1. I271B Quantitative Methods
Regression Part I

2. Administrative Merriments
Next Week: Reading and Evaluating Research
Suggest readings using regression, bivariate statistics, etc
Course Review May 5
Exam distributed May 7 in class
No lecture

3. Regression versus Correlation
Correlation makes no assumption about one whether one variable is dependent on the other– only a measure of general association
Regression attempts to describe a dependent nature of one or more explanatory variables on a single dependent variable. Assumes one-way causal link between X and Y.
Thus, correlation is a measure of the strength of a relationship -1 to 1, while regression is a more precise description of a linear relationship (e.g., the specific slope which is the change in Y given a change in X)
But, correlation is still a part of regression: the square of the correlation coefficient (R2) becomes an expression of how much of Y’s variance is explained by X.

4. Basic Linear Model Yi = b0 + b1xi + ei.
X (and X-axis) is our independent variable(s)
Y (and Y-axis) is our dependent variable
b0 is a constant (y-intercept)
b1 is the slope (change in Y given a one-unit change in X)
e is the error term (residuals)

5. Basic Linear Function

6. Slope
So...what happens if B is negative?

7. The ‘least squares’ Solution
The distance between each data point and the line of best fit is squared. All of the squared distances are added together.
Adapted from Myers, Gamst and Guarino 2006

8. The Least Squares Solution (cont)
For any Y and X, there is one and only one line of best fit. The least squares regression equation minimizes the possible error between our observed values of Y and our predicted values of Y (often called y-hat).
Adapted from Myers, Gamst and Guarino 2006

9. Important features of regression
There are two regression lines for any bivariate regression
Regression to the mean (regression effect)
Appears whenever there is spread around the SD line (1 SD increase in X  r x SDy)
Regression fallacy
Attempting to explain the regression effect through another mechanism

10. Statistical Inference Using Least Squares
We obtain a sample statistic, b, which estimates the population parameter.
b is the coefficient of the X variable (e.g., how much of a change in the predicted Y is associated with a change in X)
We also have the standard error for b, indicated by e.
We can use a standard t-distribution with n-2 degrees of freedom for hypothesis testing.
Yi = b0 + b1xi + ei.

11. Root Mean Square Error = actual – predicted
The root-mean-square (r.m.s.) error is how far typical points are above or below the regression line.
Prediction errors are called residuals.
The average of the residuals is = 0.
The S.D. of the residuals is the same as the r.m.s. of the regression line

12. Interpretation: Predicted Y = constant + (coefficient * Value of X)
For example, suppose we are examining Education (X) in years and Income (Y) in thousands of dollars
Our constant is 10,000
Our coefficient for X is 5

13. Heteroskedasticity
OLS regression assumes that the variance of the error term is constant. If the error does not have a constant variance, then it is heteroskedastic (literally, “different scatter”).
Where it comes from
Error may really change as an X increases
Measurement error
Underspecified model

14. Heteroskedasticity (continued)
Consequences
We still get unbiased parameter estimates, but our line may not be the best fit.
Why? Because OLS gives more ‘weight’ to the cases that might actually have the most error from the predicted line.
Detecting it
We have to look at the residuals (difference between observed responses from the predicted responses)
First, use a residual versus fitted values plot (in STATA, rvfplot) or the residuals versus predicted values plot, which is a plot of the residuals versus one of the independent variables.
We should see an even band across the 0 point (the line), indicating that our error is roughly equal.
If we are still concerned, we can run a test such as the Breusch-Pagan/Cook-Weisberg Test for Heteroskedasticity. It tests the null hypothesis that the error variances are all EQUAL, and the alternative hypothesis that there is some difference. Thus, if it is significant then we reject the null hypothesis and we have a problem of heteroskedasticity.

15. What to do about Heteroskedasticity?
Perhaps other variables better predict Y?
If you are still interested in current X, you can run a robust regression which will adjust the model to account for heteroskedasticity.
Robust regression modifies the estimates for our standard errors and thus our t-tests for coefficients.

16. Data points and Regression

17. Program and Data for Today:
Regress.do GSS96_small.dta