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Real and nominal factors in the cyclical behavior of interest rates, output, and money
Journal of Economic Dynamics and Control 5 (1983) 289-309. North-Holland
REAL AND NOMINAL FACTORS IN THE CYCLICAL BEHAVIOR OF INTEREST RATES, OUTPUT, AND MONEY Kenneth J. SINGLETON* Carnegie-Mellon
University,
Pittsburgh,
PA 15213,
USA
Received February 1981, final version received October 1982 Short-term nominal interest rates, output, unemployment, monetary growth, and inflation are represented as functions of two common latent indices: a ‘real’ index that underlies comovements in output and unemployment, and a ‘neutral’ index that is independent of output. A Fisherian interpretation of the real and neutral components of interest rates is provided under the assumption that anticipated inflation affects output primarily through its effect on real interest rates. Then, monthly time series on the real and neutral components of short-term treasury bill yields are constructed, and the cyclical behavior of these series is compared to the behavior of output and monetary growth.
1. Introduction Modern ‘equilibrium’ theories of the business cycle assign a potentially important role to real interest rate fluctuations in the propagation of cycles in real’ output [e.g., Lucas (1975), Barro (1980a, b), Blanchard (1980), Blinder and Fischer (1980)]. Indeed, the principle means by which anticipated changes in the money stock and inflation affect output in these models is through changes in real rates [Lucas (1975), Tobin and Brainard (1976), Fischer (1979)]. Yet, little attention has been directed toward characterizing empirically the temporal covariance structure of monetary growth, inflation, real interest rates, and output. ’ This paper represents an attempt to shed *I wish to thank the participants of workshops at several universities and of the NBER conference on inflation and financial markets, especially Robert Barro, Alex Cukierman, Jacob Frenkel, Herschel Grossman, Lars Hansen, Robert Hodrick, Bennet McCallum, Robert Shiller, Chris Sims, and the referee for helpful comments on earlier drafts of this paper. Robert Litterman kindly provided his computer software for the calculations in section 4. ‘Much attention has been given to the relationship among monetary growth and real output implied by equilibrium business cycle models, as well as the empirical relationships among ex post real rates and monetary growth. The importance of studying output, interest rates, and monetary growth simultaneously in empirical analyses is evidenced by the results obtained in Mehra (1978) and Sims (1980). Although Mehra and Sims did not address the issue of real interest rate behavior, their studies are perhaps the closest precursors of this analysis. 0165-l 889/83/$03.00 0 Elsevier Science Publishers
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some light on the nature of the co-movements among these variables for the United States during the postwar period. The framework for the investigation is a latent index mode1 [Geweke (1977), Sargent and Sims (1977), Geweke and Singleton (1981a)] of nominal interest rates, output, unemployment, monetary growth, and inflation. Nominal interest rates, the monetary growth rate, and the inflation rate are represented as functions of two common latent indices: a ‘real’ index that underlies co-movements in output and unemployment, and a ‘neutral’ index that is independent of output. Fluctuations in nominal interest rates associated with the real business cycle are assumed, in the spirit of many equilibrium business cycle theories, to represent fluctuations in real interest rates. Real rates may change in response to aggregate demand and supply shocks or to changes in the anticipated inflation rate (the Mundell-Tobin effect). The ‘neutral’ index, on the other hand, is assumed to capture the effects on interest rates of anticipated changes in the inflation and monetary growth rates that leave output unchanged. Under these assumptions, it is possible to study aspects of the cyclical behavior of real interest rates, and the nature of real and neutral variations in the monetary growth rate. This is accomplished without having to specify the information sets underlying agents’ expectations of inflation. In section 2 the latent index mode1 is presented, and the decomposition of the monetary growth rate, interest rates, and inflation rate into real and neutral components is discussed in the context of several recent business cycle models. The index model is shown to be identified in section 3. Identification is achieved without having to assume a subset of the variables in the model are econometrically exogenous. Furthermore, the latent index mode1 is shown to accommodate certain types of errors in measuring such variables as the inflation rate and money stock. The estimated parameters of the mode1 for the subperiods 1954 through 1965 and 1965 through 1976 of the postwar period are also reported in section 3. A comparison of the results for these two subperiods is of interest because of the apparent change in the monetary policy rule during the 19631965 period meftci and Sargent (1978)]. The index model is not structural in the sense of Hurwicz (1962) and, therefore, the parameter values are not invariant to changes in policy regimes. The later subperiod seems to have been characterized by relatively higher inflation rates and greater activism by the Federal Reserve, which suggests that the behavior of the real components of interest rates may have been different across subperiods. In fact, the parameter estimates indicate that the real components of both nominal interest rates and monetary growth were substantially more volatile during the later subperiod. In section 4 a procedure for extracting observable time series ‘that represent the effects of the real and neutral indices on interest rates is
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described and implemented. The graphs of these series are then compared with the ex ante real interest rate series constructed in several previous studies. The findings suggest that real rates constructed from the growth rate of the CPI may be very misleading measures of the actual real rates, especially over the recent subperiod. Finally, some concluding remarks are presented in section 5. 2. The latent index model Let y, be the 2 x 1 vector including the log of real output (43 and the unemployment rate (u,) and let n; E [ii, fit, ti,], where i, is a vector of nominal interest rates, Ejt is the inflation rate, and rri, is the rate of growth of the money stock. For the purpose of decomposing movements in n, into movements associated with the real business cycle and movements that are neutral with respect to real activity, I adopt the following latent index representation of yI and n,:
In (l), the constituents of e are mutually uncorrelated and uncorrelated with the common latent indices z1 and z2 at all leads and lags; 12,,(L), 1,,(L), I,,(L) are (possibly two-sided) lag polynomials with square summable coefficients; and the latent variables are purely indeterministic, covariance stationary time series. All variables may be serially correlated. The restriction that y, is a function only of the first index identifies zl as a ‘real’ index underlying the cyclical covariation of U, and qr; that is, covariation of output and unemployment is due entirely to the disturbances embodied in zl. These include shocks to investment, labor and capital productivity, and consumption. In addition, most equilibrium business cycle models imply that unanticipated changes in the price level and policy variables will also directly affect real economic variables [e.g., Lucas (1975), Sargent (1973)]. Although I refer to z1 as the ‘real’ index, innovations in zlt are assumed to embody innovations in these nominal variables as well. From a theoretical point of view, macroeconomic models in which Q and u, are related thro’ugh a production function, Phillips curve, and an equation describing labor force participation imply that output and unemployment will be functions of a common set of disturbances [e.g., Ando (1974), Sargent (1976) and McCallum (1979)]. The assumption that the effects of these disturbances are captured by a single latent index does not necessarily imply that the ‘primitive’ real disturbances to the economy are unit-dimensional Sargent and Sims (1977), Litterman and Sargent (1979) and Singleton (1980),
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for example, consider the following two-index model:
in which the common latent indices are the constructs unanticipated (vit= ~~-4~) and anticipated (vZt=$J ‘aggregate demand’, and where all polynominals in L are one-sided. Although (~~-4~) and fi, are not uncorrelated at all leads and lags, this model can be represented in the form (1). This is accomplished by forming the two-sided least squares projection
substituting this expression for v2t (=fiJ in (2), and letting zl,= vll=(%-ij,) and z2,= 5,. For this example, A, ,(L)=a(L) and A,,(L)=c(L) are one-sided, is two-sided, and z1 and z2 are by construction 122 l(L) = (b(L) + c(ww mutually uncorrelated at all leads and lags. Both (~-6~) and 4, are in general functions of all of the variables in agents’ information set and, hence, may embody a large number of primitive economic disturbances. The emphasis on demand assumes that the underlying disturbances to the economy which are being captured by z1 are primarily demand-related. More generally, output, unemployment, and real interest rates will be functions of a common set of supply and demand shocks. While in principle these shocks may affect qr and u, in a manner that gives rise to several real disturbances, the empirical evidence in Litterman and Sargent (1979) and Singleton (1980) is consistent with the single-index representation of real activity assumed here (see also section 3). In effect, the zero restrictions in (1) identify zi as representing the combined influence of all ‘real’ disturbances on output and, in this sense, the real business cycle is being identified with cycles in real output. The second index is a source of comovements in nominal monetary growth, inflation, and nominal interest rates, but not a source of comovements in real variables. The restrictions that zi and z2 be orthogonal at all leads and lags, together with the zero restrictions in (l), overidentify the component L,,(L)z,, of a,. They do not necessarily imply that anticipated changes in policy variables are neutral in this model. The association of (v,-IjJ and & with vlt and vlr, respectively, in (2) does imply a version of the natural rate hypothesis. However, if the broader interpretation of vl, in the last paragraph is adopted and anticipated changes in nominal variables affect real output, then vi and v2 will not equal (q,-$ and G1, and they will in general be correlated both contemporaneously and at various leads and ,lags. Nevertheless, the two-sided projection of v21 on vit can be used to transform this modified version of (2) into a representation of the form (1). Thus, (1)
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allows for potential non-neutral effects of anticipated nominal variables -on output? To further interpret the orthogonality condition and make the last observation more precise, suppose each interest rate i, satisfies the following rational version of Fisher’s (1930) equation: i,=+,+g,
(3)
where T, is the ex ante real rate of interest and $: is the optimal linear least squares forecast of dl (over the relevant horizon), based on information available at time t. Many of the equilibrium models of the business cycle assume anticipated inflation affects output primarily through its effects on real interest rates due to Mundell (1963) - Tobin (1965) effects. Under this assumption, fi; may be correlated with r at all leads and lags, but there is nevertheless a component of @: that is orthogonal to both t and 4. That is, in the absence of ‘rigidities’ that lead to direct effects of $: on yt (i.e., effects other than through r), there is a component of i, that is orthogonal to r, and y, at all leads and lags. Specifically, let
be the least squares projection of 0: onto {rt-s; s=O,+l,.. nominal interest rate it can be expressed as
.}. Then each
i,= r,+ f c&r,-, +r,, ( s=--00 > where c, is the component of anticipated inflation that is orthogonal to the real rate r at all leads and lags. Now let zr be a real index underlying the cyclical covariation of the real variables r,, qt, and I+ Also, define If,(L)z,, and Afz(L)z2, to be the subvectors of A,r(L)zr, and t22(L)~2f, respectively, corresponding to the subvector i, of n,. Then L$r(L)zIr and A;2(L)zzt can be interpreted as the terms (rt+x 6,rt-J and &, respectively, in (4). Under this interpretation, the index z2 is neutral in the strong sense that it underlies comovements in nominal interest rates, monetary growth, and inflation that do ‘Litterman and Sargent (1979) included a vector of commodity prices, instead of i0 in n, Neither Litterman and Sargent (1979) or Singleton (1980) imposed any restrictions on Lv’, Q beyond those in (1) and, hence, their representations are observationally equivalent to the following interpretatton of (1) as a model that embodies Mundell and Tobin effects on output. It follows that their evidence on the number of real indices can be used in support of a single, more broadly interpreted, real index in the analysis here.
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not alfect real variables at all leads and lags. Because 1,,(L) is allowed to be two-sided in (l), the restriction cov[zl,,zZ,] =O, for all s, does not imply that anticipated policy variables are neutral in the usual sense of the term; ni,-,(s lo), I, and y, may be correlated. Finally, the vector e captures the effects of measurement errors or disturbances that are specific to variables in y or II. The preceding motivation of the index representation in terms of recent macroeconomic theories imposes no restrictions on the covariance structure of s. To identify the model and proceed with estimation, it is necessary to assume that the constituents of E are uncorrelated at all leads and lags (they may be serially correlated. As emphasized by Sargent and Sims (1977), these conditions are imposed not because they are believed to have a foundation in the theoretical ideas motivating the model, but because some such conditions must be imposed to proceed with estimation. It should be emphasized, though, that this assumption together with the assumed two-index representation (1) imply testable restrictions that are not refuted by the data. 3. Estimated value-s of the parameters The restrictions on the covariance structure yI and n, implied by (1) can be decomposed using frequency domain techniques. Let x’=CY),n’], z’=[zi, zJ, and S,(w), S,(o), and S,(o) be the spectral density matrices of x, z, and e, respectively. Then (1) and the accompanying assumptions imply that S,(o) decomposes as [Geweke and Singleton 19gla]:
where view) is the Fourier transform of the function AL,(s) and the prime denotes transposition and conjugation. S,(o) and S,(w) are diagonal for all w, since the constituents of t and e are uncorrelated at all leads and lags. A heuristic view of the spectral density function will be useful for interpreting the estimates of the parameters in (5). Variation at frequency 27r/P corresponds to oscillations of period P. If a time series x,, recorded monthly for example, displays strong seasonal oscillations, then the spectral density function for x will be large or ‘powerful’ at frequency 27412. A series which is serially uncorrelated has a constant spectral density function, while one for which x, and x,+~ have correlation near one for small s will be very powerful at the low frequencies (those near o =0) compared to the high frequencies (near o=n). The variance of a time series can be decomposed into variation across frequencies. Thus, eq. (5) can be viewed as a representation of the covariation of the elements of x at frequency o. The integral across all frequencies gives the conventional covariance matrix of x.
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From Proposition 1 in Geweke and Singleton (1981a), it follows that the non-zero elements in the &(w) and S,(o) and S,(w) are over-identified. Some normalizations must ‘be imposed before the free parameters in (5) can be estimated, however. For the purposes of this analysis, I shall normalize the diagonal elements of S,(o) to be unity, /o( sn. To interpret this normalization, recall that every purely indeterministic, covariance stationary process has a one-sided moving average representation mold (193811. Therefore, I can write
where cov[&,, &J=O, uncorrelated. Substituting
for all t and s, and [r (6) and (7) into (1) gives
and
&
are serially
(8)
Eq. (8) expresses J+ and n, as functions of the innovations in the real and neutral indices. Now if the variances of rrt and rzl are normalized to be unity, then the frequency domain representation of (8) is S*(w) = fq(w,&o,,
+ S,(o),
l+D,
(9)
where a; r(w) = & ,(w)d( w ), L1 t(w) = 0, etc., and the spectral density matrix of [rrr, &,I is the 2 x 2 identity matrix. The moduli of the complex elements of &B) in (9), I&(e$l say, are uniquely determined by these normalizations [Geweke and Singleton (1981a)l. Each I~&# represents the total contribution of (zfi}, or equivalently of {rjt}, to variation in the Ith element of x at frequency o. Thus, the proportion of the variation in xl due to ~j at frequency o is
coht.,W= ~4,@)~z/CU41~~, where [ In denotes the Ith diagonal element of the matrix in brackets. This ratio, called a coherence, is’analogous to the coefficient of determination in regression analysis. Since the integral of S,(o) gives the covariance matrix of x,, an overall measure of variation explained by the jth latent index can be obtained by integrating the numerator and denominator of (10) and then forming the ratio of these integrals. The estimation of the model was performed with monthly data for the J.ED.C.-
F
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periods 1954:l through 1965:12 and 1965: 1 through 1976:12. The vector of short-term interest rates i, included the three-month treasury bill yield (Y3MB) and the one-year treasury bond yield (Y1 YB). For fit, I used the rate of growth of the consumer price index (GCPZ), ni, was the growth rate of Ml (GMZ), output was measured by industrial production (LY=log of industrial production), and the unemployment rate (UN) was the rate for civilian workers. Sources for the data are provided in the appendix. At the outset, linear trends were removed from each series by ordinary least squares regression. Then, to diminish any small sample bias in the spectral estimator due to leakage, each of the series was prewhitened using the estimated coefficients from a second-order autoregression. After prewhitening the data, cross-periodogram ordinates were calculated at 144 equally spaced frequencies in the interval [0,2rr]. These ordinates were smoothed with a tent filter of width eleven ordinates at the base to form a consistent estimator, g’(o), of the unrestricted spectral density matrix of x. Estimates based on the prewhitened data were recolored in the usual way. Finally, estimates of A(o) and S,(o) were obtained by substituting $(a) for the maximum likelihood estimator of S,(w) in the concentrated log-likelihood function described in Geweke and Singleton (1981a), and then maximizing the resulting objective function with respect to the free parameters. This procedure yields consistent, but not maximum likelihood, estimators3 Ten frequencies, equally spaced over the interval [O,rc), were considered. The chi-square statistics for testing the overidentifying restrictions at each frequency [see Geweke and Singleton (1981a)], based on estimates of S,(w) obtained with a rectangular window, were similar to those reported by Singleton (1980), and generally lend support to the two-index representation (1).4 Specifically, for the period 1954: 1 through 1965: 12, the values of the chisquare statistics with twelve degrees of freedom ranged from 2.76 to 13.66, and for the period 1965:l through 1976112, they ranged from 3.58 to 11.3. The parameter estimates for the period 1954: 1 through 1965:12 are displayed in table 1.5 The squared moduli of the estimated values of A(U) are ‘A tent window was used instead of a rectangular window, which gives the maximum likelihood estimator of S,(o) and leads to maximum likelihood estimates of A(o), in order to improve the resolution of the estimates for use in forming the projections &&bed in section 4. More precisely, if I(oj) denotes the cross-periodogram matrix at frequency 0,. then S,(c+) =z-,,,bJ(mk-j), where b, is the ‘tent’ moving average window bj=[l -ljl/(m+ I)‘]. In my case, m is set at 4. %ince the tests were based on maximum likelihood estimates of S,(w), the usual likelihood ratio statistics are distributed asymptotically as chi-square variates [Geweke and Singleton WWI. Overlapping sets of ordinates were used to estimate the spectral densities at the ten frequencies, so the individual estimates are not independent. Therefore, an overall’goodness-of-fit test cannot be constructed by adding the individual chi-square statistics. Singleton (1980) reports an overall test for a similar index represenlation that does not call for rejection of the model at conventional significance levels. ‘The numbers in table I are actually 271(c~)\~ and 2xdiag [S&$j and, therefore, the reported numbers must be divided by 22 in order to obtain the sample spectral densities.
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Estimated Frequency
O.OOOOR 0.1111x 0.2222n 0.3333x 0.44447t 0.555671 0.6667n 0.1778X 0.8889R l.OOOK
UN
43p15 1.2615 .0.04214 0.01631 0.00225 0.01276 0.00091 0.00674 0.01834 0.01175
Real and nominal
E 0:o 0.0 0.0 0.0 0.0 0.0 0.0 0.0
O.OOOOR 0.1111x 0.22227t 0.3333x 0.4444R 0.555611 0.666ln 0.7178~ 0.88897I 1.0007I
5.8764 0.09906 0.00700 0.03534 0.01820 O.OOOOO 0.00663 0.01184 0.0053 1 O.OOOOO
LY
Y3MB
Loadings 0.13051 0.00340 0.00067 0.00018 0.00005 0.00002 0.00003 0.00003 0.00001 0.00001
on jrst 0.33540 0.27609 0.02702 0.00369 0.01684 0.00098 0.00359 0.00815 0.00185 0.00009 on second
1954:1-1965:12
Yl YB
I
297
cycles
GCPI
GM1
3.2594 1.6947 2.029 1 1.0607 2.5559 0.11351 1.5863 0.64946 1.1671 0.07212
12.548 3.9126 0.33968 0.98554 0.29594 1.5281 0.45653 1.1010 0.11458 0.44461
index(C&J&J 0.26568 0.37388 0.03425 0.00154 0.01668 0.00698 0.00142 0.00743 o.ooo79 0.00040 index
([&I&)
0.0 0.0 0.0
4.6937 0.81124 0.05530
7.6910 0.60615 0.02805
6.4933 0.51875 0.72619
2
0.01972 0.00545
0.00994 0.01733
2.2555 0.54115
i:: 0.0 0.0 0.0
0.01150 0.00722 o.ooo99 0.00139 0.00390
0.00541 0.00154 0.00037 0.00364 0.00394
0.06909 0.45341 1.7226 0.20460 2.8610
3.9808 8.6200 0.47108 0.09237 1.1812 0.84251 1.2565 0.16132 1.6469 0.44448
6.7549 2.7142 4.0910 5.5044 0.0 3.1928 5.2742 2.2600 4.2177 7.0624
9.7879 5.8380 5.1006 3.1330 2.3914 4.0216 4.8745 3.1357 6.4229 6.7548
Spectral
densities
o.ooooO o.ooo9o 0.00021 o.ooooO 0.00005 0.00004 0.0 0.00002 0.00001 0.00003 Cumulative
0.88198
in business
Table 1 values for the period
parameter
Loadings O.OOOOR O.llllrr 0.2222x 0.3333n 0.4444X 0.5556x 0.6667~ 0.7778x 0.8889x 1.00072
factors
0.99072
overall
of specific
indices
0.0 0.0 0.0 0.0 0.0093 1 0.0 0.0 0.0 0.0 0.00253
0.79059 0.04444 0.00775 0.00951 o.ooo95 0.00315 0.00220 0.0 0.00360 0.0
proportion
of variance
0.99812
0.91325
([S,(o)],,)
explained 0.42203
0.43109
dis&yed under the headings of loadings. Thus, at the zero frequency, la’,,(o)12=43.915. Th e 1oa d ings for UN and LY on the second index are zero at all ten frequencies because of the restrictions that these variables are functions of the first index only. Both output and unemployment have most of their power concentrated at the low, business cycle frequencies, and at these frequencies, most of the variation in UN and LYis due to the corknon index zl. Overall, the coherences of UN and LYwith z1 are 0.882 and 0.991,
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respectively (table 1).6 Thus, the interpretation of z1 as a ‘real’ index that captures cyclical movements in real output is supported by the data. Most of the variation in nominal interest rates at the lowest frequency is due to variation in the nominal index. This can be seen from the large values of la&&l2 and [&@)I2 relative to /?$@I2 and lQ)l”, respectively. The overall coherence of Y3MB with the nominal index is 0.891, while the coherence with the real index is only 0.107. Similarly, the coherences of YlYB with the nominal and real indices are 0.842 and 0.071, respectively. These patterns are consistent with the findings of Nelson and Schwert (1977), Fama (1977) and Shiller (1980), among others, that price expectations played a dominant role in the determination of interest rates during the nineteenfifties and early nineteen-sixties. They also suggest that real interest rates were not constant during this period, however. The large coherences of Y3MB and YIYB with the nominal index do not, of course, necessarily imply a correspondingly large overall coherence of GCPI and zZP A portion of the variation in GCPI will be due to real disturbances, and there may be substantial errors in measurement that lead to large values of [S,(o)],, relative to the loadings. So long as measurement errors are independent of i, they will be reflected in the specific index sst and will not affect inferences about interest rates in the context of (4). In fact, at the zero frequency, where most of the power of interest rates is concentrated, the coherence of GCPZ and z2t is only 0.393. An additional 19.7% of the variation in GCPI at this frequency is explained by the real index, and the remaining 41% is due to the specific index. The overall coherences of GCPZ with z1 and z2 are 0.199 and 0.223, respectively; almost 58 percent of the variation in GCPl is independent of variation in the yields and output. Similarly, low R2’s for regression equations describing the one-month treasury bill yield were obtained by Fama (1975), Nelson and Schwert (1977) and Carlson (1977). If the low overall coherence for GCPI indicates substantial measurement error is present, then this finding has at least two important implications for previous empirical studies of business cycles. First, a common method of studying ex ante real rates is to subtract GCPI from the nominal rate and then regress this difference on lagged values of GCPZ and other variables [e.g., Shiller (1980), Mishkin (1981)]. If the measurement error in GCPZ varies systematically over time, then the coefficients in these regressions will in general be estimated inconsistently. Consequently, inferences about the behavior of ex ante real rates from such equations may be misleading. 6The zero valuesof the spectraldensity functions of the specific indices are a cokequence of a diagonal element of S,(o) getting within a distance 0.0005 of zero during the e&nation. When this happens, the parameter is fixed at 0.0005 and the estimation continues as if the parameter waszero in the population. SeeGewekeand Singleton(1980) for a discussionof the implications Of this boundary solution for the asymptotic properties of the estimators, as well as some Monte Carlo evidence that the practical consequences may be minor.
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Second, models incorporating the Lucas (1972) supply equation imply that unanticipated price movements are an important determinant of movements in output. Yet Sargent (1976) and Fair (1979) found that price surprises explained little of the variation of output. The large percentage of variation in GCPI that is independent of J+ and i, raises the possibility that price innovations. play an important role in the determination of output, but that measurement error imparted a downward bias to their estimated coefficients. The results for the period 19651 through 197612 displayed in table 2 differ from those for the earlier period in several important respects. First of Estimated Frequency
UN
Table 2 values for the period
parameter LY
Y3MB Loadings
0.00007t 0.11117r 0.2222x 0.3333n 0.44&Z 0.5556x 0.6667~ 0.777&l 0.8889x 1SKKhI
45.951 0.45621 0.03373 0.00307 0.00257 0.00471 0.00051 0.00286 0.00466 0.00003 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.00007I O.llllr 0.2222X 0.3333rr 0.444472 0.5556~ 0.6667~ 0.7778~ 0.8689x l.OOOYI
3.7837 0.11881 0.02029 0.00963 0.01157 0.00652 0.00179 0.00041 0.00317 0.01167
0.99783
index
densities
overall
index
0.01071 1.8460 1.3932 4.6370 0.48420 0.00110 6.9154 4.9713 5.6028 0.14491
12.226 5.9613 8.5615 9.1399 2.7056 1.7786 0.72923 7.0778 0.15820 2.4498
58.004 1.1070 0.53879 0.40139 0.54264 0.17809 1.7096 1.2353 0.87406 5.7796
1.5577 2.2694 2.9480 0.85946 3.5847 4.9984 1.1586 1.0104 0.44943 0.04050
([&)]12)
24.422 . 1.2698 0.46787 0.10274 0.10961 0.02534 0.00546 0.02406 0.0069 1 0.00599 indices
0.0 0.13274 0.0 0.01374 0.0 0.0 0.0 0.00703 0.0 0.00081
0.80201 0.0 0.05397 ii0659
0.99771
Gh41
([$a~)],,)
of the specijic
proportion
GCPI
1.8994 0.86879 0.04962 0.02962 0.01600 0.02316 0.00585 0.00265 0.00175 0.00018
45.535 0.76236 0.36881 0.07518 0.10251 0.03561 0.02072 0.01409 0.01034 0.00842
0.0 0.0 0.00006 0.00001 0.0 0.00001 0.00001 0.00001 0.00001 0.0 Cumulative
0.92132
on the second
0.0 0.0 0.0 0.0 0.0 0.0’ 0.0
Spectral
YIYB
19.474 0.52882 0.10691 0.02133 0.01434 0.02737 0.00628 0.00325 0.02879 0.00240
Loadings 0.00007I O.lllln 0.2222X 0.3333x 0.444472 0.5556x 0.6667~ 0.7778~ 0.8889x 1.0007t
on thefirst
0.04698 0.00333 0.00034 0.00004 0.00005 0.00001 O.OOOOO O.OOOOQ O.OOOOO 0.00001
1965:1-197612
0.01277 0.01149 0.0 0.0 0.00327 of variance 0.97055
([S,(w)],,) 8.6988 1.6271 4.8315 1.7461 3.9107 1.9677 0.0 2.4352 1.2935 1.6558
11.538 19.853 8.0042 8.3564 8.2091 4.6153 18.971 11.828 6.9903 4.3027
explained 0.77384
0.40425
300
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all, notice that the variance of GCPl increased substantially, with most of the increase in variation being associated with the nominal index. The large coherence of GCPZ with the nominal index at the lowest frequency seemed implausibly large given that during the period there were major supply shocks, droughts, etc. Therefore, I estimated a quarterly version of the model with industrial production and GCPI replaced by real GNP and the growth rate of the GNP deflator, respectively. The results for the implicit price deflator were qualitatively the same as those for GUI. Together, these findings suggest that the low frequency behavior of inflation was much more closely associated with the low frequency behavior of the neutral index than with output during this sample period. ’ Another interesting feature of the results for GCPZ is that the spectral shape of the specific index sgl is similar for the two subperiods. The specific indices seem to have behaved qualitatively and quantitatively the same during the two periods. The increased variability iof inflation is reflected in increases in the loadings of Y3MB and YZYB on the neutral index, particularly at the low frequencies. There are, however, larger percentage increases in the loadings of the nominal interest rates on the real index. Over thirty percent of the variation in Y3MB i3 now due to variation in zi, which suggests that real yields on three-month treasury bills were much more variable during this period than during Famh’s (1975) sample period. The absolute variation of Yl YB due to z1 also increased substantially from the earlier period, but only ten percent of the overall variation in the one-year bond yield is due to the real index. The relative values of the loadings for Y3MB and Yl YB suggest that real’yields on three-month bills were much more volatile than real yields on one-year bonds. The variance of GM1 also increased from the earlier period. Much of this increase can be attributed to the larger values of the loadings on the real index, especially at the low frequencies. As for the earlier period, the real index accounts for more of the variation in GM1 than the neutral index. 4. The cyclical behavior of the real and neutral components of interest rates
The spectral density matrices described in section 3 summarize the covariance structure of interest rates and output. They do not, however, provide much information about the behavior of the real and neutral components of interest rates on a month-by-month basis. In this section I describe and implement a procedure for extracting observable time series ‘A similar, but less extreme, loading pattern is obtained when GCPI is replaced by the log of the CPI (LCPI). In this level version of the model, the ratio of the loadings of LCPI on the red and neutral indices at the zero frequency is 0.488. Nevertheless, the results for the nominal interest rates in table 2 and in section 4 are qualitatively the same for the models with LCPI and GCPI.
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301
representing the cumulative effects up to time t of the latent indices on the nominal interest rates. This signal extraction problem has been discussed by Geweke and Singleton (1981b) and Litterman and Sargent (1979). For this analysis, I shall use the Litterman-Sargent approach, which in the notation of section 2 can be described as follows. Let u,(L) be the jth column of the matrix of lag polynomials in (9) and rJt be the jth latent index, j= 1,2. The objective is to estimate the 6 x 6 matrices B{ in the projections Uj{L)
=
f
@X,-i
+
4,
j=l,2,
i=O
(11)
where x is the vector of observable variables in (6) and v/ is a least-squares projection error satisfying E[v/xi -J = 0, i = 0,. . ., m. Postmultiplying (11) by x,-, and taking expectations gives the least squares normal equations
ce7,~j,x(s) =izo Wx,x(s -i)>s=O,...,m,
(12)
where C,j+(~)=E(u,((L)
Thus, a consistent estimator of CJs), s=. . ., - l,O, 1,. . ., can be obtained from a consistent estimator of S,,,(w), /o( 5 7~. Since cl= [ti,, r2J is uncorrelated with ss, for all s and t, and the spectral density matrix of <, is the 2 x 2 identity matrix, the cross-spectral density matrix of x1 and a~L)5j, can be expressed as
~x.,,,c~,=a;(~,~~~Y, j= 1,2, where &Lo) is the jth
column
of A”(U) in (9). Thus, from the consistent
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estimates of /T(w), estimates of the cross-covariances of axL)~j~ and x,-,, cx,,<,(s) = (l/274 jz ZJ@“@‘eimsd~,
can be recovered. Similarly, the spectral density matrix of x implied by the model is z(i(w)A”((o)+S,(o). By inverse Fourier transforming the estimates of S,(o), a consistent estimate of &x(s) is obtained. Upon substituting these estimates into (12), the normal equations can be solved for the B{ used in the prediction of U~L)rj,. See Litterman and Sargent (1979) for additional details. Estimates of the II{, i=O, . . ., 4, were calculated using the estimates of the spectral densities in tables 1 and 2. The implied fitted values of a,,(L)<,, and a,,(L)t,, for Y3MB and YZYB during the period 195415 through 1965:12 are presented in figs. 1 and 2 [Q(&~, and a&)5,, appear in the legend as Noise 1 and Noise 2, respectively]. Note that the fitted values from (11) are functions of the current and past histories of the observed variables only. Accordingly, the relative behavior of the series in figs. 1 and 2 at each date t reflects only what was known at date t, even though the a,,(Q,)t[jt in (9) may be two-sided.
C, 1.5
ACTUAL
X
NOISE -__. :
C
NOISE -_-___. 2
-.1.5
-2
1
20
LO
60 T!KEII
80 ‘?I MChITL.IS II/
ioa
20
Fig. 1. Decomposition of Y3MB for the period 1954:5-1965:12.
140
303
K.J. Singleton, Real and nominal factors in business cycles 2.5
Legend 2
1.5
A
ACT’JAL _--
X
NOISC ! _--.
r:
NOW ----.
2
I
+5
0.5
g n
0
-.0.5
-.I
e-l.5
-2
I
20
40
IN8tkINTHS
60
I
100
1
120
140
TIME
Fig. 2. Decomposition
of Yl YE for the period 19545-1965: 12.
Perhaps the most striking feature of these figures is that the a,,(~?)<,, I = 3,4, are nearly constant, while the neutral components tract the time paths of nominal interest rates very closely. The real components of Y3MB and Yl YB moved procyclically with industrial production. The largest movements in a,,(&$,, and a,i(&)r,, occured during the 1957-1958 recession, when these series accounted for over half of the decline in nominal rates. An interesting feature of the 1955-1960 period is that the peaks and troughs in industrial production tended to occur about three months prior to the corresponding peaks and troughs in the real components of Y3MB and YIYB. This pattern suggests that expansions (contractions) in output over the business cycle are followed after several months by increases (decreases) in real interest rates. Another interesting feature of the results is the procyclical behavior of the real component of GMI. Toward the end of the 1958 recession, for example, GAJI was positive for several consecutive months, and so was the nominal was negative component of GMI. In contrast, the real component throughout much of 1958 and its lowest value was attained when industrial production reached its lowest value. This tendency for the real component of GMZ to move procyclically with LYis consistent with models that emphasize
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the demand side of the money market, with money supply responding passively to economic events [e.g., Sims (1980)]. While the money supply may have responded passively to real economic developments prior to 1965, it is generally agreed among economists that the period 1965 through 1976 was one of active intervention by the Federal Reserve. Therefore, it should be informative to compare the time paths of the real and neutral components of interest rates for the later period to the paths just discussed. The forecasted values of a,,(L)<,, and a,,(L)<,,, I= 3,4, are displayed in figs. 3 and 4. There are several important differences between these figures and figs. 1 and 2. First, the fluctuations in the real components of interest rates were much larger during the 1965-1976 period. As before, these series move procyclically with output. Second, the turning points in the nominal components of Y3MB and YZYB during 1970 and 1974 lagged behind the turning points in- both observed interest rates and the nominal component of GMZ. The explanation for this pattern seems to lie in the large values of GCPI during 1970 and 1974, and the fact that most of this inflation was associated with the neutral index. Apparently, expectations of inflation remained high, even after the downturn in real inter&t rates, output, and monetary growth.
2-
A
ACTUAL --
X
NOISE -_--. :
I
20
40
60 T,fJE
Fig. 3. Decompositionof
Y3MB
IN8;~NTHS
‘Ofi
120
for the period 1965:5-1976:12.
l/.0
’
K.J. Singleton, Real and nominal factors in business cycles
Legend .’ :‘
/-0
305
I
NOISC -. t -. NOLC____. 2
20
,
I
40
60
I
IN8tiONTHS
0
100
120
140
TIME
Fig. 4. Decomposition
of Yl YE for the period 1965:5-1976:12.
To shed some light on the relationship among output and monetary growth during this period, consider the episode of tight credit during 1969. GM1 declined from April, 1969 through September, 1969 and most of this decline was associated with the real index. At the outset of this decline, the real components of interest rates were rising along with output. By July, 1969, however, the real components of Y3MB and YZYB were declining. LY declined in June and July, increased slightly in August and September, and then declined steadily. Thus, the initial decline in GMZ seems to have led to a fall in both output and real interest rates. From August 1969 through February 1970, however, most of the decline in GMZ was associated with the neutral index. After a prolonged decline in GMZ, further declines were not associated with real output during this period. A similar output-money relationship emerged during’the first part of 1973-1975 period. Throughout this discussion I have been interpreting movements in the real components of Y3MB and YZYB as reflecting movements in real interest rates. The series ari(L)rr,, I= 3,4, are generally not equal to the ex ante real rates given in the respective versions of eq. (3) for Y3MB and YZYB, since r, and $: will generally be correlated at some leads and lags (see section 2), and r1 and r2 are orthogonal. Nevertheless, real interest rates and the a,,(,!,)~,,
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may behave approximately the same over the business cycle, especially if Tobin effects are not too large. Therefore, it may be informative to compare at this point the behavior of the series a,,(&,,, I= 3,4, to the behavior of the ex ante real rate series constructed by others. The real components do not display the same cyclical behaviour as the ex ante real rate constructed by Elliot (1977) from a ‘neoKeynesian, monetary’ model, which is his preferred model of the several he considers. Specifically, Elliot’s real rate peaked during 1972 and then fell steadily through 1974. Also he finds that real rates are independent of current output levels. In contrast the real components of Y3MB and YlYB were increasing until late 1973 and had fallen only a small amount by the end of 1974. The ex post real rate Elliot calculates by subtracting the realized inflation rate from the nominal interest rate also does not behave like the The latter differences are not surprising to me in light of the %c&l, evidence presented in section 3 of this paper and elsewhere [e.g., Fama (1977)] suggesting that reported inflation rates are poor measures of actual inflation rates. Measurement errors in the reported values of #, imply errors in measuring ex post real rates, and it seems likely that they will also contaminate many measures of ex ante real rates. The fluctuations in Carlson’s (1977) ex ante interest rate, calculated from the Livingston data on expectations of inflation, resemble more closely the fluctuations in my real series. The local peaks in our series during 19691970 and 1973 appear to occur at about the same times, for example. However, the variation in Carlson’s series is much larger, especially during the 1954-1962 period. Finally, it is interesting to compare my real series to the series on the cost of capital calculated from individual firm data for the period 1960-1974 by Ciccolo (1975) and Tobin and Brainard (1976). In theory, their cost of capital series and the real rates on treasury bills need not be identical, because of the different risk characteristics of bonds and capital. Nevertheless, the two series should be highly correlated, since arbitrage and the systematic response of investors to new information about the economy unites all asset markets. In fact, there is a close resemblance among the series. For example, the TobinBrainard series reaches a trough in 1967, a peak in 1969, another trough in late 1971, and then it rises steadily during 1972 and 1973. A similar pattern is displayed in fig. 3. 5. Concluding remarks This paper has attempted to characterize empirically the interrelationships among monetary growth, real interest rates, and output during the postwar period. From the analysis of these series over the periods 1954-1965 and 1965-1976, several patterns of correlation emerged. First, the real
K.J.
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307
components of interest rates, which are assumed to be functions of real interest rates, were not constant during either subperiod. The volatility of these components was greatest during the second period, when over thirty percent of the variation in the three month bill rate was due to variation in the real index. Furthermore, real interest rates appeared to move procyclically with output during both subperiods, and to exhibit turning points that lagged behind those of output by several months. These aspects of the results suggest that potentially misleading inferences about future capital formation may be drawn from studying ex post real rates calculated by subtracting the inflation rate from nominal yields on financial assets. During the 1972-1973 period, for example, such ex post real rates were declining [e.g., Elliot (1977)], while Tobin and Brainard’s (1976) cost of capital variable and the real components of interest rates calculated here were rising. Theoretical explanations for the procyclical behavior of real interest rates are provided in Barro (1980a) and Cukierman (1980). Their explanations, as well as the timing of the output-real rate relationship, warrant further investigation. Data appendix Y3MB
is the market yield on three-month
treasury bills reported
in the
Federal Reserve Bulletin. Yl YB
is the yield on one-year
treasury
bonds reported
in the U.S.
l?easury Bulletin. GCPI
GM1
LY UN
is the annualized monthly growth rate of the consumer price index for all urban consumers reported by the Bureau of Labor Statistics. The data are not seasonally adjusted. is the growth rate of currency plus demand deposits (MI) reported in the Federal Reserve Bulletin. The data are monthly averages of daily figures and they are seasonally adjusted. is the log of industrial production in manufacturing reported in the Federal Reserve Bulletin. The data are seasonally adjusted. is the unemployment rate for the civilian labor force of age sixteen years and over reported by the Bureau of Labor Statistics. The data are seasonally adjusted.
References Ando, A.A., 1974, Some aspects of stabilization policies, the monetarist controversy, and the MPS model, International Economic Review 15, 541-571. Barro, R.J., 198Oa, A capital market in an equilibrium business cycle model, Econometrica 48, Sept. Barro, R.J., 1980b, Intertemporal substitution and the business cycle, National Bureau of Economic Research working paper no. 490, June.
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cycles
Blanchard, O.J., 1980, The monetary mechanism in the light of rational expectations, in: S. Fischer, ed., Rational expectations and economic policy (University of Chicago Press, Chicago, IL). Blinder, A.A. and S. Fischer, 1980, Inventories, rational expectations, and the business cycle, Journal of Monetary Economics, forthcoming. Carlson, J.A., 1977, Short-term interest rates as predictors of inflation: Comment, American Economic Review 67,469-475. Ciccolo, J., 1975, A linkage between product and financial markets: Investment and q. Essay III of unpublished Ph.D. dissertation (Yale University, New Haven, CT). Cukierman, A., 1980, Interest rates over the cycle, inventories and monetary policy: A theoretical analysis, Unpublished manuscript, Sept. (University of Tel-Aviv, Tel-Aviv). Elliot, J.W., 1977, Measuring the expected real rate of interest: An exploration of macroeconomic alternatives, American Economic Review 67,429-444. Fair, R.C., 1979, An analysis of the accuracy of four macroeconometric models, Journal of Political Economy 87, 701-718. Fama, E.F., 1975, Short-term interest rates as predictors of inflation, American Economic Review 65269-282. Fama, E.F., 1977, Interest rates and inflation: The message in the entrials, American Economic Review 67, 487-496. Fischer, S., 1979, Anticipations and the nonneutrality of money, Journal of Political Economy 87, 225-252. Fisher, I., 1930, The theory of interest (Macmillan, New York). Fishman, G., 1969, Spectral methods in econometrics (Harvard University Press, Cambridge, MA). Geweke, J.F., 1977, The dynamic factor analysis of economic time series, in: D. Aigner and A. Goldberger, eds., Latent variables in socioeconomic models (North-Holland, Amsterdam) 365-383. Geweke, J.F. and K.J. Singleton, 1980, Interpreting the likelihood ratio statistic in factor models when sample size is small, Journal of the-American Statistical Association 75, 133-137. Geweke, J.F. and K.J. Singleton. 1981a. Maximum likelihood conlirmatorv factor analvsis of economic time series, International Economic Review 22, 35-52. Geweke, J.F. and K.J. Singleton, 1981b, latent variable models for time series: A frequency domain approach with an application to the permanent income hypothesis, Journal of Econometrics 17,287-304. Hurwicx, L., 1962, On the structural form of independent systems, in: E. Nagel et al., eds., Logic, methodology, and the philosophy of science (Stanford University Press, Stanford, CA). Litterman, R. and T.J. Sargent, 1979, Detecting neutral price level changes and the effects of aggregate demand with index models, Unpublished manuscript, Sept. (University of Minnesota, Minneapolis, MN). Lucas, R.E., 1972, Expectations and the neutrality of money, Journal of Economic Theory 4, 103-124. Lucas, R.E., 1975, An equilibrium model of the business cycle, Journal of Political Economy 83, 1113-1144. McCallum, B.T., 1979, Monetarism, rational expectations, oligopolistic pricing, and the MPS econometric model, Journal of Political Economy 87, 57-73. Mehra, Y.P., 1978, Is money exogenpus in money-demand equations, Journal of Political Economy 86,21 l-228. Mishkin, F.S., 1981, The real interest rate: An empirical investigation, Carnegie-Rochester. Mundell, R.A., 1963, Inflation and real interest, Journal of Political Economy 71, 28C&283. Neftci, Salih and Thomas J. Sargent, 1978, A little bit of evidence on the natural rate hypothesis from the U.S., Journal of Monetary Economics 4, 315-320. Nelson, Charles R. and G. William Schwert, 1977, Short-term interest rates as predictors of inflation: On testing the hypothesis that the real rate of interest is constant, American Economic Review 67,478-486. Sargent, Thomas J., 1973, Rational expectations, the real rate of interest, and the natural rate of unemployment, Brookings Papers on Economic Activity 2,429-472. Sargent, Thomas J., 1976, A classical macroeconometric model for the United States, Journal of Political Economy 84, 207-237.
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and nominalfnctors
in
business
cycles
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Sargent, Thomas J. and Christopher A. Sims, 1977, Business cycle modeling without pretending to have too much a priori economic theory, in: Christopher A. Sims, ed., New methods in business cycle research: Proceedings from a conference (Federal Reserve Bank, Minneapolis, MN) 45-101. Shiller, Robert J., 1980, Can the Fed control real interest rates?, in: S. Fischer, ed., Rational expectations and economic policy (University of Chieago Press, Chicago, IL). Sims, Christopher A., 1980, Comparison of interwar and postwar business cycles: Monetarism reconsidered, American Economic Review 70, 250-257. Singleton, Kenneth J., 1980, A latent time series model of the cyclical behavior of interest rates, International Economic Review 21, 559-575. Tobin, James, 1965, Money and economic growth, Econometrica 33, 671-684. Tobin, James and William C. Brainard, 1976, Asset markets and the cost of capital, Cowles Foundation working paper no. 427 (Yale University, New Haven, CT). Weld, Herman, 1938, The analysis of stationary time series (Ahnquist and Wicksell, Uppsala).
REAL AND NOMINAL FACTORS IN THE CYCLICAL BEHAVIOR OF INTEREST RATES, OUTPUT, AND MONEY Kenneth J. SINGLETON* Carnegie-Mellon
University,
Pittsburgh,
PA 15213,
USA
Received February 1981, final version received October 1982 Short-term nominal interest rates, output, unemployment, monetary growth, and inflation are represented as functions of two common latent indices: a ‘real’ index that underlies comovements in output and unemployment, and a ‘neutral’ index that is independent of output. A Fisherian interpretation of the real and neutral components of interest rates is provided under the assumption that anticipated inflation affects output primarily through its effect on real interest rates. Then, monthly time series on the real and neutral components of short-term treasury bill yields are constructed, and the cyclical behavior of these series is compared to the behavior of output and monetary growth.
1. Introduction Modern ‘equilibrium’ theories of the business cycle assign a potentially important role to real interest rate fluctuations in the propagation of cycles in real’ output [e.g., Lucas (1975), Barro (1980a, b), Blanchard (1980), Blinder and Fischer (1980)]. Indeed, the principle means by which anticipated changes in the money stock and inflation affect output in these models is through changes in real rates [Lucas (1975), Tobin and Brainard (1976), Fischer (1979)]. Yet, little attention has been directed toward characterizing empirically the temporal covariance structure of monetary growth, inflation, real interest rates, and output. ’ This paper represents an attempt to shed *I wish to thank the participants of workshops at several universities and of the NBER conference on inflation and financial markets, especially Robert Barro, Alex Cukierman, Jacob Frenkel, Herschel Grossman, Lars Hansen, Robert Hodrick, Bennet McCallum, Robert Shiller, Chris Sims, and the referee for helpful comments on earlier drafts of this paper. Robert Litterman kindly provided his computer software for the calculations in section 4. ‘Much attention has been given to the relationship among monetary growth and real output implied by equilibrium business cycle models, as well as the empirical relationships among ex post real rates and monetary growth. The importance of studying output, interest rates, and monetary growth simultaneously in empirical analyses is evidenced by the results obtained in Mehra (1978) and Sims (1980). Although Mehra and Sims did not address the issue of real interest rate behavior, their studies are perhaps the closest precursors of this analysis. 0165-l 889/83/$03.00 0 Elsevier Science Publishers
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some light on the nature of the co-movements among these variables for the United States during the postwar period. The framework for the investigation is a latent index mode1 [Geweke (1977), Sargent and Sims (1977), Geweke and Singleton (1981a)] of nominal interest rates, output, unemployment, monetary growth, and inflation. Nominal interest rates, the monetary growth rate, and the inflation rate are represented as functions of two common latent indices: a ‘real’ index that underlies co-movements in output and unemployment, and a ‘neutral’ index that is independent of output. Fluctuations in nominal interest rates associated with the real business cycle are assumed, in the spirit of many equilibrium business cycle theories, to represent fluctuations in real interest rates. Real rates may change in response to aggregate demand and supply shocks or to changes in the anticipated inflation rate (the Mundell-Tobin effect). The ‘neutral’ index, on the other hand, is assumed to capture the effects on interest rates of anticipated changes in the inflation and monetary growth rates that leave output unchanged. Under these assumptions, it is possible to study aspects of the cyclical behavior of real interest rates, and the nature of real and neutral variations in the monetary growth rate. This is accomplished without having to specify the information sets underlying agents’ expectations of inflation. In section 2 the latent index mode1 is presented, and the decomposition of the monetary growth rate, interest rates, and inflation rate into real and neutral components is discussed in the context of several recent business cycle models. The index model is shown to be identified in section 3. Identification is achieved without having to assume a subset of the variables in the model are econometrically exogenous. Furthermore, the latent index mode1 is shown to accommodate certain types of errors in measuring such variables as the inflation rate and money stock. The estimated parameters of the mode1 for the subperiods 1954 through 1965 and 1965 through 1976 of the postwar period are also reported in section 3. A comparison of the results for these two subperiods is of interest because of the apparent change in the monetary policy rule during the 19631965 period meftci and Sargent (1978)]. The index model is not structural in the sense of Hurwicz (1962) and, therefore, the parameter values are not invariant to changes in policy regimes. The later subperiod seems to have been characterized by relatively higher inflation rates and greater activism by the Federal Reserve, which suggests that the behavior of the real components of interest rates may have been different across subperiods. In fact, the parameter estimates indicate that the real components of both nominal interest rates and monetary growth were substantially more volatile during the later subperiod. In section 4 a procedure for extracting observable time series ‘that represent the effects of the real and neutral indices on interest rates is
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described and implemented. The graphs of these series are then compared with the ex ante real interest rate series constructed in several previous studies. The findings suggest that real rates constructed from the growth rate of the CPI may be very misleading measures of the actual real rates, especially over the recent subperiod. Finally, some concluding remarks are presented in section 5. 2. The latent index model Let y, be the 2 x 1 vector including the log of real output (43 and the unemployment rate (u,) and let n; E [ii, fit, ti,], where i, is a vector of nominal interest rates, Ejt is the inflation rate, and rri, is the rate of growth of the money stock. For the purpose of decomposing movements in n, into movements associated with the real business cycle and movements that are neutral with respect to real activity, I adopt the following latent index representation of yI and n,:
In (l), the constituents of e are mutually uncorrelated and uncorrelated with the common latent indices z1 and z2 at all leads and lags; 12,,(L), 1,,(L), I,,(L) are (possibly two-sided) lag polynomials with square summable coefficients; and the latent variables are purely indeterministic, covariance stationary time series. All variables may be serially correlated. The restriction that y, is a function only of the first index identifies zl as a ‘real’ index underlying the cyclical covariation of U, and qr; that is, covariation of output and unemployment is due entirely to the disturbances embodied in zl. These include shocks to investment, labor and capital productivity, and consumption. In addition, most equilibrium business cycle models imply that unanticipated changes in the price level and policy variables will also directly affect real economic variables [e.g., Lucas (1975), Sargent (1973)]. Although I refer to z1 as the ‘real’ index, innovations in zlt are assumed to embody innovations in these nominal variables as well. From a theoretical point of view, macroeconomic models in which Q and u, are related thro’ugh a production function, Phillips curve, and an equation describing labor force participation imply that output and unemployment will be functions of a common set of disturbances [e.g., Ando (1974), Sargent (1976) and McCallum (1979)]. The assumption that the effects of these disturbances are captured by a single latent index does not necessarily imply that the ‘primitive’ real disturbances to the economy are unit-dimensional Sargent and Sims (1977), Litterman and Sargent (1979) and Singleton (1980),
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for example, consider the following two-index model:
in which the common latent indices are the constructs unanticipated (vit= ~~-4~) and anticipated (vZt=$J ‘aggregate demand’, and where all polynominals in L are one-sided. Although (~~-4~) and fi, are not uncorrelated at all leads and lags, this model can be represented in the form (1). This is accomplished by forming the two-sided least squares projection
substituting this expression for v2t (=fiJ in (2), and letting zl,= vll=(%-ij,) and z2,= 5,. For this example, A, ,(L)=a(L) and A,,(L)=c(L) are one-sided, is two-sided, and z1 and z2 are by construction 122 l(L) = (b(L) + c(ww mutually uncorrelated at all leads and lags. Both (~-6~) and 4, are in general functions of all of the variables in agents’ information set and, hence, may embody a large number of primitive economic disturbances. The emphasis on demand assumes that the underlying disturbances to the economy which are being captured by z1 are primarily demand-related. More generally, output, unemployment, and real interest rates will be functions of a common set of supply and demand shocks. While in principle these shocks may affect qr and u, in a manner that gives rise to several real disturbances, the empirical evidence in Litterman and Sargent (1979) and Singleton (1980) is consistent with the single-index representation of real activity assumed here (see also section 3). In effect, the zero restrictions in (1) identify zi as representing the combined influence of all ‘real’ disturbances on output and, in this sense, the real business cycle is being identified with cycles in real output. The second index is a source of comovements in nominal monetary growth, inflation, and nominal interest rates, but not a source of comovements in real variables. The restrictions that zi and z2 be orthogonal at all leads and lags, together with the zero restrictions in (l), overidentify the component L,,(L)z,, of a,. They do not necessarily imply that anticipated changes in policy variables are neutral in this model. The association of (v,-IjJ and & with vlt and vlr, respectively, in (2) does imply a version of the natural rate hypothesis. However, if the broader interpretation of vl, in the last paragraph is adopted and anticipated changes in nominal variables affect real output, then vi and v2 will not equal (q,-$ and G1, and they will in general be correlated both contemporaneously and at various leads and ,lags. Nevertheless, the two-sided projection of v21 on vit can be used to transform this modified version of (2) into a representation of the form (1). Thus, (1)
K.J.
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allows for potential non-neutral effects of anticipated nominal variables -on output? To further interpret the orthogonality condition and make the last observation more precise, suppose each interest rate i, satisfies the following rational version of Fisher’s (1930) equation: i,=+,+g,
(3)
where T, is the ex ante real rate of interest and $: is the optimal linear least squares forecast of dl (over the relevant horizon), based on information available at time t. Many of the equilibrium models of the business cycle assume anticipated inflation affects output primarily through its effects on real interest rates due to Mundell (1963) - Tobin (1965) effects. Under this assumption, fi; may be correlated with r at all leads and lags, but there is nevertheless a component of @: that is orthogonal to both t and 4. That is, in the absence of ‘rigidities’ that lead to direct effects of $: on yt (i.e., effects other than through r), there is a component of i, that is orthogonal to r, and y, at all leads and lags. Specifically, let
be the least squares projection of 0: onto {rt-s; s=O,+l,.. nominal interest rate it can be expressed as
.}. Then each
i,= r,+ f c&r,-, +r,, ( s=--00 > where c, is the component of anticipated inflation that is orthogonal to the real rate r at all leads and lags. Now let zr be a real index underlying the cyclical covariation of the real variables r,, qt, and I+ Also, define If,(L)z,, and Afz(L)z2, to be the subvectors of A,r(L)zr, and t22(L)~2f, respectively, corresponding to the subvector i, of n,. Then L$r(L)zIr and A;2(L)zzt can be interpreted as the terms (rt+x 6,rt-J and &, respectively, in (4). Under this interpretation, the index z2 is neutral in the strong sense that it underlies comovements in nominal interest rates, monetary growth, and inflation that do ‘Litterman and Sargent (1979) included a vector of commodity prices, instead of i0 in n, Neither Litterman and Sargent (1979) or Singleton (1980) imposed any restrictions on Lv’, Q beyond those in (1) and, hence, their representations are observationally equivalent to the following interpretatton of (1) as a model that embodies Mundell and Tobin effects on output. It follows that their evidence on the number of real indices can be used in support of a single, more broadly interpreted, real index in the analysis here.
294
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factors
in business
cycles
not alfect real variables at all leads and lags. Because 1,,(L) is allowed to be two-sided in (l), the restriction cov[zl,,zZ,] =O, for all s, does not imply that anticipated policy variables are neutral in the usual sense of the term; ni,-,(s lo), I, and y, may be correlated. Finally, the vector e captures the effects of measurement errors or disturbances that are specific to variables in y or II. The preceding motivation of the index representation in terms of recent macroeconomic theories imposes no restrictions on the covariance structure of s. To identify the model and proceed with estimation, it is necessary to assume that the constituents of E are uncorrelated at all leads and lags (they may be serially correlated. As emphasized by Sargent and Sims (1977), these conditions are imposed not because they are believed to have a foundation in the theoretical ideas motivating the model, but because some such conditions must be imposed to proceed with estimation. It should be emphasized, though, that this assumption together with the assumed two-index representation (1) imply testable restrictions that are not refuted by the data. 3. Estimated value-s of the parameters The restrictions on the covariance structure yI and n, implied by (1) can be decomposed using frequency domain techniques. Let x’=CY),n’], z’=[zi, zJ, and S,(w), S,(o), and S,(o) be the spectral density matrices of x, z, and e, respectively. Then (1) and the accompanying assumptions imply that S,(o) decomposes as [Geweke and Singleton 19gla]:
where view) is the Fourier transform of the function AL,(s) and the prime denotes transposition and conjugation. S,(o) and S,(w) are diagonal for all w, since the constituents of t and e are uncorrelated at all leads and lags. A heuristic view of the spectral density function will be useful for interpreting the estimates of the parameters in (5). Variation at frequency 27r/P corresponds to oscillations of period P. If a time series x,, recorded monthly for example, displays strong seasonal oscillations, then the spectral density function for x will be large or ‘powerful’ at frequency 27412. A series which is serially uncorrelated has a constant spectral density function, while one for which x, and x,+~ have correlation near one for small s will be very powerful at the low frequencies (those near o =0) compared to the high frequencies (near o=n). The variance of a time series can be decomposed into variation across frequencies. Thus, eq. (5) can be viewed as a representation of the covariation of the elements of x at frequency o. The integral across all frequencies gives the conventional covariance matrix of x.
295
K.J. Singleton, Real and nominal factors in business cycles
From Proposition 1 in Geweke and Singleton (1981a), it follows that the non-zero elements in the &(w) and S,(o) and S,(w) are over-identified. Some normalizations must ‘be imposed before the free parameters in (5) can be estimated, however. For the purposes of this analysis, I shall normalize the diagonal elements of S,(o) to be unity, /o( sn. To interpret this normalization, recall that every purely indeterministic, covariance stationary process has a one-sided moving average representation mold (193811. Therefore, I can write
where cov[&,, &J=O, uncorrelated. Substituting
for all t and s, and [r (6) and (7) into (1) gives
and
&
are serially
(8)
Eq. (8) expresses J+ and n, as functions of the innovations in the real and neutral indices. Now if the variances of rrt and rzl are normalized to be unity, then the frequency domain representation of (8) is S*(w) = fq(w,&o,,
+ S,(o),
l+D,
(9)
where a; r(w) = & ,(w)d( w ), L1 t(w) = 0, etc., and the spectral density matrix of [rrr, &,I is the 2 x 2 identity matrix. The moduli of the complex elements of &B) in (9), I&(e$l say, are uniquely determined by these normalizations [Geweke and Singleton (1981a)l. Each I~&# represents the total contribution of (zfi}, or equivalently of {rjt}, to variation in the Ith element of x at frequency o. Thus, the proportion of the variation in xl due to ~j at frequency o is
coht.,W= ~4,@)~z/CU41~~, where [ In denotes the Ith diagonal element of the matrix in brackets. This ratio, called a coherence, is’analogous to the coefficient of determination in regression analysis. Since the integral of S,(o) gives the covariance matrix of x,, an overall measure of variation explained by the jth latent index can be obtained by integrating the numerator and denominator of (10) and then forming the ratio of these integrals. The estimation of the model was performed with monthly data for the J.ED.C.-
F
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cycles
periods 1954:l through 1965:12 and 1965: 1 through 1976:12. The vector of short-term interest rates i, included the three-month treasury bill yield (Y3MB) and the one-year treasury bond yield (Y1 YB). For fit, I used the rate of growth of the consumer price index (GCPZ), ni, was the growth rate of Ml (GMZ), output was measured by industrial production (LY=log of industrial production), and the unemployment rate (UN) was the rate for civilian workers. Sources for the data are provided in the appendix. At the outset, linear trends were removed from each series by ordinary least squares regression. Then, to diminish any small sample bias in the spectral estimator due to leakage, each of the series was prewhitened using the estimated coefficients from a second-order autoregression. After prewhitening the data, cross-periodogram ordinates were calculated at 144 equally spaced frequencies in the interval [0,2rr]. These ordinates were smoothed with a tent filter of width eleven ordinates at the base to form a consistent estimator, g’(o), of the unrestricted spectral density matrix of x. Estimates based on the prewhitened data were recolored in the usual way. Finally, estimates of A(o) and S,(o) were obtained by substituting $(a) for the maximum likelihood estimator of S,(w) in the concentrated log-likelihood function described in Geweke and Singleton (1981a), and then maximizing the resulting objective function with respect to the free parameters. This procedure yields consistent, but not maximum likelihood, estimators3 Ten frequencies, equally spaced over the interval [O,rc), were considered. The chi-square statistics for testing the overidentifying restrictions at each frequency [see Geweke and Singleton (1981a)], based on estimates of S,(w) obtained with a rectangular window, were similar to those reported by Singleton (1980), and generally lend support to the two-index representation (1).4 Specifically, for the period 1954: 1 through 1965: 12, the values of the chisquare statistics with twelve degrees of freedom ranged from 2.76 to 13.66, and for the period 1965:l through 1976112, they ranged from 3.58 to 11.3. The parameter estimates for the period 1954: 1 through 1965:12 are displayed in table 1.5 The squared moduli of the estimated values of A(U) are ‘A tent window was used instead of a rectangular window, which gives the maximum likelihood estimator of S,(o) and leads to maximum likelihood estimates of A(o), in order to improve the resolution of the estimates for use in forming the projections &&bed in section 4. More precisely, if I(oj) denotes the cross-periodogram matrix at frequency 0,. then S,(c+) =z-,,,bJ(mk-j), where b, is the ‘tent’ moving average window bj=[l -ljl/(m+ I)‘]. In my case, m is set at 4. %ince the tests were based on maximum likelihood estimates of S,(w), the usual likelihood ratio statistics are distributed asymptotically as chi-square variates [Geweke and Singleton WWI. Overlapping sets of ordinates were used to estimate the spectral densities at the ten frequencies, so the individual estimates are not independent. Therefore, an overall’goodness-of-fit test cannot be constructed by adding the individual chi-square statistics. Singleton (1980) reports an overall test for a similar index represenlation that does not call for rejection of the model at conventional significance levels. ‘The numbers in table I are actually 271(c~)\~ and 2xdiag [S&$j and, therefore, the reported numbers must be divided by 22 in order to obtain the sample spectral densities.
K.J.
Singleton,
Estimated Frequency
O.OOOOR 0.1111x 0.2222n 0.3333x 0.44447t 0.555671 0.6667n 0.1778X 0.8889R l.OOOK
UN
43p15 1.2615 .0.04214 0.01631 0.00225 0.01276 0.00091 0.00674 0.01834 0.01175
Real and nominal
E 0:o 0.0 0.0 0.0 0.0 0.0 0.0 0.0
O.OOOOR 0.1111x 0.22227t 0.3333x 0.4444R 0.555611 0.666ln 0.7178~ 0.88897I 1.0007I
5.8764 0.09906 0.00700 0.03534 0.01820 O.OOOOO 0.00663 0.01184 0.0053 1 O.OOOOO
LY
Y3MB
Loadings 0.13051 0.00340 0.00067 0.00018 0.00005 0.00002 0.00003 0.00003 0.00001 0.00001
on jrst 0.33540 0.27609 0.02702 0.00369 0.01684 0.00098 0.00359 0.00815 0.00185 0.00009 on second
1954:1-1965:12
Yl YB
I
297
cycles
GCPI
GM1
3.2594 1.6947 2.029 1 1.0607 2.5559 0.11351 1.5863 0.64946 1.1671 0.07212
12.548 3.9126 0.33968 0.98554 0.29594 1.5281 0.45653 1.1010 0.11458 0.44461
index(C&J&J 0.26568 0.37388 0.03425 0.00154 0.01668 0.00698 0.00142 0.00743 o.ooo79 0.00040 index
([&I&)
0.0 0.0 0.0
4.6937 0.81124 0.05530
7.6910 0.60615 0.02805
6.4933 0.51875 0.72619
2
0.01972 0.00545
0.00994 0.01733
2.2555 0.54115
i:: 0.0 0.0 0.0
0.01150 0.00722 o.ooo99 0.00139 0.00390
0.00541 0.00154 0.00037 0.00364 0.00394
0.06909 0.45341 1.7226 0.20460 2.8610
3.9808 8.6200 0.47108 0.09237 1.1812 0.84251 1.2565 0.16132 1.6469 0.44448
6.7549 2.7142 4.0910 5.5044 0.0 3.1928 5.2742 2.2600 4.2177 7.0624
9.7879 5.8380 5.1006 3.1330 2.3914 4.0216 4.8745 3.1357 6.4229 6.7548
Spectral
densities
o.ooooO o.ooo9o 0.00021 o.ooooO 0.00005 0.00004 0.0 0.00002 0.00001 0.00003 Cumulative
0.88198
in business
Table 1 values for the period
parameter
Loadings O.OOOOR O.llllrr 0.2222x 0.3333n 0.4444X 0.5556x 0.6667~ 0.7778x 0.8889x 1.00072
factors
0.99072
overall
of specific
indices
0.0 0.0 0.0 0.0 0.0093 1 0.0 0.0 0.0 0.0 0.00253
0.79059 0.04444 0.00775 0.00951 o.ooo95 0.00315 0.00220 0.0 0.00360 0.0
proportion
of variance
0.99812
0.91325
([S,(o)],,)
explained 0.42203
0.43109
dis&yed under the headings of loadings. Thus, at the zero frequency, la’,,(o)12=43.915. Th e 1oa d ings for UN and LY on the second index are zero at all ten frequencies because of the restrictions that these variables are functions of the first index only. Both output and unemployment have most of their power concentrated at the low, business cycle frequencies, and at these frequencies, most of the variation in UN and LYis due to the corknon index zl. Overall, the coherences of UN and LYwith z1 are 0.882 and 0.991,
298
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Real and nominal factors
in business
cycles
respectively (table 1).6 Thus, the interpretation of z1 as a ‘real’ index that captures cyclical movements in real output is supported by the data. Most of the variation in nominal interest rates at the lowest frequency is due to variation in the nominal index. This can be seen from the large values of la&&l2 and [&@)I2 relative to /?$@I2 and lQ)l”, respectively. The overall coherence of Y3MB with the nominal index is 0.891, while the coherence with the real index is only 0.107. Similarly, the coherences of YlYB with the nominal and real indices are 0.842 and 0.071, respectively. These patterns are consistent with the findings of Nelson and Schwert (1977), Fama (1977) and Shiller (1980), among others, that price expectations played a dominant role in the determination of interest rates during the nineteenfifties and early nineteen-sixties. They also suggest that real interest rates were not constant during this period, however. The large coherences of Y3MB and YIYB with the nominal index do not, of course, necessarily imply a correspondingly large overall coherence of GCPI and zZP A portion of the variation in GCPI will be due to real disturbances, and there may be substantial errors in measurement that lead to large values of [S,(o)],, relative to the loadings. So long as measurement errors are independent of i, they will be reflected in the specific index sst and will not affect inferences about interest rates in the context of (4). In fact, at the zero frequency, where most of the power of interest rates is concentrated, the coherence of GCPZ and z2t is only 0.393. An additional 19.7% of the variation in GCPI at this frequency is explained by the real index, and the remaining 41% is due to the specific index. The overall coherences of GCPZ with z1 and z2 are 0.199 and 0.223, respectively; almost 58 percent of the variation in GCPl is independent of variation in the yields and output. Similarly, low R2’s for regression equations describing the one-month treasury bill yield were obtained by Fama (1975), Nelson and Schwert (1977) and Carlson (1977). If the low overall coherence for GCPI indicates substantial measurement error is present, then this finding has at least two important implications for previous empirical studies of business cycles. First, a common method of studying ex ante real rates is to subtract GCPI from the nominal rate and then regress this difference on lagged values of GCPZ and other variables [e.g., Shiller (1980), Mishkin (1981)]. If the measurement error in GCPZ varies systematically over time, then the coefficients in these regressions will in general be estimated inconsistently. Consequently, inferences about the behavior of ex ante real rates from such equations may be misleading. 6The zero valuesof the spectraldensity functions of the specific indices are a cokequence of a diagonal element of S,(o) getting within a distance 0.0005 of zero during the e&nation. When this happens, the parameter is fixed at 0.0005 and the estimation continues as if the parameter waszero in the population. SeeGewekeand Singleton(1980) for a discussionof the implications Of this boundary solution for the asymptotic properties of the estimators, as well as some Monte Carlo evidence that the practical consequences may be minor.
K.J.
Singleton,
Real and nominal
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299
Second, models incorporating the Lucas (1972) supply equation imply that unanticipated price movements are an important determinant of movements in output. Yet Sargent (1976) and Fair (1979) found that price surprises explained little of the variation of output. The large percentage of variation in GCPI that is independent of J+ and i, raises the possibility that price innovations. play an important role in the determination of output, but that measurement error imparted a downward bias to their estimated coefficients. The results for the period 19651 through 197612 displayed in table 2 differ from those for the earlier period in several important respects. First of Estimated Frequency
UN
Table 2 values for the period
parameter LY
Y3MB Loadings
0.00007t 0.11117r 0.2222x 0.3333n 0.44&Z 0.5556x 0.6667~ 0.777&l 0.8889x 1SKKhI
45.951 0.45621 0.03373 0.00307 0.00257 0.00471 0.00051 0.00286 0.00466 0.00003 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.00007I O.llllr 0.2222X 0.3333rr 0.444472 0.5556~ 0.6667~ 0.7778~ 0.8689x l.OOOYI
3.7837 0.11881 0.02029 0.00963 0.01157 0.00652 0.00179 0.00041 0.00317 0.01167
0.99783
index
densities
overall
index
0.01071 1.8460 1.3932 4.6370 0.48420 0.00110 6.9154 4.9713 5.6028 0.14491
12.226 5.9613 8.5615 9.1399 2.7056 1.7786 0.72923 7.0778 0.15820 2.4498
58.004 1.1070 0.53879 0.40139 0.54264 0.17809 1.7096 1.2353 0.87406 5.7796
1.5577 2.2694 2.9480 0.85946 3.5847 4.9984 1.1586 1.0104 0.44943 0.04050
([&)]12)
24.422 . 1.2698 0.46787 0.10274 0.10961 0.02534 0.00546 0.02406 0.0069 1 0.00599 indices
0.0 0.13274 0.0 0.01374 0.0 0.0 0.0 0.00703 0.0 0.00081
0.80201 0.0 0.05397 ii0659
0.99771
Gh41
([$a~)],,)
of the specijic
proportion
GCPI
1.8994 0.86879 0.04962 0.02962 0.01600 0.02316 0.00585 0.00265 0.00175 0.00018
45.535 0.76236 0.36881 0.07518 0.10251 0.03561 0.02072 0.01409 0.01034 0.00842
0.0 0.0 0.00006 0.00001 0.0 0.00001 0.00001 0.00001 0.00001 0.0 Cumulative
0.92132
on the second
0.0 0.0 0.0 0.0 0.0 0.0’ 0.0
Spectral
YIYB
19.474 0.52882 0.10691 0.02133 0.01434 0.02737 0.00628 0.00325 0.02879 0.00240
Loadings 0.00007I O.lllln 0.2222X 0.3333x 0.444472 0.5556x 0.6667~ 0.7778~ 0.8889x 1.0007t
on thefirst
0.04698 0.00333 0.00034 0.00004 0.00005 0.00001 O.OOOOO O.OOOOQ O.OOOOO 0.00001
1965:1-197612
0.01277 0.01149 0.0 0.0 0.00327 of variance 0.97055
([S,(w)],,) 8.6988 1.6271 4.8315 1.7461 3.9107 1.9677 0.0 2.4352 1.2935 1.6558
11.538 19.853 8.0042 8.3564 8.2091 4.6153 18.971 11.828 6.9903 4.3027
explained 0.77384
0.40425
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K.J.
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cycles
all, notice that the variance of GCPl increased substantially, with most of the increase in variation being associated with the nominal index. The large coherence of GCPZ with the nominal index at the lowest frequency seemed implausibly large given that during the period there were major supply shocks, droughts, etc. Therefore, I estimated a quarterly version of the model with industrial production and GCPI replaced by real GNP and the growth rate of the GNP deflator, respectively. The results for the implicit price deflator were qualitatively the same as those for GUI. Together, these findings suggest that the low frequency behavior of inflation was much more closely associated with the low frequency behavior of the neutral index than with output during this sample period. ’ Another interesting feature of the results for GCPZ is that the spectral shape of the specific index sgl is similar for the two subperiods. The specific indices seem to have behaved qualitatively and quantitatively the same during the two periods. The increased variability iof inflation is reflected in increases in the loadings of Y3MB and YZYB on the neutral index, particularly at the low frequencies. There are, however, larger percentage increases in the loadings of the nominal interest rates on the real index. Over thirty percent of the variation in Y3MB i3 now due to variation in zi, which suggests that real yields on three-month treasury bills were much more variable during this period than during Famh’s (1975) sample period. The absolute variation of Yl YB due to z1 also increased substantially from the earlier period, but only ten percent of the overall variation in the one-year bond yield is due to the real index. The relative values of the loadings for Y3MB and Yl YB suggest that real’yields on three-month bills were much more volatile than real yields on one-year bonds. The variance of GM1 also increased from the earlier period. Much of this increase can be attributed to the larger values of the loadings on the real index, especially at the low frequencies. As for the earlier period, the real index accounts for more of the variation in GM1 than the neutral index. 4. The cyclical behavior of the real and neutral components of interest rates
The spectral density matrices described in section 3 summarize the covariance structure of interest rates and output. They do not, however, provide much information about the behavior of the real and neutral components of interest rates on a month-by-month basis. In this section I describe and implement a procedure for extracting observable time series ‘A similar, but less extreme, loading pattern is obtained when GCPI is replaced by the log of the CPI (LCPI). In this level version of the model, the ratio of the loadings of LCPI on the red and neutral indices at the zero frequency is 0.488. Nevertheless, the results for the nominal interest rates in table 2 and in section 4 are qualitatively the same for the models with LCPI and GCPI.
K.J.
Singleton,
Real and nominal
factors
in business
cycles
301
representing the cumulative effects up to time t of the latent indices on the nominal interest rates. This signal extraction problem has been discussed by Geweke and Singleton (1981b) and Litterman and Sargent (1979). For this analysis, I shall use the Litterman-Sargent approach, which in the notation of section 2 can be described as follows. Let u,(L) be the jth column of the matrix of lag polynomials in (9) and rJt be the jth latent index, j= 1,2. The objective is to estimate the 6 x 6 matrices B{ in the projections Uj{L)
=
f
@X,-i
+
4,
j=l,2,
i=O
(11)
where x is the vector of observable variables in (6) and v/ is a least-squares projection error satisfying E[v/xi -J = 0, i = 0,. . ., m. Postmultiplying (11) by x,-, and taking expectations gives the least squares normal equations
ce7,~j,x(s) =izo Wx,x(s -i)>s=O,...,m,
(12)
where C,j+(~)=E(u,((L)
Thus, a consistent estimator of CJs), s=. . ., - l,O, 1,. . ., can be obtained from a consistent estimator of S,,,(w), /o( 5 7~. Since cl= [ti,, r2J is uncorrelated with ss, for all s and t, and the spectral density matrix of <, is the 2 x 2 identity matrix, the cross-spectral density matrix of x1 and a~L)5j, can be expressed as
~x.,,,c~,=a;(~,~~~Y, j= 1,2, where &Lo) is the jth
column
of A”(U) in (9). Thus, from the consistent
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cycles
estimates of /T(w), estimates of the cross-covariances of axL)~j~ and x,-,, cx,,<,(s) = (l/274 jz ZJ@“@‘eimsd~,
can be recovered. Similarly, the spectral density matrix of x implied by the model is z(i(w)A”((o)+S,(o). By inverse Fourier transforming the estimates of S,(o), a consistent estimate of &x(s) is obtained. Upon substituting these estimates into (12), the normal equations can be solved for the B{ used in the prediction of U~L)rj,. See Litterman and Sargent (1979) for additional details. Estimates of the II{, i=O, . . ., 4, were calculated using the estimates of the spectral densities in tables 1 and 2. The implied fitted values of a,,(L)<,, and a,,(L)t,, for Y3MB and YZYB during the period 195415 through 1965:12 are presented in figs. 1 and 2 [Q(&~, and a&)5,, appear in the legend as Noise 1 and Noise 2, respectively]. Note that the fitted values from (11) are functions of the current and past histories of the observed variables only. Accordingly, the relative behavior of the series in figs. 1 and 2 at each date t reflects only what was known at date t, even though the a,,(Q,)t[jt in (9) may be two-sided.
C, 1.5
ACTUAL
X
NOISE -__. :
C
NOISE -_-___. 2
-.1.5
-2
1
20
LO
60 T!KEII
80 ‘?I MChITL.IS II/
ioa
20
Fig. 1. Decomposition of Y3MB for the period 1954:5-1965:12.
140
303
K.J. Singleton, Real and nominal factors in business cycles 2.5
Legend 2
1.5
A
ACT’JAL _--
X
NOISC ! _--.
r:
NOW ----.
2
I
+5
0.5
g n
0
-.0.5
-.I
e-l.5
-2
I
20
40
IN8tkINTHS
60
I
100
1
120
140
TIME
Fig. 2. Decomposition
of Yl YE for the period 19545-1965: 12.
Perhaps the most striking feature of these figures is that the a,,(~?)<,, I = 3,4, are nearly constant, while the neutral components tract the time paths of nominal interest rates very closely. The real components of Y3MB and Yl YB moved procyclically with industrial production. The largest movements in a,,(&$,, and a,i(&)r,, occured during the 1957-1958 recession, when these series accounted for over half of the decline in nominal rates. An interesting feature of the 1955-1960 period is that the peaks and troughs in industrial production tended to occur about three months prior to the corresponding peaks and troughs in the real components of Y3MB and YIYB. This pattern suggests that expansions (contractions) in output over the business cycle are followed after several months by increases (decreases) in real interest rates. Another interesting feature of the results is the procyclical behavior of the real component of GMI. Toward the end of the 1958 recession, for example, GAJI was positive for several consecutive months, and so was the nominal was negative component of GMI. In contrast, the real component throughout much of 1958 and its lowest value was attained when industrial production reached its lowest value. This tendency for the real component of GMZ to move procyclically with LYis consistent with models that emphasize
K.J.
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Singleton, Real and nominal factors
in business
cycles
the demand side of the money market, with money supply responding passively to economic events [e.g., Sims (1980)]. While the money supply may have responded passively to real economic developments prior to 1965, it is generally agreed among economists that the period 1965 through 1976 was one of active intervention by the Federal Reserve. Therefore, it should be informative to compare the time paths of the real and neutral components of interest rates for the later period to the paths just discussed. The forecasted values of a,,(L)<,, and a,,(L)<,,, I= 3,4, are displayed in figs. 3 and 4. There are several important differences between these figures and figs. 1 and 2. First, the fluctuations in the real components of interest rates were much larger during the 1965-1976 period. As before, these series move procyclically with output. Second, the turning points in the nominal components of Y3MB and YZYB during 1970 and 1974 lagged behind the turning points in- both observed interest rates and the nominal component of GMZ. The explanation for this pattern seems to lie in the large values of GCPI during 1970 and 1974, and the fact that most of this inflation was associated with the neutral index. Apparently, expectations of inflation remained high, even after the downturn in real inter&t rates, output, and monetary growth.
2-
A
ACTUAL --
X
NOISE -_--. :
I
20
40
60 T,fJE
Fig. 3. Decompositionof
Y3MB
IN8;~NTHS
‘Ofi
120
for the period 1965:5-1976:12.
l/.0
’
K.J. Singleton, Real and nominal factors in business cycles
Legend .’ :‘
/-0
305
I
NOISC -. t -. NOLC____. 2
20
,
I
40
60
I
IN8tiONTHS
0
100
120
140
TIME
Fig. 4. Decomposition
of Yl YE for the period 1965:5-1976:12.
To shed some light on the relationship among output and monetary growth during this period, consider the episode of tight credit during 1969. GM1 declined from April, 1969 through September, 1969 and most of this decline was associated with the real index. At the outset of this decline, the real components of interest rates were rising along with output. By July, 1969, however, the real components of Y3MB and YZYB were declining. LY declined in June and July, increased slightly in August and September, and then declined steadily. Thus, the initial decline in GMZ seems to have led to a fall in both output and real interest rates. From August 1969 through February 1970, however, most of the decline in GMZ was associated with the neutral index. After a prolonged decline in GMZ, further declines were not associated with real output during this period. A similar output-money relationship emerged during’the first part of 1973-1975 period. Throughout this discussion I have been interpreting movements in the real components of Y3MB and YZYB as reflecting movements in real interest rates. The series ari(L)rr,, I= 3,4, are generally not equal to the ex ante real rates given in the respective versions of eq. (3) for Y3MB and YZYB, since r, and $: will generally be correlated at some leads and lags (see section 2), and r1 and r2 are orthogonal. Nevertheless, real interest rates and the a,,(,!,)~,,
306
K.J.
Singleton, Real and nominal factors
in business
cycles
may behave approximately the same over the business cycle, especially if Tobin effects are not too large. Therefore, it may be informative to compare at this point the behavior of the series a,,(&,,, I= 3,4, to the behavior of the ex ante real rate series constructed by others. The real components do not display the same cyclical behaviour as the ex ante real rate constructed by Elliot (1977) from a ‘neoKeynesian, monetary’ model, which is his preferred model of the several he considers. Specifically, Elliot’s real rate peaked during 1972 and then fell steadily through 1974. Also he finds that real rates are independent of current output levels. In contrast the real components of Y3MB and YlYB were increasing until late 1973 and had fallen only a small amount by the end of 1974. The ex post real rate Elliot calculates by subtracting the realized inflation rate from the nominal interest rate also does not behave like the The latter differences are not surprising to me in light of the %c&l, evidence presented in section 3 of this paper and elsewhere [e.g., Fama (1977)] suggesting that reported inflation rates are poor measures of actual inflation rates. Measurement errors in the reported values of #, imply errors in measuring ex post real rates, and it seems likely that they will also contaminate many measures of ex ante real rates. The fluctuations in Carlson’s (1977) ex ante interest rate, calculated from the Livingston data on expectations of inflation, resemble more closely the fluctuations in my real series. The local peaks in our series during 19691970 and 1973 appear to occur at about the same times, for example. However, the variation in Carlson’s series is much larger, especially during the 1954-1962 period. Finally, it is interesting to compare my real series to the series on the cost of capital calculated from individual firm data for the period 1960-1974 by Ciccolo (1975) and Tobin and Brainard (1976). In theory, their cost of capital series and the real rates on treasury bills need not be identical, because of the different risk characteristics of bonds and capital. Nevertheless, the two series should be highly correlated, since arbitrage and the systematic response of investors to new information about the economy unites all asset markets. In fact, there is a close resemblance among the series. For example, the TobinBrainard series reaches a trough in 1967, a peak in 1969, another trough in late 1971, and then it rises steadily during 1972 and 1973. A similar pattern is displayed in fig. 3. 5. Concluding remarks This paper has attempted to characterize empirically the interrelationships among monetary growth, real interest rates, and output during the postwar period. From the analysis of these series over the periods 1954-1965 and 1965-1976, several patterns of correlation emerged. First, the real
K.J.
Singleton,
Real and nominal
factors
in business cycles
307
components of interest rates, which are assumed to be functions of real interest rates, were not constant during either subperiod. The volatility of these components was greatest during the second period, when over thirty percent of the variation in the three month bill rate was due to variation in the real index. Furthermore, real interest rates appeared to move procyclically with output during both subperiods, and to exhibit turning points that lagged behind those of output by several months. These aspects of the results suggest that potentially misleading inferences about future capital formation may be drawn from studying ex post real rates calculated by subtracting the inflation rate from nominal yields on financial assets. During the 1972-1973 period, for example, such ex post real rates were declining [e.g., Elliot (1977)], while Tobin and Brainard’s (1976) cost of capital variable and the real components of interest rates calculated here were rising. Theoretical explanations for the procyclical behavior of real interest rates are provided in Barro (1980a) and Cukierman (1980). Their explanations, as well as the timing of the output-real rate relationship, warrant further investigation. Data appendix Y3MB
is the market yield on three-month
treasury bills reported
in the
Federal Reserve Bulletin. Yl YB
is the yield on one-year
treasury
bonds reported
in the U.S.
l?easury Bulletin. GCPI
GM1
LY UN
is the annualized monthly growth rate of the consumer price index for all urban consumers reported by the Bureau of Labor Statistics. The data are not seasonally adjusted. is the growth rate of currency plus demand deposits (MI) reported in the Federal Reserve Bulletin. The data are monthly averages of daily figures and they are seasonally adjusted. is the log of industrial production in manufacturing reported in the Federal Reserve Bulletin. The data are seasonally adjusted. is the unemployment rate for the civilian labor force of age sixteen years and over reported by the Bureau of Labor Statistics. The data are seasonally adjusted.
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