Geometric Ergodicity and Uniform LLN

Summary

The first half of the SME consistency machinery. Geometric ergodicity of the Markov state process ensures it converges geometrically to a stationary (ergodic) distribution independent of starting values — this neutralizes the nonstationarity of the simulated series and delivers strong $\alpha$-mixing (hence laws of large numbers). Duffie & Singleton verify it via Mokkadem’s (1985) sufficient conditions for “nonlinear AR(1)” models (irreducibility Condition B, aperiodicity, a contraction bound — Lemma 1). They then upgrade an ordinary LLN to a uniform weak LLN over the parameter set $\Theta$ (Lemma 2) using a global “Lipschitz, uniformly in probability” modulus-of-continuity condition on the simulated moments.

Overview

To prove the SME is consistent one needs (i) a LLN that holds despite the simulated series starting off its ergodic distribution, and (ii) that LLN to hold uniformly in $\beta$ over the compact parameter space $\Theta$. Section 4.1 supplies (i) via geometric ergodicity; Section 4.2 supplies (ii) via a uniform weak LLN. These feed directly into SME Consistency.

Main Content

$\rho$-ergodic and geometrically ergodic (D&S §4.1, Eq. 4.1)

Let $P_x^t$ denote the $t$-step transition probability (distribution of $X_t$ given $X_0=x$) of a time-homogeneous Markov process $\{X_t\}$. The process is $\rho$-ergodic, for $\rho \in (0,1]$, if there is a probability measure $\pi$ (the ergodic distribution) such that for every initial point $x$,

$${\rho }^{-t}\Vert P_x^t-\pi {\Vert }_v\to 0\text{as }t\to \infty ,$$

where $\Vert \cdot {\Vert }_v$ is the total-variation norm. If $\{X_t\}$ is $\rho$-ergodic for some $\rho <1$, it is geometrically ergodic.

Why it matters. Geometric ergodicity lets ergodicity substitute for stationarity in computing asymptotic distributions: the process converges geometrically to $\pi$ regardless of initial conditions, and it implies strong ($\alpha$-) mixing with mixing coefficient $\alpha (m)\to 0$ geometrically (Rosenblatt 1971; Mokkadem 1985). Notation: for an ergodic process, $X_{\infty }$ denotes a random variable with the ergodic distribution, and $\Vert X{\Vert }_q=[\mathbb{E}(\Vert X{\Vert }^q){]}^{1/q}$ is the $L^q$ norm.

Condition B — irreducibility (D&S §4.1, Eq. 4.2)

For any measurable $A\subset {\mathbb{R}}^N$ of nonzero Lebesgue measure and any compact $K\subset {\mathbb{R}}^N$, there exists an integer $t>0$ such that

$$\underset{x\in K}{\inf }{P}_x^t(A)>0.$$

This is a recurrence/irreducibility condition. It is weaker than a single-period “full support” condition — important because with endogenous state variables (e.g. capital stock) the one-step distribution can be degenerate, yet Condition B can still hold over multiple steps (see ^ex-closed-form). Aperiodicity is also required (a deterministic cycle is recurrent but not geometrically ergodic).

Lemma 1 (Mokkadem): sufficient conditions for geometric ergodicity (D&S §4.1, Eq. 4.6)

Suppose $\{Y_t\}$, defined by (3.1), is aperiodic and satisfies Condition B. Fix $\beta$ and suppose there are constants $K>0$, $\delta \in (0,1)$, and $q>0$ such that $H(\cdot ,{\epsilon }_1,\beta ):{\mathbb{R}}^N\to L^q$ is well defined and continuous with

$$\Vert H(y,{\epsilon }_1,\beta ){\Vert }_q<\delta \Vert y\Vert ,\Vert y\Vert >K.$$

Then $\{Y_t\}$ is geometrically ergodic, and $\Vert Y_t^{\beta }{\Vert }_q$ and $\Vert Y_{\infty }^{\beta }{\Vert }_q$ are uniformly bounded over $t$.

Interpretation: condition (4.6), inspired by Tweedie (1982), says that once $\{Y_t\}$ leaves a sufficiently large ball it heads back toward the ball at a uniform geometric rate — a drift/contraction-toward-the-center condition.

Lipschitz, uniformly in probability (D&S §4.2)

The family $\{f_t^{\beta }\}$ is Lipschitz, uniformly in probability if there is a sequence $\{K_t\}$ such that for all $t$ and all $\beta ,\theta \in \Theta$,

$$\Vert f_t^{\beta }-f_t^{\theta }\Vert \le K_t\Vert \beta -\theta \Vert ,\text{with }{K}^T=T^{-1}\sum _{t=1}^T{K}_t\text{ bounded in probability.}$$

This is a global modulus-of-continuity condition (over all of $\Theta$), used in place of the more usual local condition; it is what couples uniform convergence to the parameter feedback of the simulated path.

Lemma 2 (Uniform Weak Law of Large Numbers) (D&S §4.2, Eq. 4.7)

Suppose, for each $\beta \in \Theta$, that $\{Y_t^{\beta }\}$ is ergodic and $\mathbb{E}(|f_{\infty }^{\beta }|)<\infty$; suppose in addition that $\beta \mapsto \mathbb{E}(f_{\infty }^{\beta })$ is continuous and the family $\{f_t^{\beta }\}$ is Lipschitz, uniformly in probability. Then $\{f_t^{\beta }:\beta \in \Theta \}$ satisfies the uniform weak law of large numbers:

$$\underset{T\to \infty }{\lim }P\left[\underset{\beta \in \Theta }{\sup }\left|\mathbb{E}(f_{\infty }^{\beta })-\frac{1}{T}\sum _{t=1}^T{f}_t^{\beta }\right|>\delta \right]=0\text{for each }\delta >0.$$

The ergodicity hypothesis can be replaced by Mokkadem’s geometric-ergodicity conditions on $H$ and ${\epsilon }_t$ (Lemma 1) for some $q>2$. (Proof in the Appendix, following Jennrich 1969 / Amemiya 1985; cf. Newey 1991.)

Connections

• Supplies the two ingredients — ergodicity-based LLN and its uniform version — that Assumptions 1–2 of SME Consistency package, leading to Theorem 1 (weak consistency).
• Geometric ergodicity is a strictly weaker route to consistency than the AUC damping condition used for strong consistency; the conditionally heteroskedastic example is geometrically ergodic but not AUC.
• Total-variation/mixing machinery connects to the geometric ergodicity used to justify MCMC convergence and HAC (Newey–West) variance estimation.

See Also

• SME Consistency — Theorems 1–3 built on these lemmas
• Duffie-Singleton Asset-Pricing Model — Condition B checked for the growth model
• Simulated Moments Estimator Definition — the $f_t^{\beta }$, $\Theta$ this concerns