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I think this question may be better split into univariate regression and multiple regression using OLS. Here we assume $\hat{\beta}_i$ are the "slopes", where $i > 0$, and $\hat{\beta}_0$ is the intercept.


For univariate regression, if you performed OLS, and obtained $\hat{\beta}_1 = 0$ (slope), then this means the line of best fit determined by OLS is a horizontal line located at some $\hat{\beta}_0$ offset from the horizontal axis. This also means the correlation coefficient between the regressor and response is zero, indicating no linear independence.

My question regarding this univariate case is, beyond linear independence, does the zero slope and zero correlation also indicate no monotonic relationship? If the data was monotonically increasing or decreasing, then there is no way the slope could end up zero?

A slope of zero does not rule out non-monotonic relationships. For example, consider $$ y = \sin(x) + \text{constant} $$. If the regressor is distributed from $x \in [0, n\pi]$, where $n$ is some even number, then OLS will give you $\hat{\beta_1} = 0$


For multiple regression, what are some obvious and subtle characteristics of $\hat{\beta}_i = 0$? Does it still indicate that there's no linear relationship between the i-th regressor and the response? Does it say anything about the $j$-th regressor, where $j \neq i$?

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For the multivariate case, it would mean that there is no significant relationship between that particular predictor and the dependant variable

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  • $\begingroup$ Say $\hat{\beta}_1 = 0$, and you have some $X_2$ that is correlated with $X_1$. If you removed $X_2$, the $\hat{\beta}_1$ could then be non-zero? $\endgroup$ Jul 7 '20 at 22:33
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    $\begingroup$ This answer is phrased in a way that many readers might misunderstand, because "no significant relationship" (a) doesn't answer any part of the question and (b) has a meaning only in the context of the other explanatory variables. The importance of this is manifest in a situation where $X_1=Y$ and $X_2$ differs only slightly from $Y:$ the univariate regression of $Y$ on $X_2$ can exhibit a strong, significant, linear relationship while the coefficient estimate of $X_2$ in the bivariate regression of $Y$ on $(X_1,X_2)$ may be zero. $\endgroup$
    – whuber
    Jul 7 '20 at 22:46
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    $\begingroup$ Here is an R example of this phenomenon that summarizes both models and draws a scatterplot matrix to show how the three variables are related: n <- 100; y <- (x1 <- rnorm(n)) + rnorm(n); x2 <- residuals(lm(rnorm(n) ~ x1 + y)) + x1; print(summary(lm(y ~ x1 + x2)), digits=2); print(summary(lm(y ~ x2)), digits=2); pairs(cbind(y, x1, x2)) $\endgroup$
    – whuber
    Jul 7 '20 at 22:47

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