Center of Mathematical Sciences and Applications,

Harvard University

Modeling Multivariate Time Series in Economics: from Auto-Regressions to Recurrent Neural Networks

The modeling of multivariate time series in an agnostic manner, without assumptions about underlying theoretical structure is traditionally conducted using Vector Auto-Regressions. They are well suited for linear and state-independent evolution. A more general methodology of Multivariate Recurrent Neural Networks allows to capture non-linear and state-dependent dynamics. This paper takes a range of small- to large-scale Long Short-Term Memory MRNNs and pits them against VARs in an application to US data on GDP growth, inflation, commodity prices, Fed Funds rate and bank reserves. Even in a small-sample regime, MRNN outperforms VAR in forecasting: its out-of-sample predictions are about 20% more accurate. MRNN also fares better in interpretability by means of impulse response functions: for instance, a shock to the Fed Funds rate variable generates system dynamics that are more plausible according to conventional economic theory. Additionally, the paper shows how, due to its inherent non-linearity, MRNN can discover (in an unsupervised manner) different macroeconomic regimes. Utilizing its state dependence, MRNN may also be a useful tool for policy simulations under practically relevant economic conditions (such as Zero Lower Bound).

Thinking on Their Feet: Along Main Street

This paper considers the problem of learning and decision-making in a dynamic stochastic economic environment by agents subject to information processing constraints. An agent endogenously chooses to operate in terms of a simplified model of the economy, which implies: a delayed, if at all, updating of the estimates of evolving states/random variables' conditioning parameters; as well as the entropy reduction, or even its complete ``folding'' that drops the less important variables from the agent's approximating model. Specifically, parameter learning is implemented relying on computational complexity theory, which produces a constrained version of the standard Kalman filter. The latter leads to a less than one-for-one reaction to the newly observed information, without the need to postulate e.g. habit formation; which is responsible for an underreaction to permanent parameter changes (``stickiness''), as well as for an overreaction to transitory shocks (``overshooting''). In a standard stochastic growth model with government transfers, such agents may fail to realize that a fiscal expansion now necessitates a compensatory fiscal contraction later, which implies the effectiveness, in certain sense, of the fiscal stimulus policy (albeit at the expense of efficiency losses) and a violation of the Ricardian equivalence. Numerical simulations suggest high fiscal multipliers, with the effects relatively stronger at times of economic recession. Being the outcomes of endogenous choices of rational agents, these results are immune to the Lucas critique.

Ignorance and Indifference: Decision-Making in the Lab and in the Market
(Supplement)

Economic agents face an evolving, non-ergodic environment. The corresponding permanently undersampled ``population'' distribution naturally permits unseen, rare events. The principle of indifference, implemented via the methods of parameterization invariance and maximum entropy, provides a disciplined, rational approach to learning, inference and decision in this underinformed setting. Canonical economic problems of choice under uncertainty from both the micro (binary lotteries) and macro (consumption-investment) domains can be formulated in terms of dynamic online learning when new gambles and new regimes are regularly encountered. The implied Bayesian updating under invariant ignorance priors allows to reverse-engineer and rationalize, in a mutually consistent way, the Allais paradox/prospect theory's probability distortions identified in laboratory experiments as well as the equity premium/risk-free rate, non-monotone pricing kernel and portfolio underdiversification puzzles observed in financial markets.

Log, Stock and Two Simple Lotteries
(Supplement, Technical Supplement)

This paper studies the problem of decision-making under risk by agents whose information processing abilities may be limited. The constructed theoretical framework grounds on findings from economic laboratory experiments, incorporates existing neuroscience knowledge, and is implemented using information-theoretic formalism. Activation of the above information-processing constraints distorts the subjective perception of the objective stochastic environment the agent operates in, and the constrained-optimal decision-making requires appropriate adjustments. In the selected application, a general equilibrium macro-finance model, such biases of subjective perspective as overconfidence, pessimism and categorization emerge endogenously. Categorization manifests itself via dropping from consideration the less important principal dimensions (``dispersion folding'') and amplifying the random variables' (dis)similarity (``correlation inflation''). These theoretical results contribute to the understanding of such empirical regularities as the portfolio underdiversification puzzle and style investing phenomenon.

Thinking on Their Feet: Along Wall Street

This paper studies decision-making under computational complexity constraints within a dynamic non-i.i.d. stochastic environment. An application to financial markets generates underreaction (``inertia''), as well as overreaction (``momentum'') and ensuing ``excess'' volatility.

Solvency Strains and the Long-End of the Yield Curve.
With Francesco Garzarelli and Aqib Aslam. 2010. Goldman Sachs, *Fixed Income Monthly*, September: 5--7.

Finding `Fair Value' for the Ukrainian Currency.
With Malachy Meechan. 2009. Goldman Sachs, Global Viewpoint, 09/10.

Larger Supply No Threat to Bonds.
With Francesco Garzarelli. 2008. Goldman Sachs, *Global Economics Weekly*, 08/38: 2--5.

Larger Supply No Threat to Bonds.
2008. Goldman Sachs, *Fixed Income Monthly*, October: 5--7.

The Family of GS Curves.
2008. Goldman Sachs, Global Viewpoint, 08/16.

Linking the Term Structures of Interest Rates and Macroeconomic Expectations --- GS Yield Curve Valuation Model.
2007. In Jim O'Neill, Jens Nordvig, and Thomas Stolper, eds., *The Foreign Exchange Market: 2007*. Goldman Sachs: 45--55.

Sudoku Gets Bigger and Forward-Looking!
With Francesco Garzarelli and Michael Vaknin. 2007. Goldman Sachs, Global Viewpoint, 07/24.

Global Bonds: No Obvious Safe Haven.
With Francesco Garzarelli. 2007. Goldman Sachs, Global Viewpoint, 07/07.

Partisan Differences in Economic Outcomes and Corresponding Voting Behavior: Evidence from the U.S.
2004. *Public Choice*, 120(1-2): 169--189.