注册金融分析师CFA二级Quantitative method知识框架

注册金融分析师二级科目 quantitative methods, 全覆盖考纲思维导图和重点细节考点。

Quantitative Methods

Quantitative Method (1)

Linear regression

Assumptions

Linearity: Y与b1的线性关系

Homoskedasticity ---异方差问题

Independence------序列相关问题

Normality

Point eatimate

b0、b1公式

计算器计算b0、b1

Confidence interval eatimate

confidence interval计算

b1的standard error 可从 ANOVA TABLE中t.stat结果反推

Test of regression coefficient

significant test about regression coefficient

t-statistics

p-value

confidence interval method

hypothesis test about regression coefficient ---t-statistics

significant test for correlation

F-test: Ho: b1=b2=…=bk=0

ANOVA Table

SS, df, MSS,

SEE

Coefficient of determination R2， Multiple R

Estimate of Y

point estimate

confidence interval estimate

Different functional form: log

Limitations of regression

relation can change over time

public knowledge of regression

regression assumptions

Multiple regression

multiple vs linear regression区别

Partial regression coefficient：holding other constant

no exact linear relations: 若违反则multicollinearity

Hypothesis test

单个参数bi：t-test自由度不同

F-test：k不同

F vs T-test

单元：t^2=F

多元：F考虑x之间的相关性，影响t、F结果

Adjusted R-square

公式

R2>adjusted R2

may be<0

Dummy variable:X

n~n-1dummy variable

intercept和estimated coefficient解释

t-test的结论解释

Assumption violations

Heteroskedasticity

unconditional vs conditional

Effect

不影响

点估计结果b

参数估计的consistency

影响

区间估计t·Sb

检验

t-stat.--Sb

MSE偏小, P(type I error)上升

MSE偏大, P(type II error)上升

F stat.--MSE

MSE偏小, P(type I error)上升

MSE偏大, P(type II error)上升

Detecting

scatter plot

BP test

Ho: no Heteroskedasticity

Chi-square test: BP=n·R（residual）^2

Chi-square distribution, df=k, 单尾

Correcting

roubust standard errors(White-corrected standard error)

generalized least squares

Serial correlation(autocorrelation)

Postive vs Negative SC

Effect

不影响

点估计结果b

参数估计的consistency

影响

区间估计t·Sb

t/F检验结果

正SC，MSE偏小, P(type I error)上升

负SC，MSE偏大, P(type II error)上升

Detecting

scatter plot

DW test

Ho: no SC/ no positive SC

DW=2·(1-r)

a, k, n, decision rule

Correcting

Hansen method(roubust standard errors) ：注意与怀特法的区别

generalized least squares

Multicollinearity

matter of degree rather than absence or presence

Effect

不影响

参数估计的consistency

影响

点估计结果b

区间估计t·Sb

t/F检验结果

Multicollinearity，MSE偏大, P(type II error)上升

Detecting

Classic method: insignificant t+ significant F+ high R-square

Occasionally suggested method: r>0.7

Correcting: exclude one or more regression variables

Model misspecifications---inconsistent

包括

incorrect set of variables

incorrect regression equation's functional form

principle of model specifications

economic reasoning

nature of variable

parsimonious

examined for violations

useful out of sample

分类

Misspecified functional form

omitted variable

inappropriate variable scaling

inappropriate data pooling

Time-series mis specifications

lagged dependent variable as independent variable with serially correlated errors

function of a dependent variable as an independent variable

independent variables are measured with errors

Other types of time-series misspecifications(nonstationary)

relations among time series with trend

relations among time series that may be random walk

Qualitative dependent variable

probit and logit model

discrimination model: Z-score

Time series analysis

Trend model

linear trend model

Yt等差，Y changes at a constant amount: b1

散点趋近于直线

log-linear trend model

Yt等比，Y grows at an exponential rate: e^b1-1

散点呈指数趋势

limitation

time series data usually exhibit serial correlation, not appropriate for trend model.

Autoregressive model(AR)

multiperiod forecast: chain rule

Assumptions of AR

covariance stationary

强平稳vs弱平稳

3 conditions for covariance stationary

expected value constant and infinite over time

variance

covariance

性质

stationary past doesn't gurantee stationary in the future

covariance stationary time series has a finite mean-reverting level Xi=B0/(1-B1)

违反的Effect：unit root/B1=1/random walk

Random walk

Random walk with drift

features

no mean reverting level

infinite variance

Detecting

Unit root test of nonstationary: common t-test, Ho: B1=1

Dickey-Fuller test

Xt-Xt-1=bo+(b1-1)Xt-1+€

g=b1-1, Ho:g=0, Ha:g<0

revised t-table查表

Correcting：first-difference

errors are uncorrelated-违反-autocorrelation

Effect

不影响

点估计结果b

参数估计的consistency

影响

区间估计t·Sb: MSE---Sb

t/F检验结果

正SC，MSE偏小, P(type I error)上升

负SC，MSE偏大, P(type II error)上升

Detecting

DW不可用，因为误差项相关性应该为0，是显著性检验

T-test：Sr=1/(观测值个数）^0.5 观测值个数=样本容量-p df=T-k-1 reject Ho, r<>0, 存在autocorrelation

Correcting

adds seasonal lag

Homoskedasticity：ARCH

Effect

不影响

点估计结果b

参数估计的consistency

影响

区间估计t·Sb

检验

t-stat.--Sb

MSE偏小, P(type I error)上升

MSE偏大, P(type II error)上升

F stat.--MSE

MSE偏小, P(type I error)上升

MSE偏大, P(type II error)上升

Detecting

ARCH(1)

significance test for a1

t分布

Correcting: GLS

More than one time series

Cointegration

DF-EG test: reject Ho, Cointegration, can use multiple regression

Comparing Model Performance

定量

in-sample forecast errors

out-of-sample forecast errors:RMSE

定性

instability of regression coefficient

data from earlier and later

shorter and longer period of data

Quantitative Method (2)

Machine learning

machine learning V.S statistical approach

传统统计学需要假设分布

数据大小

线性/非线性

数据复杂度（维度）

X，Y的名称

hyperparameters

Types

Supervised learning

labeled training data

分类

regression model: continuous target variable

classification model

binary classification

multicategory classification

Unsupervised learning

Unlabeled data

分类

Dimension reduction

clustering

Deep learning & reinforcement learning

适用于supervised&unsupervised

based on neural network

deep learning: used for complex tasks

reinforcement: learn from its own prediction errors

overfitting

issue with Supervised Machine Learning

three nonoverlapping data sets

traning sample

validation sample--tuning

test sample--evaluate

three errors

bias error: 样本内误差, training data not fits model well, underfitting, high in sample error

variance error: 样本外误差, overfitting, high out-of-sample error

base error: residual errors, not preventable

fitting curve: optimal complexity of model

addressing method

complexity reduction: overfitting penalty

cross validation

in cross validation

k-fold cross validation

Supervised Learning Algorithms

Penalized regression-regression/continuous

Penalty term: LASSO v.s OLS--linear

Regularizition: applied to non-linear model

Support vector machine(SVM)-classification/distinct

机制: linear, 二分法, hyperplane, maximum margin, support vector, discriminant boundary

分类

hard margin: linear classifier

soft margin: not perfectly linear, trade-off between wider margin and classification error

适用：small-to-median size & complex high-dimensional data

K-nearest neighbor(KNN)-classification/distinct

机制：linear, classify a new observation by finding similarities, 少数服从多数

two concerns

hyperparameters k

k too small, high error rate

k too large, dilute the result by averaging

k is even, may no clear winner

hard to clearly define "similar"

适用：二分法/多分法

Classification and regression tree(CART)-regression&classification

机制

linear & non-linear

classification tree-categorical target variable regression tree-continuous target variable

no black box

decision tree

features, branches, cut off value

initial root node: widest separation, minimize classification error

decision node: lower within-group error

terminal node: classification error does not diminish much more from another split if Classification----majority of data points if regression----mean of labelled values

优缺点

优点: provide visual explanation

缺点: overfitting；to avoid

regularization

prune low explainatory power section

Ensemble and random forest-组合算法

ensemble learning

aggregation of heterogeneous learners

aggregation of homogenous learners: different traning data---bootstrap aggregating(bagging)反复抽样

random forest

variant of classification tree + data bagged from same data set

subset of features used in creating each tree---mitigate overfitting

determine final classification: wisdom of crowd

优点

protect against overfitting

reduce the ratio of noise to Signal--errors cancel out via different trees

缺点：black box

Unsupervised Learning Algorithms

Dimension reduction: principal components analysis

composite variable, Eigenvectors, Eigenvalue(RSS/TSS)--avoid multicollinearity

优缺点

优点：features减少，avoid overfitting

缺点：Eigenvectors是原来features组合而来, aren′t well-defined concept, could be perceived as black box

Clustering

k-means clustering

机制: hyperparameter k, k non-overlapping clusters, centroid

适用

very large data sets

high dimensional data

缺点

the choice of hyperparameters k affects outcomes

解决: using a range of values for k to find optimal number of clusters

hierarchical clustering

no predefined number of clusters

agglomerative (bottom-up) clustering

divisive (top-down) clustering

Neutral Networks

机制

artificial neural networks(ANN)

high dimensional data/ linear&nonlinear data

three types of layers

input layers: features

hidden layers: ways of transmitting data

output layer: one predict result

hyperparameters of 4-5-1

each node

summation operator---total net input

activation operator

transform total net input into final output of the node

light dimmer switch---decrease or increase the strength of input

non-linear & linear

neurons modelling

input, synaptic weights, bias term, total net input, summation operator,active function, output

forward propagation向前计算

修正调试

backward propagation: 向后计算, 调整synaptic weights

revision of hyperparameters based on out of sample performance

Applications

deep neural networks(DNNs)

more than20 hidden layers

useful in general for image, pattern and speech recognition

reinforcement learning: learns based on immediate feedback from(millions of) trials and errors-AlphaGo

Choice of ML algorithms

if data is complex(too many features)

yes

dimension reduction

no

if classification

yes

if supervised

yes

linear: KNN, SVM

nonlinear: CART, random forest, neural networks

no

linear: k-means clustering or hierarchical clustering

nonlinear: neural networks

no

linear: penalized regression

nonlinear: CART, random forest, neural networks

Big data projects

introduction

characters: volume, variety, velocity, veracity(validity), value

data analysis steps: conceptualize model task, data collection, data preparation and wrangling, data exploration, model training

structured data

1. conceptualize task/blueprint/modifiable plan

2. data collection

external data

access through API(application programming interface)

vendor: csv or other formats

internal data

3. data preparation and wrangling

data preparation(cleansing)

incompleteness error

invalidity error

inaccuracy error

inconsistency error

non-uniformity error

duplication error

outliers

trimming(truncation)

winsorization:replace with maximum or minimum value

data wrangling(preprocessing)

transformation

extraction: 生日-年龄

aggregation: salary+ revenue=total income

filtration: data rows that are not needed

selection: columns are not needed, eg.name和ID只需一个

conversion: CAD-USD

scaling

normalization

公式：归一化

优: used when data distribution unknown

缺: sensitive to outliers

standardlization

公式

优: less sensitive to outliers, as it depends on mean and standard deviation

缺: data must be normally distributed

4. data exploration

exploratory data analysis(EDA)

summary statistics

visualization

feature selection

feature engineering

5. model traning

method selection

performance evaluation

error analysis

confusion matrix

precision, recall, accuracy, F1 score

receiver operating characteristics(ROC)

shape of ROC curve

more convex curve-better

area under the curve(AUC): 0.5 random guessing

root mean square error(RMSE)-useful for regression model

model tuning

minimize total aggregate error

parameters&hyperparameters

altering the hyperparameters

each hyperparameter---confusion matrix

multiple hyperparameters

grid search: different combinations of hyperparameters

ceiling analysis: the part of the pipeline can potentially improve the performance

unstructured data

3. text preparation and wrangling

text preparation(cleansing)

remove HTML tags

remove punctuations: some need replaced with annotations

remove numbers

remove white spaces

text wrangling(preprocessing)

normalization

lowercasing

removal of stop words

stemming

lemmatization

bag-of-words(BOW) procedure: N-grams

document term matrix(DTM)

4. text exploration

EDA

text statistics: term frequency, co-occurrence

visualization

feature selection

reduction in BOW size

methods

document frequency(DF)

Chi-square

mutual information: MI=1, token更有辨识度

feature engineering

number

n-gram

name entity recognition(NER)

parts of speech(POS)