ECON 5420 - Final

discrete distribution

one of a countable number of events

continuous distribution

one of an uncountably infinite set

data-generating process

the stochastic mechanism that generates the observed sample

likelihood

L(θ|D) = f(D|θ)

f is a joint density for continuous data

f is a joint mass function for discrete data

DGP interpretation

parameters are fixed and data are random

likelihood analysis interpretation

observed data are fixed and we compare candidate parameter values

maximum likelihood estimator

choose parameters that maximize the plausibility of the observed sample under the assumed probability model

joint likelihood for continuous variables

L(θ|D) = Π (N, i=1) f(Di|θ)

joint likelihood for discrete variables

L(θ|D) = Π (N, i=1) P [Di=di|θ]

steps in maximum likelihood estimation

1. gather data (independent draws)

2. construct the likelihood using the appropriate formulation

3. maximize the likelihood (typically via log-likelihood)

monotonic transformations preserve

the location of extreme points (but not their values)

key properties of log-likelihood

products become sums

exponents become coefficients

for a local extreme value: df/dx = 0

under correct specification and standard regularity conditions, many familiar OLS properties are

special cases of general MLE results

• consistency

• asymptotic normality

• asymptotic efficiency within regular parametric models

properties of maximum likelihood estimation (MLE)

• consistency

  • the MLE converges to the true parameter value with probability as sample size grows

• asymptotic efficiency

  • MLE achieves the lowest possible variance among unbiased estimators in large samples

• pseudo-true parameter under misspecification

  • even with model misspecification, MLE converges to a meaningful parameter value

• asymptotic normality

  • allows for hypothesis testing and confidence intervals using normal approximations

• transformation invariance

  • natural plug-in estimators for transformed parameters

functional invariance interpretation

once we estimate the primitive parameters, many economically interesting objects are just plug-in transformations of θhat

OLS applies directly only when

the model is linear in parameters and has an additive error term

MLE lets us estimate models that are

nonlinear in parameters, nonlinear in probabilities, or based on non-Gaussian outcomes

hypothesis testing with MLE (single parameters)

• use the large-sample normal approximation for θhat

  • this justifies estimated standard errors, test statistics, and confidence intervals

  • results are asymptotic approximations, not exact finite-sample properties

functions of parameters

• use plug-in estimates together with delta method or bootstrap

  • transformation invariance tells us the natural plug-in estimator

multiple hypothesis tests

many ways to conduct multiple hypothesis tests with MLE

pseudo R²

• compares simple benchmark with richer model using likelihoods

• formula: 1 - [ln(Lu)/ln(L0)] where: Lu = unrestricted model being estimated and L0 = simple model with only intercept

if the true model is simple

L0 = Lu, so pseudo R² = 0

if the richer model fits better

Lu > L0, pseudo R² rises above 0

two types of binary outcomes

unconditional probability and conditional probability

linearity probability model (LPM)

• model: yi = xi’β+ui where yi ε {0,1}

• fitted conditional mean: xi’βhat

• core limitation: linearity can generate predicted values outside [0,1]

error structure problems in a LPM

• errors are definitely not normal

• error distribution depends on predicted probability values

• cannot be treated as random normally-distributed disturbances

a generalized linear model connects a linear index to the conditional mean through

a link function

in the LPM, the link is

the identity: pi = xi’β

• generally we pick a monotone function g such that g(pi) = xi’β

logit

g(p) = ln (p/(1-p))

probit

g(p) = ϕ^-1 (p)

latent index

define an unobserved latent variable: yi* = xi’β + ui

• not directly observed: the latent index exists only in the model

• interpretation: represents net utility, net benefit, or underlying propensity to choose

• threshold rule: the binary outcome records whether this latent variable crosses a threshold

threshold mapping rule

yi = {1 if yi*≥0, 0 otherwise

properties of the logit model

• very easy to compute numerically (closed-form expression)

• easy to add other terms (not just binary logit)

• because only the ratio β/σu is identified in the latent-index model, normalizing the error scale is essential

logit model

assume the error ui follows a standard logistic distribution. the logistic CDF is:

F(z) = (1/1+exp(-z))

probit model

assume the error ui follows a standard normal distribution. the standard normal CDF is:

F(z) = ∫z,-∞ ((1/√2π)exp(-0.5x²)dx = ϕ(z)

outside option

the alternative of purchasing nothing has normalized mean utility of 0

multiple inside goods

allows multiple products indexed by j in addition to the outside option

normal shocks

this gives multinomial probit, which is flexible but usually requires simulation

type 1 extreme value shocks

this gives a closed-form multinomial logit choice probability

poisson regression

we usually specify the conditional mean as

ln(Ε[yi | xi, θ]) = ln(λi) = xi’θ

in the linear fixed-effects model, we remove the individual effect by

demeaning or differencing

• the within estimator is consistent for β under standard strict-exogeneity conditions

• but the estimated λi values themselves are based on only T observations per person

• with small T, those individual effects are noisy

for random effects we need

σ²v: the variance of the individual time period shocks

σ²λ: the variance of the individual fixed effects in the population

T: number of time periods

for continuous random variables, a probability density function describes

how probability is distributed over the support

• derivative of a CDF at a given point

for a discrete random variables, a probability mass function is

the probability that a specific number is drawn

the support of a distribution

the range of possible values

censored distribution

the unit is observed, but the realized value is only known up to a threshold

truncated distribution

the unit is absent from the observed sample whenever its latent value falls outside the admissible range

tobit

censoring plus a continuous latent outcome

adverse selection

one side of the market has private information

Heckman two-step correction (heckit)

step 1: selection equation

• d*i = zi’λ + vi with observed participation indicator: di = 1[di* ≥ 0]

step 2: outcome equation

• yi = xi’β + ui