Measuring Marginal q

Abstract

Using asset prices I estimate the marginal value of capital in a dynamic stochastic economy under general assumptions about technology and preferences. The state-space measure of marginal q relies on the joint measurability of the value function, i.e. firm market value, and its underlying firm state variables. Unlike existing methodologies, the state-space marginal q requires only general restrictions on the stochastic discount factor and the firm investment technology, and it uses simple linear estimation methods. Consistently with a large class of neoclassical investment models, I construct the state-space marginal q using the firm capital stock and profitability shocks. I show that this new measure of real investment opportunities is substantially different from the conventional Tobin’s Q, it yields more plausible and robust estimates of capital adjustment costs, it increases the correlation with investment and the sensitivity of investment to fundamentals. ∗Visiting Associate Professor of Finance. The Wharton School, University of Pennsylvania. vgala@wharton.upenn.edu. I thank Andrew Abel, Hengjie Ai, Frederico Belo, Patrick Bolton, Maria Cecilia Bustamante (discussant), John Cochrane, Janice Eberly (discussant), Bob Goldstein, Francisco Gomes, Joao Gomes, Urban Jermann, Ralph Koijen, Boris Nikolov (discussant), Stavros Panageas, Nick Souleles, Martin Szydlowski, Neng Wang, Toni Whited, Amir Yaron, and seminar participants at The Wharton School, Columbia Business School, University of Minnesota, London Business School, Federal Reserve Board of Governors, EPFL Lausanne, HEC Lausanne, CAPR Workshop 2013, CEPR ESSFM 2013, UBC Summer Finance Conference 2013, EFA Meetings 2014, and AFA Meetings 2015 for valuable comments and suggestions. Tobin’s Q-theory of investment emphasizes a fundamental connection between financial markets and the real economy: marginal q i.e. marginal value of capital is a suffi cient statistic to describe investment behavior (Hayashi, 1982). As any other shadow value in economics, however, the marginal value of capital is not directly observable. To overcome such empirical limitation, researchers have thus mainly used the “observable” (average) Tobin’s Q i.e. ratio of market value of capital to its replacement cost in empirical studies. Such common practice, however, relies on a set of very restrictive underlying assumptions under which marginal q is equal or proportional to (average) Tobin’s Q.1 Despite the long-standing consensus that the restrictive underlying assumptions of perfect competition and homogeneity are misspecified, particularly at the firm level, the use of (average) Tobin’s Q remains still predominant in the empirical literature, primarily for lack of model-free and easy-to-compute alternatives.2 In this paper, I provide a new measure of marginal q to fill this gap. I propose a new methodology to measure marginal q under general assumptions regarding the nature of technology, markets, and preferences. This general procedure is both theoretically justified, and useful, empirically. Under general regularity conditions for the differentiability of the value function and the measurement of its underlying firm state variables, I show how marginal q can be easily estimated as market price elasticity of capital using a two-stage procedure, which I refer to as state-space approach. First, I project the observable market values i.e. value function onto the measurable firmlevel state-space, which includes also the firm capital stock, and then I differentiate the projected market values with respect to the firm capital stock to obtain the marginal value of capital. The key insight underlying the state-space measure of marginal q rests on the joint measurability of the value function i.e. market values and its underlying set of firm state variables. Unlike the demand side of the economy, where one get 1Hayashi (1982), and Abel and Eberly (1994) in a more general stochastic economy, showed that under the joint assumption of perfect competition and homogeneity of a firm production and adjstment cost technologies in investment and capital, marginal q is equal or proportional to (average) Tobin’s Q. 2Possible departures from perfect competition and homogeneity include market power or decreasing returns to scale in production (Gomes, 2001; Cooper and Ejarque, 2003; Abel and Eberly, 2011), and inhomogeneous costs of investment (Abel and Eberly, 1994, 1997; Cooper and Haltiwanger, 2006).