Intro to non‐rational expectations (learning) in 

macroeconomicsLecture to Advanced Macro class, 

Bristol MSc, Spring 2014

Me

• New to academia.• 20 years in Bank of England, directorate responsible for monetary policy.

• Various jobs on Inflation Report, Inflation Forecast, latterly as senior advisor, leading ‘monetary strategy team’

• Research interests:  VARs, TVP‐VARs, monetary policy design, learning, DSGE, heterogeneous agents.

• Also teaching MSc time series on VARs.

Follow my outputs if you are interested

• My research homepage• Blog ‘longandvariable’ on macro and public policy

• Twitter feed @tonyyates

This course:  overview

• Lecture 1:  learning• Lectures 2 and 3:  credit frictions• Lecture 4:  the zero bound to nominal interest rates

• Why these topics?– Spring out of financial crisis:  we hit the zero bound;  financial frictions part of the problem;  RE seems even more implausible

– Chance to rehearse standard analytical tools.– Mixture of what I know and need to learn.

Course stuff.• No continuous assessment.  Assignments not marked. All on the exam.

• But:  Not everything covered in the lectures!  And, if you want a good reference…

• Office hour tba.  Email tony.yates@bristol.ac.uk, skype:  anthony_yates.

• Teaching strategy:  some foundations, some analytics, some literature survey, intuition, connection with current debates.

• Exams won’t reach difficulty touched in the 

More course stuff

• All lectures and exercises posted initially on my teaching homepage.  In advance (ie there now) but subject to changes at short notice.

• Roman Fosetti will be taking the tutorial.• I’ll be doing 2 other talks in the evening on monetary policy issues, to be arranged.  Voluntary.  But one will cover issues surrounding policy at the zero bound.

• Feedback welcome!  Exam set, so course content set now.  But methods…

Some useful sources on learning• Evans and Honkapoja (2000) textbook.• George Evans reading list on learning and other topics

• George Evans lectures:  borrowed from here.• Bakhshi, Kapetanios and Yates, eg of empirical test of RE

• Milani – survey of RE and learning • Ellison‐Yates, mis‐specified learning by policymakers

• Marcet reading list

Motivation for thinking about non rational expectations

• Rational expectations is very demanding of agents in the model.  More plausible to assume agents have less information?

• Some non‐rational expectations models act as foundational support for the assumption of rational expectations itself, having REE as a limit.

• Some RE models have multiple REE.  We can look to non‐RE as a ‘selection device’.

Motivation for non‐RE

• RE is a‐priori implausible, but also can be shown to fail empirically….

• RE imposes certain restrictions on expectations data that seem to fail.

• Non RE can enrich the dynamics and propagation mechanisms in (eg) RBC or NK models:  post RBC macro has been the story of looking for propagation mechanisms.

Motivation 1:  RE is very demanding

• Sargent:  rational expectations = Communist model of expectations (!)

• Perhaps Communist plus Utopian• What did he mean?• Everyone has the same expectations• Everyone has the correct expectations• Agents behave as if they understand how the model works

RE in the NK model 

t Ett1 kx t e,tx t Etx t1 − it − Ett1 ex,t

it it−1 t xx t

NB the expectation is assumed to be the true, mathematical expectation

The computational demands of RE

Yt

t

x t

it

EtYt1 ASt St

e,tex,t

it−1

Method of undet coeff:  where ‘E’ appears, substitute our conjectured linear function of the state, then solve for the unknown A

REE:  when agents use a forecasting function, their use of it induces a situation where exactly that function would be the best forecast.So agents have to know the model exactly, and compute a fixed point!If you find this tricky, think of the poor grocers or workers in the model!

Motivation 1, ctd:  empirical implications and tests of RE

t1 − Ett1 eE,t

eE,t Xt

0

Compute ex post forecast errors

Regress errors on ANY information available to agents at t

Coefficients should not be statistically different from zero.  Betahat includes constant.

Take the test to the data

• Bakhshi, Kapetanios and Yates ‘Rational expectations and fixed event forecasts’

• Survey data on 70 fund managers compiled by Merril Lynch

• Regression of forecast errors on a constant, and past revisions

• Regression of revisions on past revisions• Fails dismally!

Source: BKY (2006), Empirical Economics, BoE wp version, no 176, p13.

Squares are outturns. Lines approach them in autocorrelated fashion. Revisions therefore autocorelleated.

Motivation 2:  learning as a foundation for RE

• If we can show that RE is a limiting case of some more reasonable non rational expectations model, RE becomes more plausible

• We will see that some rational expectations equilibria are ‘learnable’ and some are not.  Hence some REE judged more plausible than others.  

• Policy influences learnability, hence some policies judged better than others.

Motivation for studying learning

• Learning literature treats agents as like econometricians, updating their estimates as new data comes in.  [‘decreasing gain’]  And perhaps ‘forgetting’ [‘constant gain’].

Stability of REE, and convergence of learning algorithms

• General and difficult analysis of the two phenomena.  Luckily connected.

• Despite all the rigour, usually boils down to some simple conditions.

• But still worth going through the background to see where these conditions come from.

• Lifted from Evans and Honkapohja (2001), chs1 and 2

Technical analysis of stability and expectational dynamics

• Application to a simple Lucas/Muth model.• Find the REE• Then conjecture what happens to expectations if we start out away from RE.

• Does the economy go back to REE or not?• Least squares learning, akin to recursive least squares in econometrics.

• Studying dynamics of ODE in beliefs in notional time.

Lucas/Muth model

qt q pt − Et−1pt t

mt v t pt qt

mt m ut ′wt−1

1. Output equals natural rate plus something times price surprises

2. Aggregate demand = quantity equation.3. Money (policy) feedback rule.

Finding the REE of the Muth/Lucas model

• 1. Write down reduced form for prices.• 2. Take expectations.• 3. Solve out for expectations using guess and verify.

• 4. Done.• NB this will be an exercise for you and we won’t do it here.

Finding the REE of the Muth/Lucas model

pt Et−1pt ′wt−1 t

0 1, 1 −

1 −

pt a bwt−1 t

a 1 − ,b ′

1 −

Reduced form of this model in terms of prices.

Rational expectations equilibrium, with expectations ‘solved out’ using a guess and verify method.

Towards understanding the T‐mapping in learning that takes PLM 

to ALMpt Et−1pt ′wt−1 t

pt a1 b1′ wt−1

pt a1 b1′ ′wt−1 t

pt 2a1 2b1′ ′ ′wt−1 t

Reduced form of LM model.

Forecaster’s PLM1

ALM1 under this PLM1

ALM2 under PLM2=f(ALM1)

Repeated substitution expressed as repeated application of a T‐map

Ta1 ,b1 a1 ,b1′ ′

T a1 ,b1′ ′ 2a1 ,2b1

′ ′ ′

T2a1 ,b1 2a1 ,2b1′ ′ ′

T is an ‘operator’ on a function (in this case our agent’s forecast function) taking coefficients from one value to another

Repeated applications written as powers [remember the lag operator L?

Verbal description of T:‘take the constant, times by alpha and add mu; take the coefficient, times by alpha and add delta’

Will our theoretician ever get to the REE in the Muth‐Lucas model?

Ta1 ,b1 ?

limn→ Ta1 ,b1 ?

To work out formally whether imaginary theoretician will ever get the REE, we are asking questions of these expressions in T above.

Answer to this depends on where we start. On certain properties of the model.

Sometimes we won’t be able to say anything about it unless we start very close to the REE.

Matlab code to do repeated substitution in Muth‐Lucas

Progression of experimental theorist’s guesses in the Muth‐

Lucas model

1 2 3 4 5 6 7 8 9 100.25

0.3

0.35

1 2 3 4 5 6 7 8 9 100.8

0.9

1

1 2 3 4 5 6 7 8 9 100.4

0.5

0.6

This is for alpha=0.25<1; charts for a,b(1),b(2) respectively, green lines plot REE values.

What happens when alpha>1?

1 2 3 4 5 6 7 8 9 10-50

0

50

100

1 2 3 4 5 6 7 8 9 10-4

-2

0

2

1 2 3 4 5 6 7 8 9 10-2

0

2

4

Coefficients quickly explode. Agents never find the REE.Because coefficients explode, so will prices.Since prices didn’t explode (eg in UK) we infer not realistic.

Iterative expectational stability

• Muth‐Lucas model is ‘iterative expectational stable’ with alpha<1.

• REEs that are IE stable, more plausible, ie could be found by this repeated substitution process. (Well, a bit more!)

• Why is alpha<1 crucial?• Each time we are *by alpha.  So alpha<1 shrinks departures (caused by the adding bit) from REE.

Repeated substitution, to econometric learning

• Perhaps allowing agents to do repeated substitution is too demanding.

• Instead, let them behave like econometricians– Trying out forecasting functions– Estimating updates to coefficients– Weighing new and old information appropriately

Least squares learning recursion for the Muth‐Lucas model

Et−1pt t−1′ zt−1 , t−1 at−1

p ,bt−1p , zt−1 1,wt−1

pt Et−1pt ′wt−1 t

t t−1 t−1Rt−1zt−1pt − t−1

′ zt−1

Rt Rt−1 t−1zt−1zt−1′ − Rt−1

Beliefs stacked in phi.Decreasing gain. As t gets large, rate of change of phi gets small.Phi: ‘last period’s, plus this periods, times something proportional to the error I made using last period’s phi’R: moment matrix. Equivalent to inv(x’x) in OLS. Large elements means imprecise coefficient estimate, means don’t pay too much attention if you get a large error this period.

Close relative:  recursive least squares in econometrics

t t−1 e.t

X ′X−1X ′Y

X t−1 ,Y t, t 2, . . .T

t t−1 t−1Rt−1t−1t − t−1t−1

Rt Rt−1 t−1t−1t−1′ − Rt−1

Suppose we have an AR(1) for inflation

The OLS formula is given by this

Recursive least squares can be used to compute the same OLS estimate as the final element in this sequence

0 50 100 150 200 250 300 350 400 450 5000.845

0.85

0.855

0.86

0.865

0.87

0.875

0.88

0.885Example of recursive least squares

Periods after train samp

rhoh

at v

alue

DGP : pt pt−1 zt1

p1 . . .p500

R1 Rp1 . . .p500

0.85

z 0.2

Rhohat initialised at training sample estimateSlowly updates towards the full sample estimate, which is fractionally above the true valueConsistency properties of OLS illustrated by the slow convergence.RLS useful in computation too, saves computing large inverses over and over in update steps

Matlab code to do RLS

• %program to illustrate recursive least squares in an autogregression.• %MSc advanced macro, use in lecture on learning in macro• %formula in, eg Evans and Honkapohja, p33, 'Learning and Epectations in• %Macro', 2000, ch 2.•• %nb we don't initialise as suggested in EH as there seems to be a misprint.• %see footnote 4 page 33.•• clear all;•• %first generate some data for an ar(1): p_t=rho*p_t-1+shock• samp=1000; %set length of artificial data sample• rssamp=500; %post training sample length• p=zeros(samp,1); %declare vector to store prices in• p(1)=1; %initialise the price series• sdz=0.2; %set the variance of the shock• rho=0.85; %set ar parameter• z=randn(samp,1)*sdz; %draw shocks using pseudo random number generator in matlab•• for i=2:samp %loop to generate data• p(i)=rho*p(i-1)+z(i);• end•

RLS code/ctd…• %now compute recursive least squares.•• rhohat=zeros(rssamp,1); %vector to store recursive estimates of rho•• y=p(2:rssamp-1); %create vectors to compute ols on artificial data• x=p(1:rssamp-2);•• rhohat(1)=inv(x'*x)*x'*y; %initialse rhohat in the rls recursion to training sample• R=x'*x; %initialise moment matrix on training sample•• for i=2:rssamp %loop to produce rls estimates.• x=p(rssamp+i-1);• y=p(rssamp+i);• rhohat(i)=rhohat(i-1)+(1/i)*inv(R)*x*(y-rhohat(i-1)*x);• R=R+(1/i)*(x^2-R);• end•• y=p(2:samp); %code to create full sample ols estimates to compare• x=p(1:samp-1);•• rhohatfullsamp=inv(x'*x)*x'*y;• rhofullline=ones(rssamp,1)*rhohatfullsamp;•• time=[1:rssamp]; %create data for xaxis• plot(time,rhohat,time,rhofullline) %plot our rhohats and full sample rhohat• title('Example of recursive least squares')• xlabel('Periods after train samp')• ylabel('rhohat value')

Learning recursion in terms of the T ‐mapping

t at

bt

pt T t−1′zt−1 t, zt 1,wt′

t t−1 t−1Rt−1zt−1zt−1

′ T t−1 − t−1 t

Rt Rt−1 t−1zt−1zt−1′ − Rt−1

Crucial step here noticing that we can rewrite in terms of the T-mapping.Now dynamics of beliefs related to T()-() as before.

• Just as with our repeated substitution, and study of iterative expectational stability…

• The learning recursion involves repeated applications of the T‐mapping.

• So rate of change of beliefs will be zero if Tmaps beliefs back onto themeslves, or T()‐()=0.

Estability, the ODE in beliefs

dd

ap

bp T

ap

bp−

ap

bp

Rough translation: the rate of change of beliefs in notional time is given by the difference between beliefs in one period, and the next, ie between beliefs and beliefs operated on by the T-map, or the learning updating mechanism.

dap

d − 1ap

dbip

d − 1bip

ODE’s for our Muth/Lucas model

Estability conditions

dd

ap

bp T

ap

bp−

ap

bp T −

− 1

1 0

0 1

a1

b1

− 1 0

0 − 1

a1

b1

Stability depends on eigenvalues of transition matrix <1Transition matrix is diagonal, so eigenvalues read off the diagonal. Both=alpha-1. So alpha<1.We still need to dig in further and ask why the RLS system reduces to this condition [which looks like the iterative expecational stability condition].

Foundations for e‐stability analysis

pt T t−1′zt−1 t, zt 1,wt′

Rewrite the PLM using the T-map

t t−1 t−1Rt−1zt−1zt−1

′ T t−1 − t−1 t

Rt Rt−1 t−1zt−1zt−1′ − Rt−1

Learning recursion with PLM in terms of the T-map.If T()=(), then 1st equation is constant (in expectation), because the second term in this first equation then gets multiplied by a zero.Likewise if R is very ‘large’ [it’s a matrix], coefficients won’t change much. Here large means estimates imprecise.

Rewriting the learning recursion

t t−1 t−1St−1−1 zt−1zt−1

′ T t−1 − t−1 t

St St−1 t−1 tt 1 zt−1zt−1

′ − St−1

Here we rewrite the system by setting R_t=S_t-1It will be an exercise to show that the system is written like this and explain why.

t t−1 tQt, t−1 ,Xt t t,Rt

Xt zt−1 ,t

And we can write the system more generally and compactly like this. Many learning models fit this form. We are using the Muth-Lucas model, but the NK model could be written like this, and so could the cobweb model in EH’s textbook.

Deriving that estability in learning reduces to T()‐()

t t−1 tQt, t−1 ,Xt

dd

h h limt→ EQ

DhExpectation taken over the state variables X.Why? We want to know if stability holds for all possible starting points.Stability properties will be inferred from linearised version of the system about the REE [or whichever point we are interested in].Muth-Lucas system already linear, but in general many won’t be.

Estability

DhIf all eigenvalues of this derivative or Jacobian matrix have negative real parts, system locally stable.

Qt, t−1 ,Xt St−1−1 zt−1zt−1

′ T t−1 − t−1 t

Qst, t−1Xt vec tt 1 ztzt

′ − St−1

Unpack Q: we will show that stability is guaranteed in second equation.

h,S limt→

ESt−1−1 zt−1zt−1

′ T t−1 − t−1 t

hS,S limt→

E tt 1 ztzt

′ − St−1

Expectations taken over X, because we are looking to account for all possible trajectories.

Reducing and simplifying the estability equations

Ezt−1t 0

limt→

tt 1 1

Eztzt′ Ezt−1zt−1

′ 1 0

0 M

This period’s shock uncorrelated with last period’s data

TR 1 entry as z includes constant.Bottom right is variance of w0s because w not correlated with 1!

h,S dd St−1

−1 MT −

hS,S dSd M − S

h,S dd T −

hS,S dSd 0

We can get from the LHS pair to the RHS pair because S tends to M from any starting point.

Recap on estability

• We are done!  Doing what, again?!• That was explaining why the stability of the learning model, involving the real‐time estimating and updating, reduces to a condition like the one encountered in the repeat‐substitution exercise, involving T()‐().

• And in particular why the second set of equations involving moment matrix vanishes.

Remarks about learning literatures

• Constant gain• Stochastic gradient learning• Learnability, determinacy, monetary policy• Learning by policymakers, or two‐sided learning

• Learning using mis‐specified models, RPEs, or SCEs.

• Analogy with intra period learning, solving in nonlinear RE eg RBC models

Constant gain t t−1 Rt

−1zt−1zt−1′ T t−1 − t−1 t

Rt Rt−1 zt−1zt−1′ − Rt−1

We replace inv(t) with gamma, so weight on update term is constant.Recursion no longer converges to limiting point, but to a distribution.Has connections with kernel (eg rolling window) regression.Larger gain means more weight on the update, more weight on recent data, more variable convergent distribution.

Stochastic gradient learning

t t−1 zt−1zt−1′ T t−1 − t−1 t

Set R = I.Care: not scale or unit free.Like OLS without inv(X’X),But eliminates possibly explosive recursion for R.Loosens connection with REE limit of normal learning recursion.

Projection facilities

1. computeRt Rt−1 zt−1zt−1′ − Rt−1

2. compute tp t−1 Rt

−1zt−1zt−1′ T t−1 − t−1 t

3. if absmaxeig tp ≤ 1, t t

p , else t t−1

4. compute Et, zt

5. t t 1

6. goto 1.

LS learning recursions with constant gain can explode without PFs.Example below.Marcet and Sargent convergence results also rely on existence of suitable PFs.

So a projection facility is something that says: don’t update if it looks like exploding.

Learnability and determinacy

it t xx t, 1

See EH, Bullard and Mitra, Bullard and Schaling.This condition, the `Taylor Principle’ [note he has a rule, a curve and a principle named after him!] guarantees uniqueness of REE in the NK model.Referred to as ‘determinacy’.Means: only one value for expectations given value of fundamental shocks (like technology, monetary policy, demand).Absence implies possibility for self-fulfilling expectational shocks.We will see this in the lecture on the zero bound.Also guarantees ‘learnability’ and e-stability of REE.

Two‐sided learning

• So far we have just considered private sector learning as a replacement for the RE operator in the Muth/Lucas and the NK model.

• Obviously we could consider a policymaker learning too.

• A policy decision [eg some pseudo optimal policy] depends on knowledge of structural parameters gleaned from a regression, updated each period…

Learning with misspecification• Once we free ourselves from RE, why restrict ourselves to agents only deploying regressions that embrace the functional form of the REE?

• After all, much controversy about the functional form of the actual economy!

• Such models clearly don’t have REE.  But they can have equilibria:

• ‘Restricted perceptions’ or ‘Self‐confirming equilibria’

• Eg Cho, Sargent and Williams / Ellison‐Yates

Sargent:  Conquest of American inflation

• Fed thinks that higher inflation buys permanently lower unemployment.

• In reality, only inflation‐surprises lower unemployment.

• Fed re‐estimates mis‐specified Philips Curve each period.

• Most of time in high inflation mode, with periodic escapes to low inflation.

Ellison‐Yates• How low inflation regime also means low variance, and high inflation regime means high variance.

• When time is good for trying to lower long run u, it’s also good for using inf to smooth shocks in u

• Literature on ‘Great moderation’ (now vanished) emphasised low frequency changes in second moments of macro variables.

• Ellison‐Yates GM=temporary escape.

Inter period learning, intra‐period learning, PEA

• Intra‐period learning as an analogous way to say something about equilibrium plausibility.  See, eg, Blake, Kirsanova and Yates/Dennis and Kirsanova

• Idea:  agents behave not like econometricians, but like MSc theoreticians!

• Parameterised expectations algorithm to solve the RBC model, eg den Haan and Marcet.

Learning: contribution to business cycle/monetary policy analysis

• Cogley‐Colacito‐Sargent– Model Fed as a Bayesian learning the weights to put on competing models

– Includes model with long run trade‐off– Fed went for high inflation despite very low chance that SS model was right because payoff in event that it was right would be huge.

Learning:  contributions/ctd

• Expectations/ learning can be an extra source of propagation [explaining why cycles so large] and origin of cycles:  expectational shocks.

• Optimal policy with adaptive learning found to be more hawkish.  Stamping out persistence in inflation stops expectations rising, so stops inflation itself.

• BoC study of possible switch to price level targeting.  Didn’t go for it for reasons related to learning.

Alternative non‐rational expectations

• Sims: hyper‐rational, ‘rational inattention’– Watch things only if they r important and vary a lot

• Mankiw and Reis:  ‘sticky information’– Respond rationally, but have old info

• Brock and Hommes, King…Yates:  ‘heuristic‐switching.– Switch amongst rules of thumb according to noisy observation on performance of each

The heuristic‐switching model

t Ett x t et

et et−1 zt

x t at − T,T 0

Simple NK model. Output gap is instrument of cb. Persistent shocks to give role for dynamic forecasts. Zero inflation target.

1 : Ett1 T 0

2 : Ett1 t−1

Agents choose from two heuristic forecasts of inflation. The target, and phi*lagged inflation

Ett1 ntt−1 1 − ntT ntt−1

Epi is the weighted sum of e across different agents.

Determining % that use a heuristic

Fit −1/h∑st−ht s − Eiss2

Rolling window evaluation of forecasting performance for a heuristic.

nit expFit

∑ i1I expFit

N is the probability an agent chooses a heuristic, or, aggregated, the proportion that use it.

Theta=‘intensity of choice’ or, equivalently, noise with which F is observed.

n Tn

Think of the model as a transformation mapping an initial n into a future n.We can ask whether there is a fixed point, and what it looks like.A possible attracting point or rest point of the model

Learning:  what you need to be able to do

• Find the REE of a simple model• Verify that non‐RE expectations functions generate expectational errors that violate RE.

• Understand what properties of Eerrors violate RE.

• Understand motivation for and contribution of non‐RE models in business cycle analysis, providing examples, and comprehensible, short accounts of them.

What you need to be able to do/ctd…

• Execute test for e‐stability in simple models.• Understand and formulate least squares learning version of NK and simpler models.

• Understand where estability condition in least squares learning model comes from.  You don’t have to derive the estability test.

What you need to be able to do/ctd…

• Read and digest some examples of empirical papers on REH;  analyses of learning in macro.

• Use the papers listed in the lecture, and their bibliographies.