Given the following estimated linear model ŷ=10-2 x1-14 x2+6 x3 a. What is the change in ŷ when

Given the following estimated linear model ŷ=10-2 x1-14 x2+6...

Given the following estimated linear model
$$
\hat{y}=10-2 x_1-14 x_2+6 x_3
$$
a. What is the change in $\hat{y}$ when $x_1$ increases by 4 ?
b. What is the change in $\hat{y}$ when $x_3$ decreases by 1 ?
c. What is the change in $\hat{y}$ when $x_2$ decreases by 2 ?

Key Concepts

Linear Regression Model

A linear regression model is an equation that represents the relationship between a dependent variable and one or more independent variables through a linear combination of the predictors. The model’s coefficients quantify how much the dependent variable is expected to change with a one-unit change in an independent variable, assuming all other variables remain constant.

Interpretation of Coefficients

In linear regression, each coefficient represents the marginal (or partial) effect of a one-unit change in the corresponding independent variable on the dependent variable. This interpretation assumes that all other predictors are held constant, making it clear how each variable uniquely contributes to the outcome.

Marginal Effect Calculation

The marginal effect in a linear regression context is calculated by multiplying the change in an independent variable by its coefficient. This provides the change in the predicted value of the dependent variable attributable to the change in the independent variable, maintaining the linearity assumption.

Ceteris Paribus Assumption

The ceteris paribus assumption in regression analysis implies that when assessing the effect of a change in one independent variable, all other variables are held constant. This allows for an isolated examination of the relationship between a single predictor and the outcome variable.

Transcript

00:01 In this question, we have been given estimated linear model that looks like predicted value equals 10 minus 2x1 minus 14x2 plus 6x3.

00:16 Now, first case is about that the value of x1 increases by 4.

00:23 So our new x1 will be what? new x1 will be x1 plus 4? if we name this one as x1 dash, so it can be written as such.

00:34 So this x1 dash is equals to x1 plus 4 because it is increased by 4.

00:41 Hence, if we plug in this value of x1 dash in the equation, it will be 10 minus 2 x1 plus 4 and then here it will be minus 14x2 plus 6x3.

00:56 And hence here it will be 10 minus 2 x1 minus 8 and then it will be minus 14 x2 plus 6x3 so here we can write that predicted value is equal to 10 minus 2 x1 and then here it is minus 14 x2 plus 6x3 and then here it is minus 8 so if we compare this particular equation if we name it as equation number 2 and this one as equation number one.

01:29 So, if we compare both of these equations, we can clearly see that y cap or predicted value decreases by 8, right? so it can be written over here that y cap decreases by 8 and this is the answer for the first part of this very problem.

01:49 Now let's see what is the answer for the second part of this problem.

01:52 So here we can say that part b.

01:55 So, in the part b, we have that x3 is changing into x3, how much will it be? it will be minus 1.

02:05 So this is my new x3 value.

02:08 We can here write new x3 value.

02:12 So if we plug in this value in our equation, it will be y cap or predicted value equals 10 minus 2x1 minus 14x2 plus 6 and then x3 will be x3...

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