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.]

2. A Markov chain with state space {1, 2, 3} has transition probability matrix 00 0.3 0.1 a: 0.3 0.3 0.4 0.4 0.1 0.5 (a) Is this Markov chain irreducible? Is the Markov chain recurrent or transient? Explain your answers. (b) What is the period of state 1? Hence deduce the period of the remaining states. Does this Markov chain have a limiting distribution? (c) Consider a general three-state Markov chain with transition matrix 3011 3012 1013 P = P21 P22 P23 1031 P32 P33 Give an example of a specic set of probabilities jag-'3; for which the Markov chain is not irreducible (there is no single right answer to this1 of course l]. 2. Markov chain transitions P = [P/j] = Al- AI- NI- Al- NI- AI- NI- AI- Al- Let X1 be distributed uniformly over the states {0, 1, 2}. Let (Xill be a Markov chain with transition matrix P; thus, P(Xn+1=j \\Xn= i) = Pu , i, j E {0, 1, 2}. (a) Is the information source stationary? ~ (b) Find the stationary distribution of the Markov chain (c) Find the entropy rate of the Markov chain1. Consider the Markov chain with the following transition matrix. 0 0.5 0.5 0.5 0 0.5 0.5 0.5 0 (a) Draw the transition diagram of the Markov chain. (b) Is the Markov chain ergodic? Give a reason for your answer. (c) Compute the two step transition matrix of the Markov chain. (d) What is the state distribution *2 for t = 2 if the initial state distribution for t = 0 is no = (0.1, 0.5, 0.4)