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?

  • $\begingroup$ 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? $\endgroup$
    – jld
    Mar 26, 2018 at 0:40
  • 1
    $\begingroup$ 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. $\endgroup$
    – Jake Tyler
    Mar 26, 2018 at 0:48

1 Answer 1


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:

$$\mathbb{E}(\hat{\beta}_1) = \sum_{i=1}^n f(x_i) \mathbb{E}(Y_i) = \cdots = f(\boldsymbol{x}, \beta_1),$$

$$\mathbb{V}(\hat{\beta}_1) = \sum_{I=1}^n f(x_i)^2 \mathbb{V}(Y_i) = \cdots = f(\boldsymbol{x}, \sigma).$$

Once you have got the results, have a look to see if the estimator is biased (i.e., is its expected value equal to the thing it is used to estimate) and also have a look to see how the different data values impact the variance. This is a problem where you have heteroscedasticity, but you are using the OLS estimator despite that, so it is a good exercise to see the consequences of this kind of misspecification of the estimator. Let us know how you go.


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