I. Markov Chains A Markov chain (or process) is one in which future outcomes are determined by a current state. Future outcomes are based on probabilities. The probability of moving to a certain state depends only on the state previously occupied and does not vary with time. An example of a Markov chain is the maximum education achieved by children based on the highest educational level attained by their parents, where the states are (1) earned college degree, (2) high school diploma only, (3) elementary school only. If pj is the probability of moving from state i to state j, the transition matrix is the m × m matrix Pi1 P12 Pim ... P = LPm1 Pm2 Pmm, ... The table represents the probabilities for the highest educational level of children based on the highest educational level of their parents. For example, the table shows that the probability pz1 is 40% that parents with a high-school education (row 2) will have children with a college education (column 1). 4. If Pis the transition matrix of a Markov chain, the (i, j)th entry of P" (nth power of P) gives the probability of passing from state i to state j in n stages. What is the probability that the grandchild of a college graduate is a college graduate? 5. What is the probability that the grandchild of a high school graduate finishes college? 6. The row vector v(0) = [0.317 0.565 0.118] represents the proportion of the U.S. population 25 years or older that has college, high school, and elementary school, respectively, as the highest educational level in 2013.* In a Markov chain the probability distribution v(*) after k stages is v (k) = v(0) pk, where Pk is the kth power of the transition matrix. What will be the distribution of highest educational attainment of the grandchildren of the current population? 7. Calculate P³, P*, P³, . Continue until the matrix does not change. This is called the long-run or steady-state distribution. What is the long-run distribution of highest educational attainment of the population? Highest Educational Maximum Education That Children Achieve Level of Parents College High School Elementary College 80% 18% 2% High school 40% 50% 10% Elementary 20% 60% 20% 1. Convert the percentages to decimals. 2. What is the transition matrix? 3. Sum across the rows. What do you notice? Why do you think that you obtained this result? *Source: U.S. Census Bureau.