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I have two linear regression models: one with intercept term, and the other is without. I wanted to compare the models based on their parameters. MLE estimator for intercept model is well known to be equal to $$ \hat{\beta_1}= \frac{S_{xy}}{S_{xx}}=\frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2} $$ and MLE estimator for no intercept model is:

$$ \hat{\beta_1^0}= \frac{\sum x_i y_i}{\sum x_i^2} $$ We can also show that $\hat{\beta_1}$ is an unbiased estimator of the slope term in the case of intercept model, and $\hat{\beta_1^0}$ is an unbiased estimator of the slope term in the case of no-intercept model.

The problem arises when I try to compute the covariance of these two estimators:

$$ \mathrm{Cov}(\hat{\beta_1}, \hat{\beta_1^0}) = \mathbb{E} [(\hat{\beta_1^0}-\mathbb{E}[\hat{\beta_1^0}])(\hat{\beta_1}-\mathbb{E}[\hat{\beta_1}])]=$$

$$ =\mathbb{E}[(\frac{\sum x_i y_i}{\sum x_i^2}-\frac{x_i}{x_i^2}\mathbb{E}[y_i])(\frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2}- \frac{\sum (x_i - \bar{x})}{\sum (x_i-\bar{x})^2}\mathbb{E}[(y_i-\bar{y})])] $$

And from here on I have my doubts. Since we have to Assume that r.v. is either $y_i \sim N(\beta_0+\beta_1 x_i, \sigma^2)$ or $y_i \sim N(\beta_1^0 x_i, \sigma^2)$. But that will result in one of the estimators being biased, and this will not allow me to compare the models. Am I correct?

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