CFA数量分析（1）

数量分析（1）的内容，包括线性回归、多元线性回归、时间序列。 后续待更新

Quantative Mthods(数量分析）

• 传统(study session 2)
  • Regression
    • Linear regression
      • Model
        • 模型构成要素
          • varibles(变量）
            • corss-sectional data
            • time-series data
          • regression coefficients（回归系数）
            • 求解b1后，用均值（X,Y)求解b0
            • 估计回归系数
              • linear least squares(最小二乘法）
                • 最小值：
                  • (dependent variable – predictedvalue of dependent variable)2
              • estimated parameters or fitted parameters：
                • Note that we never observe the population parameter values b0 and b1 in a regression model.
          • error term（残差）
        • excel模型
      • Assumptions（6条）
        • 1.Y,X 是以b0,b1为参数的直线（X is linear in the parameters b0 and b1），不排除X多次幂
          • • 不可采用线性回归
          • • 可以采用线性回归分析
        • 2.X, is not random
        • 3.
          • 误差项ε是一个期望值为零的随机变量
        • 保证计算出正确的b0,b1
        • 4.
          • 同方差假设(homoskedasticity assumption)
        • 5.
          • 误差项ε相互独立
        • 6. ε, 呈正态分布( normally distributed)
        • 检测是否有特定的估计参数
      • SEE(Sdandard Error of Estimate)
        • 衡量回归模型对变量关系描述的准确程度
      • The coefficient of determination(决定系数)
        • 单变量
          • 相关系数的平方，r^2
        • 多变量
          • 决定系数的逻辑
          • total variation=
            • RSS(SSr):sum of squares of regression
          • unexplained variation=
            • SSe:sum of squares of error
          • Total variation = Unexplained variation + Explained variation
        • 决定系数越大，拟合效果越好
      • Hypoththesis testing
        • H0:ρ=0，H1:ρ≠0.H0:null hypothesis,H1:alternative hypothesis
        • confidence interval
          • • 置信区间外，拒绝H0。
        • t-test
          • • reject H0
              • t-value 越大越好
            • we can reject the hypothesis that the true parameter is equal to 0 at the 0.5 percent significance level (99.5 percent confidence).
        • p-value
          • The p-value is the smallest level of significance at which the null hypothesis can be rejected.拒绝H0的最小值。
          • p-value 越小越好。通常的参考值是0.05
          • 显著性水平（significance level ）是估计总体参数落在某一区间内，可能犯错误的概率，用α表示。——百度百科
          • error type
            • Analysts often choose the 0.05 level of significance, which indicates a 5 percent chance of rejecting the null hypothesis when, in fact, it is true (a Type I error)
      • Analysis of variance(ANOVA)
        • Analysis of variance (ANOVA) is a statistical procedure for dividing the total variability of a variable into components that can be attributed to different sources.
        • F-test
          • The F-statistic tests whether all the slope coefficients in a linear regression are equal to 0.
          • H0: b1 = 0 ， Ha: b1 ≠ 0
          • F越大越好
        • 子主题
      • Prediction Intervals
        • two sources of uncertainty
          • the error term itself contains uncertainty.
          • estimated parameters
      • limitations
        • parameter instability
          • regression relations can change over time, just as correlations can.
        • public knowledge
          • public knowledge of regression relationships may negate their future usefulness.
        • assumptions are violated
          • hypothesis tests and predictions based on linear regression will not be valid
    • Multiple Linear Regression
      • Introduction
        • t-test
        • ANOVA
          • two types of uncertainty:
            • SEE(standard error of estimate):uncertainty in the regression model itself
            • b0,b1: uncertainty about the esti mates of the regression coefcients.
          • • 随着自变量Xi的数量的增加，R^2会增加，R^2的可靠性降低，此时要对照adjusted R^2来看
        • Dummy Variables
          • Dummy variables in a regression model can help analysts determine whether a particular qualitative independent variable explains the model’s dependentvariable.一个定性的自变量能否解释因变量
          • 值(0,1)
          • 要在n个分类中确认，需要有n-1个虚拟变量
          • 截距表示被省略的分类X对应的Y平均值,斜率表示每个分类对Y的的增量效果（incremental effect ）
        • 与一个变量的线性回归类似
      • Asumptions and Violations
        • Asumtions
          • A linear relation exists between the Xj and Y.
          • Xj are not random;no exact linear relation exists between Xj,Xk
          • 同方差性
          • ε is uncorrelated across observations.
          • ε is normally distributed
        • Violations
          • heteroskedasticity
            • 误差项方差与观测值方差不同
            • the variance of the errors differs across observations
            • no conditional heteroskedasticity
            • conditional heteroskedasticity
            • Breusch–Pagan test
          • serial correlation（autocorrelated）
            • 误差项与观测值相关
            • regression errors are correlated across observations
              • 前一期间的因变量当下一期间的自变量 lag value
            • Positive serial correlation
              • 方差会减小
              • t-statistics：inflates
              • F-statistic：inflates
            • Durbin–Watson statistic （DW)
              • DW=2*(1-r)
                • DW的值介于0-4
              • 参考值:DW=2,
              • DW偏离DW=2太远，表明有序列相关问题
          • multicollinearity
            • 两个自变量之间存在线性关系
            • 1个或多个自变量X存在高度相关性
              • 不是完美相关（not perfectly）
            • t-statistics：不显著 t值小
            • F-statistic：显著，F值大
              • F检验与t检验宝墩，F大t小
            • 决定系数R^2会增大
            • 单个斜率系数的方差会增加，总体方差减小
      • Model Specification misSpecification
        • Model Specification
          • cogent economic reasoning
            • The model should be grounded in cogent economic reasoning
          • functional form .(LN,对数化)
            • The functional form chosen for the variables in the regression should beappropriate given the nature of the variables.(LN,对数化)
          • parsimonious（简约)
            • The model should be parsimonious（简约).
              • 小X，大Y，见微知著。
          • assumptions violations
            • be examined for violations of regression assumptions before being accepted.
          • useful out of sample
            • The model should be tested and be found useful out of sample before being accepted.
        • misSpecification
          • functional form
            • variables could be omitted
            • variables may need to be transformed
            • pools data from different samples
          • X correlated with the error term
            • estimated regression coefcientsto be biased and inconsistent
          • time-series misspecifcation
            • lagged dependent variables as independent
            • including a function of dependent variable as an independent variable
            • independent variables that are measured with error
      • qualitative dependent variable
        • Probit models
          • based on the normal distribution
        • logit models
          • based on the logistic distribution
        • discriminant analysis
  • Time Series
    • Trend Models
      • linear trend
        • • 随时间增长固定的金额
      • log-linear trend
        • have exponential growth(指数式增长）
          • 以固定增长率稳定增长
          • 随时间增长固定的比率
        • • In contrast, the linear trend model (Equation 1) predicts that ytgrows by a constant amount from one period to the next.
        • predicted trend value of yt
        • 增长率
          • continuous compounding连续复利
        • 线性趋势回归会有回归误差与观测值相关的问题，log会修正一些，但并没有解决。
      • Testing for Correlated Errors
        • DW-test
          • H0:没有序列相关问题，
      • 使用趋势模型的前提是协方差平稳，协方差不平稳，则模型是无效的。
    • Autoregressive (AR) Time-Series Models
      • • 0<b1<1的时候，X的均值趋向于常数，误差项满足相关条件的时候属于协方差稳定序列b1=1的时候，X属于随机游走序列 b1=0的时候，相当于对随机游走序列进行一阶差分后的协方差稳定序列b1>1的时候，属于协方差不稳定序列，稳定增长的销售业绩属于此类序列
      • We must assume that the time series we are modeling is covariance stationary
        • 以前看过一个介绍时间序列平稳性的帖子（跟楼主的问题不符），回忆如下：假设你看到两个酒鬼（即两个随机游走序列）四处流浪，醉鬼相互不认识（即他们是独立的），所以他们的路径之间没有任何有意义的关系。但假设这两个随机游走序列是醉鬼与他的狗，这时尽管每个单独的路径仍然是一个不可预知的随机游走过程，然而醉酒和狗两者之间的距离是可预见性。例如，如果狗远离于他的主人，狗会倾向于朝他的方向移动，所以这两个随机游走序列有接近的趋势）。醉鬼和他的狗组成了一对协整序列。如果两个非平稳的时间序列某些线性组合是平稳的，则可以说这两个序列具有协整关系。然后，我们就可以探索序列之间的长期均衡关系了。 作者：史晓斌链接：https://www.zhihu.com/question/21982358/answer/21491673来源：知乎著作权归作者所有。商业转载请联系作者获得授权，非商业转载请注明出处。 人大多时候心情舒畅，平静，与人合平共处，此时这个人的状态就像经济学中的均衡，时间序列的平稳性，这表现在股票价格上就是股票处于盘整状态。人有时会失去理智，情绪失控，其行为就是异常行为，此种状态下的人就是经济学中的不均衡，时间序列的非平稳性，这表现在股票价格上就是股票处于疯狂上涨与下跌中。如果情绪极度失控，价格时间序列就会呈现，高波动性与脱离时间序列平衡性的现象。 作者：黄勇链接：https://www.zhihu.com/question/21982358/answer/36915898来源：知乎著作权归作者所有。商业转载请联系作者获得授权，非商业转载请注明出处。
      • Covariance-Stationary Series
        • • the expected value of the time series must be constant and finite in all periods
        • • t = 1, 2, …, T; s = 0, ±1, ±2, …, ±Twhere λ signifies a constant
          • the covariance of the time series with itself for a fixed number of periods in the past or future must be constant and finite in all periods
          • the variance of the time series must be constant and finite in all periods
        • 怎么看是不是协方差平稳？直接看图，如果图显示出相同的均值和方差
      • 所有滞后值的自相关系数=0
        • For a specific autoregressive model to be a good fit to the data, the autocorrelations of the error term should be 0 at all lags.
    • A random walk
      • 前值+不可预测的随机项
        • revious period plus an unpredictable random error
      • 不协方差平稳
        • If the time series is a random walk, it is not covariance stationary
      • 带漂移的随机游走
        • A random walk with drift is a random walk with a nonzero intercept term.
      • 有单位根
        • All random walks have unit roots.
          • 如果一个时间序列有单位根，则不可能协方差平稳
      • 单位根的处理
        • first-differencing the time series;（一阶拆分），对一阶拆分后的序列做自回归估计。
    • Moving-Average Time-Series Models
      • 移动平均
        • • 滞后于实际，起到平滑数据的作用（如平滑季节性波动）
          • 因为滞后的原因，不能起到预测的效果
      • • MA(1) model
      • • MA(q)：A qth order moving-average model
        • ts first q autocorrelations are nonzero while autocorrelations beyond the first q are zero.
    • ARMA models
      • autoregressive moving average models
      • the parameters in ARMA models can be very unstable;
      • determining the AR and MA order of the model can be difficult;
      • ARMA models may not forecast well
    • ARCH
      • Autoregressive conditional heteroskedasticity model自回归条件异方差模型
      • If the coefficient on the squared residual is statistically significant, the time-series model has ARCH(1) errors
      • • if a time-series model has ARCH(1) errors
    • 多元时间序列问题
      • 所有时间线都没有单元根，回归可用
        • If neither of the time series has a unit root, then we can safely use linear regression.
      • 仅有其中一个时间序列有单元根，回归可不可用
        • If one of the two time series has a unit root, then we should not use linear regression
      • 所有序列都有单元根，且时间序列协整，回归可用
        • If both time series have a unit root and the time series are cointegrated, we may safely use linear regression
      • 所有序列都有单元根，且时间序列不协整，回归不可用
        • however, if they are not cointegrated, we should not use linear regression
      • (Engle–Granger) Dickey–Fuller test协整检验
        • The (Engle–Granger) Dickey–Fuller test can be used to determine if time series are cointegrated
    • 时间序列的一些问题
      • 协方差平稳会形成均值回归mean reverting
      • 比较不同回归模型的准确性
        • The root mean squared error (RMSE)均方根误差：误差平方并均值的平方根
          • 越小越好
      • 时间序列模型的参数会不稳定，在使用时间序列模型用于估计时，要检测时间序列是否稳定。
    • 时间序列预测的步骤
      • 理解你遇到的投资问题，选择一个初始的时间序列模型
        • regression model
          • 用一个变量预测另一个变量
        • time-series model
          • 用同一个变量的前期数据预测该变量
      • 如果使用时间序列模型，先画图看看是否是协方差平稳
        • 不含
          • 线性趋势a linear trend
          • 指数趋势an exponential trend
          • 季节性seasonality
          • 样本区间内数据出现闲置的偏离，均值或者协方差的的变化a significant shift
      • 步骤
        • 画图检查线性趋势或指数趋势是否最合理的
        • 估计趋势参数
        • 计算剩余residuals
        • Durbin–Watson statistic检测序列相关问题
          • 若不存在
            • 模型可用
          • 若存在
            • 使用自回归模型autoregressive model
      • autoregressive model
        • 协方差平稳的处理violations of stationarity
          • a linear trend,
            • first-difference the time series（一阶差分）.
          • exponential trend
            • take the natural log of the time series and then first-difference it 对数化时间序列，然后一阶拆分
          • shifts significantly during the sample period
            • estimate different time-series models before and after the shift
          • significant seasonality
            • include seasonal lags
        • 自回归模型的构建
          • Estimate an AR(1) model
          • Test to see whether the residuals from this model have significant serial correlation。不存在序列相关问题，则AR(1)可用。
          • 如果存在序列相关问题，使用AR(2)进行进一步估计，重复前述步骤。一直到不存在序列问题为止。
          • 检查季节性问题
            • 方法一：画图观察
            • 方法二：Examine the data to see whether the seasonal autocorrelations of the residuals from an AR model are significant (for example, the fourth autocorrelation for quarterly data)
            • To correct for seasonality, add seasonal lags to your AR model. For example, if you are using quarterly data, you might add the fourth lag of a time series as an additional variable in an AR(1) or an AR(2) model.
          • 检测异方差问题 conditional heteroskedasticity
            • ARCH(1)
            • Regress the squared residual from your time-series model on a lagged value of the squared residual.
            • Test whether the coefficient on the squared lagged residual differs significantly from 0
            • If the coefficient on the squared lagged residual does not differ significantly from 0, the residuals do not display ARCH and you can rely on the standard errors from your time-series estimates.
            • use generalized least squares or other methods to correct for ARCH
      • out-of-sample forecasting performance
      • 子主题
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