…[2.02] The hat mark over Y in [2.02] indicates that it is the fitted value of Y corresponding to X and not the actual value. In Figure 2.3 the fitted points are represented by the points R 1 – R 4 . One thing that should be accepted from the beginning is that however much care you take in drawing the line you can never discover the true values of and
b 1 and b 2 are only estimates, and they maybe good or bad. Once in awhile your estimates maybe absolutely accurate, but this can only be by coincidence and even then you will have noway of knowing that you have hit the target exactly. This remains the case even when you use more sophisticated techniques. Drawing a regression line by eye is all very well, but it leaves a lotto subjective judgment. Furthermore, as will become obvious, it is not even possible when you have a variable Y depending on two or more explanatory variables instead of only one. The question arises, is there away of calculating good estimates of and algebraically The answer is yes The first step is to define what is known as a residual for each observation. This is the difference between the actual value of Y in any observation and the fitted value given by the regression line, that is, the vertical distance between P i and R i in observation i. Which will be denoted by e i
