Introductory Econometrics: A Modern Approach - Notes

Introduction to Econometrics

Econometrics involves developing statistical methods to estimate economic relationships, test theories, and evaluate policies.
Unlike experimental data, non-experimental data are collected without controlled conditions, making causal inference more challenging.
Economic models use mathematical equations to describe relationships, often based on utility maximization.
A key question in econometrics: Have enough other factors been held fixed to establish causality?

Basic Econometric Model

A simple econometric model can be written as:

$$ Y = f(X_1, X_2, X_3, X_4, X_5, X_6, X_7) $$

where:

( Y ) = hours spent in criminal activities
( X_1 ) = “wage” for an hour spent in criminal activity
( X_2 ) = hourly wage in legal employment
( X_3 ) = income other than from crime or employment
( X_4 ) = probability of getting caught
( X_5 ) = probability of conviction if caught
( X_6 ) = expected sentence if convicted
( X_7 ) = age
The function ( f(.) ) must be specified before an econometric analysis can proceed.
The error term ( u ) captures unobserved factors affecting ( Y ).

Econometric Model of Crime

A typical econometric model formalizing the economic model of crime:

$$ \text{crime} = \beta_0 + \beta_1 \text{wage} + \beta_2 \text{othinc} + \beta_3 \text{freqarr} + \beta_4 \text{freqconv} + \beta_5 \text{avgsen} + \beta_6 \text{age} + u $$

where:

crime = measure of frequency of criminal activity
wage = wage from legal employment
othinc = other sources of income
freqarr = frequency of arrests
freqconv = frequency of convictions
avgsen = average sentence length
age = individual’s age
( u ) = error term capturing unobserved variables

Dealing with the Error Term

We can never completely eliminate ( u ), making its treatment crucial in econometrics.
The error term contains factors like unobserved individual traits, omitted variables, and measurement errors.

Types of Data in Econometrics

1. Cross-Sectional Data

Observations taken at a given point in time across different individuals, households, firms, cities, etc.
Assumed to be randomly sampled from the population.
Potential issues include sample selection bias and spatial correlation (e.g., similar businesses located near each other may not be truly independent).

2. Time Series Data

Observations of variables over time (e.g., GDP, inflation, stock prices).
Often exhibits autocorrelation, making independent observation assumptions problematic.
Should be stored in chronological order.

3. Pooled Cross Sections

Combination of two or more cross-sectional datasets at different time points.
Each sample is drawn independently in different years.

4. Panel Data (Longitudinal Data)

Observes the same cross-sectional units (e.g., individuals, firms) over multiple time periods.
Allows researchers to control for unobserved characteristics that do not change over time.
Facilitates causal inference by analyzing changes within units.

Causal Inference in Econometrics

Key Question: Have enough other factors been held fixed to make a case for causality?
Counterfactual reasoning is central to causal inference.
Example: Estimating the effect of a job training program on wages involves comparing outcomes with and without training under identical conditions.

Simple Linear Regression Model

A basic linear regression equation:

$$ Y = \beta_0 + \beta_1 X + u $$

where:

( Y ) = dependent variable
( X ) = independent variable
( \beta_0 ) = intercept
( \beta_1 ) = slope (measuring the change in ( Y ) for a one-unit change in ( X ))
( u ) = error term

The slope coefficient ( \beta_1 ) is the ceteris paribus effect of ( X ) on ( Y ).

Ordinary Least Squares (OLS) Estimation

OLS estimates ( \beta_0 ) and ( \beta_1 ) by minimizing the sum of squared residuals:

$$ \sum (Y_i - \hat{Y}_i)^2 $$

where:

( Y_i ) = actual values
( \hat{Y}_i ) = predicted values from the regression line

Assumptions of OLS Regression

Linearity in Parameters: The relationship is modeled as a linear function of the parameters:

$$ Y = \beta_0 + \beta_1 X + u $$
Random Sampling: Data are collected through a random process.
Sample Variation in ( X ): The independent variable ( X ) must vary.
Zero Conditional Mean: The error term has an expected value of zero given ( X ):

$$ E(u | X) = 0 $$
Homoskedasticity: The variance of ( u ) is constant across values of ( X ):

$$ Var(u | X) = \sigma^2 $$

If any of these assumptions fail, OLS may produce biased or inefficient estimates.

Interpreting Regression Output

The fitted regression equation provides predicted values for ( Y ) at different values of ( X ).
R-Squared measures the proportion of variance in ( Y ) explained by ( X ):

$$ R^2 = 1 - \frac{\sum (Y_i - \hat{Y}_i)^2}{\sum (Y_i - \bar{Y})^2} $$
- ( R^2 ) close to 1: Strong fit.
- ( R^2 ) close to 0: Weak fit.

Binary (Dummy) Variables in Regression

Used to capture categorical effects (e.g., gender, participation in a program).
Example:

$$ Y = \beta_0 + \beta_1 D + u $$

where ( D = 1 ) if condition applies, otherwise ( D = 0 ).

Extensions and Considerations

Log Models: Used when relationships are multiplicative:

$$ \log(Y) = \beta_0 + \beta_1 \log(X) + u $$
Heteroskedasticity: If ( Var(u | X) ) changes with ( X ), OLS standard errors may be incorrect.
Simultaneous Equations: If ( X ) is determined by ( Y ), simple OLS is inappropriate.

Conclusion

OLS is a foundational tool in econometrics, but its assumptions must be carefully considered.
Understanding data structure (cross-sectional, time-series, panel) is critical for appropriate model selection.
The ultimate goal is causal inference, ensuring that relationships reflect actual effects rather than spurious correlations.