3.4 Extend the Roll (1984) model to allow for a serially correlated order- type indicator variable. In particular, let I, be a two-state Markov with -1 and 1 as the two states, and derive expressions for the moments of AP, in terms of s and the transition probabilities of I. How do these results differ from the IID case? How would you reinterpret Roll's (1984) findings in light of this more general model of bid-ask bounce?
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- 12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction randomly choose which way to go. Figure 3.28 (a) Construct the transition matrix for the Markov chain that models this situation. (b) Suppose we start with 15 robots at each junction. Find the steady state distribution of robots. (Assume that it takes each robot the same amount of time to travel between two adjacent junctions.)The following are some applications of the Markovinequality of Exercise 29:(a) The scores that high school juniors get on the verbalpart of the PSAT/NMSQT test may be looked upon asvalues of a random variable with the mean μ = 41. Findan upper bound to the probability that one of the studentswill get a score of 65 or more.(b) The weight of certain animals may be looked uponas a random variable with a mean of 212 grams. If noneof the animals weighs less than 165 grams, find an upperbound to the probability that such an animal will weigh atleast 250 grams.Consider the problem of sending a binary message, 0 or 1, through a signal channelconsisting of several stages, where transmission through each stage is subject to a fixedprobability of error α. Suppose that X0 = 0 is the signal that is sent and let Xn, be thesignal that is received at the nth stage. Assume that {Xn} is a Markov chain with transitionprobabilities, Poo = P11 = 1- α and P01 = P10 = α, where 0 < α < 1.(a) Determine P {Xo = 0, X1 = 0, X2 = 0}, the probability that no error α occurs up tostage n = 2.(b) Determine the probability that a correct signal is received at stage 2.
- 1- The number of items produced in a factory during a week is known to be a randomvariable with mean 50● Using Markov's inequality, what can you say about the probability that this week'sproduction exceeds 75?● If the variance of one week's production is equal to 25, then using Chebyshev'sinequality, what can be said about the probability that this week's production isbetween 40 and 60?Consider the three competing grocery stores in town, T’San Grocery, 654 Grocery and Sunlight Grocery. Consider the transition probabilities below: To To To From T'San 654 Sunlight T'San 0.65 0.15 0.2 654 0.15 0.80 0.05 Sunlight 0.10 0.15 0.75 a. Compute the steady-state probabilities for this three-state Markov process.b. What market share will each store obtain?A study of pine nut crops in the American southwest from 1940 to 1947 hypothised that nut production followed a Markov chain. The data suggested that if one year's crop was good, then the probabilities that the following year's crop would be good, fair, or poor were 0.08, 0.07, and 0.85, respectively; if one year's crop was fair, then the probabilities that the follow-ing year's crop would be good, fair, or poor were 0.09, 0.11, and 0.80, respectively; if one year's crop was poor, then the probabilities that the following year's crop would be good, fair, or poor were 0.11, 0.05, and 0.84, respectively. (a) Write down the transition matrix for this Markov chain. (b) If the pine nut crop was good in 1940, find the probabilities of a good crop in the years 1941 through 1945. (c) In the long run, what proportion of the crops will be good, fair, and poor
- Based on the data that has been gathered what is the probability of disasters in July & Augustus using Markov chain ?For each of the following transition matrices, do the following: (1) Determine whether the Markov chain is irreducible. (2) Find a stationary distribution π; is the stationary distribution unique? (3)Give the period of each state. (4) Without using any software package, find P200 approximately.Data collected from selected major metropolitan areas in the eastern United States show that 3% of individuals living within the city limits move to the suburbs during a one-year period, while 1% of individuals living in the suburbs move to the city during a one-year period. Answer the following questions assuming that this process is modeled by a Markov process with two states: city and suburbs. (a) Prepare the matrix of transition probabilities. To From City Suburbs City Suburbs (b) Compute the steady-state probabilities. (Enter your probabilities as fractions.) City?1= Suburbs?2=
- What are the difficulties in estimating the following model? Use as much detail as possible in answering this question while considering the Gauss-Markov assumptions and OLS estimator. Economic productivity = β0 + β1Unemployment + β2Innovation + θiControls + ei Where unemployment is the average unemployment rate of a country and innovation is an index of R&D performance.The computer center at Rockbottom University has been experiencing computer downtime. Let us assume that the trials of an associated Markov process are defined as one-hour periods and that the probability of the system being in a running state or a down state is based on the state of the system in the previous period. Historical data show the following transition probabilities. To From Running Down Running 0.80 0.20 Down 0.30 0.70 (a) If the system is initially running, what is the probability of the system being down in the next hour of operation? (b) What are the steady-state probabilities of the system being in the running state and in the down state? (Enter your probabilities as fractions.) Running?1=Down?2=A group of high-risk automobile drivers (with three moving violations in one year) are required, according to random assignment, either to attend a traffic school or to perform supervised volunteer work. During the subsequent five-year period, these same drivers were cited for the following number of moving violations: NUMBER OF MOVING VIOLATIONS TRAFFIC SCHOOL VOLUNTEER WORK 0 26 0 7 15 4 9 1 7 1 0 14 2 6 23 10 7 8 Why might the Mann–Whitney U test be preferred to the t test for these data? Use U to test the null hypothesis at the .05 level of significance. Specify the approximate p-value for this test result.