The coefficient of explanation or quality of the regression

### As we learned in Chapter 3, if we mark 5 rows instead of 2 in an estimation using Excel, we will obtain additional information about the estimation. In this chapter, we will focus on the first value that appears in the third row. This value is called ( R^2 ), and it is always a number between 0 and 1. ( R^2 ) implies the quality of the regression, and it answers the question: How strongly does the explaining variable explain the explained variable? In the model, the closer ( R^2 ) is to 1, the better the regression is, i.e. X explains Y better. The closer ( R^2 ) is to 0, the less quality the regression is, i.e. the explanatory power that X has regarding Y is smaller. ### Example Researcher A believes that X explains Y, while researcher B believes that Z explains Y. – Researcher A’s model is ( Y = aX + b ). – Researcher B’s model is ( Y = cZ + d ). The observation data appear in the following table: | Observation No. | Y | X | Z | |——|—-|—-| | 1 | 23 | 1 | 11 | | 2 | 16 | 2 | 6 | | 3 | 5 | 3 | 1 | | 4 | 32 | 4 | 14 | | 5 | 15 | 5 | 5 | | 6 | 30 | 6 | 12 | | 7 | 21 | 7 | 7 | | 8 | 12 | 8 | 2 | | 9 | 35 | 9 | 13 | | 10 | 18 | 10 | 4 | | 11 | 41 | 11 | 15 | | 12 | 28 | 12 | 8 | | 13 | 33 | 13 | 10 | | 14 | 20 | 14 | 3 | | 15 | 33 | 15 | 9 | After the estimation, the following graphs were obtained: #### Researcher A #### Researcher B  Because in Researcher B’s graph the points are more concentrated around the line, his regression is of higher quality. Higher quality regression means that the model explains the sample better. We will check what the ( R^2 ) value is in each of the estimates:  – in Researcher A’s estimate it is 0.2087.
– In researcher B’s estimate it is 0.8145. ( R^2 ) of researcher B is greater than ( R^2 ) of researcher A, so we can claim that Z explains Y better than X. ### Note It is permissible to compare two models using ( R^2 ) only if two conditions are met: 1. Both models have the same explanatory variable.
2. Both models have the same number of explanatory variables (including intercept). The models of the above researchers meet these conditions. In both, the explained variable is Y, and in both there is one explanatory variable and intercept. ### The extreme values ​​of ( R^2 ) When ( R^2 = 1 ), then the explanation is perfect, and all the points of the sample are on the regression line. When ( R^2 = 0 ), the regression explains nothing.