Multiplicative autocorrelation in stationary Markov processes

Proposed answer to the following question(s):

Summary

A stationary process \(Z_t\) has multiplicative autocorrelation when \[\begin{eqnarray*} \operatorname{Cor}\!\left[{ Z_t}, { Z_r}\right] & = & \operatorname{Cor}\!\left[{ Z_t}, { Z_s}\right] \operatorname{Cor}\!\left[{ Z_s}, { Z_r}\right] \end{eqnarray*}\] for all \(t \le s \le r\). Autocorrelation is defined as \[\begin{eqnarray*} \operatorname{Cor}\!\left[{ Z_t}, { Z_s}\right] & := & \frac{ \operatorname{Cov}\!\left[{ Z_t}, { Z_s}\right] }{\sigma^2} \end{eqnarray*}\] with \(\sigma^2 = \operatorname{Var}({ Z_t})\).

A stationary autoregressive process has multiplicative autocorrelation [1]. However, not all stationary Markov processes have multiplicative autocorrelation. See the section below about a real-valued 3-state Markov chain for a counterexample.

Among discrete-time stationary processes, only autoregressive processes have multiplicative autocorrelation. Some Markov processes are not obviously autoregressive processes even though technically they are. For example, all stationary real-valued two-state Markov chains are autoregressive (and thus also have multiplicative autocorrelation).

Multiplicative autocorrelation implies autoregression

Consider any real-valued discrete-time stationary Markov process \(Z'_t\) and translate it to \(Z_t := Z'_t - \operatorname{E}\!\left[{ Z'_t}\right]\) without loss of generality.

Let \[\begin{eqnarray*} \sigma^2 & := & \operatorname{Var}({ Z_t}) \\ \rho & := & \operatorname{Cov}\!\left[{ Z_t}, { Z_{t+1}}\right] / \sigma^2 \end{eqnarray*}\]

Multiplicative autocorrelation implies \[\begin{eqnarray*} \operatorname{Cor}\!\left[{ Z_t}, { Z_{t+n}}\right] & = & \rho^n \\ \operatorname{Cov}\!\left[{ Z_t}, { Z_{t+n}}\right] & = & \rho^n \sigma^2 \end{eqnarray*}\]

Define what will be shown to be “white noise” of \(Z_t\) as autoregressive process: \[\begin{eqnarray*} \epsilon_t & := & Z_t - \rho Z_{t-1} \end{eqnarray*}\] By convenient translation, \[\begin{eqnarray*} \operatorname{E}\!\left[{ Z_t}\right] & = & 0 \\ \epsilon_t & = & 0 \\ \operatorname{Cov}\!\left[{ Z_t}, { Z_s}\right] & = & \operatorname{E}\!\left[{ Z_t Z_s}\right] \\ \operatorname{E}\!\left[{ Z_t^2}\right] & = & \sigma^2 \\ \operatorname{E}\!\left[{ Z_t Z_{t+1}}\right] & = & \rho \end{eqnarray*}\]

Consider any \(n > 0\). \[\begin{eqnarray*} \operatorname{E}\!\left[{ \epsilon_t \epsilon_{t+n}}\right] & = & \operatorname{E}\!\left[{ (Z_t - \rho Z_{t-1})(Z_{t+n} - \rho Z_{t+n-1})}\right] \\ & = & \operatorname{E}\!\left[{ Z_t Z_{t+n}}\right] + \rho^2 \operatorname{E}\!\left[{ Z_{t-1} Z_{t+n-1}}\right] - \rho (\operatorname{E}\!\left[{ Z_t Z_{t+n-1}}\right] + \operatorname{E}\!\left[{ Z_{t-1} Z_{t+n}}\right]) \\ & = & (1 + \rho^2) \rho^n \sigma^2 - \rho (\rho^{n-1} \sigma^2 + \rho^{n+1} \sigma^2) \\ & = & 0 \end{eqnarray*}\] thus \(\epsilon_t\) satisfies the “white noise” condition for expressing \(Z_t\) as the autoregressive process \[\begin{eqnarray*} Z_{t+1} & = & \rho Z_t + \epsilon_t \end{eqnarray*}\] QED

Real-valued 2-state Markov chain

For any stationary two-state Markov chain [1] \(Z_t\), \[\begin{eqnarray*} \operatorname{Cor}\!\left[{ Z_t}, { Z_0}\right] & = & \operatorname{Cor}\!\left[{ Z_t}, { Z_s}\right] \operatorname{Cor}\!\left[{ Z_s}, { Z_0}\right] \end{eqnarray*}\]

Proof Let \[\begin{eqnarray*} q_1 := \operatorname{P}({ Z_t = a_1 }) \\ q_0 := \operatorname{P}({ Z_t = a_0 }) \end{eqnarray*}\] Map \(Z_t\) to a more convenient \[\begin{eqnarray*} Y_t & := \frac{Z_t - a_0}{a_1 - a_0} \end{eqnarray*}\] Since \(Y_t\) only equals \(0\) or \(1\): \[ \operatorname{E}\!\left[{ Y_t}\right] = \operatorname{E}\!\left[{ Y_t^2}\right] = q_1 \] and thus \[ \operatorname{Var}({ Y_t}) = q_1 - q_1^2 = q_1 q_0 \] For convenience let \[\begin{eqnarray*} p_0 & := & \operatorname{P}({ Y_1 = 0 \mid Y_0 = 1 }) \\ p_1 & := & \operatorname{P}({ Y_1 = 1 \mid Y_0 = 0 }) \\ s & := & p_0 + p_1 \end{eqnarray*}\] Since \(Y_t\) is stationary, it follows that \(q_i = p_i/s\) for \(i \in \{0,1\}\). In preparation for induction, assume \[\begin{eqnarray*} \operatorname{P}({ Y_t = 1 \mid Y_0 = 1 }) & = & q_1 + q_0 (1-s)^t \\ \operatorname{P}({ Y_t = 1 \mid Y_0 = 0 }) & = & q_1 - q_1 (1-s)^t \end{eqnarray*}\] It must follow that \[\begin{eqnarray*} \operatorname{P}({ Y_{t+1} = 1 \mid Y_0 = 1 }) & = & \operatorname{P}({ Y_{t+1} = 1 \mid Y_1 = 1 }) (1-p_0) + \operatorname{P}({ Y_{t+1} = 1 \mid Y_1 = 0 }) p_0 \\ & = & [q_1 + q_0 (1-s)^t] (1-p_0) + [q_1 - q_1 (1-s)^t] p_0 \\ & = & q_1 + [q_0 (1-p_0) - q_1 p_0](1-s)^t \\ & = & q_1 + [q_0 (1-p_0) - (1- q_0) p_0](1-s)^t \\ & = & q_1 + [q_0 - p_0](1-s)^t \\ & = & q_1 + [q_0 - q_0 s](1-s)^t \\ & = & q_1 + q_0 (1-s)^{t+1} \\ \end{eqnarray*}\] and \[\begin{eqnarray*} \operatorname{P}({ Y_{t+1} = 1 \mid Y_0 = 0 }) & = & \operatorname{P}({ Y_{t+1} = 1 \mid Y_1 = 1 }) p_1 + \operatorname{P}({ Y_{t+1} = 1 \mid Y_1 = 0 }) (1-p_1) \\ & = & [q_1 + q_0 (1-s)^t] p_1 + [q_1 - q_1 (1-s)^t] (1-p_1) \\ & = & q_1 + [q_0 p_1 - q_1 (1-p_1)](1-s)^t \\ & = & q_1 + [(1 - q_1) p_1 - q_1 (1-p_1)](1-s)^t \\ & = & q_1 + [p_1 - q_1](1-s)^t \\ & = & q_1 + [q_1 s - q_1](1-s)^t \\ & = & q_1 - q_1 (1-s)^{t+1} \\ \end{eqnarray*}\] which completes induction, noting the base case of \(t=0\) is true.

Due to the convenient mapping to \(Y_t\), \[\begin{eqnarray*} \operatorname{E}\!\left[{ Y_t Y_0}\right] & = & \operatorname{P}({ Y_t = 1 \mid Y_0 = 1 }) \operatorname{P}({ Y_0 = 1}) \\ & = & (q_1 + q_0 (1-s)^t) q_1 \\ & = & q_1^2 + q_0 q_1 (1-s)^t \end{eqnarray*}\] thus \[\begin{eqnarray*} \operatorname{Cov}\!\left[{ Y_t}, { Y_0}\right] & = & \operatorname{E}\!\left[{ Y_t Y_0}\right] - \operatorname{E}\!\left[{ Y_t}\right] \operatorname{E}\!\left[{ Y_0}\right] \\ & = & q_1^2 + q_0 q_1 (1-s)^t - q_1^2 \\ & = & q_0 q_1 (1-s)^t \\ \operatorname{Cor}\!\left[{ Y_t}, { Y_0}\right] & = & (1-s)^t \end{eqnarray*}\] QED

Counterexample of Real-Valued 3-State Markov Chain

Let \(Z_t\) be a stationary Markov process such that \[ \operatorname{P}({ Z_t = -1}) = \operatorname{P}({ Z_t = 0}) = \operatorname{P}({ Z_t = 1}) = 1/3 \] \[\begin{eqnarray*} \operatorname{P}({ Z_{t+1} = 0 \mid Z_t = -1}) & = & 1/2 \\ \operatorname{P}({ Z_{t+1} = 1 \mid Z_t = 0}) & = & 1/2 \\ \operatorname{P}({ Z_{t+1} = -1 \mid Z_t = 1}) & = & 1/2 \end{eqnarray*}\] and for all \(i \in \{-1, 0, 1\}\), \[\begin{eqnarray*} \operatorname{P}({ Z_{t+1} = i \mid Z_t = i}) & = & 1/2 \\ \end{eqnarray*}\]

Conveniently \(\operatorname{E}\!\left[{ Z_t}\right] = 0\), thus \(\operatorname{Cov}\!\left[{ Z_t}, { Z_s}\right] = \operatorname{E}\!\left[{ Z_t Z_s}\right]\). For one time step we have \[\begin{eqnarray*} \operatorname{P}({ Z_1 = 1 \wedge Z_0 = 1 }) & = & (1/3)(1/2) \\ \operatorname{P}({ Z_1 = -1 \wedge Z_0 = 1 }) & = & (1/3)(1/2) \\ \operatorname{P}({ Z_1 = 1 \wedge Z_0 = -1 }) & = & 0 \\ \operatorname{P}({ Z_1 = -1 \wedge Z_0 = -1 }) & = & (1/3)(1/2) \end{eqnarray*}\] thus autocorrelation of one time step must be positive: \[ \operatorname{E}\!\left[{ Z_1 Z_0}\right] = (1 \cdot 1) \frac{1}{6} + (-1 \cdot 1) \frac{1}{6} + (-1 \cdot -1) \frac{1}{6} = \frac{1}{6} \] For two time steps \[\begin{eqnarray*} \operatorname{P}({ Z_2 = 1 \wedge Z_0 = 1 }) & = & (1/3) (1/2)^2 \\ \operatorname{P}({ Z_2 = -1 \wedge Z_0 = 1 }) & = & (1/3) [ (1/2)^2 + (1/2)^2 ] \\ \operatorname{P}({ Z_2 = 1 \wedge Z_0 = -1 }) & = & (1/3) (1/2)^2 \\ \operatorname{P}({ Z_2 = -1 \wedge Z_0 = -1 }) & = & (1/3) (1/2)^2 \end{eqnarray*}\] thus the autocorrelation for two time steps must be negative: \[ \operatorname{E}\!\left[{ Z_2 Z_0}\right] = (1 \cdot 1) \frac{1}{12} + (-1 \cdot 1) \frac{2}{12} + (1 \cdot -1) \frac{1}{12} + (-1 \cdot -1) \frac{1}{12} = - \frac{1}{12} \] thus \[\begin{eqnarray*} \operatorname{Cor}\!\left[{ Z_2}, { Z_0}\right] & \not= & \operatorname{Cor}\!\left[{ Z_2}, { Z_1}\right] \operatorname{Cor}\!\left[{ Z_1}, { Z_0}\right] \end{eqnarray*}\]

References

1. Hamilton JD (1994) Time series analysis. Princeton University Press, Princeton, N.J