b) We are interested in estimating and constructing a confidence interval

for the percentage change in price when a 200-square-foot bedroom is

added to a house. How can you construct this confidence interval?

What assumptions are needed to do so? State all assumptions and ex-

plain why you need the assumptions. Calculate the estimates and con-

fidence interval, and interpret them.

(c) In order to construct a confidence interval we need to estimate a stan-

dard error for the estimate in part (b). Provide the way to estimate the

standard error using another multiple regression model (Hint: Define a

parameter which represents the above estimator (in part (b)). Then one

can estimate a standard error using this relationship with another multi-

ple regression model). Based on the estimated standard error, construct

a 95% confidence interval.

(d) Suppose now u does not follow normal distribution any more. Elabo-

rate how can you calculate the confidence interval in (b) and (c). And

calculate such confidence interval using R.

3. Suppose that the model

pctstck = ?0 + ?1funds + ?2risktol + u,

satisfies the first four Gauss-Markov assumptions, where pctstck is the per-

centage of a worker's pension invested in the stock market, funds is the num-

ber of mutual funds that the worker can choose from, and risktol is some

measure of risk tolerance (larger risktol means the person has a higher tol-

erance for risk). If funds and risktol are positively correlated, what are the

bias and inconsistency in ??

1, the slope coefficient in the simple regression of

pctstck on funds? Moreover, how do you interpret the estimates you obtain

from this simple regression? [You SHOULD answer these questions crystal

clear.]

Consider a Markov chain {Xn, n = 0, 1, . .. } on the state space S = {0, 1, 2}. Suppose that the Markov chain has the transition matrix 0 1 - P = 1 0 0 N/H 1. Show that the state space is irreducible. 2. Show that the Markov chain is periodic. Find the period of the Markov chain. 3. Let h denote a stationary mass of the Markov chain. Find h(x) for all x E S.PL Which of the following statements correctly differentiates between positive and normative A. Positive economics is based on judgments, whereas normative economics is not. OB. Positive economics describes what people ought to do, whereas normative econon O C. Positive economics is descriptive, whereas normative economics is advisory. O D. Positive economics can only be applied to microeconomics, whereas normative ecoConsider a standard chessboard with an 8 x 8 grid of possible locations. We define a Markov chain by randomly moving a single chess piece on this board. The initial location Xo is sampled uniformly among the 82 = 64 squares. At time t, the piece then chooses Xt+1 by sampling uniformly from the set of legal moves given its current location Xt. For a description of legal chess moves, see: http://en. wikipedia. org/wiki/Rules_of_chess#Basic_moves. a) Suppose the chess piece is a king, which can move to any of the 8 adjacent squares. Is the Markov chain irreducible? Is the Markov chain aperiodic? b) Suppose the chess piece is a bishop. Is the Markov chain irreducible? Is the Markov chain aperiodic? c) Suppose the chess piece is a knight. Is the Markov chain irreducible? Is the Markov chain aperiodic?1. Consider the Markov chain with state space {0, 1, 2} and transition matrix 1 2 P = HO 2 (a) Suppose Xo = 0. Find the probability that X2 = 2. (b) Find the stationary distribution of the Markov chain. (c) What proportion of time does the Markov Chain spend in state 2, in the long run? (d) Suppose X5 = 1. What is the expected additional number of steps (after time 5) until the first time the Markov chain will return to state 1