Specification & Data Problemsintermediate

Units of Measurement and Scaling

Rescaling a variable changes coefficients and standard errors in a predictable way without changing the substance of the model. If a regressor $x_j$ is divided by $c$ , its coefficient and standard error are multiplied by $c$ , leaving the $t$ statistic untouched. Scaling $y$ multiplies all coefficients and standard errors by the same factor. Standardized (beta) coefficients rescale every variable to standard deviations so slopes are comparable across regressors. Quantities like the $t$ statistics, $R^2$ , and fitted values are invariant to these changes of units.

Why it matters

Measuring income in dollars or in thousands of dollars cannot change what the data say, only the labels on the numbers. So when you switch units, the coefficients move by exactly the conversion factor and everything that should not change, the fit, the $t$ statistics, the predictions, stays put. Beta coefficients take this further by putting all variables on the same standard-deviation ruler, which lets you ask which regressor moves $y$ most in relative terms.

Formulas

Rescaling a regressor

x_j^{*}=x_j/c \;\Rightarrow\; \hat{\beta}_j^{*}=c\,\hat{\beta}_j,\quad \operatorname{se}(\hat{\beta}_j^{*})=c\,\operatorname{se}(\hat{\beta}_j)

Coefficient and standard error scale together, so the

t

statistic

\hat{\beta}_j/\operatorname{se}(\hat{\beta}_j)

is unchanged.

Standardized (beta) coefficient

b_j=\hat{\beta}_j\,\frac{\hat{\sigma}_{x_j}}{\hat{\sigma}_y}

Interprets as the standard-deviation change in

y

per one standard-deviation rise in

x_j

, making magnitudes comparable across regressors.

Worked examples

Scenario

A wage regression measures earnings in dollars, and you want the coefficient per thousand dollars instead.

Solution

If `regress wage educ` gives a slope of 1500 (dollars per year of schooling) and you redefine `gen wagek = wage/1000`, then `regress wagek educ` gives 1.5. The standard error shrinks by the same factor of 1000, the $t$ statistic, $R^2$ , and fitted pattern are identical.

NoteOnly the units of the reported number change, not the economic content.

Scenario

Compare which of several regressors has the largest relative effect on test scores.

Solution

Run `regress score classize income tchexp, beta`. The `beta` option reports standardized coefficients, so you can compare a one standard-deviation change across class size, family income, and teacher experience on the same scale rather than comparing raw slopes in incompatible units.

Common mistakes

✗Thinking rescaling changes the model’s conclusions. It changes only the units of the coefficients and their standard errors, never the $t$ statistics, $R^2$ , or fitted values.
✗Comparing raw coefficients across variables measured in different units. A slope of 1500 on dollars and 0.03 on a ratio are not directly comparable; that is what beta coefficients are for.
✗Believing the $t$ statistic shifts when you rescale a regressor. The coefficient and its standard error move by the same factor, so the ratio is invariant.
✗Assuming a bigger standardized coefficient proves greater importance or causality. Beta coefficients aid comparison of magnitudes but say nothing about identification.

Revision bullets

•Dividing $x_j$ by $c$ multiplies $\hat{\beta}_j$ and its SE by $c$ .
•Scaling $y$ multiplies all coefficients and SEs by the same factor.
• $t$ statistics, $R^2$ , and fitted values are invariant to rescaling.
•Beta coefficients put variables on a standard-deviation scale for comparison.
•In Stata, the `beta` option reports standardized coefficients.

Quick check

You divide a regressor by 1000 and re-estimate. What happens to its t statistic?

Standardized (beta) coefficients are most useful for:

Connected topics

R-squared MLR model t test Logs & Elastic.Multicollin.

Sources

Wooldridge (2019), §6.1
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.
Shows how changing units of measurement rescales coefficients and standard errors and introduces standardized coefficients.
Wooldridge (2019), §6.1 (beta coefficients)
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.
Defines beta coefficients as effects in standard-deviation units for cross-variable comparison.

How to cite this page

Dr. Phil's Quant Lab. (2026). Units of Measurement and Scaling. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-units-scaling

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