A Road Map#
Authors: Lars Peter Hansen (University of Chicago) and Thomas J. Sargent (NYU)
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The Preface described where these tools came from. This Roadmap describes how they fit together. It sketches each of the thirteen chapters in turn, then traces the half dozen objects that recur across them, and closes with some routes through the book for readers who do not intend to read it front to back.
Two features of the book shape how it is best read. First, the chapters are cumulative in machinery but not in subject: a conditional expectation operator introduced to characterize ergodicity in Chapter 2 returns to extract a martingale in Chapter 4 and to price a cash flow in Chapter 8. Readers who skip a chapter will meet its apparatus again later. Second, the same mathematical object often carries two economic interpretations, one for the econometrician outside a model and one for a decision maker inside it. That doubling is deliberate. It is the book’s answer to the question posed at the end of the Preface: if a model is an approximation, what should we assume the agents inside it believe?
The problem#
A researcher has one long time series and wants to learn the probability model that generated it. A Law of Large Numbers is the only instrument for that job, and it discloses less than one might hope. Chapter 1 makes the limitation precise: what a Law of Large Numbers reveals is a probability measure conditioned on invariant events, and the book calls such a measure a statistical model (Definition 1.6). Probabilities across statistical models are never disclosed, no matter how long the sample.
That gap organizes everything that follows. Chapters 1 through 9 develop tools for working within a statistical model: how to represent one, how to extract its long-run components, how to learn it when states are hidden, how to tell two of them apart, and how to use one to value a stochastic cash flow. Chapters 10 through 13 take up what to do about the gap: how a decision maker who distrusts both the prior over models and the models themselves should act, and how an econometrician should estimate when no model is fully specified.
The chapters#
Foundations: Chapters 1 through 4#
Chapter 1 builds a stochastic process from a measure-preserving transformation \({\mathbb S}\) and a measurement function \(X\), then states a Law of Large Numbers (Theorem 1.2) whose limit points are expectations conditioned on invariant events. The ergodic decomposition (Proposition 1.2) represents any measure-preserving probability as a mixture of ergodic ones. The mixture weights are the prior; the components are the statistical models. Risk and ambiguity part company here.
Chapter 2 specializes to Markov processes and replaces the transition distribution with the conditional expectation operator \({\mathbb T}\). Chapter 1’s invariant events reappear as eigenfunctions of \({\mathbb T}\) with unit eigenvalue, and ergodicity becomes the requirement that \({\mathbb T}f = f\) have only constant solutions. The chapter also supplies the mean-zero subspace \({\mathcal N}\) and a strong contraction condition that later chapters need.
Chapter 3 turns to processes whose increments are stationary, which is what logarithms of many economic time series look like. Its central result (Proposition 3.1) splits such a process into a trend, a martingale, a stationary component, and a constant. The martingale increment is the permanent shock. The chapter’s Central Limit Theorem (Proposition 3.2) then shows that the variance governing long-horizon behavior is the variance of that martingale increment, not the variance of the increment process. Cointegration is the restriction that a linear combination kills both trend and martingale.
Chapter 4 imposes Markov structure on the increments and makes the Chapter 3 decomposition constructive. The construction needs only \({\mathbb T}\). A recursive utility example then introduces three objects the second half of the book leans on: a continuation value, a risk-adjusted certainty equivalent, and a stochastic discount factor obtained as the slope of an indifference curve.
Responses, learning, and discrimination: Chapters 5 through 7#
Chapter 5 differentiates a Markov system with respect to a date-zero perturbation and obtains stochastic responses \((\Lambda, \Delta)\). These generalize impulse responses to nonlinear systems, where the responses are themselves random. The chapter also separates these local objects from the causal readings often attached to impulse responses in applied work.
Chapter 6 hides some states. The distribution of the hidden state conditioned on the signal history is itself Markov, and the chapter constructs it for a linear-Gaussian system (the Kalman filter and smoother), for a discrete hidden state, for unknown parameters treated as invariant hidden states, and for regime switching. Treating a parameter as an invariant hidden state connects learning to Chapter 1: the parameter indexes a statistical model, so learning it is learning which invariant event occurred.
Chapter 7 asks how fast data discriminate among competing models. A likelihood ratio process is a multiplicative martingale, and its logarithm is an additive functional with a negative trend coefficient. That sign is why the data eventually select the model that generated them (Proposition 7.3) without any prior over models. Chernoff entropy measures the rate. Score processes, obtained by differentiating a log likelihood, are additive martingales whose long-run variance is Fisher information.
Valuation: Chapters 8 and 9#
Chapter 8 exponentiates. A multiplicative functional grows geometrically, and Theorem 8.1 splits it into a geometric trend, a multiplicative martingale, and a ratio of a principal eigenfunction. That eigenfunction solves (8.8), which is Chapter 2’s unit-eigenvalue problem with the eigenvalue released from one because the process grows. The factorization does the work in every application: long-term valuation, the limiting risk-return tradeoff, and a measure of how far investor beliefs can depart from a baseline before a statistician could detect the difference.
Chapter 9 perturbs a multiplicative functional and differentiates, producing shock elasticities. These separate an exposure to uncertainty from the price of that exposure, at every horizon. Responses and elasticities answer different questions about one system: a response is a derivative with respect to a state, an elasticity a derivative with respect to an exposure to a shock.
Uncertainty and estimation: Chapters 10 through 13#
Chapter 10 separates three things that expected utility merges. Risk is probability within a statistical model. Ambiguity is doubt about the prior across models. Misspecification is doubt about the models themselves. Distorting a prior and distorting a likelihood become separate minimizations with separate penalties, so the two doubts can be switched on one at a time and their consequences compared.
Chapter 11 develops recursive utility, which separates risk aversion from intertemporal substitution, and shows that the same recursion describes a decision maker who distrusts the model and evaluates plans under a penalized worst case. The two readings are observationally equivalent within a model and differ in what they say about a model’s limitations. Two approximations follow: a continuous-time limit, and a small-noise expansion modified so that uncertainty has first-order consequences.
Chapter 12 represents a partial derivative of a value function as an asset price. Chapter 5’s responses propagate a marginal change in a state; an uncertainty-adjusted expectation discounts the resulting flow. Decomposing the result by horizon and by state variable is what makes a marginal valuation legible. A climate application shows why: the social cost of global warming and the social value of research and development are each near-cancellations of much larger contributions of opposite sign.
Chapter 13 estimates without a likelihood. GMM stays available when a model is partially specified or known to be misspecified, and studying a family of estimators at once yields an efficiency bound. Misspecification is then quantified rather than assumed away, using the same divergences that appear in Chapters 8 and 10.
Threads#
Six objects recur. Following one of them through the book is an alternative to reading it in order.
The statistical model. Defined in Chapter 1 as what a Law of Large Numbers can disclose. Markov form in Chapter 2; learned from signals in Chapter 6; discriminated among in Chapter 7; the object of ambiguity in Chapter 10; only partially specified in Chapter 13.
Martingale extraction. Additive martingales carry the permanent shocks of Chapters 3 and 4. Multiplicative martingales carry the change of probability in Chapters 7 and 8. The two constructions are parallel, and the logarithm of a multiplicative martingale is an additive functional.
The operator \({\mathbb T}\) and its eigenfunctions. Introduced in Chapter 2 to characterize ergodicity through \({\mathbb T}f = f\); inverted as \(({\mathbb I}-{\mathbb T})^{-1}\) in Chapter 4 to sum forecasts over horizons; generalized to the principal eigenvalue problem of Chapter 8 once the process being averaged grows.
Derivatives. Stochastic responses in Chapter 5, shock elasticities in Chapter 9, small-noise expansions in Chapter 11, marginal valuations in Chapter 12. All are local, and all scale linearly in the size of the perturbation.
Change of measure. A likelihood ratio in Chapter 7, a distorted belief in Chapter 8, a tilted prior or likelihood in Chapter 10, a robustness adjustment in Chapter 11, an uncertainty-adjusted discount in Chapter 12. The exponential tilting that solves the minimization problems in Chapters 8, 10, and 11 is one formula; what differs is the object tilted.
Entropy and divergence. Chernoff entropy measures how fast models separate in Chapters 7 and 8. Relative entropy penalizes belief distortion in Chapters 8, 10, and 11. Divergences bound expectations under misspecification in Chapters 10 and 13. Statistical detectability and economic robustness are measured in the same units, which is what allows a bound on one to discipline the other.
Routes through the book#
Chapters 1 through 4 are prerequisites for everything else and are best read in order. After that:
Time series econometrics. Chapters 6, 7, and 13. Hidden states, likelihood-based discrimination, and estimation without a likelihood.
Asset pricing and valuation. Chapters 5, 8, 9, 11, and 12. Chapter 8 is the hinge; Chapter 5 can be read as needed.
Decision theory under uncertainty. Chapters 7, 8, 10, and 11. Chapter 7 supplies the statistical discipline that Chapter 10 imposes on how far beliefs may be distorted.
The shortest path to the book’s central argument. Chapters 1, 4, 8, and 10.
Each chapter ends with a Summary, then Exercises, then Answers in dropdowns. The exercises verify formulas stated in the text, work through derivations the chapters compress, and in several cases connect results across chapters. They are meant to be done with pencil and paper.