The 'Elusive' Capital-User Cost Elasticity Revisited

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. The Elusive Capital-User Cost Elasticity Revisited Jonathan N. Millar and Brahima Coulibaly 2007-25 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

The (cid:145)Elusive(cid:146)Capital-User Cost Elasticity Revisited Brahima Coulibaly and Jonathan Millar (cid:3) Board of Governors of the Federal Reserve System May 15, 2009 Abstract This paper sheds new light on the estimation of the long-run elasticity of the demand for businesscapital(cid:151) forameasurethatincludesbothequipmentandstructures(cid:151)tochangesinits user cost using a quarterly panel of two-digit manufacturing industries from South Africa from 1970 to 2000. For a variety of regression speci(cid:133)cations, we (cid:133)nd highly signi(cid:133)cant estimates of the user cost elasticity in the vicinity of 1:0 as implied by a Cobb-Douglas production (cid:0) function. These estimates contrast sharply with many previous studies that obtained small and/orstatisticallyinsigni(cid:133)cantestimatesoftheusercostelasticity. Thisdi⁄erencein(cid:133)ndings may owe to the fact that the capital demand curve is better identi(cid:133)ed in a small open economy because shocks to capital supply are more likely to be exogenous. The economic embargo imposedonSouthAfricafrom1985toearly1994temporarilyforceditseconomytobecomemore closed and therefore provides a unique opportunity to assess the importance of identi(cid:133)cation in theestimationoftheusercostelasticity. We(cid:133)ndthattheestimatedmagnitudeoftheusercost elasticityisconsiderablysmallerovertheembargoperiod. These(cid:133)ndingssuggestthatthetrue elasticityisinthevicinityoftheCobb-Douglasbenchmark,andthatidenti(cid:133)cationisimportant to uncovering this estimate. Keywords:usercostelasticity;(cid:133)xedinvestment;capitalaccumulation;priceofcapital;interest rate JEL classi(cid:133)cations: C23, E22, E44, E62 Coulibaly: Economist, Division of International Finance, Board of Governors of the Federal Reserve (cid:3) System,Washington,DC,20551;Millar: Economist,DivisionofResearchandStatistics,BoardofGovernors of the Federal Reserve System, Washington, DC, 20551. An earlier version of this paper was entitled, "Estimating the Long-Run User Cost Elasticity of Capital for a Small Open Economy: Evidence Using Data fromSouthAfrica." WethanktheSouthAfricanTradeandIndustrialPolicyStrategies,andCoenPretorius, Nthabiseng Molemoeng, Stefaans Walters of the Reserve Bank of South Africa for providing the data set used in the study. We would also like to thank Masao Ogaki and other seminar participants at Ohio State University, the Midwest Economics Association Meetings, and the Board of Governors of Federal Reserve System for useful comments and suggestions. The views expressed in the paper are those of the authors and do not necessarily re(cid:135)ect those of the Board of Governors or the Federal Reserve System. 1

1 Introduction Economists have long had a keen interest in knowing the degree that businesses wish to adjust capital holdings in response to shifts in the supply of capital(cid:151)the user cost elasticity of capital. The attention paid to this parameter is well justi(cid:133)ed, as its magnitude is of central importance for calibrating macroeconomic models that, in turn, are used for forecasting, for evaluating economic hypotheses, and examining (cid:133)scal and monetary policy alternatives. For example, this elasticity is relevant for assessing the e⁄ectiveness of investment tax incentives, such as the bonus depreciation allowancesenactedbytheU.S.federalgovernmentin2002, 2003and2008. Despiteitssigni(cid:133)cance, econometricianshavegenerallyfounditdi¢ culttoidentifythiselasticityinempiricalwork, leading to estimates that vary substantially in magnitude from study to study. This re(cid:135)ects the familiar econometric challenge of identifying the slope of a demand curve in an environment where both demand and supply can be shifted by sources that cannot always be isolated.1 This challenge is complicated further in this context by adjustment costs(cid:151)both internal and external. These costs prolong the period needed for capital to fully adjust to a given disturbance and tend to make the demand response depend on the anticipated persistence of the shock (Tevlin and Whelan [2003]).2 Recently, new methods have been applied to this topic that point to a fairly large aggregate response of capital demand to changes in capital supply. Much of this work has followed from the insightsofCaballero[1994],whoarguedthattheusercostelasticitycanbeidenti(cid:133)edmoree⁄ectively using a cointegration approach that emphasizes the low-frequency variation in the data, thereby de-emphasizing transitory distortions associated with adjustment costs and possibly sidestepping endogeneity problems. Using this strategy for aggregate data for equipment capital, Caballero [1994]andSchaller[2006]obtainstatisticallysigni(cid:133)cantestimatesofthelong-runusercostelasticity that are close to the Cobb-Douglas benchmark of 1:0.3 However, the estimates in these studies (cid:0) becomeinsigni(cid:133)cantwhenstructuresareincludedinthemeasureofcapital. Thisisnotanegligible omission, as structures constitute a huge fraction of the nominal stock of business capital, and 1In principle, this problem could be addressed using instrumental variables. As noted by Hassett and Hubbard [2002], conventional instrumental variables, such as lagged endogenous variables or sales-to-capital ratios, have not proven successful. 2Thepresenseof(cid:133)xed costsand thedistribution ofcapitalacross(cid:133)rmscan alsoa⁄ectalsotheresponseofcapital changes in the user cost (Caballero, Engel and Haltiwanger [1995]). 3Anumberofotherpapershavealsoreportedsigni(cid:133)cantnegativeestimatesoftheusercostelasticityfor(cid:133)rmlevel data. These include Cummins, Hassett and Hubbard [1994], Chirinko, Fazzari and Meyer [1999], Chrinko, Fazzari and Meyer [2004], Guiso et al. [2002] and, more recently, Gilchrist and Zakraj(cid:154)ek [2007]. 2

therefore should be an important component of the overall response of capital to changes in its user cost.4 More recent work by Smith [2007] uses cointegration methods for a panel of United Kingdom industries, but with a measure of capital that includes both equipment and structures. He (cid:133)nds a user cost elasticity that is substantially smaller(cid:151) around 0:40. Using a stationary (cid:0) speci(cid:133)cation and aggregate data for the United States, Tevlin and Whelan [2003] (cid:133)nd estimates of the user cost elasticity around 0:20. (cid:0) This study revisits the subject using a unique quarterly dataset of manufacturing industries from South Africa for the period between 1970 and 2000. We think that the South African experience over this period is particularly pertinent to the user cost elasticity debate. The country is su¢ ciently small and open that its economy likely takes interest rates and capital goods prices as given (Schaller [2006]), and is su¢ ciently isolated that we can conjecture that these prices are also exogenous. This allows us to bypass some of the challenges posed by endogeneity. Also, our industry level dataset has a large number of observations and a cross-sectional dimension, which allows us to control for sources of endogeneity stemming from latent aggregate and industry e⁄ects andmaymakeourresultslesssusceptibletosmallsamplebiasthansomepreviousstudies. Finally, the embargo imposed on South Africa provides a unique opportunity to assess the importance of endogeneityforestimates of theusercostelasticitysince the embargoforced the country(cid:146)seconomy to transition from open toward autarky, and then back to open. Under our working assumption that the user cost is exogenous in a small and isolated open economy, we would expect the user cost elasticity to be smaller during the embargo period(cid:151) when the endogeneity problem was likely more severe(cid:151)than in the non-embargo periods. Usingbothcointegrationanddistributedlagspeci(cid:133)cationsthatarederiveddirectlyfromoptimal investment behavior in the presence of adjustment costs, we (cid:133)nd highly signi(cid:133)cant estimates of the user cost elasticity in the range of 0:80 to 1:0. In most cases, these estimates are statistically (cid:0) (cid:0) indistinguishable from the Cobb-Douglas benchmark of 1:0. Unlike previous studies (Caballero (cid:0) [1994] or Schaller [2006]) that also (cid:133)nd user cost elasticity estimates in this range, our estimates are for measures of business (cid:133)xed capital that includes both equipment and structures. To our knowledge, this study is one of the (cid:133)rst to document such a large user cost elasticity for a broad 4According to estimates from the Bureau of Economic Analysis for the postward period, structures on average accounted for almost two-thirds of the nominal stock of private nonresidential (cid:133)xed capital and about one-third of nominal private (cid:133)xed investment in the United States. 3

measure of business capital that includes both equipment and structures, and the (cid:133)rst to show that similar estimates can be obtained using both stationary and cointegrating regression speci(cid:133)cations. Interestingly, we (cid:133)nd that controlling for the e⁄ect of the embargo period(cid:151) when the economy was less open and more likely to have interest rates and prices of capital goods that are determined endogenously(cid:151) leads to a statistically signi(cid:133)cant increase in the absolute value of the user cost elasticity estimate. The estimated elasticity during the embargo period is much smaller, in the range of estimates found in the majority of previous studies. These latter results are particularly intriguingas theysuggestthe importance of endogeneityas a possible explanation forwhyprevious studies(cid:151)which largely employ data from large economies(cid:151)have often failed to obtain estimates of the user cost elasticity of a substantial magnitude. 2 The Embargo: Some Background As mentioned above, a key feature of the South African economic history that we exploit in this study is the country(cid:146)s unique revision toward autarky that began around 1985 and ended early in 1994. During this period, the world imposed economic sanctions on South Africa to encourage an end to its apartheid regime(cid:151)a political system that granted di⁄erent rights to citizens based on race. As a result of the embargo, several foreign public and private entities operating in South Africa decided to disinvest and/or stop making new investments or reinvestments of earnings in the country.5 In addition to these restrictions on capital (cid:135)ows, several countries also restricted or banned trade with South Africa. These restrictions limited the country(cid:146)s ability to trade goods and (cid:133)nancial claims with the rest of the world. South Africa(cid:146)s trade-to-GDP ratio dropped from an average of 23 percent during the preembargo period to an average of 19 percent during the embargo, then snapped back to an average of25percentaftertheembargowaslifted. Thecountry(cid:146)scurrentaccountbalance, showninFigure 5The International Monetary Fund estimated that the embargo cost South Africa $8 billion in foregone foreign investment between 1985 and 1991, about 3 percent of the country(cid:146)s cumulative GDP from 1985 to 1991. The U.S. GeneralAccountingO¢ ce(GAO)estimatedthat$10.8billion(cid:135)owedoutofSouthAfricafromJanuary1985through June 1989, including $3.7 billion in loan repayments to banks, $7.1 billion in other debt repayments and capital (cid:135)ight(GAO1990,12,17). Similarly,TrustBank(aSouthAfricancommercialbank)calculatedthatthecountryhad forgone nearly $14 billion in loans and direct investments between 1985 and 1990 in comparison to what loans and directinvestmentswouldhavebeenhadmoney(cid:135)owedinattheratesthathadprevailedbefore1985(TheEconomist, 10 February 1990, 69). 4

1, also follows a pattern consistent with these restrictions.6 Before 1985, the country registered current account de(cid:133)cits that averaged 2 percent of GDP. However, when economic sanctions intensi(cid:133)ed between 1985 and 1993, the current account balance swung to a surplus that averaged about 2.4 percent of GDP. When economic sanctions were lifted in early 1994, the current account balance reversed again to a de(cid:133)cit as the country re-integrated into the world economy.7 For estimation purposes, we interpret the beginning of the embargo as September 1985, when o¢ cial sanctions were enacted against South Africa by the European Community and the United States, andtheendoftheembargoasApril1994, whenthecountryheldits(cid:133)rstall-racedemocratic elections.8 ThesedatesarealsoconsistentwiththeswingsinSouthAfrica(cid:146)scurrentaccountbalance discussed above. 3 Theoretical Motivation We assume that each industry can be represented by a forward-looking representative (cid:133)rm that operates in perfectly competitive markets and that faces internal costs for adjusting its capital stock. Each of these (cid:133)rms maximizes its market value by choosing its labor input for the current period and its capital stock for the following period. The investment decision balances costs of adjustment against the costs associated with deviating from capital holdings that would be optimal in the absence of internal adjustment costs. As a preliminary step, we de(cid:133)ne frictionless capital as: y i;t k y (cid:27)u +((cid:27) 1)ak = 1 (cid:27) ((cid:27) 1) u (1) i(cid:3);t i;t i;t i;t 2 i;t 3 (cid:17) (cid:0) (cid:0) (cid:0) (cid:0) h i aK 6 i;t 7 4 5 for industries i = 1;:::;N, where y and u are log of output and log of the user cost for i;t i;t frictionless capital in industry i, and (cid:27) is the user cost elasticity of capital.9 The (cid:133)nal term (cid:0) 6Detailed historical accounts of the economic embargo and the disinvestment can be obtained from the Institute forInternationalEconomicswebsiteathttp://www.petersoninstitute.org/research/topics/sanctions/southafrica.cfm. 7See Coulibaly [2009]. 8Shortly after these elections, the United Nations adopted a resolution for all of its members to end economic sanctions against the country. 9Thisequationisinspiredbythestandard(cid:133)rstorderconditionforcapitalforacasewheretherearenoadjustment frictions and where the production function takes the following constant elasticity of substitution (CES) form: (cid:27) F(AK;AL ;K ;L )= ! AKK (cid:27) (cid:0)(cid:27) 1 +(1 ! ) AL L (cid:27) (cid:0)(cid:27) 1 (cid:27) (cid:0) 1 ; i;t i;t i;t; i;t i i;t i;t (cid:0) i i;t i;t (cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21) 5

involves ak , which summarizes the extent that technology directly augments capital in production. i;t If such a term exists, we assume it is known in the current period by (cid:133)rms, but is not observed by econometricians. The frictionless user cost in each industry is given by: E (cid:1)pK U = pK 1 (cid:0) (cid:28)z t r f +(cid:16) +(cid:14) t (cid:0) 1 i;t (2) i;t i;t (cid:0) 1 (cid:18) 1 (cid:0) (cid:28) (cid:19) 2 t (cid:0) 1 i (cid:0) pK i h ;t 1 i3 (cid:0) 4 5 where (cid:28) is the corporate tax rate, pK is the real price of capital goods, z is the present value of t i;t f the depreciation allowances associated with one unit of capital, r is the risk free real interest rate, t (cid:16) is an appropriate risk premium, and (cid:14) is the depreciation rate.10 i We assume that the determinants of k in equation (1) in each industry evolve according to i(cid:3);t a joint stochastic process that competitive (cid:133)rms treat as given.11 A key issue for identi(cid:133)cation is whetherthereiscapital-augmentingtechnologicalprogresssothataK showsupasadeterminantof i;t k . Ourspeci(cid:133)cationallowsfortechnologytohaveindependente⁄ectsonallfactorsofproduction, i(cid:3);t thereby nesting as special cases a number of important alternatives. One special case is Hicks- Neutral technological progress, where a common technological factor a = aK = aL augments all it i;t i;t inputs to the same extent. A second important special case is for technology to solely augment laborsothataK = 0. AsKing,Plosser,andRebelo[1988]show,thiscaseistheoreticallyappealing i;t because it allows for the existence of a balanced growth steady state. A third possibility is that production takes a Cobb-Douglas form, which imposes the restriction that (cid:27) = 1 so that the (cid:133)nal term involving the unobserved technology factor drops out of equation (1). A (cid:133)nal, more general, possibility is a case where both capital- and labor-augmenting technology are allowed to have distinct trajectories. We assume that the representative (cid:133)rm in each industry i seeks to maximize the present value of its cash (cid:135)ows by choosing optimal trajectories of capital and labor in the face of adjustment where(cid:27)istheelasticityofinputsubstitution,K isthelevelofcapital,andL isthelevelofthevariableinput, AK i;t i;t i;t and AL represent the degree that technology augments capital and labor, respectively. For simplicity, we suppress i;t all constants and industry (cid:133)xed e⁄ects. 10We include a risk premium in the cost of capital to allow for the possibility that the (cid:133)rm(cid:146)s stakeholders are risk averse, so that they require additional compensation for variation in returns around their expected value. 11In the context of our competitive model, an individual (cid:133)rm(cid:146)s choice of output is essentially determined by its predeterminedlevelofcapitalandbythegivenmarket-determinedwagesthatdeterminesitsusageofvariableinputs. 6

costs for capital: 1 R t+j E (1 (cid:28) ) F (K ;L ) w L pk [I +J(K ;:::;K )] t t t+j i;t+j i;t+j i;t+j i;t+j i;t+j i;t+j t+1+i t+1+i M (cid:0) f (cid:0) g(cid:0) (cid:0) X j=0 n o (3) where the optimization is subject to the capital accumulation constraint K = K (1 (cid:14) )+ i;t+1 i;t+1 i;t (cid:0) I . In this formulation, w is the given real market wage, 0 < R t+j 1 is a risk-adjusted factor i;t i;t t (cid:20) pK that discounts between periods t and t + j, and pk = t+j(1 (cid:28) z ) is the real price of i;t+j pY (cid:0) t+j t+j t+j capital after deducting the present value of capital depreciation allowances. J( ) is a convex and (cid:1) linearly homogeneous function that captures internal costs associated with adjusting the capital stock. Since adjustment costs should mainly be an issue when the capital stock is not static, we restrict this function so that, for any (cid:133)xed K > 0; J(K;:::;K) = 0 and @ J(K;:::;K) = 0 for all @Kj j. This function is a generalization of the usual adjustment cost function in which M = 1, which implies that (cid:133)rms choose their path of investment in order to smooth capital growth over time. As argued by Tinsley [2002], costs could be a function of many lags of capital growth and, a priori, it is di¢ cult to rule out cases where (cid:133)rms smooth capital adjustment according to criteria that put weight on higher-order changes. The functional form shown above is su¢ ciently (cid:135)exible that it allows for these more-general forms of smoothing in capital accumulation while still allowing for the more familiar case where M = 1.12 InAppendixA, wegeneralizetheapproachdescribedinTevlinandWhelan[2003]toshowthat, up to a linear approximation, the optimal capital stock will follow a distributed lag of the form: y i;t k = 1 (cid:27) ((cid:27) 1) G(B) u (5) i;t+1 2 i;t 3 (cid:0) (cid:0) h i aK 6 i;t 7 4 5 where G(B) = G +G B +G B2 +:::; is a matrix polynomial in the backshift operator B, and 0 1 2 G is a 3 3 matrix for any whole number j.13 This equation serves as the structural motivaj (cid:2) tion for the regression speci(cid:133)cations in the remainder of the paper. The matrix lag polynomials 12For concreteness, one example of an adjustment cost function that satis(cid:133)es these properties is: J(K ;:::;K )=K M (cid:18)m (1 B)m (cid:1)Kt+1 2 ; (4) t+1 t+1 (cid:0) M t m=1 2 (cid:0) Kt h (cid:16) (cid:17)i where Bis the backshift operator. This form clearly nePsts the familiar case of capital adjustment costs (M = 1), but also allows for costs that entail higher-order smoothing(cid:151)such as adjustment costs that impose direct costs for changing the investment rate (M =2). 13Note that these matrices are not generally diagonal. This is because current and lagged values of a given 7

that multiply each of the right-hand-side variables trace out, in reduced form, how the capital stock evolves in response to innovations to each of the determinants of frictionless capital. The attributes of these responses depend not only on the long-run frictionless elasticities, but on the matrix lag polynomial G(B) which, in turn, re(cid:135)ects both the magnitude of adjustment costs and the anticipated persistence of shocks to fundamentals. These two characteristics have important implications for estimation. For example, Chirinko, Fazzari and Meyer [1999] and others have recognized that proper identi(cid:133)cation of the user cost elasticity should allow for the possibility that capital may not fully re(cid:135)ect the e⁄ects of a given shock to the user cost elasticity for quite some time. Also, (cid:133)ndings in Tevlin and Whelan [2003] suggest that long-run changes to the capital stock are driven primarily by shocks that are more persistent.14 This latter issue is particularly important for identi(cid:133)cation. To illustrate using equation (5) above, let g (B) denote element (l,h) lh of the matrix polynomial G(B). In the absence of restrictions on G(1), the long-run coe¢ cient on the user cost is g (1) (cid:27)g (1)+((cid:27) 1)g (1), so that the long run response of the user cost 12 22 32 (cid:0) (cid:0) does not identify the parameter (cid:27) unless g (1) = 1 and g (1) = g (1) = 0. In Appendix B, we 22 12 32 extend results in Tevlin and Whelan [2003] to show that when the process for a given fundamental has a unit root, the frictionless elasticity corresponding to that variable is at least partially identi(cid:133)ed. When all the frictionless fundamentals have unit roots, then G(1) = I and all three of the 3 frictionless demand elasticities(cid:151)if these variables were observable(cid:151) could be identi(cid:133)ed using their long-run responses. In this model, the potential presence of the unobserved factor aK means that cointegration i;t between capital, output and the user cost may not hold in general, but may hold for the special cases (discussed above) in which this term disappears.15 If cointegration does hold, then a levels speci(cid:133)cation could be a particularly e¢ cient identi(cid:133)cation strategy because the estimated parameters from this regression are super-consistent even in the presence of endogeneity. However, if cointegration fails to hold, then estimates using levels speci(cid:133)cations could yield spurious results.16 fundamentalmayhelppredictthefuturevaluesotherfundamentalsthatbecomerelevantdeterminantsofinvestment in the face of sluggish adjustment. 14Simulation evidence in Caballero [1994] shows that adjustment costs can be a huge source of small sample bias in such regressions for the timespans similar to what we normally observe in the data. 15It will also hold if aK is stationary. i;t 16The empirical evidence for a cointegration speci(cid:133)cation of this nature is mixed. Using aggregate U.S. data, Tevlin and Whelan [2003] cannot reject no cointegration for speci(cid:133)cations using equipment capital. By contrast, Schaller [2006] (cid:133)nds evidence for cointegration for equipment capital after adjusting their estimates to account for small sample bias. Though Caballero [1994] and Smith [2007] use levels speci(cid:133)cations, they do not formally test for 8

We regard the potential presence of such a relationship as an empirical issue, and test the null of no cointegration using formal panel cointegration tests. In addition, we estimate the user cost elasticity using di⁄erence speci(cid:133)cations that do not rely on cointegration. As one would expect, identi(cid:133)cation in these speci(cid:133)cations hinges on our ability to isolate exogenous movements in the user cost while controlling for changes in output(cid:151)the relevant shift factor for demand. We think that our South African data may be particularly useful in this regard, because the user cost is likely to be exogenous during the non-embargo portion of our dataset. Taken together, estimates using these alternative approaches should provide a fairly robust sense of the range of elasticity estimates that can be supported by the data. 4 Data Our dataset consists of a quarterly panel of 24 two-digit manufacturing industries over the period from 1970:Q1 to 2000:Q4.17 Quarterly industry-level estimates of the real capital stock (K ), it (cid:133)xed investment, gross value added (Y ), consumption of (cid:133)xed capital, and industry-speci(cid:133)c price it de(cid:135)atorsforinvestment(pK)andoutput(pY)wereobtainedfromSouthAfricaTradeandIndustrial it it Policy Strategies (TIPS). Quarterly data for prime borrowing rates (r ) and the average corporate t tax rate ((cid:28) ) were obtained from the South African Reserve Bank. The user cost of capital for each t industry in each quarter (U ) was calculated using equation (2). The cost of capital component of i;t the user cost (the bracketed term in the user cost equation) is calculated as the sum of the nominal borrowing cost in the preceding quarter (r ), a (cid:133)xed risk premium of 10 percentage points, the t 1 (cid:0) estimated depreciation rate ((cid:14) ), less a proxy for anticipated capital gains for investment goods i;t (E (cid:1)pk =pk )inthatquarter.18 Foreachindustry,weproxyforexpectedcapitalgainsusing t 1 i;t i;t 1 (cid:0) (cid:0) the cohnditioinal forecast from an OLS regression that projects the four-quarter rate of increase in the investment price de(cid:135)ator onto variables in the time t information set: Namely, current and cointegration. 17We excluded four industries from our sample (tobacco, leather products, glass products, and communications equipment) because their investment data were questionable or did not exist. Taken together, these four industries account for an average of about 31percent of quarterly nominal output and about 13percent of the nominal capital 2 4 stock for the manufacturing sector during our sample. 18Depreciationratesforeachindustryineachquarterwerecalculatedbydividingtheconsumptionof(cid:133)xedcapital by the capital stock at the end of the previous quarter, then converting this (cid:133)gure to an annual rate. 9

lagged values of the nominal interest rate and lags of the dependent variable.19 The capital price component of the user cost (pk ) is formed as the ratio of the industry(cid:146)s investment de(cid:135)ator and i;t its output de(cid:135)ator, and is multiplied by our estimates of the relevant tax terms.20 Figures2and3showtime-seriesplotsofthecapital-outputratioandusercostsforeachindustry in our panel. Note that all three of our primary variables of interest (capital, output and the user cost) vary both in the time and cross-section dimensions. In turn, since corporate tax rates and risk-free rates do not vary across industries, cross-sectional variation in the user cost owes almost entirely to di⁄erences across industries in the relative price of capital, anticipated capital gains, and capital depreciation rates.21 In reality, our user cost measure may miss some variations in the user cost stemming from changes in the risk premium, which could vary idiosyncratically over time and across industries and may not be adequately re(cid:135)ected in our measure of borrowing costs. But we think it is quite likely that risk premiums are not integrated processes, so this variation is not an issue(cid:151)at least asymptotically(cid:151)in our levels speci(cid:133)cations. In the di⁄erence speci(cid:133)cations, we attempt to limit the in(cid:135)uence of potential variations in the risk premium by di⁄erencing, and by controlling for (cid:133)xed and aggregate e⁄ects. 19We use the four-quarter rate of change, rather than the one-quarter change that matches the frequency of our sample,inordertostatecapitalgainsatanannualfrequencyandtominimizevariationsowingtopriceseasonalities. Our proxy for expected capital gains is large enough to make the cost of capital negative in some quarters for a few industries. Negative user costs are ruled out by optimization and, from a more practical standpoint, make it impossible to take logs. To deal with this problem, we chose a fairly large risk premium. This ruled out negative user costs for all observations but the furniture industry between 1980Q3 and 1981Q1(cid:150)when the investment price de(cid:135)ator grew at an annual rate of more than 30 percent. In order to maintain a balanced panel, we set the cost of capitalforthesethreeobservationstotheaverageofthevaluesin1980Q2and1981Q2. Droppingtheseobservations has no meaningful e⁄ect on our estimates. 20We calculate the the present value of future depreciation allowances in each year z using the formula: z = it t rt (cid:14) + i; (cid:14) t i;t = 1 (cid:14) i;t (1+r t )(cid:0) j(1 (cid:0) (cid:14) i;t )j (cid:0) 1 Notethatthisformulaimplicitlyassumesthat(cid:133)rmsexpectinterestrates,the j=1 X corporate tax rate, and the rate of depreciation to remain constant in the coming periods(cid:150)i.e. all tax changes are surprises. 21Itispossiblethattaxescouldstilldrivecross-sectionalvariationsintheusercostduetocross-industrydi⁄erences in the depreciation rate (which factors into z ). But industry depreciation rates vary little across time, so that a t (cid:133)rst order approximation, (cid:133)xed and aggregate e⁄ects should temper the e⁄ect of this variation on our estimators. 10

5 Estimation and Results 5.1 Unit Root Tests Before proceeding to estimation, we formally test for unit roots in our measures of capital, output, and the user cost. As mentioned earlier, unit roots are necessary to identify frictionless elasticities from long-run responses. We also test for a unit root in the ratio of capital to output, which is a precondition for a cointegrating relation between capital, output, and the user cost. Starting with the single time series tests, Figure 4 shows the results of Dickey-Fuller GLS tests by industry.22 To correct for small-sample size distortions, we augmented these equations with lag di⁄erence terms using the lag selection criterion described in Ng and Perron [2001] to choose an appropriate lag order.23 These tests fail to reject unit roots at 5 percent signi(cid:133)cance in all but six of the twentyfour industries in our panel. At 10 percent signi(cid:133)cance, we fail to reject unit roots in all but nine industries. A potential issue for our analysis is the presence of structural breaks. In our dataset, there is a strong rationale to believe that there may be structural breaks at the time that the embargo was imposed and removed. These structural breaks, if present, could make it di¢ cult to draw inferences about the existence of unit roots, an important characteristic for our estimation. To assess the e⁄ect of potential structural breaks, we conduct unit roots tests following the procedure in Clemente, Montaæes and Reyes [1998], which is robust for two structural breaks. Figure 5 shows the results of this test for each industry in our sample. The break-robust tests fail to reject a unit root for capital, output, and the capital-to-output ratio. For the user cost, the tests fails to reject a unit root at 5 percent signi(cid:133)cance for all but two of the industries in our panel(cid:151)close to what we would expect from type I error.24 We also tested for the presence of unit roots using panel tests, which have the additional bene(cid:133)t 22Theestimationequationsforeachtestincludedadrifttermandareaugmentedwithlaggeddi⁄erencestocorrect for small-sample size distortions. 23We(cid:133)tARMA(1,1)processestotheusercostofeachindustryandfoundthataboutone-halfoftheindustrieshad negative estimated MA coe¢ cients. As is well understood, this property tends to cause unit root tests to overreject the null when the estimation equation includes an insu¢ cient number of lagged di⁄erence terms. Ng and Perron [2001] show that their modi(cid:133)ed information criterion is much more e⁄ective than other lag-selection criteria(cid:150)such as the AIC and SIC(cid:150)for mitigating this problem. 24Weobtainedverysimilar(cid:133)ndingswhenweusedthetheZivot-Andrewstest(AndrewsandZivot[1992]),whichis robustforonestructuralbreak. Whenestimatingelasticitiesintheremainderofthepaper,wecontrolforaggregate e⁄ects, which should limit the e⁄ect of potential breakpoints on our estimates. 11

of exploiting information contained in the cross-sectional dimension of the dataset. Table 1 shows the results for a number of alternative speci(cid:133)cations. The (cid:133)rst line shows p-values for the panel unitroottestsdevelopedbyPesaran[2007], whichmaintainsanullhypothesisthatagivenvariable has a unit root for all industries against the alternative of no unit root in at least one industry.25 The second line of the table shows p-values obtained by applying the Hadri [1999] test to our full dataset, which maintains the null of stationarity for all 24 industries against the alternative of a unit root. For robustness, we conducted all of these tests both on our full panel of 24 industries and for an "I(1) subsample" of industries in which single-series tests failed to reject the presence of a unit root in the user cost at the 10 percent signi(cid:133)cance level. Taken together, these tests provide support for maintaining that the relevant variables in our full panel have unit roots at the 5 percent signi(cid:133)cance level. However, at the 10 percent signi(cid:133)cance level, the evidence for unit roots in the user cost for all 24 industries in the full panel is mixed. For the I(1) subsample, the panel tests do support the existence of unit roots in the user cost. Since unit roots in the user cost are preconditions for identi(cid:133)cation, we err on the side of caution by also showing elasticity estimates obtained by restricting the dataset to just the I(1) subsample. 5.2 Estimates Using Panel Cointegration Techniques We begin by estimating a cointegrating speci(cid:133)cation between capital, output and the user cost. Equation (5) can be rearranged to obtain: k y = (cid:27) u +e , (6) i;t+1 i;t i i;t i;t (cid:0) (cid:0) where y i;t e = 1 (cid:27) ((cid:27) 1) [G(B) I ] u it i i 3 2 i;t 3 (cid:0) (cid:0) (cid:0) h i aK 6 i;t 7 4 5 is an unobserved cointegrating residual. This speci(cid:133)cation follows previous studies (Caballero [1994], Schaller [2006], Smith [2007], and others) in that it restricts the capital-output elasticity to unity. We estimate the long-run user cost elasticity using this speci(cid:133)cation for the full sample and 25Thereareanumberofalternativepanelunitroottestsavailable,includingthetestsdevelopedbyLevin,Linand Chu [2002], Levin, Lin and Chu [2002], and Im, Pesaran and Shin [2003]. We chose the Pesaran [2007] test because it is more robust than these other tests for generic forms of cross-sectional correlation between the residuals across groups. 12

for our I(1) subsample.26 With output and the user cost containing unit roots, the stationarity condition for the error term is an empirical question that can be veri(cid:133)ed using cointegration tests. In our model, this condition boils down to a claim that the factor aK follows a stationary process i;t or is non-existent.27 Small sample bias remains an important econometric issue when testing for cointegration because our sample may not be su¢ ciently large to overcome (cid:133)nite-sample correlation between the regressors and the structural error term in equation (6).28 We estimate our cointegrating relationship using pooled DOLS (Kao and Chiang [2000]) which assumes homogeneity, and mean-group DOLS (Pedroni [2001]) which allows for heterogeneity in the true elasticity across industries. Both ofthesespeci(cid:133)cationsincludedynamicOLStermsthatcorrectforbiasesthatarisein(cid:133)nitesamples when there is correlation between the error term and our regressors.29 The structural form of the error in equation (6) provides some useful guidance about what variables to include as dynamic correction terms. Speci(cid:133)cally, when the conditions for cointegration hold, the error term will generally include lagged di⁄erences in both the user cost and output. For this reason, we include (cid:133)rst-di⁄erenced lags and leads of both of these variables in all of our speci(cid:133)cations.30 In addition, we include time dummies and (cid:133)xed e⁄ects in order to correct for biases that arise in the presence of contemporaneous correlation between residuals across industries (Pedroni [2001]). 26In our analysis, we found allowing the output elasticity to be freely estimated did not a⁄ect our estimates of the user cost elasticity. The user cost elasticity was also little a⁄ected when the capital-output elasticity to take a wide range of alternative values. 27These conditions ensure that e is stationary as follows. As shown in the Appendix , the (cid:133)rst two columns of i;t G(1)will be the same as the (cid:133)rst two columns of the identity matrix I if the processes for output and the user cost 3 contain unit roots and the right-hand side variables are not cointegrated. This implies that the (cid:133)rst two columns of G(1) I are zeros,so thatthe lag polynomialsin the errorterm thatmultiply the I(1)processesforoutputand the 3 (cid:0) user cost must contain unit roots. Therefore, all of these terms are I(0). By contrast, the lack of a unit root for aK ensures that the third column of G(1) I will not be nonzero, ensuring that the lag polynomials that multiply i;t (cid:0) 3 the I(0)process aK contain no unit roots. Therefore, these terms are also I(0). i;t 28Simulations in Caballero [1994] show that the degree of small-sample bias can be considerable when estimating single-equation cointegrating regressions. We repeated these simulation experiments in a panel context (not shown) and came to a similar conclusion. 29Kao and Chiang [2000] show that estimates of the coe¢ cients in a cointegrating regression from an uncorrected panel OLS estimator have a biased asymptotic distribution. Their simulations show that a pooled dynamic panel OLS(DOLS)estimatorhasonlyasmallbiasforsampleswithcross-sectionandtimedimensionssimilartoourpanel, and that this estimator outperforms alternative estimators such as pooled FMOLS. 30These terms should have no e⁄ect asymptotically. The inclusion of output had very little e⁄ect if we included many leads and lags of our error correction terms, but the estimates tended to be more stable and converged more rapidly with output included. 13

We begin by testing the validity of the cointegrating relation in equation (6) using both our full dataset and the I(1) subsample. We conduct two sets of tests. The (cid:133)rst set is a homogeneouscointegrationspeci(cid:133)cationthatusescointegratingresidualsfrompooledDOLSspeci(cid:133)cations, while the second set is a heterogenous speci(cid:133)cation that uses cointegrating residuals (cid:133)tted using our mean-group DOLS speci(cid:133)cations. All of our tests assess the null hypothesis of no cointegration using test statistics described by Pedroni [1999].31 We conduct these tests for pooled withindimension ("panel") statistics that maintain the null that the residuals in all industries have unit roots against the alternative that these residuals have a common stable autoregressive parameter, and for between-dimension ("group") statistics that maintain the null that the cointegrating residuals in all industries have a unit root against the alternative that the residuals in all industries have stable(cid:151)but not necessarily common(cid:151)autoregressive roots.32 The results, shown in Table 2, show that our tests are able reject the null of no cointegration, and provide fairly strong empirical support for both our homogeneous and heterogeneous cointegrating speci(cid:133)cations. Table 3 reports our user elasticity estimates from these two cointegration speci(cid:133)cations. The (cid:133)rst and second columns of the table report estimates using our full panel of 24 industries and for the I(1) subset of industries, respectively. Results for our pooled DOLS speci(cid:133)cations(cid:151)shown in the top portion of the table(cid:151)all point to highly signi(cid:133)cant estimates of the user cost elasticity that areintheneighborhoodoftheCobb-Douglasbenchmarkof 1:0. Allofthesepooledspeci(cid:133)cations (cid:0) include 25 leads and lags of (cid:133)rst-di⁄erences in output and the user cost, the order of which was determined using a sequential t-test procedure similar to that described by Ng and Perron [1995].33 This number of dynamic correction terms is in line with speci(cid:133)cations used by Caballero [1994] and Schaller[2006]forquarterlyequipmentcapital. Figure6showshowvaryingthenumberofincluded leads and lags a⁄ects our estimates. Estimates of the user cost elasticity(cid:151)shown in the top panels of the (cid:133)gure(cid:151)tended to increase in absolute magnitude as we added more dynamic correction terms, but generally remained in the range of 1:0 for a wide range of possible speci(cid:133)cations. The (cid:0) bottom panels of this (cid:133)gure show that our lag/lead length roughly corresponds with what would 31To calculate the statistics, we used a modi(cid:133)ed version of Pedroni(cid:146)s RATS code, which is available at: http://econpapers.repec.org/software/bocbocode/. 32The(cid:133)rstthreestatisticsareanalogoustothepanelunitroottestsdevelopedbyLevin,LinandChu[2002],while the fourth and (cid:133)fth tests are akin to the panel tests in Im, Pesaran and Shin [2003]. 33Speci(cid:133)cally, we started by estimating a speci(cid:133)cation that included 32 leads and lags of (cid:133)rst-di⁄erenced output and user costs and then tested the joint signi(cid:133)cance of the coe¢ cients on the last included lead and lag of the user cost. If they were not signi(cid:133)cant at 10 percent, we reestimated after dropping one lead and lag. 14

be suggested by the Akaike Information Criterion.34 The bottom portion of Table 3 summarizes our estimation results using group-mean DOLS. This method estimates separate DOLS speci(cid:133)cations for each industry and then forms an estimate of the aggregate elasticity using a weighted average of the industry estimates. To correct for (cid:133)nite-sample bias, we include in each DOLS regression 8 leads and 16 lags of the (cid:133)rst-di⁄erences in output and the user cost.35 We (cid:133)nd a user-cost elasticity estimate of 0:54 for the full panel, (cid:0) but the estimate rises in magnitude to 0:85 for the I(1) subsample. (cid:0) Inthefollowingsection,wepresentandestimateadistributed-lagspeci(cid:133)cation. Themotivation for this alternative speci(cid:133)cation is two-fold. First, it provides some assurance that our results are robust to alternative econometric speci(cid:133)cations. Second, it allows us to test the importance, for estimation, of the exogeneity of shocks to the user cost. We estimate the user cost elasticity during the embargo period and compare it to the elasticity in the non-embargo period. Such an analysis could not be appropriately carried out with the cointegration speci(cid:133)cation which, under the maintained assumptions, should be consistent even in the presence of such endogeneity. 5.3 Distributed Lag Speci(cid:133)cation When shocks to the user cost are exogenous, and both output and the user cost contain unit roots, the (cid:133)rst two columns of the matrix G(1) in equation (5) are identical to the (cid:133)rst two columns of a three-dimensional identity matrix I . Given these assumptions, we can estimate distributed lag 3 speci(cid:133)cations of the form: (cid:1)k = (cid:17) +T +N y (B)(cid:1)y +Nu(B)(cid:1)u +e (7) i;t i t i i;t i i;t i;t forindustries i = 1;:::;N, where (cid:17) isanindustry(cid:133)xede⁄ect, T isadummyforyeartthatcontrols i t for aggregate e⁄ects, and e is the portion of the structural residual that remains after controlling i;t for these e⁄ects. This speci(cid:133)cation is obtained by taking the (cid:133)rst di⁄erence of equation (5), where we let (cid:17) +T +e collectively denote the components of the structural residual: i t it 34We also explored the Schwarz criterion. 35Lag selection results for the homogeneous speci(cid:133)cation suggest that we might consider including more dynamic correction terms. We are reluctant to do so because the asymptotics of the industry-level estimators are solely in the time dimension, which is limited in our dataset. Speci(cid:133)cally, our chosen lead/lag length eliminates 24 of the 117 availabletimeseriesobservationsineachindustry. Theindustry-levelregressionsthenincludeabout50parameters, leaving only about 40 degrees of freedom. 15

[g (B) (cid:27) g (B)+(1 (cid:27) )g (B)](cid:1)aK: 13 i 23 i 33 i;t (cid:0) (cid:0) Under our maintained unit root assumptions, the long run lag polynomials in the above equation take the structural form: y N (B) = g (B) (cid:27) g (B)+(1 (cid:27) )g (B); and i 11 (cid:0) i 21 (cid:0) i 31 Nu(B) = g (B) (cid:27) g (B)+(1 (cid:27) )g (B); i 12 i 22 i 32 (cid:0) (cid:0) where the long-run responses are restricted such that N y (1) = 1 and Nu(1) = (cid:27) . To estimate i i (cid:0) i this equation, we assume that the terms in these lag polynomials become small beyond some (cid:133)nite lag order, so that the long run responses of capital to each fundamental can be obtained by cumulating the estimated parameters of the relevant distributed lag function.36 Though we maintain that innovations to the user cost are exogenous in our data, we take some additional precautions to guard against the possibility that the user cost may not be entirely exogenous. Our panel estimators allow us to control for latent aggregate e⁄ects, which (among other things) protects against variations in the user cost stemming from technological progress that augments capital demand to the same extent in all industries, and latent (cid:133)xed e⁄ects that (among other things) defend against (cid:133)xed di⁄erences in technological growth across industries that might be re(cid:135)ected in their user cost.37 We start by including a large number of lags in our regressions and then apply the sequential t-test procedure described in Ng and Perron [2001] to successively eliminate lags of the user cost that(cid:151)at the margin(cid:151)do not contain statistically relevant information about the overall magnitude of the response.38 Figure 7 shows the results of this lag order selection exercise for two alternative speci(cid:133)cations. The circles that denote the estimate at each lag show the outcome of our sequential t-testatthatlag: Darkcirclesdenotethatthecoe¢ cientonthelastincludedlagcanbestatistically 36Our benchmark speci(cid:133)cation shown does not include any lagged values of the dependent variable as regressors as in some speci(cid:133)cations. Including lags of the depedent variables had little e⁄ect on our results. In principle, the twoapproachesshouldprovideequivalentanswersinalargesampleifthetruedistributedlagrepresentationis"well behaved" in the sense that the lagged responses decay geometrically. Our speci(cid:133)cation has the additional bene(cid:133)ts of allowing for a more general form of the response function. 37As mentioned earlier, these controls may also eliminate measurement error in the user cost stemming from pure time-series and pure cross-sectional variation in the risk premium. 38Speci(cid:133)cally, starting from a large number of included lags, we successively eliminate the last included lag until the estimated coe¢ cient on this last lag is statistically signi(cid:133)cant at 10 percent. 16

distinguished from zero at 10 percent signi(cid:133)cance level. The outcome of these sequential t-tests suggest that we include 27 quarterly lags in our baseline speci(cid:133)cation.39 Table 4 shows estimates from our distributed lag speci(cid:133)cations for both the heterogeneity speci(cid:133)cation that assumes an identical user cost elasticity across industries ("HOM"). We also show results of a seemingly unrelated speci(cid:133)cation that estimates separate elasticities for each industry, and construct the aggregate elasticity by averaging the industry-speci(cid:133)c elasticities ("HET") using asweightseachindustry(cid:146)saverageshareofthetotalnominalcapitalstockoverthesampleperiod.40 The estimated long-run user cost elasticities for these speci(cid:133)cations range between 0:48 (het- (cid:0) erogeneous speci(cid:133)cation) and 0:62 (homogeneous speci(cid:133)cation) for the full sample, both of which (cid:0) are highly signi(cid:133)cant. For the subsample of industries for which unit root in the user cost is more robust, the estimates of user cost elasticity are larger, ranging from 0:77 to 0:83. In the (cid:0) (cid:0) following section, we assess the importance of identi(cid:133)cation for estimation of the user cost elasticity by examining the e⁄ect of embargo on our estimates. 5.4 Economic Embargo, Endogeneity, and User Cost Elasticity Presumably when the economy became less open during the embargo period (1985-1994), one wouldexpectshockstotheusercosttohavebecomelessexogenousbecausetheembargointroduced di⁄erencesbetweenSouthAfrica(cid:146)sdomesticinterestratesandtheworldinterestrates, andbetween domestic and world prices of capital goods. This presumption is borne out in the data. Figure 8 shows the aggregate user cost for (cid:133)xed business capital in the United States (a proxy for the world) and for South Africa.41 Though the composition of the business capital stock likely di⁄ers in these 39When calculating these criterion, we adjust the sample so that it remains (cid:133)xed as we add more lags. We also conduct lag order speci(cid:133)cations using the Akaike and Schwarz information criteria for each number of included lags. The results did not provide additional useful guidance. 40In order to allow for estimation error in these shares, we include in our system a second set of regressions that estimates the time average of the industry shares for each industry in the panel. The formula for each weightedaverage elasticity was then calculated using estimates from these entire set of regressions. The standard errors of the estimated aggregate elasticitywere determined using the delta method where we account for cross-correlation of residuals across the two regressions. 41ForSouthAfrica,weconstructedtheusercostusingavariantofequation(2),wherewesubstitutedtheindustryspeci(cid:133)c price de(cid:135)ators for output and investment with de(cid:135)ators formed by chain-aggregating across all 24 industries in our panel. The U.S. user cost was calculated using BEA aggregates and a quarterly average of the prime lending rate. To ensure that the levels of these two user costs were broadly comparable, we assumed that U.S. capital was also subject to a (cid:133)xed risk premium of 10 percentage points. We use the real exchange rate to converted the U.S. user cost so that it is denominated in terms of South African goods and services. 17

two countries, broad patterns in these user costs suggest that the cost of capital in South Africa became detached from the rest of the world during the embargo period. The contemporaneous correlation between the two user costs series is highly positive both before and after the embargo, butfellsigni(cid:133)cantlyduringtheembargo. Thispatternisconsistentwiththeconjecturethatshocks to the user cost during the embargo period were less in(cid:135)uenced by exogenous factors.42 In light of this, the identi(cid:133)cation problem would be heightened during the embargo period, making it more di¢ cult to estimate the user cost elasticity. We formally test this hypothesis by estimating the user elasticity during the embargo and non-embargo periods. To capture the potential e⁄ect on our estimates of this heightened endogeneity, we augment the lagged-di⁄erence speci(cid:133)cation in the preceding section to include terms that interact the observable explanatory variables with a dummy variable for the embargo period. Our formulation for this regression is: (cid:1)k = (cid:17) +T +Ny(B)(cid:1)y +My(B)(d (cid:1)y ) i;t i t i;t t i;t (8) +Nu(B)(cid:1)u +Mu(B)(d (cid:1)u )+(cid:15) , i;t t i;t i;t so that the embargo a⁄ects the entire long run relationship between capital and its fundamentals, butonlyforobservationsofcapitalgrowththatoccurwithintheembargoperiod. Asinourprevious distributedlagspeci(cid:133)cation, weestimatetheseregressionsusingcontemporaneousobservationsand 27 lags of each the fundamentals (including the interactions with the embargo dummy). Given these estimates, we determine the long-run elasticity of capital with respect to the user cost by calculating Nu(1). The marginal e⁄ect of the embargo-period data on our long-run user cost elasticity is calculated as Mu(1), while the long-run elasticity to during the embargo-period is Nu(1)+Mu(1). Results using this speci(cid:133)cation are shown in Table 5. The (cid:133)rst and second columns show estimates using our full panel of 24 industries and the I(1) subsample. We (cid:133)nd that the magnitude of the user cost elasticity outside of the embargo period increases from the 0:45 to 0:62 range (cid:0) (cid:0) (in previous section) to 0:75. For the I(1) sample, the estimate increases from the 0:77 to (cid:0) (cid:0) 0:83 range to 0:86. Interestingly, the user cost elasticity for the embargo period is signi(cid:133)cantly (cid:0) (cid:0) lower and in the 0:25 to 0:27 range, consistent with estimates in many previous studies that (cid:0) (cid:0) document a small and, often, insigni(cid:133)cant user cost elasticity. These results are consistent with the hypothesis that the user cost became more endogenous during the embargo period, leading to 42These patterns are even stronger when we exclude tax terms from the calculations of the user cost. 18

a greater role of simultaneity bias. These (cid:133)ndings may help explain why studies of the capital-user cost elasticity using stationary speci(cid:133)cations and data from large economies often (cid:133)nd very little role for the user cost in determining the size of the capital stock.43 The similarity results from this speci(cid:133)cation after correcting for the embargo period to those shown earlier from the cointegration speci(cid:133)cation is particularly intriguing and suggests that our estimates are fairly robust across econometric speci(cid:133)cations. To better describe how capital adjusts to innovations in the user cost, we show the marginal and cumulative responses of capital to a one percent increase in the user cost in (cid:133)gures 9 and 10, respectively. These responses are for our embargo-corrected estimates using the I(1) subsample(cid:151) thoseforthefullsamplehavesimilarcontours. Thesemarginalresponsesforourembargocorrected estimatesshowadistincthump-shapethatreachesmaximumresponseatabouttwelvequartersand then gradually attenuates to about zero by the twenty eighths quarter. These responses suggest a rather slow and non-monotonic adjustment process that extends over about seven years. This helps document the importance of focusing on the long-run in order to capture the full response of the capital stock to e⁄ect of a shock to the user cost, and the importance of using a regression speci(cid:133)cation that is (cid:135)exible enough to allow the shape of the response to be non-monotonic over time. Our estimation assumed that the embargo started in 1985 and ended in early 1994. The exact dating of the embargo period could be uncertain as it is with most event studies. The choice of these dates is supported by anecdotal evidence and narratives on the imposition and removal of the embargo as well as in the aggregate economic data shown earlier. Nonetheless, we conduct sensitivity analyses that vary the embargo(cid:146)s starting and ending dates within a two-year window of our chosen dates. Figure 11 shows in the bottom panels, the log-likelihood of various starting and ending dates, and the top panels show the corresponding user cost elasticities. The bottom panels show the log-likelihood of each alternative, holding all else equal, while the top and middle panels show corresponding elasticity estimates for the non-embargo and embargo periods, respectively. Our user cost elasticity estimate is not sensitive to reasonable changes in the starting and ending dates of the embargo as shown in the top panels and our starting date (1985) maximizes the log-likelihood. 43For instance, see the review by Chirinko [1993]. 19

6 Concluding Remarks In this study we estimate a statistically signi(cid:133)cant user cost elasticity between 0:80 and 1:0 (cid:0) (cid:0) which, in most cases, are statistically indistinguishable from the Cobb-Douglas benchmark of 1:0. (cid:0) This study is one of the (cid:133)rst to document such a large user cost elasticity for a broad measure of business capital that includes both equipment and structures, and the (cid:133)rst to show that similar estimates can be obtained using both stationary and cointegrating regression speci(cid:133)cations. One explanation for our ability to identify a large user cost elasticity is that exogenous shocks to the user cost are better identi(cid:133)ed in a small and isolated economy like South Africa. In a closed economy or in a large open economy, the capital stock and the user cost of capital are jointly determined by a domestic demand and supply equilibrium that equates the marginal product and marginal opportunity cost of capital services. This simultaneity introduces inconsistency into estimates of the user cost elasticity. In a small open economy like South Africa, however, shocks to the user cost are largely in(cid:135)uenced by world interest rates and prices of capital goods that the country takes as given. The economic embargo that the world imposed on South Africa between about 1985 and early 1994 forced their economy to revert toward autarky. This provides a unique opportunity to determine the extent that endogeneity attenuates estimates of the capital-user cost elasticity. We (cid:133)nd a robust correlation between the user cost for South Africa and that of the United States (a proxy for world user costs) before the embargo. This correlation falls signi(cid:133)cantly during the embargo period, and goes back up after the sanctions are lifted and South Africa is re-integrated into the world economy. We (cid:133)nd that during the embargo period, when the user cost became less in(cid:135)uenced by exogenous factors, the estimated user cost elasticity fell considerably(cid:151)to magnitudes that are more in line with those obtained in many previous studies. These results underscore the importance of identi(cid:133)cation to uncovering the (cid:145)elusive(cid:146)user cost elasticity of capital. 20

A Appendix: Derivation of Estimation Equation We start by (cid:133)nding a linearized solution for the (cid:133)rm(cid:146)s capital stock for some arbitrary M > 0 for the generalized, linearly homogeneous adjustment cost function J( ) that satis(cid:133)es the restrictions (cid:1) described in the main text. After substituting out investment using the capital accumulation constraint in equation (3), the (cid:133)rst order condition for the capital stock in this case(cid:151)after some rearranging(cid:151)is: @F(AL L ;AK K ) E (cid:1)pK E t+1 t+1 t+1 t+1 = pK r+(cid:14) t t+1 + (9) t @K t (cid:0) pK " t+i # " (cid:2) t (cid:3)# M E pK (1+r)1 mJ (K ;:::;K ) (10) t t+m (cid:0) 1+m t+1+m t+1+m M m=0 (cid:0) P (cid:2) (cid:3) whereJ ( )denotesthepartialderivativeofJ withrespecttoitslth argument,forl = 1;:::;M+1.44 l (cid:1) Noting that the (cid:133)rst term on the right-hand side of this equation is the frictionless user cost U from equation (2) and assuming that F ( ; ) takes the standard CES form with elasticity of t+1 (cid:1) (cid:1) substitution (cid:27), the equation above can be manipulated to form: E Y t+1 (cid:27) 1 AK t+1 (cid:27) (cid:0)(cid:27) 1 = E K t(cid:3)+1 (cid:27) 1 t t 2 K U 3 K (cid:18) t+1 (cid:19) (cid:0) t+(cid:1)1 " (cid:18) t+1 (cid:19) # 4 5 M Et [pk t+m ] (1+r)1 mJ (K ;:::;K ) m=0 pK t (cid:0) 1+m t+1+m t+1+m (cid:0) M = 1+ ; P r+(cid:14) Et [(cid:1)pk t+1 ] (cid:0) pK t wherewehaveusedthede(cid:133)nitionoffrictionlesscapitalfrom(1). GivenanassumptionJ (K;:::K) = l 0forany(cid:133)xedK > 0, thesecondtermontheright-hand-sidesimpli(cid:133)estozeroinano-growthstate where K remains (cid:133)xed at K . Since J( ) is homogeneous of degree one in its arguments, Euler(cid:146)s (cid:3) (cid:1) homogeneousfunctiontheoremensuresthatthepartialderivativesJ ( )arehomogeneousofdegree l (cid:1) zero. Therefore,J (K ;K ;:::;K ;:::;K ) =J Kt+1+m; Kt+m:::;1;:::; Kt+1+m M 1+l t+1+m t+m t+1 t+1+m (cid:0) M 1+l Kt+1 Kt+1 Kt+1 (cid:0) for l = 0;:::;M and m = 0;:::;M. Using this property, we can linear(cid:16)ize around a no-growth state (cid:17) where 1 = Kt+1 = Kt 1 = ::: = Kt+1 M and where E pk =pk = 1+(cid:25)k m45 to obtain: Kt K(cid:0) t Kt (cid:0) t t+m t 44For simplicity, we suppress all industry subscripts, and dro(cid:2)p time subs(cid:3)cript(cid:0)s for (cid:28), z(cid:1), (cid:14) and r. 45The steady state that we require can be described in more detail. Noting that k(cid:3) = y (cid:27)u+((cid:27) 1)aK, (cid:0) (cid:0) it can be shown that the growth rate of the frictionless stock is governed by the following approximation: (cid:1)k t(cid:3) (cid:25) (cid:1)k t +((cid:27) (cid:0) s(cid:3)L )(cid:1)aK t +s(cid:3)L (cid:1)l t +(cid:1)aL t (cid:0) (cid:1)k t (cid:0) (cid:27)(cid:1)u t , where s(cid:3)L is the previous period(cid:146)s labor share and l is the log (cid:0) (cid:1) 21

M M E k k + d (k k )+ d (k k ); (11) t t(cid:3)+1 t+1 m t+1+m t+1 m t+1 m t+1 (cid:25) m=1 (cid:0) m=1 (cid:0) (cid:0) where (cid:2) (cid:3) P P 1+r M m 1+(cid:25)K j d (cid:27) (cid:0) J ; and m (cid:17) r+(cid:14) (cid:25)k 1+r 1(cid:3)+j;m+1+j (cid:18) (cid:0) (cid:19) j=0 (cid:18) (cid:19) 1+r M P m 1+(cid:25)K j (cid:0) m d (cid:27) (cid:0) J for m = 1;:::;M; m (cid:17) r+(cid:14) (cid:25)k 1+r m(cid:3)+1+j;1+j (cid:18) (cid:0) (cid:19) j=0 (cid:18) (cid:19) P and where J is the partial derivative of J( ) with respect to its jth and kth arguments, evaluated j(cid:3);k (cid:1) at the zero-growth state.46 De(cid:133)ne (cid:26) 1+r , which amounts to a discount factor that adjusts (cid:17) 1+(cid:25)k for the risk premium embedded in the discount rate r and for the relative rate of capital goods in(cid:135)ation (cid:25)k. We assume that (cid:26) > 1. Applying this de(cid:133)nition and noting that J = J by j(cid:3);k k(cid:3);j Young(cid:146)s Theorem, it can be shown that d = (cid:26) md , for m = 1;:::;M. Using this fact and m (cid:0) m collecting terms, equation (11) simpli(cid:133)es to the following 2Mth order di⁄erence equation: M M E k d (cid:26) mk +d k + d k (12) t t(cid:3)+1 m (cid:0) t+1+m 0 t+1 m t+1 m (cid:25) m=1 m=1 (cid:0) (cid:2) (cid:3) P P = D(B)k (13) t+1 where we have de(cid:133)ned the following polynomial in the backshift operator: D(B) = d ((cid:26)B) M +:::+d ((cid:26)B) 1+d +d B+:::+d BM M (cid:0) 1 (cid:0) 0 1 M and let d 1 M d (1 + (cid:26)m). This function satis(cid:133)es the restriction that D(1) = 1, so 0 (cid:17) (cid:0) m=1 m that the optimal caPpital stock tracks its frictionless target in the long run. In addition, since this backshift polynomial D(B) is symmetric in B and ((cid:26)B) 1, each backward-stable root (cid:21) will have (cid:0) m a corresponding forward-stable root (cid:21) (cid:26) 1. Taken together, these properties imply that D(B) m (cid:0) can be restated as the following product of lead and lag polynomials: a(B)a (cid:26) 1B 1 (cid:0) (cid:0) D(B) = ; a(1) a((cid:26) 1) (cid:0) (cid:0) (cid:1) of L. Given this condition, the steady state we describe for capital requires (cid:27)(cid:1)u=((cid:27) (cid:0) s(cid:3)L )(cid:1)aK+s(cid:3)L (cid:1)l+(cid:1)aL , which ensures that the supply and demand for capital shift in tandem to keep the equilibrium quantity of capital (cid:0) (cid:1) constant. 46Note that all terms involving r and Et [(cid:1) p p k t k t+1 ] drop out in the vicinity of the steady state since J j(cid:3) =0. 22

where we have de(cid:133)ned a(x) M (1 (cid:21) x) = 1+a x+:::+a xM. Using this representation, (cid:17) m=1 (cid:0) m 1 M the di⁄erence equation for theQcapital stock in equation (12) simpli(cid:133)es to: E t a(B)k t+1 a(1)a (cid:26) (cid:0) 1 a (cid:26) (cid:0) 1B (cid:0) 1 (cid:0) 1 k t(cid:3)+1 = 0. (14) (cid:0) h i (cid:0) (cid:1) (cid:0) (cid:1) Note that this the same general form as the solutions derived by Tinsley [2002] in which (cid:133)rms minimize a quadratic loss objective in which adjustment costs are a function of M lags of capital, and much of the remainder of this derivation closely resembles his setup and results.47 Atthisstage,weletf t denotea (cid:26) (cid:0) 1 E t a (cid:26) (cid:0) 1B (cid:0) 1 (cid:0) 1 k t(cid:3)+1 ,sothata(B)k t+1 = a(1)f t . This factor, which constitutes a moving(cid:0)targ(cid:1)et tohwa(cid:0)rd which(cid:1) capitalierror-corrects over time, amounts to a weighted average of anticipated future frictionless stocks where the weights are determined implicitly by the discount factor and the eigenvalues that are embedded in in the lead polynomial a (cid:26) 1B 1 .48 Letting g denote the (M 1) 1 lead vector [f ;:::;f ], the forward motion (cid:0) (cid:0) t t+M 1 t 0 (cid:0) (cid:2) (cid:0) of(cid:0)this targ(cid:1)et can be described by the companion system: g = E Ag +a (cid:26) 1 k ; (15) t t t+1 (cid:0) t(cid:3)+1 = a (cid:26) (cid:2) 1 E 1(cid:0) Ai(cid:19) (cid:1) k (cid:3) (cid:0) t M t(cid:3)+1+i (cid:20)i=0 (cid:21) (cid:0) (cid:1) P where A as the M M bottom row companion matrix of the lead polynomial a (cid:26) 1B 1 : (cid:0) (cid:0) (cid:2) 47Note that for the M =1 case, this solution is fully consistent with what Tevlin and Whelan(cid:0)[2003] ob(cid:1)tain using a much simpler reduced-form approach in which (cid:133)rms chose capital holdings to minimize the quadratic loss function of the form: 1 (cid:18)lE t (cid:13)((cid:1)k t+l )2+(k t+l (cid:0) k t(cid:3)+l )2 : X l=1 h i Indeed, the solution using this formulation is identical to our model if we allow d 1 =(cid:13) and (cid:26)(cid:0) 1 =(cid:18). 48This can be seen more clearly by noting from equation (15) that f t =(cid:19)0 M g t =a (cid:26)(cid:0) 1 E t 1(cid:19)0 M Ai(cid:19) M k t(cid:3)+1+i ; (cid:20)i=0 (cid:21) (cid:0) (cid:1) P so that the weight on the anticipated capital stock at on the capital stock at horizon 1+i is (cid:19)0 M Ai(cid:19) M . The matrix Ai can be decomposed as T(cid:3)iU, where (cid:3) is a diagonal matrix composed of the M forward eigenvalues (cid:26)(cid:0) 1(cid:21) m , T is theM M matrixoftheM correspondingeigenvectors,and U =T(cid:0) 1. Applyingtherulesofmatrixalgebra,itcan (cid:2) be shown that (cid:19)0 M Ai(cid:19) M = M m=1 ((cid:26)(cid:0) 1(cid:21) m )iT M;m U m;M . Note that the sum M m=1 T M;m U m;M is the M;M element of theidentitymatrixTU,sothattheweightgiventofrictionlesscapitalathorizoniamountstoaweightedaverageof P P the M "discount factors" ((cid:26)(cid:0) 1(cid:21) m )i. 23

0 I M 1 A = (cid:0) ; (16) " a M (cid:26) (cid:0) M a M 1 (cid:26) (cid:0) (M (cid:0) 1) ::: a 1 (cid:26) (cid:0) 1 # (cid:0) (cid:0) (cid:0) (cid:0) and (cid:19) is an M 1 selection vector that has one as its Mth element and zeros elsewhere. M (cid:2) Tevlin and Whelan [2003] show that the forward-looking nature of the capital target is crucial for empirical estimation because the long-run response of capital to an unanticipated change in fundamentals is determined by the degree that this new information a⁄ects the e⁄ective capital target f .49 In turn, this response is closely related to the anticipated persistence of the disturt bances. To allow for these expectation e⁄ects, we assume that (cid:133)rms forecast the determinants of the frictionless capital stock in equation (1) using the following VAR(p) process: p 1 (cid:0) v = C(B)v +e ; where C(B) = C Bj, (17) t+1 t t+1 j+1 j=0 X wherev isthevectorofq determinantsofk ,C isaq q matrixofVARcoe¢ cientsforlagj,and t t(cid:3)+1 j (cid:2) e is a serially uncorrelated vector of covariance-stationary forecast errors such that E [e ] = 0 t t t+i q 1 (cid:2) foranywholenumberi.50 De(cid:133)ningthepq (cid:2) 1vectorz t = [x0 t ;:::;x0 t (cid:0) p+1 ]0, thisVAR canberestated in companion form as z = Hz +[e 0 ]0, where t+1 t 0t+1 1 (p 1)q (cid:2) (cid:0) C C 1 p C = ; (18) " I (p 1)q (cid:1)(cid:1)(cid:1) 0 (p 1)q p # (cid:0) (cid:0) (cid:2) is the pq pq companion matrix for the VAR, so that E [z ] = Hiz for any whole number i.51 t t+i t (cid:2) To rule out explosive dynamics, we assume that q of the eigenvalues of matrix C are exactly equal 0 to one, and that the remaining pq q eigenvalues have modulus less than one. 0 (cid:0) After restating the frictionless capital stock in vector form as k = bv , equation (14) can t(cid:3)+1 0 t+1 be solved to yield that: f = bDz (19) t 0 t 49 The VAR form used here allows for greater generality than Tevlin and Whelan [2003], who assumed that each frictionless fundamental follows its own AR(p) process. 50For brevity, we abstract from constant terms. Allowing for them does not substantively a⁄ect our results. 51We allow our notation to accomodate an arbitrary number of frictionless fundamentals to account for the possibility that the capital-augmenting technology term does not exist. In addition, this allows for the possibility that the VAR could be augmented to include non-fundamental factors that improve the forecasts of these relevant fundamentals. 24

where we have de(cid:133)ned the q pq matrix: (cid:2) D a (cid:26) (cid:0) 1 [I q 0 q (p 1)q ]1 (cid:19)0 M Ai(cid:19) M Ci+1; (20) (cid:17) (cid:2) (cid:0) i=0 (cid:16) (cid:17) (cid:0) (cid:1) P where D is composed of p adjacent q q matrices such that D =[D D ]. Using this partition, 1 p (cid:2) (cid:1)(cid:1)(cid:1) and the fact that z t = [x0 t ;:::;x0 t p+1 ]0, equation (19) can be restated in the following lag polynomial (cid:0) representation: f = bD(B)x ; where D(B) D +D B+:::+D Bp 1: (21) t 0 t 1 2 p (cid:0) (cid:17) Inserting this representation into equation (14), it can be shown that the optimal capital stock is determined by the equation: a(B)k = a(1)bD(B)x ; (22) t+1 0 t or, after inverting the backshift polynomial a(B), by the MA representation: k = bG(B)x , where G(B) a(1)a(B) 1D(B): (23) t+1 0 t (cid:0) (cid:17) Since G(1) = D(1) by construction, both the forward-looking target f and the optimal capital t stock k share the same long-run sensitivity to shocks. This shows that the long-run sensitivity t of capital to an unanticipated change in frictionless fundamentals is governed by the anticipated persistence of this e⁄ect, which is embodied in the polynomial D(B). B Appendix: First Proof Recall from the previous appendix the de(cid:133)nition of the VAR matrix polynomial C(L) shown in equation (17). We begin by showing that if C(1) = I (cid:151)so that the VAR(p) in equation (17) can q be restated as a VAR(p-1) system in (cid:1)v (cid:151)then G(1) = I . A necessary condition to establish that t q C(1) = I is that all q of the variables in the VAR(p) contain unit roots.52 Once we establish this q result, we turn our attention to cases with q unit roots, where 1 q < q. 0 0 (cid:20) 52Su¢ ciency requires that we rule out the existence of cointegrating relations between the q variables in the VAR. It is well known that a cointegrated system can never be represented by a (cid:133)nite-order VAR in (cid:133)rst di⁄erences (for instance, see Hamilton [1994]). Since we can think of no good theoretical argument that would impose a long-run relation between the determinants of frictionless capital, this assumption seems reasonable. 25

Begin by de(cid:133)ning the matrix S as a pq q matrix of stacked identity operators [I q I q ]0. (cid:2) (cid:1)(cid:1)(cid:1) Using this de(cid:133)nition, equation (20), and the fact that G(1) = D(1) from equation (23), this means that: G(1) = DS =a (cid:26) (cid:0) 1 [I q 0 q q(p 1) ]1 (cid:19)0 M Ai(cid:19) M Ci+1S (24) (cid:2) (cid:0) i=0 (cid:16) (cid:17) (cid:0) (cid:1) P Straightforward matrix algebra using equation (18) establishes that: CS = [C(1) I I ]0 = S; (25) q q (cid:1)(cid:1)(cid:1) when C(1) = C +:::+C = I , a fact that can be applied iteratively to show that Ci+1S = S for 1 p q all non-negative integers i. Noting that (cid:19)0 M Ai(cid:19) M is a scalar, we can then simplify equation (24) to become: (cid:16) (cid:17) G(1) = a (cid:26) (cid:0) 1 [I q 0 q (p 1)q ]S1 (cid:19)0 M Ai(cid:19) M = I q a (cid:26) (cid:0) 1 1 (cid:19)0 M Ai(cid:19) M ; (26) (cid:2) (cid:0) i=0 i=0 (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) P (cid:0) (cid:1)P so our desired result boils down to proving that 1i=0 (cid:19)0 M Ai(cid:19) M = a (cid:26) (cid:0) 1 (cid:0) 1 : Straightforward calculations show that the summation in this exprPession(cid:16)simpli(cid:133)e(cid:17)s to: (cid:0) (cid:1) 1(cid:19)0 M Ai(cid:19) M = (cid:19)0 M 1Ai (cid:19) M = (cid:19)0 M (I A) (cid:0) 1(cid:19) M ; (27) (cid:0) i=0 (cid:18)i=0 (cid:19) P P since all the roots of the bottom row companion matrix are stable. Note that the expression on the right hand side of this equation is pre- and postmultiplied by the selection vector (cid:19) , so M this summation is simply the bottom right-hand element of the inverted matrix (I A) 1, Using M (cid:0) (cid:0) equation (16), this matrix takes the form: 1 1 1 0 0 0 (cid:0) (cid:0) (cid:1)(cid:1)(cid:1) 2 0 1 1 0 0 3 (cid:0) (cid:1)(cid:1)(cid:1) (I M A) (cid:0) 1 = 6 . . . . . . . . . ... . . . . . . 7 : (cid:0) 6 7 6 0 0 0 1 1 7 6 (cid:1)(cid:1)(cid:1) (cid:0) 7 6 6 a M (cid:26) (cid:0) M a M 1 (cid:26) (cid:0) (M (cid:0) 1) a M 2 (cid:26) (cid:0) (M (cid:0) 2) a 2 (cid:26) (cid:0) 2 1+a 1 (cid:26) (cid:0) 1 7 7 6 (cid:0) (cid:0) (cid:1)(cid:1)(cid:1) 7 4 5 The bottom diagonal element of this matrix can be calculated by block inversion. Dividing (I A) 1 into the blocks: M (cid:0) (cid:0) 26

1 1 0 0 0 (cid:0) (cid:1)(cid:1)(cid:1) 0 1 1 0 0 2 3 2 3 N (cid:0) (cid:1)(cid:1)(cid:1) ; N ; 1 (cid:17) 6 . . . . . . . . . ... . . . 7 2 (cid:17) 6 . . . 7 6 7 6 7 6 0 0 0 1 7 6 1 7 6 (cid:1)(cid:1)(cid:1) 7M 1 M 1 6 (cid:0) 7M 1 1 4 5 (cid:0) (cid:2) (cid:0) 4 5 (cid:0) (cid:2) N a (cid:26) M a (cid:26) (M 1) a (cid:26) (M 2) a (cid:26) 2 3 M (cid:0) M 1 (cid:0) (cid:0) M 2 (cid:0) (cid:0) 2 (cid:0) (cid:17) (cid:0) (cid:0) (cid:1)(cid:1)(cid:1) 1 M 1 h i (cid:2) (cid:0) and N = 1+a (cid:26) 1, and noting that: 4 1 (cid:0) 1 1 1 1 (cid:1)(cid:1)(cid:1) 2 0 1 1 1 3 (cid:1)(cid:1)(cid:1) (N 1 ) (cid:0) 1 6 . . . . . . ... . . . . . . 7 ; (cid:17) 6 7 6 0 0 1 1 7 6 (cid:1)(cid:1)(cid:1) 7 6 7 6 0 0 0 1 7 6 (cid:1)(cid:1)(cid:1) 7 4 5 1 thestandardformulaforblockinversionsuggeststhatthebottomdiagonalelementis N N (N ) 1N (cid:0) : 4 3 1 (cid:0) 2 (cid:0) Straightforward calculations using this formula show that (cid:19)0 M (I (cid:0) A) (cid:0) 1(cid:19) M = a (cid:16)(cid:26) (cid:0) 1 (cid:0) 1 : This (cid:17) proves the (cid:133)rst result. (cid:0) (cid:1) We now turn our attention to cases where the VAR(p) in equation 17 has fewer than q unit roots. Without loss of generality, assume that the (cid:133)rst q < q of the variables in vector v have 0 t unit roots. Given this setup, we need to show that the columns of the matrix G(1) that multiply the (cid:133)rst q elements of v are the same as the (cid:133)rst q columns of the identity matrix I . As a (cid:133)rst 0 t 0 q step, Appendix C shows that, in the absence of cointegrating relations, the (cid:133)rst q columns of the 0 summed VAR polynomial C(1) must be the same as the (cid:133)rst q columns of an identity matrix I . 0 q Now we proceed to calculate G(1) = D(1) using equation (24). Equation (25) still holds, and the matrix product CS can be decomposed as follows: C(1) I q CS = S+ (cid:0) ; " 0 (p 1)q # (cid:0) where the (cid:133)rst q columns of the matrix in brackets must contain only zeros. By successively 0 pre-multiplying this matrix by C, one can show that H (C(1) I ) Ci+1S = S+ i+1 (cid:0) q ; " 0 (p 1)q # (cid:0) 27

for any non-negative integer i, where we have de(cid:133)ned H = I +C H for any whole number i, i q 1 i 1 (cid:0) and let H 0 : Inserting this expression into equation (24) and simplifying, we obtain that: 0 q q (cid:17) (cid:2) G(1) = I q a (cid:26) (cid:0) 1 1 (cid:19)0 M Ai(cid:19) M +a (cid:26) (cid:0) 1 1 (cid:19)0 M Ai(cid:19) M H i+1 (C(1) I q ): (28) (cid:0) i=0 (cid:20)i=0 (cid:21) (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1)P (cid:0) (cid:1) P Finally, using our results for the q unit root case (above) for the (cid:133)rst term of this equation, we can simplify this expression to become: G(1) = I q +a (cid:26) (cid:0) 1 1 (cid:19)0 M Ai(cid:19) M H i+1 (C(1) I q ): (29) (cid:0) (cid:20)i=0 (cid:21) (cid:16) (cid:17) (cid:0) (cid:1) P Since the (cid:133)rst q columns of the second set of terms of the matrix C(1) I contain only zeros, 0 q (cid:0) one can con(cid:133)rm that(cid:151)regardless of the form of the q q matrix in brackets(cid:151)the (cid:133)rst q columns 0 (cid:2) of the second expression on the right-hand-side of this equation must also contain only zeros. By implication, the (cid:133)rst q columns of D(1) must be the (cid:133)rst q columns of the identity matrix I . 0 0 q C Appendix: Second Proof Partitioning the vector v so that the (cid:133)rst q variables are I(1) and the remaining q q variables t 0 0 (cid:0) areI(0). Wewishtoprovethat,whenanonstationaryVAR(p) representationoftheformshownin equation (17) exists, the (cid:133)rst q columns of the matrix C(1) must be the same as the corresponding 0 columns of the identity matrix I . q We begin by forming a basis of the space of q 1 vectors a such that av is I(0). Since the 0 t (cid:2) (cid:133)rst q elements of v are assumed to be I(1) and there are no cointegrating relations linking these 0 t variables, one such basis is the (q q ) q matrix: 0 (cid:0) (cid:2) A = 0 I : (30) 0 (q (cid:0) q0)xq0 q (cid:0) q0 (cid:2) (cid:3) Therefore, the space of I(0) linear combinations av is spanned by the space of vectors hA for 0 t 0 0 any nonzero q 1 vector h. Since the elements of (cid:1)v must be stationary, it can always be written t (cid:2) using a Wold representation of the form: (1 B)v = W(B)e ; where W(B) = W +W B+W B2+:::; (31) t t 0 1 2 (cid:0) andW isaq q matrixforj = 0;1;2;:::. Usingthisrepresentation,onecanperformamultivariate j (cid:2) Beveridge-Nelson decomposition to determine that: 28

t v = v +W(1) e +(cid:17) (cid:17) ; where (cid:17) 1 (W +W +:::)e t 0 s t 0 t s+1 s+2 t s s=1 (cid:0) (cid:17) (cid:0) s=0 (cid:0) is a stationary variable.53P Multiplying this relation through b P y A, it is evident that, in order to 0 ensure that the linear combinations of v in the space spanned by A are stationary, it must be true t 0 that the basis A satis(cid:133)es: 0 AW(1) = 0 ; (32) 0 q q (cid:2) so that any rotation of A must also satisfy this condition. 0 Now note that the VAR(p) can be written as [I C(B)B]v = e . Premultipling both sides q t t (cid:0) of equation (31) by the expression [I C(B)B] and simplifying, one obtains the restriction that: q (cid:0) (1 x)I = [I C(x)x]W(x) q q (cid:0) (cid:0) for all values x. Evaluating this expression at x = 1, we (cid:133)nd that: [I C(1)]W(1) = 0 ; q q q (cid:0) (cid:2) which shows that all the rows of [I C(1)] are in the space spanned by the basis A. From q 0 (cid:0) equation (30), it is clear that all the 1 q vectors spanned by this basis must have zeros in their (cid:2) (cid:133)rstq columns, whichimpliesthatthe(cid:133)rstq columnsofthematrix[I C(1)]mustbecomposed 0 0 q (cid:0) of zeros. In turn, this requires that the (cid:133)rst q columns of C(1) are identical to the corresponding 0 columns of the identity matrix I . q 53We assume the normality condition that the sequence f sW s g 1s=0 is absolutely summable. 29

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Figure 1: Ratio of Current Account to Gross Domestic Product for South Africa, 1970 to 2001. 34

Figure2: RatioofrealcapitaltooutputfortwentyfourSouthAfricanmanufacturingindustries,1970Q1to2000Q4. Figure 3: Real user cost for twenty four South African manufacturing industries, 1970q1 to 2000q4. 35

Figure 4: Bars for each industry denote test statistics for a Dickey-Fuller GLS test, where we controlfor aggregate e⁄ects using a preliminary regression. The number of included lag correction terms was chosen by the lag selection criterion in Ng and Perron [2001]. 36

Figure 5: Bars for each industry denote test statistics for a Clemente-Montanes-Reyes additive outlier test, which maintains the null of a unit root and allows for two structural breaks. 37

Figure 6: Sensitivity of user cost elasticity estimates from homogeneous panel DOLS to number of included lags and leads, for the I(1) subsample. The top panel show the user cost elasticity estimate for given number of DOLS leads/lags. The bottom panel shows the Information criteria for each lag/lead speci(cid:133)cation. Figure 7: Long-run user cost elasticity estimate from the HOM and HET-W distributed lag speci(cid:133)cations as a function of the number of included lags. Dots denote the point estimate at each lag, and bars denote a 95 percent con(cid:133)denceinterval. Shadeddotsdenotethatat-testforthejointsigni(cid:133)canceofthelastincludedlagoftheusercost is signi(cid:133)cant at 10 percent. 38

Figure 8: Comparative analysis of the user cost elasticity for the United States and South Africa. Figure 9: Estimatedmarginalresponseofcapitaltoa1percentincreaseintheusercostwith95percentcon(cid:133)dence band. 39

Figure 10: Estimated cumulative response of capital to a 1 percent increase in the user cost with 95 percent con(cid:133)dence band. Start Date End Date eta m0 eta m0 itsE yticits 5. itsE yticits 5. alE ts 1 alE ts 1 oC resU 1 5.1 983q1 1984q1 1985q1 1986q1 1987q1 oC resU 5.1 1991q4 1992q4 1993q4 1994q4 1995q4 1996q4 tc effE 5.1 tc effE 5.1 o o g g ra1 ra1 b b m m E E d eta 5. d eta 5. m m itsE1 0 983q1 1984q1 1985q1 1986q1 1987q1 itsE 0 1991q4 1992q4 1993q4 1994q4 1995q4 1996q4 0 0 0 4 max 0 4 max d o o hilekiL g 0 0 0 0 2 3 5 5 restriction d o o hilekiL g 0 0 0 0 2 3 5 5 restriction o5 o5 L0 L0 0 0 1 1 15983q1 1984q1 1985q1 1986q1 1987q1 5 1991q4 1992q4 1993q4 1994q4 1995q4 1996q4 Start Date End Date Figure 11: Sensitivity of results from embargo speci(cid:133)cation to beginning and end dates of the embargo, for speci- (cid:133)cations that are restricted so that the long-run capital-output elasticity is one. Left panels show sensitivity to the begin date, while right panels show sensitivity to the ending date. Top panels: Estimates of corrected user user cost elasticity with 95 percent con(cid:133)dence interval. Middle panels: Estimated e⁄ect of embargo on elasticity with 95 percent con(cid:133)dence interval. Bottom panels: Log-likelihood for given start or end date. 40

stseT ytiranoitatS dna tooR tinU lenaP :1 elbaT latipaC latipaC tuptuO tsoC resU H H tesataD tuptuO A 0 429:0 000:1 584:0 390:0 )0(I eno tsael tA )1(I llA elpmaS lluF 000:0 000:0 000:0 000:0 )1(I llA )0(I llA 309:0 000:1 101:0 363:0 )0(I eno tsael tA )1(I llA elpmasbuS )1(I 000:0 000:0 000:0 000:0 )1(I llA )0(I llA nidesivedtsettoortinulenapFDACehtfonoisrevagnisutoortinuafosisehtopyhllunehtrofeulav-pehtstroperenilpoteht,elpmashcaeroF detalumisgnisu,seirtsudnissorcascitsitats-tdezilamronfoegarevaehtgnikatybcitsitats"rabtZ"adetcurtsnocew,tsetruonI .)7002(naraseP taht tcaf a , )]7002[ nihS dna naraseP ,mI ees( llun eht rednu detubirtsid yllamron si citsitats siht ,yllacitotpmysA .stnemom dnoces dna tsr(cid:133) .]7002[naraseP ni seulav lacitirc eht esu daetsni ew nehw detce⁄a ylufgninaem ton era stluseR ..llun eht rednu eulav-p eht etupmoc ot esu ew tce⁄a ton seod os gniod ,sisehtopyh evitanretla eht rednu yranoitats-dnert rof wolla ot "dnerT" a edulcni ew ,noitac(cid:133)iceps evitanretla na nI rof secnere⁄id deggal dedulcni fo rebmun ehT .noitalerroc laires dna ecnedneped lanoitces-ssorc rof tsubor era stset esehT .stluser stset ruo naht erom rof tnecrep 01 ta tnac(cid:133)ingis saw gal taht litnu deppord erew sgal hcihw ni noiretirc tset-t laitneuqes a yb nesohc saw yrtsudni hcae fo llun eht sniatniam hcihw ,tset lenap )9991( irdaH eht rof eulav-p eht stroper tesatad hcae rof enil mottob ehT .seirtsudni eht fo tnecrep 01 lanoitces-ssorcfosmrofcirenegdnanoitalerroclairesroftsuborsitsetnarasePehT .stce⁄eetagerggadnadex(cid:133)roflortnocstsetllA .ytiranoitats .yticitsadeksoreteh dna srorre detalerroc yllaires ot tsubor era tset irdaH eht fo stluser ehT .noitalerroc rorre 41

Table 2: Panel Cointegration Tests (p-value under H of No Cointegration) 0 Full I(1) Test Panel Panel HOM: Pooled DOLS panel t-stat (p) 0:000 0:000 group t-stat (p) 0:000 0:000 HET: Group-Mean DOLS panel t-stat (p) 0:000 0:000 group t-stat (p) 0:000 0:000 Listed p-values are for the null of no cointegration for the statistics described in Pedroni(2001). For the homogeneous speci(cid:133)cation, residuals are from a cointegrating vector that was (cid:133)tted using a pooled DOLS speci(cid:133)cation that included 25 leads and lags of the (cid:133)rst-di⁄erence of the independent variables. The heterogeneous speci(cid:133)cations were estimated using a group-mean DOLS estimator that included 8 leads and 16 lagsofthe(cid:133)rst-di⁄erencedindependentvariables. Allestimatorscontrolforgroup(cid:133)xede⁄ectsandandtime e⁄ects. The results of these tests were unchanged when we restricted the output elasticity to unity. 42

Table 3: Estimates from Panel Cointegration Speci(cid:133)cations Long Run Full I(1) Elasticity of Panel Panel Capital to HOM: Pooled DOLS 0:965 1:000 User Cost (cid:0) (cid:0) (0:064) (0:072) Sample Size 1;560 975 HET: Group Mean DOLS 0:536 0:845 User Cost (cid:0) (cid:0) (0:039) (0:056) Sample Size 2;208 1;380 Pooled DOLS: Speci(cid:133)cations include 25 leads and lags of changes in output and the user cost, along with current values. Standard errors are robust for cross-sectional correlation in the error term and autocorrelation. Mean-Group DOLS: Speci(cid:133)cations include 8 leads and 16lagsofthe(cid:133)rst-di⁄erencesin theusercostand output. Standard errorsarerobust to cross-sectional correlation and autocorrelation. All speci(cid:133)cations control for (cid:133)xed and aggregate e⁄ects. 43

)rorre dradnats tsubor( noitac(cid:133)icepS ecnere⁄iD gnisu noissergeR lenaP SLO morf setamitsE :4 elbaT )1(I lluF nuR gnoL lenaP lenaP fo yticitsalE W-TEH MOH W-TEH MOH ot latipaC 628:0 767:0 774:0 516:0 (cid:0) (cid:0) (cid:0) (cid:0) ))1(uN( tsoC resU )450:0( )970:0( )840:0( )760:0( 91 52 91 72 dedulcnI sgaL 053;1 053;1 061;2 061;2 eziS elpmaS yrtsudniotnoitiddaniselbairavtnednepedniehtfosgaldnaelbairavtnednepedehtfonoitavresbosuoenaropmetnocehtedulcnisnoissergerllA detoned setamitse ehT .eno eb ot deniartsnoc si yticitsale tuptuo latipac nur-gnol eht ,snoitac(cid:133)iceps lla nI .stce⁄e etagergga dna,stce⁄e dex(cid:133) neht ,yrtsudni hcae rof seiticitsale tsoc resu etarapes setamitse "W-TEH" ehT .yticitsale emas eht evah seirtsudni lla taht os tcirtser "MOH" revo kcots latipac lanimon latot eht fo erahs egareva sti yb etamitse s(cid:146)yrtsudni hcae sthgiew taht yticitsale etagergga eht fo etamitse na smrof htgnelgalehthcihwnierudecorptsetTlaitneuqesagnisudenimretedsawnoitac(cid:133)icepshcaenisgaldedulcniforebmunehT .doirepelpmaseht htob ot tsubor era srorre dradnatS .tnac(cid:133)ingis )yltnioj erew( saw )s(gal dedulcni tsal eht no )s(tneic ¢eoc eht litnu denetrohs ylevisseccus saw .slaudiser eht ni yticitsadeksoreteh dna noitalerroc lanoitces-ssorc 44

)rorre dradnats tsubor( noitac(cid:133)icepS ecnere⁄iD gnisu noissergeR lenaP SLO morf setamitsE :5 elbaT )1(I lluF nuR gnoL lenaP lenaP fo yticitsalE doireP ot latipaC 858:0 947:0 (cid:0) (cid:0) ))1(uN( ograbmE-noN )580:0( )470:0( 562:0 052:0 (cid:0) (cid:0) ))1(uM+)1(uN( ograbmE tsoC resU )431:0( )801:0( 395:0 994:0 ))1(uM( ecnere⁄iD )241:0( )511:0( 72 72 dedulcnI sgaL 053;1 061;2 eziS elpmaS htiw gnola ,sgal 72 dna elbairav tnedneped eht fo noitavresbo suoenaropmetnoc eht edulcni snoisserger llA gnimrofrepybdenimretedsawsgaldedulcniforebmunehT .stce⁄eetagerggadna,stce⁄edex(cid:133)yrtsudnidex(cid:133) -ssorc htob rof tnuocca srorre dradnats tsubor ehT .tesatad eritne eht rof erudecorp tset-T laitneuqes a eht taht os detcirtser era snoitac(cid:133)iceps llA .slaudiser eht ni yticitsadeksoreteh dna noitalerroc lanoitces xirtam ecnairavoc eritne eht gnisu detaluclac erew srorre dradnatS .eno si yticitsaletuptuolatipac nur-gnol .sretemarap detamitse lla rof 45