Nonlinear response of firm investment to Q:

Journal of Monetary Economics 42 (1998) 261—288

Nonlinear response of firm investment to Q: Testing a model of convex and non-convex adjustment costs1 Steven A. Barnett!, Plutarchos Sakellaris",* ! International Monetary Fund, Washington, DC 20431, USA " Department of Economics, University of Maryland, College Park, MD 20742, USA Received 16 October 1995; received in revised form 20 May 1997; accepted 15 September 1997

Abstract Abel and Eberly (1994) study optimal investment behavior in the presence of flow fixed costs, proportional costs and convex costs. A clear prediction is that investment will alternate between regimes of insensitivity and responsiveness to q separated by unknown threshold levels of q. At the firm level, we find evidence for different regimes of sensitivity to q but not for a regime of zero sensitivity. Our finding that investment has a nonlinear relationship to q is important because it implies an elasticity of aggregate investment to q (and fundamentals) that is high and variable over time. ( 1998 Elsevier Science B.V. All rights reserved. JEL classification: E22; E32 Keywords: Investment; Tobin’s Q; Adjustment costs; Nuisance parameters

1. Introduction The bulk of the empirical literature on firm investment in recent years has been along the lines of Tobin’s q. This was introduced by Brainard and Tobin

* Corresponding author. E-mail: [email protected] 1 The paper was written while Steven A. Barnett was a graduate student at the Department of Economics, University of Maryland at College Park. 0304-3932/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 9 3 2 ( 9 8 ) 0 0 0 2 8 - 2

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(1968), and Tobin (1969) as an arbitrage theory of investment and was shown by Hayashi (1982) to be consistent with the neoclassical theory of investment when investment is reversible and there are convex costs to adjusting the capital stock. Despite its strong theoretical underpinnings, the q theory has explained investment rather poorly at both the aggregate and the firm level.2 Various researchers, partially prompted by the apparent insensitivity of investment to q, have dropped the assumptions of convex adjustment costs and of investment reversibility. A large body of literature has thus emerged that studies irreversible investment in the presence of fixed and proportional costs.3 These theories do not predict that investment responds to q continuously. Instead, investment should be insensitive to movements in q for large periods of time followed by discrete adjustments to the firm’s desired level of capital. These are, then, theories of lumpy investment. Abel and Eberly (1994) gave a characterization of optimal investment behavior when there are convex costs as well as fixed costs in flow form, and investment could be irreversible.4 In this framework, there are regions where investment in a homogeneous capital good is insensitive to q as well as regions where it is responsive to q. These regions are completely characterized by the level of q. In the original version of their model three regions may arise. For low values of q investment may be negative, for intermediate values of q it may be zero, and for high values of q it may be positive. We show in Section 1 of this paper that extensions of their framework can lead to other regions of insensitivity to q at observed investment rates greater than zero. In this paper we estimate the relationship between investment and q at the firm level allowing this relationship to vary across regimes defined by the level of q. We allow for the breakpoints between regimes to be endogenously picked as the theory implies that they depend on the specification of the augmented adjustment cost function, which must be estimated. A crucial task of this paper is to test for the validity of the kind of nonlinear behavior of investment that is implied by the Abel and Eberly (1994) framework. Thus, it is important to have a procedure capable of testing the null hypothesis of only one regime against the

2 Various attempts have been made to address empirical shortcomings of the q approach and associated econometric problems. Among the issues addressed are those of endogeneity, measurement error, multiple capital goods and imperfect competition. One of the most convincing attempts to establish a strong response of firm investment to q is by Cummins et al. (1994). They use tax reforms as natural experiments. 3 For further discussion and references see Dixit and Pindyck (1994) and for empirical applications see Bertola and Caballero (1994), Caballero and Engel (1994) and Cooper et al. (1995) among others. 4 Note that since the fixed costs are specified in flow rather than stock form this model does not nest (S,s) models of investment.

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alternative hypothesis of several regimes. The fact that the thresholds between regimes in the alternative hypothesis are unknown poses the problem of developing appropriate asymptotic distributions for test statistics as the conventional ones do not apply. Hansen (1996) provides the statistical framework to carry out such testing. We find that the null of a single regime is rejected in favor of the alternative of three regimes. This is true even when we allow for other forms of nonlinearity in the investment—q relationship by including higher-order q terms in the estimation. There are regimes of high investment sensitivity to q as well as ones of low sensitivity but no regime of zero sensitivity. Investment as a function of q is convex for low values of q and concave for intermediate and high values of q. While this shape is not consistent with a homogeneous-capital framework as in Abel and Eberly (1994), it is consistent with the investment function reported by Abel and Eberly (1996) in a framework where the firm invests in heterogeneous capital goods. The nonlinear relation between firm investment and q documented in this paper has been ignored by previous studies using this empirical framework.5 A consequence of this nonlinearity is that it is necessary to consider the entire distribution of q across firms in evaluating the responsiveness of aggregate investment to shocks. We find that, compared to the linear specification that has been standard to date, the elasticity of aggregate investment to q is higher (by a factor of about 2.3) and dramatically more variable over time. Finally, we find that the estimated investment—q relation is not altered when we use investment expenditures net of capital sales, which is a conceptually appropriate but unconventional measure for gross investment. Thus, the standard practice in the literature of using the investment expenditures measure is empirically valid. In Section 2 of this paper we present the Abel and Eberly (1994) model as well as an extension of it. In Section 3 we describe the firm level data that we use. Section 4 contains a brief exposition of the econometric and testing methodology (more details are given in Appendix A). In Section 5 we present the results and discuss them. We present our conclusions in the final section.

2. The model We first develop the standard neoclassical model of a firm when investment is reversible and there are convex costs to adjusting the capital stock. Then, we

5 The paper by Abel and Eberly (1996), however, which was written contemporaneously with our paper, also emphasizes the nonlinear relation between investment and q.

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describe Abel and Eberly’s (1994) extension to include (partial) irreversibility and fixed costs of investment. Finally, we expand this framework to allow the possibility that investment beyond some ‘usual’ or ‘normal’ level is subject to dramatically different costs. The fundamental behavioral assumption of the neoclassical theory is that managers maximize the expected discounted value of the firm’s stream of net cash flows, that is the fundamental value of the firm. Further assumptions that we maintain here are that the marginal shareholder is risk-neutral, that capital is the only quasi-fixed factor of production, and that the firm faces increasing and convex costs of installing new capital. The fundamental value of the firm in the absence of taxes is » "max E M[%(K , g )!t(I , K , u )!p I ] t M N= t t t t t t t t Is s/t s = < b [%(K , g )!t(I , K , u )!p I ]N, (1) # + j s s s s s s s s/t`1 j/t`1 where b is the period j discount factor, E is the expectations operator condij t tional on information known at time t, K is the beginning-of-period capital, I is s s gross investment in physical units. The profit function %(K , g ) is increasing and s s concave in K , and g is a firm-specific profitability shock.6 The unit purchase s s cost of capital is p and the installation cost function t (I , K , u ) is continuously s s s s differentiable, decreasing in I for I(0, increasing in I for I'0, decreasing in K, and convex in both arguments. The exogenous shock to installation costs, u , is t observed by the firm at time t but not by the econometrician. We assume that t ( ) is independent of u . The firm managers maximize (1) subject to the Is s accumulation equation for the capital stock:

A

B

K "(1!d)K #I t`1 t t All variables are expressed in real terms. The first-order condition for investment is

(2)

t (I , K )#p "E (j ), (3) I t t t t t where Mj N is the sequence of Lagrange multipliers on the capital accumulation s equation. Along the optimal path the shadow price of installed capital, j , is s = i j" + < b (1!d)i~s~1[% (K , g )!t (I , K , u )], (4) s j K i i K i i i i/s`1 j/s`1

A

B

6 This function incorporates the optimal choices of any variable factors of production.

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and the firm chooses investment so as to equate the cost of investment with its expected benefit on the margin. The cost of investment includes both direct purchasing and installation costs and the benefit includes additions to profit and reductions in installation costs. If the installation cost function is linearly homogeneous in I and K and t ( ) is I invertible then Eq. (3) can be rewritten as: I t "f (E (j )!p )"f (p (qM!1)), (5) t t t t t K t where qM"E (j )/p is Tobin’s marginal q and f ( ) is the inverse function of t ( ). t t t t I Since the installation cost function, t( ), is convex f ( ) is increasing in its argument. Furthermore, provided that p is constant, marginal q is a sufficient t statistic for investment. Abel and Eberly (1994) introduced into this framework partial irreversibility by allowing the purchase price of capital goods (p`) to be higher than the resale price (p~). They also accommodated fixed costs of investment in the form of a flow cost of fixed size incurred at each instant that investment is undertaken. We simplify the presentation by ignoring fixed costs here, though we later discuss their impact. We define the augmented adjustment cost function: c(I , K )"t(I , K , u )#p`I d #p~I (1!d ), (6) t t t t t t t t t where d "1 if the firm undertakes positive investment, zero otherwise. This t introduces a non-differentiability of the adjustment cost function at I"0, which implies that there is a range of values of E (j ) for which the optimal investment t t for the firm is zero and Eq. (3) does not hold. For values of E (j ) outside this t t range the FOC for investment holds with p "p` for high E (j ), which implies t t t positive investment, and p "p~ for low E (j ), which implies negative investt t t ment. The range of values of E (j ) for which investment is zero can be shown to t t be between c~,t (0, K )#p~ and c`,t (0, K )#p`. I I t I I t We define marginal q as qM"E (j )/p`; if p` and p~ are constant then it is t t t not important which one we use to normalize the shadow price of capital. There is a range of marginal q’s for which the firm’s investment is zero and insensitive to variations in marginal q:

G

f (qMp`!p~)(0 if qM(c~/p` I t t I t" 0 (5@) if c~/p`)qM)c`/p` I t I K t f (qMp`!p`)'0 if qM'c`/p` I t t Thus, firm investment behavior is characterized by three regimes. These regimes are completely defined by the level of marginal q of the firm: low, medium, or high. Note also that the sensitivity of I/K to qM could differ across the low and

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the high regime depending on the curvature of t( ) and the relative magnitude of p` and p~.7 A straightforward extension of this model introduces more regimes for the firm’s investment behavior. Suppose that the firm incurs an additional proportional cost f when it invests beyond a threshold rate of investment I/K"h. This additional cost may be due to a jump in the purchase price of capital. Such a situation would arise, for example, if h"d, all investment up to dK is for replacement whereas beyond that it is for expansion, and the price of expanding is higher than that of replacing capital. Alternatively, if h captures the firm’s ‘usual’ rate of investment then the additional proportional cost may arise from having to pay overtime wage to a permanent crew of firm employees working as installation technicians. This kink in the augmented adjustment cost function at I/K"h introduces another region of insensitivity of investment to qM: I t "h if t (hK , K )/p`(qM!1(t (hK , K )/p`#f/p`.8 I t t I t t K t One can also incorporate the effect of taxation on investment. In this case, the cash flow term in Eq. (1) is amended as follows (1!q )[%(K , g )!t(I , K , u )]!(1!C )p I , s s s s s s s s s where q is the corporate tax rate and C is the tax benefit of investing, which incorporates the investment tax credit and depreciation allowance. The following quantity is, then, a sufficient statistic for investment QM"[E j !(1!C )p`]/ t t t t t (1!q ) and the region of insensitivity to tax-adjusted Q, when p`'p~, is t defined by 1!C t(p`!p~)(QM(t (0, K ). t (0, K )! I t t t I t 1!q t t 2.1. Empirical implications To summarize, the fundamental empirical implication of the Abel and Eberly (1994) model is that when fixed costs and partial irreversibility of investment are

7 A fixed cost of investment when the latter differs from zero has the effect of making the firm require a stronger profitability signal in order to take an action. This enlarges the region of inactivity and insensitivity to qM. In addition, this makes it more likely that the low region corresponds to negative qMs only, in which case negative investment will not be observed. See Abel and Eberly (1994) for the details. 8 Note that f has to be positive in order to preserve convexity of the augmented adjustment cost function and maintain a strict one to one mapping from qM to regimes of investment behavior.

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incorporated in the neoclassical model with convex adjustment costs the relationship between investment and fundamentals becomes nonlinear. Firm investment alternates between regimes of zero and positive sensitivity to qM. These regimes are completely identified by threshold levels of qM. These threshold levels, however, are unobservable to the econometrician since they depend on the adjustment cost function whose parameters are to be estimated. Since marginal q is unobservable, further steps are necessary to make the theory operational empirically. Hayashi (1982) provided conditions justifying the use of financial market valuation of the firm and of Tobin’s average q, qA, in t place of marginal q in Eq. (5) or Eq. (5@).9 Tobin’s average q is qA"» /(p K ) with t t t t the fundamental value of the firm » captured by its financial market valuation.10 Tax-adjusted average q is defined as QA"[» /K !(1!C )p ]/(1!q ). t t t t t t 3. The data We use firm level data on publicly traded corporations contained in Standard and Poor’s Compustat. These data originate in the Annual Reports and 10-K forms filed by the corporations with the Securities and Exchange Commission. The NBER maintains a dataset that contains many different years of Compustat releases (See Hall, 1990). An attractive feature of this dataset is that it is an unbalanced panel and includes firms with relatively short appearances. Thus, some of the sample selection problems associated with balanced panels are avoided. 3.1. Sample selection The original panel contains 49,225 observations for 2,726 firms from 1959 to 1987. We deleted non-manufacturing firms as well as observations after the occurrence of a significant merger or acquisition for the remaining firms. This

9 These conditions are that the firm is a price-taker in output and input markets, its production function is linearly homogeneous in its inputs, its installation cost function is linearly homogeneous in I and K, and that financial markets are efficient. Abel and Eberly (1994) show that if instead P(K, g) and t(I, K) are homogeneous of degree o in K and (I, K) respectively, the two measures of q are proportional and qM"oqA. 10 This empirical approach has been adopted by numerous studies, see e.g. Summers (1981) and Hayashi (1982). There are other ways of obtaining empirical investment models from the standard neoclassical framework with convex adjustment costs. Abel and Blanchard (1986) estimate E j using t t forecasts of the marginal revenue product of capital and of the discount rate. An alternative approach is to estimate the Euler equation written in terms of observables alone (see Shapiro, 1986, Hubbard and Kashyap, 1992; Whited, 1992 among others).

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left a sample of 2,489 firms and 37,682 observations. Of these, 2,118 firms (36,742 observations) had at least five continuous years of data. We deleted observations with missing data on investment, capital stock, and the components of market valuation necessary for the construction of average q. This left 1,793 firms with 24,492 observations. We deleted 44 firms with inconsistent stock market data and kept firms with at least five years of data. The resulting sample, which we term the ‘Full sample’, is an unbalanced panel of 1,561 firms with 23,207 observations. The yearly coverage ranges between 263 and 1,216 firms. The first year is 1960 because the specification includes some lagged values. For some tests we use a modified measure of gross investment, NI, for which we require data on the sales of property, plant and equipment (see Section 3.2). We have valid data for the property, plant and equipment (PPE) sales variable for years 1971—87 only. In addition, for some observations this variable is missing. The resulting ‘Restricted’ sample has 13,252 observations. We chose not to follow the common practice of deleting outlying observations of investment to capital ratios, average qs, or other variables used in the empirical work. We deleted only observations where we could identify coding errors or other inconsistencies. We believe that the practice of dropping ‘outliers’ begs the question of explaining such observations. We did, however, verify that the exclusion of ‘outliers’ did not alter our estimation and testing results. 3.2. Variable definitions The gross investment (I) measure that we use is capital expenditures on property, plant, and equipment. This is standard in the empirical literature on firm investment. However, many firms are observed to sell and purchase capital assets simultaneously. One could argue, then, that for the purpose of testing a theory that assumes that capital is homogeneous it is appropriate to use a modified measure of gross investment (NI) that subtracts sales of property, plant and equipment from purchases.11 We show in Section 5 that the conclusions are robust to using this second definition. The numerator of average Tobin’s q is the sum of the market value of common stock, the liquidating value of preferred stock, the market value of long-term debt, and the book value of short-term debt. The denominator of average q is the sum of the replacement values of property, plant, and equipment and of inventories. We found that alternative definitions of average q did not affect our results significantly. Cash

11 This illustrates the need for models of heterogeneous capital. One possibility is for different vintages of capital to be heterogeneous due to their ‘putty-clay’ nature (see, e.g., Sakellaris, 1997). Alternatively, different ‘putty-putty’ capital goods could be heterogeneous if there is limited substitutability among them in production (see Hayashi and Inoue, 1991) or if they are subject to different costs (see Abel and Eberly, 1996).

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flow is the sum of depreciation and income before extraordinary items and discontinued operations. 3.3. Summary statistics Summary statistics for the variables used in the regressions as well as other variables of interest are included in Table 1. It is interesting to note that there are only three observations of zero investment expenditures. Similarly, only about 2% of the observations have investment rates (I/K) less than 0.025. A word of caution is in order here. Even though zero investment is rarely observed at the firm level aggregation may be masking periods of zero investment. At the plant level, there is substantial evidence of zero investment. Cooper et al. (1995) (Fig. 6) report that about 10% of their plant-level observations involve no investment. Similarly, aggregation over different types of capital goods could lead to observing positive firm investment at all times even though the firm undertakes no investment in some types (see Abel and Eberly, 1996). Finally, lags in ordering and building capital and time aggregation may also conceal inaction. Fig. 1 is the histogram of the investment—capital ratio (I/K) for the Full sample. About 15% of the observations have an investment rate greater than 0.3, which corresponds to a rather high growth rate for capital. Fig. 2 compares

Table 1 Descriptive statistics

Variable

25th percentile

Median

75th percentile

Average

Standard deviation

Full sample I/K Tobin’s Q Tax adjusted Q K (82$ millions) Cash flow/K Sales/K

0.09 0.84 1.02 21.80 0.14 2.10

0.15 1.23 1.79 72.80 0.22 3.38

0.23 1.95 3.21 345.10 0.34 5.19

0.20 1.79 2.89 747.10 0.26 4.37

0.24 2.18 4.28 2918.60 0.33 4.63

0.00 0.07 0.08 0.76

0.00 0.12 0.13 1.03

0.02 0.19 0.20 1.54

0.02 0.16 0.18 1.47

Restricted sample Capital sales/K NI/K I/K Tobin’s Q

0.06 0.22 0.21 1.97

Notes: The full sample size is 23,207 over the years 1960—1987. The restricted sample size is 13,252 and covers the years 1971—1987. ‘I’ is investment expenditure (Compustat variable d30), Capital Sales is the revenue from the sale of property, plant and equipment (Compustat variable d107), and NI is investment expenditure (‘I’) minus capital sales.

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Fig. 1. Histogram of I/K for the full smaple.

the histogram of I/K with that for the modified investment—capital ratio, NI/K, for the Restricted sample. The distribution of this modified measure is quite similar to that of I/K, the main difference being that more of its mass is in the lower end. It is noteworthy that there are negative observations of NI/K (3% of the total number).

4. Econometric methodology Our methodology is a direct application of the technique developed by Hansen (1996) to test models where there are nuisance parameters that are not identified under the null hypothesis. He develops a testing strategy based on the critical value from a simulated distribution. In particular, we estimate a threshold model of the type discussed in Hansen (1996). The basic idea in a singlethreshold model is that the relationship between the dependent variable and the independent variables depends on whether an exogenous variable (that may or may not be included in the regression) is above or below an unknown threshold. This threshold separates the two regimes. The objective is to estimate the threshold, as well as the coefficients in both regimes. The threshold is the nuisance parameter, since it is not identified under the null hypothesis that there

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Fig. 2. Histogram of I/K for the restricted sample.

is only one regime. The presence of this nuisance parameter invalidates standard testing techniques such as the Likelihood ratio, Wald, and Lagrange multiplier. The technique developed by Hansen (1996) and applied in this paper allows this type of model to be tested for the presence of two regimes versus the null of a single regime. It is also applicable for multiple thresholds corresponding to more than two regimes. The main model of interest for the paper is I/K"¹a#F/#Xb l(q)c )#Xb l(c (q)c ) 1 l 2 l h #Xb l(q'c )#e, (7) 3 h which is formally the alternative model. ¹ is a set of time dummies, F is a set of firm dummies, X is the matrix of independent variables, l( ) is the indicator function, and the disturbance e is i.i.d. (0, p2). The null model is that there are no thresholds, I/K"¹a#F/#Xb #e. (8) 0 Let C"(c , c ) be the threshold parameters, h"(a, /) the coefficients that are l h constant across regimes and B"(b , b , b ) the coefficients that vary over the 1 2 3 regimes. The objective is twofold: 1) estimate C, B, h and 2) test whether b "b "b . 1 2 3

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Note that under the null hypothesis, the parameters in C are not identified. If C were known the estimation would simply be OLS and standard testing theory would apply. However, the presence of the nuisance parameters C makes the model non-linear and invalidates the standard theory. The methodology basically involves the following steps: 1) Discretize the parameter space for C into N points, defined as C , where (C "(ci, ci ): i"1, 2,2,N). 2) For each point in a a l h C construct the design matrix defined as Z(c), where Z(c)"(¹, F, X , X , X ). a l m h Estimate the remaining parameters h(c) and B(c) by OLS. Calculate the Sum of Squared Residuals, SSR(c), and the appropriate test statistics, ¹(c), which are based on the Wald principle. 3) Construct a test statistic which is a function of the ¹(c) defined as H(C ). In practice, we calculate the average (H ) and the a !7' supremum (H ) of ¹(c) over the parameter space C . 4) Generate a simulated 461 a distribution for H(C ) and define this F(C ). Based on this simulated distribution a a accept or reject the null hypothesis. The Appendix describes each of these steps in detail. It is useful to give an intuitive motivation for the procedure. The idea is to scan over the threshold parameter space looking for the point that minimizes the sum of squared residuals. This is facilitated by the fact that for each point in C, the remaining parameters may be estimated with OLS. The point of C, c* which minimizes SSR yields the parameter estimates for the model h(c*) and B(c*). The difficulty in testing the restrictions that b (c*)"b (c*)"b (c*) is 1 2 3 that a normal chi-square test (Wald or LM) is valid only if c* is known. However, c* is estimated thus the researcher has considerably more degrees of freedom than suggested by the number of restrictions. This may lead to the null hypothesis being rejected more frequently than suggested by the size of the test. Hansen (1996) demonstrates that a correct distribution for a test statistic which takes account of nuisance parameters may be simulated. His technique provides better guidance than any sort of rule of thumb such as doubling the degrees of freedom.

5. Empirical results 5.1. Specification The specification of the empirical model is dictated by Eq. (5@), which prescribes three regimes for investment. As was described in Section 3, however, there are few observations of zero investment and, by definition of the gross investment variable I, no negative observations.12 The extension of the

12 The modified investment variable (NI), however, does include negative observations.

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framework that we developed in the end of Section 2 predicts three regimes for positive values of investment. In order to accommodate testing of both versions of the framework we run specifications that allow for three regimes as the alternative to a single regime. The next choice involves the specification of f ( ) in (5@), the inverse of the marginal adjustment cost function t ( ). It is common for researchers to assume I a quadratic form for t( ) and obtain a linear relationship between I/K and q. We allow for the possibility that f ( ) is nonlinear in q, i.e. that t( ) has terms in I of order higher than quadratic. This is important beyond the fact that there is no evidence of quadratic against other forms of convex costs. The objective in this paper is to detect non-linearities arising from fixed costs and kinks in the adjustment cost function. These were shown in Section 2 to correspond to specific regimes for investment behavior, which are different from nonlinearities due to the curvature of the adjustment cost function. Since f (q) is unknown we take Taylor series expansions of various orders either on q or on ln(q). We present the results of the expansions on the level of Tobin’s q in Table 2. We also discuss results based on ln(q) terms. All estimated equations include fixed firm and year effects. We also repeat the analysis incorporating the effect of taxation. We present the results of threshold regressions on Taylor series expansions of tax-adjusted q in Table 3. 5.2. Discussion of results Model 1 contains only linear terms in Tobin’s q. The coefficient on q in the restricted estimation is 0.04, which is in the range of estimates of past studies. The unrestricted estimation picks a middle region confined by q levels of 1.95 and 3.24. These correspond to the 75th and the 90th percentiles of observed qs. The null of a single regime is rejected at the 1% level regardless of the statistic used (H or H ).13 However, none of the picked regimes is characterized by 461 !7' insensitivity of investment to q as would be implied by the theoretical model with homogeneous capital. The coefficient on q falls monotonically as q rises. Table 5 contains the R2s for the whole sample and for the observations in the three regions separately under the alternative hypothesis of three regimes. The number in brackets is the increase in the R2 over the null of a single regime. To ensure that we are not merely detecting the effects of omitted higher order terms in q we also run specifications including quadratic terms in q (see

13 For statistical inference we concentrate on the heteroskedasticity-robust version of the tests. We tested for heteroskedasticity in all specifications that we ran using a variant of the Breusch—Pagan test that is robust to non-normality of the disturbances. The alternative hypothesis for the test was heteroskedasticity that was a function of K, firm capital. In every case the null of homoskedasticity was rejected at the 1% level.

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Table 2 Threshold regressions for Tobin’s Q Model 1 Restricted Q

Q

Q

0.0391* (0.0010)

0.057* (0.0015) !0.0006* (0.00004)

!0.1047* (0.0307) 0.1036* (0.0057)

0.1369* (0.0322) !0.1176* (0.0403) 0.1486* (0.0268)

!0.2464* (0.0226) 0.1064* (0.0052)

!0.1861* (0.0252) 0.0938* (0.0161) 0.0053 (0.0060)

0.0115 (0.0213) -

0.0669* (0.0237) -

— 0.0572* (0.0123)

Q2

Regime 3 Constant Q

0.1403* (0.0371) 0.0281* (0.0038)

Q2

Threshold 1 (percentile) Threshold 2 (percentile)

Model 2R

0.057* (0.0015) !0.0006* (0.00004)

Q2

Regime 2 Constant

Model 1R

0.0391* (0.0010)

Q2

Regime 1 Constant

Model 2

1.95 (75) 3.24 (90)

— 0.178* (0.0307) !0.0213* (0.0077)

0.2526* (0.0365) 0.0409* (0.0053) !0.0003* (0.0001) 1.13 (45) 3.24 (90)

-

—

—

0.0281* (0.0038)

0.0407* (0.0053) 0.0003* (0.0001)

2.23 (80) 3.24 (90)

2.23 (80) 3.24 (90)

Notes: *Significant at the 1% level; **significant at the 5% level; ***significant at the 10% level; - restricted to zero. All regressions include a complete set of firm and year dummies. Reported in parentheses are the standard errors of estimates except for the threshold estimates where it is the corresponding percentile of q. All standard errors are from White’s heteroskedasticity consistent variance—covariance matrix, except for the estimates under the null model of a single regime which use the OLS variance—covariance matrix.

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Model 2). The coefficient on this term under a single regime is negative, which is consistent with the monotonically declining coefficient of I/K on q across regimes in Model 1. The null of a single regime is again rejected at the 1% level, though the middle regime is now much wider (qs between 1.13 and 3.24). Once again no regime is characterized by insensitivity of investment to q.14 Fig. 3 contains the graph of the marginal response of I/K to q. The responsiveness of a firm’s investment rate to q increases with q for low values of q then decreases for intermediate values of q and eventually flattens out (solid line for unrestricted Model 2). This is on average 2.5 to 3 times higher for the low- and intermediate-q regimes than the estimate of responsiveness of 0.04 in the specification that is common in the literature (one regime, linear q, i.e. restricted Model 1).15 Investment as a function of q is estimated in our data to be convex for low values of q and concave for high values of q. This nonlinear relation is demonstrated in Fig. 4. The top panel uses the estimates of Model 2, which includes quadratic terms in Tobin’s q, whereas the bottom panel uses estimates from regressions that include quadratic terms in log of q.16,17 Comparison of the restricted and the unrestricted estimates in these graphs facilitates an explanation of the rejection of the null. The shape of the relation is such that it is fit better by local polynomial expansions around three different values of q (under the alternative of three regimes) than with a global expansion around one value of q (as is required by the null of one regime). Table 3 contains the results of threshold regressions using the tax-adjusted q measure. The R2s are contained in Table 5. The null of a single regime is rejected at the 1% level in favor of the alternative of three regimes regardless of

14 An anonymous referee has pointed out to us that aggregation across types of (heterogeneous) capital may hide non-responsiveness of investment to q in the middle range. It is possible that variations in a firm’s q in this middle range push investment in specific types of capital beyond the range of inaction while leaving others within it. At the firm level, then, one would observe that investment is responsive to q. 15 This nonlinear relation does not seem to be a reflection of systematic differences in the characteristics of firms across different levels of q. The number of firms that have only high qs or only low qs is limited. For the regimes estimated for Model 2, 88 percent of firms contain observations in at least two regimes and 26% in all three regimes. The high regime, which encompasses 10% of observations only, is visited by 42% of the firms. The low and middle regimes are visited by 81 and 91% of the firms respectively. 16 The latter estimates are available upon request. Note that these graphs portray accurately the shape of the predicted relation only. To obtain the exact predicted level of I/K corresponding to a particular value of q one needs to add a firm-specific and a year-specific effect. 17 Abel and Eberly (1996) find similar nonlinearities in the relation between investment and q and demonstrate that their nonlinear model tracks average annual investment more successfully than does the linear model.

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Fig. 3. Derivative of I/K with respect to Q.

whether we include linear terms only (Model 3) or additional quadratic ones (Model 4).18 The thresholds picked in these regressions correspond to the same percentiles as those in the Tobin’s q regressions, implying that the regions picked are similar. The estimated responsiveness of investment is lower in this set of results. This is due to the fact that tax-adjusted q has a higher mean than Tobin’sq (see Table 1). In summary, the results on the non-linear relationship between investment and q are qualitatively the same with either measure of q. The nonlinear relation between investment and q implies that any policy conclusions about the responsiveness of investment to fundamental factors that

18 In this case, the lower threshold for QA varies over time with the term X "[(1!C )/ t t (1!q )](p`!p~). To garner how important this variation may be we compare it to the variation in t t t the observed values of QA. Since we do not have data on p~ we assume that it is 3/4 of p`. Then, t t the difference between the high and the low values of X in our sample is 0.07. Taking the estimated t lower threshold for Model 4 of Q "1.61 as the mid-point of this range, this produces a range for A QA of 1.57 to 1.64, which correspond to the 44th to 46th percentiles of the distribution. This range is less than 5 percentage points, which is the space between grid points used in the threshold regressions. Thus, it seems a good first approximation to assume that X is constant. t

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Fig. 4. Investment as a function of Tobin’s Q.

are based on the restricted, linear estimate or that are evaluated at mean or median values of I/K and q will be misleading. Instead, the entire distribution of I/K and q has to be considered. To demonstrate this we calculate the responsiveness of aggregate I/K to q and the elasticity of aggregate investment to q. The results are vastly different depending on whether one uses the standard linear specification or a nonlinear one together with the information in the distribution of q across firms. Figs. 5 and 6 display the estimates for each year in the sample. The solid lines correspond to non-linear specifications whereas the dashed lines to restricted linear specifications. In both cases, the thin lines refer to Tobin’s q estimates and the thick lines to tax-adjusted q estimates. The responsiveness of aggregate I/K to q varies over time and is higher than the linear estimate (by up to 3 times in certain years). The variability of this magnitude is due to changes over time in the cross-sectional distribution of firms’ q. The elasticity of aggregate investment to Tobin’s q is about 0.84 on average in the nonlinear model as

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Table 3 Threshold regressions for tax-adjusted Q Model 3 Restricted Q

0.0195* (0.0005)

Q2

Model 4

0.0287* (0.0007) !0.0002* (9.7e!06)

Model 3R

Model 4R

0.0195* (0.0005)

0.0287* (0.0007) !0.0001* (9.7e!06)

!0.2294* (0.0212) 0.0551* (0.0027)

!0.181* (0.0225) 0.0559* (0.0064) 1.0e!05 (0.0015)

0.0017 (0.0201) -

0.0533** (0.0222) -

Regime 1 Constant Q

!0.0819* (0.0250) 0.0555* (0.0030)

Q2

0.0533* (0.0187) !0.0007 (0.0114) 0.0345* (0.0064)

Regime 2 Constant

—

Q

0.0325* (0.0058)

Q2

— 0.085* (0.0124) !0.0056* (0.0018)

-

Regime 3 Constant Q

0.1493* 0.0316) 0.0140* (0.0019)

Q2

Threshold 1 (percentile) Threshold 2 (percentile)

3.21 (75) 5.76 (90)

0.2182* (0.0286) 0.0204* (0.0026) !8.6e!05* (2.8e!05) 1.61 (45) 5.76 (90)

— 0.0140* (0.0019)

3.77 (80) 5.76 (90)

— 0.0203* (0.0026) !8.5e!05* (2.8e!05) 3.77 (80) 5.76 (90)

Notes: *Significant at the 1% level; **significant at the 5% level; ***significant at the 10% level; - restricted to zero. All regressions include a complete set of firm and year dummies. Reported in parentheses are the standard errors of estimates except for the threshold estimates where it is the corresponding percentile of q. All standard errors are from White’s Heteroskedasticity consistent variance— covariance matrix, except for the estimates under the null model of a single regime which use the OLS variance—covariance matrix. The notation 9.7e!06 signifies 9.7]10~6.

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Fig. 5. Responsiveness of aggregate I/K to Q.

opposed to 0.37 in the linear one. More significantly, this elasticity is much more variable in the nonlinear model.19 5.3. Robustness checks We have checked the robustness of our findings to a number of modifications. The qualitative results remain the same in specifications that include up to cubic terms in ln(q) or q. The estimated nonlinear relation is similar and the null of one regime is always rejected. Our results are not due to our not following the common practice of excluding ‘outliers’. Dropping observations with extreme values of I/K, q, or K does not change the qualitative results of estimation or testing. 19 The elasticity of aggregate investment to aggregate q for Fig. 6 was calculated by increasing the qs of all firms by 1% and observing the implied percentage change in total sample investment. Under the assumption that the increase in firm qs reflects increase in firm value (the numerator of q) alone, the aggregate q increases by 1% also. If one is not willing to maintain this assumption the levels plotted in Fig. 6 are not valid; however, the relative size of the two elasticities is valid as both have the same percentage change in aggregate q in their denominator.

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Fig. 6. Elasticity of aggregate investment to Q.

An alternative test of the modified Abel and Eberly (1994) model is possible. We restrict the middle regime under the alternative hypothesis to display insensitivity of investment to q. We then allow the thresholds corresponding to such a regime to be chosen endogenously and test the null of a single regime against this modified alternative hypothesis (models 1R, 2R in Tables 2 and 3R, 4R in Table 3). The null is rejected again. However, in sensitivity analysis we found that the estimated threshold levels of q are unstable across different specifications that impose the restriction of zero sensitivity in the middle regime.20 This indicates that the imposed restriction is not valid. One might be concerned that the reduced responsiveness of I/K to q for high values of q is due to measurement error in q. High stock market values (and qs) may be associated with assets of the firm other than physical capital (e.g. R&D or intangible capital) and, thus, q’s may contain substantial noise regarding the investment decision. We believe that this source of concern should not be

20 For instance, in a specification that contains up to quadratic terms in ln(q) the threshold qs were estimated to be at the 40th and the 55th percentile. The results are available upon request.

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Table 4 Threshold regressions for Tobin’s Q using the restricted sample NI-1 Restricted Q

NI-2

I-2

0.0352* (0.0013)

0.0549* (0.0021) !0.0005* (0.00004)

0.0548* (0.0020) !0.0005* (0.00004)

!0.0018 (0.0158) 0.0713* (0.0132)

0.152*** (0.0786) !0.0574 (0.0434) 0.1021* (0.0270)

0.1214*** (0.0735) !0.0378 (0.0404) 0.0931* (0.0247)

0.0844* (0.0090)

0.2118** (0.0958) !0.0332 (0.0279)

0.1853** (0.0895) !0.0248 (0.0261)

0.1699* (0.0281) 0.0282* (0.0060)

0.2343* (0.085) 0.0434* (0.0079) !0.0003** (0.0001)

0.2168* (0.0801) 0.0430* (0.0077) !0.0003* (0.0001)

Q2

Regime 1 Constant Q Q2

Regime 2 Q Q2

Regime 3 Constant Q Q2

Threshold 1 (percentile) Threshold 2 (percentile)

0.97 (45) 2.52 (90)

1.19 (60) 2.52 (90)

1.19 (60) 2.52 (90)

Notes: *Significant at the 1% level; **significant at the 5% level; ***significant at the 10% level. The dependent variable is Tobin’s q (not tax-adjusted). Columns labeled ‘NI’ or ‘I’ report results using NI/K or I/K (respectively) as the dependent variable. The number refers to the model specification. All regressions include a complete set of firm and year dummies. Reported in parentheses are the standard errors of estimates except for the threshold estimates where it is the corresponding percentile of q. All standard errors are from White’s Heteroskedasticity consistent variance—covariance matrix, except for the estimates under the null model of a single regime which use the OLS variance—covariance matrix.

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Table 5 R-squares Model

Sample

Regime 1

Regime 2

Regime 3

1

0.3587 [0.0133] 0.3606 [0.0071] 0.3582 [0.0141] 0.3601 [0.0077] 0.3582 [0.0127] 0.3599 [0.0063] 0.3578 [0.0136] 0.3524 [0.0070] 0.3358 [0.0095] 0.3385 [0.0045] 0.3704 [0.0050]

0.2293 [0.0074] 0.1852 [0.0009] 0.2304 [0.0082] 0.1947 [0.0006] 0.2425 [0.0117] 0.2423 [0.0083] 0.2442 [0.0112] 0.1652 [!0.0041] 0.2234 [0.0012] 0.2402 [0.0025] 0.2767 [0.0035]

0.2446 [0.0080] 0.2368 [0.0079] 0.2508 [0.0100] 0.2354 [0.0091] 0.2522 [0.0034] 0.2511 [0.0013] 0.2495 [0.0073] 0.2003 [0.0063] 0.1964 [0.0077] 0.1850 [0.0041] 0.2120 [0.0046]

0.3263 [0.0265] 0.3309 [0.0175] 0.3296 [0.0278] 0.3342 [0.0185] 0.3262 [0.0265] 0.3307 [0.0173] 0.3296 [0.0278] 0.3336 [0.0124] 0.3781 [0.0197] 0.3839 [0.0136] 0.4014 [0.0138]

2 3 4 1R 2R 3R 4R NI-1 NI-2 I-2

Notes: The R-squared is defined as the square of the correlation coefficient; specifically the covariance of the actual values with the estimated ones divided by the square root of the product of the variances of the actual and estimated values. This whole quantity is then squared. The first column provides the R-squared for the whole sample under the alternative of three regimes. The number in brackets is the increase in the R-squared over the null of a single regime. The next three columns provide the same comparison for the three parts of the sample that correspond to each regime.

important in the intermediate region where the values of q are, approximately, between 1 and 3. Thus, the finding of concavity of the investment function in this intermediate region should not be due to measurement error. The robustness of the S-shape when we use logarithmic expansions of q suggests that measurement error is also not important in finding reduced responsiveness to q in the high region. Our general conclusions on the shape of the non-linear relation between the investment rate and q do not change when we consider the modified gross investment rate (NI/K). As described in Section 2, NI is constructed by subtracting any capital sales from purchases of capital assets (I). We present the estimates for two model specifications in Table 4 and the R2s in Table 5. The estimates of the coefficients are similar regardless of the dependent variable used

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Table 6 Threshold regressions including cash flows and capital Tobin’s Q

Tax-adjusted Q

Restricted Q Q2 Cash Flows Ln(K)

0.0326* !0.0003* 0.173* !0.1042*

(0.0015) (3.6e!05) (0.0047) (0.0034)

0.0161* 9.0e!05* 0.1741* !0.1042*

(0.0007) (9.6e!06) (0.0047) (0.0034)

ºnrestricted Cash flows Ln(K)

0.1670* !0.1050*

(0.0208) (0.0091)

0.1674* !0.105*

(0.0208) (0.0091)

0.0686* 0.1102* !0.0159

(0.0472) (0.0286) (0.0163)

!0.0395*** 0.0380* 0.0053

(0.0238) (0.0099) (0.0048)

0.1760* !0.0294*

(0.0464) (0.0109)

0.0728* !0.0064*

(0.0154) (0.0023)

0.2268* 0.0223* !0.0002**

(0.0528) (0.0051) (7.4e!05)

0.1687* 0.0112* !4.5e!05**

(0.0325) (0.0025) (2.0e!05)

Regime 1 Constant Q Q2 Regime 2 Q Q2 Regime 3 Constant Q Q2 Threshold 1 Threshold 2 R-squared sample

1.22 3.18 0.4189

(50) (90) [0.0038]

1.76 5.62 0.4183

(50) (90) [0.0040]

Notes: *Significant at the 1% level; **significant at the 5% level; ***significant at the 10% level. The cash flows and Ln(K) variables are each lagged one period. This required excluding the first year of observations leaving a sample with 22,532 observations. See also notes to Tables 2 and 5. The notation 3.6e!05 signifies 3.6]10~5.

even in the regime of low qs where the two measures are most likely to be different. The null is always rejected at the 1% level. It seems, then, that the standard practice in the Tobin’s q literature of using ‘I’ rather than NI as the gross investment variable is valid. This exercise also provides some evidence of stability of our estimates across time. Model 2 in Table 2 is the same as model I-2 in Table 4. The sample, however, for the latter specification is substantially smaller as it does not cover the first 11 years of the Full sample. The estimated coefficients and regime thresholds are similar for each specification across samples.

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5.4. Some caveats We conclude this section with a note of caution. The neoclassical theory of the firm with convex adjustment costs and its extension by Abel and Eberly (1994) implies that marginal q is a sufficient statistic for firm investment (see Eqs. (5) and (5@)). Empirically, however, variables such as cash flows and sales are significantly related to firm investment in linear regressions that control for q (see Fazzari et al., 1988 among others). A natural question is whether these variables are simply proxying for non-linear interaction between investment and q. This does not seem to be the case. We find that even when the relation between investment and q is allowed to be non-linear variables other than q are important as can be seen in Table 6. Cash flows are positively related to investment. Additionally, the size of the firm’s capital stock is negatively related to their investment—capital ratio. This evidence is a simple rejection of the theory together with the assumptions employed to make it operational empirically. It also indicates a possible violation of some identifying assumptions that we maintain in order to make the theory operational empirically. Two assumptions guarantee that investment responds to Tobin’s (average) q and that the unknown and unobservable thresholds are given by values of Tobin’s q. These are that: (A) t ( ) be independent of K and u; and (B) there be a one-to-one I mapping between Tobin’s (average) q and marginal q. To satisfy (B) and the first part of (A) we assumed, as is common, linear homogeneity of the %( ) and t( ) functions. If, for example, (A) does not hold then the thresholds are also a function of K or u, as well as of q. This may explain the variability of threshold qs across specifications.

6. Conclusions Abel and Eberly (1994) gave a characterization of optimal investment behavior when there are convex as well as fixed costs and investment could be irreversible. In this framework, there is a range of q where investment is insensitive to q surrounded by ranges where it is responsive to q. In this paper, we estimate the relationship between investment and q at the firm level allowing it to vary across regimes, which are defined by threshold levels of q. We find evidence in favor of three regimes in the investment—q relationship and significant nonlinearities even within regimes. Investment, however, is sensitive to q in all three regimes. The relationship between firm investment and q is S-shaped, i.e. the responsiveness of investment to q increases with q for low values of q then decreases for intermediate values of q and eventually flattens out. At the median q, as well as most other values, the estimated magnitude of responsiveness is considerably higher than that obtained in the linear specification that is standard in the

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literature. In order to evaluate the responsiveness of aggregate investment to fundamental factors within this empirical framework one needs to consider the entire distribution of I/K and q across firms. This leads to estimates of the elasticity of aggregate investment that are highly variable from year to year due to variations in the distribution of q over time. The reduced-form finding of a non-linear S-shaped relationship between investment and q should be useful in formulating and evaluating structural models of firm investment. Abel and Eberly (1996) report that the S-shaped investment function is consistent with a model of firm investment in heterogeneous capital goods each subject to separate fixed costs and convex costs. Future work could be directed at developing and estimating models of firm investment in different plants where the structure of adjustment costs differs across plants and the investment function is S-shaped at the firm level. Finally, the methodology we employ in this paper can be applied to test other investment models with thresholds. For instance, it can be used to detect whether different regimes of responsiveness of investment to fundamentals arise from the presence of financing constraints or non-convex costs.

Acknowledgements We would like to thank seminar participants at Duke, the Universities of Maryland, and Virginia, the Applied Micro Workshop at the Federal Reserve Bank of Cleveland, the Federal Reserve Board, and the 1995 NBER Summer Institute (EFCCL), John Haltiwanger, Kevin Hassett, and an anonymous referee for helpful comments. We are grateful to Kevin Hassett for providing us with data. The views expressed are those of the authors and do not necessarily represent those of the Fund. We take responsibility for any errors.

Appendix A. Estimation and testing procedure A.1. Choice of parameter space for C The threshold variable in our application is Tobin’s q. Three regimes are to be estimated implying that there are two threshold variables, c and c . The first l h problem is to choose the parameter space for these variables. The parameter space for C is two dimensional and consists of all couples (c , c ) between the l h tenth and the ninetieth percentile of the empirical distribution of q in increments of 5%, such that c is at least ten percentiles greater than c . This results in h l C containing 120 elements. In smaller samples it may be feasible to try every possible point in the empirical distribution of the threshold variable. For example, if there were two regimes and 100 observations the entire empirical

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distribution would involve only 100 points. However, Hansen (1996) cites a recommendation of Andrews (1993) that at least 15% of the observations be left in each tail to avoid problems with singularities, implying that only 70 points need to be ‘scanned’. Due to the large number of observations in our sample we believe that leaving 10% in each tail is reasonable. At the same time, it is computationally infeasible to scan over every point in the data thus we resort to a grid search. All the results are still valid when a discrete approximation to C is made, however the results become conditional on the subspace of C chosen. We believe that the chosen space is sufficiently ‘dense’ in C, however we cannot rule out the possibility that there may be a ‘peak’ at a particular point in C which our space excludes.

A.2. Estimation and calculation of test statistics The estimation of the model for each point in C is relatively straightforward. First the design matrix Z(c) is constructed, leaving a model which is linear in the remaining parameters. These parameters h(c) and B(c) are estimated with OLS and the SSR(c) is calculated. In addition, at each point three statistics, M¼)0.(c), ¼8-$(c), ¼-. (c)N, are also calculated and stored. Each statistic is constructed as the standard Wald statistic, with the latter two using a heteroskedasticity-robust VC matrix (White, 1980) based on the unrestricted and restricted residuals respectively. These are used as part of the testing procedure. The above steps are repeated for each point in C, the parameter estimates corresponding to the point in C that minimizes the SSR are selected. The variance of the disturbance is estimated and the standard errors and p-values for individual parameters (with the exception of the nuisance ones) are calculated using a heteroskedasticityrobust VC matrix. We construct two types of test statistics, H and H , utilizing the three !7' 461 statistics calculated for each point in C. H (H ) is the average (supremum) !7' 461 over the parameter space C for each of the three Wald-type tests mentioned earlier. Let C consist of N elements and define ¼ to be a N]3 matrix, where the ith row is [¼)0. (c ), ¼8-$ (c ), ¼-. (c )]. H and H are 3]1 matrices i i i !7' 461 consisting of the column averages and column maxima respectively. These statistics are then compared to the critical values which are generated as described below. There is an additional complication caused by the presence of firm specific dummies. There are over 1500 firms in the sample. This makes the matrix Z(c)@Z(c) more than 1500 by 1500 which is difficult (if possible) to invert. To get around this problem, the model is transformed by taking deviations from firm means. It is easy to verify that this does not alter the calculation of the remaining parameters, standard errors, or test statistics.

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A.3. Calculation of the empirical distribution The empirical distribution for the test statistics corresponding to H and !7' H , F and F respectively are calculated following Hansen (1996). p-values 461 !7' 461 are then estimated from these distributions. The larger the empirical distribution of F the closer it comes to its asymptotic distribution. A major advantage of this method is that the number of elements in F is under the control of the researcher. The distributions of F composed of J points are calculated in the following manner. A vector v the same length as the sample (23,207) of j standardized normal (pseudo-) random numbers is generated. Then for each point in C a statistic ¹(c) is calculated, where ¹(c) equals ¹(c)"v@ K@MR[Z(c)@Z(c)]~1»C(c)[Z(c)@Z(c)]~1R@N~1Kv , j j where K"R[Z(c)@Z(c)]~1Z(c)@ and R is the matrix of restrictions such that R[h(c), B(c)]"0. The rank of R equals the number of restrictions. »C(c) is Z(c)@Z(c)pL 2 in the homoskedastic case. For the heteroskedasticity-robust cases »C(c)"Z(c)@[diag(u2, u2, 2, u2 )]Z(c), NT 1 1 where the u are based on the residuals from the unrestricted or restricted models i as required. Let ¹ be defined as ¼, such that the rows correspond to the points j in C and the columns to the corresponding statistics, [¹)0.(c ), ¹8-$(c ), ¹-.(c )]. i i i The F and F are calculated as the average and supremum of the columns of !7' 461 ¹ , respectively. This procedure is repeated J times to generate the correspondj ing distributions for F and F . Estimates of the p-values can then be !7' 461 generated from this distribution. In other words if J"1000, the 950th highest element of F and F would represent an approximate 95% significance !7' 461 critical value. There is intuitive appeal to this procedure. Suppose that c* were known with certainty. Then the asymptotic distribution of F and of F is Chi-square. To !7' 461 see this note that ¹(c) can be written as v@ Xv . It is easy to verify that X is j j idempotent and symmetric with rank equal to the rank of R (i.e., the number of restrictions) which means that repeated draws of the vector v will converge in j distribution to a random variable with a chi-square distribution. This suggests, that if this technique were followed for a known breakpoint, the researcher would simply be simulating the appropriate chi-square distribution. When C contains more than a single point, the simulated distribution is a function (either the average or supremum) of the points in C for each draw of v . Hansen j (1996) demonstrates that p-values constructed from the simulated distribution approximate the true p-values under the null, with the approximation improving the larger the number of v vectors drawn. j

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