Hey there! Let's unpack this since you're deep-diving into endogeneity—specifically omitted variable bias (OVB)—and your R code isn't showing the expected correlation between $x_1$ and the error term.

A Quick OVB Refresher #

First, let's recap to make sure we're on the same page:

• Endogeneity occurs when your explanatory variable is correlated with the error term in your regression model. This breaks the Gauss-Markov assumptions and leads to biased coefficient estimates.
• The most common cause is omitted variable bias: this happens when a confounding variable affects both your outcome $y$ and your key explanatory variable $x_1$, but you leave it out of your model. That variable gets absorbed into the error term, creating the correlation you're trying to observe.

Why Your R Code Might Not Be Showing the Correlation? #

Let's build a concrete, reproducible R example that will demonstrate this correlation, so you can compare it to your code and spot where things might be off:

# Set seed for consistent results
set.seed(123)

# Step 1: Simulate our omitted confounding variable x2
x2 <- rnorm(1000, mean = 0, sd = 1)

# Step 2: Generate x1 such that it's correlated with x2 (x2 affects x1)
x1 <- 0.7*x2 + rnorm(1000, mean = 0, sd = 0.5)

# Step 3: Generate y, which depends on both x1 and x2
y <- 1.2*x1 + 0.9*x2 + rnorm(1000, mean = 0, sd = 0.3)

# Step 4: Run the misspecified model (we leave out x2)
misspecified_model <- lm(y ~ x1)

# Extract the error terms from the misspecified model
error_terms <- residuals(misspecified_model)

# Calculate correlation between x1 and error terms
cor(x1, error_terms)

When you run this, you’ll get a correlation around 0.25 (the exact number might shift slightly with different seeds), which confirms the endogeneity caused by OVB.

Common Fixes for Your Original Code #

If your code isn’t producing this correlation, here are the most likely issues:

• No causal link between omitted variable, x1, and y: You need to explicitly generate $x_1$ with a dependence on the omitted variable, and $y$ with a dependence on both $x_1$ and the omitted variable. If either of these relationships is missing, the correlation won’t exist.
• Noise is too strong: If the random error terms in your $x_1$ or $y$ generation have a large standard deviation, they can drown out the underlying correlation. Try reducing the sd value in your rnorm() calls to make the relationships clearer.
• Accidentally included the omitted variable: Double-check your lm() formula—if you included the confounding variable in your model, you’ve eliminated the OVB by design, so no correlation will show up.

Key Takeaway #

Omitted variable bias only creates endogeneity when two conditions are met:

1. The omitted variable is correlated with your explanatory variable ($x_1$).
2. The omitted variable directly affects your outcome ($y$).

If either condition is missing, you won’t see the correlation between $x_1$ and the error term—so that’s almost certainly where your code went off track!

内容的提问来源于stack exchange，提问作者Benykō-Zamurai