What is the assumption underlying the linear regression models
i.Derive the least -squares estimates by any parameters of your choice
The assumption underlying linear regression models is that there is a linear relationship between the independent variable(s) and the dependent variable. This means that the relationship can be represented by a straight line.
To derive the least-squares estimates, you need to minimize the sum of the squared differences between the observed values of the dependent variable and the predicted values from the regression equation.
Here is the step-by-step process for deriving the least-squares estimates:
1. Start with a dataset that includes both the independent variable(s) and the dependent variable.
2. Choose the parameters that will be estimated in the regression model. These parameters are the coefficients of the independent variable(s) in the regression equation.
3. Define the regression equation, which represents the linear relationship between the independent variable(s) and the dependent variable. The equation should be of the form Y = β0 + β1X1 + β2X2 + ... + βnXn, where Y is the dependent variable, β0 is the intercept, β1, β2, ..., βn are the coefficients of the independent variables X1, X2, ..., Xn.
4. Estimate the regression coefficients by minimizing the sum of the squared differences between the observed values of Y and the predicted values from the regression equation. This can be done using mathematical techniques like the method of ordinary least squares (OLS), which finds the values of β0, β1, β2, ..., βn that minimize the sum of squared residuals.
5. Once you have estimated the regression coefficients, you can use them to predict the values of the dependent variable based on the values of the independent variable(s).
Note that the choice of parameters and the specific technique used to minimize the sum of squared differences may vary depending on the specific linear regression model and software you are using.