Question Two Consider the following statements: a) A matrix Amxn is nonsingular ar b) Amxn is square c) Amxn has linear independence d) The rank of Am×n is r(A) = min{m, n} Use "necessary condition”, “sufficient condition”, or “necessary and sufficient condition” to describe how Statement a) is related to Statements b), c), and d). Here are the conditions • Necessary Condition: q constitutes a necessary condition of p; mathematically, pq, read as: if p, then q; or, p implies q • Sufficient Condition: q is a sufficient condition for p; mathematically, p = q, read as: p if q; or, if q, then p; or, q implies p • Necessary and Sufficient Condition: q is both necessary and sufficient for p; mathematically, pq, read as: p if and only if q

Which of the following statements is true under the Gauss-Markov assumptions? a. Among a certain class of estimators, OLS estimators are best linear unbiased but are asymptotically inefficient. b. Among a certain class of estimators, OLS estimators are biased but asymptotically efficient. c. Among a certain class of estimators, OLS estimators are best linear unbiased and asymptotically efficient. d. The LM test is independent of the Gauss-Markov assumptions. O a. a O b. b О с. с O d. d

f(x1,x2) 4 2 0 -2 -4 −1 2 0 1 1 0 2 -1 X1 3-2 x2 Figure 2: Graph of the function f(x1, x2) = 2x1 = x²-x+3 Consider the following function: f(x1, x2) = 2x1 - x-x+3. - For your visual convenience, a portion of the function is depicted in Figure 2. (a) Compute the gradient vector of f. (b) Compute the Hessian matrix of f. (c) Determine all points that yield stationary values of f. (d) Use the second-order conditions to determine whether each of the stationary val- ues of f is a maximum, a minimum, or neither.

Consider the following linear program: Maximize 30X + 10Y Subject to: 3X + Y ≤ 300 X + Y ≤ 200 X ≤ 100 Y ≥ 50 X,Y ≥ 0 (a) Solve the problem graphically. Is there more than one optimal solution? Explain. (b) Are there any redundant constraints?

Consider the following linear programming model: Max: X₁² + X₂ + 3X3 O certainty O proportionality O linearity O divisibility Subject to: X₁ + X₂ £ 3 X₁ + X₂ £ 1 X₁, X₂ ³0 This problem violates which of the following properties or assumptions?

1. Write this system of equations in matrix form. X1 + X2 + X3 = 1; x1 – X2 = 4; X2 + X3 = 7.

Solve each of the following equations for x x+7=14 5x+4=24 12−4x=−20   Organize each of the following equations to express Pas a function of Q. QQ as a function of P PP as a function of Q Q=25−P      Q=12−3P      6Q=14−2P

Consider the linear spline function f(x) = 1 + x + h1(x, 2) where h1(x,2) = x-2 if x>2 and 0 otherwise. What is the slope of f at the point x=-1?

C. Suppose your neighbor is a physics major who also races motorcycles, and she tells you that horsepower can be expressed in terms of the following equation: H = MDA/S where M=mass, D-distance, A=acceleration and S and H are as defined above. Based on this conversation with your neighbor, you decide to change the functional form of the relationship to include 1/H rather than H as an explanatory variable because that's the appropriate theoretical relationship between the two variables. Estimate the above equation, substituting 1/H for H. Which of the two models do you prefer and why?

Given that f(x) = b^x for b > 0, determine the values of b that will make the function f an increasing or a decreasing function.