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個體計量 - Coggle Diagram

- - - - bivariate normal distribution
      - inverse mill's ratio
        
        pdf(z'delta)/cdf(z'delta)
        
        E(eta|di=1)
      - yi = x'b + lambda(covariance)+vi
  - - - profit selection
      - logit selection
      - endogenous switch regression
        
        different kind of outcome depends on selection equation
        
        compared to matching (with CIA) : with joint distribution
- - - - 動差條件
      - H0: E(mi(\theta))=0
      - number of L>#K
      - 卡方分配
      - property
        
        consistent
        
        asymptotically normal
    - - X1 is not orthogonal to \epsilon
        
        selection biaes
        
        measurement error
      - condition
        
        L>=K
        
        cov(zx)!=0
        
        con(z\epsilon)=0
      - method
        
        GMM: \hat{beta}=(Z'X)^-1(Z'Y)
        
        2SLS: \hat{beta}=(\hat{X}'\hat{X})^-1(\hat{X}'Y)
      - hausman test
        
        拒絕H0時用IV
    - - 假設母體服從的分配：mean, variance
      - maximizing jointed probability
      - y is i.i.d.
      - property
        
        consistent
        
        asymptotically normal
        
        asymptotically efficient
  - - - \epsilon is i.i.d., LSDV or FE
    - - 拒絕H0時，ci is not orthogonal to x, use FE
      - always use FE first
    - - between variation
      - low variance of x may leed to insignificant since the variance of beta hat increase
      - estimate Ci in fixed model
        
        take Ci as dummy
        
        LSDV, get \beta FE
        
        then y-(y hat FE), regress residuals on Ci
      - 無法做out of sample prediction
    - - vs OLS, as GLS is fixing 變異數齊一性
    - - ((t=1,T=1)-(t=0,T=1)) - ((t=1,T=0) - (t=0,T=0))
      - issues
        
        y的趨勢像不像
        
        time window, donut domain
        
        whether people easily move to the other group or not
- - - - E(yi)=xi'\beta
      - P(yi=1)-E(yi)
      - \hat{y} could be outside of [0,1]
      - beta is consistent but not efficient
        
        hetroskedasticity
    - - Probit
        
        隨機干擾項服從常態分配
        
        E(yi)=cdf（x'b）
        
        use MLE to find joint probability
        
        choose \beta to maximized joint probability
      - Logit
        
        隨機干擾項分配與probit不同
        
        as probit
      - beta as marginal effect
        
        d P(y=1)/ dx
        
        pdf(x'b)b
      - MLE is more efficient than GMM
        
        GMM have distribution assumption in this field
    - - outcome(y*)的機率分配
      - 假設y=0的機率分配
      - 選0的機率為一塊面積，while>0是density
      - non-linear model
        
        MLE:choose beta to maximized joint probability
    - - 對選擇模型中的隨機干擾項假設(1st part probability)
      - regardless of yi<0(2nd expectation)
      - vs
        
        Tobit: no distribution assumption on outcome
        
        simultaneous equation: no exclusive variable
        
        selection bias: ignore y<0, which have meanings
  - - - as standard FE,RE
    - - probit
        
        FE, beta inconsistent
        
        yi* is unobservable
        
        can not remove Ci by differencing
        
        T不夠，C inconsistent，影響beta
        
        RE
        
        condition on Ci
      - logit
        
        FE
        
        condition on summation yit
        
        given 該item 為哪一種狀況的機率，在與其他item 連乘
        
        每個人全部要不要做的機率（compared to probit每個人各期要不要做的幾率）
        
        RE
        
        as probit
- - - - v(beta) increase