Longitudinal data and conditioning

Suppose you have a predictor \(X_{it}\) on person \(i\) at time \(t\), and an outcome \(Y_{it}\), where the outcome is generated by \[Y_{it}\mid X_{it}=x = a_i +1\times x+\epsilon\] where \(\epsilon\) are iid \(N(0,1)\) errors and \(a_i\) are iid \(N(0,1)\) random intercepts. I will show you the data generation code later, but this is honestly Further suppose that you fit a linear model by least squares lm(y~x, data=the_data) ## ## Call: ## lm(formula = y ~ x, data = the_data) ## ## Coefficients: ## (Intercept) x ## -0.