Long-run Uncertainty and Value Lars Peter Hansen University of Chicago and NBER Thomas J. Sargent New York University and NBER Jose Scheinkman Princeton University and NBER July 9, 2009 Contents Contents vii List of Figures ix Preface xi Acknowledgements xiii 1 Stochastic Processes 1 1.1 Constructing a Stochastic Process: I . 1 1.2 StationaryStochasticProcesses . 3 1.3 Invariant Events and the Law of Large Numbers . 4 1.4 Constructing a Stochastic Process: II . 7 1.5 Stationarityreconsidered . 9 1.6 LimitingBehavior. .. .. 10 1.7 Finite-stateMarkovchain . 14 1.8 VectorAutoregression . 16 2 Additive Functionals 19 2.1 Construction .......................... 19 2.2 Limiteddependence. 20 2.3 MartingaleExtraction . 22 2.4 Cointegration .......................... 24 2.5 Identifying Shocks with Long-Run Consequences . 25 2.6 CentralLimitTheory. 30 2.7 Growth-rateRegimes . 33 2.8 AQuadraticModelofGrowth . 34 3 Multiplicative Functionals 37 vii viii CONTENTS 3.1 MultiplicativeGrowth . 37 3.2 FourGrowthSpecifications. 40 3.3 Multiplicative Martingale Extraction . 46 4 Likelihood-ratio Processes 49 4.1 LikelihoodProcesses . 49 4.2 Log-Likelihoods ......................... 52 4.3 Scoreprocess .......................... 53 4.4 Limiting Behavior of the Likelihood Ratio . 55 4.5 Twisted Probability Measures . 57 Bibliography 59 List of Figures 1.1 This figure depicts the state evolution as a function of sample point ω induced by the transformation S. The oval shaped region is the collection Ω of all sample points. 2 2.1 The top panel plots the logarithm of consumption (smooth blue series) and logarithm of corporate earnings (choppy red series). The bottom panel plots the difference in the logarithms of con- sumptionandcorporateearnings. 26 2.2 This figure plots the response of the logarithm of consumption to a permanent shock (solid) and and to a temporary shock (dashed line). The permanent shock is identified as the increment to the common martingale component of the logarithm of consumption and the logarithm of corporate earnings. The figure comes from Hansenetal.(2008). .. .. 28 2.3 This figure plots posterior histograms for the magnitudes of the short-term and long-term responses of the logarithm of consump- tion to the shocks. The magnitude is measured as the absolute value across the contributions from the two shocks. The up- per panel depicts the histogram for the immediate response and the lower panel depicts the histogram for the long-term limiting response. The figure comes from Hansen et al. (2008). 29 2.4 This figure plots the impulse responses for the logarithm of out- put obtained by estimating a three-variate vector autoregression. The shocks are identified as in Fisher (2006). The top panel gives the response to an investment-specific technology shock, the middle panel gives the response to neutral technology shock and the bottom panel gives the response to a transient shock that is uncorrelated with the other two shocks. 31 ix x List of Figures 2.5 This figure plots the impulse responses for the logarithm of hours obtained by estimating a three-variate vector autoregression. The shocks are identified as in Fisher (2006). The top panel gives the response to an investment-specific technology shock, the middle panel gives the response to neutral technology shock and the bottom panel gives the response to a transient shock that is uncorrelated with the other two shocks. 32 3.1 This figure illustrates a particular additive functional and its multiplicativecounterpart. 38 3.2 The top panel plots a simulation of the multiplicative process studied in example 3.2.2, while the bottom panel plots a simu- lation of the Cecchetti et al. (2000) process for annual aggregate U.S. mentioned in example 3.2.3. 43 4.1 The logarithmic function is the concave function in this plot. This function is zero when evaluated at unity. By forming aver- ages using the two endpoints of the straight line below the loga- rithmic function, we are lead to some point on the line segment depending upon the weights used in the averaging. Jensen’s in- equality in this case is just a statement that the line segment must lie below the logarithmic function. 52 4.2 The vertical axis is implied asymptotic bound on the decay rate for the mistake probabilities for distinguishing between two mod- els. The bound is scaled by 100 an thus expressed as a percent. This bound is reported for alternative choices of α as depicted on the horizontal axis. The difference between the two curves reflects the choice of models in the numerator and denominator of the likelihood ratio. The best bound is given by the maximum of the two curves. The maximizing value for each curve is the same and given a maximizing choice α∗ of one curve, the other curve is maximized at 1 α∗. ................... 57 − Preface This manuscript started off as the Toulouse Lectures given by Lars Pe- ter Hansen. Our aim is to explore connections among topics that relate probability theory to the analysis of dynamic stochastic economic systems. Martingale methods have been a productive way to identify shocks with long-term consequences to economic growth and to characterize long-run dependence among macroeconomic time series. Typically they are applied by taking logarithms of time series such as output or consumption in order that growth can be modeled as accumulating linearly over time, albeit in a stochastic fashion. Martingale methods applied in this context have a long history in applied probability and applied time series analysis. We review these methods in the first part of this monograph. In the study of valu- ation, an alternative martingale approach provides a notion of long-term approximation. This approach borrows insights from large deviation the- ory, initiated in part to study the behavior of likelihood ratios of alternative time series models. We show how such methods provide characterizations of long-term model components and long-term contributions to valuation. Large deviation theory and the limiting behavior of likelihood ratios has also been central to some formulations of robust decision making. We de- velop this connection and build links to recursive utility theory in which investors care about the intertemporal composition of risk. Our interest in “robustness” and likelihood ratios is motivated by our conjecture that the modeling of the stochastic components to long-term growth is chal- lenging for both econometricians and the investors inside the models that econometricians build. More technical developments of some of these themes are given in Hansen and Scheinkman (1995), Anderson et al. (2003), Hansen and Scheinkman (2009) and Hansen (2008). xi Acknowledgements The author wishes to thank Jarda Borovicka, Adam Clemens, Serhiy Kokak, Junghoon Lee and Kenji Wada for their comments and assistance. These notes give some of the underlying support material for the Toulouse Lectures with the listed title. xiii Chapter 1 Stochastic Processes In this chapter, we describe two ways of constructing stochastic processes. The first is one that is especially convenient for stating and proving limit theorems. The second is more superficial in the sense that it directly speci- fies objects that are outcomes in the first construction. However, the second construction is the one that is most widely used in modeling economic time series. We shall use these constructions to characterize limiting behav- ior both for stationary environments and for environments with stochastic growth. 1.1 Constructing a Stochastic Process: I We begin with a method of constructing a stochastic process that is con- venient for characterizing the limit of points of time series averages.1 This construction works with a deterministic transformation S that maps a state of the world ω Ω today into a state of the world S(ω) Ω tomorrow. The state of the∈ world itself is not observed. Instead, a vect∈ or X(ω) that contains incomplete information about ω is observed. We assign probabili- ties over states of the world ω, then use the functions S and X to deduce a joint probability distribution for a sequence of X’s. In more detail: The probability space is a triple (Ω, F,Pr), where Ω is a set of sam- • ple points, F is an event collection (sigma algebra), and Pr assigns 1A good reference for the material in this section and the two that follow it is Breiman (1968). 1 2 CHAPTER 1. STOCHASTIC PROCESSES . . S3(2) S( 2) 2. S2(2) Figure 1.1: This figure depicts the state evolution as a function of sample point ω induced by the transformation S. The oval shaped region is the collection Ω of all sample points. probabilities to events. The (measurable) transformation S : Ω Ω used to model the evo- • lution over time has the property that for→ any event Λ F, ∈ S−1(Λ) = ω Ω : S(ω) Λ { ∈ ∈ } is an event. Notice that St(ω) is a deterministic sequence of states of the world in Ω. The vector-valued function X : Ω Rn used to model observations • → is Borel measurable. That is for any Borel set b in Rn, Λ= ω Ω : X(ω) b F. { ∈ ∈ } ∈ In other words, X is a random vector. 1.2. STATIONARY STOCHASTIC PROCESSES 3 The stochastic process Xt : t =1, 2, ... used to model a sequence of • observations is constructed{ via the formula:} t Xt(ω)= X[S (ω)] or X = X St. t ◦ The stochastic process X : t =1, 2, ... is a sequence of n-dimensional { t } random vectors, and the probability measure Pr allows us to make proba- bilistic statements about the joint distribution of successive components of this sequence. It will sometimes be convenient to extend this construction to date zero by letting X0 = X. While this construction of a stochastic process may at first sight appear special, it is not, as the following example illustrates.