# Unit of measure in the OLS regression

Suppose I estimate by OLS the following linear model $$Y_i=\beta_0+\beta_1 X_i+U_i$$ where $$Y_i$$ denotes the weight of individual $$i$$ (in pounds), and $$X_i$$ denotes the height of individual $$i$$ (in inches).

Let $$\hat{\beta}_0$$ and $$\hat{\beta}_1$$ be the OLS estimates. Let $$R^2$$ be the R-squared from the regression . Let $$SER$$ be the Root Mean Square Error.

Question: Suppose that instead of measuring weight and height in pounds and inches, the variables are measured in centimeters and kilograms. What are the regression estimates (coefficients, $$R^2$$, and SER) from this new centimeter-kilogram regression? \begin{aligned} & \text{1 Pound = 0.453 Kilogram}\\ & \text{1 Inch= 2.54 Centimeter}\\ \end{aligned}

Thoughts: I found similar questions in this forum (e.g., here and here) but I'm unable to clear up my mind on how to proceed.

(A) With regards to the coefficients, I would suggest to proceed by noticing that \begin{aligned} Y_i=\beta_0+\beta_1X_i+U_i & \Leftrightarrow Y_i*0.453=0.453*\beta_0+0.453*\beta_1X_i+0.453*U_i,\\ & \Leftrightarrow Y_i*0.453=0.453*\beta_0+\frac{0.453}{2.54 }*\beta_1(X_i*2.54)+0.453*U_i \end{aligned} Therefore, $$\hat{\beta}^{new}_0=0.453*\hat{\beta}_0$$ and $$\hat{\beta}^{new}_1=\frac{0.453}{2.54 }*\hat{\beta}_1$$

(B) $$R^2$$ does not change because it has no unit of measure.

(C) Let $$\hat{U}_i=Y_i-\hat{\beta}_0-\hat{\beta}_1X_i$$ and $$SER=\sqrt{\frac{1}{n-2}\sum_{i=1}^n \hat{U}_i^2}$$. Note that $$\hat{U}^{new}_i=Y_i^{new}-\hat{Y}^{new}_i=0.453 *Y_i-\Big[\frac{0.453}{2.54 }*\hat{\beta}_1(X_i*2.54)+0.453\hat{\beta}_0\Big]=0.453* \hat{U}_i$$ Hence, $$SER^{new}=0.453*SER$$

Are A,B,C correct?

• Yes, A, B, and C are correct. The coefficients change according to rescaling, R-squared stays the same and the residuals are rescaled with the DV. Aug 26, 2021 at 9:42