Couple of questions about multiple linear regression

Question

I have questions about 2 true/false statements.

1. 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 the answer for this is true because I think it uses x1, x2... as predictors.

2. In the multiple linear regression model,

   $$y_i = β_0 + β_1x_{i1} + β_2x_{i2} + β_3x_{i3} + ε_i$$

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

   I don't think the above statement is true but am trying to understand why.

All help would be greatly appreciated

Answer 1

Your doubts about the first statement are justified: it is the whole model not just a single predictor.

I would not have used the same language for the second one but it seems to me to be correct: unit change in the predictor corresponds to change of $\beta$ in the response.