Linearity-Generating Processes: A Modelling Tool Yielding Closed Forms for Asset Prices

This methodological paper presents a class of stochastic processes with appealing properties for theoretical or empirical work in finance and macroeconomics, the "linearity-generating" class. Its key property is that it yields simple exact closed-form expressions for stocks and bonds, with an arbitrary number of factors. It operates in discrete and continuous time. It has a number of economic modeling applications. These include macroeconomic situations with changing trend growth rates, or stochastic probability of disaster, asset pricing with stochastic risk premia or stochastic dividend growth rates, and yield curve analysis that allows flexibility and transparency. Many research questions may be addressed more simply and in closed form by using the linearity-generating class.

The 6D Bias and the Equity Premium Puzzle

If decision costs lead agents to update consumption every D periods, then econometricians will find an anomalously low correlation between equity returns and consumption growth (Lynch 1996). We analytically characterize the dynamic properties of an economy composed of consumers who have such delayed updating. In our setting, an econometrician using an Euler equation procedure would infer a coefficient of relative risk aversion biased up by a factor of 6D. Hence with quarterly data, if agents adjust their consumption every D = 4 quarters, the imputed coefficient of relative risk aversion will be 24 times greater than the true value. High levels of risk aversion implied by the equity premium and violations of the Hansen-Jagannathan bounds cease to be puzzles. The neoclassical model with delayed adjustment explains the consumption behavior of shareholders. Once limited participation is taken into account, the model matches most properties of aggregate consumption and equity returns, including new evidence that the covariance between ln(Ct+h/Ct) and Rt+1 slowly rises with h.

The great diversification and its undoing

We investigate the hypothesis that macroeconomic fluctuations are primitively the results of many microeconomic shocks, and show that it has significant explanatory power for the evolution of macroeconomic volatility. We define “fundamental” volatility as the volatility that would arise from an economy made entirely of idiosyncratic microeconomic shocks, occurring primitively at the level of sectors or firms. In its empirical construction, motivated by a simple model, the sales share of different sectors vary over time (in a way we directly measure), while the volatility of those sectors remains constant. We find that fundamental volatility accounts for the swings in macroeconomic volatility in the US and the other major world economies in the past half century. It accounts for the “great moderation” and its undoing. Controlling for our measure of fundamental volatility, there is no break in output volatility. The initial great moderation is due to a decreasing share of manufacturing between 1975 and 1985. The recent rise of macroeconomic volatility is due to the increase of the size of the financial sector. We provide a model to think quantitatively about the large comovement generated by idiosyncratic shocks. As the origin of aggregate shocks can be traced to identifiable microeconomic shocks, we may better understand the origins of aggregate fluctuations.

Log(Rank-1/2): A Simple Way to Improve the OLS Estimation of Tail Exponents

A popular way to estimate a Pareto exponent is to run an OLS regression: log (Rank) = c - blog (Size), and take b as an estimate of the Pareto exponent. Unfortunately, this procedure is strongly biased in small samples. We provide a simple practical remedy for this bias, and argue that, if one wants to use an OLS regression, one should use the Rank -1/2, and run log (Rank- 1/2) = c-b log (Size). The shift of 1/2 is optimal, and cancels the bias to a leading order. The standard error on the Pareto exponent is not the OLS standard error, but is asymptotically (2/n)^{1/2}b. To obtain this result, we provide asymptotic expansions for the OLS estimate in such log-log rank-size regression with arbitrary shifts in the ranks. The arguments for the asymptotic expansions rely on strong approximations to martingales with the optimal rate and demonstrate that martingale convergence methods provide a natural and conceptually simple framework for deriving the asymptotics of the tail index estimates using the log-log rank-size regressions.

Shrouded Attributes, Consumer Myopia, and Information Suppression in Competitive Markets

Bayesian consumers infer that hidden add-on prices (e.g. the cost of ink for a printer) are likely to be high prices. If consumers are Bayesian, firms will not shroud information in equilibrium. However, shrouding may occur in an economy with some myopic (or unaware) consumers. Such shrouding creates an inefficiency, which firms may have an incentive to eliminate by educating their competitors' customers. However, if add-ons have close substitutes, a "curse of debiasing" arises, and firms will not be able to profitably debias consumers by unshrouding add-ons. In equilibrium, two kinds of exploitation coexist. Optimizing firms exploit myopic consumers through marketing schemes that shroud high-priced add-ons. In turn, sophisticated consumers exploit these marketing schemes. It is not possible to profitably drive away the business of sophisticates. It is also not possible to profitably lure either myopes or sophisticates to non-exploitative firms. We show that informational shrouding flourishes even in highly competitive markets, even in markets with costless advertising, and even when the shrouding generates allocational inefficiencies.

Why Has CEO Pay Increased So Much?

This paper develops a simple equilibrium model of CEO pay. CEOs have different talents and are matched to firms in a competitive assignment model. In market equilibrium, a CEO%u2019s pay changes one for one with aggregate firm size, while changing much less with the size of his own firm. The model determines the level of CEO pay across firms and over time, offering a benchmark for calibratable corporate finance. The sixfold increase of CEO pay between 1980 and 2003 can be fully attributed to the six-fold increase in market capitalization of large US companies during that period. We find a very small dispersion in CEO talent, which nonetheless justifies large pay differences. The data broadly support the model. The size of large firms explains many of the patterns in CEO pay, across firms, over time, and between countries.

Rank-1/2: A Simple Way to Improve the OLS Estimation of Tail Exponents

Despite the availability of more sophisticated methods, a popular way to estimate a Pareto exponent is still to run an OLS regression: log(Rank)=a-b log(Size), and take b as an estimate of the Pareto exponent. The reason for this popularity is arguably the simplicity and robustness of this method. Unfortunately, this procedure is strongly biased in small samples. We provide a simple practical remedy for this bias, and propose that, if one wants to use an OLS regression, one should use the Rank-1/2, and run log(Rank-1/2)=a-b log(Size). The shift of 1/2 is optimal, and reduces the bias to a leading order. The standard error on the Pareto exponent zeta is not the OLS standard error, but is asymptotically (2/n)^(1/2) zeta. Numerical results demonstrate the advantage of the proposed approach over the standard OLS estimation procedures and indicate that it performs well under dependent heavy-tailed processes exhibiting deviations from power laws. The estimation procedures considered are illustrated using an empirical application to Zipf's law for the U.S. city size distribution.

Why Has CEO Pay Increased So Much?

This paper develops a simple competitive model of CEO pay. It appears to explain much of the rise in CEO compensation in the US economy, without assuming managerial entrenchment, mishandling of options, or theft. CEOs have observable managerial talent and are matched to assets in a competitive assignment model. The marginal impact of a CEO's talent is assumed to increase with the value of the assets under his control. Under very general assumptions, using results from extreme value theory, the model determines the level of CEO pay across firms and over time, and the pay-sensitivity relations. We predict that the level of CEO compensation should increase one for one with the average market capitalization of large firms in the economy. Therefore, the eight-fold increase of CEO pay between 1980 and 2000 can be fully attributed to the increase in market capitalization of large US companies. The model predicts the cross-section Cobb-Douglass relation between pay and firm size and can be used to study other large changes at the top of the income distribution, and offers a benchmark for calibratable corporate financeExecutive compensation, wage distribution, Pareto distribution, wage inequality, assignment, incentives, pay performance sensitivity