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. Water consumption example 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

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