# Maximum Likelihood Estimate (MLE) equivalent to finding $\hat y$ in linear regression with i.i.d. Gaussian noise distribution

In an assignment I need to show that for linear regression, with the noise i.i.d. Gaussian distributed $\epsilon_i \sim N(0,\sigma^2)$, that finding the Maximum Likelihood Estimate (MLE) is equivalent to finding $\hat{y}$, which minimizes the sum of the least square errors to the $x$'s.

From Wikipedia's page on linear regression I saw a section stating that the MLE is identical to the OLS (Ordinary Least Squares) estimate when the noise has the above-mentioned distribution. Could this help solve the problem? I would greatly appreciate a step-by-step guide to solving this problem.

• If it's an assignment please add the self-study tag. – Christoph Hanck Apr 2 '15 at 4:34
• I edited your question - please check if it is ok. I changed $\epsilon_i - N(0,\sigma^2)$ to $\epsilon_i \sim N(0,\sigma^2)$ since I imagine it was a mistake but if not feel free to revert my edit. – Tim Apr 2 '15 at 6:18

Let $\mu_i = \alpha + x_i^T\beta$ where $x_i$ is a p-dimensional vector of observed covariates, and $\beta$ is a p-dimensional vector of regression coefficients. Then let $y_i \overset{ind}{\sim} N(\mu_i,\sigma^2)$. Write the joint distribution of the $y_i$'s and find the log-likelihood. You will see that the objective function for $\alpha$ and $\beta$ is the sum of squares $\sum_{i=1}^n (y_i - \mu_i)^2$ as it is for a least squares problem.