# Regression with noises in X. Should I use the unbiased estimator or the OLS estimator for forecasting?

I am working with a dataset that includes variables $$Y$$ and $$X$$. I assume that

$$Y = \beta X + \epsilon$$

satisfies all the assumptions of OLS. Based on industry knowledge, I know that theoretically $$\beta = 1$$. However, I am aware that $$X$$ contains noise, meaning I have observed $$\hat{X} = X + u$$ where $$u$$ represents the noise. When I run the OLS estimate, I find that the estimated $$\beta$$ is less than 1 due to the noise in $$X$$, which inflates the variance of $$X$$.

I have different pairs of $$(Y, X)$$ from various groups. I can verify my assumption by observing that in larger groups (where data variability is reduced by the Central Limit Theorem), the fitted $$\beta$$ is closer to 1.

Given this, my question is: If I want to make an out-of-sample forecast, should I use 1 as my $$\beta$$ or should I use the OLS $$\beta$$ estimate, which is biased?

On one hand, it seems that using the unbiased $$\beta$$ of 1 is preferable since it is theoretically unbiased. However, if I assume the same noise $$u$$ is present in my out-of-sample $$X$$, my residual term will have larger variance because it includes the term $$\beta^2 \text{Var}(u)$$. Using a smaller, biased $$\beta$$ might help reduce the residual variance, despite the bias.

Is this a bias-variance trade-off scenario? What is the best approach here if my goal is to minimize the out-of-sample residual MSE?

• where data variability is reduced by the Central Limit Theorem What does this mean? $//$ How do you know that $\beta = 1?$ What about an intercept term? $//$ should I use the OLS β estimate, which is biased? Why would the OLS estimator be biased?
– Dave
Commented Jul 10 at 21:59
• OLS is biased because it doesn’t assume noise in X. So if you do ols estimate with cov(x_observed, y)/var(x_observed), it will become cov(x,y)/(var(x)+var(u)), assuming u is independent from y. So basically your beta coefficient will be shrinked. For the data I can’t disclose too much since it is confidential. But it is some physical data and we know that beta should be 1 in theory Commented Jul 10 at 22:27
• Then it sounds like you have a good starting point to craft an informative prior distribution and approach this with Bayesian estimation.
– Dave
Commented Jul 11 at 3:06