# coefficient magnitude

"An interpretation based on coefficient magnitude also suggests that an increase of x (independent variable) by one standard deviation is associated with the increase of y (dependent variable by 2.63% of its standard deviation."

Could somebody explain to me the formula for calculating 2.63%? The coefficient of the x is 0.05 and I have other information like mean, std, and standard error

Thanks

You have some variable $$x$$ in the regression. Calculate the standard deviation of $$x$$, which we will call $$s_x$$.

You have the coefficient on $$x$$ in the regression, which we will call $$\hat\beta_x$$.

To determine how much changes in $$y$$ is associated with a change in $$x$$ of $$s_x$$, multiply $$s_x\times\hat\beta_x$$.

Now we know by how much $$y$$ changes when $$x$$ changes by one standard deviation. To determine how many standard deviations of $$y$$ that is, divide by the standard deviation of $$y$$. If you then want this in terms of percent, multiply by $$100\%$$.

Overall, this corresponds to $$\dfrac{s_x\hat\beta_x}{s_y}\times 100\%$$.

Note that all of this assumes a linear model. If you have a nonlinear model or a linear model that acts on nonlinear functions of the original variables (e.g., $$x$$ and $$x^2$$ are in the model), then this does not work, as one characteristic of nonlinearity is that the rate of change (related to $$\hat\beta_x$$) is not constant.

• Thank you so much. Yes, I have a linear regression. But the dependent variable (y) is in Ln or log format. Is this change the formula? @Dave Feb 19 at 19:24
• You would take the standard deviation of whatever you feed into the models, so if that’s a logged value, take the standard deviation of the logged $y$. However, taking a logarithm usually implies an interest in percent changes. How to consider percent standard deviation changes when the model is of percent changes in $y$ probably warrants its own posted question.
– Dave
Feb 19 at 19:32
• Thanks. I added a question here. I would appreciate it if you could explain it to me stats.stackexchange.com/questions/605960/… Thanks @Dave Feb 20 at 16:51