Let a sample $(x,y)\in {\mathbb{R}}^{2n}$ be given, where $y$ only attains the values $0$ and $1$. We can try to model this data set by either linear regression

${y}_{i}={\alpha}_{0}+{\beta}_{0}{x}_{i}$

with the coefficients determined by the method of least squares or by logistic regression

${\pi}_{i}=\frac{\mathrm{exp}({\alpha}_{1}+{\beta}_{1}{x}_{i})}{1+\mathrm{exp}({\alpha}_{1}+{\beta}_{1}{x}_{i})},$

where ${\pi}_{i}$ denotes the probability that ${y}_{i}=1$ under the given value ${x}_{i}$ and the coefficients are determined by the Maximum-Likelihood method. My question is whether the following statement holds true.

Claim: If ${\beta}_{0}>0$ (${\beta}_{0}<0$), then ${\beta}_{1}>0$ (${\beta}_{1}>0$).

I figure this could be due to the sign of the correlation coefficient.

${y}_{i}={\alpha}_{0}+{\beta}_{0}{x}_{i}$

with the coefficients determined by the method of least squares or by logistic regression

${\pi}_{i}=\frac{\mathrm{exp}({\alpha}_{1}+{\beta}_{1}{x}_{i})}{1+\mathrm{exp}({\alpha}_{1}+{\beta}_{1}{x}_{i})},$

where ${\pi}_{i}$ denotes the probability that ${y}_{i}=1$ under the given value ${x}_{i}$ and the coefficients are determined by the Maximum-Likelihood method. My question is whether the following statement holds true.

Claim: If ${\beta}_{0}>0$ (${\beta}_{0}<0$), then ${\beta}_{1}>0$ (${\beta}_{1}>0$).

I figure this could be due to the sign of the correlation coefficient.