Linear Regression with One Regression Chapter 4 Linear Regression with One Regression

Linear Regression with One Regressor (SW Chapter 4) Linear regression allows us to estimate, and make inferences about, population slope coefficients. Ultimately our aim is to estimate the causal effect on Y of a unit change in X – but for now, just think of the problem of fitting a straight line to data on two variables, Y and X.

Confidence intervals: The problems of statistical inference for linear regression are, at a general level, the same as for estimation of the mean or of the differences between two means. Statistical, or econometric, inference about the slope entails: Estimation: How should we draw a line through the data to estimate the (population) slope (answer: ordinary least squares). What are advantages and disadvantages of OLS? Hypothesis testing: How to test if the slope is zero? Confidence intervals: How to construct a confidence interval for the slope?

Linear Regression: Some Notation and Terminology (SW Section 4.1)

The Population Linear Regression Model – general notation

This terminology in a picture: Observations on Y and X; the population regression line; and the regression error (the “error term”):

The Ordinary Least Squares Estimator (SW Section 4.2)

Mechanics of OLS

The OLS estimator solves:

Application to the California Test Score – Class Size data

Interpretation of the estimated slope and intercept

Predicted values & residuals:

OLS regression: STATA output

Measures of Fit (Section 4.3)

The Standard Error of the Regression (SER)

Example of the R2 and the SER

The Least Squares Assumptions (SW Section 4.4)

The Least Squares Assumptions

Least squares assumption #1: E(u|X = x) = 0.

Least squares assumption #1, ctd.

Least squares assumption #2: (Xi,Yi), i = 1,…,n are i.i.d.

Least squares assumption #3: Large outliers are rare Technical statement: E(X4) <  and E(Y4) < 

OLS can be sensitive to an outlier:

The Sampling Distribution of the OLS Estimator (SW Section 4.5)

Probability Framework for Linear Regression

The Sampling Distribution of 1 ˆ b

The mean and variance of the sampling distribution of

Now we can calculate E( ) and var( ):

Next calculate var( ):

What is the sampling distribution of ?

Large-n approximation to the distribution of :

The larger the variance of X, the smaller the variance of

The larger the variance of X, the smaller the variance of

Summary of the sampling distribution of :

Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals (SW Chapter 5)

But first… a big picture view (and review)

Object of interest: 1 in,

Hypothesis Testing and the Standard Error of (Section 5.1)

Formula for SE( )

Summary: To test H0: 1 = 1,0 v. H1: 1  1,0,

Example: Test Scores and STR, California data

Confidence Intervals for 1 (Section 5.2)

A concise (and conventional) way to report regressions:

OLS regression: reading STATA output

Summary of Statistical Inference about 0 and 1:

Regression when X is Binary (Section 5.3)

Interpreting regressions with a binary regressor

Summary: regression when Xi is binary (0/1)

Heteroskedasticity and Homoskedasticity, and Homoskedasticity-Only Standard Errors (Section 5.4)

Homoskedasticity in a picture:

Heteroskedasticity in a picture:

A real-data example from labor economics: average hourly earnings vs A real-data example from labor economics: average hourly earnings vs. years of education (data source: Current Population Survey):

The class size data:

So far we have (without saying so) assumed that u might be heteroskedastic.

What if the errors are in fact homoskedastic?

We now have two formulas for standard errors for

Practical implications…

Heteroskedasticity-robust standard errors in STATA

The bottom line:

Some Additional Theoretical Foundations of OLS (Section 5.5)

The Extended Least Squares Assumptions

Efficiency of OLS, part I: The Gauss-Markov Theorem

The Gauss-Markov Theorem, ctd.

Efficiency of OLS, part II:

Some not-so-good thing about OLS

Limitations of OLS, ctd.

Inference if u is Homoskedastic and Normal: the Student t Distribution (Section 5.6)

Practical implication:

Summary and Assessment (Section 5.7)