Adaptive Learning and Applications in Monetary Policy. Noah Williams

Transcription

1 Adaptive Learning and Applications in Monetary Policy Noah University of Wisconsin - Madison Econ 899

2 Motivations J. C. Trichet: Understanding expectations formation as a process underscores the strategic interdependence that exists between expectations formation and economics. (Zolotas lecture, 2005) Ben S. Bernanke: In sum, many of the most interesting issues in contemporary monetary theory require an analytical framework that involves learning by private agents and possibly the central bank as well. (NBER, July 2007).

3 Introduction Since Lucas (1972, 1976) and Sargent (1973) the standard assumption in the theory of economic policy is rational expectations (RE). This assumes, for both private agents and policymakers: knowledge of the correct form of the model knowledge of all parameters, and knowledge that other agents are rational & know that others know.... RE assumes too much and is therefore implausible. We need an appropriate model of bounded rationality What form should this take?

4 Learning A general answer is given by the Cognitive Consistency Principle: economic agents should be about as smart as (good) economists. Economists forecast economic variables using. econometric techniques, so a good starting point: model agents as econometricians. Neither private agents nor economists at central banks do know the true model. Instead economists formulate and estimate models. These models are re-estimated and possibly reformulated as new data becomes available. Economists engage in processes of learning about the economy. Do such processes of learning create new tasks for monetary policy? This presentation gives an overview of the rapidly growing literature on learning and monetary policy.

5 Starting Points Forecasts (including private forecasts) of future inflation and output have a key role in monetary policy: Empirical evidence e.g. by (Clarida, Gali and Gertler 1998). Bank of England and ECB discuss private forecasts in addition to internal macro projections. The private sector is forward-looking (e.g. investment, savings decisions). Private agents and/or policy-makers are learning.

6 Fundamental Issues There may be multiple equilibria, depending on interest rate policy. Policy may lead to expectational instability if expectations are not always rational. These problems necessitate careful design of interest rate rules. Central message: Policy should facilitate learning by private agents. Learning may also have additional implications for outcomes: increased persistence or amplification, large movements with small shocks. Study positive role of learning in explaining inflation dynamics in US and Latin America.

7 A Muth/Lucas-type Model Before looking at issues of policy, we review the learning approach and the E-stability technique, beginning with a simple univariate reduced form: p t = µ + αe t 1p t + δ w t 1 + η t. (RF) E t 1 p t denotes expectations of p t formed at t 1, w t 1 is a vector of exogenous shocks observed at t 1, η t is an exogenous unobserved iid shock, and w t follows an exogenous stationary VAR process. Muth s cobweb model of supply (with lag) and demand yields this reduced form, as does a simple Lucas-type supply model.

8 Rational Expectations First consider the model under RE: p t = µ + αe t 1 p t + δ w t 1 + η t. The model has a unique RE solution since E t 1 p t = µ + αe t 1 p t + δ w t 1 E t 1 p t = (1 α) 1 δ + (1 α) 1 δ w t 1 Hence the unique REE is p t = ā + b w t 1 + η t, where ā = (1 α) 1 δ and b = (1 α) 1 δ.

9 Least Squares Learning Under learning, agents have the beliefs or perceived law of motion (PLM) p t = a + bw t 1 + η t, a, b are unknown. At the end of time t 1 they estimate a, b by LS (Least Squares) using data through t 1, i.e. Then they use the estimated coefficients to make forecasts E t 1 p t. Here the standard least squares (LS) formula are ( at 1 b t 1 ) = z i = ( t 1 ) 1 ( t 1 ) z i 1 z i 1 z i 1 p i, where i=1 i=1 ( ) 1 w i.

10 The Model under Learning The timing is: End of t 1: w t 1 and p t 1 observed. Agents update estimates of a, b to a t 1, b t 1 using {p s, w s 1 } t 1 s=1. Agents make forecasts Et 1p t = a t 1 + b t 1w t 1. Period t: (i) The shock η t is realized, p t is determined as p t = (µ + αa t 1 ) + (δ + αb t 1 ) w t 1 + η t and w t is realized. (ii) agents update estimates to a t, b t using {p s, w s 1 } t s=1 and make forecasts E t p t+1 = a t + b tw t. The system under learning is a fully specified dynamic system under learning. Question: Will (a t, b t ) (ā, b) as t?

11 Recursive Learning LS updating and the system can be set up recursively. Letting ( ) ( ) at 1 φ t = and z t = the system is: b t Et 1p t = a t 1 + b t 1w t 1 p t = µ + αet 1p t + δ w t 1 + η t, φ t = φ t 1 + t 1 Rt 1 z t 1 (p t φ t 1z t 1 ) R t = R t 1 + t 1 (z t 1 z t 1 R t 1 ), This recursive formulation is useful for (a) theoretical analysis, and (b) numerical simulations. w t

12 Learning Dynamics Recall the dynamics of beliefs: φ t = φ t 1 + t 1 Rt 1 z t 1 (p t φ t 1z t 1 ) R t = R t 1 + t 1 (z t 1 z t 1 R t 1 ), Note that φ t 1 determines E t 1 p t and hence observed p t, which then influences φ t. System is self-referential, unlike standard estimation problems. Here the weight on new information is 1 t, which is known as the gain. Decreases at t. In some cases interested in settings where gain is constant ε > 0. Then perpetual learning. Equivalent to discounting past observations. We are interested in whether φ t φ as gain decreases to zero.

13 Convergence Results Theorem: Consider model (RF) with Et 1 p t = a t 1 + b t 1 w t 1 and with a t 1, b t 1 updated over ( ) ( ) at ā time using least-squares. If α < 1 then b t b with probability 1. If α > 1 convergence occurs with probability 0. Thus the REE is stable under LS learning both for Muth model (α < 0) and Lucas model (0 < α < 1).. Possible to have unstable REE in Muth model for some parameters (Giffen good)

14 E-Stability Proving the theorem is not easy. However, there is an easy way of deriving the stability condition α < 1 that is quite general. Start with the PLM p t = a + b w t 1 + η t, and consider what would happen if (a, b) were fixed at some value possibly different from the RE values (ā, b). The corresponding expectations are E t 1p t = a + b w t 1, which would lead to the Actual Law of Motion (ALM) p t = µ + α(a + b w t 1 ) + δ w t 1 + η t.

15 Temporary Equilibrium The implied ALM gives the mapping T: PLM ALM: ( ) ( ) a µ + αa T =. b δ + αb The REE ā, b is a fixed point of T. Expectational-stability ( E-stability) is defined by the differential equation ( d a dτ b ) = T ( a b ) ( a b Here τ denotes artificial or notional time. ā, b is said to be E-stable if it is stable under this differential equation. ).

16 E-Stability in the Example In the current case the T-map is linear. Component by component we have da dτ db i dτ = µ + (α 1)a = δ + (α 1)b i for i = 1,..., p. It follows that the REE is E-stable if and only if α < 1. This is the stability condition, given in the theorem, for stability under LS learning. Intuition: under LS learning the parameters a t, b t are slowly adjusted, on average, in the direction of the corresponding ALM parameters.

17 Numerical simulation of learning in Muth model. iid iid µ = 5, δ = 1 and α = 0.5. w t N (0, 1) and η t N (0, 1/4). Initial values a 0 = 1, b 0 = 2 and R 0 = I 2. Convergence to the REE ā = 10/3 and b = 2/3 is rapid.

18 The E-Stability Principle To study convergence of LS learning to an REE, specify a PLM with parameters φ. The REE is the PLM with φ = φ. Compute the ALM for this PLM. This gives a map φ T(φ), with fixed point φ. E-stability = local asymptotic stability of φ under dφ dτ = T(φ) φ. The E-stability condition is that all eigenvalues of DT( φ) have real parts less than 1. The E-stability principle: E-stability governs local stability of an REE under LS and closely related learning rules. E-stability can be used as a selection criterion in models with multiple REE.

19 E-Stability and Learning Recall the dynamics of beliefs with gain ε t : φ t = φ t 1 + ε t R 1 t z t 1 (p t φ t 1z t 1 ) But p t = T(φ t 1 ) z t 1 + η t, so we can write: φ t φ t 1 ε t = Rt 1 ( zt 1 z t 1[T(φ ) t 1 ) φ t 1 ] + z t 1 η t R t R t 1 ε t = z t 1 z t 1 R t 1 As ε t 0 these converge to the ODEs: φ = R 1 E(zz )[T(φ) φ] Ṙ = E(zz ) R Clearly R(t) R = E(zz ) so locally the dynamics of φ governed by E-stability ODE: φ = T(φ) φ

20 Learning Dynamics Results So if REE φ is E-stable then if beliefs (φ t, R t ) are initialized in a neighborhood of ( φ, R) then as t and ε t 0 we beliefs will converge. Stability results are the same for different gain sequences, but notion of convergence is different. If ε t = 1/t then this convergence with probability 1 as t. Beliefs eventually stabilize at REE. If ε t = ε > 0 then this is weak convergence. Distribution of beliefs centered on φ but beliefs continually fluctuate over time. Distribution shrinks as ε 0. Constant gain models have been widely used more recently to analyze fluctuations resulting from belief dynamics.

21 The New Keynesian Model Log-linearized New Keynesian model (CGG 1999, Woodford 2003 etc.): x t = ϕ(i t E t π t+1 ) + E t x t+1 + g t π t = λx t + βe t π t+1 + u t, where x t =output gap, π t =inflation, i t = nominal interest rate. E t x t+1, E t π t+1 are expectations. Parameters ϕ, λ > 0 and 0 < β < 1. Observable shocks follow ) ) ( gt u t = F ( gt 1 u t 1 + ( gt ũ t ), F = ( µ 0 0 ρ where 0 < µ, ρ < 1, and g t iid(0, σ 2 g), ũ t iid(0, σ 2 u). ),

22 Policy Rules Interest rate setting by a standard Taylor rule, e.g. i t = χ π π t + χ x x t where χ π, χ x > 0 or i t = χ π π t 1 + χ x x t 1 or i t = χ π E t π t+1 + χ x E t x t+1 Taylor principle: χ π > 1 ensures determinacy of equilibrium under rational expectations. What conditions on policy rule (χ π, χ x ) ensure stability of the economy when agents do not have rational expectations? Under learning we treat the IS and PC Euler equations as behavioral equations. Explicit infinite-horizon formations also have been studied: Preston (IJCM, 2005 and JME, 2006), Evans, Honkapohja & Mitra (JME, 2009).

23 Determinacy Combining IS, PC and the i t rule leads to a bivariate reduced form in x t and π t.letting y t = (x t, π t ) and v t = (g t, u t ) the model can be written ( ) ( ) ( ) ( ) xt E = M t x t+1 xt 1 gt π t Et + N + P, π t+1 π t 1 u t y t = ME t y t+1 + Ny t 1 + Pv t. If the model is determinate there exists a unique stationary REE of the form y t = by t 1 + cv t. Determinacy condition: compare # of stable eigenvalues of matrix of stacked first-order system to # of predetermined variables. If indeterminate there are multiple solutions, which include stationary sunspot solutions.

24 Learning in the NK Model Under learning, agents have beliefs or a perceived law of motion (PLM) y t = a + by t 1 + cv t, where we now allow for an intercept, and estimate (a t, b t, c t ) in period t based on past data. - Forecasts are computed from the estimated PLM. - New data is generated according to the model with the given forecasts. - Estimates are updated to (a t+1, b t+1, c t+1 ) using least squares. Question: when is it the case that (a t, b t, c t ) (0, b, c)? We determine this using E-stability

25 T-Map in the NK Model Reduced form y t = MEt y t+1 + Ny t 1 + Pv t. Under the PLM (Perceived Law of Motion) y t = a + by t 1 + cv t. Et y t+1 = (I + b)a + b 2 y t 1 + (bc + cf)v t. This ALM (Actual Law of Motion) y t = M (I + b)a + (Mb 2 + N )y t 1 + (Mbc + NcF + P)v t. This gives a mapping from PLM to ALM: T(a, b, c) = (M (I + b)a, Mb 2 + N, Mbc + NcF + P).

26 E-Stability in the NK Model The optimal REE is a fixed point of T(a, b, c). If d/dτ(a, b, c) = T(a, b, c) (a, b, c) is locally asymptotically stable at the REE it is said to be E-stable. See EH, Chapter 10, for details. The E-stability conditions can be stated in terms of the derivative matrices DT a = M (I + b) DT b = b M + I M b DT c = F M + I M b, where denotes the Kronecker product and b denotes the REE value of b. E-stability governs stability under LS learning.

27 Results for Taylor-rules in NK model(bullard & Mitra, JME 2002) i t = χ π π t + χ x x t yields determinacy and stability under LS learning ifλ(χ π 1) + (1 β)χ x > 0. Note that χ π > 1 is sufficient.

28 Lagged Data in the Taylor Rule With i t = χ π π t 1 + χ x x t 1, determinacy & E-stability for χ π > 1 and χ x > 0 small. Also an explosive region (χ π > 1 and χ x large) and a determinate E-unstable region (χ π < 1 and χ x moderate).

29 Forecast-based Taylor Rule For i t = χ π E t π t+1 + χ x E t x t+1, determinacy & E-stability for χ π > 1 and χ x > 0 small. Indeterminate & E-stable for χ π > 1 and χ x large. Recent work has shown stable sunspot solutions in that region.