Graph E(y) versus x1 for values of x2=0,1, and 2 on the same...

Graph $E(y)$ versus $x_{1}$ for values of $x_{2}=0,1,$ and 2 on the same sheet of paper. Now graph $E(y)$ versus $x_{2}$ for values of $x_{1}=0$, l, and 2 on another sheet of paper. What relationships do the three lines on the graphs have to one another?
$$E(y)=1-2 x_{1}+3 x_{2}$$

Key Concepts

Intercept Shifts with Fixed Values

While the slopes remain constant in these partial plots, the different fixed values of the other independent variable lead to different intercepts. For instance, in the E(y) versus x1 plot, varying x2 changes the intercept by a multiple of its coefficient (3x2), and similarly for the E(y) versus x2 plot, the intercept shifts according to -2x1. These intercept shifts reflect the additive contribution of the fixed values to the overall expected value of y.

Parallel Lines in Partial Regression Plots

When a regression model is additive (i.e., there are no interaction terms), plotting the relationship of the dependent variable with one independent variable for different fixed values of another results in lines that have the same slope. In this case, the slopes remain constant (e.g., -2 for x1 plots and 3 for x2 plots) across the different lines, resulting in a series of parallel lines.

Multiple Linear Regression

Multiple linear regression is a statistical method used to model the relationship between one dependent variable and two or more independent variables. In this context, the equation E(y) = 1 - 2x1 + 3x2 expresses how the expected value of y changes linearly with changes in x1 and x2, with each coefficient indicating the unique contribution of its corresponding variable when the others are held constant.

Ceteris Paribus (Holding Other Variables Constant)

The ceteris paribus principle involves examining the relationship between the dependent variable and one specific independent variable while keeping the other independent variable(s) fixed. Graphically, this is represented by plotting E(y) versus one independent variable for fixed values of the other, thereby isolating the effect of that one variable.

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