Econometrics I

econometrics

using economic theory and statistical techniques to analyze economic data

examines the relationship between two or more variables

Examples of use:

• testing of economic theories

• forecasting 

• fitting economic models to real-world data

• policy recommendations

Causality

action—> effect

• causal effects best estimated using randomized controlled experiments

Forecast

knowledge of causal relationships not necessary

Sources of data

• experimental data

• observational data

Types of data

• cross-sectional data

• time series data

• panel data

Linear regression model with a single regressor

Yi = β0 + β1Xi + ui

Interpretation of Yi = β0 + β1Xi + ui :

β1 . . . slope: by how much Y changes when X changes by one unit

β0 . . . intercept: what is the value of Y when X = 0

• An intercept does not always have a real-world meaning (like in the class size example)

• The intercept has a mathematical meaning

ui… error term: incorporates factors that influence Y but are not included in the model

ordinary least squares (OLS) estimators

values of b0 and b1 that minimize this sum

but minimizing the sum using calculus gives us explicit formulas:

βˆ 1 = ∑ n i=1 (Xi-but minimizing the sum using calculus gives us explicit formulas:

βˆ 1 = ∑ n i=1 (Xi X¯ ) (Yi Y¯ ) ∑ n i=1 (Xi X¯ ) 2 = sXY/s 2 X

βˆ 0 = Y¯ βˆ 1X

Why use OLS estimators?

• widely used

• software available

• good theoretical properties

Two Measures of Fit

R²- fraction of variation in Y explained by X

SER- standard error of regression- how far Yi typically is from its predicted value

R²

Write: Yi=Yi^+ui^
Variation of Yi: TSS= ∑ (Yi-Y)²

Variation of Yi^: ESS= ∑(Yi^-Y)²  

R² is the ratio of explained variation to total variation: R²= ESS/TSS

alterntaively, we can take variation in residuals, SSR ∑ ui^²

since TSS=ESS+SSR it is also R²=1-SSR/TSS

Properties of R²

0<=R²<=1

small R²: poor fit

large R²: good fit - X is good at predicting Y

R²= 0 when βˆ 1 = 0, ESS = 0 (i.e. fitted line is horizontal)

R² = 1 when Xi explains all variation in Yi (i.e. all datapoints lie on the fitted line)

SER

estimator of the standard deviation of ui

measure of the spread of the observations around the regression line

definition:

su^²= 1/n-2 ∑ui^²= SSR/ n-2

SER= su^=sqr(SSR/n-2)

Assumption 1

the conditional distribution of ui given Xi has mean zero

E (ui jXi) = 0

"regression line correct on average"

assumption satisfied in many cases, not satisfied in other cases

E (ui jXi) = 0 implies cov (Xi , ui) = 0

therefore cov (Xi , ui) =/ 0 implies E (ui jXi) =/ 0

if ui is correlated with Xi , assumption 1 is violated

Assumption 2

(Xi , Yi) are independent and identically distributed (i.i.d)

satisfied if observations drawn randomly from a population (in a survey)

example of non-independent data: time series

Assumption 3

Large outliers are unlikely

values outside the usual range are not very likely

outliers make OLS estimators misleading

finite fourth moments: 0 < E(Xi^4) < ∞ and 0 < E (Yi^4) < ∞ (funite kurtosis)

most distributions have finite fourth moments

source of outliers: data entry errors (typos etc.)

under homoskedasticity

βˆ 0 and βˆ 1 are efficient (have smallest variance in a class of all linear unbiased estimators)

Two sided test (T-statistic) 

t = (estimator - hypothesized value)/standard error of the estimator

p-value

probability of obtaining a statistic at least as different from the null hypothesis value as is the actual statistic

the smallest significance level at which the null hypothesis could be rejected

When to use a one-sided test?

only use one-sided test when there is a good reason

• economic theory

• empirical evidence

when unsure, use two-sided test

95% confidence interval for β1

an interval that contains the true value of β1 with a 95% probability

the set of values of β1 that cannot be rejected by a 5% two-sided hypothesis test
βˆ 1 +- 1.96SE βˆ1

If for predicted change multiply by change

Binary variable

takes on only two values, 0 or 1

test statistic: t = βˆ 1 /SE(βˆ 1 )

95% confidence interval for β1 : βˆ 1  1.96 SE(βˆ 1 

homoskedastic

if the variance of the conditional distribution of ui is constant for i = 1, . . . , n and in particular does not depend on Xi . Otherwise, the error term is heteroskedastic.

Properties of OLS estimators:

if OLS assumptions 1-3 hold, then OLS estimators are

• unbiased

• consistent

• asymptotically normal

• if in addition the errors ui are homoskedastic, the OLS estimators are also efficient (have smallest variance among all unbiased linear estimators)

Omitted Variable Bias

occurs when two conditions are true:

1. the omitted variable is correlated with the included regressor

corr(X, u)=/ 0

therefore E(ui|Xi)=/0 and the first least squares assumption is incorrect

OLS estimator is thus biased and inconsistent

2. the omitted variable is a determinant of the dependent variable

Formula: 

pxu=Corr(Xi, ui) 

suppose the second and theird least squares assumption hold

βˆ1-p→ β1 + ρXu (σu/σX)

if pxu=/0, then even in large samples βˆ1 does not converge in probability of β1

the size of the bias depends on teh size of pxu

the direction of the bias depends on the sign of pxu

Multiple Regression Model

Yi=B0+B1X1i+B2X2i+…_ui, i=1,…,n

B1…slope of coefficient on X1: by how much Y is expected to change holding X2,…, Xk constant

Standard Error of Regression for Multiple regression

SER= su^

where su^²= SSR/ (n-k-1)

k is the number of regressors excluding the constant 

division by n-k-1 is the degrees of freedom adjustment

Adjusted R Sqaured

adjusted R² = 1- (n-1/n-k-1)(SSR/TSS)

adding a regressor → two opposite effects

• SSR decreases (better fit) 

• n-1/ n-k-1 increases (because k increases)

Properties

• adjusted R²<=R²

• adjusted R² can increase or decrease with k 

• adjusted R² can be negative

• adjusted R²= 1 - su²/ sY² 

Least Squares Assumptions for Multiple Regressions

1. conditional distribution of ui given X1i, X2i, …, Xki has zero mean: E(ui|X1i, X2i,…, Xki) =0 

2. (X1i, X2i, …, Xki, Yi), i=1,…, n are iid

3. large outliers are unlikely, 0<E(X1i^4) <inf, …, 0 < E (Xki^4< inf and 0 < E(Yi^4) < inf

4. there is no perfect multicollinearity

Perfect Multicollinearity

Regressors are said to be perfectly multicollinear if one of the regressors is a perfect linear function of other regressors

Dummy Variable Trap

occurs where there are G binary variables, each observation falls into one and only one category, there is an intercept in the regression and all G binary variables are included as regressors

Imperfect Multicollinearity

two or more regressors are highly correlated (there is a linear function of regressors that is highly correlated with another regressor) 

different from perfect multicollinearity

under imperfect multicollinearity, one or more coefficients will be imprecisely estimated

px1, x2=corr(X1, X2) 

in linear regression with regressors X1 and X2, sigmaB^1²= (1/n)(1/1-p²)(sigma u²/ Sigma x²)

Properties of OLS Estimators

B^0, B^1, …, B^k are random variables

under least squares assumptions 1-4, B^0, B^1, …, B^k are unbiased and consistent

in large samples, B^0, B^1, …, B^k are jointly normally distributed and B^j ~N (Bj, sigmaBj²), j=0,…, k

B^0, B^1, …, B^k are usually correlated

F statistic

hypothesis: 

• H0: B1=0, B2=0, …, Bk=0 

• H1: Bj=/0 for at least one j, j=1,…k

under the null, none of the regressors explain Y 

also called test for significance of regression 
k restrictions (q=k) 

F~Fk,inf

Homoskedasticity-only F-statistic

F= (SSRr-SSRur)/q/SSRur/(n-kur-1) = (R²ur-R²r)/q/(1-R²ur)/(n-kur-1)

Confidence Set

a 95% confidence set for two or more coefficients is a set that contains the true population values of these coefficients in 95% of randomly drawn samples

Quadratic Regression Model 

TestScorei= B0 + B1 Incomei + B2 Incomei² + ui

model is nonlinear in variables but linear in parameters

we can test for the presence of nonlinearity formally: 

H0: B2=0 (regression is linear) 

H1: B2=/ 0 (regression is quadratic) 

Population Regression Function

E(TestScorei|Incomei)=B0+B1Incomei+B2Incomei²

population coefficients are unknown→ need to be estimated

Effect on Y of a change in a regressor

Expected change in Y associated with a change in X1, holding X2,…, Xk constant

Linear model: ∆ Y= B1∆X1

nonlinear model: ∆Y=f(x1+∆X1,X2,…,Xk)-f(X1,X2,…,Xk)

in nonlinear model, expected change depends on the value of X1,  X2, …, Xk (whereas in linear models it does not) 

Standard errors of estimated effects

Approach 1:

• compute F-statistic for testing that B1+21B2=0

• because q=1

• F=t²=(B^1+21B2^/SE(B1^+21B2^))²=(∆Y^/SE(∆Y^))²

• therefore: SE(∆Y^)=|∆Y|/sqrtF

Approach 2: 

• transform the regression model in such a way that one of the coefficients in the transformed regression is B1+21B2

• denote γ= B1+21B2

• then SE(B1^+21B2^)=SE(γ^)

sequential hypothesis testing

Step 1: pick a maximum value of r and estimate the polynomial regression of this r

Step 2: test H0: Br = 0; if this is rejected, X^r belongs to the regression, so use polynomial of degree r

Step 3: if H0: Br=0 is not rejected, eliminate X^r from the regression and estimate a polynomial regression of degree r-1; test whether Br-1=0; if this is rejected, use polynomial of degree r-1

Step 4: continue this procedure until the coefficient on the highest power is statistically significant

Important property of logarithm:

ln(x+∆x)-ln(x)~=∆x/x (when ∆x/x is small)

Logarithmic Regression Models

1. linear-log model: Yi=B0+B1+lnXi+ui

2. log-linear model: ln(Yi)= B0+B1Xi+ui

3. log-log model: ln(Yi) = B0+B1ln(Xi) +ui

all three models linear in parameters→ OLS methods can be used to estimate unknown values of parameters

Linear-Log model

1% change in X is associated with a change in Y of 0.01B1

Log-linear model

one-unit change in X is associated with a 100x B1% change in Y

Log-log model

1% change in X is associated with a B1% change in Y

Comparing Logarithmic Specifications

Adjusted R² can be used to compare log-linear and log log model

adjusted R² can be used to compare linear log and linear model

Adjusted R² cannot be used to compare linear-log and log-log models

Internal Validity

A statistical analysis is internally valid if the statistical inferences about causal effects are valid for the population being studied

• estimator of causal effect should be unbiased and consistent

• hypothesis tests should have desired significance level confidence intervals should have desired confidence level

Threats: 

• unbiasedness and consistency of B^ coefficients

  • omitted variable bias

  • misspecification of functional form

  • errors-in variables

  • sample selection

  • simultaneous causality

• correct estimation of SE (B^)

  • heteroskedastic errors

  • errors correlated across observations

External Validity

A statistical analysis is externally valid if its inferences and conclusion can be generalized from the population and settings studied to other populations and settings

Threats:

• differences in population

• differences in settings

Threat: Omitted Variable Bias

Omitted variable bias arises when omitted variable

• determines Y

• is correlated with one or more included regressors

OLS estimator is biased and inconsistent

Solutions to omitted variable bias:

1. omitted variable is observed

   1. including omitted variable reduces possible bias

   2. however if the variable does not belong (its true coefficient is zero), variance of other estimated coefficients increases

   3. variance-bias trade-off

2. omitted variable is not observed

   1. use instrumental variables (IV) regression (discussed later)

   2. use randomized controlled experiments (discussed later)

   3. use panel data (not discussed in this course)

Threat: Misspecification of Functional Form

Functional form is not specified correctly

e.g. true population function is nonlinear but the estimated regression is linear

OLS estimator biased and inconsistent

a type of omitted variable bias

Threats: Errors-In-Variables

Data on regressors may be recorded with error:

respondents in survey give wrong answers

typos

wrong data downloaded

this is called measurement error

leads to error-in-variables bias

error term is correlated with regressor→ B1 biased and inconsistent

Solutions:

get an accurate measure of X

use instrumental variables (correlated with Xi but uncorrelated with measurement error) - discussed later

develop a model of the measurement error and estimate parameters

• requires knowledge of type of measurement error

• ad hoc

• not discussed here

Threats: Sample Selection

when availability of data is influenced by a selection process that is related to the value of the dependent variable

introduces correlation between error term and regressor → selection bias

Threats Simultaneous Causality

in our models so far, X was causing Y

but the causality can also run the other way round (Y causes X)

simultaneous causality

OLS estimators biased and inconsistent

simultaneous causality bias

Solutions: 

use instrumental variables regression - discussed later

use randomized controlled experiments - discussed later

Threats: Heteroskedasticity

solution: use heteroskedasticity-robust standard error formula

Threats: serially correlated

e.g. if a school performs better than average one year, it will probably do so also next year

arises mostly in time series data and panel data

second least squares assumption violated

solution: use heteroskedasticity- and serial correlation- robust formula for standard error

Average Causal Effect

Causal effect can be different for each individual

• the effects of a drug can depend on

  • age

  • whether smoking

  • other health conditions

solution: estimate mean causal effect in population

often sufficient

Ideal Randomized Control Experiment

1. select individuals at random from population

   1. distribution of the potential outcomes and causal effects is from the same distribution as in the population

   2. so the expected value of the causal affect in the sample is the same as the average causal effect in the population

2. assign individuals randomly to treatment or control group

   1. an individual’s treatment status is independent of their potential outcomes

   2. so the expected value of the outcome for those treated minus the expected value of the outcome for those not treated equals the expected value of the causal effect

Checking for Balance

Characteristics of people in the treatment group similar to those in control group

differences not significant

experiment well designed

Notation

X_i… treatment indicator variable

• X_i=1…treatment

• X_i=0… no treatment

Y_i…observed outcome

Y_1i…potential outcome when treatment received

Y_0i…potential outcome when no treatment received

Y_1i-Y_0i…causal effect of treatment

E(Y_1i-Y_0i)…average causal effect

if observations on Y_i and X_i come from an ideal randomized control trial

• E(Y_1i-Y_0i)=E(Y_i|X_i=1)-E(Y_i|X_i=0)

Differences Estimator

the difference in sample averages for treatment and control groups

can be computed by regressing the outcome variable Y_i on binary treatment indicator X_i:

• Y_i=B₀+B₁X_i+u_i, i=1,…,n

if X_i is randomly assigned then E(u_i|X_i)=0 and the OLS estimator of causal effect B₁ is unbiased and consistent

Differences Estimator with Additional Regressors

adding regressors→ improved efficiency

including control variables W:

• Y_i=B₀+B₁X_i+B₂W_1i+…+B_1-rW_ri+u_i, i=1,…,n

variables W must be such that u_i satisfies

• E(u_i|X_i, W_i) = E(u_i|W_i)

• conditional mean independence

satisfied if W_i are pretreatment characteristics and X_i is randomly assigned

variables W_i do not have causal interpretation

Threats to Internal Validity: Failure to Randomize

treatment not assigned randomly

based on the characteristics or preferences of subject

nonrandom assignment leads to correlation of X_i and u_i → biased estimator of the treatment effect

Threats to Internal Validity: Failure to Follow the Treatment Protocol

people do not always follow the treatment

failure to follow the protocol completely: partial compliance

element of choice in whether the subject receives a treatment

X_i can be correlated with u_i even with initial random assignment

bias in OLS estimator

Threats to Internal Validity: Attrition

Subjects dropping out after their assignment

dropping out may be unrelated to the treatment program

• e.g. leave to care for a sick relative

this does not cause a bias

but dropping out may be related to the treatment

Threats to Internal Validity: Experimental Effects

merely being in the experiment can change behavior of subjects

hawthorne effect

double blind protocol can mitigate this effect

• neither the subject nor the experimenter know whether the subject receives the treatment or not

economics: double blind experiments often infeasible

• both experimenter and subject know in which group the subject is

Threats to Internal Validity: Small Sample Sizes

experiments with human subjects are expensive

sample sizes sometimes small

small size does not cause bias

but causal effects are estimated imprecisely

inference can be misleading

Threats to External Validity: Nonrepresentative Sample

population studied and population of interest must be similar

e.g. training program with former prison inmates does not generalise to workers who have never committed a crime

Threats to External Validity: Nonrepresentative Program or Policy

the policy or program of interest must be similar to program studied

example:

• program studied: small scale, tightly monitored experiment

• program implemented: scaled-up, not the same quality control, less well funded→ not as effective

another difference in programs: duration

Threats to External Validity: General Equilibrium Effects

turning a small, temporary experimental program into a widespread, permanent program might change economic environment

the results cannot be generalized

small program: internally valid, measure causal effect holding constant the market or policy environment

large program: these factors are not held