Questions about multiple linear regression

Question

I have a couple true/false questions basically, one of them is this

In the multiple linear regression model the coefficient of multiple determination gives the proportion of total variability due to the effect of a single predictor

I know the coefficient of multiple determination indicates the amount of total variability explained by the model, but I'm not sure about the single predictor part, I don't think this is true because it uses x1, x2... as predictors no?

The other question is this;

In the multiple linear regression model

$$y_i = β_0 + β_1x_{i,1} + β_2x_{i,2} + β_3x_{i,3} + ε_i$$

the parameter $β_1$ represents the variation in the response corresponding to a unit increase in the variable $x_1$

I don't think this question is true but can't really explain why

All help would be greatly appreciated

Answer 1

1. You are right. The $R^2$ is a (kind of) generalization of the Pearson correlation coefficient for multiple covariates.

2. True. Note that $\frac{\partial}{\partial x_k}\mathbb{E}[y|x_1,..,x_K]= \beta_k$ that is approximated by $\Delta \hat{y} = \Delta x_k \hat{\beta}_k$, where $\Delta x_k = 1$.