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ECON6300/7320 Features of microeconometrics

创建日期：2024-06-12 17:17:40

所属分类：经济Economics

标签： ECON6300

ECON6300/7320 Features of microeconometrics

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ECON6300/7320

Features of microeconometrics (1)
▶ Data pertain to firms, individuals, households, etc
▶ Focus on "outcomes", and relationships linking outcomes
to actions of individuals
▶ earnings = f(hours worked, years of education, gender,
experience, institutions)
▶ Heteroegeneity of economic subjects’ preferences,
constraints, goals etc. explicitly acknowledged (no
"representative agent" assumption)
▶ Noisy data, large samples,
▶ Economic factors supplemented by social, spatial,
temporal interdependence
2 / 44
Features of microeconometrics (2)
▶ Sources of data:
▶ Surveys (Govt/private); cross section or longitudinal (panel)
▶ Census
▶ Administrative data (by-product: Tax related, health related,
program related)
▶ Natural experiments
▶ Designed experiments
▶ Randomized trials with controls
▶ Type of data impacts method and model used in analysis
3 / 44
Features of microeconometrics (3)
▶ Measures of "outcomes"
▶ Continuous (e.g. earnings)
▶ Discrete (binary or multinomial chpoice as in discrete
choice) or integers-valued (number of doctor visits)
▶ Partially observed/censored (hours of work)
▶ Proportions or intervals
▶ Type of measure may affect the choice of model used
▶ Many types of regression models
4 / 44
Objectives of econometric models
1. Data description and summary of associations between
variables
2. Conditional prediction
3. Estimation of causal ("structural") parameters
– Inference about structural parameters and
interdependence between endogenous variables
4. Policy analysis, prospective and retrospective
– Simulation of counter-factual scenarios to address "what
if" type questions
– Analysis of interventions, both actual and hypothetical
5. Empirical confirmation or refutation of hypotheses
regarding microeconomic behavior.
5 / 44
An example of Mincerian earnings regression
lnE = β0 + β1yreduc + β3age + β3occ + w′γ + ε
1. Does this regression equation (with perhaps a small
number of regressors) provide a good fit to the sample
data? Is the fit improved by adding age2 to the regression?
[Data description]
2. Is the regression equation a good predictor of earnings at
different ages and occupations? [Conditional prediction]
3. What does the regression say about the rate of return to an
extra year of education? [Causal parameter]
4. Can the regression be used to explain the sources of
earnings differential between male and female workers?
[Counterfactual analysis]
▶ These seemingly different objectives are connected, but
may imply differences in emphasis on various aspects of
modeling
6 / 44
Regression decomposition - an example of
counterfactual analysis
▶ Consider the problem of explaining male-female earnings
differential
Y gi = β
g
0 +
∑
xikβ
g
k + ε
g
i , g = M,F
∆̂ = Y¯ F − Y¯M
= (β̂F0 − β̂M0 ) +
K∑
k=1
(β̂Fk − β̂Mk )x¯Fk +
K∑
k=1
(x¯Fk − x¯Mk )β̂Mk + R
▶ This is counterfactual analysis as it answers the question:
What if certain differentials were equalized?
7 / 44
Structural vs Reduced form models
▶ Very highly structured models, derived from detailed
specification of: underlying economic behavior; institutional
set-up, constraints and administrative information;
statistical and functional form assumptions, assumptions of
agent’s optimizing behavior.
▶ Structural models can be preferable for modelling
objectives 3-5 (causal parameters, policy analysis,
confirmation/refutation of microeconomic theory)
▶ Reduced form studies which aim to uncover correlations
and associations among variables.
▶ Reduced form models can be preferable for modelling
objectives 1-3 (data description, prediction, causal
parameters)
8 / 44
General set-up and notation
▶ Data: (y : (N × 1),X : (N × K ))
▶ A joint unknown population distribution of data: f (y,X;θ0),
where both f and θ0 are unknown
▶ Three approaches:
1. Fully parametric: assume f is given, θ0 is finite dimensional
but unknown
2. Semi-parametric: assume that θ0 is finite dimensional but
unknown we can specify some moment functions for y , e.g.
E[y |X ], or Var[y |X ] and we do not want to make
assumptions about the distribution f (.)
3. Nonparametric: assume that θ0 is infinite dimensional, and
we want to estimate the relation between y and X without
making a parametric assumption about f (.)
9 / 44
▶ θ0 : vector of mean and variance parameters in the
relationships to be estimated
▶ θ̂ : the estimator of θ0 based on sample of observations
from the population of interest.
▶ In general θ̂ ̸= θ0; (θ̂ − θ0) : sampling error has a statistical
distribution
▶ Ideally the distribution of θ̂ is centered on θ0 (unbiased
estimator) with high precision (efficiency property), and
a known distribution, to support statistical inference
(probability statements and hypothesis testing).
▶ Consistency means θ̂ p→ θ0.
10 / 44
General approach to estimation and inference
▶ Model specification and identification
▶ Which specification/restrictions are reasonable?
▶ Can the parameter θ0 be recovered given infinite data?
▶ Correct model specification or correct specification of key
components of the model given the data we have available
is necessary for consistency
▶ Qualification: All models are necessarily misspecified as
they are simplifications
11 / 44
▶ Under additional assumptions the estimators are
asymptotically normally distributed,
▶ i.e. the sampling distribution is well approximated by the
multivariate normal in large samples:
θ̂
a∼ N [θ,V [θ̂]]
where V[θ̂] denotes the (asymptotic) variance-covariance
matrix of the estimator (VCE).
▶ Efficient estimators have small variance
12 / 44
▶ In many (most) cases large sample (normal) distribution of
θ̂ is the best we can do. Hence inference on θ̂ is based on
distributions derived from the normal
▶ Test statistics based on (asymptotic) normal results include
z-test, t-test, Wald test, F-test,...
▶ Standard errors of the parameter estimates are obtained
from V̂ [θ̂].
▶ Different assumptions about the data generating process
(DGP), such as heteroskedasticity, can lead to different
VCE.
13 / 44
OLS
▶ Linear regression estimated by least squares can be
regarded as semi-parametric
▶ Goal: to estimate the linear conditional mean function
E [yi |xi ] = x′iβ = β1xi1 + β2xi2 + · · ·+ βKxiK , (1)
where usually an intercept is included so xi1 = 1.
▶ E[yi |xi ] is of direct interest if goal is prediction based on x′iβ
▶ Econometrics interested in marginal effects (e.g. price
change on quantity transacted): ∂ E [yi |xi ]∂xij = βj .
▶ The linear regression has two components, conditional
mean and the error
yi = E[ yi |xi ] + ui (2)
yi = x′iβ + ui , i = 1, . . . ,N. (3)
14 / 44
OLS (1)
▶ Recall: y : N × 1 column vector with i th entry yi , X : N × K
regressor matrix X to have i th row x′i .
▶ Convention is that all vectors as column vectors, with
transpose if row vectors are desired.
▶ In matrix notation:
y = Xβ + u
▶ The objective function is the sum of squared errors,
QN(β) = (y− Xβ)′(y− Xβ) ≡
N∑
i=1
(yi − x′iβ)2
which is minimized with respect to β
▶ Solving FOC (first oder conditions) using calculus methods
yields the OLS solution: X′(y− Xβ) = 0
▶ Matrix notation provides a very compact way to represent
estimator and variance matrix formulas that involve sums
of products and cross-products.

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