# Can the parameters be identified separately in this model?

Suppose we have a model

$$y_i=\epsilon_i^1+\epsilon_i^2$$

$$\epsilon_i^1\sim N(0,\sigma_1^2)$$

$$\epsilon_i^2\sim N(0,\sigma_2^2)$$

$$\epsilon_i^1\perp\epsilon_i^2$$

Assume $$y_i$$ is i.i.d. I have derived the log-likelihood function to be

$$-\frac{n}{2}\ln(\sigma_1^2+\sigma_2^2)-\frac{n}{2}\ln 2\pi-\frac{1}{2(\sigma_1^2+\sigma_2^2)}\sum_{i=1}^n y_i^2$$

Can $$\sigma_1^2$$ and $$\sigma_2^2$$ be identified separately in this model?

• Because $y_i \sim N(0, \sigma_1^2+\sigma_2^2),$ you can only hope to identify the sum $\sigma_1^2+\sigma_2^2.$ That is apparent in the log likelihood, which clearly is a function of the sum only. BTW, the notation is confusing, because it's natural to interpret the superscripts as powers rather than indices -- and the power interpretation makes some sense until one contemplates the meaning of the independence condition. Most readers will therefore have to change their initial interpretations and re-read this question to understand what you mean. That's asking a lot.
– whuber
Commented Sep 20, 2022 at 12:28
• @whuber Thank you very much for your comment. Commented Sep 20, 2022 at 12:31