Multiple Regression

1. Multiple Regression Fundamentals Basic Interpretations

2. Statistical Models • E(Y) is a conditional mean, a ‘regression’ • A ‘linear’ regression is: • Then usually we have: • And the other assumptions about the errors

3. The fitted values • Where the residual sum of squares • Is made as small as possible (least squares)

4. Analysis of Variance • Source SS df MS • Regression ESS K-1 • Residual RSS n-K MSE • Total TSS n-1 • The main purpose of such a display is to present the MSE • The ‘Omnibus F test’ is rarely used as it tests: • This null hypothesis is rarely of scientific interest • (It is given in most regression output. So what!)

5. Interpretation • The meaning of the ‘coefficients’ is different for every model. • Be careful! We tend to use the same symbols to conceptualize the models but the coefficients can mean very different things EVEN when they are coefficients for the same variables

6. Y is water81 is income is water80 We write: And: But any one coefficient is interpreted in light of the others in the model. See Hamilton for the details Water consumption example

7. Notice that: • In the second model, • But in the first model, • This looks complicated, but it is central to understanding and interpreting

8. For example, if a household has • Then the second model says that the expected water consumption for this household is: • If another household has: • Then: • The difference in expected water consumption is:

9. But! • This is true only if the previous water consumption was the SAME in the 2 households • This addition part to the statement is only required with the second model, but not with the first simpler model that did not involve previous water consumption