## Quantitative Methods II: Econometrics Mr. Moody

Text
Pindyck and Rubinfeld, Econometric Models and Economic Forecasts (PR) .
Kennedy, Peter, A Guide to Econometrics, Second Edition (K)

References on reserve:
Maddala, G.S. Introduction to Econometrics (M)
Kelejian and Oates, Introduction to Econometrics (KO) HB139.K44
Rao and Miller, Applied Econometrics (RM) HB74.M3R32

## Review: Correlation and Regression.

Describing the relationship between two variables

Scatter diagrams
Correlation
Simple regression

Estimation and the Theory of Least Squares
Gauss-Markov assumptions

linear function of Y
betahat is random variable with a mean and a variance
betahat is an unbiased estimator of beta
deriving the variance of beta
Gauss-Markov theorem (ols is BLUE)
ols is a maximum likelihood estimator

Properties of estimators

small sample properties: bias, efficiency, mean square error, relative efficiency, robustness
large sample (asymptotic) properties: consistency, mse, consistency, asymptotic distributions
consistency "carries over" transformations while unbiasedness does not
References: M ch 2.6, PR pp. 27-30, RM ch 3, K ch 2,3

Inference and Hypothesis Testing

Assume the error term is distributed normally, then the sampling distribution of betahat is also normal with
mean = beta (truth)
variance = var(u)/Sum of x2
however, we must estimate var(u)
testing hypotheses concerning beta
confidence intervals
testing the goodness of fit: F test
t, F, and chi-square distributions
Reference: K ch 4

Multiple Regression

Why? Because life is complicated: omitted variable bias
three variable regression model
interpretation of formulas
omitted variables and irrelevant variables
goodness of fit: R2
M ch 4, PR ch 4-5, KO ch 4

dummy variables
References: PR pp. 104-108, 121-123, M pp. 251-266, KO ch 5.2, RM pp. 88-104, 138-159, K ch 13

Useful Tests

F-test
Chow test
J-test for non-nested hypotheses
Pe test for log-linear vs linear models
Granger causality test
References: PR 110-112, 115-117, 216-219; M 329-331, 443-446

Maximum likelihood and the likelihood ratio test

References: M pp. 83-86, K ch 4.4

Digression: torturing the data until it tells you what you want to hear:
Leamer, "Let's Take the Con out of Econometrics" American Economic Review, March, 1983, 31-43.

Econometrics: What if the Gauss-Markov Assumptions are Violated?

Heteroskedasticity

Definition: nonconstant error variance, a common problem in cross sections
Tests: plot residuals, White
Effects: (1) ols estimates remain unbiased, but (2) inefficient, (3) standard errors and t-scores are incorrect
Cure: weighted least squares
1.known variances: weighted least squares
2.unknown variances: assume that the error variance is a function of an observable variable (the usual case)
White's heteroskedastic robust standard errors
References: M ch 5, PR ch 6.1, KO ch 6.3, K ch 7

Specification Bias

Rule: if one or more of the explanatory variables in a regression are correlated with the error term, the resulting ols estimates are biased and inconsistent

Causes of correlation between X and u
incorrect functional form
omitted variables
errors of measurement in the independent variables
simultaneous equations

Errors in variables

Definition
Effects: ols is biased and inconsistent
Cure: instrumental variables (two stage least squares)
Problem: choice between a biased but efficient estimator (ols) and an unbiased but inefficient estimator (IV) References: M ch 11.1-11.3, 11.5-11.7, PR ch 7

Simultaneous equations

When an equation is part of a simultaneous equation system, such that causation runs from Y to X as well as X to Y, then X is correlated with the error term and ols is biased and inconsistent.

Example: the consumption function
Endogenous and exogenous variables, structural versus reduced form
Cure: instrumental variables (2sls)
The identification problem
the order condition for identification
Types of equation systems: general, recursive, block recursive
Strategies: ols, ols with lags, reduced form, 2sls, VAR
Basmann test for over-identification restrictions
Hausman test for mis-specification
System estimation methods: ZELS, 3SLS
References: M ch 9, M ch 12.10, PR ch 11; KO ch 7, K ch 9.

## Time Series Analysis

Linear Dynamic Models

Autocorrelation

Definition: ut correlated with ut-1 (and/or ut-2, etc.)
Effects
ols remains unbiased
variance of betahat will not be minimum (loss of efficiency)
standard errors will be underestimated and t-scores overestimated
predictions will be inefficient
if regressors include a lagged dependent variable, then ols estimators will be biased and inconsistent as well as inefficient.
Tests: Durbin-Watson statistic
Breush-Godfrey (Lagrange Multiplier) test

There are two reasons for autocorrelation (1) serial correlation in the error term and (2) omitted variables with time components.

If the autocorrelation is due to omitted lagged variables, then we can't fix it with Cochrane-Orcutt. We need to test to see if we have serial correlation or mis-specified dynamics.

Testing for mis-specified dynamics: likelihood ratio test.

References: M ch 6,PR ch 6.2, RM ch 3.3, KO ch 6.2, K ch 7.4
Granger and Newbold, "Spurious Regressions in Econometrics" Journal of Econometrics 2, (1974) 111-120.

#### Analysis of non-stationary data

Random walks and unit root tests
Cointegration and long run equilibrium
Testing for cointegration
Estimating the cointegrating regression
Spurious regressions
Nonsense and unbalanced regressions
References: Granger "Introduction," P&R ch 15.3, 15.4

# Pooling Time Series and Cross Section Data

motivation: cure for one kind of omitted variable bias, efficient use of data, increases degrees of freedom
digression: estimating production functions
least squares dummy variables (LSDV) or fixed effects
variance-components or random effects
autocorrelation and heteroskedasticity
Reference: PR ch 9.4

Omitted variable bias

review: multiple regression formula
determining the direction of bias
There is only one way to be right and many ways to be wrong.
It is wrong to include an irrelevant variable (inefficiency)
and it is wrong to leave out a relevant variable (bias).
However, omitting a relevant variable whose value is less than its standard error will decrease mse's.
proxy variables, M ch 11.6
Wallace noncentral F test for mse
References: PR ch 7.3, 7.5.1; KO ch 6.4, RM pp. 29-67. Goodnight and Wallace "Operational Techniques and Tables for Making Weak MSE Tests for Restrictions in Regression." Econometrica (7/72) 699-709

Regression Diagonistics

Influential Observations

Multicollinearity

References: M ch 7, KO ch 6.1