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I am confused about how the two following models are different.

Using least squares, we fit the first model:

$\eta = \beta_0 + \beta_1 x$.

We also fit a second model:

$\eta = \alpha + \beta (x-\bar{x})$.

Wouldn't $\hat{\alpha}$ be the mean of the response variable whereas $\hat{\beta_0}$ would be the y-intercept? Then wouldn't it be true that $\hat{\beta} \neq \hat{\beta_0}$?

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The only difference in these models is that the second is parameterised so that the "intercept" term refers to the expected response when the explanatory variable is equal to its sample mean value (as opposed to zero). Since $\bar{x}$ is a scalar constant that is fixed by your explanatory variables, you have $\alpha = \beta_0 + \beta_1 \bar{x}$ and $\beta = \beta_1$. This relationship should also flow through to the OLS estimators.

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