In the context of a controlled experiment, consider the simple linear regression formulation Y_i = ?₀ + ?₁X_i + u_i. Let the Y_i be the outcome, X_i the treatment level, and u_i contain all the additional determinants of the outcome. Then A)the OLS estimator of the slope will be inconsistent in the case of a randomly assigned X_i since there are omitted variables present. B)X_i and u_i will be independently distributed if the X_i be are randomly assigned. C)?₀represents the causal effect of X on Y when X is zero. D)E(Y | X = 0)is the expected value for the treatment group.

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In the context of a controlled experiment where the treatment level $X_{i}$ is randomly assigned, $X_{i}$ and the error term $u_{i}$ are independent of each other. This independence is crucial for the Ordinary Least Squares (OLS) estimator of the slope to be unbiased and consistent, as it ensures that the treatment assignment does not systematically correlate with other determinants of the outcome captured in $u_{i}$. Random assignment is a key feature of controlled experiments designed to establish causal relationships by eliminating confounding variables.

In the Context of a Controlled Experiment, Consider the Simple