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?

  • 6
    $\begingroup$ 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. $\endgroup$ Aug 26, 2021 at 9:42


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