What is the relationship between simple linear and multiple linear regression coefficients?

Question

For simplicity, let's restrict multiple linear regression case to 2 predictors, $x_1, x_2$. You regress $y$ on each individually and get $\hat{\beta}_1, \hat{\beta}_2$. Now you regress $y$ on both and get $\hat{\gamma}_1, \hat{\gamma}_2$.

So I know if $(x_1 - \bar{x}_1 \perp \bar{x}_2$, then $\hat{\beta}_i = \hat{\gamma}_i$, but if they're not orthogonal, what can be said about the relationship between them?

If in each of the simple linear regression cases, the slope was positive, i.e., $\hat{\beta}_1, \hat{\beta_2} > 0$, can we expect $\hat{\gamma}_1, \hat{\gamma}_2 > 0$?

I just asked a similar question on the math SE (https://math.stackexchange.com/questions/3791992/relationship-between-projection-of-y-onto-x-1-x-2-individually-vs-projecti), but I'm looking to gain more of a linear algebra intuition from answers to that question. Here, I am hoping for a broader intuition (both statistical and theoretical) of the relationship between simple and multiple linear regression coefficients.

Answer 1

Here is a simple example that provides insight.

y = c(5.8,5.2,4.7,8.7,8.1,7.7,10.2,9.6,9.0)
x1 = c(1,1.5,2,1.8,2.7,3.5,3,4,4.5)
x2 = c(1,1,1,2,2,2,3,3,3)

summary(lm(y~x1))
summary(lm(y~x2))
summary(lm(y~x1+x2))

plot(x1,y,col=x2)
legend("topleft", c("x2=1", "x2=2", "x2=3"), pch=1, col=1:3)

The simple regressions have significant positive relationships, but the multiple regression shows that the effect of x1 is significant and negative. The graph gives the intuition clearly:

Ignoring x1, there are generally higher values of y for larger x2. Similarly, ignoring x2, there are generally larger values of y for larger x1. These observations explain the simple regression results.

In the multiple regression model, the slope coefficients are estimates of the effect of one x while the other is held fixed. And you can easily see in the graph that the values of y are smaller as x1 increases within any of the three groups where x2 is held fixed (at either 1,2, or 3).