I want to compute the probability $P_{ij}$ to move from decile $i$ in one period to decile $j$ in the next period in the distribution of a stationary AR(1) process

$$Y_t = \rho Y_{t - 1} + \upsilon_t,$$

where $\upsilon_t$ is an i.i.d. shock with distribution $N(0, \sigma^2)$ and $|\rho| < 1$. The stationary distribution of $Y$ is given by $N(0, \tau^2)$, where $\tau \equiv \sigma / \sqrt{1 - \rho^2}$.

Is the following expression what I should be trying to compute? Or is there some more straightforward way to do it?

\begin{align} P_{ij} &= \int_{(i - 1) / 10}^{i / 10} \underbrace{\Pr\biggl(\Phi^{-1}\left(\frac{j - 1}{10}\right) \leq \frac{Y_t}{\tau} < \Phi^{-1}\left(\frac{j}{10}\right)\biggr)}_{\text{probability to move to decile $j$ in time $t$}} \underbrace{\frac{\phi\left(\Phi^{-1}(x)\right)}{0.1}}_{\substack{\text{share of decile $i$} \\[0.5ex] \text{at time $t - 1$}}} d x, \end{align} where $\Phi^{-1}$ is the inverse cumulative distribution function of a standard normal variable and $\phi$ its probability density function.


1 Answer 1


Letting $Z_t=Y_t/\tau$ where $\tau^2=\sigma^2/(1-\rho^2)$ and $d_i=\Phi^{-1}(\frac i{10})$, $i=1,2,\dots,9$ denote the corresponding deciles, we have, from the definition of conditional probability, $$ \begin{align} P_{ij}&=\Pr(d_{j-1}<Z_t\le d_j|d_{i-1}<Z_{t-1}\le d_i) \\&=\frac{\Pr(d_{j-1}<Z_t\le d_j \cap d_{i-1}<Z_{t-1}\le d_i)} {\Pr(d_{i-1}<Z_{t-1}\le d_i)} \\&=\frac{\Phi_2(d_i,d_j)-\Phi_2(d_i,d_{j-1})-\Phi_2(d_{i-1},d_j)+\Phi_2(d_{i-1},d_{j-1})}{1/10} \end{align} $$ where $\Phi_2$ is the cdf of the bivariate standard normal distribution with correlation $\rho$ (available as the function mvtnorm::pmvnorm in R).

It does not follow that the process is a first order Markov chain with the $P_{ij}$'s as the transition probabilities. This is demonstrated numerically below by computing higher order transisition probabilities based on the joint multivariate normal distribution of $Z_t,Z_{t-1},\dots,Z_{t-n}$ using a straightforward generalisation of the above expression. Indeed, the resulting Markov chain does not appear to be of any finite order, analogous to how an for example an AR(1) process plus white noise becomes a non-Markovian ARMA(1,1) process with partial autocorrelation function only tailing off.

# Compute the probability of transitioning from state i to j
# If i is a vector, the probabability of transitioning to state j
# given that the previous states are i[1], i[2], ..., i[n] is
# computed.
p <- function(i, j, rho) {
  order <- length(i)
  num <- pmvnorm(lower = qnorm(c(j - 1, i - 1)/10), 
          upper = qnorm(c(j, i)/10), 
          sigma = toeplitz(rho^(0:order)), 
          keepAttr = FALSE)
  den <- pmvnorm(lower = qnorm((i - 1)/10), 
          upper = qnorm(i/10), 
          sigma = toeplitz(rho^(0:(order-1))), 
          keepAttr = FALSE)
  names(den) <- NULL
# Probability of remaining in state 1
p(1, 1, .5)
#> [1] 0.3240152
# Probability of remaining in state 1 given that the processes
# has been in state 1 for up to five previous time points
p(c(1, 1), 1, .5)
#> [1] 0.3476454
p(c(1, 1, 1), 1, .5)
#> [1] 0.3520791
p(c(1, 1, 1, 1), 1, .5)
#> [1] 0.3533796
p(c(1, 1, 1, 1, 1), 1, .5)
#> [1] 0.353544

Created on 2023-12-30 with reprex v2.0.2

  • 1
    $\begingroup$ I've never seen this derivation before so it was neat. But could you explain why the resulting probabilities are not ( atleast approximately ) the transition probabilities from decile to decile of a first order markov chain. $\endgroup$
    – mlofton
    Dec 29, 2023 at 14:58
  • 1
    $\begingroup$ Frederik: It seems like a first order markov chain to me also but, given that I would have not have come up with Jarle's derivation, I could be missing something. Jarle: Is it because the probabilities change each time you come back to an old state that you were at earlier ? $\endgroup$
    – mlofton
    Dec 30, 2023 at 5:24
  • 1
    $\begingroup$ @JarleTufto Ah! That sounds right. But in that case perhaps I misphrased (or you misunderstood) my original question. I’m after an expression for the share of the probability mass in one decile that moves to another. So, the information about $Z_t$ should not be incomplete. In fact, I know the exact distribution of $Z_t$ in each decile. $\endgroup$
    – Fredrik P
    Dec 30, 2023 at 16:19
  • 2
    $\begingroup$ @JarleTufto You might be correct that question lacks in clarity. I think that I perhaps shouldn’t have called what I’m after a “stochastic matrix”. And after thinking about it a bit more I realize that you are perfectly right saying the probabilities are not those of a first order Markov chain. $\endgroup$
    – Fredrik P
    Dec 30, 2023 at 18:54
  • 1
    $\begingroup$ I'm not seeing justification for coarsening the data to deciles. And deciles are manipulated by how the data are sampled, so analyses based on deciles are hard to interpret. $\endgroup$ Dec 31, 2023 at 7:52

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