(a) Derive the conditions under which the standard OLS estimator is unbiased. Clearly state the assumptions and provide a step-by-step derivation. Provide an example of a data situation where the OLS estimator is not appropriate, and explain which assumption is violated and why. [math: [4]] (b) Explain the role of [math: λ] in the Ridge and Lasso objective functions, and describe what happens to the bias and variance of the estimator as [math: λ→0] and as [math: λ→∞]. Discuss how [math: λ] can be selected in practice. [math: [3]] (c)In a simulation study with [math: 10,000] replications, three estimators -- OLS, Ridge, and Lasso -- are used to estimate the same regression coefficient. The figure below displays the resulting histograms of the coefficient estimates from the three methods. The true parameter value is [math: 3]. Using the shape, spread, and location of the three histograms, identify which colour corresponds to each method: OLS, Ridge, and Lasso. Justify your answer. [math: [3]][Fill in the blank]多项填空题