致性要求一个估计量是渐近无偏的，因此任何有限样本偏倚都必须随着n的增加而减小

8.3. The simplest Central Limit Theorm(CLT) 中心极限定理 Is known as the Lindberg-Levy CLT.This CLT states that if｛Xi｝are iid with known population mean μ and variance σ², for sufficiently large sample sizes n (n≥30),the distribution of the sample mean is approximated by Xi是独立同分布且来自总体μ，σ²，当样本足够大的时候，样本的均值逼近于正态分布，且这个正态分布均值μ，方差σ² /n

8.3.1. Will be approximately normal

8.3.2. Has mean equals to the population mean μ

8.3.3. Has variance equal to the population variance of variable divided by sample size, which equals σ² / n

8.4. Standard Error of the sample mean

8.4.1. Known population variance已知总体方差 标准误，越小，样本统计量估计总体函数精度越高

8.4.2. Unknown population variance 未知总体方差 不知道总体方差用样本方差代替，服从t分布

9. Multivariate moments 多元变量的矩

9.1. The sample analog can be extended from univariate statistics to multivariate statistics 样本模拟量可以从单变量统计扩展到多变量统计

9.2. The CLT for a pair of mean estimators is virtually identical to that of a single mean estimator, where the scalar variance is replaced by the covariance matrix. In practice, this CLT is applied by treating the mean estimators as a bivariate normal random variable 一对均值估计器的CLT 实际上与单个均值估计器的CLT 相同，后者的标量方差被协方差矩阵代替。在实践中，通过将均值估算器视为二元正态随机变量来应用此CLT

10. Estimating the covariance and correlation 估计协方差和相关性

10.1. The sample covariance estimator uses the sample analog to the expectation operator

10.2. Dividing by n-1,rather than n, produces an unbiased estimate of the covariance n-1 变成无偏估计

10.3. The correlation is estimated by dividing the sample covariance by the product of the sample standard deviation of each asset 通过将样本协方差除以每种资产的样本标准差的乘积来估算相关性

11. Coskewness & cokurtosis 斜偏度和斜峰度 (衡量单个变量，偏度→对称性，峰度→聚集性) 斜→两个

11.1. Like variance, skewness and kurtosis can be extended to pairs of random variables. The two coskewness Measures are

11.2. The three Measures of cokurtosis are defined as

12. Constent

12.1. Estimating the mean,the variance,the skewness,and the kurtosis of a population

12.2. Bias of estimator 估计误差

12.3. BLUE & LLN & CLT

12.4. Estimating mean of two series

12.5. Estimating covariance of two series

12.6. Coskewness and cokurtosis

B. Hypothesis Testing假设检验

1. Learning objectives

1.1. Construct an appropriate null hypothesis and alternative hypothesis and distinguish between the two 构造适当的原假设和替代假设，并区分两者

1.2. Construct and apply confidence intervals for one-sided and two-sided hypothesis Tests, and interpret the results of hypothesis Tests with a specific level of confidence 构建和应用单侧和双侧假设检验的置信区间，并以特定置信度解释假设检验的结果

1.3. Explain the difference between Type l and Type ll errors and how these relate to the size and power of a test 解释类型l 和类型ll错误之间的区别，以及它们样本容量和解释力度之间的关系

1.4. Understand how a hypothesis test and a confidence interval are related 了解假设检验和置信区间之间的关系

1.5. Explain what the p-value of a hypothesis test measures 解释假设检验的p 值是什么

1.6. Identify the steps to test a hypothesis about the difference between two population means 确定检验关于两个总体均值之间差异的假设的步骤

2. Confidence Interval 置信区间(点估计两侧设立区间)

2.1. A range of vaules constructed from sample data so the parameter occurs within that range at a specified probability 由样本数据构造的值的范围，因此参数以指定的概率出现在该范围内

2.2. Interval Estimation

2.2.1. Confidence interval = Point estimate ± Reliability factor × Standard error 点估计值± 依赖因子× 标准误

2.2.2. Level of significance (α) 显著性水平 (总体参数不落在区间内的概率，小概率)

2.2.3. Degree of Confidence (1-α) 置信度

2.3. Assume that a population distribution follows a normal distribution with a mean of μ and a standard deviation of σ. Calculate that confidence interval for the population mean with the degree of confidence 1-α 假设总体分布遵循正态分布，均值为μ ，标准差为σ 。计算总体的置信区间平均值，置信度为1-α

2.4. Characteristics of Confidence intervals 置信区间的特征

2.4.1. In order to make estimates as reliable as possible, the degree of confidence 1-α needs to be as large as possible 为了使估计尽可能可靠，置信度1-α 必须尽可能大

2.4.2. To make the Estimation as accurate as possible, the confidence internal should be as short as possible 为了使估算尽可能准确，置信区间α 应尽可能短

2.4.3.

2.5. Construction of confidence interval of population mean 总体均值置信区间的构建

2.5.1. With known population Variance σ 已知总体方差 假设置信度1-α，显著性水平α Z分布

2.5.2. With unknown population Variance σ 不知道总体方差 t分布(尾巴厚点)

Where the number of degrees of freedom is n-1

2.5.3. Reliability Factors for Confidence Intervals Based on the Standard Normal Distribution 基于标准正态分布的置信区间的可靠性因子(t分布比z分布大一点)

①90% confidence intervals: Z0.05= 1.65

②95% confidence intervals: Z0.025= 1.96

③99% confidence intervals: Z0.005= 2.58

2.6. Example

2.6.1.

3. Hypothesis Tests 假设检验

3.1. Hypothesis Testing

3.2. Null and Alternative Hypotheses 原假设和备择假设

3.2.1. The null Hypothesis (H0) is a statement about the population values of one of more parameters, '=' 原假设（H0 ）是有关一个或多个参数'=' 的总体值的声明

3.2.2. The Alternative Hypothesis (H1), which determines the values of the population parameter where the null hypothesis should be rejected, '≠',' ＜',' ＞' 备择假设（H1 ），用于确定应拒绝零假设的总体参数的值，“≠ ”，“＜”，“＞”

3.3. One-tailed test vs. Two-tailed test 单尾，双尾检验

3.3.1.

3.4. Test Statistic 检验统计量

3.4.1. A quantity calculated based on a sample 根据样本计算的数量

3.4.2. When the t rue value of the mean (μ) is equal to the value assumed by the Null (μ0), then the asymptotic distribution leads to the test statistic

3.5. Critical value 临界值

3.5.1. The distribution of the test statistic 检验统计量的分布，Z分布，T分布

3.5.2. One-tailed or two-tailed test 单尾或双尾检验

3.5.3. Significance level ( α ): In hypothesis testing, the significance level is the minimum allowable probability events in an experiment 显着性水平（α ）：在假设检验中，显着性水平是实验中的最小允许概率事件

3.6. Decision Rule判决准则

3.7. The p-value of a test( 近几年基本没考了！)

3.7.1. p-value: Is measures the probability of observing a test statistic that is more extreme than the one computed from the observed data when the null hypothesis is true 是当原假设成立时测量观察到的检验统计量比从观察到的数据计算出的检验统计量更为极端的概率

3.7.2. Rejecting the null hypothesis if the p-value is less or equal to level of significance, α ,or (p-value≤α)

3.8. Type l Error, Type ll Error,Level of Significance, Power of test第一类错误，第二类错误，检验力

3.8.1. Type l error

Rejecting null hypothesis when it is true 当原假设是真的，但你拒绝了

P( Type l Error )=α.≤α (level of significance of the test)

3.8.2. Type ll Error

Failing to Reject the null hypothesis when it is false 当原假设是错误的时候，你没有拒绝

The probability of making a Type ll error is equal to β, P(Type ll Error)=β

3.8.3. Power of Test检验力

Rejecting the null hypothesis when it is false 当原假设错误，拒绝原假设

Power of test is equal to (1-β)

3.8.4.

3.8.5. With other conditions unchanged, either error probability arises at the cost of the other error probability decreasing 在其他条件不变的情况下，任何一个错误概率都以另一个错误概率降低为代价而产生(跷跷板效应) 通常情况更关注第一类错误

3.8.6. How to reduce both errors? Increase the sample size，∵ σ/√n是标准误

3.8.7. 例题

3.9. Testing the equality of two means 比较两个总体均值的相等性（配对检验）

3.9.1. The two populations are not independent 数据不是独立

3.9.2. The t-test in this section is based on data arranged in paired observations. Paired observation are observations that are dependent because they have something in common. A paired comparison test is a statistical test for differences in dependent items 本节中的t检验基于成对观察中排列的数据。 配对观测是依赖的观测，因为它们具有某些共同点。 配对比较检验是对相关项目差异的统计检验

3.9.3.

3.9.4.

4. Constant

4.1. Confidence interval Estimate

4.2. Hypothesis Tests: z-test, t-test

4.3. Type l and Type ll errors and power of a test

4.4. p-value of a hypothesis test

4.5. Interpret the results of hypothesis Tests with a specific level of confidence

4.6. Identify the steps to test a hypothesis about the difference between teo population means

C. Linear Regression 一元线性回归

1. Learning objectives

1.1. Describe the models that can be estimated using linear regression and differentiate them from those which cannot 描述可以使用线性回归估计的模型，并将其与那些不能进行线性回归的模型区分开来

1.2. Interpret the results of an OLS regression with a single explanatory variable 用单个解释变量解释OLS 回归的结果

1.3. Describe the key assumption of OLS parameter estimation 关键假设

1.4. Characterize the properties of OLS estimators and their sampling distributions 参数性质服从的分布

1.5. Construct, apply, and interpret hypothesis tests and confidence intervals for a single regression coefficient in a regression 理解一元回归系数的假设检验，置信区间

1.6. Describe the relationship between t-statistic, it's p-value, and a confidence interval t 统计量，p-value, 置信区间的关系

2. Linear Regression parameter线性回归参数

2.1. Linear regression assumes a linear relationship between and explanatory variable X and a dependent variable Y so that 线性回归假设解释变量X 和因变量Y 之间存在线性关系，因此

2.1.1. Y （因变量，被解释变量）= α+ βX （自变量，解释变量）+ ε

β :commonly called the slope or the regression coefficient 回归系数，斜率项

α,β回归系数

α: commonly called the intercept, is a constant 截距项

ε :commonly called the shock/innovations/error, represents a component in Y that cannot be explained by X 随机误差项，代表影响Y变量不影响X变量的东西, 服从N(0,σ²)正态分布

2.2. So that we get a true regression line 真实回归直线，回归模型两边取期望，把ε的影响去掉

2.2.1. E(Y)=α +βX

2.2.2. The sample points scatter around the true regression line 样本点分散在真实回归线附近

2.3. The purpose of collecting samples is to estimate the true regression line 收集样本的目的是估计真实的回归线

2.3.1.

2.4. Y^ is the fitted value of Y. Then the estimated regression Model is △ 有ε→ 模型 △ 没有ε→ 直线

2.4.1.

3. Dummy Variable 虚拟变量

3.1. There are occasions when the independent Variable is binary in nature-it is either "on" or "off". Quantify the impact of qualitative events 在某些情况下，独立变量本质上是二进制的- 它可以是“on ”或“off ”。量化定性事件的影响 不能定量处理的变量透过虚拟变量 又称虚设变量、名义变量或哑变量，用以反映质的属性的一个人工变量，是量化了的自变量，通常取值为0或1。引入哑变量可使线形回归模型变得更复杂，但对问题描述更简明，一个方程能达到两个方程的作用，而且接近现实。

3.1.1. Y=α+β1X+β2D+ε

3.1.2. Y=α+β1X+β2(X × D)+ε

3.1.3. D=0 or 1

3.2. Indicates the difference in the dependent variable for the category represented by the dummy variable 指示由虚拟变量表示的类别的因变量之间的差异

3.3. Are often used to quantify the impact of qualitative events 通常用于量化定性事件的影响

4. Linear Regression 线性回归

4.1. Ordinary Least Squares （OLS ）最小二乘法 Estimation is a process that estimates a line with the intercept α^ and slope β^ that best fits the observed data because it minimizes the squared deviations between the line Y^=α^+β^X and the realizations of the dependent variable Y

4.1.1. 例题

4.2. OLS estimation has the following assumption 最小二乘法假设(三＋1大假设)

4.2.1. Shock are mean zero conditional on X, so that E(ε|X)=0 残差项期望值等于0

4.2.2. Data are realizations from iid random variables. Formally, it is assumed that the pairs (Xi,Yi) are iid draws from their joint distribution Xi,Yi 独立同分布，采样独立

4.2.3. No outliers. The probability of large outliers in X should be small 没有异常值。( 回归斜率受影响)X 中较大的离群值的概率应该很小

4.2.4. Constant Variance of Shocks 残差项方差是常数 V ［∈ i | X ］ =σ² ＜∞

This assumption is known as homoskedasticity 同方差

Heteroskedasticity 异方差refers to that the variance of the error Term is notconstant 递增异方差在金融业常见

4.3. Implications of OLS Assumption

4.3.1. The estimators are unbiased so that 无偏估计

4.3.2. α^ and β^ obey normal distribution

4.4. Significance test for a regression coefficient 回归系数的显着性检验

4.4.1. The hypothesis that is tested is whether the true slope is zero (β=0). The appropriate test structure for the null and alternative hypotheses is 检验的假设是真实斜率是否为零（β= 0 ）。原假设和备择假设的适当检验结构是

H0: β=0, Hα: β≠0

4.4.2. Decision rule: Reject H0 if t ＞t critical or t ＜-t critical

4.4.3. Rejection of null hypothesis means the regression coefficient is different from the hypothesized value given a level of significance α

4.4.4. p-value can be understood as the "level of significance" of "Test statistic"

If p-value＜α, Reject H0

A small p-value more tends to Reject H0

4.4.5. 例题

4.5. Confidence interval for regression coefficients回归系数的置信区间

4.5.1. Confidence interval for the regression coefficient, β^, is calculated as

tc: the critical two-tailed t-value for the selected confidence level with the appropriate number of degrees of freedom, which is equal to the number of sample observations Minus 2 (i.e. n-2)

Sβ^: the standard error of the regression coefficient

4.5.2. 例题

5. Constant

5.1. OLS regression with a single explanatory variable

5.2. The key assumption of OLS parameter estimation

5.3. The properties of OLS estimators and their sampling distributions

5.4. Hypothesis Tests and confidence intervals for a single regression coefficient in a regression

D. Linear regression with Multiple Explanatory Variables 多元(多个解释变量)线性回归

1. Learning objectives

1.1. Distinguish between the relative assumptions of single and multiple regression 区分单因素回归和多元回归的相对假设

1.2. Interpret regression coefficients in a multiple regression 解释多元回归中的回归系数

1.3. Interpret goodness of fit measures for single and multiple regressions, including R² and adjusted R² 解释单次和多次回归的拟合优度，包括R²和调整后的R²

1.4. Construct,apply, and interpret joint hypothesis Tests for multiple coefficients in a regression 构造，应用和解释联合假设检验回归中的多个系数

2. The Basics of Multiple Regression 多元回归的基础

2.1. Multiple regression is regression analysis with more than one independent variable 多元回归是具有多个自变量的回归分析

2.2. Yi= α+ βi X1i +β2 X2i +......βKi ki+ εi

2.2.1. The intercept Term is the value of the dependent variable when the independent variables are all equal to zero 截取项是自变量全为零时因变量的值

2.2.2. Each slope coefficient is the estimated change in the dependent variable for a one unit change in that independent variable, holding the other independent variables constant. That's why slope coefficients in a multiple regression are sometimes called partial slope coefficient 每个斜率系数是该自变量中一个单位变化的因变量的估计变化，而其他自变量则保持不变。 这就是为什么多元回归中的斜率系数有时称为部分斜率系数的原因

2.3. OLS Estimator

2.3.1.

3. The assumption of the Multiple linear regression多元线性回归的假设

3.1. There is no exact linear relation between any two or more explanatory variables 任何两个或多个解释变量之间没有确切的线性关系

3.2. The random variables (X1i, X2i...,Ski) are assumed to be iid 随机变量来自于独立同分布

3.3. εi has conditional mean zero given X1i ,X2i ,...,Ski that is E(εi | X1i, X2i...,Ski）=0 条件均值=0

3.4. Variance of error terms is constant foe all observations: E(εi² | X1i, X2i...,Ski)=c 条件方差=常数

3.5. The assumption of no outliers is extended to hold for each explanatory variable 异常值的点不能太多

4. Measure if fit TSS=ESS+RSS 拟合优度(拟合度，结合度)

4.1. TSS: The Total Sum of Squares 总平方和 which is defined as the sum of the squared deviations of Y around the sample mean 衡量样本Y 的分散程度(样本值减样本均值的平方)

4.1.1. ESS: The explained sum of squares 拟和值的分散程度, 解释平方和

4.1.2. RSS: The residual sum of squares 残差平方和(样本值减拟合值的平方）

4.2. Coefficient of Determination, R² and Adjusted R²

4.2.1.

In multiple regression, the R² Increases whenever an independent variable is added 在多元回归中，每添加一个自变量，R²就会增加

4.3. The adjusted R² is a modified version of the R² that does not necessarily Increase with a new independent variable is added.Adjusted R² is given by 调整后的R² 是R² 的修改版本，并不一定会添加新的自变量来增加。调整后的R² 由下式给出

4.3.1.

① Adding an additional explanatory variable may decreases adjusted R² 添加附加的解释变量可能会降低调整后的R²

② The adjusted R² may produce negative values 调整后的R² 可能会产生负值

③ Adjusted R²≤ R²

5. Joint Hypothesis Testing 联合零假设检验

5.1. Test whether at least one slope coefficient is significantly different from zero 测试至少一个斜率系数是否明显不同于零

5.1.1. H0: β1=β2=β3=......=βk=0; Ha: at least one βi≠0 ( j=1 to k )

5.2. F-Statistic

5.2.1.

5.3. Decision rule: Reject H0, if

5.3.1.

6. Hypothesis test for a single slope coefficient 单个斜率系数的假设检验

6.1. Significance test for a regression coefficient

6.1.1.

6.2. Regression coefficient confidence interval

6.2.1.

7. Testing parameters in regression models 同方差条件下F 统计量

7.1. Implementing an F-test requires estimating two models. The first is the unrestricted Model an RSS denoted by RSS U. The second Model, called the restricted Model, imposes the null hypothesis on the unrestricted Model and it's RSS is denoted RSS R. The F-test compares The fit of these two models: 实施F 检验需要估计两个模型。第一个是RSS U 表示的无限制模型RSS ，第二个模型称为限制模型，它在无限制模型上施加了零假设，其RSS 表示为RSSR 。F 检验比较这两个模型的拟合度：

7.1.1.

Where q is the number of restrictions imposed on the unrestricted Model to produce the restricted Model and k U is the number of explanatory variables in the unre 其中q 是施加于非限制模型以产生限制模型的限制数量，而k U 是非限制模型中的解释变量的数量

7.2. 例题

7.2.1.

8. Application: Muti-factor Risk Models

8.1. Fama-French Three-factor model

9. Constant

9.1. The assumption of multiple regression 多元回归假设条件

9.2. Interpret regression coefficients in a multiple regression 解释多元回归中的回归系数

9.3. Interpret goodness of fit measures, including R² and adjusted R² 拟合优度，调整的拟合优度( 考虑自由度的损失)

9.4. Construct, apply, and interpret joint hypothesis Tests and confidence intervals for multiple coefficient 三个假设检验，整体F 检验，( 解释除残差越大越好) ，单个t 检验，同方差条件F 检验

E. 新Regression Diagnostocs 回归诊断

1. Learning objectives

1.1. Explain how to test whether a regression is affected by heteroskedasticity 异方差 说明如何测试回归是否受异方差影响

1.2. Describe approaches to using heteroskedastic data 如何使用有异方差的数据

1.3. Characterize multicollinearity and its consequences, distinguish between multicollinearity and perfect collinearity 表征多重共线性及其后果，区分多重共线性和完美共线性

1.4. Describe the consequences of excluding a relevant explanatory variable from a model and contrast those with the consequences of including an irrelevant regressor 描述从模型中排除相关解释变量的后果，并将其与包括不相关的回归变量的后果进行对比

1.5. Explain two model selection procedures and how these relate to the bias-variance tradeoff 解释两种模型选择程序以及它们与偏差方差权衡的关系(由大到小，数据分组)

1.6. Describe the various methods of visualizing residuals 对于残差合适化，画残差图

1.7. Describe methods for identifying outlier and their impact 描述识别异常值的方法及其影响

1.8. Determine he conditions unexpected which OLS is the best linear unbiased estimator 确定他的条件是否意外，哪个OLS 是最佳线性无偏估计量

2. Omitted Variable Bias 遗漏变量

2.1. Omitted variable is one that has a non-zero coefficient but is not included in a model 遗漏变量是具有非零系数但不包含在模型中的变量

2.2. When the omitted variable is non-correlated with the independent variable in the model 当遗漏的变量与模型中的自变量不相关时

2.2.1. The residuals contain any effect of the omitted variable,which may Increase the variance of residuals and may affect the accuracy of OLS estimation 残差包含被遗漏变量的任何影响，这可能会增加残差的方差并可能影响OLS 估计的准确性

2.3. When the omitted variable is correlated with the independent variable in the model 当遗漏变量与其他解释变量相关时

2.3.1. This changes the regression coefficients on the included variables so that they do not consistently estimate. This is called omitted variable bias. The OLS estimators will be biased and inconsistent 这将更改所包含变量的回归系数，以使它们不会一致地估计。这称为遗漏变量偏差。OLS 估计量将有偏差且不一致

2.4.

3. Extraneous included Variables 包含无关变量

3.1. The true model is Yi= α + β1 X1i + εi

3.2. An extraneous variable is one that is included in the model but is not needed 加了一个无关变量回会减少残差 Yi=α+ β1X1i +β2X2i +μi ,where μi=εi-β2X2i

3.3. This Type of variable has a true coefficient of 0 (β2=0) 此类型的变量的真实系数为0 （β2= 0 ）趋近于0

3.3.1. An extraneous variable reduces R² adjusted 一个无关的变量减小了调整后的R²

3.3.2. Because an extraneous variable may make the standard error of the regression coefficient to be larger, the absolute value of the test statistics may be reduced 由于外来变量可能会使回归系数的标准误变大，因此可能会降低检验统计量的绝对值

3.4.

4. Omitted variables & Extraneous included Variables

4.1. ①理论：方程中的变量是否有明确的理论意义？

4.2. ②t检验：变量的估计系数是否在预期方向上是显著？

4.3. ③调整的R²拟合系数: 当变量加入模型后，调整的R²是否增加？

4.4. ④偏度：当变量加入模型后，其他变量的回归系数是否显著的改变？

4.5. 如果满足上述四个条件，那么该变量应当包含进入模型中。 如果没有任何一个条件被满足，模型中不能包含该变量。 如果仅满足部分条件，那么就需要研究者做出决定。

5. The Bias-Variance Tradeoff 偏差与方差的权衡

5.1. The choice between omitting a relevant variable and including an irrelevant variable is ultimately a trade-off between Bias and variance 忽略相关变量和包含不相关变量之间的选择最终是偏差与方差之间的权衡

5.2. 当遗漏一个变量时，出现bias偏差，模型回归系数的variance减少。 当包含无关变量时，bias不变，β系数趋近于0，但会导致variance增加

5.3. 两种方法可以权衡Two of these methods are general-to-specific model selection and m-fold cross-validation

5.3.1. Gerenal-to-specific model:

GtS model selection begins by specifying a large model that includes all relevant Variables. If there are coefficients in this estimated model that are statistically insignificant, the variable with the coefficient with the smallest absolute t-statistic is removed.Then, the model is re-estimated using the remaining explanatory variables.These two steps(remove and re-estimate) are repeated until the model contains no coefficients that are statistically 尽可能由大到小搜集可能的解释变量，一个个检验，大点好，先剔除最不显著，计算解释变量直到每个系数都显著，至少没有偏差，估计较准

5.3.2. M-fold cross-validation 数据分组交叉验证

Cross-validation begins by randomly splitting the data into m equal sized blocks.Parameters are then estimated using m-1 of the m blocks in turn, and the sum of squared errors are computed using these estimated parameter values on the data in the excluded block. This process is repeated a total of m tomes 把数据分为m 组，将训练集分组，取一组当测试组，其他训练组，求回归方程

Finally, the preferred model is models by selecting the model with the smallest out-of-sample sum of squared residuals 最后，首选模型是通过选择具有最小残差平方和的样本外模型的模型

6. Heteroskedasticity 异方差

6.1. Homoskedasticity ：同方差 The variance of the conditional distribution of εi given Xi is constant for i=1,....n 给定Xi 的εi 条件分布的方差对于i = 1 ，.... n 是恒定的

6.2. Heteroskedasticity: 异方差 Refers to the situation that the variance of the error Term is not constant 指误差项方差不恒定的情况

6.3.

6.4. Effect of heteroskedasticity on regression analysis 异方差性在回归分析中的影响

6.4.1. The coefficient estimators are not affected (remain unbiased and consistent) 系数估计不受影响（保持无偏且一致）

6.4.2. The standard errors (bias) are usually unreliable estimates 标准误差（偏差）通常是不可靠的估计

6.4.3.

6.5. White's estimator 怀特检验

6.5.1. Regress the squared residuals on a constant, all explanatory variables,and the cross-product of all explanatory variables 假设三元回归，回归常数，所有解释变量和所有解释变量的叉积的平方残差

6.5.2. Compute the squared the residuals using the OLS parameter estimators, and then regresses the squared residuals by a regression model: 使用OLS 参数估算器计算残差的平方，然后通过回归模型对残差的平方进行回归：

6.5.3. If the data are homoscedastic, then ei² should not be explained by any the variables on the right-hand side and the null 如果数据是同质的，则ei² 不应由右侧的任何变量和null 来解释。

6.5.4. The test statistic is computed as nR², where the R² is computed in the second regression 测试统计量计算为nR² ，其中R² 在第二次回归中计算

6.5.5. Where k is the number of explanatory variables in the first-stage model, n is the sample sizes n 是样本个数，R² 拟合优度

6.6. Approaches to Modeling Heteroskedastic Data

6.6.1. The first (and simplest) is to ignore the heteroskedasticity when estimating the parameters and then use the heteroskedasticity-robust(White) covariance Estimator. 第一个（也是最简单的）是在估计参数时忽略异方差，然后使用异方差鲁棒（怀特）协方差估计器。

6.6.2. The second approach is to transform the data(i.e. the natural log) 第二个方法是去自然对数

6.6.3. The final(and most complicated) approach is to use weighted least squares (WLS) 用加权二乘法

7. Multicollinearity 多重共线性

7.1. Perfect multicollinearity means that one of the independent variables is a perfect linear combination of the other.One explanatory variable can be completely explained by movements in another explanatory variable 完全多重共线性是指一个自变量是另一个的完美线性组合。一个解释变量可以通过另一个解释变量的移动来完全解释

7.2. Such a case between two independent variables would be X1i= α0 + λX2i, where the X are independent variables: Yi=α +β1X1i+ β2X2i+...+βkXki+ εi

7.2.1. Perfect multicollinearity can usually be avoided by careful screening of the independent variables before a regression is run 通常可以通过在运行回归之前仔细筛选自变量来避免完美的多重共线性

7.3. Imperfect multicollinearity means multicollinearity without perfect correlation: X1i= α0+λx2i+ui, where ui is a stochastic error them 不完全多重共线性，X1X2 存在一定的相关性

7.4. The major consequences of multicollinearity are: 对回归系数造成什么影响？

7.4.1. Estimates will remain unbiased 无偏

7.4.2. The variances of the estimates will increase 方差增加

7.4.3. The computed t-scores will fall T 检验值降低，增加不能拒绝原假设的概率，第二类错误增加

7.5. In practice, multicollinearity is often a matter of degree rather than absence / presence 多重共线性不是存在不存在的问题，而是程度多少的问题，

7.5.1. Important question is how much exists

7.5.2. The severity can change from sample to sample

7.6. Researchers a develop a general feeling for the severity of multicollinearity by examining a number of characteristics 研究人员通过检查许多特征，对多重共线性的严重性有了一般的认识

7.6.1. Variance inflation factors 方差膨胀因子

Ri² is coefficient of determination that comes from a regression of Xi on the other independent variables in the model Ri² 是确定系数，该系数来自模型中其他自变量的Xi 的回归

Values above 10(i.e., which indicate 90% of the variation in Xi can be explained by the other variables in the model) are considered excessive 大于10 的值（即表示Xi 的90 ％的变化可以由模型中的其他变量解释）被认为共线性严重

7.7. Residual Plots 残差图

7.7.1. An ideal model would have residuals that are not systematically related to any of the included explanatory variables 理想模型的残差与所包含的任何解释变量均不系统相关

7.7.2. The basic residual plot compares e(y-axis) against the realization of the explanatory Variables X 基本残差图将e （y 轴）与解释变量X 的实现进行比较 用残差为纵坐标，解释变量当横坐标，做出来的闪点图。 考察模型假设的合理性，直观效果好 OLS的图形图a左上角，均值为0，方差常数。

7.8. Outliers 异常点。＞1 ，异常点

7.8.1. Outliers are values that, if removed from the sample,produce large changes in the estimated coefficients 异常值是一些值，如果将其从样本中删除，则会在估计系数中产生较大的变化 用cook 距离检查异常的点

8. Strengths of OLS 最小二乘法的优势

8.1. OLS is the Best Linear Unbiased Estimator(BLUE).Best indicates that the OLS estimator that is linear and unbiased 最小二乘法是最佳线性无偏估计，n→∞ 满足一致性 但是有两个假设前提，残差满足同方差，很多估计不是线性的

8.2. It comes with two important caveats: 它带有两个重要警告：

8.2.1. The BLUE property of OLS depends crucially on the homoscedasticity of the residuals

8.2.2. Many estimators are not linear 许多估计量都不是线性的

8.3. The assumption that the errors are normally distributed strengthens BLUE to BUE. So that OLS is the Bear(in the sense of having the smallest variance) Unbiased Estimator among all linear and nonlinear estimators 去掉线性会是最佳的吗BUE ？, 残差要满足正态分布

8.4. The only requirements on the errors are the assumptions that the residua is have conditional mean zero and that there are no outliers 对误差的唯一要求是假设残差的条件均值为零，并且不存在离群值

9. Constant

9.1. Omitted variable and extraneous included variable

9.2. The Bias-Variance tradeoff

9.3. heteroskedasticity

9.4. multicollinearityand perfect collinearity

9.5. Residuals Plots

9.6. Outliers

9.7. Determine the conditions under which OLS is BLUE

Stationary Time Series 平稳时间序列(二阶宽平稳过程，协方差平稳，均值期望方差都是常数，协方差跟时间起点无关）

1. Learning objectives

1.1. Describe the requirements for a series to be covariance-stationary 描述斜方差平稳时间序列

1.2. Define the autocovariance Function and the autocorrelation function 定义自斜方差，自相关函数

1.3. Define white noise, describe independent white noise and Gaussian white noise 白噪声，独立白噪声，高斯白噪声的含义

1.4. Explain how a lag operator works 延迟算子

1.5. Define and describe the properties of autoregressive(AR) processes 自回归模型

1.6. Define and describe the properties of moving average(MA) processes 移动平均模型

1.7. Define and describe the properties of autoregressive moving average(ARMA) processes 自回归移动平均模型

1.8. Describe the Box-Pierce Q-statistic and the Ljung-Box Q statistic 两个Q 统计量

1.9. Describe the role of mean reversion in long-horizon forecasts 长期预测中均值回归现象

1.10. Explain how seasonality is modeled in a covariance stationary ARMA 解释用斜方差平稳对季节性时间序列建模

2. Data type 数据类型

2.1. Cross-section data 横切面数据

2.1.1. Cross-section data is data that is collect by analyzing different sets of data from different sources at the particular time. For instance, we're studying the GDP of 3 devoloping countries in year 1999. 横截面数据是通过在特定时间分析来自不同来源的不同数据集而收集的数据。例如，我们正在研究1999 年三个发展中国家的GDP 。 同一对象，不同时间

2.2. Time series data 时间数列数据

2.2.1. Time series data is data that is measured using a sequence of certain points at particular times. The Dow Jones index is an example of data that is measured using time series data, as the data collected is listed at a certain time on each day. 时间序列数据是在特定时间使用一系列特定点测量的数据。道琼斯指数是使用时间序列数据测量的数据示例，因为收集的数据在每天的特定时间列出。

2.3. Panel data 面板数据

2.3.1. Panel date contain observations of multiple phenomena obtained over multiple time periods for the same firms or individuals. 面板数据包含对同一公司或个人在多个时间段内获得的多种现象的观察结果。

2.3.2. Time series and cross-sectional data can be thought of as special cases of panel data that are in one dimension only (one panel member for the former,one time point for the latter. 时间序列和横截面数据可以认为是面板数据的特殊情况，它们仅在一维（前者为一个面板成员，后者为一个时间点）。

3. Covariance Stationarity 协方差平稳时间序列

3.1. Stationarity is a key concept that formalizes the structure of a time series and justifies the use of historical data to build models 平稳性是一个关键概念，可以正式化时间序列的结构并证明使用历史数据来构建模型是合理的

3.2. At time series is covariance stationary if it satisfies the following three conditions: 如果满足以下三个条件，则时间序列上的协方差是平稳的： ( 期望值，方差，斜方差不随时间变化而变化）

3.2.1. ① Constant and finite Expected value : E(yt)=μ

3.2.2. ② Constant and finite variance : Var(yt)=σ

3.2.3. ③ Constant and finite covariance between values at any given lag:

3.2.4. Symmetric: γ(h)=γ(-h) 对称

3.2.5. γ(0)=cov(yt,yt)=various(yt)

3.3. 协方差平稳等于二阶宽平稳

3.4. The autocorrelation Function(ACF) 自相关函数

3.4.1.

4. The Lag Operator 延迟算子

4.1. The lag operator L shifts the time Index of an observation,so that LY t= LY t-1

4.2.

5. White noise 白噪声

5.1. A White noise process is denoted as→ Where σ² is the variance of the shock WN→ 白噪声缩写

5.2. White noise processes ｛εt｝have three properties

5.2.1. Mean zero 0 均值

5.2.2. Constant and finite variance 常数的方差

5.2.3. No autocorrelation or autocovariance 不存在相关性

5.3. If the White noise is independent, it is called independent White noise 独立白噪声

5.4. If White noise obeys Normal distribution, it is called Gaussian White noise 如果白噪声服从正态分布，则称为高斯白噪声

5.5. Wold's Theorm 伍德定理

5.5.1. Let yt be any zero-mean covariance-stationary process.Then we can write it as 对任意0 均值协方差平稳的过程，都可以表现写成白噪声加权组合

6. Autoregressive (AR) Models 自回归模型

6.1. autoregressive model: Current value of series is linearly related to its past values, plus an additive stochastic shock 自回归模型：系列的当前值与其过去的值线性相关，外加随机冲击

6.2. AR(1) process

6.2.1.

6.2.2.

6.3. In AR(p) models, the current value of yt is a linear combination of its lagged values of the following type

6.3.1. AR(p) process Innovations 新息，t 时刻产生的新信息

6.3.2. ACF of AR(1) model

6.4. The partial autocorrelation Function(PACF) of AR(1) model 偏自相关函数

7. Moving Average (MA) Models 移动平平均过程

7.1. Obvious approximation to the World representation; Modeling time series as distributed lags of current and past shocks 伍德定理告诉我们，任何协方差平稳时间序列，都可以写成无穷多项白噪声线性组合，现实中没有无穷多项组合，构成时间序列组合用有限项移动平均过程来模拟平稳过程

7.2. MA(1) model

7.3. MA(q) model

7.4. ACF of MA(1) model 自相关函数，常数项

7.4.1. The ACF is always zero for lags larger than 1

7.4.2. The PACF of an MA(1) is Exponential decay and has non-zero values at all lags 指数衰减

8. Autoregressive Moving Average (ARMA) Models 在回归移动平均模型

8.1. ARMA(p,q) model

8.2. ARMA(p,q) models have ACFs and PACFs that decay slowly to zero as the lag Increases ( while possibly oscillating ) ARMA （p ，q ）模型具有ACF(自相关函数) 和PACF(偏相关函数)，都是拖尾的，它们随着滞后的增加而缓慢衰减至零（同时可能会振荡）

8.3. An ARMA(p,q) process is covariance-stationary if |ø| ＜1. The MA coefficient plays no role in determining whether the process is covariance-stationary, because any MA is covariance stationary 如果|ø| ＜1 ，则ARMA （p ，q ）过程是协方差平稳的。MA 系数在确定过程是否是协方差平稳的过程中不起作用，因为任何MA 都是协方差平稳的

9. Model Selection 建模

9.1. The first step in model building is to inspect the sample autocorrelation and sample PACFs 建立模型的第一步是检查样本自相关和样本PACF

9.2. The next step is to measure their fit. Most model selection criteria attempt to find model with the smallest out-of-sample 1-step-ahead mean square prediction error(the smaller, the better), also known as the Mean Squared Error. 下一步是衡量它们的适合度。大多数模型选择标准都试图找到样本外提前一阶均方误差最小（越小越好）的模型，也称为均方误差。

9.2.1. Mean Squared Error(MSE) 最小均方差 希望残差项平方越小越好，R²拟合优度最大

9.2.2. Akaike Information Criterion (AIC)

Where T is the sample size and k is the number of parameters

9.2.3. Bayesian Information Criterion (BIC)

10. Model Diagnosis 模型诊断

10.1. If the selected model is correct,the prediction ero of the model should be White noise sequence.How to judge whether a time series is White noise or not, we need to use Q-test quantity 假设模型选择是正确的，预测的残差项就是白噪声，用Q 检验统计量检查是不是白噪声，是的话选择正确，

10.2. It's basic idea is: because White noise is sequence uncorrelated, the autocorrelation function of any order should be 0 at the same time 它的基本思想是：由于白噪声是序列不相关的，因此任何阶数的自相关函数应同时为0

10.3.

11. Forecasting 预测预报

11.1. Forecasts use current information to predict the future 预测使用当前信息来预测未来

11.2. The one-step forecast 超前一步预测

11.3. The two-step forecast 超前两步预测

11.4. These steps repeat for any horizon h so that: 超前h 步预测，等于AR(1)的期望值

11.4.1. This Limit is the same as the mean reversion level of an AR(1)

11.5. Example

12. Seasonality 季节性时间序列

12.1. The specification of Seasonal ARMA model is denoted

12.1.1. Where p and q are orders of the short -run lag polynomials, ps and as are the orders of the seasonal lag polynomials, and f is the seasonal horizon(E.g.,4 or 12) 其中p 和q 是短期滞后多项式的阶数，ps 和季节性滞后多项式的阶数一样，f 是季节性时域（例如4 或12 ）

12.2.

13. Constant

13.1. Requirements for a covariance-stationary series 协方差平稳序列的要求

13.2. Define the auto correlation Function(ACF) 定义自相关函数

13.3. Define White noise,Gaussian White noise 定义白噪声，高斯白噪声

13.4. Explain how a lag operator works 延迟算子

13.5. AR processes, MA processes, ARMA processes

13.6. Box-Pierce Q-statistic and the Ljung-Box Q statistic

13.7. Describe the role of mean reversion in long-horizon forecasts

13.8. Explain hoe seasonality is modeled in a covariance stationary ARMA

Non-Stationary Time Series非平稳时间序列

1. Learning objectives

1.1. Describe linear and nonlinear time trends 描述线性和非线性时间趋势

1.2. Explain how to use regression analysis to model seasonality 说明如何使用回归分析对季节性进行建模

1.3. Describe a random walk and a unit root 描述随机行走，及单位根

1.4. Explain the challenge of modeling time series containing unit roots 解释如何应对有单位跟的事件序列

1.5. Explain how to construct an h-step-ahead point forecast for a time series with seasonality 解释如何构建具有季节性的时间序列的h 提前点预测

1.6. Calculate the estimated trend value and form and interval forecast for a time series 计算估计的趋势值以及时间序列的形式和间隔预测

2. Time Trends

2.1. Polynomial Trend 多项式趋势

2.1.1. The most basic example of a non-stationary time series is a process with a linear time trend: 非平稳时间序列的最基本示例是具有线性时间趋势的过程：

yt= δ0 + δ1t +εt

Where εt~WN(0,σ²)

This process is non-stationary because the mean depends on time: E ［ Yt ］ = δ0 + δ1t 该过程是不稳定的，因为均值取决于时间

In this case, the time trend δ1 Measures the average change in Y across subsequent observations. The mean grows at a constant amount. 在这种情况下，时间趋势δ1 度量随后观察到的Y 的平均变化。平均值以恒定量增长。

2.1.2. The linear time trend model generalizes to a polynomial time trend model by including higher powers of time 线性时间趋势模型通过包含更高的时间幂，将其推广为多项式时间趋势模型

2.1.3. In practice, most time trend models are limited to first (i.e.,linear) or second-degree (i.e.,quadratic) polynomials

3. Non-stationary Time Series

3.1. Log-linear trend 对数线性模型

3.1.1. Constant growth rates are more plausible constant growth in most financial and macroeconomic variables. These growth rates can be examined using a log-linear models: 在大多数金融和宏观经济变量中，恒定增长率是更合理的恒定增长率。 可以使用对数线性模型检查这些增长率：

3.2. Seasonality 季节性

3.2.1. Definition: a seasonal time series exhibits deterministic shifts throughout the year 定义：一个季节性的时间序列在一年中表现出确定性的变化 长度固定的，周期波动，同期相关性进行拟合

3.2.2. Many financial assets(products) exhibit seasonality that results from the features of markets or regulatory structures 许多金融资产（产品）表现出季节性，这是由市场特征或监管结构导致的

3.3. Seasonal Dummy Variables 季节性虚拟变量

3.3.1. Deterministic seasonalities produce differences in the mean of a time series that are simple to model using dummy variables 确定性季节性会产生时间序列平均值的差异，使用虚拟变量可以轻松地对其进行建模

3.3.2. The seasonality repeats each periods, and the model is: 加入二值变量，R 值变量

3.4. Time Trends 时间趋势模型, Seasonalities 季节性模型, And Cycles 循环超过一年，时间不固定

3.4.1. Example: a trend-stationary process is: yt= δ0 + δ1t + εt

It indicates that the residual (process removed by the deterministic trend) is covariance-stationary 它表明残差（确定性趋势消除的过程）是协方差平稳的

If the residual is not a white noise but follows an AR(1), adding an AR(1) term should augment the model: 如果残差不是白噪声而是遵循AR （1 ），则添加AR （1 ）项应可增强模型：

If there is a seasonal component to the data, then seasonal dummies can be added as well: 如果数据中包含季节性成分，则还可以添加季节性虚拟变量：

3.5. Random walk 随机行走

3.5.1. Random walks are the most important source of non-stationary in economic time series 随机游走是经济时间序列中非平稳的最重要来源

3.5.2. Repeating the substitution until observation zero is reached 重复替换，直到达到观测值零 走到最后回回到起点，方差大，时间不确定，没有均值回归

It can be seen that Yt depends equally on every shock between period 1and t, as well as on an initial value Y0

V［Yt］=tσ²

3.5.3. A unit root process is usually described using a lag polynomial; taken and AR(2) process as an example ： 单位根 通常使用滞后多项式描述过程； 以AR（2）过程为例：

Can be written with a lag polynomials as

The roots of the characteristic equation equals to 1, which is the evidence of the unit root 特征方程的根等于1 ，这是单位根的证明

3.6. Unit roots 单位根

3.6.1. There are three key challenges when modeling a time series that has a unit root: 对具有单位根的时间序列进行建模时，存在三个主要挑战：

Parameter estimators In ARMA models fitted to time series containing a unit root have what is called Dickey-Fuller(DF) distribution 参数估计量在适合包含单位根的时间序列的ARMA 模型中，具有Dickey-Fuller （DF ）分布非(对称样本依赖分布)

Spurious regression: when modeling the relationship between two or more non-stationary time series, the statistics for the model are significant, but the relation between the two series are unreliable 虚假回归：在对两个或多个非平稳时间序列之间的关系进行建模时，该模型的统计数据很重要，但两个序列之间的关系并不可靠

A unit root process does not mean revert 单位根进程并不能导致均值回归

3.6.2. The solution to all three problems is to difference a time series that contains a unit root. Because φ(L)(1-0.8L) is a lag polynomials of a stationary process, then the random variable defined by the difference must be stationary 解决这三个问题的方法是对包含单位根的时间序列求差。因为φ （L ）（1-0.8L ）是平稳过程的滞后多项式，所以由差定义的随机变量必须是平稳的

3.7. Dicky-Fuller(DF) Test β=1, 随机行走，存在单位根，非平稳

3.8. H-step-ahead point forecast H 提前点预测

3.8.1. Constructing forecasts from models with time trends, seasonalities, and cyclical components is no different from constructing forecasts from stationary ARMA models 从具有时间趋势，季节性和周期性成分的模型中构建预测与从固定ARMA 模型中构建预测没有什么不同

The forecast is the time T Expected value of Y t+h

For a linear time trend models:

3.9. Confidence Intervals 置信区间

3.9.1. In some application, it is useful to construct both the forecasts and confidence intervals to express the uncertainty of the future value 在某些应用中，构造预测和置信区间以表示未来价值的不确定性是有用的

3.9.2. The forecast confidence interval depends on the Variance of the forecast Error which is defined as 预测置信区间取决于预测误差的方差，其定义为

When the Error is Gaussian white noise N(μ,σ²),then the 95% confidence interval for the future value is

In practice, σ is not known. Howerver, it can be estimated as the square root of the residual Variance from the estimated regression 实际上，σ 是未知的。但是，可以通过估计回归将其估计为残差方差的平方根

4. Constant

4.1. Linear and nonlinear time trends 知道线性非线性时间趋势

4.2. How to use regression analysis to model seasonality 如何使用回归分析对季节性进行建模

4.3. Describe a random walk and a unit root 叙述随机行走与单位根

4.4. Describe how to test if a time series contains a unit root 描述如何测试时间序列是否包含单位根

4.5. Explain how to construct an h-step-ahead point forecast for a time series 解释如何构建时间序列的h 提前点预测

4.6. Form an interval forecast for a time series 形成时间序列的间隔预测

Measuring Return, Volatility, and Correlation测量收益率，波动率，相关性

1. Learning objectives

1.1. Calculate and convert between simple and continuously compounded returns 单利连续复利定义计算及相互转换

1.2. Define and distinguish between volatility, variance rate, and implied volatility 区分波动率方差跟隐含波动率

1.3. Describe how the first two moments may be insufficient for non-normal distributions 描述前两个时刻对于非正态分布可能是不足的

1.4. Explain how to use Jarque-Bera test to the determine normally distributed returns 说明如何使用Jarque-Bera 检验来确定正态分布的收益

1.5. Describe the power Law and it's use for non-normal distributions 描述幂律及其在非正态分布中的使用

1.6. Define correlation and difference between correlation and dependence 定义相关性和相关性与依赖性之间的区别

2. Measuring returns 衡量回报

2.1. All estimators of volatility depend on returns, and there are two common methods used to construct returns 所有波动率的估计量都取决于收益，并且有两种常用的方法来构造收益

2.1.1. 简单收益率 The usual definition of a returns on an Asset bought at time t-1 and sold at time t is 在时间t-1( 前一期) 购买并在时间t 出售的资产收益的通常定义是

2.1.2. Returns computed using this formula are called simple returns and are traditionally expressed with an uppercase letter (i.e.,Rt) 使用此公式计算的收益称为简单收益，传统上用大写字母（即Rt ）表示

2.2. 对数收益率 There are also continuously compound return, also known as log returns. These are computed as the difference of the natural logarithm of the price: 还存在连续复利收益率，也称为对数收益率。这些计算为价格自然对数的差：

2.2.1. The log return is traditionally denoted with lower case letter(i.e.,rt) 对数收益率传统上用小写字母（即rt）表示

2.3. The Total return over Multiple periods is the sum of the single period log returns: 多个期间的总收益是单个期间对数收益率的总和：

2.4. Converting between the simple and log return uses the relationship: 在简单收益和对数收益之间进行转换使用以下关系：

2.4.1. The log return is always less than the simple return rt ＜Rt 1 ＋Rt= e ʳᵗ

3. Measuring volatility and risk 波动率及风险

3.1. The volatility of a financial asset is usually measured by the standard deviation of its returns 金融资产的波动率通常通过其连续收益的标准偏差来衡量 波动率：单位时间内rt连续复利收益率的标准差σ 风险：实际收益率与期望收益率的偏离度

3.1.1. This measure is the volatility of the returns over the time span where the returns are measured, and so if returns are computed using daily closing prices, this measure is the daily volatility 该度量是衡量回报的时间跨度内的回报波动率，因此，如果使用每日收盘价计算回报，则此度量值就是每日波动率

3.1.2. Is simply the square of volatility

3.1.3. Rule of square root:

3.1.4. Business day: 252 days per year.

3.2. Implied volatility is an alternative measure that is constructed using option prices 隐含波动率是使用期权价格构建的一种替代度量

3.3. All values in the BSM model including the call price, are observable the volatility: 期权定价模型 Ct 看涨期权价格 由哪几个因素决定？ ① Rf 无风险利率 ② T 期限 ③ Pt 到期价格 ④ K 执行价格 ⑤ σ² 波动率

3.4. The value of σ that equates the observed call option price with the four other parameters in the formula is known as the implied volatility 使观察到的看涨期权价格与公式中的其他四个参数相等的σ 值被称为隐含波动率 交易活跃的隐含动率估算不活跃的期权波动率价格 隐含波动率＞实际波动率

3.5. The VIX Index is another measure of implied volatility Volatility Index 标普500 看涨看跌期权隐含波动率加权求和，恐慌指数

4. Non-normal distribution 非正态分布

4.1. A normal distribution is symmetric and thin-tailed, and so has no skewness or excess kurtosis 正态分布是对称且细尾的，因此没有偏斜或峰度过大

4.1.1. Skewness=0,kurtosis=3

4.2. However ，many return series are both skewed and fat-tailed. For example, consider the returns of gold, the S&P 500,and the JOY/USD exchange rate: 实际上，有偏且肥尾

4.3. The Jarque-Beta test statistic is used to formally test whether the sample skewness and kurtosis are compatible with an assumption that the returns are normally distributed Jarque-Bera 检验统计量用于正式检验样本偏度和峰度是否与收益呈正态分布的假设兼容

4.4. H0: S=0 and K=3, where H1: S≠0 or K≠3

4.5. The test statistic is

4.5.1. Where T is the sample size, S^is the skewness and K^is the kurtosis of the sample 其中T 是样本大小，S ^ 是偏度，K ^ 是样本的峰度

5. Power Law 幂律

5.1. An alternative to assuming Normal distribution 替代正态分布

5.2. In practice the distribution of asset price changes is more likely to exhibit fatter tails than the normal distribution,The power Law states that when is large, the value of a variable K has the following property 实际上，资产价格变化的分布比正态分布更容易出现尾巴。幂定律指出，变量K 的值大时，具有以下属性： 负指数 第一跟第二落差很大

5.3.

6. Correlation Versus Dependence 相关性和相依性

6.1. The dependence between assets plays a key role in portfolio diversification and tail risk 资产之间的依存关系在投资组合多元化和尾部风险中起着关键作用

6.1.1. Recall that two random variables, X and Y, are independent if their joint density is equal to the product of their marginal densities:

6.1.2. Any random variables that are not independent are dependent 相依事件就是非独立事件

6.2. Financial are highly dependent and exhibit both linear and nonlinear dependence 金融资産高度依赖，并且表现出线性和非线性依赖

6.2.1. The linear correlation estimator (also known as Pearson's correlation) measures linear dependence 线性相关估计器（也称为Pearson 相关）可测量线性相关性

6.2.2. Previous chapters have shown that correlation (i.e., p) and the regression slope (i.e.,β) are intimately related 前面的章节已经表明相关性（即p ）和回归斜率（即β ）密切相关

6.3. Nonlinear dependence takes many forms and cannot be summarized by a single statistic 非线性相依性

6.3.1. For example, many Asset returns have common heteroskedasticity (i.e., the volatility across Asset is simultaneously high or Low). However,linear correlation does not capture thus type of dependence between assets. 例如，许多资产收益具有共同的异方差性（即资产的波动率同时高或低）。但是，线性相关不能捕获资产之间的依存关系。

7. Spearman's correlation 秩相关系数

7.1. Rank correlation is the linear correlation estimator applied to the ranks of the observations 秩相关是应用于观测值秩的线性相关估计器

7.1.1. Rankxi is the order for Xi in the series X

7.1.2. Rank Yi is the order for Yi in the series Y

7.2. When variables have a linear relationship, rank and linear correlation are usually similar in magnitude. The rank correlation estimators is less efficient than the linear correlation estimator 当变量具有线性关系时，秩和线性相关通常在大小上相似。秩相关估计器的效率低于线性相关估计器，xy都要排序，较没效率

7.3. Significant differences in the two correlations indicate an important nonlinear relationship. Rank correlation has two distinct advantages over linear correlation 两种相关性的显着差异表明重要的非线性关系。与线性相关相比，秩相关具有两个明显的优势

7.3.1. It is robust to outliers because only the ranks, not the values of X and Y, are used 由于仅使用等级而不使用X 和Y 的值，因此对异常值具有鲁棒性

7.3.2. Linear correlation is only invariant with respect to increasing linear transformations 线性相关仅相对于线性变换的增加是不变的

7.3.3. Rank correlation is invariant with respect to any monotonic increasing transformation 秩相关对于任何单调递增的变换都是不变的

8. Kendall's τ( 分类，有序变量之间求相关性)

8.1. Kendall's τ is defined as the difference between the probability of concordance and probability of discordance ： 肯德尔的τ 定义为一致性概率和不一致率之间的差：

8.1.1. Where nc is the count of the concordant pairs

8.1.2. Concordant pairs: sign (Xi-Xj) =sign (Yi-Yj), i≠j

8.1.3. nd is the count of the discordant pairs

8.1.4. Discordant pairs：sign(Xi-Xj) ≠sign(Yi-Yj)≠j

9. Comparison between three measures

9.1. The relationship between linear (Pearson) correlation, rank (Spearman) correlation, and Kendall's r for a bivariate normal 二元正态线性（Pearson ）相关，秩（Spearman ）相关和Kendall's r 之间的关系

10. Constant

10.1. Simple and continuously compounded returns 简单利率，连续复利

10.2. Volatility, Variance rate, and implied volatility 波动率，方差率和隐含波动率

10.3. Explain the Jarque-Bera test

10.4. Describe the power Law and its use for non-normal distributions

10.5. Spearman's correlation

10.6. Kendall's τ

Simulation and Bootstrapping仿真测试

1. Learning objectives

1.1. Describe the basic steps to conduct a Monte Carlo simulation 蒙特卡罗模拟的基本过程

1.2. Explain the the use of antithetic and control variates in reducing Monte Carlo sampling error 说明使用对立变量和控制变量来减少蒙特卡洛模拟的抽样误差

1.3. Describe the bootstrapping method and its advantage over Monte Carlo simulation 描述自举方法及其相对于蒙特卡洛模拟的优势

1.4. Describe pseudo-random number generation 描述伪随机数生成

1.5. Describe situations where the bootstrapping method is ineffective 描述自助方法无效的情况

2. Monte Carlo Simulation 蒙特卡罗模拟

2.1. Monte Carlo wae derived from the name of a famous casino in Monaco

2.2. The Monte Carlo method is a tool for transforming the problems of nature into statistical computations using the numerous inputs sampled over and over again. It offers a wide scale of possible outcomes and chances and shows all the possibilities in order to come to the correct decision 蒙特卡洛方法是一种工具，它可以使用大量重复采样的输入将自然问题转化为统计计算。它提供了广泛的可能结果和机会，并显示了所有可能性，以便做出正确的决定

2.3. Four basic steps to conduct a Monte Carlo simulation

2.3.1. Step ① : Specify the data generating process (DGP) 指定数据生成过程p-model(ε)风险因子

2.3.2. Step②: Estimate an unknown variable or parameter

2.3.3. Step③: Save the estimate from step②

2.3.4. Step④: Go back to step① and repeat this process N times

2.4. Example: The process of simulation: △St= St(μ△t+σε√△t) where ε is a standard normal random variable, St is stock price at time t, μ is expected return on stock,σ is volatility of the stock,T is time to maturity,△t= T-t0 /n simulating a price path 示例：模拟过程：其中ε 是标准正态随机变量，St是时间t 的股票价格，μ是预期的股票收益率，σ 是股票的波动性，T 是到期时间，模拟了价格路径

3. Applications of Monte Carlo Simulation in Financial 蒙特卡罗模拟在金融中的应用

3.1. Monta Carlo is used in Corporate finance to model components of project cash flow, which are impacted by uncertainty 蒙特卡洛（Monte Carlo ）用于公司融资中，以建模受不确定性影响的项目现金流组成

3.2. Monte Carlo is used for option pricing where numerous random paths for the price of an underlying Asset are generated 蒙特卡洛用于期权定价，其中生成了基础资产价格的众多随机路径

3.3. It is similarly used for Pricing fixed income securities and Interest rate derivatines. But the Monte Carlo simulation is used most extensively in portfolio management and personal financial planning 类似地，它也用于固定收益证券和利率衍生产品的定价。但是蒙特卡洛模拟在投资组合管理和个人理财规划中得到了最广泛的应用

4. Monte Carlo Simulation

4.1. Reducing Monte Carlo Sampling Error 减少蒙特卡洛采样误差

4.1.1. The Sampling variation in a Monte Carlo study is measured by the standard Error estimate 蒙特卡洛研究中的抽样变化是通过标准误差估计来衡量的

Where var(x) is the Variance of the estimates of the quantity of interest over the N replications 其中var （x ）是N 次复制中感兴趣数量的估计值的方差

To reduce the Monte Carlo standard error by a factor of 10, the number of replications must be increased by a factor of 100 要将蒙特卡洛标准误差减少10 倍，必须将复制次数增加100 倍

5. Variance Reduction Techniques 减少方差的技术

5.1. Antithetic variates: The method attempts to reduce variance by introducing negative correlation between pairs of observation 对立变量：该方法试图通过在观察对之间引入负相关来减少方差

5.2. Suppose to estimate μ=E(X) ,we have generated two samples X1,X2

5.2.1.

5.3. Control variates: involves employing a variable similar to that used in the simulation, but whose properties are known prior to the simulation 控制变量法：涉及使用与模拟中使用的变量类似的变量，但其属性在模拟之前已知

5.3.1.

5.4. Random Number Re-Usage across Experiments: 公共随机数法(比较两个系统方案，采用完全相同随机数序列，)

5.4.1. Using the same sets of draws across experiments can greatly reduce the variability of the difference in the estimates across those experiments. However, the accuracy of the actual estimates in each case will not be increased, of course. 在整个实验中使用相同的抽奖集可以大大减少这些实验中估计差异的可变性。 但是，在每种情况下，实际估算的准确性当然不会提高。

5.5. Random Number Generation 随机数生成

5.5.1. Computer-generated random number draws are known as pseudo-random numbers, since they are in fact not random at all, but entirely deterministic, since they have been derived from an exact formula. 计算机生成的随机数抽取被称为伪随机数(真随机数不可预测，随机发生的物理事件)，因为它们实际上根本不是随机的，而是完全确定性的，因为它们是从精确的公式中得出的。

5.5.2. By carefully choosing the values of the user-adjustable parameters, it is possible to get the pseudo-random number generator to meet all the statistical properties of true random number. 通过仔细选择用户可调参数的值，可以使伪随机数生成器满足真实随机数的所有统计特性。

5.5.3. Pseudo random number generators produced deterministic sequences of numbers that appear stochastic, and match closely the desired probability distribution. 伪随机数生成器产生了确定性的数字序列，这些序列看起来是随机的，并且与所需的概率分布非常匹配。

5.5.4. The simplest class of Numbers to generate are from a uniform (0,1) distribution 要生成的最简单的方法，生成（0,1）分布

5.5.5. Computers generate continuous uniform random number draws. The initial value is called the seed(like y0 in the following example) 同余法 计算机生成连续一致的随机数绘制。 初始值称为随机种子（如以下示例中的y0）

5.5.6. For some standard distributions,e.g., uniform and Normal, the standard software such as MATLAB® provides built-in random number generators 对于某些标准分布，例如统一和正态分布，标准软件（例如MATLAB®）提供了内置的随机数生成器

5.5.7. The inverse Transform Method逆变换法

To simulate from other distributions,Then apply the inverse transform method for generating random variables from a specified distribution. 为了从其他分布进行模拟，然后应用逆变换方法从指定的分布生成随机变量。

Consider a random variable X with a continuous , strictly increasing CDF Function F(x) 考虑随机变量X，其CDF函数F（x）连续且严格增加

We can simulate X according to X=F⁻¹(U), U~Unif［0,1］

5.6. Sampling techniques

5.6.1. Bootstrapping: repeatedly draws data from a sample set 自助法：反复从样本集中提取数据 有放回的抽样

5.6.2. Advantage

No strong distribution assumptions 没有模型假设

Support fat fails支持肥尾现象

5.6.3. Bootstrapping method may be ineffective失效

The first limitation arises due to state changes so that the current state is different from its normal state. 第一个限制是由于状态变化而引起的，因此当前状态不同于其正常状态。

The second limitation arises due to structural changes in markets so that the present is significantly different from the past. 第二个限制是由于市场结构的变化而引起的，因此现在与过去有很大的不同。

5.7. Comparing Simulation and Bootstrapping

5.7.1. A Monte Carlo Simulation uses a full statistical model that includes an assumption about the distribution of the shocks 蒙特卡洛模拟使用完整的统计模型，其中包括有关冲击分布的假设

5.7.2. The bootstrap method avoids the specification of a model and and instead makes the key assumption that the present resembles the past 引导程序方法避免了模型的规范，而是做出了关键假设，即现在类似于过去

5.7.3. Both Monte Carlo simulation and bootstrapping suffer from the "Black Swan" problem 蒙特卡洛模拟和自举都遭受“黑天鹅”问题的困扰

6. Constant

6.1. Conduct a Monte Carlo simulation

6.2. Antithetic and control variates

6.3. bootstrapping method

6.4. Pseudo-random number generation

6.5. Comparing Simulation and Bootstrapping

CFA 1,2级浓缩