Notes on Large Deviations in Economics and Finance. Noah Williams

Transcription

1 Notes on Large Deviations in Economics and Finance Noah Williams Princeton University and NBER noahw

2 Notes on Large Deviations 1 Introduction What is large deviation theory? Loosely: a theory of rare events. Typically provide exponential bound on probability of such events and characterize them. Large growth area in math/applied probability in last 20 years. Seeing some applications in economics. Why is it useful? Often interested in characterizing extreme events in themselves (crashes, failures, busts). Bounding probability of extreme events can characterize likelihood of typical events.

3 Notes on Large Deviations 2 Formal Definition of LDP Consider {Z ɛ } on (Ω, F, P ), taking values in X. A rate function S : X [0, ] has the property that for any M < the level set {x X : S(x) M} is compact. Definition: A sequence {Z ɛ } satisfies a large deviation principle (LDP) on X with rate function S and speed ɛ if: 1. For each closed subset F of X,: lim sup ɛ 0 2. For each open subset G of X,: ɛ log P {Z ɛ F } inf x F S(x). lim inf ɛ 0 ɛ log P {Zɛ G} inf x G S(x).

4 Notes on Large Deviations 3 Interpretation Under regularity, upper & lower hold with equality: lim ɛ log P ɛ 0 {Zɛ F } = inf x F i.e. log P {Z ɛ F } C exp( S/ɛ). S(x) = S. Note F is independent of ɛ. If Z ɛ Z / F then P 0. Compare w/clt. Let ɛ = 1/n, Z n = i X i/n, X i, i.i.d., a > 0. ( CLT : P Z n Z a ) 2Φ( a) n LDP : P ( Z n Z a ) C exp( S a n)

5 Notes on Large Deviations 4 Contexts & Applications Performance of decision rules (and estimators, robustness) Performance of portfolios (outperforming benchmarks, large losses) Proofs of convergence (implies weak LLN) Rare events: extreme or non-equilibrium outcomes (large business cycles, escape dynamics) Transitions between equilibria and equilibrium selection (evolutionary games)

6 Notes on Large Deviations 5 Chebyshev P (X i [a, )) E [exp (λ (X i a))] exp(λ(x a)) 1 Pr(X>= a) 0 a

7 Notes on Large Deviations 6 Basic idea: Cramer s theorem Take Z n = X i /n, X i i.i.d. Fix a > E(X). Define: H(λ) = log E exp( λ, X ) S(x) = sup λ { λ, x H(λ)} Then Z n satisfies an LDP with rate function I and speed 1/n. Sketch of proof for upper bound: Consider scalar case, and fix a > EX. P (Z n [a, )) = E [1{Z n a 0}] (Chebyshev) E [exp (nλ (X i a))] for all λ 0 = n exp( nλa) E exp(λx i ) log P (Z n [a, )) n sup [λa log E exp(λx i )] = ns(a) λ>0 i=1

8 Notes on Large Deviations 7 Illustration: Bivariate N(0, Σ) 1 S = min x 2 x Σ 1 x s.t. x a 1 Escape Set Iso Cost Lines 0.5 y y1

9 Notes on Large Deviations 8 Some notes Lower bound is also key: uses (exponential) change of measure. Low probability event deterministic optimization problem. Let x = arg min x F S(x). Then for all δ > 0: lim P ( Z n x < δ Z n F ) = 1. n Tom s favorite quotes: 1. Rare events are exponentially rare. 2. If an unlikely event occurs, it is very likely to occur in the most likely way.

10 Notes on Large Deviations 9 Some Extensions Gärtner-Ellis: varying statistics. Let Z ɛ µ ɛ. Define: H(λ) = lim ɛ 0 ɛ log E exp ( λ/ɛ, Z ɛ ) Provided limit exists, rate function S is Legendgre of H. Sanov s Theorem: Gives LDP for empirical measures of i.i.d. sample. Rate function is relative entropy w.r.t. original meas. Donsker-Varadhan: extend to empirical measures of Markov

11 Notes on Large Deviations 10 Some Relationships Entropy: information theory, statistical mechanics Perturbation methods and sharp LD: log P C + ɛ S 1 + ɛ 2 S Risk sensitivity and (asymptotic) robustness via Varadhan. Let L be a (continuous) loss function L : X R. lim ɛ log ɛ 0 E[exp(L(Zɛ )/ɛ)] = sup{l(x) S(x)} x X Idempotent probability/possibility theory/fuzzy measures. Puhalskii: Deviabilities Π(F ) = sup x F exp( S(x)). A nonadditive measure: Π(A B) = Π(A) Π(B). ( LDP : lim ɛ 0 X h(x) 1 ɛ dµ ɛ (x)) ɛ = sup x X h(x)π(x)

12 Notes on Large Deviations 11 Applications: i.i.d. or cross-section Hypothesis testing: Say accept hypothesis when (sample) log likelihood ratio above a cutoff. Asymptotics of type I and II errors for fixed cutoffs determined by LDP (Chernoff). Krasa-Villamil (1992) - Suppose a bank pools (i.i.d.) risks. Probability of default decreases exponentially with size. If cost increases slower w/size, interior optimal size. Dembo-Deuschel-Duffie (2003) - Portfolio consists of positions subject to losses Z i at exposures U i. Evaluate portfolio losses L n = i Z iu i, characterize P (L n nx). Stutzer (2003): Minimize probability that portfolio will grow at rate lower than some benchmark. Use LDP to derive long run decay rate. Show that yields result similar to max utility of wealth with endogenous preference power.

13 Notes on Large Deviations 12 Convergence of Exchange Economies Random excess demands z i (ω, p) : Ω R d R d. Aggregate: Z n (ω, p) = n i=1 z i(ω, p). Prices: π n (ω) = {p : Z n (ω, p) = 0}. Expectation Economy : z(p) = lim n Z n (ω, p)/n. Prices π. Define H(λ, p), S(x, p) as in Gärtner-Ellis. Then under some continuity conditions, Nummelin (2000) shows that {π n } satisfies an LDP with speed 1/n and rate function S(0, p). Some simple conditions insure π is nonempty, gives asymptotic existence (a.s. equilibrium exists eventually) and convergence to expectation economy.

14 Notes on Large Deviations 13 Sample Path Large Deviations So far: LDP for single realization or sample average. Now: Realization of event in a (dynamic) sample path. Most complete theory: Freidlin-Wentzell for diffusions Simplest to present: discrete time analogues x σ t+1 = g σ (x t ) + σw t+1. W t N(0, 1), g σ g 0, x = g 0 (x ). Gärtner-Ellis: given x t = x, 1-Step LDP at rate σ 2 for x t+1 with rate function: S 1 (x, y) = 1 2 (y g0 (x)) 2 For multi-step rate function, sum up the one-step transitions: S(x, y, T ) = inf {x t } T t=0 1 2 T 1 (x t+1 g 0 (x t )) 2 s.t. x 0 = x, x T = y t=0

15 Notes on Large Deviations 14 Multi-Step Sample Path LDP LDP conditional on x σ 0 = x is then: ( ) x σ t x 0 t > a lim σ 0 σ2 log P sup 1 t T = inf {y: y x 0 t >a} S(x, y, T ) Define escape time from stable point x : τ σ = min {t > 0 : x σ t x a, x σ 0 x < a} Let S = inf {y x ±a,t < } S(x, y, T ), y the minimizer. Then for η > 0 we have: ( ( ) S η lim P exp σ 0 σ 2 ( )) S + η τ σ exp σ 2 = 1. lim P σ 0 ( xσ τ σ y < η) = 1.

16 The Exit/Escape Problem I Average Motion/ Mean Dynamics * F

17 The Exit/Escape Problem I When pushed away, tend to return to equilibrium Sample Path Average Motion/ Mean Dynamics * F

18 Rare Motion/ Escape Dynamics The Exit/Escape Problem I Average Motion/ Mean Dynamics * F Occasionally, escape to boundary of set

19 The Exit/Escape Problem II: Multiple Equilibria 2 stable equilibria with basins of attraction Separatrix * * F 1 F 2 Average Motion/ Mean Dynamics

20 The Exit/Escape Problem II: Multiple Equilibria Occasionally escape from one eq to the other Separatrix Rare Motion/ Escape Dynamics * * F 1 F 2 Average Motion/ Mean Dynamics

21 The Exit/Escape Problem II: Multiple Equilibria Separatrix Rare Motion/ Escape Dynamics * * F 1 F 2 Average Motion/ Mean Dynamics Generates a Markov chain over equilibria

22 Notes on Large Deviations 15 Application: Large Business Cycles Small Noise Asymptotics... studies stochastic growth model. Characterize prob. and freq. of large movements away from deterministic steady state. Nonlinearity in policy function determines expected direction of large movements. Convex: positive movements more likely. Concave: negative more likely. The way state, policy defined in model implies sharp recessions (slightly) more likely than large booms.

23 Notes on Large Deviations 16 Intuition for Asymmetry Consider 2 step for simplicity. Say x = 0, x 1 = 0.5, a = x 2 = 1. Start of date 1: at x = 0. End of date 1: at 0.5. Start of date 2: at g 0 (0.5). End of date 2: at Linear Convex Concave 45 Degree State Evolution Distance From Stable Point

24 Notes on Large Deviations 17 Application: Escape Dynamics In some learning models, recurrent escapes from a stable limit point are a key feature of time series. Example: Single agent learning a vector γ. Truth depends on agent s action, and beliefs. Belief : y in = γ i x in + η in, i = 1, 2, Action : x n = b n + W n, W n N(0, σ 2 I) Decision : b n = b(γ) Truth : y in = f i (b n ) + γ i x in Updating : γ n+1 = γ n + ε(y n γx n )x n Specify so for σ > 0 unique stable equilibrium γ.

25 Notes on Large Deviations 18 An Example Convergence: as ε 0, γ n γ = [.75,.25]. Recurrent escapes. 1 Simulated Time Path of Beliefs, ε= Simulated Time Path of Beliefs, ε=

26 Notes on Large Deviations 19 Overview: Escape Dynamics Similar type of LDP applies, gives predictions of probability and frequency of escape, location of dominant escape path. With multiple equilibria, can calculate limit transition rates between equilibria. Limit distribution over equilibria collapses on stochastically stable or long-run equilibrium. With unique equilibrium, can still have interesting escape dynamics (as in the example). Often have escapes toward a near equilibrium.

27 Notes on Large Deviations 20 Analysis: Near Equilibrium As noise 0, there are 2 (stable) equilibria, 1 of which gets discovered by escape dynamics. 1 Level Sets of Rate Function 0.08 Rate Function Along Diagonal γ 1 * γ 1 bar γ γ γ 1

28 Notes on Large Deviations 21 Comparing the Predictions to Simulations Exit Distribution γ 2 11 Log Mean Escape Time Mean Predicted 1 SD Band Location 0.5 Mean Predicted 1 SD Band γ Gain (ε) /Gain (1/ε)

29 Notes on Large Deviations 22 Some References Books: Dembo-Zeitouni (1998, Springer, 2nd ed.), Freidlin-Wentzell (1998, Springer, 2nd ed.), Puhalskii (2001, Chapman), Dupuis-Ellis (1997, Wiley) Econ - decisions etc.: Krasa-Villamil (1992 JET, 1992 Oxford Ec. Papers), Nummelin (2000, Ann. App. Prob.), Williams (2003, Small Noise ) Econ - multiple eq.: Foster-Young (1990, J. Pop Bio), Kandori-Mailath-Rob (1993, Ec.), Young (1993, Ec.), Ellison (1993, Ec., 2000 ReStud), Kasa (2004, IER), Williams (2002, unpub Stochastic Ficititious Play ), Benaïm-Weibull (2003, Ec.) Econ - escape dynamics: Sargent (1999, Conquest book), Cho-Williams-Sargent (2002, ReStud), Williams (2004, unpub, Escape Dynamics ), Bullard-Cho (2002, unpub), Cho-Kasa (2003, unpub) Finance: Stutzer (2003, J. Econometrics), Dembo-Deuschel-Duffie (2004, Fin. & Stoch.), Pham (2003, Fin. & Stoch.), Callen-Govindaraj-Xu (2000, Ec. Theory)