What does “least squares” estimates mean? What is being estimated? What is being squared? In what sense are the squares “least”?

TASK

1. Consider the following two least-squares estimates of the relationship between interest rates and the federal budget deficit in the United States:

Model 1:

N = 56, R2 = 0.00

where:         = Interest rate on AAA corporate bonds

= Federal budget deficit as percentage of GNP

Model 2:   .369

N = 38, R2 = 0.40

where:         = Interest rate on 3 – month Treasury bills

= Federal budget deficit in billions of dollars

= Rate of inflation (in percent)

1. What does “least squares” estimates mean? What is being estimated? What is being squared? In what sense are the squares “least”?
2. What does it mean to have an R2 of 0.00? Is it possible for R2 to be negative?  What about ?
3. R2 tells us what percent of variation in the dependent variable from its mean can be explained by a regression model (regression equation). How does an increase in R2 impact TSS, ESS, and RSS for a given regression equation?
4. What signs would you have expected for the estimated slope coefficients of the two models?
5. Comparing the two equations, which model has estimated signs corresponding to your prior expectations?
6. Can one argue that model 2 is preferred on the basis of fit?