Two-dimensional sufficient statistic for OLS model

Suppose that the random variables $$Y_1,...,Y_n$$ satisfy $$Y_i=\beta x_i + \epsilon_i, i=1,...,n$$ where $$x_1,...,x_n$$ are fixed constants, and $$\epsilon_1,...,\epsilon_n$$ are i.i.d. $$N(0,\sigma^2), \sigma^2$$ unknown.

(a) Find a two-dimensional sufficient statistic for $$(\beta, \sigma^2)$$.

I have no idea where to even start with this one. Prof didn't do any examples like this in class. Any help would be appreciated.

• Check out our policy on self-study questions. Mar 15, 2020 at 8:24
• Jul 21, 2020 at 15:06

Let's compute the likelihood function for this model. Since $$Y_i=\beta x_i+\epsilon_i\sim\mathcal{N}(\beta x_i,\sigma^2)$$: $$\mathcal{L}(\beta,\sigma^2)=\prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma²}}\exp\left(-\frac{(y_i-\beta x_i)^2}{2\sigma^2}\right)\\ =\frac{1}{({2\pi\sigma^2})^\frac{n}{2}}\exp\left(\frac{1}{2\sigma^2}\sum_{i=1}^n (y_i-\beta x_i)^2\right)\\ =\frac{1}{({2\pi\sigma^2})^\frac{n}{2}}\exp\left(\frac{1}{2\sigma^2}\sum_{i=1}^n y_i^2-\frac{1}{2\sigma^2}\sum_{i=1}^n2\beta x_iy_i+\frac{1}{2\sigma^2}\sum_{i=1}^n\beta^2x_i^2\right)\\ =\frac{1}{({2\pi\sigma^2})^\frac{n}{2}}\exp\left(\frac{1}{2\sigma^2}\sum_{i=1}^n y_i^2\right)\exp\left(-\frac{\beta}{\sigma^2}\sum_{i=1}^nx_iy_i\right)\exp\left(\frac{\beta^2}{2\sigma^2}\sum_{i=1}^nx_i^2\right)$$ Now, since the likelihood function can be factorized in terms that depend on the sample only through $$\sum_{i=1}^ny^2$$ and $$\sum_{i=1}^nx_iy_i$$, we can affirm that $$T(Y)=\left(\sum_{i=1}^nx_iY_i,\sum_{i=1}^nY_i^2\right)$$ is a sufficient statistic.