# OLS estimate of $\beta$

The random variables $Y_i$, $i = 1,...,n$ are normally and independently distributed with $E(Y_i) = \beta x_i$ and $Var(Y_i) = \sigma^2 \times x_i$. Find the ordinary least squares estimate of $\beta$. What are the expected value and variance of this estimate?

• please add the self-study tag and read its wiki: stats.stackexchange.com/tags/self-study/info what have you tried? What are you stuck on?
– jld
Mar 26, 2018 at 0:40
• Thank you for the link. So far I have attempted to find the OLS estimate for β and got: (sum from i=1 to n of x_i * Y_i) / (sum from i=1 to n of (x_i)^2). I have also tried to compute the variance of the OLS and got: σ^2*x_i / (sum from i=1 to n of (x_i)^2) but I am not sure if this is correct. Also, forgive me for my formatting. I will attempt to learn more about this as I continue to post. Mar 26, 2018 at 0:48

The OLS estimator in linear regression has a standard matrix expression that is well-known. In the case of a model with no intercept term and a single explanatory variable, you have a single vector of explanatory values $\boldsymbol{x}$ and a vector of response values $\boldsymbol{y}$. The OLS coefficient estimator is:
$$\hat{\beta}_1 = (\boldsymbol{x}^\text{T} \boldsymbol{x})^{-1} (\boldsymbol{x}^\text{T} \boldsymbol{y}) = \cdots = \sum_{i=1}^n f(x_i) y_i,$$
where $f$ is some function you can figure out. See if you can simplify the matrix expression into scalar form to get your estimator. Once you have done that, you have the estimator as a linear function of the response values, and so you can get the expected value and variance of the estimator using standard rules for linear functions:
$$\mathbb{E}(\hat{\beta}_1) = \sum_{i=1}^n f(x_i) \mathbb{E}(Y_i) = \cdots = f(\boldsymbol{x}, \beta_1),$$
$$\mathbb{V}(\hat{\beta}_1) = \sum_{I=1}^n f(x_i)^2 \mathbb{V}(Y_i) = \cdots = f(\boldsymbol{x}, \sigma).$$