If we have a linear regression of the form
$$ Y = \beta_0 + \beta_1X_1 + \beta_2X_2 $$
is it valid to interpret the coefficient $\beta_1$ as the associated change in $Y$ when $X_1$ increases by a unit of 1, when $X_2=0$?
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$\begingroup$ The fact that rate of change of (expected) $Y$ wrt $X_i$ is the same no matter what the values of the $X$ variables are is essentially our assumption that the relationship is linear. $\endgroup$– John MaddenSep 13, 2022 at 22:06
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Just omit the last caveat and your interpretation is basically accurate, though I will add one small edit based on dipetkov's helpful comment:
$\beta_1$ is the associated change in the expected $Y$ when $X_1$ increases by a unit of 1.
This holds whatever the value of $X_2$ (conditional on the model being accurate, though this caveat is also covered by the addition of 'expected' to the definition). Variation in $X_2$ is irrelevant here because there is no interaction term in your model.
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1$\begingroup$ I'd like to be explicit and say "the associated change in the expected Y" or "the expected change in Y associated with". $\endgroup$– dipetkovSep 13, 2022 at 5:35
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$\begingroup$ Thanks, is what I have a special case or is what I have incorrect as it stands? $\endgroup$ Sep 13, 2022 at 6:24
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$\begingroup$ @dipetkov Thanks, that is a good idea. I have edited to reflect that. $\endgroup$– mktSep 13, 2022 at 6:49
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1$\begingroup$ @Stef: it is a linear regression model - typically not all points fit the equation exactly, but there is assumed to be an error term with zero mean and independent of the $X_i$. $\endgroup$– HenrySep 13, 2022 at 13:17
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2$\begingroup$ @Stef Regression is a model for the conditional mean E(Y | X = x). To understand this better, consider the Difference between confidence intervals and prediction intervals Also take a look at Section 10.2 in Regression and Other Stories. It's freely available online. $\endgroup$– dipetkovSep 13, 2022 at 13:27