Explanation Calculation: The relationship between the dependent variable y and independent variable x 1 . y = { β 0 + β 1 x 1 + ε ; if x 1 ≤ x ˜ β 0 * + β 1 * x 1 + ε ; if x 1 > x ˜ Here, x ˜ is a known constant. The new independent variable x 2 is defined as: x 2 = { 0 ; if x 1 ≤ x ˜ 1 ; if x 1 > x ˜ Furthermore, it is given that β 2 = β 0 * − β 0 and β 3 = β 1 * − β 1 . Multiple linear regression model: A multiple linear regression model is given as y i = β 0 + β 1 x 1 i + ... + β k x k i + ε i where y i is the response variable, and x 1 i , x 2 i , ... , x k i are the k predictor variables. The quantities β 0 , β 1 , ... , β k are the slopes corresponding to x 1 i , x 2 i , ... , x k i respectively. β ^ 0 is the estimated intercept of the line, from the sample data. A multiple regression equation describes the combined effect of all the predictors in the model. Even when the effect of a particular predictor is being studied from a multiple regression equation, it assumes that the effects of all the other predictors are accounted for. Indicator variable : An indicator variable is a variable assigned to a qualitative predictor variable that has more than one category, which takes values 1 or 0. The value 1 denotes the presence of the particular category for which the indicator variable is created and the value 0 denotes the absence of that category. Here, the independent variable x 2 is an indicator variable, since it takes values 1 or 0. That is, x 2 = { 0 ; if x 1 ≤ x ˜ 1 ; if x 1 > x ˜ From the coefficients β 2 and β 3 , it can be written that, β 2 = β 0 * − β 0 β 0 * = β 2 + β 0 And β 3 = β 1 * − β 1 β 1 * = β 3 + β 1 . The relationship between dependent variable y and the independent variable x 2 is given as: y = { β 0 + β 1 x 1 + ε ; if x 1 ≤ x ˜ β 0 * + β 1 * x 1 + ε ; if x 1 > x ˜

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