Introduction to Markov Chain Variance Calculation

In this article, we will explore how to calculate the variance of period-to-period change in a Markov chain. A Markov chain is a mathematical system that undergoes transitions from one state to another according to certain probabilistic rules. The concept of variance in a Markov chain refers to the spread or dispersion of changes in income levels over time.

Background and Definitions

A Markov chain is typically represented by a transition matrix P, where each row represents the probability distribution of transitioning from one state to another. The states are usually denoted as i, where i = 1,…,n.

For this problem, we assume that there are n states in our Markov chain. Each state corresponds to a log level of income. Let α be the vector representing the log levels of income for each state. Our goal is to calculate the variance of an individual’s change in income over some number of periods given their starting state.

The Variance Calculation Process

To calculate the variance, we first need to compute the probability distribution of transitioning from one state to another after n periods. This can be achieved by raising the transition matrix P to the power of n (P^n). We then compute the dot product of each state’s basis vector with the resulting matrix.

Let’s define a function e(i) that returns the standard basis vector for state i:

e <- function(i) {
  e_i = rep(0, length(alpha))
  e_i[i] = 1
  return(e_i)
}

The variance is then calculated as the difference between the squared Euclidean distance of the state’s change and its original log level, multiplied by the probability of transitioning from that state:

p2p_variance <- function(n, alpha, P) {
  variance = list()
  pi_n = list()
  
  # Compute the probability distribution after n periods
  for (i in 1:length(alpha)) {
    pi_n[[i]] = e(i) %*% (P ^ n)
    
    # Calculate the squared Euclidean distance of the state's change and its original log level
    beta = (t(alpha) - t(alpha)[i])^2
    
    # Calculate the variance
    variance[[i]] = abs((pi_n[[i]] %*% t(beta)) - (((pi_n[[i]] %*% alpha) - alpha[i]) %^% 2))
    
    # Store the computed variance in a vector
  }
  
  return(t(variance))
}

Example Usage

To calculate the variance of an individual’s change in income over 100 periods, we can use the following code:

# Define the log levels of income for each state (alpha)
alpha = c(3.4965, 3.5835, 3.6636, 3.7377, 3.8067, 3.8712, 3.9318, 
          3.9890, 4.0431, 4.0943, 4.1431)

# Define the transition matrix P
P = rbind(c(0.9004, 0.0734, 0.0203, 0.0043, 0.0010, 0.0003, 
           0.0001, 0.0001, 0.0000, 0.0000, 0.0000),
         c(0.3359, 0.3498, 0.2401, 0.0589, 0.0115, 0.0026, 
           0.0007, 0.0003, 0.0001, 0.0001, 0.0000),
         c(0.1583, 0.1538, 0.3931, 0.2346, 0.0481, 0.0090, 
           0.0021, 0.0007, 0.0003, 0.0001, 0.0001),
         c(0.0746, 0.0609, 0.1600, 0.4368, 0.2178, 0.0397, 
           0.0073, 0.0019, 0.0006, 0.0002, 0.0001),
         c(0.0349, 0.0271, 0.0559, 0.1724, 0.4628, 0.2031, 
           0.0344, 0.0067, 0.0018, 0.0006, 0.0003),
         c(0.0155, 0.0122, 0.0230, 0.0537, 0.1817, 0.4870, 
           0.1860, 0.0316, 0.0066, 0.0018, 0.0009),
         c(0.0066, 0.0054, 0.0100, 0.0204, 0.0529, 0.1956, 
           0.4925, 0.1772, 0.0307, 0.0064, 0.0023),
         c(0.0025, 0.0023, 0.0043, 0.0084, 0.0186, 0.0530, 
           0.2025, 0.4980, 0.1760, 0.0275, 0.0067),
         c(0.0009, 0.0009, 0.0017, 0.0035, 0.0072, 0.0168, 
           0.0490, 0.2025, 0.5194, 0.1721, 0.0260),
         c(0.0003, 0.0003, 0.0007, 0.0013, 0.0029, 0.0061, 
           0.0142, 0.0430, 0.2023, 0.5485, 0.1804),
         c(0.0001, 0.0001, 0.0002, 0.0003, 0.0008, 0.0017, 
           0.0032, 0.0068, 0.0212, 0.1079, 0.8578))
}

# Calculate the variance of an individual's change in income over n periods
variance = p2p_variance(100, alpha, P)

# Print the calculated variances
for (i in 1:length(variance)) {
  print(paste("Variance for state", i, ":", variance[i, 1]))
}

Conclusion

In this article, we have explored how to calculate the variance of period-to-period change in a Markov chain. The p2p_variance function takes as input the number of periods n, the log levels of income for each state alpha, and the transition matrix P. It computes the probability distribution after n periods and calculates the squared Euclidean distance between each state’s change and its original log level.

The calculated variance is stored in a vector where each element corresponds to a state. The p2p_variance function can be used to calculate the variance of an individual’s change in income over multiple periods.

Last modified on 2023-08-11