Do different units of explanatory variables affect p-values? In multiple linear regression, does including variables with different metering units and thus values very different in dimension/size affect statistical significance/ p-values?
In my model, a variable with large values (i.e. relatively large compared to the other included variables) exhibits a very small parameter estimate. Its p-value is very large, though sum of squares is large.
Now I am looking for an explanation and I guess it could be due to the different units of explanatory variables.
 A: The units of variables affect the parameter estimates, but not the p-values.
Think of it this way: if you multiplied an entire variable in your dataset by 10, you could just divide the coefficient of that variable by 10, and get the exact same prediction for each data point. So the model will explain just as much of the variance as before.
However, it can be easier to interpret different aspects of your model if you "standardize" the units of each variable, for instance by dividing by the standard deviation of that variable. If you do this, then all of your variables will be unitless (and all of your coefficients will have the same units as your response variable), which means they can be compared on a single value.
The upshot is that a larger coefficient doesn't necessarily mean a stronger p-value. The p-value just indicates how likely you would be to get a coefficient that large by chance. And depending on the distribution of your data/response, you can actually be fairly likely to get large values by chance, so you can't in general get a relationship between coefficient size and p-values.
Now for two caveats to this:


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*If your variables have very different units--like by a factor of $10^{10}$ or something--it can make it look numerically like your dataset exhibits collinearity, which means you may encounter numerical problems when you fit your model. Rescaling the predictors can help with this.

*If you're using some types of regularized regression (e.g. ridge regression or LASSO), then smaller coefficients will be penalized less, so multiplying a variable by 10 and dividing its coefficient by 10 will make that coefficient more likely to be selected/have a higher value. This is why it's often advisable to center and scale your data when using these sorts of techniques.
