Regression Coefficient

Purposes of Estimating Regression Coefficients

The estimated regression coefficients or “estimators”

(the “hat” on the beta, i.e., ‘⁁’, refers to an estimated regression coefficient), calculated from the regression analysis, can then serve multiple purposes:

1. Evaluating the importance of individual predictor variables. A common approach here would be to use standardized response and predictor variables in the regression analysis. From the standardized variables, one can estimate standardized regression coefficients, which measure in standard deviations the change in the response variable that follows a standard unit change in a predictor variable. The standardization procedure allows one to assess the importance of individual predictor variables in situations in which the predictor variables are measured in different units.

2. Analyzing the changes in the response variable following a one-unit change in the corresponding explanatory variables X1, X2;…;Xk. For instance,

1 indicates the change in the response variable following a one-unit change in the predictor variable X1, while holding all other predictor variables, X2, X3; …; Xk, constant.

3. Forecasting values of the response for selected sets of predictor variables. In this case, using all the estimated regression coefficients,

0,1;2; …;k, together with a given set of predictor variables X1, X2;…; Xk, one would be able to forecast the expected value of the response variable.

Besides the absolute magnitudes of the estimated regression coefficients, the signs of the estimators are of major importance because they indicate direct or inverse relationships between the response variable and the corresponding predictor variables. It is usual to have prior expectations of the expected outcomes of the signs of the estimators. In other words, it is conceptually predetermined whether the relationship between the response variable and a predictor variable should be of positive or negative nature.

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Example of Use of Estimated Regression Coefficients

In real estate appraisal, an analyst might be interested in how building characteristics such as floor space in square feet (X1), number of bedrooms (X2), and the number of bathrooms (X3) help in explaining the average market values (Y) of real estate properties. In this oversimplified example, the estimated regression coefficients may be used for several purposes. Using standardized regression coefficients, one could determine whether the floor space in square feet, the number of bedrooms, or the number of bathrooms is most important in determining market values of real estate properties. In addition, the unstandardized, estimated regression coefficient

1 would indicate the expected change in the average market value of real estate properties for a 1-square-foot change in floor space while the number of bedrooms and bathrooms remain constant. Analogously,2 or3 would indicate how much one more bedroom or one bathroom would potentially add to the expected market value of real estate properties. And finally, using0,1,2, …,k, one could estimate the market value of a hypothetical 2,200-square-foot real estate property with three bedrooms and two bathrooms. Of course, one would expect all the estimated regression coefficients to have positive signs.

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