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I have a function $$l(\beta_0, \beta_1, \sigma^2) = -\frac{n \log(2\pi)}{2} - n \log \sigma - \frac{1}{2 \sigma^2} \sum_{i=1}^{n} (y-\beta_0 - \beta_1 x_i)^2$$ which is the log-likelihood function of MLE for SLR (Simple Linear Regression) for a regression model of $y = \beta_1x + \beta_0$.

Taking the partial derivative with respect to $\beta_1$ we have: $$\frac{\partial}{\partial \beta_1} = \frac{1}{\sigma^2} \sum_{i=1}^{n} x_i(y-\beta_0-\beta_1x_i)$$

So to find the maximum of this function we have to set this function to 0, which I got to the point of:

$$\sum_{i=0}^{n} x_i(y-\beta_0-\beta_1x_i) = 0$$

I am kind of at a loss on how to proceed from here when trying to find a root for this equation, it seems to be very difficult trying to find factors, I'm wondering if anyone can give me hints to proceed from this.

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1 Answer 1

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You can try expanding it out:

$$\displaystyle \sum_{i=0}^n x_iy - \beta_0\sum_{i=0}^nx_i-\beta_1\sum_{i=0}^nx_i=0$$

And then solve for $\beta_1$ to get:

$$\displaystyle \beta_1=\frac{\sum_{i=0}^nx_iy-n\beta_0}{\sum_{i=0}^n x_i}=y-\frac{n}{\sum_{i=0}^nx_i}\beta_0\space\space\space\space---(A)$$

But you still have a $\beta_0$ in the expression, so you need to find the partial derivative with respect to $\beta_0$ as well and solve that first. Then use the MLE you found for $\beta_0$ and plug it into what you found in (A).

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