# What happens when there are errors in the predictor variables?

I asked this question yesterday but I can't log into my account anymore: Using a Response Variable as a Predictor Variable in a Future Model?

After thinking about this question (estimate net cost-benefit impact of hiring a new person on factory productivity), I thought of a new way to look at this problem (i.e. what do when predictor variables have errors?). I thought that maybe Instrumental Variables can be used?

Approach 1: Analyzing Net Benefit Directly with Instrumental Variables

1. Model the relationship between employees and processed orders:

$$\hat{P}_t = \hat{\alpha} \hat{E}_t + \hat{\epsilon}_t$$

Where $$\hat{E}_t$$ is the instrumented version of $$E_t$$. We use a first-stage regression:

$$E_t = \gamma_0 + \gamma_1 Z_t + \nu_t$$

$$\hat{E}_t = \hat{\gamma}_0 + \hat{\gamma}_1 Z_t$$

Here, $$Z_t$$ is an instrumental variable (e.g., lagged budget allocations for hiring).

2. The rest of the model remains the same, but use $$\hat{E}_t$$ instead of $$E_t$$ in all equations.

I think the instrumental variable $$Z_t$$ (or $$Z_{t-1}$$) should be correlated with the number of employees but not directly with the error term in the main equation. This might help address potential endogeneity issues?

Is this how instrumental variables are used??