Completing the imperfect-information model of the forward premium puzzle: risk aversion, stochastic output, generalized signal extraction, regime switches, and empirical validation of the Kalman-implied premium

Albuquerque (2008, Economics Letters) showed that in a two-country monetary model where agents imperfectly observe money supplies and filter them with a Kalman filter, the forward premium regression acquires a time- and country-specific deterministic term $\varphi_t$ and conditionally heteroskedastic forecast errors, rationalizing the panel evidence of Mayfield and Murphy (1992) — but only under risk neutrality ( $\gamma=\delta=0$ ) and deterministic output ( $\mu_{Y_i}=1$ ). The open problem is the completion of this account: characterize the implications for $\varphi_t$ and the conditional variance of forecast errors when these simplifications are relaxed, when the signal-extraction problem is generalized (dividends, government spending, jointly imperfectly observed interacting monetary policies), and under non-anticipated finite-date regime switches, and test empirically whether estimated country fixed effects equal the country-specific Kalman-implied $\varphi_t$ time-averaged over the sample.

Background

Uncovered interest parity (UIP) is the textbook arbitrage condition stating that the expected depreciation of a currency equals its interest-rate disadvantage vis-à-vis another currency; equivalently, the forward exchange rate (the rate agreed today for delivery at a future date, fixed by covered interest parity) should be an unbiased predictor of the future spot rate. The empirical test, usually called the Fama or forward premium regression, regresses realized depreciation on the forward premium (the forward rate minus the current spot rate, in logs). UIP predicts a slope of one; since Fama (1984) the robust finding is a slope below one and frequently negative — currencies with high interest rates tend to appreciate rather than depreciate. This is the forward premium puzzle, one of the longest-standing anomalies in international finance. Mayfield and Murphy (1992) documented its panel dimension: estimating the regression with country-specific intercepts (fixed effects) across countries, they found systematic cross-country differences in intercepts and in the variability of residuals. Albuquerque (2008) proposed an informational account. The model is a two-country monetary model in the tradition of Lucas (1982) — two countries, each with its own money and output, whose exchange rate is pinned down by relative money demands — with one departure: agents do not perfectly observe the current money supplies (the monetary policy shocks); instead they observe noisy signals and estimate the hidden state with a Kalman filter, the standard recursive linear minimum-mean-square-error estimator of an unobserved state from noisy observations (Anderson and Moore, 1979). Because the filter's error covariance matrix starts away from its steady state $P$ and converges to it over time, the model-implied deterministic term $\varphi_t$ in the forward premium regression is a function of time and of the country pair rather than a constant: its variation over time and across country pairs generates the fixed effects, and the filter's changing precision generates conditional heteroskedasticity (time-varying variance) in forecast errors, matching the patterns Mayfield and Murphy found. All of this is derived only under two simplifications chosen to isolate the informational mechanism: utility is risk-neutral in both countries (the risk-aversion parameters are set to $\gamma=\delta=0$ , so risk premia play no role) and output growth is deterministic ( $\mu_{Y_i}=1$ , so endowments are non-stochastic). The author states that relaxing these simplifications, formalizing richer signal-extraction environments, characterizing the behavior of $\varphi_t$ around regime switches, and validating the mechanism empirically are left open.

Problem Statement

Complete the imperfect-information account of the forward premium puzzle given by Albuquerque (2008), by answering, within a common objective of establishing whether the mechanism survives outside the stripped-down model: (1) Theoretical robustness: in the same two-country monetary model with imperfectly observed money supplies, re-derive the equilibrium and characterize the forward premium regression deterministic term $\varphi_t$ and the conditional variance of forecast errors when risk aversion is nonzero ( $\gamma,\delta\neq 0$ ) and output growth is stochastic ( $\mu_{Y_i}$ random rather than identically equal to 1), stating which qualitative implications (deterministic time and cross-country variation of $\varphi_t$ , fixed effects, conditional heteroskedasticity) survive and how quantitative implications change. (2) Generalized signal extraction: formalize richer signal-extraction environments in which the hidden fundamentals are multidimensional, and determine how the dimensionality and cross-country correlation structure of the latent processes shape the deterministic term in $\varphi_t$ and the residual heteroskedasticity. (3) Regime switches: characterize analytically or quantitatively the path of $\varphi_t$ after non-anticipated but finite-date regime switches — the transient of the filter covariance away from and back toward its steady state $P$ — including the speed of convergence and the conditions under which the induced deterministic variation is empirically detectable. (4) Empirical validation: construct panel estimates of the forward premium regression across country pairs, specifying the identification scheme used to isolate monetary innovations and the sample and data construction, and stating the calibration or estimation of the model's money-process and signal-noise parameters; test the model's distinguishing prediction that the estimated country fixed effects equal the country-specific Kalman-implied $\varphi_t$ (computed from the filter's one-step-ahead forecast-error covariance under those parameters) time-averaged over the sample. The question comprises four linked sub-questions (1)-(4). The parts are independently refutable, so a demonstration that a specific claim inside one part fails refutes that part while leaving the others open.

References

Albuquerque, Rui (2008). The forward premium puzzle in a model of imperfect information. Economics Letters 99(3), 461-464. DOI: 10.1016/j.econlet.2007.09.016. https://doi.org/10.1016/j.econlet.2007.09.016
Mayfield, E. Scott and Robert G. Murphy (1992). Interest rate parity and the exchange risk premium: Evidence from panel data. Economics Letters 40(3), 319-324. DOI: 10.1016/0165-1765(92)90012-N. https://doi.org/10.1016/0165-1765(92)90012-N
Fama, Eugene F. (1984). Forward and spot exchange rates. Journal of Monetary Economics 14(3), 319-338. DOI: 10.1016/0304-3932(84)90046-1. https://doi.org/10.1016/0304-3932(84)90046-1
Lucas, Robert E. (1982). Interest rates and currency prices in a two-country world. Journal of Monetary Economics 10(3), 335-359. DOI: 10.1016/0304-3932(82)90032-0. https://doi.org/10.1016/0304-3932(82)90032-0
Anderson, Brian D. O. and John B. Moore (1979). Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall.