What Happens After A Technology Shock?

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 768 June 2003 What Happens After A Technology Shock? Lawrence J. Christiano, Martin Eichenbaum, and Robert Vigfusson NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/.

What Happens After A Technology Shock? ∗ Lawrence J. Christiano, Martin Eichenbaum and Robert Vigfusson † ‡ § Abstract We provide empirical evidence that a positive shock to technology drives up per capita hours worked, consumption, investment, average productivity and output. This evidence contrasts sharply with the results reported in a large and growing literature thatargues,onthebasisofaggregatedata,thatpercapitahoursworkedfallafterapositivetechnologyshock. Wearguethatthedifferenceinresultsprimarilyreflectsspecificationerrorinthewaythattheliteraturemodelsthelow-frequencycomponentofhours worked. Keywords: productivity, long-run restriction, hours worked, weak instruments. ChristianoandEichenbaumthanktheNationalScienceFoundationforfinancialassistance. Theviewsin ∗ thispaperaresolelytheresponsibilityoftheauthorsandshouldnotbeinterpretedasreflectingtheviewsof the Board of Governors of the Federal Reserve System or of any person associated with the Federal Reserve System. We are grateful for discussions with Susanto Basu, Lars Hansen, Valerie Ramey, and Harald Uhlig. Northwestern University and NBER. † Northwestern University and NBER. ‡ Board of Governors of the Federal Reserve System. (Email robert.j.vigfusson@frb.gov) §

1 Introduction Standard real business cycle models imply that per capita hours worked rise after a permanent shock to technology. Despite the a priori appeal of this prediction, there is a large and growing literature that argues it is inconsistent with the data. This literature uses reduced form time series methods in conjunction with minimal identifying assumptions that hold across large classes of models to estimate the actual effects of a technology shock. The resultsreportedinthisliteratureareimportantbecausetheycallintoquestionbasicproperties of many structural business cycle models. Consider, for example, the widely cited paper by Gali (1999). His basic identifying assumption is that innovations to technology are the only shocks that have an effect on the long run level of labor productivity. Gali (1999) reports that hours worked fall after a positivetechnologyshock. Thefall issolongandprotractedthat, accordingtohisestimates, technology shocks are a source of negative correlation between output and hours worked. Because hours worked are in fact strongly procyclical, Gali concludes that some other shock or shocks must play the predominant role in business cycles with technology shocks at best playing only a minor role. Moreover, he argues that standard real business cycle models shed little light on whatever small role technology shocks do play because they imply that hours worked rise after a positive technology shock. In effect, real business cycle models are doubly dammed: they address things that are unimportant, and they do it badly at that. Other recent papers reach conclusions that complement Gali’s in various ways (see, e.g., Shea (1998), Basu, Kimball and Fernald (1999), and Francis and Ramey (2001)). In view of the important role attributed to technology shocks in business cycle analyses of the past two decades, Francis and Ramey perhaps do not overstate too much when they say (p.2) that Gali’s argument is a ‘...potential paradigm shifter’. Not surprisingly, the result that hours worked fall after a positive technology shock has attractedagreatdealofattention. Indeed,thereisagrowingliteratureaimedatconstructing general equilibrium business cycle models that can account for this result. Gali (1999) and others have argued that the most natural explanation is based on sticky prices. Others, like Francis and Ramey (2001) and Vigfusson (2002), argue that this finding is consistent with real business cycle models modified to allow for richer sets of preferences and technology, such as habit formation and investment adjustment costs.1 We do not build a model that can account for the result that hours fall after a technology shock. Instead, we challenge the result itself. Using the same identifying assumption as Gali (1999), Gali, Lopez-Salido, and Valles (2002), and Francis and Ramey (2001), we find that a positive technology shock drives hours worked up, not down.2 In addition, it leads to a rise in output, average productivity, investment, and consumption. That is, we find that a permanent shock to technology has qualitative consequences that a student of real business cycles would anticipate.3 At the same time, we find that permanent technology shocks play 1Other models that can account for the Gali (1999) finding are contained in Christiano and Todd (1996) and Boldrin, Christiano and Fisher (2001). 2Chang and Hong (2003) obtain similar results using disaggregated data. 3That the consequences of a technology shock resemble those in a real business cycle model may well reflectthattheactualeconomyhasvariousnominalfrictions,andmonetarypolicyhassuccessfullymitigated those frictions. See Altig, Christiano, Eichenbaum and Linde (2002) for empirical evidence in favor of this 1

a very small role in business cycle fluctuations. Instead, they are quantitatively important at frequencies of the data that a student of traditional growth models might anticipate. Since we make the same fundamental identification assumption as Gali (1999), Gali, Lopez-Salido, and Valles (2002) and Francis and Ramey (2001), the key questions is: What accounts for the difference in our findings? By construction, the difference must be due to different maintained assumptions. As it turns out, a key culprit is how we treat hours worked. For example, if we assume, as do Francis and Ramey, that per capita hours worked is a difference stationary process and work with the growth rate of hours (the difference specification), then we too find that hours worked falls after a positive technology shock. But if we assume that per capita hours worked is a stationary process and work with the level of hours worked (the level specification), then we find the opposite: hours worked rise after a positive technology shock. Standard, univariate hypothesis tests do not yield much information about which specification is correct. They cannot reject the null hypothesis that per capita hours worked are difference stationary. They also cannot reject the null hypothesis that hours worked are stationary. This is not surprising in light of the large literature that documents the difficulties that univariate methods have in distinguishing between a difference stationary stochastic process and a persistent stationary process.4 So we have two answers to the question, ‘what happens to hours worked after a positive technology shock?’ Each answer is based on a different statistical model, depending on the specification of hours worked. Each model appears to be defensible on standard classical grounds. To judge between the competing specifications, we assess their relative plausibility. To this end, we ask, ‘which specification has an easier time explaining the observation that hours worked falls under one specification and rises under the other?’ Using this criterion, we find that the level specification is preferred. We now discuss the results that lead to this conclusion. First, the level specification encompasses the difference specification. We show this by calculating what an analyst who adopts the difference specification would find if our estimated level specification were true. Forreasonsdiscussedbelow,bydifferencinghoursworkedthisanalystcommitsaspecification error. We find that such an analyst would, on average, infer that hours worked fall after a positive technology shock even though they rise in the true data-generating process. Indeed the extent of this fall is very close to the actual decline in hours worked implied by the estimated difference specification. In addition, the level specification easily encompasses the impulse responses of the other relevant variables. Second, the difference specification does not encompass the level specification. We calculate what an analyst who adopts the level specification would find if our estimated difference specification were true. The mean prediction is that hours fall after a technology shock. So, focusing on means alone, the difference specification cannot account for the actual estimates associated with the level representation. However, the difference specification predicts that the impulse responses based on the level representation vary a great deal across repeated samples. This uncertainty is so great that the difference specification can account for the level results as an artifact of sampling uncertainty. As it turns out, this result is a Pyrrhic interpretation. 4See, for example, DeJong, Nankervis, Savin, and Whiteman (1992). 2

victory for the difference specification. The prediction of large sampling uncertainty stems from the difference specification’s prediction that an econometrician working with the level specification encounters a version of the weak instrument problem analyzed in the literature (see, for example, Staiger and Stock, 1997). In fact, a standard weak instrument test finds little evidence in the data. To quantify the relative plausibility of the level and difference specifications, we compute the type of posterior odds ratio considered in Christiano and Ljungqvist (1988). The basic idea is that the more plausible of the two specifications is the one that has the easiest time explaining the facts: (i) the level specification implies that hours worked rises after a technology shock, (ii) the difference specification implies that hours worked falls, and (iii) the outcome of the weak instruments test. Focusing only on facts (i) and (ii), we find that the odds are roughly 2 to 1 in favor of the level specification over the difference specification. However, once (iii) is incorporated into the analysis, we find that the odds overwhelmingly favor the level specification. This finding may seem strange in light of the literature which argues that it is hard to determine whether a time series is stationary or contains a unit root.5 The resolution of this apparent contradiction is that the literature in question relies on univariate methods, while we rely on multivariate methods. Hansen (1995) shows that incorporating information from related time series has the potential to enormously increase the power of unit root tests (see also Elliott and Jansson, 2003). This phenomenon is what underlies our encompassing results. We assess the robustness of our results against alternative specifications of the low frequency component of per capita hours worked. In particular, we consider the possibility of a quadratic trend in hours worked. We show that there is a trend specification that has the implication that hours worked drops after a positive shock to technology. Using the methodology described above, we argue that the preponderance of the evidence favors the level specification relative to this alternative trend specification. The remainder of this paper is organized as follows. Section 2 discusses our strategy for identifyingtheeffectsofapermanentshocktotechnology. Section3presentstheresultsfrom a bivariate analysis using data on hours worked and the growth rate of labor productivity. Later we show that on some dimensions inference is sensitive to only including two variables in the analysis. But the bivariate systems are useful because they allow us to highlight the basic issues in a simple setting and they allow us to compare our results to a subset of the results in the literature. Section 4 reports our encompassing results and the posterior odds ratioforthebivariatesystems. InSection5weexpandtheanalysistoincludemorevariables. Here, we establish the benchmark system that we use later to assess the cyclical effects of technology shocks. Section 6 explores the robustness of our analysis to the possible presence of deterministic trends. In addition, we examine the subsample stability of our time series model. In Section 7 we report our findings regarding the overall importance of technology shocks in cyclical fluctuations. Section 8 contains concluding remarks. 5For example, see Christiano and Eichenbaum (1990). 3

2 Identifying the Effects of a Permanent Technology Shock In this section, we discuss our strategy for identifying the effects of permanent shocks to technology. We follow Gali (1999), Gali, Lopez-Salido, and Valles (2002) and Francis and Ramey(2001)andadopttheidentifyingassumptionthattheonlytypeofshockwhichaffects the long-run level of average labor productivity is a permanent shock to technology. This assumption is satisfied by a large class of standard business cycle models. See, for example, the real business cycle models in Christiano (1988), King, Plosser, Stock and Watson (1991) andChristianoandEichenbaum(1992)whichassumethattechnologyshocksareadifference stationary process.6 As discussed below, we use reduced form time series methods in conjunction with our identifying assumption to estimate the effects of a permanent shock to technology. An advantageofthisapproachisthatwedonotneedtomakealltheusualassumptionsrequiredto construct Solow-residual based measures of technology shocks. Examples of these assumptions include corrections for labor hoarding, capital utilization, and time-varying markups.7 Ofcoursethereexistmodelsthatdonotsatisfyouridentifyingassumption. Forexample, the assumption is not true in an endogenous growth model where all shocks affect productivity in the long run. Nor is it true in an otherwise standard model when there are permanent shocks to the tax rate on capital income. These caveats notwithstanding, we proceed as in the literature. We estimate the dynamic effects of a technology shock using the method proposed in Shapiro and Watson (1988). The starting point of the approach is the relationship: ∆f = µ+β(L)∆f +α˜(L)X +εz. (1) t t 1 t t − Here f denotes the log of average labor productivity and α˜(L), β(L) are polynomials of t order q and q 1 in the lag operator, L, respectively. Also, ∆ is the first difference operator − and we assume that ∆f is covariance stationary. The white noise random variable, εz, t t is the innovation to technology. Suppose that the response of X to an innovation in some t non-technologyshock, ε ,ischaracterizedbyX = γ(L)ε ,whereγ(L)isapolynomialinnont t t negativepowersofL. Weassumethateachelementofγ(1)isnon-zero. Theassumptionthat non-technology shocks have no impact on f in the long run implies the following restriction t on α˜(L) : α˜(L) = α(L)(1 L), (2) − where α(L) is a polynomial of order q 1 in the lag operator. To see this, note first that − the only way non-technology shocks can have an impact on f is by their effect on X , while t t the long-run impact of a shock to ε on f is given by: t t α˜(1)γ(1) . 1 β(1) − 6If these models were modified to incorporate permanent shocks to agents’ preferences for leisure or to government spending, these shocks would have no long run impact on labor productivity, because labor productivity is determined by the discount rate and the underlying growth rate of technology. 7See Basu, Fernald and Kimball (1999) for an interesting application of this alternative approach. 4

The assumption that ∆f is covariance stationary guarantees 1 β(1) < . This assumpt | − | ∞ tion, together with our assumption on γ(L), implies that for the long-run impact of ε on f t t to be zero it must be that α˜(1) = 0. This in turn is equivalent to (2). Substituting (2) into (1) yields the relationship: ∆f = µ+β(L)∆f +α(L)∆X +εz. (3) t t 1 t t − We obtain an estimate of εz by using (3) in conjunction with estimates of µ, β(L) and α(L). t If one of the shocks driving X is εz, then X and εz will be correlated. So, we cannot t t t t estimate the parameters in β(L) and α(L) by ordinary least squares (OLS). Instead, we apply the standard instrumental variables strategy used in the literature. In particular, we use as instruments a constant, ∆f and X , s = 1,2,...,q. t s t s − − Givenanestimateoftheshocksin(3),weobtainanestimateofthedynamicresponseoff t and X to εz as follows. We begin by estimating the following qth order vector autoregression t t (VAR): Y = α+B(L)Y +u , Eu u = V, (4) t t 1 t t 0t − where ∆f Y = t , t X t µ ¶ and u is the one-step-ahead forecast error in Y . Also, V is a positive definite matrix. The t t parameters in this VAR, including V, can be estimated by OLS applied to each equation. In practice, we set q = 4. The fundamental economic shocks, e , are related to u by the t t following relation: u = Ce , Ee e = I. t t t 0t Without loss of generality, we suppose that εz is the first element of e . To compute the t t dynamicresponseofthevariablesinY toεz,werequirethefirstcolumnofC. Weobtainthis t t column by regressing u on εz by ordinary least squares. Finally, we simulate the dynamic t t response of Y to εz. For each lag in this response function, we computed the centered 95 t t percent Bayesian confidence interval using the approach for just-identified systems discussed in Doan (1992).8 3 Bivariate Results Thissectionreportsresultsbasedonasimple, bivariateVARinwhichf isthelogofbusiness t labor productivity. Thesecondelement inY isthe logof hours workedinthe businesssector t divided by a measure of the population.9 Our data on labor productivity growth and per capita hours worked are displayed in the first row of Figure 1. We consider two sample periods. The longest period for which data are available on the variables in our VAR is 1948Q1-2001Q4. We refer to this as the long sample. The start 8This approach requires drawing B(L) and V repeatedly from their posterior distributions. Our results are based on 2,500 draws. 9OurdataweretakenfromtheDRIEconomicsdatabase. Themnemonicforbusinesslaborproductivityis LBOUT.ThemnemonicforbusinesshoursworkedisLBMN.Thebusinesshoursworkeddatawereconverted to per capita terms using a measure of the civilian population over the age of 16 (mnemonic, P16). 5

of this sample period coincides with the one in Francis and Ramey (2001) and Gali (1999). Francis and Ramey (2001) and Gali, Lopez-Salido, and Valles (2002) work, as we do, with per capita hours worked, while Gali (1999) works with total hours worked. Since much of the business cycle literature works with post-1959 data, we also consider a second sample period given by 1959Q1-2001Q4. We refer to this as the short sample. We choose to work with per capita hours worked, rather than total hours worked, since this is the object that appears in most general equilibrium business cycle models. There are two additional reasons for this choice. First, for our short sample period, there is evidence against the difference stationary specification of log total hours worked. We found this evidence using a version of the covariates adjusted Dicky-Fuller test proposed in Hansen (1995).10 Specifically, weregressedthegrowthrateoftotalhoursworkedonaconstant, time, the lag level of log total hours worked and 4 lags of the growth rate of total hours worked and 4 lags of productivity growth. We then performed an F test for the null hypothesis that the coefficient on the lag level of log total hours worked and the coefficient on time are jointly zero. This amounts to a test of the null hypothesis that log total hours worked is difference stationary, against the alternative that it is stationary about a linear trend. The F statistics for the long and short sample periods are 5.72 and 9.07, respectively. According totabulatedcritical values, the F statisticfor thelong sample exceeds the 10percent critical value. However, the F statistic for the short sample exceeds the 1 percent critical value.11 Because the short sample plays an important role in our analysis, we are uncomfortable adopting the difference stationary specification. Second, suppose we assume, as in Gali (1999), that the log of hours is stationary about a linear trend. We find this specification unappealingbecauseitimpliesthatpermanentshocks, originatingfromdemographicfactors, to total hours and total output are ruled out. Note that by working with per capita hours, we do not exclude the possibility that demographic shocks have permanent effects on total hours worked and total output. We now turn to our results. Panel A of Figure 2 displays the response of log output and log hours to a positive technology shock, based on the long sample. A number of interesting resultsemergehere. First,theimpacteffectoftheshockonoutputandhoursispositive(1.17 percent and 0.34 percent, respectively) after which both rise in a hump shaped pattern. The responses of both output and hours are statistically significantly different from zero over the 20quartersdisplayed. Second, in thelongrun, outputrises by1.33 percent. Byconstruction the long run effect on hours worked is zero. Third, since output rises by more than hours does, labor productivity also rises in response to a positive technology shock. Panel B of Figure 2 displays the analogous results for the short sample period. As before, the impact effect of the shock on output and hours is positive (0.94 and 0.14 percent, 10Other tests have been proposed by Elliott and Jansson (2003). We work with a version of Hansen’s CADF test for two reasons. First, Elliott and Jansson show in simulations that the CADF test can have better size properties but weaker power than their test. We are particularly concerned that the size of our testis correct. Second, theCADFtestis essentially thesame as our test for weakinstruments, and so using the CADF test enhances consistency of the test statistics used in the paper. 11Weusedthetabulatedcriticalvaluesin‘Case4’, TableB.7, ofHamilton(1994, p. 764). Tocheckthese, we also computed bootstrap critical values by simulating a bivariate, 4-lag VAR fit to data on the growth rate of productivity and the growth rate of total hours. The calculations were performed using the short and long sample periods. The results of these experiments coincide with what is reported in the text. 6

respectively), after which both rise in a hump-shaped pattern. The long run impact of the shock is to raise output by 0.96 percent. Again, average productivity rises in response to the shock and there is no long run effect on hours worked. The rise in output is statistically different from zero at all horizons displayed. The rise in hours is statistically significantly different from zero between one and three years after the shock. So regardless of which sample period we use, the same picture emerges: a permanent shock to technology drives hours, output and average productivity up. The previous results stand in sharp contrast to the literature according to which hours worked falls after a positive technology shock. The difference cannot be attributed to our identifying assumptions or the data that we use. To see this, note that we reproduce the bivariate-based results in the literature if we assume that X in (1) and (3) corresponds to t the growth rate of hours worked rather than the level of hours worked. The two panels in Figure 3 display the analogous results to those in Figure 2 with this change in the definition of X . t AccordingtothepointestimatesdisplayedinPanelsAandBofFigure3, apositiveshock totechnologyinducesariseinoutput, butapersistentdeclineinhoursworked.12 Confidence intervals are clearly very large. Still, the initial decline in hours worked is statistically significant. This result is consistent with the bivariate analysis in Gali (1999) and Francis and Ramey (2001). The question is: Which results are more plausible, those based on the level specification or the difference specification? We turn to this question in the next section. 4 Analyzing the Bivariate Results The previous section presented conflicting answers to the question: how do hours worked respond to a positive technology shock? Each answer is based on a different statistical model, corresponding to whether we assume that hours worked are difference stationary or stationary in levels. To determine which answer is more plausible, we need to select between the underlying statistical models. The first subsection below addresses the issue usingstandardclassicaldiagnostictestsandshowsthattheydonotconvincinglydiscriminate between the competing models. The following sections address the issue using encompassing methods. 4.1 Standard Classical Diagnostic Tests We begin by testing the null hypothesis of a unit root in hours worked using the Augmented Dickey Fuller (ADF) test. For both sample periods, this hypothesis cannot be rejected at the 10 percent significance level.13 Evidently we cannot rule out the difference specification, 12For the long sample, the contemporaneous effect of the shock is to drive output up by 0.56 percent and hours down by 0.31 percent. The long run effect of the shock is to raise output by 0.84 percent and hours worked by 0.06 percent. For the short sample, the contemporaneous effect of the shock is to raise output 0.43 percent and reduce hours worked by 0.30 percent. The long run effect of the shock is to raise output by 0.74 percent and hours worked by 0.05 percent. 13For the long and short sample, the ADF test statistic is equal to 2.46 and 2.49, respectively. The − − critical value corresponding to a 10 percent significance level is 2.57. In Appendix C, we compute the − 7

at least based on this test. Of course it is well known that standard unit root tests have very poor power properties relative to the alternative that the time series in question is a persistent stationary stochastic process. So while it is always true that failure to reject a null hypothesis does not mean we can reject the alternative, this caveat is particularly relevant in the present context. To test the null hypothesis that per capita hours is a stationary stochastic process (with no time trend) we use the KPSS test (see Kwiatkowski et al. (1992)).14 For the short sample period, we cannot reject, using standard asymptotic distribution theory, the null hypothesis at the five percent significance level.15 For the long sample period, we can reject the null hypothesis at this level. However, it is well known that the KPSS test (and close variants like the Leybourne and McCabe (1994) test) rejects the null hypothesis of stationarity too often if the data-generating process is a persistent but stationary time series.16 It is common practice touse size-correctedcritical values that areconstructed usingdata simulated froma particular data-generating process.17 We did so using the level specification VAR estimated over the long sample. Specifically, using this VAR as the data-generating process, we generated 1000 synthetic data sets, each of length equal to the number of observations in the long sample period, 1948-2001.18 For each synthetic data set we constructed the KPSS test statistic. In 90 and 95 percent of the data sets, the KPSS test statistic was smaller than 1.89 and 2.06, respectively. The value of this statistic computed using the actual data over the period 1948-2001 is equal to 1.24. Thus we cannot reject the null hypothesis of stationarity at conventional significance levels. 4.2 Encompassing Tests: A Priori Considerations The preceding subsection showed that conventional classical methods are not useful for selecting between the level and difference specifications of our VAR. An alternative way to select between the competing specifications is to use an encompassing criterion. Under this criterion, a model must not just be defensible on standard classical diagnostic grounds. It must also be able to predict the results based on the opposing model. If one of the two views fails this encompassing test, the one that passes is to be preferred. In what follows we review the impact of specification error and sampling uncertainty on critical values based on bootstrap simulations of the estimated difference model based on thelongand short samples. The 10 percent critical values are -2.87 and -2.78, respectively. These critical values also result in a failure to reject at the 10 percent significance level. 14In implementing this test we set the number of lags in our Newey-West estimator of the relevant covariance matrix to eight. 15The value of the KPSS test statistic is 0.4. The asymptotic critical values corresponding to ten and five percent significance levels are 0.347 and 0.46, respectively. 16See Table 3 in Kwiatkowski et al. (1992) and also Caner and Kilian (1999) who provide a careful assessment of the size properties of the KPSS and Leybourne and McCabe tests. 17CanerandKilian(1999)providecriticalvaluesrelevantforthecaseinwhichthedatageneratingprocess is a stationary AR(1) with an autocorrelation coefficient of 0.95. Using this value we fail to reject, at the five percent significance level, the null hypothesis of stationarity over the longer sample period. 18The maximal eigenvalue of the estimated level specification VAR is equal to 0.972. We also estimated univariateAR(4)representations forhours workedusing thesyntheticdatasets and calculated themaximal roots for the estimated univariate representations of hours worked. In no case did the maximal root exceed one. Furthermore, 95 percent of the simulations did not have a root greater than 0.982. 8

theabilityofeachspecificationtoencompasstheother. Otherthingsequal, thespecification, that will do best on the encompassing test, is the one that predicts the other model is misspecified. This consideration leads us to expect the level specification to do better. This is because the level specification implies the first difference specification is misspecified , while the difference specification implies the level specification is correctly specified. This consideration is not definitive because sampling considerations also enter. For example, the difference specification implies that the level specification suffers from a weak instrument problem. Weak instruments can lead to large sampling uncertainty, as well as bias. These considerations may help the difference specification. 4.2.1 Level Specification Suppose the level specification is true. Then the difference specification is misspecified. To see why, recall the two steps involved in estimating the dynamic response of a variable to a technology shock. The first involves the instrumental variables equation used to estimate the technology shock itself. The second involves the vector autoregression used to obtain the actual impulse responses. Supposetheeconometricianestimatestheinstrumental variables equationunderthemistaken assumption that hours worked is a difference stationary variable. In addition, assume that the only variable in X is log hours worked. The econometrician would difference X t t twice and estimate µ along with the coefficients in the finite-ordered polynomials, β(L) and α(L), in the system: ∆f = µ+β(L)∆f +α(L)(1 L)∆X +εz. t t 1 t t − − Suppose that X has not been overdifferenced, so that its spectral density is different from t zero at frequency zero. Then, in the true relationship, the term involving X is actually t α¯(L)∆X , where α¯(L) is a finite ordered polynomial. In this case, the econometrician comt mits a specification error because the parameter space does not include the true parameter values. The only way α(L)(1 L) could ever be equal to α¯(L) is if α(L) has a unit pole, i.e., − if α(L) = α¯(L)/(1 L). But, this is impossible, since no finite lag polynomial, α(L), has − this property. So, incorrectly assuming that X has a unit root entails specification error. t We now turn to the VAR used to estimate the response to a shock. A stationary series that is first differenced has a unit moving average root. It is well known that there does not exist a finite-lag vector autoregressive representation of such a process. So here too, proceeding as though the data are difference stationary entails a specification error. Of course, it would be premature to conclude that the level specification is likely to encompass the difference specification’s results. For this to occur, the level specification has to predict not just that the difference specification entails specification error. It must be that the specification error is enough to account quantitatively for the finding one obtains when adopting the difference specification. 4.2.2 Difference Specification Suppose the difference specification is true. What are the consequences of failing to assume a unit root in hours worked, when there in fact is one? To answer this question, we must 9

address two sets of issues: specification error and sampling uncertainty. With respect to the former, note that there is no specification error in failing to impose a unit root. To see this, first consider the instrumental variables regression: ∆f = µ+β(L)∆f +α(L)∆X +εz. (5) t t 1 t t − Here, the polynomials, β(L) and α(L), are of order q and q 1, respectively. The econo- − metrician does not impose the restriction α(1) = 0 when it is, in fact, true. This is not a specification error, because the parameter space does not rule out α(1) = 0. In estimating the VAR, the econometrician also does not impose the restriction that hours worked is difference stationary. This also does not constitute a specification error because the level VAR allows for a unit root (see Sims, Stock and Watson (1990)). We now turn to sampling uncertainty. Recall that the econometrician who adopts the level specification uses lagged values of X as instruments for ∆X . But if X actually has a t t t unit root, this entails a type of weak instrument problem. Lagged X ’s are poor instruments t for ∆X because ∆X is driven by relatively recent shocks while X is heavily influenced by t t t shocks that occurred long ago. At least in large samples, there is little information in lagged X ’s for ∆X .19 t t Results in the literature suggest that weak instruments can lead to substantial sampling uncertainty. This uncertainty could help the difference specification encompass the level results simply as a statistical artifact. In addition, weak instruments can lead to bias, which could also help the difference specification. The implications of the literature (see, for example, Staiger and Stock (1997)) for the weak instrument problem are suggestive, though not definitive in our context.20 Since the precise nature of the problem is somewhat different here, we now briefly discuss it.21 First, we analyze the properties of the instrumental variables estimator. We then turn to the impulse response functions. Suppose the instrumental variables relation is given by (5) with µ = 0. Let the predetermined variables in this relationship be written as: z¯ = [∆f ,...,∆f ,∆X ,...,∆X ]. t t 1 t q t 1 t q − − − − So, the right hand side variables in (5) are given by x = [z¯,∆X ]. The econometrician who t t t adopts the level specification uses instruments composed of q lagged ∆f ’s and q+1 lagged t 19To see this, consider the extreme case in which X is a random walk. In this case, X is the sum of t t 1 shocks at date t 1 and earlier, while ∆X is a function only of date t shocks. In this c−ase, there is no t − overlapbetween∆X andX .Moregenerally,when∆X iscovariancestationary,itisasquaresummable t t 1 t function of current and past−shocks, while X is not. In this sense, the weight placed by X on shocks t 1 t 1 in the distant past is larger than the weight p−laced by ∆X on those shocks. − t 20For a discussion of this in the context of instrumental variables regressions of consumption growth on income, see Christiano (1989) and Boldrin, Christiano and Fisher (1999). 21A similar weak instrument problem is studied in dynamic panel models. This literature considers the case when the lagged level of a variable is used to instrument for its growth rate and the variable is nearly a unit root process. The literature studies the consequences of the resulting weak instrument problem when the panel size increases, holding the number of time periods fixed (see Blundell and Bond 1998, and Hahn, Hausman, and Kuersteiner 2003.) Our focus is on what happens as the number of observations increases. 10

X ’s. This is equivalent to working with the instrument set z = [z¯,X ]. Relation (5) can t t t t 1 − be written as: ∆f = x δ +εz. t t t The instrumental variables estimator, δIV, expressed as a deviation from the true parameter value, δ, is 1 1 1 δIV δ = z x − z εz . (6) − T t0 t T t0 t µ ¶ µ ¶ X X Here signifies summation over t = 1,...,T. To simplify notation, we also do not index the estimator, δIV, by T. Relation (6) implies P δIV δ = T 1 z¯ t0 z¯ t T 1 z¯ t0 ∆X t − 1 T 1 z¯ t0 εz t − · T 1 P X t − 1 z¯ t T 1 P X t − 1 ∆X t ¸ · T 1 P X t − 1 εz t ¸ 1 L P Q z¯z¯ Q z¯∆X P− 0 , P → ϕ ζ % · ¸ µ ¶ L where ‘ ’ signifies ‘converges in distribution’. Here, ϕ, ζ and % are well defined random → variables, constructed as functions of integrals of Brownian motion (see, e.g., Proposition 18.1 in Hamilton, 1994, pages 547-548). According to the previous expression, δIV δ has − a non-trivial asymptotic distribution. By contrast, suppose ‘strong’ instruments, such as ∆X , s > 0, are used. Then, the asymptotic distribution of δIV δ collapses onto a single t s − − point and there is no sampling uncertainty. This is the sense in which our type of weak instruments lead to large sampling uncertainty. See Appendix B for an analytic example. Nowconsiderthelargesampledistributionofourestimatorofimpulseresponsefunctions. Denote the contemporaneous impact on h of a one-standard deviation shock to technology t by Ψ = E(u εz)/σ . Here, u denotes the disturbance in the VAR equation for ∆X . We 0 t t εz t t denote the estimator of Ψ by ΨIV: 0 0 1 1/2 ΨIV = ρIV uˆ2 , 0 T t · ¸ X 1 uˆ εz,IV ρIV = T t t . 1/2 2 1 uˆ2 1/2P1 εz,IV T t T t · ¸ ³ ´ £ P ¤ P Here, uˆ is the fitted value of u and εz,IV is the instrumental variables estimator of the t t t technology shock:22 εz,IV = ∆f x δIV = x δ δIV +εz. t t t t t − − The formulas provided by Hamilton (1994, Theor¡em 18.1)¢ can be used to show that the asymptotic distribution of ΨIV exists and is a function of the asymptotic distribution of 0 δ δIV (see Appendix B for an illustration). This result follows from two observations. − First, the parameter estimates underlying uˆ converge in probability to their true value. So, t 22Here,uˆ isthefittedresidualcorrespondingtou ,theseconddisturbancein(4). Wedeletethesubscript, t 2t 2, to keep from cluttering the notation. 11

1 uˆ2 converges in probability to σ2, the variance of u . This is true even when the VAR is T t u t estimated using the level of X (see Sims, Stock and Watson, 1990). Second, by assumption t P bothx andεz arestationaryvariableswithwell-definedfirstandsecondmoments. Itfollows t t that the asymptotic distribution of ΨIV is non-trivial because the asymptotic distribution of 0 δIV isnon-trivial. TheexactasymptoticdistributionofΨIV canbeworkedoutbyapplication 0 of the results in Hamilton (1994, theorem 18.1). The previous reasoning establishes that the weak instrument problem leads to high sampling uncertainty in ΨIV. In addition, there is no reason to think that the asymptotic 0 distribution of ΨIV is even centered on Ψ . Appendix B presents an example where ΨIV is 0 0 0 centered at zero. The previous analysis raises the possibility that the moments of estimators of interest to us may not exist. In fact, it is not possible to guarantee that the asymptotic distribution of δIV has well-defined first and second moments. For example, in numerical analysis of a specialcasereportedinAppendixB,wefindthattheasymptoticdistributionofδIV resembles a Cauchy distribution, which has a median, but no mean or variance. For the simulation methodologythatweusebelow, itiscrucialthatdistributionsofimpulseresponseestimators have first and second moments. Fortunately, all the moments of the asymptotic distribution ofΨIV arewell defined. ThisfollowsfromthefactsthatρIV isacorrelationandσˆ converges 0 u in probability to σ . These two observations imply that the asymptotic distribution of ΨIV u 0 has compact support, being bounded above by σ and below by σ . u u − Tosummarize,inthissubsectionweinvestigatedwhathappenswhenananalystestimates an impulse response function using the level specification when the difference specification is true. Our results can be summarized as follows. First and second moments of the estimator are well defined. However, the estimator may be biased and may have large sampling uncertainty. 4.3 Does the Level Specification Encompass the Difference Specification Results? To assess the ability of the level specification to encompass the difference specification, we generated two groups of one thousand artificial data sets from the estimated VAR in which the second element of Y is the log level of hours worked. In the first and second t group, the VAR corresponds to the one estimated using the long and short sample period, respectively. So in each case the data generating mechanism corresponds to the estimated level specification. The number of observations in each artificial data set of the two groups is equal to the corresponding number of data points in the sample period. In each artificial data sample, we proceeded under the (incorrect) assumption that the difference specification was true, estimated a bivariate VAR in which hours worked appears in growth rates, and computed the impulse responses to a technology shock. The mean impulse responses appear as the thin line with circles in Figure 4. These correspond to the prediction of the level specification for the impulse responses that one would obtain with the (misspecified) difference specification. The lines with triangles are reproduced from Figure 3 and correspond to our point estimate of the relevant impulse response function generated from the difference specification. The gray area represents the 95 percent confidence interval 12

of the simulated impulse response functions.23 From Figure 4 we see that, for both sample periods, the average of the impulse response functions emerging from the ‘misspecified’ growth rate VAR are very close to the actual estimated impulse response generated using the difference specification. Notice in particular that hours worked are predicted to fall after a positive technology shock even though they rise in the actual data-generating process. Evidently the specification error associated with imposingaunitrootinhoursworkedislargeenoughtoaccountfortheestimatedresponseof hoursthatemergesfromthedifferencespecification. Thatis,ourlevelspecificationattributes the decline in hours in the estimated VAR with differenced hours to over-differencing. Note also that in all cases the estimated impulse response functions associated with the difference specification lie well within the 95 percent confidence interval of the simulated impulse response functions. We conclude that the level specification convincingly encompasses the difference specification. 4.4 Does the Difference Specification Encompass the Level Results? To assess the ability of the difference specification to encompass the level specification, we proceeded as above except now we take as the data-generating process the estimated VAR’s in which hours appears in growth rates. Figure 5 reports the analogous results to those displayed in Figure 4. The thick, solid lines, reproduced from Figure 2, are the impulse responses associated with the estimated level specification. The thin lines with the triangles are reproduced from Figure 3 and are the impulse responses associated with the difference specification. ThethinlineswithcirclesinFigure5arethemeanimpulseresponsefunctionsthatresult from estimating the level specification of the VAR using the artificial data. They represent the difference specification’s prediction for the impulse responses that one would obtain with the level specification. The gray area represents the 95 percent confidence interval of the simulated impulse response functions. This area represents the difference specification’s prediction for the degree of sampling uncertainty that an econometrician working with the level specification would find. Two results are worth noting. First, the thin line with triangles and the thin line with circles are very close to each other. Evidently, the mean distortions associated with not imposing a unit root in hours worked are not very large. In particular, the difference specification predicts - counterfactually - that an econometrician who adopts the level specification will find that average hours fall for a substantial period of time after a positive technology shock. Notice, however, the wide confidence interval about the thin line, which includes the thick, solid line. So, the difference specification can account for the point estimates based on the level specification, but only as an accident of sampling uncertainty. Atthesametime,thepredictionoflargesamplinguncertaintyposesimportantchallenges to the difference specification. First, the prediction of large sampling uncertainty rests fundamentally on the difference specification’s implication that the econometrician working 23Confidence intervals were computed point wise as the average simulated response plus or minus 1.96 times the standard deviation of the simulated responses. 13

with the level specification encounters a weak instrument problem. As we show below, when we apply a standard test for weak instruments to the data, we find little evidence of this problem. Second, the estimated confidence intervals associated with impulse responses from the estimated level specification are relatively narrow (see Figure 2). We suspect that this is hardtoreconcilewiththedifferencespecification’simplicationoflargesamplinguncertainty. To assess whether there is evidence of weak instruments in the data, we examined a standard F test for weak instruments. We regressed ∆X on a constant, X , and the t t 1 − predetermined variables in the instrumental variables regression, (5). These are ∆X and t s − ∆f , s = 1,2,3. Our weak instruments F statistic is the square of the t statistic associated t s − with the coefficient on X . In effect, our F statistic measures the incremental information t 1 in X about ∆X .24 If t − he difference specification is correct, the additional information is t 1 t − zero. For the sample periods, 1948-2001 and 1959-2001, the value of our test statistic is 10.94 and 10.59, respectively. To assess the significance of these F statistics, we proceeded using the following bootstrap procedure. For each sample period, we simulated 2,500 artificial data sets using the corresponding estimated difference specification as the data-generating process. For the 1948-2001 sample, we found that 2.3 percent of the simulated F statistics exceed 10.94. For the shorter sample, the corresponding result is 0.84 percent. So, in the shortsample, theweakinstrumenthypothesisisstronglyrejected. Theevidenceissomewhat more mixed in the longer sample. The evidence against the difference specification reported here is stronger than we obtained using the ADF test in section 4.1. This is consistent with the analysis of Hansen (1995) and Elliott and Jansson (2003), who show that incorporating additional variables into unit root tests can dramatically raise their power. Monte Carlo studies presented in Appendix C make, in our context, this power gain concrete. 4.5 Quantifying the Relative Plausibility of the Two Specifications The results of the previous two subsections indicate that the level specification can easily accountfortheestimatedimpulseresponsefunctionsobtainedwiththedifferencespecification. The difference specification has a harder time. While it can account for the level results, its ability to do so rests fundamentally on its implication that the level specification is distorted by a weak instrument problem. In this section we quantify the relative plausibility of the two specifications. We do so using the type of posterior odds ratio considered in Christiano and Ljungqvist (1988) for a similar situation where differences and levels of data lead to very different inferences.25 The basic idea is that the more plausible of the two VAR’s is the one that has the easiest time explaining the facts: (i) the level specification implies that hours worked rise after a technology shock, (ii) the difference specification implies that hours 24Our F test is equivalent to a standard ADF test with additional regressors. In the unit root testing literature, this test is referred to as the covariate ADF test (Hansen 1995). 25EichenbaumandSingleton(1988)found,inaVARanalysis,thatwhentheyworkedwithfirstdifferences of variables, there was little evidence that monetary policy plays an important role in business cycles. However,whentheyworkedwithatrendstationaryspecification,monetarypolicyseemstoplayanimportant role in business cycles. Christiano and Ljungqvist argued that the preponderance of the evidence supported the trend stationary specification. 14

worked falls, and (iii) the value of the weak instruments F statistic. We use a scalar statistic - the average percentage change in hours in the first six periods after a technology shock - to quantify our findings for hours worked. The level specification estimates imply this change, µ ,is equal to 0.89 and 0.55 for the long and short sample h period, respectively. The analogous statistic, µ , for the growth specification is 0.13 and ∆h − 0.17 in the long and short sample period, respectively. − To evaluate the relative ability of the level and difference specification to simultaneously account for µ and µ , we proceed as follows. We simulated 1,000 artificial data sets using h ∆h each of our two estimated VARs as the data generating mechanism. In each data set, we calculated (µ ,µ )using the same method used to compute these statistics in the actual h ∆h data. To quantify the relative ability of the two specifications to account for the estimated values of (µ ,µ ), we computed the frequency of the joint event, µ > 0 and µ < 0. For h ∆h h ∆h the long sample period, the level and difference specifications imply that this frequency is 65.2 and 34.2, respectively. That is, P(Q A) = 0.65 | P(Q B) = 0.34, | where Q denotes the event, µ > 0 and µ < 0, A indicates the level specification, B h ∆h indicates the difference specification and P denotes the percent of the impulse response functions in the artificial data sets in which µ > 0 and µ < 0. Suppose that our priors h ∆h over A and B are equal: P(A) = P(B) = 1/2. The unconditional probability of Q, P(Q), is 0.65 0.5+0.34 0.5 = 0.495. The probability of the two specifications, conditional on × × having observed Q, is: P(A,Q) P(Q A)P(A) P(A Q) = = | = 0.657 | P(Q) P(Q) P(B,Q) P(Q B)P(B) P(B Q) = = | = 0.343. | P(Q) P(Q) So, we conclude that, given these observations, the odds in favor of the level specification relative to the difference specification are 1.9 to 1. Similar results emerge for the short sample period. Here the percent of impulse response functions in the bottom right hand quadrant is 52.4 in the artificial data generated by the level specification, while it is 25.6 for the difference specification. The implied values of P(Q A) and P(Q B) are 0.672 and 0.328. So, the odds in favor of the level specification | | relative to the difference specification are slightly larger than two to one. We now incorporate into our analysis information about the relative ability of the two specifications to account for the weak instruments F statistic. We do this by redefining Q to be the event, µ < 0, µ > 0, and F > 10.94, for the long sample. Recall that 10.94 is ∆h h the value of the F statistic obtained using the actual data from the long sample. We find that P(Q A) = 0.38 and P(Q B) = 0.01. This implies that the odds in favor of the level | | specification relative to the difference specification are 26.08 to one. The analogous odds based on the short sample period are 67.67 to one. Evidently, the odds ratio jumps enormously when the weak instruments F statistic is incorporated into the analysis. Absent the F statistic, the difference specification has some 15

ability to account for the impulse response function emerging from the level specification. But, this ability is predicated on the existence of a weak instrument problemassociated with hours worked. In fact, our F test indicates that there is not a weak instrument problem. We conclude that, based on these purely statistical grounds, the level specification and its implications are more plausible than those of the difference specification. Of course the odds in favor of the level specification would be even higher if we assigned more prior weight to the level specification. For reasons discussed in the introduction this seems quite natural to us. Our own prior is that the difference specification simply cannot be true because per capita hours worked are bounded. 5 Moving Beyond Bivariate Systems In the previous two sections we analyzed the effects of a permanent technology shock using a bivariate system. In this section we extend our analysis to allow for a richer set of variables. We do so for two reasons. First, the responses of these other variables are interesting in their own right. Second, there is no a priori reason to expect that the answers generated from small bivariate systems will survive in larger dimensional systems. If variables other than hours worked belong in the basic relationship governing the growth rate of productivity, and these are omitted from (1), then simple bivariate analysis will not generally yield consistent estimates of innovations to technology. Our extended system allows for four additional macroeconomic variables: the federal funds rate, the rate of inflation, the log of the ratio of nominal consumption expenditures to nominalGDP,andthelogoftheratioofnominalinvestmentexpenditurestonominalGDP.26 The last two variables correspondto theratioof real investment andconsumption, measured in units of output, to total real output. Standard models, including those that allow for investment-specific technical change, imply these two variables are covariance stationary.27 Data on our six variables are displayed in Figure 1. 5.1 Level and Difference Specification Results To conserve on space we focus on the 1959 - 2001 sample period.28 Figure 6 reports the impulse response functions corresponding to the level specification, i.e., the system in which the log of per capita hours worked enters in levels. As can be seen, the basic qualitative 26Ourmeasuresofthegrowthrateoflaborproductivityandhoursworkedarethesameasinthebivariate system. Wemeasured inflation using the growth rate of the GDP deflator, measured as the ratio of nominal outputtorealoutput(GDP/GDPQ).Consumptionismeasuredasconsumptiononnondurablesandservices and government expenditures: (GCN+GCS+GGE). Investment is measured as expenditures on consumer durablesandprivateinvestment: (GCD+GPI).ThefederalfundsseriescorrespondstoFYFF.Allmnemonics refer to DRI’s BASIC economics database. 27See for example Altig, Christiano, Eichenbaum and Linde (2002). This paper posits that investment specific technical change is trend stationary. See also Fisher (2003), which assumes investment specific technical change is difference stationary. Both frameworks imply that the consumption and investment ratios discussed in the text are stationary. 28Data on the federal funds rate is available starting only in 1954. We focus on the post 1959 results so thatwecancompareresultstothebivariateanalysis. Wefoundthatour6variableresultswerenotsensitive to using data that starts in 1954. 16

results from the bivariate analysis regarding hours worked and output are unaffected: both rise in hump-shaped patterns after a positive shock to technology.29 The rise in output is statistically significant for roughly two years after the shock, while the rise in hours worked is statistically significant at horizons roughly two to eight quarters after the shock. Turning to the other variables in the system, we see that the technology shock leads to a prolonged, statistically significant fall in inflation and a statistically insignificant rise in the federal funds rate. Both consumption and investment rise, with a long run impact that is, by construction, equal to the long run rise in output.30 The rise in consumption is estimated with much more precision than the rise in investment. Figure 7 reports the impulse response functions corresponding to the difference specification, i.e. the system in which the log of per capita hours enters in first differences. Here a permanent shock to technology induces a long lived decline in hours worked, and a rise in output.31 In the long run, the shock induces a 0.55 percent rise in output and a 0.25 percent decline in hours worked. Turning to the other variables, we see that the shock induces a rise in consumption and declines in the inflation rate and the federal funds rate. Investment initially falls but then starts to rise. Perhaps the key thing to note is the great deal of sampling uncertainty associated with the point estimates. For the horizons displayed, none of the changes in hours worked, output, consumption, investment or the federal funds rate are statistically significant. The only changes that are significant are the declines in the inflation rate. Evidently, if one insists on the difference specification, the data are simply uninformativeabouttheeffectofapermanenttechnologyshockonhoursworkedoranything else except the inflation rate. 5.2 Encompassing Results We now turn to the question of whether the level specification can encompass the difference specification results. As with the bivariate systems, we proceeded as follows. First, we generated one thousand artificial data sets from the estimated six-variable level specification VAR. The number of observations in each artificial data set is equal to the number of data points in the sample period, 1959 - 2001. In each artificial data sample, we estimated a six-variable VAR in which hours worked appears in growth rates and computed the impulse responses to a technology shock. The mean impulse responses appear as the thin line with circles in Figure 8. These responses correspond to the impulse responses that would result from the difference specification VAR being estimated on data generated from the level specification VAR. The thin lines with triangles are reproduced from Figure 7 and correspond to our point estimate of the relevant impulse response function generated from the difference specification. The gray area repre- 29Thecontemporaneouseffectoftheshockistodriveoutputandhoursworkedupby0.51percentand0.11 percent, respectively. The long run effect of the shock is to raise output by 0.97 percent. By construction the shock has no effect on hours worked in the long run. 30The contemporaneous effect of the shock is to drive consumption and investment up by 0.42 and 0.90 percent, respectively. The long run effect of the shock is to raise both consumption and investment by 0.97 percent. 31The contemporaneous effect of the shock is to drive output up by 0.12 percent and hours worked down by 0.27 percent. − 17

sents the 95 percent confidence interval of the simulated impulse response functions.32 The thick black line corresponds to the impulse response function fromthe estimated six-variable level specification VAR. The average impulse response function emerging from the ‘misspecified’ difference specification is very close to the actual estimated impulse response generated using the difference specification. As in the bivariate analysis, hours worked are predicted to fall after a positive technology shock even though they rise in the actual data-generating process. Also, in all casestheestimatedimpulseresponsefunctionsassociatedwiththedifferencespecificationlie well within the 95 percent confidence interval of the simulated impulse response functions. So, as before, we conclude that the specification error associated with imposing a unit root in hours worked is large enough to account for the estimated response of hours that emerges from the difference specification. Wenowconsiderwhetherthedifferencespecificationcanencompassthelevelspecification results. To do this we proceed as above except that we now take as the data-generating process the estimated VARs in which hours appears in growth rates. Figure 9 reports the analogous results to those displayed in Figure 8. The thick, solid lines, reproduced from Figure6, aretheimpulseresponsefunctionsassociatedwiththeestimatedlevelspecification. The thin line with the triangles are reproduced from Figure 7 and correspond to our point estimate of the impulse response function generated from the difference specification. The gray area represents the 95 percent confidence interval of the simulated impulse response functions. The thin line in Figure 9 with circles is the mean impulse response function associated with estimating the level specification VAR on data simulated using, as the data-generating process, the difference specification VAR. Notice that the lines with triangles and circles are very similar. So, focusing on point estimates alone, the difference specification is not able to account for the actual finding with our estimated level VAR that hours worked rise. Still, in theendthedifferencespecificationiscompatiblewithourlevelresultsonlybecauseitpredicts so much sampling uncertainty. As discussed earlier, this reflects the difference specification’s implicationthatthelevelmodelhasweakinstruments. Asinthebivariatecase, thereislittle empirical evidenceforthis. Sincetherearemorepredeterminedvariablesintheinstrumental variables regression, the weak instrument F statistic now has a different value, 21.68. This rejects the null hypothesis of weak instruments at the one percent significance level. 5.3 The Relative Plausibility of the Two Specifications Asinthebivariatesystem, wefirstquantifytherelativeplausibilityofthelevelanddifference specifications with a scalar statistic: the average percentage change in hours in the first six periods after a technology shock. The estimated level specification implies this change, µ , is equal to 0.31. The statistic for the difference specification, µ , is 0.29. We then h ∆h − incorporate the weak instrument F statistic into the analysis. We simulated 1,000 artificial data sets using each of our two estimated VARs as data generating mechanisms. In each data set, we calculated (µ ,µ )using the same method h ∆h 32These confidence intervals are computed in the same manner as the intervals reported for the bivariate encompassing tests. The interval is the average simulated impulse response plus or minus 1.96 times the standard deviation of the simulated impulse responses. 18

used to compute these statistics in the actual data. Using each of our two time series representations, we computed the frequency of the joint event, µ > 0 and µ < 0. This h ∆h frequencyis66.7acrossartificialdatasetsgeneratedbythelevelspecification, whileitis36.7 in the case of the difference specification. The implied odds in favor of the level specification over the difference specification are 1.8 to one. Next, we incorporate the fact that the weak instrument F statistic takes on a value of 21.68. Incorporating this information into our analysis implies that the odds in favor of the level specification relative to the difference specification jumps dramatically to a value of 333.0 to one. So as with our bivariate systems, we conclude on these purely statistical grounds that the level specification and its implications are more ‘plausible’ than those of the difference specification. 6 Sensitivity Analysis In this section we investigate the sensitivity of our analysis along three dimensions: the choiceof variablestoincludeintheanalysis, allowingfordeterministictrendsandsubsample stability. 6.1 Sensitivity to Choice of Variables While the qualitative effects of a permanent shock to technology are robust across the bivariate and six-variable systems, the quantitative effects are quite different. One way to see this is to compare the relevant impulse response functions (see Figures 2 and 6). A different way to do this is to assess the importance of technology shocks in accounting for aggregate fluctuations using the bivariate and six-variables systems. In the next section, we show that technology shocks are much less important in the larger system. To help us analyze the sources of this sensitivity, we now briefly report results from two four variable systems. In the first, the CI system, we add two variables to the benchmark bivariate system: the ratio of consumption expenditures to nominal GDP and the ratio of investment expenditures to nominal GDP. In the second, the Rπ system, we add the federal funds rate and the inflation rate to the benchmark bivariate system. Figure 10 reports the point estimates of the impulse response functions from the level specification six-variable system(depicted by the thick line), the CI system(depicted by the linewith‘*’)andtheRπ system(depictedbythelinewith‘X’).Tworesultsareworthnoting. First, the six-variable and the CI systems generate very similar results for the variables that are included in both. Second, the six-variable and the Rπ systems generate qualitatively different responses of hours worked. In both the six-variable and the CI systems, the impact effect of a positive technology shock on hours worked is positive after which they continue to rise in a hump shaped pattern. But in the Rπ system, hours worked falls for roughly 3 quarters after a positive technology shock. The most natural interpretation of this result is specification error. Both the CI and Rπ systems are misspecified relative to the six-variable system. But the quantitative effect of the specification error associated with omitting consumption and investment from the analysis (the Rπ system) is sufficiently large to affect qualitative inference about the effect 19

of a technology shock on hours worked. Of course, if the six-variable system is specified correctly, it should be able to rationalize the response of hours worked in the Rπ system. To see if this is the case, we proceeded as follows. First, we generated one thousand artificial data sets from the estimated six-variable VAR. The number of observations in each artificial data set is equal to the number of data points in the short sample period. In each artificial data sample, we estimated a VAR for the four variable Rπ system and computed theimpulseresponsestoatechnologyshock. Themeanimpulseresponsesappearasthethin line with circles in Figure 11. These correspond to the prediction of the six-variable VAR for the impulse responses one obtains using the Rπ system VAR. The thin line with the ‘X’ are reproduced from Figure 10 and correspond to our point estimate of the relevant impulse response function generated fromthe Rπ system. The gray area represents the 95 confidence interval of the simulated impulse response functions. The thick black line corresponds to the impulse response function from the estimated six-variable VAR. Note that the average impulse response functions emerging from the ‘misspecified’ Rπ system are very close to the estimated impulse responses generated using the actual Rπ system. Sothespecificationerrorassociatedwithomittingconsumptionandinvestmentislarge enough to account for the estimated response of hours that emerges from the Rπ specification. In all cases the estimated impulse response functions associated with the misspecified Rπ specification lie well within the 95 percent confidence interval of the simulated impulse response functions.33 We conclude that it is important to include at least C and I in our analysis. While it may be desirable to include R and π on a priori grounds, the results of central interest here seem to be less sensitive to omitting them. 6.2 Quadratic Trends From Figure 1 we see that per capita hours worked seem to follow a U shaped pattern. This suggests the possibility that hours worked may be stationary around a quadratic trend. If so, then the systems considered above are misspecified and may generate misleading results. With this in mind, we investigate two issues. First, is the response of hours worked to a technology shock sensitive to imposing a quadratic trend in hours worked? Second, to the extent that the results are sensitive, which set of results is most plausible? We begin by redoing our analysis of the six-variable system with two types of quadratic trends. In case (i), we allow for a quadratic trend in all the variables of the VAR. This seems natural since other variables like inflation and the interest rate also exhibit U shaped behavior (see Figure 1). In case (ii), we allow for a quadratic trend only in per capita hours worked. Except for these trends the other variables enter the system as in the level specification. Figure 12 reports our results. The dark, thick lines correspond to the impulse responsefunctionsimpliedbythesix-variablelevel specification. Thelinesindicatedwith0’s and x’s correspond to the impulse response functions generated fromthis systemmodified as 33For completeness, werepeated the analysis for thesystems in which hours enter in growth rates. Again, the six-variable and the CI systems are more similar to each other than the Rπ system. However, the response of consumption is much smaller in the CI system than in the six-variable system. Finally, we computed the analogous results to those in Figure 14 and again found that the six-variable system can encompass the CI growth rate system. 20

described in (i) and (ii) above. The grey area is the 95 percent confidence interval associated with the lines indicated with x’s. We report only this confidence interval, rather than all three, in order to give some sense of sampling uncertainty while keeping the figure relatively simple. Three things are worth noting. First, if we allow for a quadratic trend in all of the variables in the VAR, after a small initial fall, hours worked rise as in the level specification in response to a positive technology shock. Second, if we allow for a quadratic trend only in hours worked, then hours worked do in fact fall in a persistent way after a positive shock to technology. Third, in either case, the impulse response function of hours worked is estimated with very little precision. One cannot reject the views that hours worked rise, fall or do not change. If one insists on allowing for quadratic trends, then there is simply very little information in the data about the response of hours worked to a technology shock. Still, focusing on the point estimates alone, the estimated response of hours worked to a technology shock is sensitive to whether we include a quadratic trend in hours worked. We now turn to the question of which results are more plausible: those based on our 6-variable level specification, or those based on the quadratic trend specifications. We begin by performing a classical test of the null hypothesis of no trend in per capita hours worked. Specifically, we regress the log of per capita hours worked on a constant, time and time-squared. We then compute the t statistic for the time-squared term allowing for serial correlation in the error term of the regression using the standard Newey-West procedure.34 The resulting t statistic is equal to 8.13. Under standard asymptotic distribution theory, this has probability value of essentially zero under the null hypothesis that the coefficient on the time-square term is zero. So, on the basis of this test, we would reject our level specification. But, it is well-known that the asymptotic theory for this t statistic is quite poor in small samples, especially when the error terms exhibit high degrees of serial correlation. This is exactly the situation we are in according to our level model, since its eigenvalues are quite large.35 To address this concern, we adopt the following procedure. We simulate 1,000 synthetic time series on per capita hours worked using our estimated level model. The disturbances used in these simulations were randomly drawn from the fitted residuals of our estimated level model. The length of each synthetic time series is equal to the length of our sample period. We found that 13.3 percent of these t statistics exceed 8.13. So, from the perspective of the level model, a t statistic of 8.13 is not particularly unusual. We conclude that our t test fails to reject the null hypothesis that the coefficient on the time-squared term is equal to zero. This result may at first seem surprising in view of the U shape of the per capita hours worked data in Figure 1. Actually, such shapes are at all not unusual in a time series system with eigenvalues that are close to unity. This is why the apparent evidence of a U-shaped trend in the hours data is not evidence against our level model. Evidently classical methods cannot be used to convincingly discriminate between the level model and the quadratic trend model. We now turn to the encompassing and posterior odds approach. 34We allow for serial correlation of order 12 in the Newey-West procedure. 35The two largest eigenvalues of the determinant of [I B(L)] in (4) are 0.9903 and 0.9126. − 21

6.2.1 Encompassing Results Appendix A discusses our encompassing results. In discussing our results we refer to the two quadratic trend models as the Trend in All Equations and the Trend in Hours Only models. Our main results can be summarized as follows. The Level model easily accounts for the results obtained using the two quadratic trend models. This is true even if we focus on point estimates alone. In particular, the Level model successfully accounts for the fact that one quadratic trend model implies a fall in hours after a technology shock, while the other implies a rise. The encompassing result is even stronger when we take sampling uncertainty into account. Focusing on the point estimates alone, the Trend in Hours Worked model is unable to encompass the results of either of the other two models. Specifically, it cannot account for the fact that hours worked rise in each of the other two models. However, once sampling uncertainty is taken into account, this encompassing test also does not reject the Trend in Hours Only model. Two things are worth noting regarding the Trend in All Equations model. First, focusing on the point estimates alone, this model can encompass the results based on the Trend in Hours Only model. But, it does not encompass the results based on the Level model. In particular, the Trend in All Equations model predicts, counterfactually, that the Level model produces a fall in hours worked after a positive technology shock. Second, even when sampling uncertainty is taken into account, the encompassing test rejects the Trend in All Equations model vis a vis the Level model. We conclude that the encompassing analysis allows us to exclude the Trend in All Equations model. However, it does not allow us to discriminate between the Level and the Trend in Hours Only model. With this motivation, we turn to the posterior odds ratio. 6.2.2 The Relative Plausibility of the Two Specifications We quantify the relative plausibility of the three models with a scalar statistic: the average percentage change in hours in the first six periods after a technology shock. The estimated Level, Trend in All Equations, and Trend in Hours Only models imply this change is equal to µ = 0.31, µ = 0.12, and µ = 0.16, respectively. 1 2 3 − We simulated 1,000 artificial data sets using each of our three estimated VARs as data generating mechanisms. In each artificial data set, we calculated (µ ,µ ,µ ) using the same 1 2 3 method used to compute these statistics in the actual data. For each data generating mechanism, we computed the frequency of the joint event, µ ,µ > 0, µ < 0. This frequency is 1 2 3 19.30, 3.50 and 5.60 for the Level, Trend in All Equations, and Trend in Hours Only models, respectively. So the posterior odds in favor of the Level model relative to the Trend in All EquationsandTrendinHoursOnlymodelisroughly5.5and3.4, respectively. Onthisbasis, we conclude that the Level model and its implications are more ‘plausible’ than those of the two quadratic trend models. 6.3 Subsample Stability In this subsection we briefly discuss subsample stability, focusing on the six-variable level specification. Authors such as Gali, Lopez-Salido, and Valles (2002), among others, have 22

argued that monetary policy may have changed after 1979, and that this resulted in a structural change in VAR’s. Throughout our analysis, we have assumed implicitly that there has been no structural change. This section assesses the robustness of our conclusions to the possibility of subsample instability. Figures 13 and 14 display the estimated impulse responses of the variables in our system to a technology shock, for the pre-1979Q4 and post-1979Q3 sample periods, respectively. In each case, the thick, solid line is the impulse response implied by the full-sample estimated VAR.Thethinlineswith‘*’representtheestimatedimpulseresponsefunctionsbasedonthe indicatedsub-sample. Thethinlineswithboldstarsrepresentthemeanimpulseresponsesfor the indicated subsample implied by the full-sample VAR. The gray areas are the associated 95percentconfidenceintervals. Boththethinlineswithboldstarsandassociatedconfidence intervals were generated using the methods discussed above. The key results are as follows. First, according to the point estimates, in the early period hours worked fall for roughly three quarters before rising sharply in a hump-shaped pattern. In the late period, the estimated response of hours worked is similar to the estimates based on the full sample period. Second, the point estimates for each sample period lie well within the 95 percent confidence intervals. This is consistent with the view that the responses in the subperiods are the same as they are for the full sample.36 The evidence is also consistent with the view that there is no break in the response of consumption and investment. Third, there is some evidence of instability in the response of the interest rate and inflation. In particular, in the first subsample the drop in inflation and in the interest rate are sufficiently large that portions of their impulse response functions lie outside their respective confidence intervals. These drops are sufficiently large that if one applies a conventional F test for the null hypothesis of no sample break in the VAR, the hypothesis is rejected at the one percent significance level. This rejection notwithstanding, the key result from our perspective is that inference about the response of hours worked to a technology shock is not affected by subsample stability issues.37 7 How Important Are Permanent Technology Shocks for Aggregate Fluctuations? In Section 4 and Section 5, we argued that the weight of the evidence favors the level specification relative to the difference specification. Here, we use the level specification to assess the role of technology shocks in aggregate fluctuations. We conclude that (i) technology shocks are not particularly important at business cycle frequencies but they do play an important role at relatively low frequencies of the data, and (ii) inference based on 36We also computed confidence intervals using the estimated VAR’s for the subsamples as the data generating processes. We found that the full sample estimated impulse response functions lie well within these confidence intervals. 37WealsoinvestigatedsubsamplestabilityusingourfourvariableRπsystem. Consistentwiththeresultsin Gali,Lopez-Salido,andValles(2002),hoursworkedfallssharplyandpersistentlyafterapositivetechnology shock. In addition, output also falls briefly. We found that our full sample, six variable VAR encompasses theseimpulseresponsefunctions,aswellastheresponseoftheinterestrate. But,thereismarginalevidence against its ability to encompass the response of inflation in the early period. 23

bivariate systems greatly overstates the cyclical importance of technology shocks. 7.1 Bivariate System Results We begin by discussing the role of technology shocks in the variability of output and hours worked based on our level specification bivariate VAR. Table 1 reports the percentage of forecasterrorvarianceduetotechnologyshocks, athorizonsof1, 4, 8, 12, 20and50quarters. By construction, permanent technology shocks account for all of the forecast error variance of output at the infinite horizon. Notice that technology shocks account for an important fraction of the variance of output at all reported horizons. For example, they account for roughly 80 percent of the one step ahead forecast error variance in output. In contrast, they account for only a small percentage of the one step forecast error variance in hours worked (4.5percent). Buttheyaccountforalargerpercentageof theforecasterrorvarianceinhours worked at longer horizons, exceeding forty percent at horizons greater than two years. The first row of Table 3 reports the percentage of the variance in output and hours worked at business cycle frequencies due to technology shocks. This statistic was computed as follows. First we simulated the estimated level specification bivariate VAR driven only by the estimated technology shocks. Next we computed the variance of the simulated data after applying the Hodrick-Prescott (HP) filter. Finally we computed the variance of the actual HP filtered output and hours worked. For any given variable, the ratio of the two variances is our estimate of the fraction of business cycle variation in that variable due to technology shocks. The results in Table 3 indicate that technology shocks appear to play a significant role for both output and hours worked, accounting for roughly 64 and 33 percent of the cyclical variance in these two variables, respectively. AdifferentwaytoassesstheroleoftechnologyshocksispresentedinFigure15. Thethick line in this figure displays a simulation of the ‘detrended’ historical data. The detrending is achieved using the following procedure. First, we simulated the estimated reduced form representation (4) using the fitted disturbances, uˆ , but setting the constant term, α, and t the initial conditions of Y to zero. In effect, this gives us a version of the data, Y , in which t t any dynamic effects from unusual initial conditions (relative to the VAR’s stochastic steady state) have been removed, and in which the drift has been removed. Second, the resulting ‘detrended’ historical observations on Y are then transformed appropriately to produce the t variables reported in the top panel of Figure 15. The high degree of persistence observed in outputreflectsthatourprocedureforcomputingoutputmakesittherealizationofarandom walk with no drift. The procedure used to compute the thick line in Figure 15 was then repeated, with one change, to produce the thin line. Rather than using the historical reduced form shocks, uˆ , t the simulations underlying the thin line use Ceˆ, allowing only the first element of eˆ to be t t non-zero. This first element of eˆ is the estimated technology shock εz, obtained from (3). t t The results in the top panel of Figure 15 give a visual representation of what is evident in Table 1 and the first row of Table 3. Technology shocks appear to play a very important role in accounting for fluctuations in output and a smaller, but still substantial role with respect to hours worked. Weconcludethissectionbybrieflynotingthesensitivityofinferencetowhetherweadopt the level or difference specification. The bottom panels of Tables 1 and 3 and the bottom 24

panel of Figure 15 report the analogous results for the bivariate difference specification. Comparing across the Tables or the Figures the same picture emerges: with the difference specification, technology shocks play a much smaller role with respect to output and hours worked than they do in the level specification. For example, the percentage of the cyclical variance in output and hours worked accounted for by technology shocks drops from 64 and 33 percent in the level specification to 11 and 4 percent in the difference specification. So imposing a unit root in hours worked, not only affects qualitative inference about the effect of technology shocks, it also affects inference about their overall importance. 7.2 Results Based on the Larger VAR We now consider the importance of technology shocks when we incorporate additional variablesintoouranalysis. Table2reportsthevariancedecompositionresultsforthesix-variable level specification system. Comparing the first two rows of Table 1 and 2, we see that technology shocks account for a much smaller percent of the forecast error variance in both hours and output in the six-variable system. For example, in the bivariate system, technology shocks account for roughly 78 and 24 percent of the 4 quarter ahead forecast error variance in output and hours, respectively. In the six-variable system these percentages fall to 40 and 15 percent respectively. Still technology shocks continue to play a major role in the variability of output, accounting for over 40 percent of the forecast error variance at horizons between four and twenty quarters. Technology shocks do play an important role in accounting for the forecast error variance in hours worked at longer horizons, accounting for nearly 30 percent of this variance at horizons greater than 4 quarters, and more than 40 percent of the unconditional variance. The decline in the importance of technology shocks is much more pronounced when we focus on cyclical frequencies. Recall from Table 3 that, based on the bivariate system, technology shocks account for roughly 64 and 33 percent of the cyclical variation in output and hours worked. In the six-variable systems, these percentages plummet to ten and four, respectively. Interestingly,asimilarresultemergesfromthefourvariableCI andRπ systems. For example, in the latter system, technology shocks account for roughly 64 and 33 percent of the cyclical variation in output and hours worked. Turningtotheothervariables, Table2indicatesthattechnologyshocksplayasubstantial role in inflation, accounting for over 60 percent of the one step ahead forecast error variance and almost 40 percent at even the 20 quarter horizon. Technology shocks also play a very important role in the variance of consumption, accounting for over 60 percent of the one step ahead forecast error variance and almost 90 percent of the unconditional variance. These shocks also play a substantial, if smaller, role in accounting for variation in investment. These shocks, however, do not play an important role in the forecast error variance for the federal funds rate. Turning to business cycle frequencies, two results stand out in Table 3. First, technology shocks account for a very small percentage of the cyclical variance in output, hours worked, investment and the federal funds rate (10, 4, 1 and 7 percent respectively). Second, technology shocks account for a moderately large percentage of the cyclical variation in consumption (16.7 percent) and a surprisingly large amount of the cyclical variation in inflation (32 percent). 25

Figure 16 presents the historical decompositions for the six-variable level specification VAR. Technology shocks do relatively well at accounting for the data on output, hours, consumption, inflation and to some extent investment at the lower frequencies. While not reported here, the results are similar for the six-variable difference specification VAR. 8 Conclusions A theme of this paper is that the treatment of the low frequency component of per capita hours worked has an important impact on inference about the response of hours worked to a technology shock. We explored the impact on inference of treating per capita hours as difference stationary, stationary, or stationary about a deterministic trend. We also investigated the impact of omitted variables on inference. We conclude that the evidence overwhelmingly favors specifications which imply that per capita hours worked rises in response to a technology shock. Throughout, we assume that only one shock affects productivity in the long run and we refer to it as a ‘technology shock’. We do this because it is the standard interpretation in the literature. But, other interpretations are possible too. For example, the shock that we identify could in principle be any permanent disturbance that affects the rate of return on capital, such as the capital tax rate, the depreciation rate, or agents’ discount rate. If some or all of these shocks are operative and have permanent effects on productivity, then our inferences may be distorted. To explore this possibility requires making additional identifyingassumptionsandincorporatingnewdataintotheanalysis. Fisher(2002)doesthis by considering two types of technology shocks. He argues that investment-specific shocks play a relatively important role at cyclical frequencies in driving aggregate fluctuations. Significantly, he finds that our key result is robust to the presence of a second shock: both of the technology shocks that Fisher identifies lead to an increase in hours worked. 26

A Encompassing Analysis for Level and Quadratic Trend Models This appendix provides additional details to the general discussion about encompassing that appears in Section 6.2. We discuss the ability of each model in Section 6.2 to encompass the response of hours from the other two models. As in the text, the three models are the ‘Levels’ model, the ‘Trend in All Equations’ model, and the ‘Trend in Hours Only’ model. In Figure A, each of Panels A, B and C report encompassingresults forthe particularmodel indicatedintheassociatedpanel header. Each panel has two columns. Each column focuses on the ability of the model to encompass the empirical results obtained using one of the other two models. Panel A evaluates the Level model’s ability to account for the results based on the Trend in All Equations model and the Trend in Hours Only model. To do this, we simulated 1,000 synthetic time series, each of length equal to our sample period. Using each of these time series, we estimated the Trend in All Equations model and the Trend in Hours Only model. Wethencomputedtheimpulseresponsefunctionofinterest. Thestarredlineineach column indicates the mean response across the 1,000 time series. The grey area indicates the associated 95 percent confidence interval. The dark, thick line indicates the estimated impulse response function based on the Level model. The line with circles represents the estimated impulse response function based on the Trend in All Equations model. The line with x’s represents the estimated impulse response function of the Trend in Hours Only model. Note in Panel A how all the impulse responses lie well inside the grey area. This implies that the level model encompasses the two quadratic trend models. Since these models are not misspecified when the level model is true, this result reflects the effects of small sample uncertainty. We verified this by doing the calculations reported in Figure A on much longer synthetic data sets. We found that the resulting average impulse response nearly coincided with the Level model’s estimated impulse response. Panel B evaluates the ability of the Trend in Hours Only model to account for the results based on the Level and the Trend in All Equations models. The labeling convention on the lines is the same as in Panel A. Focusing on the point estimates alone, the Trend in Hours Worked model is unable to encompass the results of either of the other two models. Specifically, it cannot account for the fact that hours worked rise in each of the other two models. However, once sampling uncertainty is taken into account, this encompassing test does not reject the Trend in Hours Only model. PanelCevaluatestheabilityoftheTrendinAllEquationsmodeltoaccountfortheresults based on the Level and the Trend in Hours Only models. Again, the labeling convention on the lines is the same as in Panel A. Two things are worth noting here. First, focusing on the point estimates alone, the Trend in all Equations model can encompass the results based on the Trend in Hours Only model, but it does not encompass the results based on the Level model. In particular, the Trend in All Equations model predicts, counterfactually, that the Level model produces a fall in hours worked after a positive technology shock. Second, even when sampling uncertainty is taken into account, the encompassing test rejects the Trend in All Equations model vis a vis the Level model. 27

B Asymptotic Distribution of Impulse Response Estimators When Difference Specification is True, But Level Specification is Adopted This appendix analyzes a special case of our environment to illustrate the results in Section 4.2.2. We derive a closed-form representations of the asymptotic distribution of the instrumental variables estimator and of the estimator of a technology shock’s contemporaneous impact on hours worked. We discuss the bias in these estimators. We consider the case, µ = 0, β(L) = 0 and q = 2, and ∆X = θ∆X +u , where θ < 1, t t 1 t u = ψεz +ε and Eεzε = 0. Here, ψ is the contemporaneous impact o − f a one unit sh | o | ck to t t t t t technology, εz. The formulas in Hamilton (1994, Theorem 18.1) can be used to deduce: t ρ+ σvω δIV δ L σu δ . − → θ ρ+ σvω ≡ ∗ " − σu # ³ ´ Here, δ = (δ , δ ) and δ , δ correspond to the coefficients on ∆X and ∆X , respectively. ∗ ∗0 ∗1 ∗0 ∗1 t t 1 − Also, ψσ2 1 W(r)dW˜ (r) ρ = εz, ω = 2 0 , σ2 = σ2 ρ2σ2, σ2 [W(1)]2 1 v εz − u u R − ˜ and W(r) and W(r), 0 r 1, are independent Brownian motions. ≤ ≤ Using graphical analysis, we found that the cumulative distribution function of ω resembles that of the zero-median Cauchy distribution, with cumulative density, arctan ω P(ω) = 0.5+ 0.835 . π ¡ ¢ We simulated 100 artificial sets of observations, each of length 11,000, on ω. We computed the median in each and found that the mean of the 100 medians was 0.0015. The standard − deviation across the 100 artificial data sets is 0.0138. So, under the null hypothesis that the true median is zero, the mean of 0.0015 is a realization from a normal distribution with − standard deviation, 0.0138/√100 = 0.00138. The probability of a mean less than 0.0015 − under the null hypothesis exceeds 10 percent. So, we fail to reject. This, taken together with our graphical analysis, is consistent with the notion that the above zero-median Cauchy distribution is a good approximation of the distribution of ω. Regardingthelargesampledistributionoftheestimatorofthecontemporaneousresponse of hours to technology, Ψ , we find, after tedious algebra 0 ρ δ ΨIV L σ − ∗0 . 0 → u × 1/2 (δ )2 2δ ρ+ ρ ∗0 − ∗0 ψ h i This illustrates the observation in the text, that the asymptotic distribution of ΨIV is a 0 function of the asymptotic distribution of δIV δ. − The median of the asymptotic distribution of ΨIV is obtained by setting δ to its median 0 ∗0 value, which we argued above is ρ. Hence, the median of the asymptotic distribution of 28

ΨIV is zero, regardless of the true value of Ψ . The intuition for this result is simple. 0 0 It is easily verified that the median of an instrumental variables regression’s estimators correspondstotheprobabilitylimit of the correspondingOLSestimators. Butinminimizing residual variance, ordinary least squares chooses the residuals to be uncorrelated with the right hand variables. These residuals are the OLS estimates of the technology shocks. The disturbance in the VAR equation for ∆X is a linear function of the right hand variables in t the instrumental variables equation. As a result, it is not surprising that the OLS estimate of the technology shock is uncorrelated with the disturbance in the VAR equation for ∆X . t This lack of correlation is what underlies ΨIV being centered on zero. 0 C Impact of Covariates on the Power of Unit Root Tests A key factor driving our finding that level specifications are more plausible than difference specifications is the large value of our weak instruments F statistics. Though the level specifications have little difficulty accounting for a large F, the difference specifications have considerable difficulty doing this. Our finding is consistent with recent findings in the literature on testing for unit roots. In particular, the weak instruments F statistic turns out to be a variant of the multivariate extension to the ADF test proposed by Hansen (1995) (see also and Elliott and Jansson, 2003). Because this test introduces additional variables, i.e., ‘covariates’, into the analysis, Hansen refers to it as the covariates ADF (CADF) test. An important finding in the literature is that the CADF test has considerably greater power than the ADF test. This appendix reports the power gain from using the CADF rather than the ADF test in our context. We compute critical values for sizes 0.01, 0.05 and 0.10 using each of our three difference specifications (the bivariate models based on the short and long sample, and the six-variable model based on the short sample). Critical values are computed based on the type of bootstrap simulations used throughout our analysis. The critical values are for t statistics used to test the null hypothesis that the coefficient on lagged, log per capita hours worked is zero in a particular ordinary least squares regression. In the case of the ADF test, the regression is of hours growth on the lagged level of log, per capita hours and three lags of hours growth. Three sets of critical values are computed for the ADF t statistic, one for each our three difference specifications. Corresponding to each critical value, we compute power using bootstrap simulations of the relevant estimated level VAR. The results are reported in Table A1. To understand the table, note, for example, that the difference specification estimated usingthe longsamplehas the propertythattheADFt statisticisless than 3.8 in1percent − of the artificial samples. When we simulated the bivariate level specification estimated using the long sample, we found that 4.8 percent of the time the simulated t statistics are smaller than 3.8. Thus, the power of the 1 percent ADF t statistic is 4.8 percent based on the − long sample bivariate VAR. Interestingly, power is nearly twice as great in the short sample as in the long sample. Conditional on the long sample, there is little difference between the bivariate and six-variable results. 29

We turn now to an assessment of the impact on power of adding covariates. Our CADF t statistic resembles theADFt statistic, except that the underlying regressionalsoincludes all the predetermined variables in the instrumental variables regression, (3). Since the number of predetermined variables is different in the bivariate and six-variable systems, we have two CADFt statistics. The first corresponds to our bivariate analysis. It is based on a regression like the one underlying the ADF test, except that it also includes three lags of productivity growth. The second corresponds to our six-variable analysis. In particular, it adds three lags of each of the federal funds rate, the rate of inflation, the log of the ratio of nominal consumption expenditures to nominal GDP, and the log of the ratio of nominal investment expenditures to nominal GDP. We compute critical values for our two CADF t statistics in the same way as for the ADF statistic. In particular, we compute two sets of critical values for our bivariate CADF statistic,onecorrespondingtoeachoftheshortandlongsampleestimateddifferencespecifications. Thecriticalvaluesforthesix-variableCADFt statisticarebasedonbootstrapsimulationsof the estimated six-variable difference VAR. Corresponding to each critical value, we compute power using bootstrap simulations of the relevant estimated level difference VAR. Corresponding to each critical value, we also computed the power of the statistic when the level specification is true. This was done by bootstrap simulation of the relevant level specification VAR. Results are reported in Table A2. Comparing Tables A1 and A2, power increases substantially with the introduction of covariates. With a 1 percent size, power jumps by an order of magnitude in the short sample. 30

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Figure 1: Data Used in VAR Labor Productivity Growth Average Hours 0.04 -7.4 0.02 -7.45 0 -7.5 -7.55 -0.02 -7.6 -0.04 1949 1959 1969 1979 1989 2001 1949 1959 1969 1979 1989 2001 Inflation Consumption to Output Ratio 0.03 -0.25 0.02 -0.3 0.01 0 -0.35 -0.01 1949 1959 1969 1979 1989 2001 1949 1959 1969 1979 1989 2001 Investment to Output Ratio Federal Funds -1.2 15 -1.3 10 -1.4 5 -1.5 1949 1959 1969 1979 1989 2001 1949 1959 1969 1979 1989 2001 34

Figure 2: Response of Log-output and Log-hours to a Positive Technology Shock Level Specification Panel A: Sample Period 1948Q1-2001Q4 Output Hours 1.5 2 1.8 1.6 1.4 1 1.2 1 0.8 0.5 0.6 0.4 0.2 0 0 0 5 10 15 0 5 10 15 Periods After Shock Periods After Shock Panel B: Sample Period 1959Q1-2001Q4 Output Hours 2 1.2 1.8 1 1.6 1.4 0.8 1.2 1 0.6 0.8 0.4 0.6 0.4 0.2 0.2 0 0 0 5 10 15 0 5 10 15 Periods After Shock Periods After Shock Thick Line: Impulse Responses from Level Specification Gray Area: 95 percent Confidence Intervals 35

Figure 3: Response of Log-output and Log-hours to a Positive Technology Shock Difference Specification Panel A: Sample Period 1948Q1-2001Q4 Output Hours 0.6 1.4 1.2 0.4 1 0.2 0.8 0 0.6 -0.2 0.4 -0.4 0.2 -0.6 0 0 5 10 15 0 5 10 15 Periods After Shock Periods After Shock Panel B: Sample Period 1959Q1-2001Q4 Output Hours 1.4 0.6 1.2 0.4 1 0.2 0.8 0 0.6 -0.2 0.4 0.2 -0.4 0 -0.6 0 5 10 15 0 5 10 15 Periods After Shock Periods After Shock Line with Triangles: Impulse Responses from Difference Specification Gray Area: 95 percent Confidence Intervals 36

Figure 4: Encompassing with Level Specification as the DGP Panel A: Sample Period, 1948Q1-2001Q4 Output Average Hours 1.6 1 1.4 0.8 1.2 0.6 1 0.4 0.8 0.2 0.6 0 0.4 -0.2 0.2 -0.4 0 -0.6 0 5 10 15 0 5 10 15 Panel B: Sample Period, 1959Q1-2001Q4 Output Average Hours 0.8 1.4 0.6 1.2 0.4 1 0.2 0.8 0 0.6 0.4 -0.2 0.2 -0.4 0 5 10 15 0 5 10 15 Thick Line: Impulse Responses from Level Specification Line with Triangles: Impulse Responses from Difference Specification Circles: Average Impulse Response for Simulations from given DGP Gray Area: 95 percent Confidence Intervals For Simulations for given DGP 37

Figure 5: Encompassing with Difference Specification as the DGP Panel A: Sample Period,1948Q1-2001Q4 Output Average Hours 2 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 0 5 10 15 0 5 10 15 Panel B: Sample Period, 1959Q1-2001Q4 Output Average Hours 1 1.5 0.5 1 0.5 0 0 -0.5 -0.5 -1 0 5 10 15 0 5 10 15 Thick Line: Impulse Responses from Level Specification Line with Triangles: Impulse Responses from Difference Specification Circles: Average Impulse Response for Simulations from Difference Specification DGP Gray Area: 95 percent Confidence Intervals For Simulation Impulse Responses 38

Figure 6: Six-variable System, Level Specification,Sample Period 1959-2001 Output Hours 1.5 0.6 1 0.4 0.5 0.2 0 0 0 5 10 15 0 5 10 15 Inflation Fed Funds 0 60 40 -0.1 20 0 -0.2 -20 -40 -0.3 0 5 10 15 0 5 10 15 Consumption Investment 3 1.5 2 1 1 0.5 0 -1 0 0 5 10 15 0 5 10 15 Thick Line: Impulse Responses from Level Specification Gray Area: 95 percent Confidence Intervals 39

Figure 7: Six-variable System, Difference Specification, Sample Period 1959-2001 Output Hours 1.5 0.5 1 0 0.5 0 -0.5 -0.5 -1 0 5 10 15 0 5 10 15 Inflation Fed Funds 0 0 -0.1 -50 -0.2 -100 0 5 10 15 0 5 10 15 Consumption Investment 1.5 2 1 1 0 0.5 -1 0 -2 -3 0 5 10 15 0 5 10 15 Line with Triangles: Impulse Responses from Difference Specification Gray Area: 95 percent Confidence Intervals For Simulation Impulse Responses 40

Figure 8: Encompassing Test with the Level Specification as the DGP, 1959-2001 Output Hours 1 0.4 0.2 0.5 0 -0.2 0 -0.4 -0.6 -0.5 -0.8 0 5 10 15 0 5 10 15 Inflation Fed Funds 0.05 20 0 0 -0.05 -20 -0.1 -40 -0.15 -60 -0.2 0 5 10 15 0 5 10 15 Consumption Investment 2 1 1 0.5 0 -1 0 -2 0 5 10 15 0 5 10 15 Thick Line: Impulse Responses from Level Specification Line with Triangles: Impulse Responses from Difference Specification Circles: Average Impulse Response for Simulations from Difference Specification DGP Gray Area: 95 percent Confidence Intervals For Simulation Impulse Responses 41

Figure 9: Encompassing Test with the Difference Specification as the DGP, 1959-2001 Output Hours 1 0.5 0.5 0 0 -0.5 -0.5 0 5 10 15 0 5 10 15 Inflation Fed Funds 40 0 20 -0.05 0 -0.1 -20 -0.15 -40 -0.2 -60 0 5 10 15 0 5 10 15 Consumption Investment 1 2 1 0.5 0 -1 0 -2 0 5 10 15 0 5 10 15 Thick Line: Impulse Responses from Level Specification Line with Triangles: Impulse Responses from Difference Specification Circles: Average Impulse Response for Simulations from Level Specification DGP Gray Area: 95 percent Confidence Intervals For Simulation Impulse Responses 42

Figure 10: Comparing the Six-Variable Specification to 2 different Four-Variable, Level Specification Output Hours 1.5 0.6 1 0.4 0.5 0.2 0 0 -0.2 0 5 10 15 0 5 10 15 Inflation Fed Funds 0 60 40 -0.1 20 0 -0.2 -20 -40 -0.3 0 5 10 15 0 5 10 15 Consumption Investment 3 1.5 2 1 1 0.5 0 -1 0 0 5 10 15 0 5 10 15 Thick Line: has all six variables, Gray Area: 90 percent Confidence Intervals For Six-variable System ‘X’: has hours, labor productivity, inflation and the federal funds rate. ‘*’: has hours, labor productivity, consumption and investment 43

Figure 11: Encompassing Four-variable systems with Six-variable systems Level Specification 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 0 5 10 15 Thick Line: has all six variables, Circles: Average Response from Simulations Using Six-variable System as DGP Gray Area: 95percent Confidence Intervals For Simulations ‘X’ has hours, labor productivity, inflation and the federal funds rate. 44

Figure 12: The Effect of Adding A Quadratic Trend Output Hours 0.4 1 0.2 0 0.5 -0.2 -0.4 0 0 5 10 15 0 5 10 15 Inflation Fed Funds 50 0 -0.1 0 -0.2 -50 0 5 10 15 0 5 10 15 Consumption Investment 1.4 2 1.2 1 1 0.8 0.6 0 0.4 -1 0.2 0 5 10 15 0 5 10 15 Thick Line: Hours, ‘X’s Detrended Hours, Circles Quadratic Trend estimated in the VAR. Gray Area: 95 percent Confidence Intervals For Detrended Hours 45

Figure 13: Encompassing pre-1979Q4 Period Output Hours 0.6 1 0.4 0.2 0.5 0 -0.2 0 -0.4 0 5 10 15 0 5 10 15 Inflation Federal Funds 0.05 40 0 20 -0.05 0 -0.1 -20 -0.15 -40 -60 -0.2 0 5 10 15 0 5 10 15 Consumption Investment 2 1 1 0.5 0 -1 0 0 5 10 15 0 5 10 15 Thick Line: Full Sample Response, Thin Line: Subsample Response, Stars Subsample Response Using Full Sample as DGP Gray Area Confidence Interval for Subsample Response Using Full Sample as DGP 46

Figure 14: Encompassing post-1979Q3 Period Output Hours 0.6 1 0.4 0.2 0.5 0 -0.2 0 -0.4 0 5 10 15 0 5 10 15 Inflation Federal Funds 0.05 40 0 20 -0.05 0 -0.1 -20 -0.15 -40 -0.2 -60 0 5 10 15 0 5 10 15 Consumption Investment 1 2 0.8 1 0.6 0.4 0 0.2 -1 0 0 5 10 15 0 5 10 15 Thick Line: Full Sample Response, Thin Line: Subsample Response, Stars Subsample Response Using Full Sample as DGP Gray Area Confidence Interval for Subsample Response Using Full Sample as DGP 47

Figure 15: Historical Decomposition: Bivariate System, Level Specification Output Hours 0.1 0.06 0.08 0.04 0.06 0.02 0.04 0.02 0 0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 -0.08 -0.08 1965 1970 1975 1980 1985 1990 1995 2000 1965 1970 1975 1980 1985 1990 1995 2000 Difference Specification Output Hours 0.04 0.1 0.02 0 0.05 -0.02 -0.04 0 -0.06 -0.08 -0.05 -0.1 -0.12 -0.1 -0.14 1965 1970 1975 1980 1985 1990 1995 2000 1965 1970 1975 1980 1985 1990 1995 2000 Thick Line: Historical Decomposition Using All Shocks Thin Line: Historical Decomposition Using Just Technology Shocks 48

Figure 16: Historical Decomposition: Six-Variable System , Level Specification Output Hours 0.04 0.02 0.05 0 0 -0.02 -0.04 -0.05 -0.06 -0.08 -0.1 1970 1980 1990 2000 1970 1980 1990 2000 -3 Inflation Fed Funds x 10 10 15 10 5 5 0 0 -5 1970 1980 1990 2000 1970 1980 1990 2000 Consumption Investment 0.05 0.1 0 0 -0.1 -0.05 -0.2 1970 1980 1990 2000 1970 1980 1990 2000 Thick Line: Historical Decomposition Using All Shocks Thin Line: Historical Decomposition Using Just Technology Shocks 49

Figure A: Encompassing Analysis for Level and Quadratic Trend Models Panel A: DGP Levels Trend in Hours Only Trend in All Equations 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 -0.2 -0.3 -0.2 -0.4 -0.3 -0.5 -0.4 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 Panel B: DGP Trend in Hours Only Levels Trend in All Equations 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0 -0.1 0 -0.2 -0.2 -0.3 -0.4 -0.4 -0.6 -0.5 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 Panel C: DGP Trend in All Equations Levels Trend in Hours Only 0.4 0.4 0.3 0.2 0.2 0.1 0 0 -0.1 -0.2 -0.2 -0.3 -0.4 -0.4 -0.5 -0.6 -0.6 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 Thick Line: Estimated Levels Model Stars: Predicted Mean Response X’s: Estimated Trend in Hours Only Circles: Estimated Trend in All Equations Gray Area: 95% Confidence Interval Around Predicted Mean Response 50

Table 1: Contribution of Technology Shocks to Variance, Bivariate System Level Specification Forecast Variance at Indicated Horizon Variable 1 4 8 12 20 50 Output 81.1 78.1 86.0 89.1 91.8 96 Hours 4.5 23.5 40.7 45.4 47.4 48.3 Difference Specification Forecast Variance at Indicated Horizon Variable 1 4 8 12 20 50 Output 16.5 11.7 17.9 20.7 22.3 23.8 Hours 21.3 6.4 2.3 1.6 1.0 0.5 Table 2: Contribution of Technology Shocks to Variance, Six-variable System Level Specification Forecast Variance at Indicated Horizon Variable 1 4 8 12 20 50 Output 31.2 40.3 44.6 41.5 44.8 70 Hours 3.6 15.4 28.8 28.4 28.8 43.9 Inflation 60.2 47.0 43.2 41.1 39.5 47.7 Fed Funds 1.6 1.4 1.7 1.7 3.7 23.3 Consumption 61.6 64.2 67.3 66.8 71.8 88.4 Investment 10.3 20.1 24.1 20.9 20.4 25.3 Difference Specification Forecast Variance at Indicated Horizon Variable 1 4 8 12 20 50 Output 1.7 0.6 2.6 6.4 17.2 35.5 Hours 20.8 11.9 8.0 7.1 5.7 2.3 Inflation 58.5 54.7 55.6 52.4 47.4 33.8 Fed Funds 0.0 7.5 10.5 13.7 17.2 16.9 Consumption 7.9 4.1 8.7 14.3 25.3 34.3 Investment 1.1 2.0 1.1 1.3 3.7 13.8 51

Table 3: Contribution of Technology Shocks to Cyclical Variance (HP Filtered Results) Level Specification Variables in VAR Output Hours Inflation Federal Funds Consumption Investment Y,H 63.8 33.4 Y,H,∆P,R 17.8 17.9 53.2 11.2 Y,H,C,I 19.9 18.5 20.1 20.7 Y,H,∆P,R,C,I 10.2 4.1 32.4 1.3 16.8 6.7 Difference Specification Variables in VAR Output Hours Inflation Federal Funds Consumption Investment Y,∆H 10.6 7.0 Y,∆H,∆P,R 6.8 8.5 48.4 8.1 Y,∆H,C,I 1.3 6.3 0.32 5.5 Y,∆H,∆P,R,C,I 1.6 6.1 35.2 4.9 3.7 2.6 Table A1: Power of Standard ADF t Test Bivariate Specification Six-Variable Specification Long Sample Short Sample Short Sample Size Critical Value Power Critical Value Power Critical Value Power 0.01 -3.835 0.048 -3.705 0.108 -4.290 0.045 0.05 -3.253 0.184 -3.109 0.353 -3.410 0.223 0.10 -2.870 0.363 -2.780 0.548 -2.963 0.400 Table A2: Power of CADF t Test Bivariate Specification Six-Variable Specification Long Sample Short Sample Short Sample Size Critical Value Power Critical Value Power Critical Value Power 0.01 -3.588 0.396 -3.266 0.589 -4.184 0.689 0.05 -2.908 0.784 -2.686 0.864 -3.350 0.888 0.10 -2.616 0.895 -2.403 0.938 -2.879 0.946 52