Deriving B1 and B2 in Multiple Regression Model

Step 1: Define the Multiple Regression Model
- The multiple regression model can be defined as: Y = B0 + B1X1 + B2X2 + ... + BnXn + ε
- Where Y is the dependent variable, Xi represents the independent variables, Bi are the coefficients to be estimated, and ε is the error term.

Step 2: Estimate the Coefficients B1 and B2
- To estimate the coefficients B1 and B2, we can use the least squares method.
- The least squares method minimizes the sum of the squared differences between the observed values of the dependent variable and the values predicted by the model.

Step 3: Calculate B1 and B2
- B1 and B2 can be calculated using the following formulas:
- B1 = Σ(X1i - X1̅)(Yi - Ȳ) / Σ(X1i - X1̅)²
- B2 = Σ(X2i - X2̅)(Yi - Ȳ) / Σ(X2i - X2̅)²
- Where X1i and X2i are the values of the independent variables, Y is the dependent variable, X1̅ and X2̅ are the mean values of the independent variables, and Ȳ is the mean value of the dependent variable.

Step 4: Example
- Let's consider a multiple regression model where Y is the sales revenue, X1 is the advertising budget, and X2 is the price of the product.
- We have the following data:
- X1 = [100, 200, 150, 180]
- X2 = [10, 15, 12, 14]
- Y = [500, 700, 600, 650]
- Calculating B1 and B2 using the formulas mentioned above will give us the estimated coefficients for the model.
By following these steps and calculations, you can derive the coefficients B1 and B2 in a multiple regression model.