Question

Consider the following linear regression model 1. For any X = x, let Y = xB+U, where B erk. 2. X is exogenous. 3. The probability model is {f(u; ) is a distribution on R: Ef [U] = 0, VAR; [U] = 62,0 >0}. 4. Sampling model: {Y}}}=1 is an independent sample, sequentially generated using Y; = xiß +Ui, where the U; are IID(0,62). (i) Let K > 0 be a given number. We wish to estimate B using least-squares subject to the constraint BTB < K? Write down the Lagrangian for this optimization problem. Show all of your derivations. (ii) What is the constraint qualification for this problem ? Does this optimization problem satisfy it? If so, then what is its consequence ? Explain your answers. (iii) What are the Karush-Kuhn-Tucker conditions for this optimization problem. Explain your answer. (iv) Compute the optimal solution of this optimization problem. Show all of your derivations. (v) What is the relationship of the optimal solution in part (iv) of this question with the ridge regression estimator ? How does the optimal value of the Lagrange multiplier relate to the value of K ?

Answer 1

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