Help with MLE regression I have a data set containing two variables x and y. I want to estimate the parameters for a regression model. 
The regression model is:

The maximum likelihood function that estimates the parameters is:

Here is a link to the data set: http://www.filedropper.com/data_14
Is there a way to estimate the parameters of this function? I've tried and spent many hours with R but can't get it to work. Matlab will do just fine as well.
Thanks in advance!
 A: Your log-likelihood is wrong:
A t-density with location $\mu$, scale $\sigma$ and degrees of freedom $\nu$ is given by:
$f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\sigma\,\Gamma(\frac{\nu}{2})} \left(1+\frac{(t-\mu)^2}{\nu\sigma^2} \right)^{-\frac{\nu+1}{2}}\!.$
setting $\nu=3$ and replacing $\sigma$ with $\sigma x^\lambda$.
$f(t) = \frac{2} {\sqrt{3}\pi\,\sigma x^\lambda\,} \left(1+\frac{(t-\mu)^2}{3\sigma^2x^{2\lambda}} \right)^{-2}\!.$
Writing the likelihood (using $y_i$ for $t$ and $\beta_0+\beta_1 x_i$ for $\mu$)
$$\cal{L} = \prod_{i=1}^{n} \frac{2} {\sqrt{3}\pi\,\sigma x^\lambda\,} \left(1+\frac{(y_i-\beta_0+\beta_1 x_i)^2}{3\sigma^2x^{2\lambda}} \right)^{-2}\!.$$
So the log-likelihood is
$$\cal{l} = k+ \sum_{i=1}^{n} -\log(\sigma x^\lambda\,) -2\log\left(1+\frac{(y_i-\beta_0+\beta_1 x_i)^2}{3\sigma^2x^{2\lambda}} \right)\!.$$
(though the $k$ may freely be dropped; it's still the log of the likelihood)

With pre-specified df, as here, MLE might be used*, but you may have to take some care with the heteroskedasticity parameter $\lambda$. 
One possibility is to start with a robust line, estimate an approximate $\lambda$ from the residuals, and then cycle between estimating the parameters (e.g. possibly via the package heavy) of the line and the value of $\lambda$ (via univariate maximization). 
Alternatively, you could maximize all the parameters simultaneously, for example, via something like maxLik::maxLik (again, starting with a robust line). 
Note that if you use a function which doesn't have weights, you'd need to use a transformation to achieve the weighting (which in turn would require the ability 
to fit a regression without a constant term).
* see, for example, http://davegiles.blogspot.com.au/2013/12/more-on-student-t-regression-models.html
