The Inflation Persistence of Staggered Contracts

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 734 August 2002 THE INFLATION PERSISTENCE OF STAGGERED CONTRACTS Luca Guerrieri 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to International Finance Discussion Papers (other than an acknowledgement that the writer has had access to unpublished material) should be cleared with the author or authors.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/.

The In(cid:176)ation Persistence of Staggered Contracts Luca Guerrieri ⁄ Abstract One of the criticisms routinely advanced against models of the business cycle with staggered contracts is their inability to generate in(cid:176)ation persistence. This paper flnds that staggered contracts (cid:181)a la Taylor are, in fact, capable of reproducing the in(cid:176)ation persistence implied by U.S. data. Following Fuhrer and Moore, I capture the moments that the contract speciflcation needs to replicate by using the correlograms from a small vector autoregression (VAR) that includes in(cid:176)ation among the endogenous variables. A simple structural model substitutes the in(cid:176)ation equation from the VAR with the contract speciflcation. I estimate the contract parameters in the structural model by maximum likelihood. The correlogram for the endogenous variables from the estimated structural model, including that for in(cid:176)ation, are very close to the correlograms from the VAR (and are contained within their 90% confldence intervals). By the same metric, where Taylor contracts do not fare well is in reproducing the cross-correlations between in(cid:176)ation and output. Keywords: maximum likelihood, Phillips curve. ⁄Correspondence: BoardofGovernorsoftheFederalReserveSystem,WashingtonD.C.20551-0001. Telephone(202) 452 2550. Fax (202) 872 4926. E-mail Luca.Guerrieri@frb.gov. I am grateful to David Bowman, Joe Gagnon, Dale Henderson and Andy Levin for advice and encouragement. While writing this paper, I had useful conversations with Paul Bergin, Jason Brown, Chris Erceg, Jefi Fuhrer, Chris Gust, Peter Ireland, John Roberts, Argia Sbordone, Ralph Tryon, and Jonathan Wright. I also need to thank the participants of the International Finance Workshop for their insightful comments. Remaining errors are my own. The views in this paper are solely the responsibility of the author, and should not be interpreted as re(cid:176)ecting the views of the Board of Governors of the Federal Reserve System, or of any other person associated with the Federal Reserve System.

1 Introduction The study of the Phillips curve has been central to macroeconomics since Phillips (1958) identifled a negative correlation between in(cid:176)ation and unemployment. King and Watson (1994) give a comprehensive discussion of the evolution of the traditional empirical literature. Nominal rigidities have become the standard theoretical underpinning of what Gal¶‡ and Gertler(1999)called\thenewPhillipscurve". Theyusedlimitedinformationestimationto show that a standard contracting speciflcation provides a good description of the U.S. data. Sbordone (2002) validated the results of Gal¶‡ and Gertler (1999) by using an alternative estimation method that follows Campbell and Shiller (1988). Gal¶‡, Gertler, and Lo¶pez- Salido (2001a) suggested that the staggered contract mechanism flts the European data possibly even better than the U.S. data. Rudd and Whelan (2001) question the power of tests employed by Gal¶‡ and Gertler (1999) and Gal¶‡, Gertler, and Lo¶pez-Salido (2001a). Gal¶‡, Gertler, and Lo¶pez-Salido (2001b) address these concerns. The results of Fuhrer and Moore (1995) stand out in the growing empirical literature on the new Phillips curve. They showed that the staggered price mechanism of Taylor (1980) (henceforth, referred to as \standard") is not capable of generating the in(cid:176)ation persistence that they observed in the U.S. data. Fuhrer and Moore showed that an alternative contracting speciflcation (henceforth, referred to as \relative"), flrst introduced by Buiter and Jewitt (1981), fares much more favorably in fltting the U.S. data. This alternative speciflcation postulates that, when choosing a contract wage, workers care about the relative remuneration with respect to other outstanding contracts. This has the practical efiect of introducing an extra lag of in(cid:176)ation in the implied Phillips curve, which accounts for its ability to generate greater persistence. The results of Fuhrer and Moore (1995) continue to be greatly in(cid:176)uential. Representative recent papers that list it as a motivation are Calvo, Celasun, and Kumhof (2001), who postulate sticky in(cid:176)ation at the onset, and Mankiw and Reis (2002), who build on the work of Roberts (1997) in assuming sticky information. The evaluation procedure of Fuhrer and Moore (1995) has two steps. First, a simple 1

statistical model captures the properties of the data that the contracting speciflcation needs to reproduce. The statistical model takes the form of an unconstrained vector autoregression (VAR) with output per person, in(cid:176)ation and the short-term interest rate as the endogenous variables. Then, the equation for in(cid:176)ation in the VAR is replaced with a contracting speciflcation, thus generating a structural model where only the parameters in thecontractingspeciflcationareunknown. Second, thestructuralparametersareestimated via maximum likelihood. Coenen and Wieland (2000) followed the methodology of Fuhrer and Moore (1995) to calibrate a general equilibrium model of the Euro area. In line with Gal¶‡, Gertler, and Lo¶pez-Salido (2001a), they found that both the standard and the relative contracting speciflcation \flt euro area data reasonably well." The sample in Fuhrer and Moore (1995) spans the period from 1965 to 1993, thus including the oil crises of the 70s, as well as the Volcker disin(cid:176)ation. Evans and Wachtel (1977), Taylor (2000), and Cogley and Sargent (2000) documented that a high degree of in(cid:176)ation persistence is a characteristic of the late 1960s and 1970s, but not necessarily of the remaining postwar period. Erceg and Levin (2001) developped a model with standard contracts where agents use optimal flltering to disentangle persistent and transitory shifts in monetary policy. They attributed the observed persistence in in(cid:176)ation, following the Volcker disin(cid:176)ation, to uncertainty over monetary policy. The purpose of this paper is to test whether or not the lower persistence of in(cid:176)ation found by other authors in the U.S. data for the 1980s and 1990s translates into signiflcantly difierent estimates of the parameters in the standard and relative contract model. As a byproduct, this is also a test of whether the results of Fuhrer and Moore (1995) still hold true when using the additional data that have become available since the original publication of their study. Not only are longer time series available, but the series have been revised. The data that Fuhrer and Moore (1995) used come from the productivity release of the Bureau of Labor Statistics. Duke and Usher (1998) document the latest improvements to these series. 2

Using the sample from 1980 to 2001, I flnd that relative contracts are able to reproduce the in(cid:176)ation persistence observed in the data. This is still true if the estimation sample starts in 1965q1 or in 1960q1. The results concerning relative contracts reported by Fuhrer and Moore (1995) still hold with updated data and longer series, and are resilient to introducing breaks in the linear detrending of output, as well as reestimation over smaller subsamples. More surprisingly, I also flnd that standard contracts perform very well. The metric that I use to make these claims is the distance between correlograms for in(cid:176)ation, the interest rate and output coming from the VAR and the structural models. I compare the correlograms from the unrestricted VAR with the correlograms from the estimated structural model with standard contracts and with relative contracts. I flnd that the correlogram from the VAR for each of the three endogenous variables is close to the two structural counterparts across all the subsamples I consider. I compute the Monte-Carlo 90% confldence interval for the correlograms from the VAR. The correlograms for the two structural models invariably lie within the confldence bands. This is true not only for the benchmark sample 1980 to 2001, but also for extended samples going back to 1960. WhereTaylorcontractsdonotperformwellisinreproducingthecross-correlationsbetween in(cid:176)ation and output. Focusing on the 1980s and 1990s, I estimate a change in the structural parameters of the two contracting models I consider. This shift is consistent with a lower persistence of the in(cid:176)ation series, but is not statistically signiflcant. In previous work, Guerrieri (2001), found that staggered contracts set up following Calvo (1983), produced a better flt to the U.S. data than staggered contracts of one single flxed duration a(cid:181) la Taylor. Similarly, Jadresic (1999) found that a trimodal distribution of contract durations flt the U.S. data better than a flxed contract duration. The contracts in this paper, by allowing the coexistence of multiple contract durations, follow more closely the setup of Taylor (1980). Yun (1996) showed how to reconcile contracts a(cid:181) la Calvo with a flrst order condition coming from a proflt maximization problem. Chari, 3

Kehoe, and McGrattan (2000), transferred the setup of Yun (1996) to contracts of flxed duration. In this paper I show how to allow for multiple contracts of flxed duration a(cid:181) la Taylor, in a way that can be mapped into a proflt maximization exercise, and that is still parsimonious in terms of the size of the implied state space. This reinterpretation, can then be mapped into the setup of Fuhrer and Moore (1995). Allowing for a distribution of contract durations makes Taylor staggered contracts closer to the Calvo counterparts. The single contract duration is rejected by the data, substantiating that this development has empirical relevance. The plan for the rest of the paper is as follows: in Section 2, I build some intuition for the difierence between standard contracts and relative contracts; in Section 3, I lay out the methodology I used in the VAR estimation. In Section 4, I describe the structural estimation, and report the estimation results; and in section 5, I conclude. 2 Comparing Standard and Relative Contracts Gal¶‡ and Gertler (1999) gave a good review of the recent state of the literature. I will only attempt to summarize the salient points. The structure behind the new Phillips curve is an environment of monopolistically competitive flrms that are faced with a constraint on price adjustment. Following Taylor (1980), flrms are allowed to reset their contract price every n periods. Firms are otherwise symmetric in every other respect. At any period, n overlapping contracts are in force. Chari, Kehoe, and McGrattan (2000) showed that proflt maximization implies a flrst order condition for a flrm resetting its price at time t, that, by log-linearizing, leads to: n¡1 1 P = E P„ +(cid:176)Y~ (1) t t t+i t+i n Xi=0 ‡ · P is the log of the contract price set at time t, Y~ is an output measure and E denotes t t t expectations conditional on the information set available at time t. This also happens to be be the contracting speciflcation chosen by Taylor (1980)1. The log of the aggregate price, 1This is indeed a special case of that model, for a particular set of contract weights. Taylor’s paper focused on 4

P„ , is then given by: t n¡1 1 P„ = P (2) t t¡i n Xi=0 combining equation (1) and equation (2), setting n = 2, allowing for the fact that under rationalexpectationsE P = P ¡† (where†isaforecasterror), andflnallyreworkingthe t¡1 t t t price equation in terms of in(cid:176)ation, denoted by … , one obtains the Phillips curve equation, t which as shown in Appendix A, is given by 1 … = E … +(cid:176)(Y~ +E Y~ +Y~ +E Y~ )¡ † (3) t t t+1 t t t+1 t¡1 t¡1 t t 4 2.1 The relative contract model Fuhrer and Moore (1995) argued that the persistence imparted to in(cid:176)ation by the standard contracting speciflcation does not flt the U.S. in(cid:176)ation data as well as their relative speciflcation. Their alternative model can be summarized by the following equations, where each variable is to be thought in log deviation from steady state. The contract equation is the following: n¡1 1 P ¡P„ = E V +(cid:176)Y~ (4) t t t t+i t+1 n Xi=0 ‡ · where P is the price contract that starts in period t, P„ is the aggregate price level, Y~ is an t t t output measure. The aggregate price level is still governed by equation (2). V is a relative t price index, that takes following form n¡1 1 V = P ¡P„ (5) t t¡i t¡i n Xi=0 ‡ · Then, for n=2, the Phillips curve equation implied by this contracting speciflcation takes the form: 1 1 … = (… +E … )+(cid:176)(Y~ +E Y~ +Y~ +E Y~ )¡ † (6) t t¡1 t t+1 t t t+1 t¡1 t¡1 t t 2 4 staggeredwages. Thesubsequentliteratureshiftedthesetuptostaggeredprices. HuangandLiu(2002)showedthatthe staggered wage interpretation allows one to escapethe criticism of staggered contracts of Chari, Kehoe, and McGrattan (2000). 5

Comparing equations (3) and (6), one can immediately see that the relative contract speciflcation, for any given contract length, appends an extra lag of in(cid:176)ation to the Phillips curve equation. 2.2 Allowing for multiple contract lengths Rather than maintaining that all contracts last n periods, a more (cid:176)exible setup would allow for a distribution of contract durations. Following Blinder (1994), one could also brand such a setup as more plausible. A simple way to model this distribution is to assume that when flrms set a price, they face uncertainty over the contract duration. The price they set might be in force for any length of time between 1 and n periods. Firms do know, however, the relevant probabilities. Then, let (cid:181) be the probability that a contract will be in force 1 only one period, let (cid:181) be the probability that a contract will be in force for two periods, 2 and so on. Let the vector (cid:181) summarize the relevant contract weights. The elements of (cid:181) are all non-negative and sum to 1. Fixing the longest contract duration at four periods (n = 4), the aggregate price level becomes 1 1 1 2 1 3 P„ = (cid:181) P +(cid:181) P +(cid:181) P +(cid:181) P (7) t 1 t 2 t¡i 3 t¡i 4 t¡i 2 3 4 Xi=0 Xi=0 Xi=0 The setup of Fuhrer and Moore (1995) can be reinterpreted to conform to this setup. One way to impose some structure on the distribution of contract lengths would be to pick a functional form for the weights on contract prices in equation (7). Letting f denote the i weight on the contract price with lag i, equation (7) would then be rewritten as 3 P„ = f P (8) t i t¡i Xi=0 Fuhrer and Moore (1995) imposed that f = 0:25+(1:5¡i)s (9) i where s is the only parameter governing the shape of the distribution of contract durations. To be able to match a choice for s into a vector (cid:181), s needs to be contained in the interval 6

Table 1: Mapping s into (cid:181) s (cid:181) (cid:181) (cid:181) (cid:181) f f f f 1 2 3 4 0 1 2 3 0 0 0 0 1 0.25 0.25 0.25 0.25 0.06 0.06 0.12 0.18 0.63 0.34 0.28 0.22 0.16 0.08 0.08 0.16 0.24 0.52 0.37 0.29 0.21 0.13 1 1 1 1 0 1 1 1 0 6 6 3 2 2 3 6 between2 0 and 1. In Appendix B, I show how to map a choice for s into a set of contract 6 weights (cid:181) to (cid:181) . In Table 1, I perform this mapping for selected values of s. As shown, as 1 4 s decreases, the weight on the longer contracts increases. In this stochastic contract setup, the contract price rule for the standard model becomes 3 P = f E P„ +(cid:176)Y~ (10) t i t t+i t+i Xi=0 ‡ · while, for the relative contract setup, following Fuhrer and Moore (1995) 3 P ¡P„ = f E V +(cid:176)Y~ (11) t t i t t+i t+i Xi=1 ‡ · where V can be substituted into (11) from equation (5). t 2.3 Nesting the two models Let – govern the fraction of agents using relative contracts. Then the standard and the relative contract models can be nested by letting the contract price equation become 3 P ¡–P„ = f E Vn +(cid:176)Y~ (12) t t i t t+i t+i Xi=1 ‡ · where the index Vn is given by t n¡1 1 Vn = P ¡–P„ (13) t n t¡i t¡i Xi=0 ‡ · 2This is equivalent to the condition imposed by Fuhrer and Moore (1995) that the polynomial in the lag operator used to rewrite the aggregate price equation be invertible. 7

3 VAR estimation In order to assess the properties of the data that the contracting speciflcation needs to reproduce I rely on a simple statistical model that takes the form of a VAR. Detrended output and in(cid:176)ation are the series of interest. Following Bernanke and Blinder (1992), FuhrerandMoore(1995)andCoenenandWieland(2000), Iincludetheshorttermnominal interest rate in the VAR to help in the formation of output expectations. Thus the three endogenous variables in the VAR are detrended log of output, in(cid:176)ation and the short term interest rate. Just as Fuhrer and Moore (1995) the series for the above variables come from the productivity release of the Bureau of Labor Statistics. While the interest rate series goes back to 1934, the output and price series start in the flrst quarter 1947. For the VAR estimation I discard the flrst part of the sample and take the flrst quarter of 1960 as the starting date for the analysis. I keep this flrst portion of the data as a presample, that I exploit later in the maximum-likelihood estimation of the structural parameters. The measure of output that I consider is log of the nonfarm business output per person. Themeasureofin(cid:176)ationcomesfromaquarterlydifierenceinthelogofthenonfarmbusiness output de(cid:176)ator. Finally the interest rate series is the 3 month treasury bill rate from the secondary market quoted on a discount basis. I linearly detrend the output measure 3 In the detrending, I have considered both single as well as multiple trends. I have considered breaks in 1983q1, which coincides with the end of Volcker’s disin(cid:176)ation program, as an alternative, as well as 1992. The additional trends do not appear to afiect the results. Here, I report only the results using one trend. To decide the number of lags for the endogenous variables in the VAR equations I followed the general-to-speciflc approach. I started with a speciflcation that included eight lags. I reduced this number, until the parameters on the longest lag were jointly signiflcant across equations, and the residuals were uncorrelated. To test for correlation, I used a 3While one-sided flltering would be more rigorous, I use linear detrending procedure to ensure comparability with the results of Fuhrer and Moore (1995). I reserve the one-sided-flltering reflnement to possible extensions of this paper. 8

Portmentau test on lag 12. I settled on a VAR speciflcations that included three lags of all the endogenous variables. The VAR structure on which I settle has the form 3 Y~ = C Y~ +C r +C … +† (14) t y;1;i t¡i r;1;i t¡i …;1;i t¡i y;t Xi=1 3 r = C Y~ +C r +C … +† (15) t y;2;i t¡i r;2;i t¡i …;2;i t¡i r;t Xi=1 3 … = C Y~ +C r +C … +† (16) t y;3;i t¡i r;3;i t¡i …;3;i t¡i …;t Xi=1 where r is the short-term interest rate. The intercept term is excluded from the VAR t structure to ensure a zero-in(cid:176)ation steady state, consistent with the two contracting speciflcations in this paper4. When varying the sample length, I kept the VAR structure flxed. For reasons of space, I do not report all the coe–cient estimates over the various subsamples I consider. I show the correlograms for the endogenous variables in Figure 1(this flgure also includes the correlograms from the structural estimation described below). The correlogram has the advantage over impulse response functions of not requiring an identiflcation scheme. I also report a 90% confldence interval around the correlograms. This is calculated using the Monte Carlo procedure described by Christiano, Eichenbaum, and Evans (1999). 4 Structural estimation In order to estimate the structural parameters in the standard and in the relative contracting speciflcation, I replace the the in(cid:176)ation equation in the VAR described in equations 4The parameter estimates for the VAR are relegated to an appendix. A likelihood ratio test conflrms the validity of the restriction that the constant term be zero. The conclusions reported below are resilient to reintroducing a constant in the VAR. Excluding the constant does afiect the shape of the correlogram for in(cid:176)ation and the interest rate from the VAR,depictedinFigure1. Withoutaconstant,thein(cid:176)ationpersistenceimpliedbytheVARappearstobehigher,thus making the task for the structural model harder, given the prior, from Fuhrer’s and Moore’s work, that the structural model is not capable of reproducing the in(cid:176)ation persistence in the data. 9

(29) to (31) with the relevant contract equations. I link prices to in(cid:176)ation by using … = 4(P„ ¡P„ ) (17) t t t¡1 Therefore, in the case of standard contracts, I call structural model the system of equations (10), (8) and (17), plus (29) and (30) from the VAR. In the case of relative contracts, I call structural model the system of equations (11), (8) and (17), plus (29) and (30) from the VAR. For the purposes of estimation, I augment the contract price equation in both structural models with an observational error that I call † . P;t In both cases, the state space is given by X · (P„;… ;P ;y~;r )0. For any choice of t t t t t t the parameters (cid:176) and s, by standard methods, I can flnd the AR(1) representation for the variables in the state space, which can then be rewritten as X = A X +A X +A X +C† (18) t 1 t¡1 2 t¡2 3 t¡3 t where † = († ;† ;† )0, while A ;A ;A , and B are conformable matrices of coe–cients t y;t r;t P;t 1 2 3 (which can be thought of as functions of s and (cid:176)). This system of equations, however, still holds two identities. I then split the state space X into two parts S and Z . S is deflned t t t t as S · (P„;P )0, while Z is deflned as Z · (Y~;r ;… )0. I can then rewrite equation (18) t t t t t t t t as Z = A~ Z +A~ Z +A~ Z +B~ S +B~ S +B~ S +C~† (19) t 1 t¡1 2 t¡2 3 t¡3 1 t¡1 2 t¡2 3 t¡3 t To form the maximum likelihood function, I follow Harvey (1981), and condition on the flrst observation. I use the innovation representation of equation (19) , assuming that † is t identically and independently distributed across time as normal. To form the likelihood, the last hurdle to overcome is that the contract price P is unobserved. To remedy this, I t adopt the following procedure. I assume that P , prior to 1947, is in steady state. Given t a choice for (cid:176) and s, I use equation (19) to back out † . Using equations (18) and (19) I t can then dynamically generate a series for P and † . In order to dilute the assumption t t that P be in steady state prior to 1947, I use data for the period between 1947 and 1960 t 10

as a presample, with the sole purpose of initializing the value of P . I have used Monte t Carlo experiments to conflrm that after a period of 13 years, the initial value of P becomes t irrelevant. I maximize the likelihood using a Newton-Raphson based algorithm. To verify that the output of the algorithm maximizes the likelihood function, I use a linear-search procedure. 4.1 Estimation Results The estimation results are summarized in Tables 2 to 4. Table 2 reports the estimates for the relative contract model. Regressions 1 to 3 difier by the starting date of the sample. In regression 1, whose sample starts in 1980, s, the parameters governing the distribution of contract durations, is estimated at 0.0460, with a standard error of 0.0149. The implied distribution of contract durations is the following: 5% of contracts last one quarter, 9% two quarters, 14% three quarters, 72 % four quarters. The weight of the output measure in the contract equations, (cid:176), is estimated at 0.0425 with a standard error of 0.0161. Both estimates are highly statistically signiflcant. The variation in s over difierent samples is not statistically signiflcant. The estimate of (cid:176) drops in Regression 2, when the sample starts in 1965q1, and in Regression 3, when the sample starts in 1960q15. This is consistent with a higher persistence of the in(cid:176)ation process in the 1960s and 1970s. A Portmenteau test on the residuals of this regression (one equation at a time), whose Q(12) statistics are reported in Table 5, rejects the null hypothesis of white noise disturbances at conventional signiflcance levels. Table 3 reports the estimates for the relative contract speciflcation. For Regression 1, whosesamplespans1980q1-2001q4, theestimateforsis0.0895, withastandarddeviation of 0.0298. Over the longer samples, the estimates for s and gamma, are not statistically signiflcantly difierent. 5Using the sample from 1965 to 1993 (as Fuhrer and Moore were constrained to do), I can obtain estimates of the parameters for the relative contract model, but not for the standard model. These estimates are in line with the ones originally reported by Fuhrer and Moore (1995). 11

Table 4 reports the estimates for a contracting speciflcation that nests both the relative and the standard model. The fraction of agents adopting relative contracts, –, is estimated in the order of 80% regardless of the start of the sample. The estimate is statistically signiflcant at standard confldence levels. Figure 1 compares the correlograms for in(cid:176)ation, the output measure and the interest rate obtained from three sources: the VAR, the estimated structural model with standard contracts, and the estimated structural model with relative contracts. The sample period used is that of regression 1, from 1980q1 to 2001q4. Figures 2 and 3 repeat the comparisons respectively for the sample period 1965q1-2001q4 and for 1960q1-2001q4. Across samples, one can see that the correlogram for in(cid:176)ation for both relative and standard contracts is close to the correlogram for the VAR and is contained within the Monte-Carlo 90% confldence interval for the correlogram for the VAR. Figures 4 to 6 compare the fltted values for in(cid:176)ation from the structural model with relative contracts, and from the structural model with standard contracts with the actual values for in(cid:176)ation. Each of these flgures focuses on a difierent sample. Both models appear to perform satisfactorily across samples. Especially for the sample 1980q1-2001q4, it is hard to distinguish the performance of the two models. Using the information in Tables 2 to 4, one can set up a likelihood ratio test for the restrictionthatthenestedstructuralmodelonlyincludeseitherthestandardortherelative contract speciflcation. The standard model is rejected, while the relative model fails to be rejected at conventional signiflcance levels. At flrst, in light of the comparisons of the correlograms in Figures 1 and 3, this flnding appears surprising. In those flgures, the performance of standard contracts appeared hardly distinguishable from the performance of relative contracts. Figure 7 provides an explanation for the results of the likelihood ratio tests. In the case of the cross-correlogram for in(cid:176)ation on lagged output, and for output on lagged in(cid:176)ation, the standard model at lags 1 to 5, lies well outside the 90% confldence interval. It is on these dimensions that the likelihood test is penalizing the standard contract speciflcation. This provides an explanation for why the proportion of 12

relative contracts in the standard model is estimated as being so high, as well as why the standard model is rejected when performing a likelihood-ratio test. Figures 8 and 9 conflrm that the same flnding applies when the sample starts in 1965q1 or 1960q1, respectively. In the light of flgures 7 to 9, the Standard model performs satisfactorily in terms of reproducing the in(cid:176)ation persistence implied by the correlogram from the VAR. Where it is not performing as well as the relative contract model is in reproducing the comovements between output and in(cid:176)ation. 4.2 Comparing Impulse Response Functions Fuhrer and Moore (1995) closed their model by estimating a VAR in output, in(cid:176)ation and the interest rate. This is the way I proceed for the purposes of estimating the unknown parameters in the contract equations. To understand the difierences in the standard and relativecontracts, insteadofpursuingthisroute, onecouldmoresimplycompletethemodel by following Taylor (1980), specifying equations for the demand and supply of money. It is easier to examine the difierences imparted by the choice of contracting speciflcation when the response of money is kept constant. This could not be achieved with an interest rate reaction function. Thus, let the demand for nominal money balances, M take the form t M = P +y (20) t t t And let money supply be described by M = M +„ (21) t t¡1 t where „ , the rate of growth of money supply, is given by „ = ‰„ +† and † is an i.i.d. t t t¡1 t t error term6. One is now in a position to simulate the efiects of shocks in the two models so as to assess the persistence properties of each speciflcation. An area where one would expect the difierence between the two contracts to emerge is in the response to monetary shocks. 6The money supply equation adopted here comes from Christiano, Eichenbaum, and Evans (1998) who argue that this is a good approximation to money supply for both M1 and M2 in the U.S., as long as ‰ is chosen to be close to 0.5. 13

I have performed a battery of tests, using temporary and permanent, announced and unannounced shocks to the rate of growth of money supply as well as to the level of money. In Figure 10 and 11, I report the impulse response functions for an unannounced shock to the rate of growth of money supply. The intuition gained in this case holds true for all the other shocks I considered. Holding the distribution of contract durations constant, the choice of (cid:176), the weight on output in the contract equation, governs the persistence of in(cid:176)ation that the two contracting speciflcations can yield. Figures 10 and 11 difier by the choice of values for (cid:176). Comparing Figures 10 and 11 one can see that the lower the value of (cid:176), the greater the persistence. The path for in(cid:176)ation does look difierent whether one uses standard of relative contracts, however, it seems hard to draw any conclusions about the relative persistence. Varying the value of s, not surprisingly, also afiect the path for in(cid:176)ation. Lower values of s, by placing a greater weight on longer contracts, yield a more persistent response of in(cid:176)ation. In the light of this analysis, for the purposes of generating greater in(cid:176)ation persistence, given the choice of n, and (cid:176), one would then replace standard contracts with relative contracts if lowering s did not produce enough extra persistence. 5 Conclusion I have used a simple VAR to capture the properties of the data that a contract model needs to reproduce. My estimation results indicate that the contract model of Taylor (1980) performs as well as the relative contract model featured in Fuhrer and Moore (1995) at reproducing the in(cid:176)ation persistence observed in the data. Both types of contract speciflcations come close to replicating the second moments captured by a simple, nonstructural VAR. Overall, the relative contract model does flt the data better than the standard contract model. However, the capacity to generate in(cid:176)ation persistence does not appear to be the major difierence driving the results. The cross-correlograms for in(cid:176)ation and output, at 14

small lags, are where I observe a better performance for the relative contract model. When limiting the estimation sample to the 1980s and 1990s, I flnd that parameters for both contract models shift consistently with lower in(cid:176)ation persistence. However, this shift is statistically signiflcant only for the case of standard contracts. I read the estimation results in this paper as supporting that the standard staggered contract model of Taylor (1980) is perfectly adequate to capture the in(cid:176)ation persistence in the U.S. data. To explain the in(cid:176)ation behavior observed in the late 1960s and 1970s, it seems more appropriate to build extra structure to the model, rather than requiring that the contract model be able to explain a higher degree of in(cid:176)ation persistence. 15

)0 = atled( stcartnoC dradnatS rof setamitsE .II elbaT )21(Q :scitsitats uatnemtroP slaudiser fo .d.s srorre dradnats dna stneiciffeoc rre.y rre.r rre.p rre.y rre.r rre.p .lekil gol atled ammag s etad dne etad trats noisserger 4.61 9.51 2.23 89500.0 97600.0 3310.0 2.5201 0 5240.0 0640.0 4q1002 1q0891 1 detcirtser 1610.0 9410.0 5.71 9.63 221 02800.0 27700.0 6020.0 8.8461 0 5410.0 7530.0 4q1002 1q5691 2 detcirtser 3010.0 3600.0 9.61 3.04 19 35800.0 43700.0 7910.0 8.6481 0 5610.0 5630.0 4q1002 1q0691 3 detcirtser 3300.0 7500.0 )1= atled( stcartnoC evitaleR rof setamitsE .III elbaT )21(Q :scitsitats uatnemtroP slaudiser fo .d.s srorre dradnats dna stneiciffeoc rre.y rre.r rre.p rre.y rre.r rre.p .lekil gol atled ammag s etad dne etad trats noisserger 4.61 9.51 9.12 89500.0 97600.0 5110.0 1.7301 1 9710.0 5980.0 4q1002 1q0891 1 detcirtser 2010.0 8920.0 5.71 9.63 8.24 02800.0 27700.0 5610.0 2861 1 6500.0 8280.0 4q1002 1q5691 2 detcirtser 0300.0 6310.0 9.61 3.04 2.15 35800.0 43700.0 0610.0 3.1881 1 1400.0 8380.0 4q1002 1q0691 3 detcirtser 1200.0 8210.0 )detcirtsernu atled( ledoM gnitseN rof setamitsE .VI elbaT )21(Q :scitsitats uatnemtroP slaudiser fo .d.s srorre dradnats dna stneiciffeoc rre.y rre.r rre.p rre.y rre.r rre.p .lekil gol atled ammag s etad dne etad trats noisserger 4.61 9.51 4.02 89500.0 97600.0 4110.0 6.7301 398428.0 3210.0 2980.0 4q1002 1q0891 1 744221.0 6700.0 2710.0 5.71 9.63 3.04 02800.0 27700.0 3610.0 1.3861 968.0 3300.0 1280.0 4q1002 1q5691 2 4170.0 0200.0 6410.0 9.61 3.04 1.84 35800.0 43700.0 8510.0 8.3881 648.0 2300.0 3380.0 4q1002 1q0691 3 90450.0 3100.0 7310.0

Figure 1: Correlogram for Regression 1 (1980q1 { 2001q4) Correlogram for Inflation 1 0.8 0.6 0.4 0.2 0 2 4 6 8 10 12 Correlogram for Output 1 VAR 0.5 VAR Conf. Int Relative Contracts Standard Contracts 0 −0.5 0 2 4 6 8 10 12 Correlogram for the Interest Rate 1 0.9 0.8 0.7 0.6 0.5 0.4 0 2 4 6 8 10 12 Figure 2: Correlogram for Regression 2 (1965q1 { 2001q4) Correlogram for Inflation 1 0.8 0.6 0.4 0.2 0 2 4 6 8 10 12 Correlogram for Output 1 VAR 0.8 VAR Conf. Int 0.6 Relative Contracts Standard Contracts 0.4 0.2 0 0 2 4 6 8 10 12 Correlogram for the Interest Rate 1 0.9 0.8 0.7 0.6 0.5 0 2 4 6 8 10 12 17

Figure 3: Correlogram for Regression 3 (1960q1 { 2001q4) Correlogram for Inflation 1 0.8 0.6 0.4 0.2 0 2 4 6 8 10 12 Correlogram for Output 1 VAR 0.8 VAR Conf. Int 0.6 Relative Contracts 0.4 Standard Contracts 0.2 0 0 2 4 6 8 10 12 Correlogram for the Interest Rate 1 0.9 0.8 0.7 0.6 0.5 0 2 4 6 8 10 12 18

Figure 4: Fitted Values for In(cid:176)ation from Regression 1 (1980q1{2001q4) Regression 1: 1980q1 to 2001q4 Inflation (+/-) 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Solid: Data Dotted: Fitted Values from Structural mode with Relative Contracts Dashed: Fitted Values from Structural model with Taylor Contracts 19

Figure 5: Fitted Values for In(cid:176)ation from Regression 2 (1965q1 { 2001q4) Regression 2: 1965q1 to 2001q4 Inflation (+/-) 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 Solid: Data Dotted: Fitted Values from Structural mode with Relative Contracts Dashed: Fitted Values from Structural model with Taylor Contracts 20

Figure 6: Fitted Values for In(cid:176)ation from Regression 3 (1960q1 { 2001q4) Regression 3: 1960q1 to 2001q4 Inflation (+/-) 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 -0.02 -0.04 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 Solid: Data Dotted: Fitted Values from Structural mode with Relative Contracts Dashed: Fitted Values from Structural model with Taylor Contracts 21

Figure 7: Comparing cross-correlograms for Regression 1 (1980q1 { 2001q4) Output, lagged Output Interest Rate, lagged Output Inflation, lagged Output 1 0.6 1 0.4 0.5 0.5 0.2 0 0 0 −0.2 −0.5 −0.4 −0.5 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Output, lagged Interest Rate Interest Rate, lagged Interest Rate Inflation, lagged Interest Rate 1 1 1 0.9 0.5 0.8 0.8 0 0.7 0.6 VAR 0.6 −0.5 0.4 0.5 VAR Conf. Int Relative Contracts −1 0.4 0.2 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8St1a0nda1rd2 Contracts Output, lagged Inflation Interest Rate, lagged Inflation Inflation, lagged Inflation 1 1 1 0.9 0.8 0.5 0.8 0.7 0.6 0 0.6 0.4 0.5 −0.5 0.4 0.2 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 22

Figure 8: Comparing cross-correlograms for Regression 2 (1965q1 { 2001q4) Output, lagged Output Interest Rate, lagged Output Inflation, lagged Output 1 1 1 0.5 0.5 0.5 0 0 0 −0.5 −0.5 −0.5 −1 −1 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Output, lagged Interest Rate Interest Rate, lagged Interest Rate Inflation, lagged Interest Rate 1 1 1 0.9 0.8 0.5 0.8 0.6 VAR 0 0.7 0.4 VAR Conf. Int −0.5 Relative Contracts 0.6 0.2 Standard Contracts −1 0.5 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Output, lagged Inflation Interest Rate, lagged Inflation Inflation, lagged Inflation 1 1 1 0.9 0.5 0.8 0.8 0 0.7 0.6 0.6 −0.5 0.4 0.5 −1 0.4 0.2 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 23

Figure 9: Comparing cross-correlograms for Regression 3 (1960q1 { 2001q4) Output, lagged Output Interest Rate, lagged Output Inflation, lagged Output 1 1 1 0.5 0.5 0 0.5 0 −0.5 0 −1 −0.5 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Output, lagged Interest Rate Interest Rate, lagged Interest Rate Inflation, lagged Interest Rate 1 1.2 1 0.8 0.5 1 0.6 VAR 0 0.8 0.4 VAR Conf. Int −0.5 0.6 0.2 Relative Contracts Standard Contracts −1 0.4 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Output, lagged Inflation Interest Rate, lagged Inflation Inflation, lagged Inflation 1 1 1 0.8 0.8 0.5 0.6 0.6 0 0.4 0.4 −0.5 0.2 0.2 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 24

Figure 10: Response to an unannounced shock to the rate of growth of money supply. Gamma = 0.004, S =0.08 Percentage Change from Baseline Aggregate Price (Pbar) (+/-) Output (ytilde) (+/-) 6 2.5 5 2.0 4 1.5 3 1.0 2 0.5 1 0.0 0 -0.5 1960 1965 1970 1975 1980 1960 1965 1970 1975 1980 Contract Price (P) (+/-) Money balances (M) (+/-) 6 5.5 5 5.0 4.5 4 4.0 3 3.5 2 3.0 1 2.5 0 2.0 1960 1965 1970 1975 1980 1960 1965 1970 1975 1980 Real Money Balances (+/-) Inflation (Pbar - Pbar(-1)) (+/-) 5 0.4 4 0.3 3 0.2 2 0.1 1 0.0 0 -1 -0.1 1960 1965 1970 1975 1980 1960 1965 1970 1975 1980 Solid: Standard Contracts Dotted: Relative Contracts 25

Figure 11: Response to an unannounced shock to the rate of growth of money supply. Gamma = 0.04, S =0.08 Percentage Change from Baseline Aggregate Price (Pbar) (+/-) Output (ytilde) (+/-) 6 2.0 5 1.5 4 1.0 3 0.5 2 0.0 1 0 -0.5 1960 1965 1970 1975 1980 1960 1965 1970 1975 1980 Contract Price (P) (+/-) Money balances (M) (+/-) 6 5.5 5 5.0 4.5 4 4.0 3 3.5 2 3.0 1 2.5 0 2.0 1960 1965 1970 1975 1980 1960 1965 1970 1975 1980 Real Money Balances (+/-) Inflation (Pbar - Pbar(-1)) (+/-) 4 0.7 0.6 3 0.5 2 0.4 0.3 1 0.2 0.1 0 0.0 -1 -0.1 1960 1965 1970 1975 1980 1960 1965 1970 1975 1980 Solid: Standard Contracts Dotted: Relative Contracts 26

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proposaltoreplacethenewkeynesianphillipscurve.Manuscript, HarvardUniversity; forthcoming in Quarterly Journal of Economics. Phillips, A. (1958). The relation between unemployment and the rate of change of money wages in the united kingdom, 1861{1957. Economica 25, 283{299. Roberts, J. M. (1997). Is in(cid:176)ation sticky. Journal of Monetary Economics 39, 173{196. Rudd, J. and K. Whelan (2001). New-Keynesian Phillips curve. Board of Governors of the Federal Reserve System: International Finance Discussion Papers. Sbordone, A. M. (2002). Prices and unit labor costs: A new test of price stickiness. Journal of Monetary Economics 49(2), 265{292. Taylor, J. B. (1980). Aggregate dynamics and staggered contracts. Journal of Political Economy 88(1), 1{22. Taylor, J. B. (2000). Low in(cid:176)ation, pass-through, and the pricing power of flrms. European Economic Review 44, 1389{1408. Yun, T. (1996). Nominal price rigidity, money supply endogeneity, and business cycles. Journal of Monetary Economics 37, 345{370. 29

A The equation for in(cid:176)ation under the setup of Taylor (1980) In a symmetric two-period setup, the log of the aggregate price level, P„ , is given by: t 1 P„ = (P +P ) (22) t t t¡1 2 where P is the contract price. Equation (4), that governs the contract price, for a two t period setup, can be rewritten as 1 1 P = P + E P +(cid:176)(y~ +E y~ ) (23) t t¡1 t t+1 t t t+1 2 2 where y~ adjusts the contract for excess demand. Combining equation 22 and equation 23, t one obtains: 1 1 1 1 1 (cid:176) P„ = ( P + E P + P + E P )+ (y~ +E y~ +y~ +E y~) (24) t t¡1 t t+1 t¡2 t¡1 t t t t+1 t¡1 t¡1 t 2 2 2 2 2 2 Using equation 22, equation 24 can be rewritten as: 1 (cid:176) 1 P„ = (E P„ +P„ )+ (y~ +E y~ +y~ +E y~)¡ † (25) t t t+1 t¡1 t t t+1 t¡1 t¡1 t t 2 2 4 where † is a forecast error such that E w = w ¡† . t t¡1 t t t To reformulate equation 25 in terms of in(cid:176)ation, notice that since the price level, P„ , is t in log form, the in(cid:176)ation at time t, … , is given by … = P„ ¡P„ . Therefore, using equation t t t t¡1 25, subtracting P„ from both sides: t¡1 1 (cid:176) 1 P„ ¡P„ = (E P„ ¡P„ )+ (y~ +E y~ +y~ +E y~)¡ † (26) t t¡1 t t+1 t¡1 t t t+1 t¡1 t¡1 t t 2 2 4 Rearranging the terms in the equation above, and adding and subtracting 1P : 2 t 1 … = (E P„ ¡P„ +P„ ¡P„ ) t t t+1 t t t¡1 2 (cid:176) 1 + (y~ +E y~ +y~ +E y~)¡ († ) t t t+1 t¡1 t¡1 t t 2 4 which, in turn, can be rewritten as: 1 (cid:176) 1 … = (E … +… )+ (y~ +E y~ ¡y~ +E y~)¡ († ) (27) t t t+1 t t t t+1 t¡1 t¡1 t t 2 2 4 Therefore, collecting terms in equation 27 yields: 1 … = E … +(cid:176)(y~ +E y~ +y~ +E y~)¡ † (28) t t t+1 t t t+1 t¡1 t¡1 t t 2 30

B Mapping s into contract weights Expanding equation (7), one obtains P„ = (cid:181) P t 1 t (cid:181) 2 + (P +P ) t t¡1 2 (cid:181) 3 + (P +P +P ) t t¡1 t¡2 3 (cid:181) 4 + (P +P +P +P ) t t¡1 t¡2 t¡3 4 But (cid:181) = 1¡(cid:181) ¡(cid:181) ¡(cid:181) . Using equation (8), combined with the equation above, one can 4 1 2 3 see that 1 1 1 f = (cid:181) + (cid:181) + (cid:181) + (1¡(cid:181) ¡(cid:181) ¡(cid:181) ) 0 1 2 3 1 2 3 2 3 4 1 1 1 f = (cid:181) + (cid:181) + (1¡(cid:181) ¡(cid:181) ¡(cid:181) ) 1 2 3 1 2 3 2 3 4 1 1 f = (cid:181) + (1¡(cid:181) ¡(cid:181) ¡(cid:181) ) 2 3 1 2 3 3 4 Which leads to ¡1 (cid:181) 3 1 1 f 1 0 1 1 0 4 4 12 1 00 0 1 0 11 1 (cid:181) = ¡1 1 1 f ¡ 1 B B 2 C C B B 4 4 12 C C B B B B 1 C C 4 B B C C C C B C B C BB C B CC B (cid:181) C B ¡1 ¡1 1 C BB f C B 1 CC B 3 C B 4 4 12 C BB 2 C B CC @ A @ A @@ A @ AA where f = 0:25+(1:5¡i)s, for 0 < s • 1 i 6 C VAR estimation results The VAR for detrended output, the interest rate and in(cid:176)ation, takes the form: 3 Y~ = C + C Y~ +C r +C … +† (29) t c;1 y;1;i t¡i r;1;i t¡i …;1;i t¡i y;t Xi=1 3 r = C + C Y~ +C r +C … +† (30) t c;2 y;2;i t¡i r;2;i t¡i …;2;i t¡i r;t Xi=1 3 … = C + C Y~ +C r +C … +† (31) t c;3 y;3;i t¡i r;3;i t¡i …;3;i t¡i …;t Xi=1 31

The estimation results for the sample 1980 { 2001 are reported in Table 5. Table 6 reports the restricted estimates, over the same sample excluding the constant term from each equation. A likelihood ratio test conflrms the validity of the restriction. The log likelihood for the unrestricted VAR is 903, while for the restricted VAR the log likelihood is 901. The null hypothesis that the restriction is valid fails to be rejected at standard signiflcance levels. 32

Table 5: VAR Parameter Estimates (constant included), Regression 1 (1980q1 { 2001q4) Parameter Estimate Error t-statistic P-value C .110E-02 .197E-02 .558 [.577] c;1 C .913 .101 9.01 [.000] y;1;1 C .197 .134 1.47 [.142] y;1;2 C -.230 .0968 -2.38 [.017] y;1;3 C .212 .0974 2.17 [.030] r;1;1 C -.500 .133 -3.77 [.000] r;1;2 C .310 .0915 3.39 [.001] r;1;3 C -.132 .0726 -1.82 [.068] …;1;1 C .044 .0684 .647 [.518] …;1;2 C -.306E-02 .0692 -.0443 [.965] …;1;3 C .409E-02 .220E-02 1.86 [.063] c;2 C 1.11 .109 10.3 [.000] y;2;1 C .0905 .150 .605 [.545] y;2;2 C -.228 .108 -2.10 [.035] y;2;3 C 1.21 .0805 15.1 [.000] r;2;1 C -.569 .148 -3.84 [.000] r;2;2 C .245 .102 2.40 [.016] r;2;3 C .0193 .0810 .238 [.812] …;2;1 C .392 .0760 5.14 [.000] …;2;2 C -.103 .0771 -1.34 [.180] …;2;3 C .156E-02 .298E-02 .524 [.600] c;3 C .125 .153 .812 [.416] y;3;1 C -.513 .200 -2.55 [.011] y;3;2 C .150 .146 1.03 [.303] y;3;3 C .600 .147 4.08 [.000] r;3;1 C -.675 .227 -2.98 [.003] r;3;2 C -.0135 .138 -.0978 [.922] r;3;3 C .391 .110 3.57 [.000] …;3;1 C .198 .103 1.92 [.055] …;3;2 C .147 .104 1.40 [.160] …;3;3 Number of observations = 85 Log likelihood = 903.188 Equation: y~ Variance of residuals = .356228E-04 Std. error of regression = .596849E-02 R-squared = .846797 Equation: r Variance of residuals = .443094E-04 Std. error of regression = .665653E-02 R-squared = .943060 Equation: … Variance of residuals = .812260E-04 Std. error of regression = .901254E-02 R-squared = .787872 33

Table 6: VAR Parameter Estimates (constant excluded), Regression 1 (1980q1 { 2001q4) Parameter Estimate Error t-statistic P-value C .917142 .101471 9.03844 [.000] y;1;1 C .192108 .134050 1.43310 [.152] y;1;2 C -.235178 .096710 -2.43179 [.015] y;1;3 C .228172 .092938 2.45510 [.014] r;1;1 C -.511142 .131610 -3.88378 [.000] r;1;2 C .325152 .087388 3.72080 [.000] r;1;3 C -.132020 .072761 -1.81443 [.070] …;1;1 C .041451 .068358 .606379 [.544] …;1;2 C -.014888 .065960 -.225713 [.821] …;1;3 C .263326 .115230 2.28523 [.022] y;2;1 C .072040 .152226 .473242 [.636] y;2;2 C -.243371 .109823 -2.21604 [.027] y;2;3 C 1.17428 .105539 11.1265 [.000] r;2;1 C -.608304 .149455 -4.07016 [.000] r;2;2 C .302533 .099237 3.04859 [.002] r;2;3 C .020595 .082627 .249247 [.803] …;2;1 C .382025 .077626 4.92132 [.000] …;2;2 C -.147313 .074903 -1.96671 [.049] …;2;3 C .129185 .153191 .843291 [.399] y;3;1 C -.226571 .202375 -1.11956 [.263] y;3;2 C .144489 .146003 .989633 [.322] y;3;3 C .623527 .140308 4.44400 [.000] r;3;1 C -.526763 .198690 -2.65117 [.008] r;3;2 C .834549E-02 .131929 .063257 [.950] r;3;3 C .391898 .109847 3.56766 [.000] …;3;1 C .193988 .103200 1.87973 [.060] …;3;2 C .129899 .099579 1.30448 [.192] …;3;3 Number of observations = 85 Log likelihood = 901.429 EQ1 : y~ Variance of residuals = .357534E-04 Std. error of regression = .597941E-02 R-squared = .846296 Equation: r Variance of residuals = .461063E-04 Std. error of regression = .679017E-02 R-squared = .942303 Equation: … Variance of residuals = .814885E-04 Std. error of regression = .902710E-02 R-squared = .787436 34