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Consider the following simple linear regression model: $$ y = \beta_{0} + \beta_{1}x + u $$ Let's say that the dependent variable, $ y $, is yearly salary, for which the unit is dollars per year. Let's say that the independent variable, $ x $, is education, whose unit is years.
What is the unit of $ \beta_{1} $?
Is it dollars/(year)^2, so that upon multiplying $ \beta_{1} $ and $ x $, the units cancel out to become dollars/year, which is the unit of the dependent variable?
More generally, can you comment on whether this reasoning applies to the units of the regression coefficients in all regression models? Does dimensional homogeneity hold for regression equations? Thank you!

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    $\begingroup$ In this particular case, I would say that in my reading salary means (annual) salary unless otherwise qualified (as in monthly salary) and informally at least its units would be acceptable if expressed as dollars or whatever other currency was in use. Spelling out that the units are dollars/year might gain you small points of esteem for precision or lose you small points of esteem for pedantry. For any international readership, it does no harm to spell out US dollars, unless again it's obvious that you are talking about some other country. $\endgroup$
    – Nick Cox
    Commented Jun 4, 2023 at 7:31
  • $\begingroup$ I understand, I shall more clearly specify the currency in subsequent questions, and be more formal when defining the regression variables. Thank you! $\endgroup$
    – Spectral
    Commented Jun 4, 2023 at 20:40

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You’re on the right track, but you should be careful about what kind of “year” you mean. One of your “years” refers to years of education, while the other refers to the period over which income is measured. Being a bit formal, I would write something like:

$$ \dfrac{\left( \dfrac{ \text{ Dollars earned } }{ \text{ Year of employment } }\right)}{ \text{ Years of education } } $$

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  • $\begingroup$ I understand, thank you so much! Just as a small follow-up question, let's say the dependent variable's unit is kilograms, and the independent variable's unit is Celsius, for temperature. The coefficient's unit would then be kilograms per Celsius, right? Thank you again in advance, I appreciate the support as I begin to learn Econometrics. $\endgroup$
    – Spectral
    Commented Jun 4, 2023 at 20:35
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    $\begingroup$ kg/$^\circ$C would be the units if the outcome were measured in kg and a predictor temperature were measured in on the Celsius scale. $\endgroup$
    – Nick Cox
    Commented Jun 4, 2023 at 23:39
  • $\begingroup$ Thank you so much! I really appreciate it. $\endgroup$
    – Spectral
    Commented Jun 7, 2023 at 3:07

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