Quant Methods 2

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General Likelihood function of Binary DVs

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Logistic vs. Logit

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Cumulative normal vs. Probit

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Formula: Logistic

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Formula: Logit

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Link function for Logit Regression

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Link function for Probit Regression

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Formula for Odds in Logit Regression

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Formula for P in Logit Regression

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Odds vs. P in Logit Regression

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Marginal effects: Logit Regression

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What does it mean to identify b?

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Write b as a function of observable variables

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Central Limit Theorem

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Asymptotic Normality

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Two assumptions of OLS

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Formula for A and B

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cov(x, y) =

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E(xy) - E(x)E(y)

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Var(x) =

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E(x^2) - E^2(x)

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GLS vs. OLS

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GLS has

1. nonconstant variances

2. nonzero covariances

in the variance-covariance matrix (u)

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FGLS: Omega

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(SOLS residuals)

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FGLS: Beta

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FGLS: Fully robust sandwich FGLS variance

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FGLS: Non-robust "usual" FGLS variance

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Assumption SGLS3:

When is the "usual" FGLS variance valid?

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SIV: Visualize Z_i and Dimensions

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SIV (Simultaneous equation) vs. SOLS

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SIV: Estimator beta hat

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Assumption SIV.1

Moments condition

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Pooled OLS

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OLS on panel data

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STATA: Pooled OLS

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reg y x, cluster(id)

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STATA: Random effects

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xtreg y x, re cluster(id)

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STATA: Fixed effects

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xtreg y x, fe cluster(id)

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Pooled OLS vs. RE

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Pooled OLS assume independence between v_{it} (error terms)

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RE vs. FE

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RE only solve the problems of autocorrelation between v_{it} (due to c_i)

FE solve endogeneity as well

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Fully robust RE vs. Usual RE

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RE solve autocorrelation between v_{it}, but

Usual RE assume constant variance and independence across u_{it}

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Wide vs long form data

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Wide: Years are wide (var2000, var2001, etc.)

Long: "Year" is a column

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IIA Assumption - Independence from Irrelevant Alternatives

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An assumption of the multinomial logit model

E.g. Cars & red bus (50-50)

IIA suggests that after blue bus is introduced, the share is (33-33-33). However, the rate should actually be (50-25-25).

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The assumption that RE makes but FE does not

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E(c|X) = E(c) = 0

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FE makes fewer assumptions than RE. Why not FE all the time?

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Time-invariant characteristics.

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Pooled OLS: Assumptions

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RE: Assumptions

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RE: Omega

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E[Y_i] vs. Avg[Y_i]

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E[Y_i]: Population parameter: There is only one

Avg[Y_i]: Sample statistics: There are many

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What is sampling variance?

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Variance of sample means from REPEATED sampling

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What is a standard error?

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The standard deviation of a sample statistic