Limits of the two regression coefficients

The regression coefficients are important statistical measures used in regression analysis. They are used to estimate the relationship between two variables, where one is the dependent variable and the other is the independent variable. The regression coefficients have limits, and the correct answer to the question is option 'D', which states that the product of the regression coefficient must be numerically less than unity. Let's discuss this in more detail.

Regression coefficients

Regression coefficients are also known as beta coefficients or regression weights. They are used to determine the relationship between two variables. In simple linear regression, there is only one independent variable, and the regression coefficient is represented by the symbol 'b'. In multiple regression, there are two or more independent variables, and the regression coefficients are represented by the symbols 'b1', 'b2', 'b3', etc.

Limits of the regression coefficients

The regression coefficients have limits, which are as follows:

a) No limit - In theory, the regression coefficients can take any value. However, in practice, they are constrained by the data and the model used.

b) Must be positive - In some cases, the regression coefficients must be positive. For example, in a linear regression model where the dependent variable is the number of sales, the regression coefficient for the price of the product must be positive, as an increase in price should lead to an increase in sales.

c) One positive and the other negative - In some cases, the regression coefficients may have opposite signs. For example, in a linear regression model where the dependent variable is the number of ice cream sales and the independent variables are temperature and rainfall, the regression coefficient for temperature may be positive, while the regression coefficient for rainfall may be negative.

d) Product of the regression coefficient must be numerically less than unity - This is the correct answer to the question. In some cases, the product of the regression coefficients must be numerically less than unity to avoid problems with multicollinearity. Multicollinearity occurs when there is a high correlation between two or more independent variables in a multiple regression model. This can lead to unstable and unreliable estimates of the regression coefficients. To avoid this problem, the product of the regression coefficients must be less than one.

Conclusion

In conclusion, the limits of the regression coefficients depend on the data and the model used. In some cases, the regression coefficients must be positive, negative, or have opposite signs. In other cases, the product of the regression coefficients must be numerically less than unity to avoid problems with multicollinearity.