What is the assumption underlying the linear regression models
i.Derive the least -squares estimates by any parameters of your choice
ii.What is the Regression through the Origin
The assumption underlying linear regression models is that there is a linear relationship between the independent variable(s) and the dependent variable. This assumption means that the relationship can be represented by a straight line.
To derive the least-squares estimates in a linear regression model, 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. This minimization process is done by adjusting the parameters (coefficients) of the regression equation.
Here's how you can derive the least-squares estimates:
1. Start with a linear regression model equation: Y = β0 + β1X1 + β2X2 + ... + βnXn + ɛ, where Y is the dependent variable, X1, X2, ..., Xn are the independent variables, β0, β1, β2, ..., βn are the parameters (coefficients), and ɛ is the error term.
2. Define the objective function: Minimize the sum of squared errors (SSE) between the observed Y values and the predicted Y values: SSE = Σ(Yi - Ŷi)², where Yi is the observed value of Y and Ŷi is the predicted value of Y based on the regression equation.
3. Differentiate the SSE with respect to each parameter βj (j=0,1,...,n): Set the partial derivatives of SSE with respect to each βj equal to zero. This will give you a system of equations that can be solved simultaneously to find the least-squares estimates.
4. Solve the system of equations: Solve the equations to find the values of β0, β1,..., βn that satisfy the condition of setting the partial derivatives equal to zero. This will give you the least-squares estimates for the parameters.
Regression through the origin, also known as a zero-intercept model, is a special case of linear regression where the regression line is forced to pass through the origin (0,0). In this case, the equation becomes Y = βX, where Y is the dependent variable, X is the independent variable, and β is the coefficient. The assumption is that there is a linear relationship between the variables, but the intercept term is assumed to be zero.
To estimate the parameter β in a regression through the origin model, you can use the same least-squares method described above but with the constraint that β0 = 0. This means that you are only estimating the slope of the regression line, and the intercept is fixed at zero. The goal is still to minimize the sum of squared errors between the observed values of Y and the predicted values from the regression line.