Intro to non-rational expectations (learning) in macroeconomics

1. Intro to non-rational expectations (learning) in macroeconomics Lecture to Advanced Macro class, Bristol MSc, Spring 2014

2. 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.

3. Follow my outputs if you are interested • My research homepage • Blog ‘longandvariable’ on macro and public policy • Twitter feed @tonyyates

4. 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.

5. 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 not as hard as hardest exercises.

6. 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…

7. 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

8. 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’.

9. 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.

10. 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

11. RE in the NK model NB the expectation is assumed to be the true, mathematical expectation

12. The computational demands of RE 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!

13. Motivation 2, ctd: empirical tests of implications of RE often fail Collect data on inflation expectations; 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.

14. Take the test to the data • Bakhshi, Kapetanios and Yates ‘Rational expectations and fixed event forecasts’ • Huge literature, going back to early exposition of theory by Muth, so this is but one example. • 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!

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

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

17. 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.

18. 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’]. • So drop the requirement that agents have to solve for fixed points! And require them only to run and update regressions. • A small step in direction of realism and plausibility. • Cost: getting lost in Sims’ ‘wilderness’.

19. 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

20. 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.

21. Lucas/Muth model • Output equals natural rate plus something times price surprises • Aggregate demand = quantity equation. • Money (policy) feedback rule.

22. 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.

23. Finding the REE of the Muth/Lucas model Reduced form of this model in terms of prices. Rational expectations equilibrium, with expectations ‘solved out’ using a guess and verify method.

24. Towards understanding the T-mapping in learning that takes PLM to ALM Reduced form of LM model. Forecaster’s PLM1 ALM1 under this PLM1 ALM2 under PLM2=f(ALM1)

25. Repeated substitution expressed as repeated application of a T-map 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’

26. Will our theoretician ever get to the REE in the Muth-Lucas model? 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.

27. Matlab code to do repeated substitution in Muth-Lucas

28. Progression of experimental theorist’s guesses in the Muth-Lucas model This is for alpha=0.25<1; charts for a,b(1),b(2) respectively, green lines plot REE values.

29. What happens when alpha>1? 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, hoping for the best

30. 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.

31. 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

32. Least squares learning recursion for the Muth-Lucas model 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.

33. Close relative: recursive least squares in econometrics 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

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

35. 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

36. 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')

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

38. 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 T maps beliefs back onto themeslves, or T()-()=0.

39. Estability, the ODE in beliefs 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. ODE’s for our Muth/Lucas model

40. Estability conditions Stability depends on eigenvalues of transition matrix <1 Transition 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].

41. Foundations for e-stability analysis Rewrite the PLM using the T-map 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.

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

43. Deriving that estability in learning reduces to T()-() Expectation 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.

44. Estability If all eigenvalues of this derivative or Jacobian matrix have negative real parts, system locally stable. Unpack Q: we will show that stability is guaranteed in second equation. Expectations taken over X, because we are looking to account for all possible trajectories.

45. Reducing and simplifying the estability equations This period’s shock uncorrelated with last period’s data TR 1 entry as z includes constant. Bottom right is variance of w 0s because w not correlated with 1! We can get from the LHS pair to the RHS pair because S tends to M from any starting point.

46. 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.

47. 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

48. Constant gain 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.

49. Stochastic gradient learning 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.

50. Projection facilities 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.