in the Presence of Measurement Error

Transcription

1 The Effects of and Cash Flow on Investment in the Presence of Measurement Error Andrew B. Abel Wharton School of the University of Pennsylvania National Bureau of Economic Research January 25, 2017 Abstract Ianalyzeinvestment,, andcashflow in a tractable stochastic model in which marginal and average are identically equal. I introduce classical measurement error and derive closed-form expressions for the coefficients in regressions of investment on and cash flow. The cash-flow coefficient is Previous versions of this paper circulated under the title "The Analytics of Investment,, and Cash Flow". I thank Anna Cororaton, Mehran Ebrahimian, Joao Gomes, Christian Goulding, Richard Kihlstrom, Stavros Panageas, Michael Roberts, Colin Ward, and Toni Whited for helpful discussion, and I thank participants in seminars at University of California at Santa Barbara, University of Southern California, Columbia University, and the Penn Macro Lunch Group for helpful comments. Finally, I thank the editor, Toni Whited (in her editorial role in addition to discussion above) and two anonymous referees for comments that helped guide the revision. Corresponding author: Andrew B. Abel, Department of Finance, The Wharton School of the University of Pennsylvania, 3620 Locust Walk, Philadelphia, PA Phone: ; Fax: ; abel@wharton.upenn.edu. 1

2 positive and larger for faster growing firms, yet there are no financial frictions in the model. I develop the concepts of bivariate attenuation and weight shifting to interpret the estimated coefficients on and cash flowinthepresenceof measurement error. JEL codes: Keywords: E22, G3 investment, cash flow,, measurement error

3 Empirically estimated investment equations typically find that Tobin s has a positive effect on capital investment by firms, and that even after taking account of the effect of Tobin s on investment, cash flow has a positive effect on investment. Intepretations of these results, of course, rely on some theoretical model of investment. Typically, the theoretical model that underlies the relationship between Tobin s and investment is based on convex capital adjustment costs. 1 In such a framework, marginal is a sufficient statistic for investment, and, as a consequence, other variables, in particular, cash flow, should not have any additional explanatory power for investment, once account is taken of marginal. The fact that cash flow has a positive impact on investment, even after taking account of, isoften interpreted as evidence of financing constraints facing firms. That interpretation is bolstered by the finding that the cash-flow coefficient is larger for firms, such as rapidly growing firms, that are likely to be financially constrained. In this paper, I develop and analyze a tractable stochastic model of investment,, and cash flow and use it to interpret the empirical results described above. As in Lucas (1967), I specify the net profit ofthefirm, after deducting all costs associated with investment, to be a linearly homogeneous function of capital and investment. Lucas showed that this linear homogeneity implies that the growth rate of the firm is independent of its size. More relevant to the analysis in this paper, Hayashi (1982) showed that this linear homogeneity implies that Tobin s, often called average, is identically equal to marginal. This equality of marginal and average is particularly powerful, because average, which is in principle observable, can be 1 Lucas and Prescott (1971) and Mussa (1977) first analyzed the link between investment and securities prices, which are related to Tobin s, inanadjustmentcostframework. 1

4 used to measure marginal, which is the appropriate shadow value of capital that determines the optimal rate of investment. In addition, this linearly homogeneous framework relates the investment-capital ratio to and most empirical analyses, in fact, use the investment-capital ratio as the dependent variable in regressions. To analyze the response of investment to requires a framework with variation in andinoptimalinvestment. Inthispaper,Idevelopamodelinwhichan exogenous Markov regime-switching process generates stochastic variation in the marginal operating profit of capital, which leads to stochastic variation in andinthe optimal investment-capital ratio. With this stochastic specification and the linear homogeneity of the profit function described above, the model is tractable enough to permit straightforward analysis of the effects on and investment of changes in the marginal operating profit for a particular firm. As mentioned earlier, a common feature of adjustment cost models of investment is that marginal is a sufficient statistic for investment. Since average and marginal are identically equal in the linearly homogeneous framework used here, average is alsoasufficient statistic for investment. In particular, cash flow should not add any explanatory power for investment after taking account of average. This feature holds in the model I present here and might appear to be an obstacle to accounting for the empirical cash-flow effect on investment described above. To overcome that obstacle, I introduce classical measurement error in Section 4 and derive a closedform expression for the plim of the cash-flow coefficient in an investment regression. That expression indicates that if is measured with error, then, as found empirically by Devereux and Schiantarelli (1990) and Bushman, Smith, and Zhang (2011), the 2

5 cash-flow coefficient can be larger for firms that grow more rapidly. However, because the model has perfect capital markets, without any financial frictions, the finding of positive cash-flow coefficients that are larger for faster-growing firmscannotbetaken as evidence of financing constraints. The interpretation of cash-flow coefficients and the role of measurement error in this paper build on and tie together two strands of the literature. One strand, represented by Cooper and Ejarque (2003), Gomes (2001), Alti (2003), and Abel and Eberly (2011), develops formal theoretical models of capital investment decisions and uses the formal model as an environment in which to analyze the cash-flow coefficient. The second strand, represented by Erickson and Whited (2000) and Gilchrist and Himmelberg (1995), focuses on the role of measurement error in inducing a spuriously positive cash-flow coefficient. All of the aforementioned papers in the first strand provideexamplesinwhichthecash-flow coefficient can be positive even when capital markets are frictionless. Thus, the finding of a positive cash-flow coefficient cannot be viewed as evidence of financial constraints. In all of these studies, however, average is not identically equal to marginal, so investment regressions using average in place of marginal suffer from misspecification. In the current paper I avoid this specification error by adhering to a generalized Hayashi condition so that average is identically equal to marginal. Therefore, average is an appropriate regressor in a linear investment equation when the adjustment cost function is quadratic. In this case, a nonzero cash-flow coefficient will arise if average is measured with error, as emphasized in the second strand of the literature. Relative to that strand, the contributions of the current paper are (1) to use a formal model of investment to 3

6 show analytically that cash-flow coefficients arising from measurement error will be larger for firms that grow more rapidly; (2) to provide analytic expressions for the plims of the coefficients on and cash flow in investment regressions in the presence of measurement error; and (3) to interpret these estimated coefficients in terms of a "bivariate attenuation" factor and a "weight shifting" factor between the coefficients on andoncashflow. Because the analysis of the model relies on the equality of marginal and average, I begin, in Section 1, by re-stating, and extending to a stochastic framework, the Hayashi condition under which average and marginal are equal. Section 2 introduces the model of the firm and analyzes the valuation of a unit of capital and the optimal investment decision in the case in which the marginal operating profit of capital is known to be constant forever. More than simply serving as a warm up to the stochastic model, Section 2 introduces a function that facilitates the analysis of the stochastic model that follows in later sections. I introduce a Markov regime-switching process for the marginal operating profit of capital in Section 3 to generate stochastic variation in and optimal investment. The optimal investmentcapital ratio is an increasing function of, whichisasufficient statistic; cash flow has no additional explanatory power for the investment-capital ratio. In order to account for the positive impact of cash flow on investment, even after taking account of, I introduce classical measurement error in Section 4. I derive the plims of the coefficients on and cash flow in investment regressions and interpret these coefficients in terms of bivariate attenuation bias and weight shifting between coefficients. Section 5 leaves the confines of the formal model and posits that the 4

7 investment-capital ratio is a linear function of cash flow as well as of. The rationale for this analysis is to demonstrate that the interpretations of bivariate attentuation bias and weight shifting do not rely on a zero cash-flow coefficient in the true model. Concluding remarks are in Section 6. The proofs of lemmas, propositions, and corollaries are in the Appendix. 1 The Hayashi Condition Before describing the specific framework that I analyze in this paper, it is useful to begin with a simple, yet more general, description of the conditions under which average and marginal are equal. Consider a competitive firm with capital stock at time, where time is continuous. The firm accumulates capital by undertaking gross investment at time, and capital depreciates at rate, so the capital stock evolves according to = (1) The firm uses capital, 0, andlabor, 0, to produce and sell output at time. I assume that the price of capital goods is constant and normalize it to be one. Define ( )=max ( ),where ( ) is revenue net of wage payments to labor and net of any capital adjustment costs. For now, I will simply assume that ( ) is concave in and. Letting ( ) be the stochastic discount factor used to discount cash flows at time back to time, the value of the firm at time is ( )= max { } ½Z 5 ¾ ( ) ( ) (2)

8 subject to equation (1). The following proposition presents conditions for the equality of average and marginal, which are essentially the same as in Hayashi (1982), though the method of proof is different from Hayashi s proof and the framework is generalized to include uncertainty and possible non-separability of costs of adjustment from other components of the revenue function. Proposition 1 (extension of Hayashi) If ( ) is linearly homogeneous in and, then for any 0, ( )= ( ), i.e., the value function is linearly homogeneous in,sothataverage, (),andmarginal, 0 ( ),areidentically equal. For the remainder of this paper, I will assume that ( ) is linearly homogeneous in and so that average and marginal are equal. 2 Model of the Firm Consider a competitive firm that faces convex costs of adjustment that are separable from the production function. The firm uses capital, 0, andlabor, 0, to produce non-storable output,,attime according to the production function = ( ),where ( ) is linearly homogeneous in and,and is the exogenous level of total factor productivity. If the amount of labor is costlessly and instantaneously adjustable, the firm chooses at time to maximize instantaneous revenue less wages ( ),where is the price of the firm s output at time and is the wage rate per unit of labor at time. The linear homogeneity of ( ) together with the assumption that the firm is a price-taker in the markets for its output and labor imply that the maximized value of revenue less wages is 6

9 ,where max [ (1) ]. The marginal (and average) operating profit of capital,, is a deterministic function of,,and,allofwhichare exogenous to the firm and possibly stochastic. Therefore, is exogenous to the firm and, henceforth, I will treat as the fundamental exogenous variable facing the firm, comprising the effects of productivity, output price, and the wage rate. Because of the linear homogeneity of the net revenue function, the marginal operating profit of capital equals the average operating profit of capital, which equals cash flow (before subtracting investment costs) per unit of capital hence, the rationale for the notation. Define to be the investment-capital ratio at time. Therefore, equation (1) implies that the growth rate of the capital stock,,is 1 =, (3) so that for µz = exp (4) Finally, I will specify the stochastic discount factor, ( ),tobesimplyexp ( ( )), so that expected future net cash flows are discounted at the constant rate. At time, thefirm chooses gross investment,. The cost of this investment has two components. The first component is the cost of purchasing capital at a price per unit that I assume to be constant over time and normalize to be one. Thus, this component of the cost of gross investment at rate is simply =,which,of course, would be negative if the firm sells capital so that 0. The second component is the cost of adjustment, ( ), which is linearly 7

10 homogeneous in and. I assume that ( ) 0 is at least twice differentiable and that 00 () for some positive constant so that () is strictly convex. I assume that () attains its minimum value, which is zero, at 0,sothat 0 ( 0 )= ( 0 )=0. The most common specifications of 0 are 0 =0and 0 =. 2 Here I will simply assume that 0 +, which implies 0 ( + ) 0. In addition, I assume that for some, 0 ( )= 1. The strict convexity of ( ) implies that 0. The total cost of investment, which comprises the purchase cost of capital and the cost of adjustment, is [ + ( )]. For a given, the total cost of investment is strictly convex in and attains its minimum value at. After choosing the optimal usage of labor, the amount of revenue less wages and less the total cost of investment is ( ) [ ( )]. (5) 2.1 Constant Consider the case in which the marginal operating profit of capital,,isknowntobe constant forever. The analytic apparatus developed in the case of certainty will prove to be useful in later sections when evolves according to a Markov regime-switching process. Ibeginbydefining, which is an admissible set of values for, the constant 2 When 0 =0, the adjustment cost function is viewed as a function of gross investment relative to the capital stock; when 0 =, the adjustment cost function is viewed as a function of net investment relative to the capital stock. Cooper and Ejarque (2003) and Zhang (2005) choose 0 = 0; Alti (2003) and³ Treadway (1969) choose 0 = (though Treadway s formulation of adjustment costs is simply which is minimized at =0, i.e., when = ); Hall (2004) considers both 0 =0and 0 = ; and Chirinko (1993) and Summers (1983) consider arbitrary 0. 8

11 marginal operating profit of capital, as 3 { : ( )+ ( + )+ + } (6) The lower bound on ensures that there is a value of such that ( ) 0 when 0, sothatthevalueofthefirm is positive. 4 The upper bound on keeps the value of the firm finite when it is positive The Value of a Unit of Capital With a constant marginal operating profit ofcapital,, constant depreciation rate,, and constant discount rate,, the optimal investment-capital ratio,,isconstant also. In this case, the value of the firm in equation (2) can be written as ( )=max Z [ ()] ( ) (7) 3 Since 0 ( )= 1 and 0 ( + ) 0, the strict convexity of ( ) implies that + and that ( )+ is strictly increasing in for all. Therefore, ( + )++ ( )+ so that is non-empty. 4 Since minimizes + (), it maximizes (). The restriction ( )+ implies that () 0 for some, whereasif were less than or equal to ( )+,then () would not be positive for any. 5 If (1) () 0 and (2), the value of the firm, with 0, wouldbe positive and infinite because the growth rate of [ () ],whichis, would exceed. The upper bound on implies that if +, then () + ( + ) + +,so () 0. 9

12 Dividing both sides of equation (7) by and using equation (4) with = yields an expression for the average value of a unit of capital, (),whichis 6 =max () + (8) The value of a unit of capital shown in equation (8) equals ( ) divided by the excess of the interest rate,, over the growth rate,. Differentiatethemaximandontheright-handsideofequation(8)withrespect to and set the derivative equal to zero to obtain 7 Rewriting equation (9) yields 1+ 0 () = () + (9) ( + ) () ( + ) 0 () =0 (10) To characterize the optimal investment-capital ratio, it will be useful to define a function () as () () ( ) 0 () (11) where is an arbitrary constant greater than or equal to +. The optimal value 6 The expression in equation (8) holds only if the integral in equation (7) is finite, which requires the growth rate of capital,, tobelessthan. The optimal value of is smaller than +, so that. 7 Define () () + and observe that 0 1 () = ()+() and that 00 1 () = + 0 () ()+ 0 (). Therefore, if 0 (b) = 0, then 00 (b) = (b). If 0 (b) =0and b +, then 00 (b) 0 so (b) is a local maximum. However, if 0 (b) =0and b +, then 00 (b) 0 so (b) is a local minimum. 10

13 of, characterized by equation (10), satisfies ( + ) =0 (12) wherethevalueof in () is set equal to +. The following lemma presents several useful properties of (). Lemma 1 Define () () ( ) 0 () and assume that and +. Then: 1. () is an increasing, linear function of. 2. () is a decreasing, linear function of for,where 0 ( )= () is strictly quasi-convex in. 4. ( 0 )=, where 0 arg min (). 5. ( ) 0, where 0 ( )= min () = ( ) = () There is a unique ( ) and a unique ( ) such that ( )= ( )=0. Also, ( ) 0 ( ). Figure 1 illustrates () as a function of for given values of and +. 8 This figure shows that ( ) 0 ( ), and illustrates the two roots,,of () =0. Since ( ) 0 (Statement 7 of Lemma 1), the optimal investment-capital ratio cannot equal. 9 8 If the adjustment cost function () is quadratic, then () is quadratic in and is a convex function of. In general, however, () need not be convex in, but it is strictly quasi-convex in. 9 Statement 7 of Lemma 1 implies that if =, then ( ) = ( ) = [ ( )] 0 and hence the value of the firm would be negative. Therefore, consistent with footnote 6, the optimal value of cannot equal +. 11

14 Figure 1: () When the marginal operating profit,, is known to be constant forever, the first-order condition for the optimal investment-capital ratio is the smaller root of ( + ) =0,where () () ( ) 0 (), () is a strictly convex adjustment cost function, and is an arbitrary constant greater than or equal to the sum of the interest rate and the depreciation rate. The roots of () =0are and. The value of at which 0 () = 1 is. 12

15 The following proposition characterizes the optimal investment-capital ratio, where the superscript "" indicates the optimal value of under certainty with a constant value of. Proposition 2 If is known with certainty to be constant and equal to forever, then the optimal investment-capital ratio, ( + ), istheuniquevalueof ( + ) that satisfies ( + ) =0. Define ( + ) as the common value of marginal and average if is known with certainty to be constant and equal to. Corollary 1 (+) = and (+) = 1 (+ ) 00 ( ) 0. Corollary 1 states that a firm with a higher deterministic constant value of marginal operating profit of capital,, will have a higher common value of marginal and average and a higher optimal value of the investment-capital ratio.. 3 Markov Regime-Switching Process for In this section I develop and analyze a model of a firm facing stochastic variation in the marginal operating profit of capital,, governed by a Markov regime-switching process. Specifically, a regime is defined by a constant value of. If the marginal operating profit of capital at time,,equals, it remains equal to until a new regime arrives. The arrival process for new regimes is a Poisson process with probability of a new arrival during a short interval of time. When a new regime arrives, a new value of the marginal operating profit of capital,, isdrawn from a distribution with c.d.f (), where the support of () is in, defined in 13

16 equation (6). () can be continuous or not continuous, so the random variable can be continuous, discrete, or mixed. The values of are drawn independently across regimes. The Markovian nature of implies that the value of the firm at time depends only on the capital stock at time,, and the value of the marginal operating profit at time,. The value of the firm ( ) is Z + ( ) = max [ ( )] ( ) (13) + ( + ) + 1 Z ( + ) () which is the maximized sum of three terms. The first term is the present value of ( )=[ ( )] over the infinitesimal interval of time from to +, which, ignoring the infinitesimal probability that and hence change during that interval, is [ ( )]. The second term is the present value of the firm at time +, conditional on remaining equal to at time +, weightedbythe probability,,that + =. The third term is the present value of the expected value of the firm at time + conditional on a new regime for at time +, weighted by the probability that a new regime will arrive by time +. The Hayashi conditions in Proposition 1 hold in this framework so that the value of the firm is proportional to the capital stock. Therefore, the average value of the capital stock, ( ), is independent of the capital stock and depends only on. I define () ( ) to be Tobin s, or equivalently, the average value of capital. Since average and marginal are identically equal in this framework, () is also 14

17 marginal. Use the definition () (), ignore the infinitesimal probability that = changes during the interval [ + ] so that = ( )( ) for +, andperformthefirstintegrationontheright-handsideofequation(13)to obtain () = max[ ( )] 1 (+ ) + (14) + ( ) () + 1 Z ( ) () () In the limit as approaches zero, equation (14) becomes 10 where 0=max () ( + + ) ()+ (15) Z () () (16) is the unconditional expected value of a unit of capital, which is also the unconditional expected value of both average and marginal. The maximization in equation (15) has the first-order condition 1+ 0 () = () (17) 10 Use the definition of in equation (16) and the fact that for small, = 1+ to rewrite equation (14) as () = max [ ( )] +(1 ( + + ) ) () + ()(1 ( + ) ). Next subtract () from both sides of the equation, and then divide both sides of the resulting equation by to obtain 0 = max [ ( )] ( + + ) ()+(1 ( + ) ). Taking the limit as approaches zero, and replacing by simply yields equation (15). 15

18 Thus, the optimal value of equates the marginal cost of investment, comprising the purchase price of capital and the marginal adjustment cost, with marginal and average. 3.1 Marginal and Average In this subsection I present alternative expressions for marginal and average. Because the model presented here is a special case of Proposition 1, marginal and average are identically equal. Nevertheless, it is helpful to examine different expressions for marginal and average and to understand why these expressions, which at first glance look different, are equivalent. Marginal at time is commonly expressed as the expected present value of the stream of contributions to revenue, less wages and adjustment costs, of the remaining undepreciated portion of a unit of capital installed at time, whichis ½Z ¾ ( ) () = (+)( ) = (18) Average at time is the value of the firm at time divided by. Dividing both sides of equation (2) by, using the linear homogeneity of ( ),and using equation (4) and ( ) =exp( ( )) yields ½Z () = µ (1 )exp Proposition 3 The value of marginal is Z () = ()+0 () + + ¾ ( ) = (19)

19 where is the optimal value of when =, and R () () is the unconditional expected value of marginal q. The value of average is () = () where R () () is the unconditional expected value of average. Proposition 1 implies that () (). To see that the expressions for () and () in Proposition 3 are equivalent, multiply () by + + and subtract the result from () multiplied by + + to obtain ( + + )[ () ()] + () = [1 + 0 ()] + ( ) (20) Since Proposition 1 implies that () () and hence =, equation (20) implies () =1+ 0 () if 6= 0,whichisthefirst-order condition in equation (17) Optimal Investment In this section I exploit the first-order condition for optimal investment in equation (17) to analyze several properties of optimal. The optimal value of depends on (), which depends on. For now, I will treat as a parameter and defer further analysis of to subsection 3.3. To analyze optimal, substitute the first-order condition for optimal from equation (17) into equation (15) to obtain 0= () ( + + ) ( + + ) 0 ()+ (21) 11 If =0, the expressions for () and () in Proposition 3 become () = (0)+ ++ () = (0)+ ++, respectively, so = and () and () are consistent with each other. and 17

20 Using the definition of () in equation (11), rewrite equation (21) as ( + + ) = (22) Equation (22) characterizes the optimal value of when there is a constant instantaneous probability,, of a regime switch. Of course, when =0,thisequation is equivalent to equation (12), which characterizes the optimal value of under certainty. The optimal value of when =0is shown in Figure 2 as point A where ( + ) =0. The introduction of a positive value of, which introduces stochastic variation in the future values of (1 ) and (1 ), has two opposing effects on optimal in equation (22). First, the introduction of a positive value of increases, from + to + +, which reduces the value of () by (1 + 0 ()) at each value of, which induces the downward shift of the curve shown in Figure 2. This downward shift of the curve reduces the value of for which () =0, as illustrated by the movement from point A to point B. The value of for which ( + + ) =0is the optimal value of that would arise if the firm were to disappear, with zero salvage value, when the regime switches. Thus, not surprisingly, the introduction of the possibility of a stochastic death of the firm reduces the value of a unit of capital and reduces the optimal investmentcapital ratio. However, if the new regime does not eliminate the firm, there is a second impact on optimal of the introduction of a positive value of. Specifically, if the firm receives a new draw of from the unconditional distribution () when the regime changes, then is the expected value of a unit of capital in the new regime. With 0, theterm on the right-hand side of equation (22) is 18

21 Figure 2: Optimal investment under uncertainty The optimal investment-capital ratio is the smaller root of ( + + ) =, where ( + + ) () ( + + ) ( + + ) 0 (), is the marginal operating profit of capital, () is a strictly convex adjustment cost function, is the interest rate, is the depreciation rate, is the instantaneous probability of arrival of a new value of, and is the unconditional expected value of a unit of capital when a new value of arrives. Theoptimalvalueof is a function ( + ) of,, +, and. If is known to be constant forever, then =0and the optimal value of is ( + 0), shownaspointa. Ifthe firm becomes worthless when a new value of arrives, then =0and the optimal value of is ( 0+ ), shown as point B. In general, if the unconditional average value of unit of capital is 0when a new value of arrives, the optimal value of is ( + ), shownaspointc. 19

22 negative, so that ( + + ) 0 at the optimal value of. Reducing the value of ( + + ) from zero to a negative value requires an increase in, as shown in Figure 2 by the movement from point B to point C. To summarize, the introduction of stochastic variation in future has two opposing effects on the optimal value of. Define ( + ) to be the optimal value of for given values of + and if = and =. 12 Formally, ( + ) is defined by ( ( + ) + + ) = (23) and ( + ) + +. if min ( + + ). Of course, this definition is meaningful only The following lemma identifies an interval of non-negative values of for which this definition is meaningful. Lemma 2 If ( + ), 0, and, thenthereexistsaunique ( + ) ( + + ) for which ( ( + ) + + ) =. The following proposition describes the impact of an increase in on marginal and average, () (), and on the optimal investment-capital ratio,. Proposition 4 Define + +. If and if ( + ), then 1. (+) = 1 ( ) 00 () 0, 2. 0 () = 0 () = 0 ((+)) = 1 0, 3. (+) = ( ) 00 () 0, 12 Note that ( 0+ 0) = ( + ), which is the optimal value of the investment-capital ratio,, in the case in which = with certainty forever. 20

23 4. 0 ((+)) = 0 Proposition 4 states that ( + ) and 0 ( ( + )) are both increasing functions of and. Most noteworthy is Statement 2, which provides a simple expression for the impact of an increase in the current marginal operating profit,, on Tobin s. Specifically, the derivative of (both average and marginal ) with respect to is simply 1, which is an increasing function of the growth rate of the capital stock,. This result is the foundation of subsection 4.2, which shows that the cash-flow coefficient in investment regressions is higher for firms that grow more rapidly. Because this result is so important, I will describe three ways to derive it. The first derivation is to differentiate the expression for () in Proposition 3 with respect to to obtain 0 () = 1+00 () (+) + + = 1 (24) where the second equality follows from the expression for (+) in Statment 1 of Proposition 4 and using + +. The second derivation is to differentiate the expression for () in Proposition 3 with respect to to obtain 0 () = 1 [1 + 0 ()] (+) () (+) + + = 1 (25) where the second equality follows from the first-order condition in equation (17), 1+ 0 () = (), andthedefinition + +. The third derivation is to differentiate the first-order condition in equation (17), 21

24 1+ 0 () = (), toobtain 0 () = 00 () ( + ) = 1 (26) where the second equality follows from Statement 1 in Proposition The Unconditional Expectation of a Unit of Capital Equation (22) is a simple expression that characterizes the optimal value of. However, this expression depends on R () (), which is the unconditional ex- pectation of the optimal value of a unit of installed capital. In this subsection, I prove that is the unique fixed point of a particular function and describe a simple algorithm to compute. Define Z () 1+ 0 ( ( + )) () (27) as the unconditional expectation of the marginal cost of investment, comprising the purchase cost of capital and the marginal adjustment cost, where ( + ) is defined in equation (23) as the optimal value of the investment-capital ratio if =. Since the value of a unit of capital is () =1+ 0 ( ( + )), optimal behavior by the firm implies that satisfies () =. Lemma 3 Suppose that the support of the distribution () is contained in. The function () 1+ R 0 ( ( + )) () has the following three properties: (1) (0) 0; (2) (1 + 0 ( + )) 1+ 0 ( + ); and (3) 0 0 () 1 for [ ( + )]. Lemma 3 together with the continuity of () leads to the following proposition. 22

25 Proposition 5 Suppose that the support of the distribution () is contained in. Then is the unique value of ( ( + )) that satisfies () =. Lemma 3 and Proposition 5 lead to a simple algorithm to compute : Choose an arbitrary 0 [ ( + )] and generate the sequence +1 = ( ), = The limit of this sequence is. Lemma 3 has the following corollary, which is useful in analyzing the effects of changes in the distribution (Φ) and hence in proving Proposition 6 later. Corollary 2 For any [ ( + )], [ ( ) ]= [ ]. To determine the impact on of changes in the distribution (), let 0 be the initial value of before the change in (). Then any change that increases ( 0 ) will increase, and any change that decreases ( 0 ) will decrease. 4 Measurement Error and the Cash-Flow Effect on Investment The model developed in this paper focuses on three variables that are often used in empirical studies of investment, specifically, the investment-capital ratio (), Tobin s ( () ()), and cash flow per unit of capital (). The first-order condition for optimal investment, 1+ 0 () = () (equation (17), with () replacing ()), implies that is a sufficient statistic for, meaning that if an observer knows the adjustment cost function and the value of, then the optimal value of can be computed in a straightforward manner without any additional information or knowledge of the values of any other variables. Indeed, if the adjustment cost function, (), is 23

26 quadratic, the marginal adjustment cost function is linear, and optimal is a linear function of. The empirical literature has a long history of finding that investment is related to cash flow even after taking account of. In particular, at least since the work of Fazzari, Hubbard, and Petersen (1988), researchers have found that in regressions of on and, estimated coefficients on both and tend to be positive and statistically significant. The finding of a significant positive coefficient on cash flow,, is often interpreted as evidence that firms face financing constraints or some other imperfection in financial markets. This interpretation of financial frictions, as they are sometimes known, is bolstered by the finding that the cash-flow effect tends to be more substantial for firms that seem more likely to face these frictions. For instance, as the argument goes, firms that are growing rapidly may encounter more substantial financial frictions, and it turns out that the cash-flow coefficient is often larger for such firms. In this section, I offer a different interpretation of estimated cash-flow coefficients. I demonstrate that if is observed with classical measurement error, then the coefficient on is biased toward zero and, more importantly, the coefficient on cash flow,, will be positive, even though the coefficient on would be zero in the absence of measurement error in. The fact that measurement error in can affect the coefficient estimates in this way been pointed out by Erickson and Whited (2000), Gilchrist and Himmelberg (1995), and Roberts and Whited (2012), though the particular simple expressions I present in this paper appear to be new. More novel, however, are (1) the analytical demonstration that the cash-flow coefficient will be larger for firms 24

27 that grow more rapidly, and (2) the analytic description of the estimated coefficients on and cash flow in terms of a bivariate attenuation factor and weight shifters. The demonstration that measurement error in can lead to a positive cash-flow coefficient does not use the particular model in this paper, other than the result that and are positively correlated with each other. I present this demonstration in subsection 4.1. Then in subsection 4.2, I use the model in this paper to show that cash-flow coefficients are larger for firms that have higher growth rates. Since the model I use here has no financial frictions whatsoever, the empirical finding of positive cash-flow coefficients, with larger coefficients for firmsthataregrowingmore rapidly, does not necessarily show that financial frictions are important or operative. 4.1 Coefficient Estimates under Measurement Error In this subsection I analyze the impact of measurement error on the estimated coefficients on and cash flow in investment regressions. To isolate measurement error from specification error that might arise by fitting a linear function to a nonlinear relationship, I assume that the adjustment cost function is quadratic so that optimal is a linear function of. In particular, the adjustment cost function is ( )= 1 2 ( 0 ) 2 (28) where 0 and, as discussed earlier, 0 +. The first-order condition for optimal in equation (17), using () in place of (), implies that = 0 + ( 1) (29) 25

28 Assume that the manager of the firm observes without error, computes without error, and chooses, but people outside the firm, including the econometrician, observe these variables with classical measurment error. 13 Specifically, the econometrician observes the value of a unit of capital as e = +, the investment-capital ratio as e = + = 0 + ( 1)+,andcashflow as e = +, where the observation errors,,and, are mean zero, mutually independent, and independent of,, and. Erickson and Whited (2000) offer a useful taxonomy of reasons for measurement error in, and except for differences between marginal and average (which are non-existent in the model presented here), those reasons could apply here. Consider a linear regression of e on e and e, after all variables have been demeaned. Let and be the probability limits of the estimated coefficients on e and e, respectively,so = (e) (ee) (ee) (e) 1 (e e) (30) (e e) The variance-covariance matrix,, of(ee e) conveniently displays the variances and covariances in equation (30), where ()+( ) ( ) () = ( ) ()+( ) ( ) (31) () ( ) 2 ()+( ) 13 For notational simplicity, I suppress the time subscript for the remainder of this paper. This notational simplication does not imply that the values of,, and are constant over time. 26

29 Substituting the relevant second moments from equation (31) into equation (30), and performing the indicated matrix inversion and matrix multiplication yields = [()+( )] [()+( )] [ ( )] 2 (32) [()+( )] () ( ) ( ) [()+( )] ( ) ( ) () Define 2 () () asthevarianceofthemeasurementerrorine normalized by (), which is the variance of the true value of ; 2 ( ) asthevarianceof () the measurement error in cash flow normalized by the variance of the true value of cash flow; and 2 = [()]2 ()() of and cash flow. I assume that as the squared correlation between the true values (33) which rules out perfect correlation between e and e. 14 Dividing the numerators and denominators of and in equation (32) by () () yields = (34) 2 () () [()] 2 14 The squared correlation between e and e is [(e e)] 2 = = (1+ 2 )()(1+ 2 )() 2 Equation (33) implies ,so[ (e e)] 2 1. If, (1+ 2 )(1+ 2 ). contrary to equation (33), ,then 2 =1and 2 = 2 =0and hence [(ee)] 2 =1. 27

30 Equation (34) shows the impact of measurement error in. If is perfectly measured, then 2 =0and, regardless of whether cash flow is measured with error, equation (34) immediately yields = and =0. Thus, if is perfectly measured, equals the derivative of the optimal value of with respect to in the firstorder condition in equation (29). In addition, the estimated effect of cash flow on investment,, is zero. Erickson and Whited (2000) use measurement-error consistent GMM estimators and find empirically that the cash-flow coefficient is zero and that investment is well explained by, whenproperlyremovingtheeffects of measurement error. If is measured with error, so that 2 0, then,, the estimated coefficient on, issmallerthan, the true derivative of with respect to. Moreover, if 2 0, then, the estimated coefficient on cash flow can be nonzero; in fact, if and cash flow are positively correlated, the estimated cash-flow coefficient,, is positive. A significantly positive coefficient on cash flow in a regression of investment on and cash flow is often viewed as evidence of financing constraints. 15 Yet equation (34) demonstrates that measurement error in will lead to a positive coefficient on cash flow, provided that and cash flow are positively correlated, even if there are no financial frictions. This argument is not restricted to the particular specification of the firm in this model, and has been made by, for example, Gilchrist and Himmelberg (1995) and Roberts and Whited (2012) Notable exceptions include Abel and Eberly (2011), Alti (2003), Bushman, Smith, and Zhang (2011), Cooper and Ejarque (2003), Erickson and Whited (2000), Gilchrist and Himmelberg (1995), and Gomes (2001). 16 Gilchrist and Himmelberg, p. 544, state "More generally, anything that systematically reduces the signal-to-noise ratio of Tobin s Q (for example, measurement error or excess volatility of stock prices) will shift explanatory power away from Tobin s Q toward cash flow, thus making such firms 28

31 I now rewrite the regression coefficients in equation (34) in a form that facilitates the interpretation of the coefficient estimates in terms of a bivariate attentuation factor and a factor that shifts weight from one coefficient to the other. It is straightforward to rewrite equation (34) as 1 = (35) 2 where (36) and ( ) () (37) is the regression coefficient in a univariate regression of true on true cash flow,. The statement in equation (36) that 0 follows directly from the assumption in equation (33). The factor definedinequation(36)helps measure the impact on coefficient estimates of 2 and 2, individually (through 2 and 2 ) 17 and jointly (through 2 2 ). The factor is symmetric in the stochastic properties of and cash flow. More precisely, is symmetric in 2 and 2,and the squared correlation 2 is symmetric in true and true cash flow,. The factor appear to be financially constrained when in fact they are not." Roberts and Whited, p. 498, state that because of the high positive correlation between Tobin s and cash flow, "the coefficient on cash flow,, is biased upwards. Therefore, even if the true coefficient on cash flow is zero, the biased OLS coefficient can be positive." 17 The individual impact of measurement error in cash flow, reflected in 2, does not appear in equation (35) because true cash flow does not affect the optimal-investment capital ratio. In Section 5, which posits a model in which true cash flow has a direct impact on the investment-capital ratio, the impact reflected in 2 appears in equation (45). 29

32 is increasing in 2 because an increase in 2 effectively reduces the amount of independent variation in true and true cash flow, thereby increasing the role of measurement error in providing independent variation, and increasing the impact of 2, 2,and 2 2 on the estimated coefficients. If is measured without error, so that 2 =0, then it follows immediately from equation (35) that = and =0. If is measured with error, then 2 0 and 1 equation (35) shows that = The attentuation factor in this 1 estimated coefficient is the product of, which is a common attenuation factor for both and,and1 2, which is an attenuation factor that is specific to 1. I will refer to as the bivariate attentuation factor, since it is a common factor that applies to both estimated coefficients and. If at least one of or cash flow is measured without error, that is, if at least one of 2 or 2 equals zero, 1 then the bivariate attenuation factor,, equals1. Nevertheless, the estimated coefficient is attenuated by the factor if is measured with error. If both and cash flow are measured with error, so 2 0 and 2 0, the bivariate attenuation factor is less than 1, which further attenuates the estimated coefficient. However, even if the bivariate attenuation factor is less than 1, the estimated cash-flow coefficient is not biased toward zero; indeed, the estimated cash-flow coefficient is biased away from its true of zero when 6=0. In addition to their joint impact on the bivariate attenuation factor, 2 and 2 individually affect the coefficient estimates through 2 and 2,where0 2 1 and I will refer to 2 and 2 as weight shifters. The rationale 18 Since 0 in equation (36), both 2 and 2 are non-negative. Use the definition of 30

33 for this terminology is that the fitted value of using the regression coefficients in equation (35) is b = e + 2 e (38) The term in square brackets in equation (38) is a weighted average, with weights 1 2 and 2,ofe and e, eachofwhichcanbeviewedasaproxyfor. The value of b in equation (38) is a weighted average of these two proxies for multiplied bytheimpactof on, whichis, and multiplied by the bivariate attenuation 1 factor Three special cases are noteworthy. First, as noted above, if 2 =0,sothat 1 is measured without error, then the bivariate attentuation factor,,equals The plim of, the estimated coefficient on, equals, andtheplimof, the estimated coefficient on cash flow,equals zero,which is the correct value under the model, since is a sufficient statistic in the model. Second, if 2 =0,sothat 1 cash flow is measured without error, then the bivariate attentuation factor,, equals 1 and 2 = 2. In this case, there is no bivariate attenuation bias, but the weight shifter attenuates to 1 2 and increases to a positive value, provided that true and true cash flow are positively correlated. Third, if 2 =1, so that true and true cash flow are perfectly correlated, then 2 = In this case, measured and measured cash flow are both noisy measures of the same (up to a linear transformation) true variable. The shifter 2 is an increasing function of 2 2 because an increase in 2 2 makes observed cash flow a relatively better measure in equation (36) to obtain since since , andsimilarly 2 31

34 of the true variable, thereby shifting more weight toward. Indeed, if 2 =0 when 2 =1, then the weight shifter 2 =1,sothatalloftheweightisshifted to perfectly measured cash flow e and mismeasured e has an estimated coefficient of zero. The analysis of measurement error presented thus far does not rely on the particular model of investment presented in this paper. Therefore, this analysis is more general than that model. In the following subsection, I will use the model developed in this paper to carry the analysis one step further and account for differences in the estimated cash-flow coefficients for firms with different growth rates Larger Cash-Flow Coefficients for More Rapidly Growing Firms The estimated cash-flow coefficients in investment regressions are typically positive and tend to be higher for firms that grow more rapidly. Fazzari, Hubbard, and Petersen (1988) categorize firms into three classes. Class I firms have relatively low ratios of dividends to income and are deemed to be financially constrained, Class III firms have relatively high ratios of dividends to income and are deemed to financially unconstrained, while Class II firms are in between. Fazzari, Hubbard, and Petersen find cash-flow coefficients for firms in Class I to be significantly significantly higher than the cash-flow coefficients of firms in Class III. They also report that firms in Class I grow much more quickly than firms in Class III. As shown in their Table 2, the average growth of sales is 13.7 percent per year for firms in Class I and only 19 Erickson and Whited (2000) derive an alternative expression for the estimated cash-flow coefficient and demonstrate that to the extent that measurement error in impartsdownwardbiasin,thecash-flow coefficient will be higher for firms with more variable cash flows. 32

35 4.6 percent per year for firms in Class III. Also the gross growth rate of the capital stock, that is, the average investment-capital ratio, is 0.26 for firms in Class I and only 0.12 for firms in Class III. Similarly, Devereux and Schiantarelli (1990) report "The perhaps surprising result from table 11.7 is that the coefficient on cash flow is greater for firms operating in growing sectors." (p. 298). More recently, Bushman, Smith, and Zhang (2011) show (in their Table 8) that the sensitivity of investment to cash flow (which they call ICFS for investment-cash flow sensitivity) is positively related to firm growth, regardless of whether firm growth is measured by sales growth, earnings growth, or growth in the number of employees. They argue that this relation between ICFSandgrowthreflects complementarity between physical capital and working capital so that capital investment is related to investment in working capital, which is a component of cash flow. They also argue that investment-cash flow sensitivity does not reflect financing constraints. In this subsection, I use the model developed in Sections 2 and 3 to provide a simple analytic explanation of the finding that cash-flow coefficients are higher for firms that grow more rapidly. Equation (35) shows that the cash-flow coefficient is proportional to (), which is the population regression coefficient of true () on true. The analog of this coefficient in the model is 0 (). 20 Statement 2 of Proposition 4 is that 0 1 () =, which is increasing in the growth rate of + ( ) capital,, forgiven +. Therefore, the cash-flow coefficient is increasing in the growth rate of the firm. To use the model to compare the investment behaviors of a slowly growing firm 20 More precisely, equals 0 () in the limit as the second and higher moments of () shrink to zero. 33

36 and a rapidly growing firm, I will consider firms that face different unconditional distributions, (), of, that endogenously lead to different growth rates for a given level of. The following proposition states that the firm with the more favorable () in the sense of strict first-order stochastic dominance will grow more rapidly and will have the higher cash-flow coefficient, which is proportional to 0 (), even though there are no financial frictions in the model. Proposition 6 Consider two firms with identical quadratic adjustment cost functions but with different unconditional distributions of, 1 () and 2 (), whichimply different unconditional values of capital, 1 and 2, respectively. If 2 () strictly first-order stochastically dominates 1 (), then , 2. ( 2 + ) ( 1 + ), 3. R ( 2+ ) 2 () R ( 1+ ) 1 (), () 0 1 (), and 5. R 0 2 () 2 () R 0 1 () 1 (). Proposition 6 states that the firm with distribution 2 () is the faster-growing firm, whether the speed of growth is measured by the investment-capital ratio at any given value of (Statement 2) or by the unconditional expectation of the investmentcapital ratio (Statement 3). This proposition also states that the firm with the distribution 2 () has the higher value of 0 () for a given value of (Statement 4) and the higher unconditional expected value of 0 () (Statement 5). Therefore, the firm with distribution 2 () has the higher value of () and hence the () higher cash-flow coefficient. To summarize, the firm that grows more rapidly has 34