How do Canadian Hours Worked Respond to a Technology Shock?

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 774 September 2003 How do Canadian Hours Worked Respond to 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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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/.

How do Canadian Hours Worked Respond to a Technology Shock? ∗ Lawrence J. Christiano, Martin Eichenbaum and Robert Vigfusson † ‡ § Abstract This paper investigates the response of hours worked to a permanent technology shock. Based on annual data from Canada, we argue that hours worked rise after a positive technology shock. We obtain a similar result using annual data from the United States. These results contradict a large literature that claims that a positive technology shock causes hours worked to fall. We find that the different results are due to the literature making a specification error in the statistical model of per capita hours worked. Finally, we present results suggesting that Canadian monetary policy has accommodated technology shocks. Keywords: productivity, long-run restriction, hours worked, weak instruments. This paper was prepared for the Bank of Canada’s November 2002 conference, “Price Adjustment and ∗ Monetary Policy”, and is forthcoming in its conference volume. Christiano and Eichenbaum thank theNational Science Foundation for financial assistance. The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any person associated with the Federal Reserve System. Northwestern University and NBER.(Email l-christiano@northwestern.edu) † Northwestern University and NBER. (email eich@northwestern.edu) ‡ Board of Governors of the Federal Reserve System. (Email robert.j.vigfusson@frb.gov) §

1 Introduction This paper investigates the response of hours worked to a permanent technology shock. Based on annual data from Canada and the United States, we argue that hours worked rises after a positive technology shock. While this result is consistent with the analysis in Christiano, Eichenbaum, and Vigfusson [CEV] (2003), it stands in sharp contrast to a large and growing literature, according to which hours worked falls after a positive technology shock (see for example Gali (1999) and Francis and Ramey (2001)).1 Theassumptionthatwemaketoidentifyatechnologyshockisthesameasintheliterature. Specifically we assume that the only type of shock which affects the long-run level of average labor productivity is a permanent shock to technology. So the difference between our results cannot be attributed to the nature of our identifying assumptions. Instead it is due to the way hours worked are incorporated into our statistical analysis. Using quarterly U.S. time series data, CEV (2003) make the following argument. Suppose that the analyst assumes that per capita hours worked is a stationary stochastic process and works with the level of hours. Using this ‘level specification’, the analyst would find that hours worked in the U.S. rises after a technology shock. On the other hand suppose that the analyst assumed that hours worked is a difference stationary process and works with the growth rate of hours worked. Using this ‘difference specification’, the analyst would find that hours worked in the U.S. falls after a technologyshock. InthispaperweshowthatexactlythesameresultholdsforannualCanadian and U.S. data. The question is: Which set of results is correct, those based on the level specification or those based on the difference specification? Not surprisingly, standard, univariate hypothesis tests do not yield much information about whether per capita hours worked has a unit root or not. These tests cannot reject either the null hypothesis that per capita hours worked are difference stationary or the null hypothesis that they are stationary. As in CEV (2003), we assess the relative plausibility of the two hypotheses and their associated implications by asking the following encompassing question: ‘Which specification has an easier time explaining the findings that emerge when the analyst proceeds using the competing specification?’ As discussed below and in CEV (2003), we expect that the specification that will do best on the encompassing criterion is the one that predicts that the other model is misspecified. These considerations lead us to think that the level specification will do better than the difference specification. This is because, if the level specification is true, then an analyst who adopts the difference specification is committing a specification error. But, if the difference specification is true, an analyst who adopts the level specification is not committing a specification error. While important, this consideration is not definitive because sampling considerations also enter. Specifically, when the difference specification is true, then an econometrician who adopts the level specification will encounter a weak instrument problem that implies large sampling uncertainty and bias in the estimates of the response of hours worked to a technology shock. To choose between the level and difference specifications, we use the kind of posterior odds ratio considered in Christiano and Ljungqvist (1988) and CEV (2003). The prefered specification is the one that can most easily explain three facts: (i) the level specification implies that hours worked rises after a technology shock, (ii) the difference specification implies that hours workedfalls,and(iii)theoutcomeofaweakinstrumentstestthatweimplement. Focusingonly 1CEV (2003) base their analysis on quarterly U.S. time series data. 1

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. Indeed in the case of Canada, the odds in favor of the level specification are over nine to one. After establishing the case for the level specification, we analyze how money growth, the interestrateandinflationinCanadarespondtoatechnologyshock. Focusingonpointestimates we find that in response to such a shock, money growth rises while the interest rate and inflation drops. Sampling uncertainty aside, these findings suggest that Canadian monetary policy makers have accommodated technology shocks. This raises a key question: exactly what role has monetary policy played in the expansion of aggregate economic activity and the fall in inflation that follow in the wake of a positive technology shock? To answer this question requires that we know how the economy would have reactedhadthemonetaryauthorityacteddifferently. Theonlyplacewherewecanperformsuch a counterfactual experiment is in a structural economic model. Such an analysis lies beyond the scope of this paper, which relies on reduced form time series methods. The methods used in this paper can deliver estimates of how monetary policy actually reacted to technology shocks. These methods, however, cannot be used directly to ask how the economy would have reacted under alternative policy rules. For an analysis of that issue we refer the reader to Altig, Christiano, Eichenbaum and Linde (ACEL) (2003). Using an estimated dynamic general equilibrium model embodying wage and price frictions, ACEL assess the response of the economy to a technology shock under alternative monetary policy rules, e.g. a k per cent growth rate rule for money. Their key conclusion is that, had policy makers not accommodated technology shocks, hours worked would have fallen for a prolonged period of time after a positive technology shock. Somewhat paradoxically, actual hours worked respond positively as in a real business cycle model, because of the systematic way monetary policy makers respond to technology shocks. Since ACEL (2003) estimate their model using U.S. data, we cannot claim that this result holds for Canada. But we suspect that it does. The remainder of this paper is organized as follows. Section 2 discusses our strategy for identifying the effects of a permanent shock to technology. In Section 3 we present the results from a bivariate analysis using data on hours worked and the growth rate of labor productivity. Section 4 discusses our encompassing strategy, the results of which are reported in Section 5. In Section 6, we report results for how in Canada inflation, the growth rate of money and the interest rate respond to a technology shock. Finally, Section 7 contains concluding remarks. 2 Identifying The Response to A Permanent Technology Shock. In this section, we discuss how we identify the effect of a permanent shock to technology. As in Gali (1999), Gali, Lopez-Salido, and Valles (2002), Francis and Ramey (2001) and CEV (2003) we assume that the only type of shock which affects the long-run level of average labor productivity is a permanent shock to technology.2 As discussed in CEV (2003) this assumption is satisfied by a large class of standard business cycle models. Still it is important to recognize 2There is now a large literature in which the long-run identifying assumption is adopted. See for example Vigfusson (2002), Altig, Christiano, Eichenbaum and Linde (2003) and Fisher (2003). 2

that there exist models in which this assumption is not satisfied.3 We estimate the dynamic effects of a technology shock using a variant of the Shapiro and Watson (1988) procedure for long-run identifying assumptions. Our description of this procedure borrows heavily from the relevant portion of CEV (2003). Our starting point 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 order t q and q 1 in the lag operator, L, respectively. Also, ∆ is the first difference operator and we − assumethat∆f iscovariancestationary. Thewhitenoiserandomvariable,εz,istheinnovation t t to technology. Suppose that the response of X to an innovation in some non-technology shock, t ε , is characterized by X = γ(L)ε , where γ(L) is a polynomial in non-negative powers of L. t t t We assume that each element of γ(1) is non-zero. The assumption that non-technology shocks have no impact on f in the long run implies the following restriction on α˜(L) : t 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 affect f is by their effect on X , while the long-run impact t t of a shock to ε on f is given by: t t α˜(1)γ(1) . 1 β(1) − The assumption that ∆f is covariance stationary guarantees 1 β(1) < . This, together t | − | ∞ with our assumption on γ(L), implies that for the long-run impact of ε on f to be zero it must t t be that α˜(1) equals zero. 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). If t oneoftheshocksdrivingX isεz, thenX andεz willbecorrelated. So, wecannotestimatethe t t t t 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 − − Given an estimate of the shocks in (3), we obtain an estimate of the dynamic response of f 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 Ã t ! 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 3Forexample,theassumptionisnottrueinanendogenousgrowthmodelwhereallshocksaffectproductivity 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. 3

practice, we set q equal to 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 dynamic response of the variables in Y to εz, we require the first column of C. We obtain this t t by regressing u on εz by ordinary least squares. Finally, we simulate the dynamic response of t t Y to εz. For each lag in this response function, we computed the centered 95 percent Bayesian t t confidence interval using the approach for just-identified systems discussed in Doan (1992).4 3 Empirical Results AkeyissuewhenworkingwithCanadiandataisthelimitedspanoftherelevantquarterlydata. Many studies of the effects of technology on the U.S. economy use quarterly data on Business Labor Productivity, a time series which is available starting in 1947. The analogous Canadian series starts only in 1987.5 Similarly, Canadian quarterly data on hours worked in the business sector are available starting in 1987.6 Since we are interested in identifying shocks that have a long run effect on productivity, we work with annual data which are available from 1961.7 Our measure of the Canadian population is for all people between the ages of 15 and 64.8 We measured output as the average level of Canadian real GDP over the year.9 GDP is a broader measure of output than the measure considered in Francis and Ramey (2002) or CEV (2003), namely private sector output. Using a broader measure, however, seems reasonable given that we also use a broader measure of hours worked. We are interested in comparing the effects of a technology shock in Canada and the United States. The U.S. data that we use are the annual version of the data used in CEV (2003). The relevant series are business labor productivity and hours.10 Our data on labor productivity growth and per capita hours worked in the United States and Canada are displayed in the first row of Figure 1 and Figure 2, respectively. For the United States, the average growth rate of these variables are 0.04 and 2.19 percent, respectively. For Canada, the average growth rate − of per capita hours and labor productivity is 0.03 and 1.80 percent, respectively. The other − variables in Figure 2 will be discussed in the multivariate section. 4This approach requires drawing B(L) and V repeatedly from their posterior distributions. Our results are based on 2,500 draws. 5The CANSIM mnemonic for Business Sector Labor Productivity is V1409153. 6The CANSIM series V1409155 measures quarterly hours worked in the business sector but starts only in 1987. TheCANSIM seriesV159660measurestotal hoursworkedin allsectorsand ismonthlybutstartsonlyin 1976. 7Annual total hours worked is measured using the CANSIM series V719842. 8Dataonthispopulationmeasureisavailablefrom1971asCANSIMseriesV466971. Weconstructpopulation data for the 1960s using the growth rate of the population aged 14 and over from the Canadian Historical Statistics. 9Real GDP is measured using CANSIM series v1992067. 10These series have DRI Economics mnemonic LBOUT and LBMN, respectively. 4

3.1 Impulse Responses We first consider results for a bivariate VAR for Y . The first element inY is the growth rate of t t the log of labor productivity f . The second element in Y is the log of per capita hours worked, t t h . Figure 3 reports the response of labor productivity and average hours to a one standard t deviationlong-runtechnologyshock. IntheUnitedStates, bothlaborproductivityandaverage hours rise by roughly one percent in the first year of the shock. Labor productivity continues to rise for the next eight years. Hours worked rises in a hump shaped pattern, reaching a peak response of almost two percent two years after a positive technology shock. Hours, then, slowly return to the pre-shock level. Using Canadian data, we find similar results. In the year of the shock, labor productivity and hours worked both rise. Labor productivity rises by roughly one percentwhileaveragehoursrisesby0.4percent. Themaximalriseinhoursworked,0.8percent, occurs a year after the shock. In Canada, it takes roughly six years for hours to return to its pre-shock level. For both countries, the rise in average productivity is statistically significant for a prolonged period of time. In contrast, confidence intervals about the estimated impulse response functions for hours worked are wide. Still, the rise in U.S. hours worked is statistically differentfromzeroforthefirstthreeyearsaftertheshock, ForCanada, theriseinhoursworked is statistically different from zero only in the first year after the shock. As with the benchmark results in CEV (2003), our findings stand in sharp contrast to the literature according to which hours worked in the United States falls after a positive technology shock (see for example Gali (1999) and Francis and Ramey (2001)). The question is what accountsfor thisdifference? Thedifference cannotbeattributedto ouridentifying assumptions since these are the same as in the literature. Using quarterly U.S. date, CEV (2003) argue that the key difference has to do with how hours worked are incorporated into the analysis. There we show that when we include ∆h in Y , then hours worked falls after a technology shock. In t t contrast when we include h in Y , then hours worked rises after a technology shock. For future t t reference we refer to the specification when h is included in Y as the level specification. We t t refer to the specification when ∆h is included in Y as the difference specification. t t We find similar results for the annual Canadian and U.S. data. Specifically suppose that X in (1) and (3) corresponds to the growth rate of hours worked rather than the level of hours t worked. Figure 4 reports our results for the United States and Canada. In both countries, a positive technology shock leads to a sharp, prolonged rise in labor productivity. In contrast to the results above, we now find that hours worked falls after a positive technology shock. Indeed, in both countries, according to our point estimates, hours worked never returns to the pre-shocklevel. Granted, confidenceintervalsarevery large. Butatleastfor theUnited States, CEV (2003) show that the initial fall in hours worked is statistically significant when the effect of a technology shock is estimated using quarterly data. Insum,whenweworkwiththelevelspecification,apositivetechnologyshockinducesalarge temporary increasein hoursworked, both in Canada and the United States. Butwhen we work with difference specification, a positive technology shock leads to a persistent decline in hours worked. In the next section we address the question: Which of the competing results is more plausible, those based on the level specification or those based on the difference specification? 5

4 Choosing Between the two Specifications The level and difference specifications are based on different statistical models, corresponding towhether we assume thathours workedare difference stationary or stationary in levels. As we saw, these specifications generated different answers to the question of what happens to hours workedafter apositivetechnology shock. Toassesswhichanswer ismoreplausible, we mustselect between the statistical models underlying the two specifications. We first address this issue using standard classical diagnostic tests. Since these do not convincingly discriminate between the competing specifications we then turn to the type of encompassing methods employed in CEV (2003). 4.1 Tests for Unit Roots and Stationarity Inthissubsectionwereporttheresultsfortwowell-knownstatistical testsofwhetheraunivariate time series has a unit root. The first test is the Augmented Dicky Fuller (ADF) test, which tests the null hypothesis that hours worked has a unit root. The second test is the KPSS test (fromKwiatkowskietal. 1992), whichteststhenullhypothesisthathoursworkedisstationary. For the United States hour series, the ADF test fails to reject, at the 10 percent significance level, the null hypothesis that per capita hours worked has a unit root.11 At the same time the KPSS test fails to reject the null hypothesis, at the 10 percent significance level, that per capita hours worked are stationary.12 For Canada, the results are somewhat more supportive of the level specification. The ADF test rejects the unit root hypothesis at the 2.5 percent significance level but fails to reject it at the 1 percent level.13 The KPSS test fails to reject the null hypothesis of stationarity at the 5 percent significance level.14 Based on these results we conclude that conventional standard classical diagnostic tests cannot be used to convincingly discriminate between our two competing statistical models of per capita hours worked, either in Canada or the United States. 4.2 Encompassing Tests In the preceding section we 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. 11The ADF test statistic (with two lags) equals 1.6014. The critical value corresponding to a 10 percent − significance level is 2.57. 12The value of the − KPSS test statistic is 0.3221. The asymptotic critical value at the ten percent significance level is 0.347. In implementing this test we set the number of lags in our Newey-West estimator of the relevant covariance matrix to two. 13TheADF test statistic hasa valueof-2.5768 with threelags while thesmall samplecriticalvalues are -2.26 and-2.66atthe10percentand5percentsignificancelevel,respectively. AsymptoticcriticalvaluesfortheADF test statistic are -2.23 and -2.58 at the 10 percent and 5 percent significance level, respectively. 14With two lags the KPSS test statistic has a value of 0.3577. Therefore one would not reject the null at the 5 percent significance level using the asymptotic critical value of 0.463. 6

In what follows we review the impact of specification error and sampling uncertainty on the ability of each specification to encompass the other. Our discussion here closely parallels the analysis in CEV (2003) who argue that, other things equal, the specification that will do best on the encompassing test is the one that predicts that the other specification is misspecified. We show in the next section that the level specification predicts that the difference specification is misspecified. We,therefore, expect that the level specification will do better than the difference specification. But as noted in CEV (2003), this consideration is not definitive because sampling considerations also enter. After discussing these issues we present our bivariate encompassing results. 4.2.1 A Priori Considerations when the Level Specification is True If the level specification is true and the econometrician adopts the difference specification, he is committing a specification error. 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. Suppose the econometrician estimates the instrumental variables equation under the mistaken 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 twice t t 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 − − SupposethatX hasnotbeenoverdifferenced,sothatitsspectraldensityisdifferentfromzeroat t frequencyzero. Then,inthetruerelationship,theterminvolvingX isactuallyα¯(L)∆X ,where t t α¯(L) is a finite ordered polynomial. In this case, the econometrician commits 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). This, however, 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. 4.2.2 A Priori Considerations when the Difference Specification is True Suppose the difference specification is true but the econometrician works with the level specification. Here the econometrician is not committing a specification error. To see this, first consider the instrumental variables regression: ∆f =µ+β(L)∆f +α(L)∆X +εz, (5) t t 1 t t − where the polynomials, β(L) and α(L), are of order q and q 1, respectively. The econome- − trician does not impose the restriction α(1) equals zero when it is, in fact, true. This is not 7

a specification error, because the parameter space does not rule out α(1) equal to zero. 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)). The fact that the econometrician is not committing a specification error does not necessarily imply that the level specification can encompasses the difference results. This is because sampling considerations must be taken into account. CEV (2003) stress that 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 explain the results of the level specification. To see why a weak instrument arises, recall that the econometrician who adopts the level specification uses lagged values of X as instruments for ∆X . If X , however, actually has a t t t unit root, this results in a weak instrument problem. Lagged X ’s are poor instruments for t ∆X because ∆X is driven by relatively recent shocks while X is heavily influenced by shocks t t t that occurred long ago. At least in large samples, there is little information in lagged X ’s for t ∆X . t A different way to see why a weak instrument problem arises when the econometrician mistakenly adopts the level specification is as follows. Consider the regression ∆h =a+Πh +p(L)∆h +q(L)∆f +εz. (6) t t 1 t 1 t 1 t − − − AtestofthehypothesisthatΠisequaltozerohastwointerpretations. First,itisthecovariates ADF test of Hansen (1995) for whether h has a unit root (Π = 0). Second, it is the standard t F-test for weak instruments discussed in Staiger and Stock (1997) for whether the lagged level of hours is a weak instrument for ∆h (Π = 0). So here testing for the presence of a unit t root in h is the same as testing for whether lagged hours is a weak instrument for the ∆h . t t If the difference specification is true, then at least asymptotically one could not reject either hypothesis. Tosummarize,whenthelevelspecificationistrue,thedifferencespecificationismisspecified. When the difference specification is true, the level specification is not misspecified but the econometricianwillencounteraweakinstrumentproblemandtherewilllikelybelargesampling uncertainty, as well as bias, associated with parameter estimates. 5 Encompassing Results We base our encompassing tests on the ability of the level and difference specifications to match three observations. The first two observations come from the empirical hours response that arises from the two different specifications. For the level specification, the average hours response following a technology shock is positive. For the difference specification, the average hours response is negative. The encompassing test compares the ability of each specification to account for both of these findings. The third observation is the empirical value of the weak instrument F-test. Specifically, we assess the ability of the level and difference specification to account for the observed F-test (6). If the level specification is true then the lagged level of hours will be a good instrument. If the difference specification is true, then the lagged level of hours will be a poor instrument. So the two specifications ought to have different implications for the weak instrument F test. 8

5.1 Does the Level Specification Encompass the Difference Specification Results? To study whether the level specification can encompass the difference specification, we proceed as in CEV (2003). For each country, we use the estimated level specification VAR as the data generating process [DGP]. With this DGP, we simulated by bootstrap 1000 artificial data sets, each of length equal to our actual sample size. For each simulated data set, we then (incorrectly) assume that the difference specification is true, and estimate a bivariate VAR in whichhoursworkedappearsingrowthrates,andcomputetheimpulseresponsestoatechnology shock. Panel A of Figure 5 reports the distribution of impulse responses that arise from the simulation. ThemeanimpulseresponsesappearasthethinlinewithcirclesintheFigure. 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 4 and correspond to our point estimate of the relevant impulse response function generated from the difference specification. The thick lines are reproduced from Figure 3 and correspond to the point estimate of the relevant impulse response function generated from the level specification. The gray area represents the 95 percent confidence interval of the simulated impulse response functions.15 From Figure 5 we see that, for both countries, the average of the impulse response functions emergingfromthe‘misspecified’growthrateVARareveryclosetotheactualestimatedimpulse response generated using the difference specification. In particular, 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 incorrectly adopting the difference specification can explain the estimated fall in hours found using the difference specification. In other words, the level specification attributes the decline in hours in the estimated VAR with differenced hours to over-differencing. We conclude that the level specification convincingly encompasses the difference specification. 5.2 Does the Difference Specification Encompass the Level Results? To assess the ability of the difference specification to encompass the level specification, we proceed as above except we now take as the data-generating process the estimated VARs in which hours worked appears in growth rates. Panel B in Figure 5 reports results analogous to those displayed in Panel A. The thick, solid lines, reproduced from Figure 3, are the impulse responses associated with the estimated level specification. The thin lines with the triangles are reproduced from Figure 4 and are the impulse responses associated with the difference specification. The thin lines with circles in Panel B are the mean impulse response functions that result 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. Two results are worth noting. First, for the United States the hours response is nearly zero. 15Confidence intervals were computed point-wise as the average simulated response plus or minus 1.96 times the standard deviation of the simulated responses. 9

This result is closer to the difference specification than the level specification and therefore suggests that the distortion associated with not imposing a unit root in hours worked is not very large. The Canadian data have results that are somewhat different. Here the mean impulse response for hours is actually close to the level specification’s result.16 In part this reflects the small sample bias issues discussed in CEV (2003). Apparently the small sample used in the current paper implies that the level specification has some difficultly recovering the true impulse responses. Consistent with this, there is a great deal of sampling uncertainty associated with the estimated impulse response function. Indeed the confidence intervals in Panel B of Figure 5 are substantially much wider than those reported in the other Figures. Recall from our discussion above that if the difference specification is true, then an econometrician who works with the level specification ought to encounter large sampling uncertainty. This prediction faces a basic problem: it rests fundamentally on the difference specification’s implication that there is a weak instrument problem. But as we show below, when we apply a standard test for weak instruments to the data, we find little evidence of this problem. Moreover, the actual estimated confidence intervals associated with impulse responses obtained using the level specification are relatively narrow (see Figure 3). 5.3 Testing for Weak Instruments Toassesswhetherthereisevidenceofweakinstrumentsinthedata,weuseastandardF testfor weak instruments. Specifically, we regressed ∆H on a constant, H , and the predetermined t t 1 − variables in the instrumental variables regression, (5). These are ∆H and ∆f , s=1,2,3. t s t s − − Our weak instruments F statistic is the square of the t statistic associated with the coefficient on H . In effect, this F statistic measures the incremental information in H about ∆H .17 t 1 t 1 t − − If the difference specification is correct, the additional information is zero. For the United States, the weak instrument F-test is 5.95. This is below Stock and Staiger’s recommended threshold value of 10 which suggests that, for the U.S. data, there may be a weak instrument problem. In contrast CEV (2003) report that there is little evidence of a weak instrument problem with the U.S. quarterly data. For Canada, the weak-instrument F-test is 11.60, which suggests that there is no weak instrument problem with the Canadian data.18 As discussed above, this calls into question a basic implication of the view that hours worked have a unit root. 5.4 The Relative Odds of the Two Specifications The results of the previous subsections indicate that the level specification can easily account for the estimated impulse response functions obtained with the difference specification. The differencespecificationhasasomewhathardertimeaccountingforthelevelspecificationresults. As in CEV (2003), we quantify the relative plausibility of the two specifications, by using the type of posterior odds ratio considered in Christiano and Ljungqvist (1988). Christiano and 16These results are somewhat different than those reported for the quarterly U.S. data in CEV (2003). 17Asnotedabove,ourF testisequivalenttoastandardADFtestwithadditionalregressors. Intheunitroot literature, this test is referred to as the covariate ADF test (Hansen 1995). 18The 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. 10

Ljungqvist developed their statistic for a similar situation where differences and levels of data lead to very different inferences.19 In our context, we claim that the more plausible of the two VARs 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 value of the weak instruments F statistic. The odds ratio that we use is calculated as follows. We simulated 1,000 artificial data sets using each of our two estimated VARs as the data generating mechanism. For an event Q we asked what was the probability of observing that event for the level specification, P(QA), | where A denotes the level specification being true, and what was the probability of observing that event for the difference specification, P(QB), where B denotes the difference specification | being true. The relative plausibility of the two different specifications can then be assessed as the odds ratio of the level specification being true versus the difference specification being true given the observed event P(AQ) P(QA)P(A) | = | . P(B Q) P(Q B)P(B) | | If one had a prior distribution that put equal weight on A and B then the odds ratio is just the ratio of the conditional probabilities P(QA) | P(Q B) | We consider five definitions of the event Q : (i) the difference specification is true and the impact effect of a technology shock on hours worked is negative, (ii) the level specification is true and the impact effect of a technology shock on hours worked is positive, (iii) both (i) and (ii) are true, (iv) the weak instrument F statistic test is greater than or equal to the F statistic obtained with the actual data, (v) events (i), (ii) and (iv) occur. Table 1 reports the frequency with which the different events were observed in the two simulated data sets. For the U.S. case, the difference specification does slightly better at predicting event (i). But the level specification does much better at predicting events (ii) and (iv). The overall plausibility of the two specifications can be most easily assessed in terms of event (v), in which case, the odds ratio favors the level specification by over five to one. Similar results hold for the Canadian case. Surprisingly, the fall in hours associated with the difference specification is actually observed more frequently when the DGP corresponds to the 19Eichenbaum and Singleton (1986) found, in a VAR analysis, that when they worked with first differences of variables, there was little evidence that monetary policy plays an important role in business cycles. However, when they worked with a trend stationary specification, monetary policy seems to play an important role in business cycles. Christiano and Ljungqvist argued that the preponderance of the evidence supported the trend stationary specification. 11

level specification. Focusing on event (v), the odds ratio favors the level specification by over nine to one. So the odds in favor of the level specification are even higher for the Canadian case than for the U.S. case. This is consistent with our F-test results indicating less of a weak instrument problem for Canada than for the United States. Table 1: Simulations Results and Odds Ratios United States Canada Percent True Odds Percent True Odds Specification Level Difference Level Difference Event (i) Impact Negative, Difference Specification 67.7 71.4 0.949 91.6 81.1 1.130 (ii) Impact Positive, Level Specification 97.1 54.3 1.788 96.9 66.8 1.451 (iii) Both Impact Responses 65.6 37.3 1.757 88.5 53.3 1.660 (iv) F-test 70.1 22.9 3.066 65.0 10.7 6.070 (v) Events (i),(ii) and (iv) True 47.6 8.9 5.343 57.6 6.2 9.290 6 Multivariate Results for Canada In this section, we discuss how other Canadian variables (money growth, inflation, and the interest rate) respond to a technology shock.20 To estimate these response functions one could proceed as in CEV (2003) and estimate the large simultaneous system. However, given our small sample size and the large number of parameters that would have to be estimated, we are reluctant to do so. For example a five variable VAR with four lags requires estimating 110 coefficients with only forty years worth of annual data. Instead we adopt the following sequential approach. Suppose we are interested in estimating the dynamic response functions of a set of variables, X ,X ,X .... For a given variable X , we estimate the technology shocks 1 2 3 i εz,i using the the following version of our basic IV equation: t ∆f =µ+β(L)∆f +α (L)∆H +α (L)∆X +εz,i. (7) t t 1 H t Xi it t − We then estimate the VAR, ∆f ∆f t t 1 H =B(L) H − +γε z,i +v  t   t 1  t t − X X it it 1    −      Finally we derive the impulse response of X to a technology shock from the estimated VAR. it Given the results of the previous section we confine our attention to the level specification. Figure 2 displays the time series on Canadian money growth, inflation, and the interest rate.21 Figure 6 reports the estimated response of these variables to a technology shock. For convenience the hours responses are repeated from Figure 3 as well as the output response implied by the bivariate VAR reported in Figure 3. Two key results emerge here. First, our point estimates indicate that inflation and the interest rate fall after an expansionary 20See CEV (2003) and ACEL (2003) for multivariate results based on quarterly U.S. data. 21The Bank Rate is measured using CANSIM series B14006. The money supply is M2, with CANSIM mnemonic B1630. Inflation is measured as the growth rate of the GDP deflator D15612. 12

technologyshockwhilemoneygrowthrises. ThissuggeststhattheBankofCanada, onaverage, accommodated technology shocks over our sample period. Second, the confidence intervals aroundtheestimatedimpulseresponsefunctionsarequitewide. Still, theinitialfallininflation is statistically significant.22 7 Conclusions Using annual Canadian and U.S. data, this paper argues that a positive technology shock leads to a rise in Canadian output and hours worked, as well as a fall in inflation. In CEV (2003), using similar methods and quarterly U.S. data, we argued that a technology shock also leads to a rise in aggregate consumption and investment. On the face of it, these findings are consistent with the predictions of the standard parameterization of a real business cycle model. But, in our view, the rise in hours worked and overall expansion in aggregate activity that follows in the wake of a technology shock reflects how monetary policy makers reacted to the technology shock. While our empirical results are suggestive on this point, they are not definitive. To make the case convincingly requires that we know how the economy would have reacted had the monetary authority acted differently. The only place where we can perform such a counterfactual experiment is in a structural economic model. ACEL (2003) conduct this type of experiment in a dynamic general equilibrium model embodying wage and price frictions. They argue that an estimated version of the model can account for how the U.S. economy reacted to monetary policy and technology shocks in the postwar era. They use this model as a laboratory to investigate how the U.S. economy would havereactedtoatechnologyshockwithadifferentmonetarypolicy. Forexample,theyconsider what would have happened under the assumption that the Federal Reserve had not accommodated technology shocks but rather had followed a k percent money growth rule. The key conclusion in ACEL is that, with this counter-factual policy, hours worked would have fallen for a prolonged period of time after a positive technology shock. In addition, compared to the actualoutcomes,outputwouldhaverisenbyfarlessandinflationwouldhavefallenbyfarmore. We suspect that the same result would be true for Canada. Based on these results and similar findingsinGaliLopez-SalidoandValles(2003), itisclearthatpolicymakersoughttobevitally interested in the supply side developments of the economy. Simple formulations of policy, like Taylor rules, often push discussions of the output gap into the background. Knowing why the gap moves is a critical input into policy decisions. Taken together, the results of this paper, CEV (2003) and ACEL (2003), suggest that policy makers have, on average, been successful at identifying technology shocks and have reacted in a way that has improved aggregate economic performance. 22Notethattheconfidenceintervalsbecomeextremelywideinthelongrun. Thismayreflectanearunitroot in our measure of the Canadian interest rate. 13

References [1] Altig, David, LawrenceJ. Christiano, MartinEichenbaumandJesper Linde, 2003, ‘AnEstimated Dynamic, General Equilibrium Model for Monetary Policy Analysis,’ Manuscript. [2] Christiano, Lawrence J., and Lars Ljungqvist, 1988, ‘Money Does Granger Cause Output in the Bivariate Money-Output Relation,’ Journal of Monetary Economics. 22(2), 217-35. [3] Christiano, Lawrence J., MartinEichenbaumand RobertVigfusson, 2003, ‘WhatHappens After A Technology Shock?’ Board of Governors of the Federal Reserve System International Finance Discussion Papers 768. [4] Doan, Thomas 1992. Rats Manual Estima Evanston, IL. [5] Eichenbaum, Martin, andKennethJ.Singleton1986.‘DoEquilibriumRealBusinessCycle TheoriesExplainPostwarU.SBusinessCycles?’ NBERMacroeconomicsAnnual 1986, 91- 135. [6] Elliott, Graham, and Michael Jansson, 2003, ‘Testing for Unit Roots with Stationary Covariates,’ Journal of Econometrics, 115, 75-89. [7] Fisher,Jonas,2002,‘TechnologyShocksMatter’,FederalReserveBankofChicagoWorking Paper WP-02-14.. [8] Francis, Neville, and Valerie A. Ramey, 2001, ‘Is the Technology-Driven Real Business Cycle Hypothesis Dead? Shocks and Aggregate Fluctuations Revisited,’ Manuscript, UCSD. [9] Gali, Jordi, 1999, ‘Technology, Employment, and the Business Cycle: Do Technology Shocks Explain Aggregate Fluctuations?’ American Economic Review, 89(1), 249-271. [10] Gali, Jordi, J. David Lopez-Salido, and Javier Valles, 2002, ‘Technology Shocks and Monetary Policy: Assessing the Fed’s Performance’, National Bureau of Economic Research Working Paper 8768. [11] Hansen, BruceE., 1995, ‘RethinkingtheUnivariateApproachtoUnitRootTesting: Using Covariates to Increase Power,’ Econometric Theory, December, 11(5), 1148-71. [12] Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., and Shin, Y. 1992, ‘Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root,’ Journal of Econometrics, 54, 159- 178. [13] Shapiro, Matthew, and Mark Watson, 1988, ‘Sources of Business Cycle Fluctuations,’ NBER,Macroeconomics Annual, 111-148. [14] Sims Christopher, James Stock and Mark Watson, 1990. ‘Inference in Linear Time Series Models with Some Unit Roots,’ Econometrica 58(1), 113-144. [15] Staiger, Douglas, and James Stock, 1997, ‘Instrumental Variables Regression with Weak Instruments,’ Econometrica, 65(3), 557-586. [16] Vigfusson, Robert J. 2002 ‘Why Does Employment Fall After A Positive Technology Shock,’ Manuscript. 14

A Figures Figure 1: Data Used in VAR, United States Labor Productivity Growth Log of Per-capita Average Hours -7.46 0.04 -7.48 0.03 -7.5 -7.52 0.02 -7.54 0.01 -7.56 0 -7.58 -7.6 -0.01 1959 1969 1979 1989 2001 1959 1969 1979 1989 2001 15

Figure 2: Data Used in VAR, Canada Labor Productivity Growth Log of Per-capita Average Hours -5.3 0.04 -5.32 0.03 -5.34 0.02 -5.36 -5.38 0.01 -5.4 0 1970 1980 1990 2000 1970 1980 1990 2000 Inflation Bank Rate 15 0.1 10 0.05 5 0 1970 1980 1990 2000 1970 1980 1990 2000 Money Growth 0.2 0.15 0.1 0.05 0 1970 1980 1990 2000 16

Figure 3: Impulse Responses Using the Level Specification Panel A) United States Labor Productivity Average Hours 3.5 4.5 3 4 2.5 3.5 3 2 2.5 1.5 2 1 1.5 0.5 1 0 0.5 0 -0.5 -0.5 -1 0 2 4 6 8 0 2 4 6 8 Panel B) Canada Labor Productivity Average Hours 1.6 1.4 2.5 1.2 1 2 0.8 0.6 1.5 0.4 1 0.2 0 0.5 -0.2 -0.4 0 0 2 4 6 8 0 2 4 6 8 Thick Line: Impulse Responses from Level Specification Gray Area: 95 percent Confidence Intervals 17

Figure 4: Impulse Responses Using the Difference Specification Panel A) United States Labor Productivity Average Hours 3 1 2.5 0.5 0 2 -0.5 1.5 -1 1 -1.5 -2 0.5 -2.5 0 0 2 4 6 8 0 2 4 6 8 Panel B) Canada Labor Productivity Average Hours 2.5 0.5 2 0 1.5 -0.5 -1 1 -1.5 0.5 -2 0 0 2 4 6 8 0 2 4 6 8 Line with Triangles: Impulse Responses from Difference Specification Gray Area: 95 percent Confidence Intervals For Simulated Impulse Responses 18

Figure 5: Encompassing Results Panel A: Level Specification As the DGP United States Canada 1.5 0.5 1 0 0.5 -0.5 0 -0.5 -1 -1 -1.5 -1.5 -2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 Periods After Shock Periods After Shock Panel B: Growth Rate Specification As the DGP United States Canada 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -2 -1.5 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 Periods After Shock Periods After Shock Thick Line: Impulse Responses from Level Specification Line with Triangles: Impulse Responses from Difference Specification Circles: Average Impulse Response for Simulations from Designated DGP Gray Area: 95 percent Confidence Intervals For Simulated Impulse Responses 19

Figure 6: Sequential Analysis Output Hours Worked 3 1.5 2.5 1 2 1.5 0.5 1 0 0.5 -0.5 0 0 2 4 6 8 0 2 4 6 8 Inflation Bank Rate 200 1 100 0.5 0 0 -0.5 -100 -1 -200 -1.5 -300 -2 0 2 4 6 8 0 2 4 6 8 Money Growth 5 4 3 2 1 0 -1 0 2 4 6 8 Thick Line: Impulse Responses from Level Specification Gray Area: 95 percent Confidence Intervals For Simulated Impulse Responses 20