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If i have a regression model with the independent variable as continuous, while my dependent variable is a dummy variable (only one independent variable) then will the linearity assumption hold completely ?

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    $\begingroup$ If your DV is 0-1-valued, Ordinary Least Squares is not appropriate any more. Look at logistic regression, where the linearity assumption is needed via the log link. Whether it holds or not will depend on your data. $\endgroup$ Feb 4, 2018 at 9:07

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Under those conditions, linearity can only hold if the range of the independent variable is bounded at both ends, or if there is no relationship between the variables (where linearity holds trivially). To see this, suppose we have two distinct points $x_0 \neq x_1$ with:

$$\mathbb{E}(Y|X = x_0) = 0,$$

$$\mathbb{E}(Y|X = x_1) = 1.$$

The condition of linearity would then require:

$$\mathbb{E}(Y|X = x) = \frac{x - x_0}{x_1 - x_0}.$$

This gives $0 \leqslant \mathbb{E}(Y|X = x) \leqslant 1$ only over the range $\min (x_0, x_1) \leqslant x \leqslant \max(x_0, x_1)$, so if the range of the independent variable extends beyond this, you would have an impossible value.

To deal with this, models that have a dummy variable as the response generally use a transformation of the independent variable to ensure that the resulting regressor variable occurs on a bounded range. Common models include logistic regression (which uses a logistic transformation and binomial response), and other forms of categorical regresssion.

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