Learning in Macroeconomic Models

Transcription

1 Learning in Macroeconomic Models Wouter J. Den Haan London School of Economics c by Wouter J. Den Haan

2 Overview A bit of history of economic thought How expectations are formed can matter in the long run Seignorage model Learning without feedback Learning with feedback Simple adaptive learning Least-squares learning Bayesian versus least-squares learning Decision theoretic foundation of Adam & Marcet 2 / 95

3 Overview continued Topics Learning & PEA Learning & sunspots 3 / 95

4 Why are expectations important? Most economic problems have intertemporal consequences = future matters Moreover, future is uncertain Characteristics/behavior other agents can also be uncertain = expectations can also matter in one-period problems 4 / 95

5 History of economic thought adaptive expectations: Ê t [x t+1 ] = Ê t 1 [x t ] + ω very popular until the 70s ( ) x t Ê t 1 [x t ] 5 / 95

6 History of economic thought problematic features of adaptive expectations: agents can be systematically wrong agents are completely passive: Ê t [ xt+j ], j 1 only changes (at best) when xt changes = Pigou cycles are not possible = model predictions underestimate speed of adjustment (e.g. for disinflation policies) 6 / 95

7 History of economic thought problematic features of adaptive expectations: adaptive expectations about x t+1 = adaptive expectations about x t+1 (e.g. price level versus inflation) why wouldn t (some) agents use existing models to form expectations? expectations matter but still no role for randomness (of future realizations) so no reason for buffer stock savings no role for (model) uncertainty either 7 / 95

8 History of economic thought rational expectations became popular because: agents are no longer passive machines, but forward looking i.e., agents think through what could be consequences of their own actions and those of others (in particular government) consistency between model predictions and of agents being described randomness of future events become important [ ] e.g., E t c γ = (E t [c t+1 ]) γ t+1 8 / 95

9 History of economic thought problematic features of rational expectations agents have to know complete model make correct predictions about all possible realizations on and off the equilibrium path costs of forming expectations are ignored how agents get rational expectations is not explained 9 / 95

10 History of economic thought problematic features of rational expectations makes analysis more complex behavior this period depends on behavior tomorrow for all possible realizations = we have to solve for policy functions, not just simulate the economy 10 / 95

11 Expectations matter Simple example to show that how expectations are formed can matter in the long run See Adam, Evans, & Honkapohja (2006) for a more elaborate analysis 11 / 95

12 Model Overlapping generations Agents live for 2 periods Agents save by holding money No random shocks 12 / 95

13 Model max c 1,t,c 2,t ln c 1,t + ln c 2,t c 2,t 1 + s.t. P t P e t+1 (2 c 1,t ) no randomness = we can work with expected value of variables instead of expected utility 13 / 95

14 Agent s behavior First-order condition: 1 c 1,t = Solution for consumption: P t P e t+1 1 = 1 c 2,t π e t+1 c 1,t = 1 + π e t+1 /2 Solution for real money balance (=savings): 1 c 2,t m t = 2 c 1,t = 1 π e t+1 /2 14 / 95

15 Money supply M s t = M 15 / 95

16 Equilibrium Equilibrium in period t implies M = M t M = P t ( 1 π e t+1 /2 ) P t = M 1 π e t+1 /2 16 / 95

17 Equilibrium Combining with equilibrium in period t 1 gives π t = P t = 1 πe t /2 P t 1 1 π e t+1 /2 Thus: π e t & πe t+1 = money demand = actual inflation π t 17 / 95

18 Rational expectations solution Optimizing behavior & equilibrium: P t P t 1 = T(π e t, π e t+1 ) Rational expectations equilibrium (REE): π t = π e t = π t = T(π t, π t+1 ) = π t+1 = 3 2 π t π t+1 = R (π t ) 18 / 95

19 Multiple steady states There are two solutions to π = 3 2 π = there are two steady states π = 1 (no inflation) and perfect consumption smoothing π = 2 (high inflation), money has no value & no consumption smoothing at all 19 / 95

20 Unique solution Initial value for π t not given, but given an initial condition the time path is fully determined π t converging to 2 means m t converging to zero and P t converging to 20 / 95

21 Rational expectations and stability πt o π t 21 / 95

22 Rational expectations and stability π 1 : value in period 1 π 1 < 1 : divergence π 1 = 1 : economy stays at low-inflation steady state π 1 > 1 : convergence to high-inflation steady state 22 / 95

23 Alternative expectations Suppose that π e t+1 = 1 2 π t πe t still the same two steady states, but π = 1 is stable π = 2 is not stable 23 / 95

24 Adaptive expectations and stability πt 0.85 Initial conditions: π e 1 =1.5, π e 2 = time 24 / 95

25 Learning without feedback Setup: 1 Agents know the complete model, except they do not know dgp exogenous processes 2 Agents use observations to update beliefs 3 Exogenous processes do not depend on beliefs = no feedback from learning to behavior of variable being forecasted 25 / 95

26 Learning without feedback & convergence If agents can learn the dgp of the exogenous processes, then you typically converge to REE They may not learn the correct dgp if Agents use limited amount of data Agents use misspecified time series process 26 / 95

27 Learning without feedback - Example Consider the following asset pricing model P t = E t [β (P t+1 + D t+1 )] If then lim j βt+j D t+j = 0 [ ] P t = E t β j D t+j j=1 27 / 95

28 Learning without feedback - Example Suppose that D t = ρd t 1 + ε t, ε t N(0, σ 2 ) (1) REE: P t = D t 1 βρ (note that P t could be negative so P t is like a deviation from steady state level) 28 / 95

29 Learning without feedback - Example Suppose that agents do not know value of ρ Approach here: If period t belief = ρ t, then P t = D t 1 β ρ t Agents ignore that their beliefs may change, [ ] [ ] D i.e., Ê t+j t Pt+j =Et 1 β ρ is assumed to equal t+j [ ] 1 1 β ρ E t Dt+j t 29 / 95

30 Learning without feedback - Example How to learn about ρ? Least squares learning using {D t } T t=1 & correct dgp Least squares learning using {D t } T t=1 & incorrect dgp Least squares learning using {D t } T t=t T & correct dgp Least squares learning using {D t } T t=t T & incorrect dgp Bayesian updating (also called rational learning) Lots of other possibilities 30 / 95

31 Convergence again Suppose that the true dgp is given by D t = ρ t D t 1 + ε t } ρ t {ρ low, ρ high { ρhigh w.p. p(ρ ρ t+1 = t ) ρ low w.p. 1 p(ρ t ) Suppose that agents think the true dgp is given by D t = ρd t 1 + ε t = Agents will never learn (see homework for importance of sample used to estimate ρ) 31 / 95

32 Recursive least-squares time-series model: least-squares estimator y t = x tγ + u t where γ T = RT 1 X T Y t T X T = [ x 1 x 2 x T ] Y T = [ y 1 y 2 y T ] R T = X T X T/T 32 / 95

33 Recursive least-squares R T = R T 1 + (x Tx T R T 1) T γ T = γ T 1 + R 1 T x T (y T x T γ T 1) T 33 / 95

34 Proof for R X T X T T ( T 1 T? X T 1 X T 1 + x Tx T = (T 1) T ) X? T X T = X T 1 X T 1 + T 1 X T X T X T X T T X T 1 X T 1 + x T x T X T 1 X T 1+x T x T T? T X T 1 X T 1 T(T 1) x Tx T X T 1 X T 1 T = X T 1 X T 1 + x T x T x Tx T T? X T 1 = X T 1 + x T x T x Tx T +X T 1 X T 1 T X T 1 X T 1 T 34 / 95

35 Proof for gamma (X T X T) 1 X T Y T X T Y T X T Y T? =? =? = ( ) X 1 T 1 X T 1 X ( T 1 Y T 1 + ) (X T X T) 1 x T y T x T x T ( ) X 1 T 1 X T 1 X T 1 Y T 1 ( X T 1 X T 1 + x T x T ) ( ) X 1 T 1 X T 1 X ( T 1 Y ) T 1 x T y T + x T x T ( ) X 1 T 1 X T 1 X T 1 Y ( T 1 I + x T x T ( ) ) X 1 T 1 X T 1 X T 1 Y T 1 ( ) x T y T + x T x T ( ) X 1 T 1 X T 1 X T 1 Y T 1 35 / 95

36 Reasons to adopt recursive formulation makes proving analytical results easier less computer intensive, but standard LS gives the same answer there are intuitive generalizations: R T = R T 1 + ω(t) ( x T x T R ) T 1 γ T = γ T 1 + ω(t)rt 1 x ( T yt x T γ T 1) ω(t) is the "gain" 36 / 95

37 Learning with feedback 1 Explanation of the idea 2 Simple adaptive learning 3 Least-squares learning E-stability and convergence 4 Bayesian versus least-squares learning 5 Decision theoretic foundation of Adam & Marcet 37 / 95

38 Learning with feedback - basic setup Model: p t = ρê t 1 [p t ] + δx t 1 + ε t RE solution: p t = δ 1 ρ x t 1 + ε t = a re x t 1 + ε t 38 / 95

39 What is behind model Model: p t = ρê t 1 [p t ] + δx t 1 + ε t Stories: Lucas aggregate supply model Muth market model See Evans and Honkapohja (2009) for details 39 / 95

40 Learning with feedback - basic setup Perceived law of motion (PLM) at t 1: p t = â t 1 x t 1 + ε t (2) Actual law of motion (ALM): p t = ρâ t 1 x t 1 + δx t 1 + ε t = (ρâ t 1 + δ) x t 1 + ε t (3) 40 / 95

41 Updating beliefs I: Simple adaptive ALM: p t = (ρâ t 1 + δ) x t 1 + ε t Simple adaptive learning: â t = ρâ t 1 + δ could be rationalized if agents observe x t 1 and ε t t is more like an iteration and in each iteration agents get to observe long time-series to update 41 / 95

42 Simple adaptive learning: Convergence or in general â t = ρâ t 1 + δ â t â t 1 = T (â t 1 ) 42 / 95

43 Simple adaptive learning: Convergence Key questions: 1 Does â t converge? 2 If yes, does it converge to a? Answers: If ρ < 1, then the answer to both is yes. 43 / 95

44 Updating beliefs: LS learning Suppose agents use least-squares learning â t = â t 1 + R 1 t x t 1 (p t x t 1 â t 1 ) t = â t 1 + R 1 t x t 1 ((ρâ t 1 + δ) x t 1 + ε t x t 1 â t 1 ) t R t = R t 1 + (x t 1x t 1 R t 1 ) t 44 / 95

45 Updating beliefs: LS learning â t = â t t R 1 t x t 1 (p t x t 1 â t 1 ) = â t t R 1 t x t 1 ((ρâ t 1 + δ) x t 1 + ε t x t 1 â t 1 ) R t = R t t (x t 1x t 1 R t 1 ) To get system with only lags on RHS, let R t = S t 1 â t = â t t S 1 t 1 x t 1 ((ρâ t 1 + δ) x t 1 + ε t x t 1 â t 1 ) S t = S t t (x tx t S t 1 ) t t / 95

46 Updating beliefs: LS learning Let θ t = Then the system can be written as [ at S t ] θ t = θ t t Q( θ t 1, x t, x t 1, ε t ) or θ t = T( θ t 1, x t, x t 1, ε t, t) Note that T ( ) = 1 t Q ( ) 46 / 95

47 Key question If θ t = 1 t Q( θ t 1, x t, x t 1, ε t, t) then what can we "expect": about θ t? In particular, can we "expect" that lim t ât = a re 47 / 95

48 Corresponding differential equation Much can be learned from following differential equation dθ dτ = h (θ (τ)) where h (θ) = lim t E [Q(θ, x t, x t 1, ε t )] 48 / 95

49 Corresponding differential equation In our example h (θ) = lim E [Q(θ, x t, x t 1, ε t )] t [ S = lim E 1 x t 1 ((ρa + δ) x t 1 + ε t x t 1 a) t (x t x t S) t+1 t [ MS = 1 ] ((ρ 1) a + δ) M S ] where [ ] M = lim E x 2 t t 49 / 95

50 Analyze the differential equation dθ dτ = h (θ (τ)) = [ MS 1 ((ρ 1) a + δ) M S ] dθ dτ = 0 if M = S & a = δ 1 ρ Thus, the (unique) rest point of h (θ) is the rational expectations solution 50 / 95

51 E-stability θ t θ t 1 = 1 ) ( θ t Q t 1, x t, x t 1, ε t, t Limiting behavior can be analyzed using dθ dτ = h(θ(τ)) = lim t E [Q(θ, x t, x t 1, ε t )] A solution θ, e.g. [a RE, M], is "E-stable" if h(θ) is stable at θ 51 / 95

52 E-stability h(θ) is stable if real part of the eigenvalues is negative: Here: h(θ) = [ (ρ 1) a + δ M S = convergence of differentiable system if ρ 1 < 0 = convergence even if ρ < 1! ] 52 / 95

53 Implications of E-stability? Recursive least-squares: stochastics in T ( ) mapping = what will happen is less certain, even with E-stability 53 / 95

54 General implications of E-stability? If a solution is not E-stable: = non-convergence is a probability 1 event If a solution is E-stable: the presence of stochastics make the theorems non-trivial in general only info about mean dynamics 54 / 95

55 Mean dynamics See Evans and Honkapohja textbook for formal results. Theorems are a bit tricky, but are of the following kind: If a solution f is E-stable, then the time path under learning will either leave the neighborhood in finite time or will converge towards f. Moreover, the longer it does not leave this neighborhood, the smaller the probability that it will So there are two parts mean dynamics: convergence towards fixed point escape dynamics: (large) shocks may push you away from fixed point 55 / 95

56 Importance of Gain γ T = γ T 1 + ω(t)rt 1 x ( T yt x T γ T 1) Gain in least squares updating formula, ω (T), plays a key role in theorems ω (T) 0 too fast: you may end up in somthing that is not an equilibrium ω (T) 0 too slowly:,you may not converge towards it So depending on the application, you may need conditions like ω(t) 2 < and t=1 ω(t) = t=1 56 / 95

57 Special cases In simple cases, stronger results can be obtained Evans (1989) shows that the system of equations (2) and (3) with standard recursive least squares (gain of 1/t) converges to rational expectations solution if ρ < 1 (so also if ρ < 1). 57 / 95

58 Bayesian learning LS learning has some disadvantages: why "least-squares" and not something else? how to choose gain? why don t agents incorporate that beliefs change? Beliefs are updated each period = Bayesian learning is an obvious thing to consider 58 / 95

59 Bayesian versus LS learning LS learning = Bayesian learning with uninformed prior at least not always Bullard and Suda (2009) provide following nice example 59 / 95

60 Bayesian versus LS learning Model: p t = ρ L p t 1 + ρ 0 Ê t 1 [p t ] + ρ 1 Ê t 1 [p t+1 ] + ε t (4) Key difference with earlier model: two extra terms 60 / 95

61 Bayesian versus LS learning The RE solution: p t = bp t 1 + ε t where b is a solution to b = ρ L + ρ 0 b + ρ 1 b 2 61 / 95

62 Bayesian learning - setup PLM: p t = b t 1 p t 1 + ε t and ε t has a known distribution plug PLM into (4) = ALM but a Bayesian learner is a bit more careful 62 / 95

63 Bayesian learner understands he is learning ] Ê t 1 [p t+1 ] = Ê t 1 [ρ L p t + ρ 0 Ê t [p t+1 ] + ρ 1 Ê t [p t+2 ] ] = ρ L p t + Ê t 1 [ρ 0 Ê t [p t+1 ] + ρ 1 Ê t [p t+2 ] [ ] = ρ L p t + Ê t 1 ρ 0 bt p t + ρ 1 bt p t+1 and he realizes, for example, that b t and p t are both affected by ε t! 63 / 95

64 Bayesian learner understands he is learning Bayesian learner realizes that Ê t 1 [ bt p t+1 ] = Ê t 1 [ bt ] Êt 1 [p t+1 ] ] and calculates Ê t 1 [ bt p t+1 explicitly LS learner forms expectations thinking that Ê t 1 [ bt p t+1 ] = Ê t 1 [ bt 1 p t+1 ] = b t 1 Ê t 1 [( ρ L p + ρ 0 bt 1 + ρ 1 bt 1 ) p t ] 64 / 95

65 Bayesian versus LS learning Bayesian learner cares about a covariance term Bullard and Suda (2009) show that Bayesian is simillar to LS learning in terms of E-stability Such covariance terms more important in nonlinear frameworks Unfortunately not much done with nonlinear models 65 / 95

66 Learning what? Model: P t = βe t [P t+1 + D t+1 ] Learning can be incorporated in many ways. Obvious choices here: 1 learn about dgp D t and use true mapping for P t = P (D t ) 2 know dgp D t and learn about P t = P (D t ) 3 learn about both 66 / 95

67 Learning what? 1 Adam, Marcet, Nicolini (2009): one can solve several asset pricing puzzles using a simple model if learning is learning about E t [P t+1 ] (instead of learning about dgp D t ) 2 Adam and Marcet (2011): provide micro foundations that this is a sensible choice 67 / 95

68 Simple model Model: P t = βe t [P t+1 + D t+1 ] D t+1 D t = aε t+1 with E t [ε t+1 ] = 1 ε t i.i.d. 68 / 95

69 Model properties REE Solution: P t /D t is constant P t /P t 1 is i.i.d. P t = βa 1 βa D t 69 / 95

70 Adam, Marcet, & Nicolini 2009 PLM: ALM: P t [ ] Pt+1 Ê t = γ P t t = 1 βγ t 1 aε t = P t 1 1 βγ ( t γ t+1 = a + aβ γ ) t 1 βγ t ( a + aβ γ ) t ε t 1 βγ t 70 / 95

71 Model properties with learning Solution is quite nonlinear especially if γ t is close to β 1 Serial correlation. in fact there is momentum. For example: γ t = a & γ t > 0 = γ t+1 > 0 γ t = a & γ t < 0 = γ t+1 < 0 P t /D t is time varying 71 / 95

72 Adam, Marcet, & Nicolini 2011 Agent i does following optimization problem max Ê i,t [ ] Ê i,t is based on a sensible probability measure Ê i,t is not necessarily the true conditional expectation 72 / 95

73 Adam, Marcet, & Nicolini 2011 Setup leads to standard first-order conditions but with Ê i,t instead of E t For example Key idea: P t = βê i,t [P t+1 + D t+1 ] if agent i is not constrained price determination is diffi cult agents do not know this mapping = they forecast Ê i,t [P t+1 ] directly = law of iterated expectations cannot be used because next period agent i may be constrained in which case the equality does not hold 73 / 95

74 Topics - Overview 1 E-stability and sunspots 2 Learning and nonlinearities Parameterized expectations 3 Two representations of sunspots 74 / 95

75 E-stability and sunspots Model: x t = ρe t [x t+1 ] x t cannot explode no initial condition Solution: ρ < 1 : x t = 0 t ρ 1 : x t = ρ 1 x t 1 + e t t where e t is the sunspot (which has E t [e t ] = 0 75 / 95

76 Adaptive learning PLM: x t = â t x t 1 + e t ALM: x t = â t ρx t 1 = â t+1 = â t ρ thus divergence when ρ > 1 (sunspot solutions) 76 / 95

77 Adaptive learning PLM: x t = â t x t 1 + e t ALM: x t = â t ρx t 1 = â t+1 = â t ρ thus divergence when ρ > 1 (sunspot solutions) 77 / 95

78 Stability puzzle There are few counter examples and not too clear why sunspots are not learnable in RBC-type models sunspot solutions are learnable in some New Keynesian models (Evans and McGough 2005) McGough, Meng, and Xue 2011 provide a counterexample and show that an RBC model with negative externalities has learnable sunspot solutions 78 / 95

79 PEA and learning Learning is usually done in linear frameworks PEA parameterized the conditional expectations in nonlinear frameworks = PEA is a natural setting to do adaptive learning as well as recursive learning 79 / 95

80 Model [ P t = E β ( Dt+1 D t X t : state variables ) ν (P t+1 + D t+1)] = G (X t ) 80 / 95

81 Conventional PEA in a nutshell Start with a guess for G (X t ), say g(x t ; η 0 ) g ( ) may have wrong functional form x t may only be a subset of X t η 0 are the coeffi cients of g ( ) 81 / 95

82 Conventional PEA in a nutshell Iterate to find fixed point for η i 1 use η i to generate time path {P t } T t=1 2 let ˆη i = arg min η (y t+1 g (x t ; η)) 2 t where y t+1 = β 3 Dampen if necessary ( Dt+1 D t ) ν (P t+1 + D t+1) η i+1 = ω ˆη i + (1 ω) η i 82 / 95

83 Interpretation of conventional PEA Agents have beliefs Agents get to observe long sample generated with these beliefs Agents update beliefs Corresponds to adaptive expectations no stochastics if T is large enough 83 / 95

84 Recursive PEA Agents form expectations using g (x t ; η t ) Solve for P t using P t = g (x t ; η t ) Update beliefs using this one additional observation Go to the next period using η t+1 84 / 95

85 Recursive methods and convergence Look at recursive formulation of LS: γ t = γ t ( t R 1 t x t yt x t γ ) t 1!!! ˆγ t gets smaller as t gets bigger 85 / 95

86 General form versus common factor represenation sunspot literature distinquishes between: 1 General form representation of a sunspot 2 Common factor representation of a sunspot 86 / 95

87 First consider non-sun-spot indeterminacy Model: k t+1 + a 1 k t + a 2 k t 1 = 0 or (1 λ 1 L) (1 λ 2 L) k t+1 = 0 Also: k 0 given k t has to remain finite 87 / 95

88 Multiplicity Solution: k t = b 1 λ t 1 + b 2λ t 2 k 0 = b 1 + b 2 Thus many possible choices for b 1 and b 2 if λ 1 < 1 and λ 1 < 2 88 / 95

89 Multiplicity What if we impose recursivity? k t = dk t 1 Does that get rid of multiplicity? No, but it does reduce the number of solutions from to 2 ( d2 ) + a 1 d + a2 k t 1 = 0 t ( d2 ) = + a 1 d + a2 = 0 the 2 solutions correspond to setting either λ 1 or λ 2 equal to 0 89 / 95

90 Back to sunspots Doing the same trick with sunspots gives a solution with following two properties: 1 it has a serially correlated sunspot component with the same factor as the endogenous variable (i.e. the common factor) 2 there are two of these 90 / 95

91 General form representation Model: E t [k t+1 + a 1 k t + a 2 k t 1 ] = 0 or E t [(1 λ 1 L) (1 λ 2 L) k t+1 ] = 0 General form representation: k t = b 1 λ t 1 + b 2λ t 2 + e t k 0 = b 1 + b 2 + e 0 where e t is serially uncorrelated 91 / 95

92 Common factor representation Model: E t [k t+1 + a 1 k t + a 2 k t 1 ] = 0 or E t [(1 λ 1 L) (1 λ 2 L) k t+1 ] = 0 Common factor representation: k t = b 1 λ t i + ζ t ζ t = λ i ζ t 1 + e t k 0 = b i + ζ 0 λ i {λ 1, λ 2 } where e t is serially uncorrelated 92 / 95

93 References Adam, K., G. Evans, and S. Honkapohja, 2006, Are hyperinflation paths learnable, Journal of Economic Dynamics and Control. Paper shows that the high-inflation steady state is stable under learning in the seignorage model as discussed in slides. Adam, K., A. Marcet, and J.P. Nicolini, 2009, Stock market volatility and learning, manuscript. Paper shows that learning about endogenous variables like prices gives you much more "action" than learning about exogenous processes (i.e. they show that learning with feedback is more interesting than learning without feedback). 93 / 95

94 References Adam, K., and A. Marcet, 2011, Internal rationality, imperfect market knowledge and asset prices, Journal of Economic Theory. Paper motivates that the thing to be learned is the conditional expectation in the Euler equation. Bullard, J. and J. Suda, 2009, The stability of macroeconomics systems with Bayesian learners, manuscript. Paper gives a nice example on the difference between Bayesian and Least-Squares learning. 94 / 95

95 business cycle models with factor-generated externalities, manuscript. Paper provides a nice example of a sunspot in a linearized RBC-type model that 95 / 95 References Evans, G.W., and S. Honkapohja, 2001, Learning and expectations in macroeconomics. Textbook on learning dealing with all the technical stuff very carefully. Evans, G.W., and S. Honkaphja, 2009, Learning and macroeconomics, Annual Review of Economics. Survey paper. Evans, G.W., and B. McGough, 2005, Indeterminacy and the stability puzzle in non-convex economies, Contributions to macroeconomics. Paper argues that sunspot solutions (or indeterminate solutions more generally) are not stable (not learnable) in RBC-type models. Evans, G.W., and B. McGough, 2005, Monetary policy, indeterminacy, and learning, Journal of Economic Dynamics and Control. Paper shows that the NK model with some Taylor rules has learnable sunspot solutions McGough, B., Q. Meng, and J. Xue, 2011, Indeterminacy and E-stability in real