I have one regression equation describing the relationship between interest rate $i$ and interest rate $c$, and another regression equation describing the relationship between house purchase volume and interest rate $c$.
The coefficient for interest rate $i$ for the regression between interest rate $i$ and interest rate $c$ is $0.5$. So, a $1\%$ increase in interest rate $i$ translates to a $0.5\%$ increase in interest rate $c$.
The coefficient for home purchase volume for the regression between home purchase volume and interest rate $c$ is $0.0000001$. So, a $\$10{,}000{,}000$ increase in home purchase volume translates to a $1\%$ increase in interest rate $c$.
How would one determine which variable, home purchase volume or interest rate $i$, has a stronger effect on interest rate $c$? I can’t seem to make sense of this due to the different units. If both independent variables were interest rates, then we could simply compare the magnitude of the coefficients and determine the larger coefficient impacts interest rate $c$ to a greater extent.
You can scale the data for both models, and then you can compare the coefficients or their $t$ statistics as they are on the same scale.