of Journal Monetary Economics 10 (I 982) 139-162. North-Holland Publishing Company
TRENDS AND RANDOM WALKS IN MACROECONMIC TIME SERIES &me Evidence and hqtkations Charles R. NELSON
CJhtmitp of Wahington, Seattle, WA 98195, US 4
Charles I. PLOSSER*
University of Rochester, Rochester, NY 14627, USA This paper investigates whether macroeconomic time series are better characterized as stationary fluctuations around a deterministic t&reador as non-stationary processes that have no :cndency to return to a dhrministic path. Using long historical time series for the U.S. we ale unable to reject the hypotit ls that these series are non-stationary stochastic processes with no tendency to return to a trend be. Based on these findings and an unobserved components model for output that decomposes fluctuations into a secular or growth component and a cyclical component we infer that shocks to the former, which we associate with real disturbances, contribute substantially to the variation in observed output. We conclude that macroeconomic models that focus on monetary disturbances as a source of purely transitory fluctuations may never be successful in explaining a large fraction of output variation and that stochastic variation due to real factors is an essential element of any model of macroeconomic fluctuations.
1. Introduction
It is common practice in macroeconomics to decompose real variables such as output, and sometimes nominal variables, inlo a secular or growth component and a cyclical component. In the case of output, the secuk component is viewed as being in the domain of growth theory with real . factors such as capital accumulation, population growth, and technological change as the primary determinants. The cyclical component, on tbe other hand, is assumed to be transitory (stationary) in nature with monetary and, Q a lesser extent, real factors being featured as primary causes. Since cyclical tluctuations are assumed to dissipate over time, any long-run or permanent
*WC have received helpful comments from Robert Barro, Stephen BcverKge, David Dickey, Robert King, Levis Kochin, Robert Lucas, John Makin, Robert Mendeilsohn, William Schwert, Robert Shiiicr, George Tiao, an anonymous referee, and others, but responsibility for all errors is entirely, ours. Hecjocm Kang, Garden, McDonald and Nejat Seyhun assisted in programming and computation. Nelson's participation in this research was supported by the Nation,al Science Foundation under grant SCtC-7906948 and Plosser's by the Center for Research in Government Policy and Business at the LJniversity of Roches'er.
0304--3902~82/000O-OOOO/$02~75 1982 North-Hclland 0
140
C.R. Nelson and C.i. Plosser, Trends and random walks in macroecGsomic time series
movement (non-stationh,lrity) is neclessarily attributed tc the secular component. The notion that the secular component does not fluctuate much over short periods of time, such as a year c\ara quarter, but rather moves slowly and smoothly relative to the cyclical component has led to the practice of `detrending' time series by regression on time (or*perhaps a polynomial in time). The residuals are then interpreted as the cyclical component to be explained by business cycle theory. 1 For example, Bodkin (1969), L,ucas (1973), Barro (1978), Sargem (1978), Taylor (1979), Hall (l980), and Kydland and Prescott (1980) all implicitly OT explicitly regard residuals from fitted linear or quadratic tim e trends as the relevant data. for business cycle analysis.* Secular movement, however, need not be modeled by a deterministic trend, For example, the class of integrated stochastic processes exemplified by the random walk, also exhibit secular movement but do not follow a deterministic path. If the secular movement in macroeconomic time series is of a stochastic rather than deterministic nature, then models based on time trend residuals arc misspecified.3 The types of misspecification that arise from ina,ppropriate detrending can be illustrated by considering the properties of residuals from a regression of a random walk on time. These properties are investigated in recent papers by Chan, Hayya and Ord (1977) and Nelson and Kang (1981). The autocorrelation function of the residuals is shown to be a statistical artifact in the sense that it is determined entirely by sample size and it implies strong positive autocorrelation at low lags with pseudo-periodic behavior at long lags.4 Empirical investigations of output fluctuations that do not consider the: possible source of this autocorrelation might be led to over-estimate both the persistence and variance of the business cycle. Conversely, the importance of real factors that influence the secular component would be under-estimsltt 4 In this paper we investigate whether macroeconomic times .series act. consistent with the time trend decomposition usually employed. Section 2 discusses the statistical issues involved in t,esting for deterministic trends and section 3 presents the results of formal and informal tests using long historical time series for the U.S. We are unable to reject the hypothesis that `Or equivalently, time is included as an explanatory variable. 2Btxns and Mitchell (1946):in their pioneering empirical investigation of business cycles, were concerned with the method of trend removal and went to great lengths to justify their procedures (see pp. 37-41, and ch. 7). 3HaU's(1980) use of the time trend model for real GNF is particularly pw::ling since in previous work @Iall (1978)] he argues that aggregate consumption behaves like a random waik. Without some rather implausible restrictions on the ether components of GNP, aggregate GNP wiii then incUe random walk characteristics and linear d&rending is likely to be inappropriate. 41t is interesting to note that McChlloch (1975) finds evidence of periodicity in logs of real income investment, and consumption after fitting a linear trend, but does not find ~riod~~jty in their fixi differences.
C.R. Nelson and C.I. Plosser, Rends and random walks in macroeconmic time series
141
these series are non-stationary stochastic processes with no tendency to return to a trend line. The implications of this finding are explored in sections 4 and 5. Assuming that any stochastic fluctuations in output of a permanent variety must be associated with secular movements, and thus real factors, the evidence presented in section 3 leads us to the inference that (i) real shocks associated with the secular component contribute substantially to the variation in observed output, and (ii) either these shocks are correlated with the innovations in the cyclical component or the secular component contains transitory fluctuations (or both). We conclude that macroeconomic models that focus on monetary disturbances as a source of purely transitory (stationary) fluctuations may never be successful in explaining a very lsrge fraction of output fluctuations and that stochastic variation due to real factors is an essential element of .any model of economic fluctuations. Some recent efforts in this direction inJude the equilibrium stochastic growth models studied by Black (1979), Lzjng and Plosser (1380), King and Plosser (1981) and Kydland and Prescott (`.981). 2. Statistical background The basic statistical issue is the appropriate representation of nonstationarity in economic time series. We are primarily concerned with ilonstationarity in the mean of the series. Such behavior implies that the series lacks a fixed long-term mean, or put positively, has a tendency to move farther away Tram any given initial state as time goes on. We consider :npo fundamentally diffe,en: classes of non-stationary processes as aitermtive hypotheses. The first class of processes consists of thaw that can be expressed as a deterministic function of time, called a trend, plus a stationary stochasrlc process wit!1 mean zero. We refer to these m trend-stationary ." processes. The tendency of economic time series to , rS) exhibit variation that increases in mean and dispersion in proportion to &solute level motivates the transformation to natural logs and the ~~su~~ptio~ that trends are linear in the transformed data. We also assume that the deviations from trend have a representation as a stationary and ~~~v~rt~b~~ ARMA process. Denoting the natural logs of the series by z, and t.he deviations from trend by c,, the linear TS class has the form
4(fI)c, = O(L)u,; w i.i.d.(O, u, rr,"), where 61and p arc fixed parameters, L is the lag operator, and &k) and 8(1-J arc p~ly~~ornia~s in L ihat satisfy the conditions for stationarity and invertibj~ity~ ental Merminism c'^the process is captured in the properties The fun
142
CR. Nelson and C.I. Plosser, T&endsand random walks in macroeconomic time x&s
of k!ng-term forecasts and uE:certainty around such forecasts. While autocorrelation in c, can be exploited in short-term forecasting, it is clear that over long horizons the only information about a future z is its mean (CII +/It). Therefore neither current nor past events will alter long-term expectations. Further, the long-term forecast error must be c which has fir&e variance. Thus uncertainty is bounded, even in the indefinitely distant future. The second class of non-saationary process considered in this paper is that class for which first or higher order differences is a stationery and kvertible ARMA process (DS processes). The counterpart of the linear TS process is the first-order DS process in natural logs written as (1 -L)z,=/C&, (2 6(L)d,= A(L)u,; m i.i.d.(O, U, ai), where (1 -L) is the difference operator and 6(L) and A(L) are polynomials satisfying the stationarity and invertibility conditions. The simplest member of the class is the random walk for which the changes are serially uncarrelated, that is rf,=u,. To see the fundamental difference betweell the TS and DS classes it is useful to express z, as the value at some reference point in the past, time zero;, plus all subsequent changes,
z,=z,+pt+ i j=l
d,.
(3)
Eqs. (3) and (1) indicate that the two types of processes can be written as a linear function of time: plus the deviation from it. The intercept in (1), however, is a fixed parameter while in (3) it is a fimction of historical events, and the deviations from trend in (1) are stationary while in (3) they are accumulations of stationary changes. The accumulation in (3) is not stationary but rather its variance increases without bound as t gets large. It is not difficult to see that the long-term forecast of a DS process will nlwnys be influenced by historical events and the variance of the forecast error will increase without bound.. The DS class is purely stochastic in nature while the TS class is fundamentally deterministic. When one assumes the latter class is appropriate one is implicitly bounding uncertainty and greatly restricting the relevanfce of the past to the future. Empirical tests may be quite sensitive to this distinction. For example, Shiller (1979) finds that the variance of holding period returns on long-term bonds is larger than would be consistent with a particuiar efficient markets (rational expectations) version of term structure
C.R. Nelson and C.I. Pfosser. `l8end.vand random wulks in macroeconomic time series
143
theory if short rates are assumed to be stationary a: q?lnd a fixed mean. They are not too large, however, if short-term rates are assumed stationary only after differencing. The crucial factor causing the discrepzqcy is that under the DS assumption any movement in short rates will have some impact on the long-term ex tions embodied in long rates9 but will have very little impact under the TS assumption.' The fundamental difference between the two classes of processes can also be expressed in terms of the roots of the AR and MA polynnmials. If we first-difkrmce the finear TS model the result is
where &IL==1) k a constant obtained by evaluating the poivnonkl 4(L) at L= 1. Eq. (4) indicates that a unit root will be present in the MA part of the ARMA process describing the first differences [( 1- L)z,J. The simplest example would be the case of a linear tr :nd ply random noise (c, = u,). The presence of the unit root implies that tire process is not invertible; that is, it does not have a convergent autoregressive representation. Recall that the first differences of a DS process are both stationary and invertible. Correspondingly, when we write the DS in terms of levels we obtain from (3 s(L)(l -L)z,=/?4L= l)+II(L)u,, (9 which contains a unit root in the AR polynomial. It wouid appear then that if a series is generated by a member of the linear TS subclass we should fail to reject the hypothesis of a unit MA root in the PaRMA model for its !kt difference, and if i,: is generated by a member of the first order DS subclass we should fail to reject the hypothesis of a unit AR root in the ARMA model for its IevekP Unfartunateiy, the standard asymptotic theory developed for stationary and invertible ARMA models is not valid for testing t.he hypotheses that either polynomial contains a unit root. To get some idea of the problem, consider the simplest version of (5) where the null hypothesis is that t, is a random walk with drift,
so p= 1 is the hypothesis we wish to test. The standard expression for the `In B sutiluent pwpr !P&r (1981) finds the vatian= of linearly d&ended stock returns is
sitnilap-ly CXCX&Wif dividennb BID alao as~med to be stationary around a linear trend (both but he does not report the impact of the I'S assumy;tion on the vllriables d&ted asld in lo results.. gPiex-c (1975) discus H technique for distinguishing between deterministic anA stochastrc non-stationarity by inspeccxtionof snmpk autocorrelation TuncGons.
144
C.R. Nelson and C.I. Plosser. l'k.nds and random walks in mucroecono~ ric time series
large sample variance of the least square estimator $ is [(I -p2)/7'l which would be zero under the null hypothesis. The true variance of course is not zero; the problem is that the conventional asymptotic theory is inappropriate in this case. Dickey (1976) and Ful!er (1976) develop the limiting distribution of j! and the conventionally calcuiated least squares t-stztistic, which we denote z for the null llzypothesis p= 1, and tabulate the distributions. They demonstrate that if p=O then the distribution of b is biased towards zero and skewed to the left, that is towards stationarity.' In addition, Dickey and Fuller provide a set of results that allow us to test the DS hypothesis against the TS hypothesis as long as we are willing to assume that only AR terms are required to obitain satisfactory representations. Tile szategy is to embed both hypotheses in a common model. The simpleiqt alternatives are a TS process with first order i\R deviations and a rarrdom walk (DS procers) with drift which are both special cases of
z, = o!+ pt + *Q/( - c/L), 1
or cqu.ivalently, after multiplication by (I- c#L),of
If the TS hypothesis is correct then 141 1. if the DS hypothesis is correct c then C#J I and (6) reduces to = z,=z,1
-I-b+u,.
It would appear then that one would want to run the regression
z,=p++z~-~+yt+u~
(7)
and test the null hypothesis y = 1, y q which 1s equivalent to $ = 1 in (6). =0 Under this null hypori&.- the usual t-ratios are not t-distributed but Dickey and Fuller provide tabuhtions of the distribution of the t-ratio for JJ again denoted T, for testing the null hypothesis cl= 1.' Dickey and Fuller (1979)
-The proMem of testi-lg for unit roots in MA polynomials is more discouraging. MA PIare only idntified under the restriction that the roots lie on or outside the unit circle; therefore'estimates will be bounded away from the unit root. Plosser and Schwert (1977) have demonstrated in the In&order MA case that application of r-tests o- likelihood rrttio tests, which would be appropriate for null hypotheses within the invertrbrhty region, lead to rcjeclinn in the vast majority of `instanceswhen the null hypothesis of a unit roct is true. `They do not develop z statistic for the+,t test or for y alone. H %wever, test on p alone is a sufficient given that. we do not consider a pr&xsswith p" 1 and y#@.a~ part of the `model spa=. A process with p= 1 and y#O would be one in which'differences in logs (rates of change) followed a deterministic path, implying ever increasing (7~0) or even decreasing (~-CO) rates of change. We rule out this kind of behavior in economx time s&z., on a priori grounds. Similarly we rule lout quadratic or higher degree time polynomial trends,
state that the distributions of /; and a) is not .-iffectedby whether ~1is zero or not, but T@) would be normal if y #O (the case we have excluded). To illustr;ite these properties we have conducted a Monte Car!0 experiment which is summarized in table 1. The sample length is 1tX observations with !500 on% In I the ting process is a random walk with a ro do@ in Case II them is non-zert) drift. As indicated by Dickey and u&r, customary testing procedm rejtit the null in favor of stationarity far too often in both cases. The distribution of /; is centered around 0.9 instead of 1.0 in both cases. In :.ddition, it is clear that the t-ratio for testing the hypothesis that y=O [denoted t(f)], is biased towards indicating a trend. Thus, standard t&kg procedures are strongly biased towards finding stationarity around a trend; they tend to reject the hypothesis p= 1 when it is true in favor of p-c 1 and "tend to reject the hypothesis y =0 when it is true. ReWs of FuRer (1976) allow us to use the distributions of fi xrd r(b) in higher order cafe In generai we want to distinguish between a TS process with AR component of crder k and a DS procx-s with AR representation of order (k- 1). In levels we write the DS model as an AR of order k with one
Table 1 Srunpling disttibusbns for tire estimators in regression model (7),1 :,=~+~z,_,+~r+Ii,, CW I: /r=O, p= 1.Q. y=o. f"l.0, i'"0. Stucientized range Percent rejections Case Ila c("1.0, -PMefin G&Q? f $1 c
;1,1,
Standard deviation
SLewness .WW
Excess Kurtosis PsP-^-PI.-v
-
-0*049 --WI7 0.895 =`T" \ - O.ao 2.26 -0.093
0x5 1.84 8.064
a013 WI63
0.879 0.490 -1.11 - 0.212 0.237 1).124
-0.615 4.12
1 K-3
13.?1 s.C.7
6.15 7.91 7.4 I
x.8 65.2 35.6
0.6SS 2.27 - O.BU7
t(3,
I.84
5.08
146
C.R. Nelson and C.I. Plosser, 7fends and randomwalks in macroeconomic time setie;r
root on ahe uni: circle. The alternatives are imbedded in the model
where c +i= 1 and y =0 if the DS hypothesis is true. The terms in lagged Z'S can be rearranged in the format
+-+(--(pJ(zt++:
-z,-&+p+yt+jft.
Pb
Fuller (1976, p. 374) shows that if the coefficient of z,- 1 is unity in (9), as it would be under the DS hypothesis, then the leasl: squares estimator of that coeffxcient has the same large sample distribution as fi in model (7), and similarly for its r-ratio.g We note that in the Dickey-Fuller procedure the null hypothesis is the DS specification while the alternative is the TS specification. As usual, acceptance of the null hypothesis is not disproof of the alternative hypothesis. It is important therefore to have a check on the power of the test. To provide this check we include in our data set a series that on a priori grounds is likely to be a member of the TS class (albeit with zero slope) rather than the DS class. namely the unemployment rate. 3. Analysis of U.S. historical data We turn now to the analysis of the U.S. historical time series listed in table 7 which include measures of output, spending, money, prices, and interest ..I rates.`* The data are annual, generally averages for the year, with start' dates from 1860 to 1909 and ending in 1970 in all cases. All series except bond yield are transformed to natural logs. Sample autocorrelations of the levels are tabulated in table 2 a typically start at around 0.96 at lag one and decay slowly with increasi Jag. This is consistent with the behavior of sample autocorrelntions from a random walk as indicated by the values calculated from 8 formula due to Wichern {1973). One exception to this characterization is the ~nem~l~yrn~nt rate which exhibits mare rapid decay as would be expected of a stationary
`Dickey and Futler (1981)have recently extended their analysis to likelihood raticl statistics. `*Data cources are as Iollows. GNP setics, industrial production, employment 1929-1970, unemployment rate, consumer prices, snd stock prices: &on& %rn Eeoaamic Growth, (1973). X1.`, money stock, and bond yield: Historica Statistics of the U.S., Colonid Times to &YQ, ~CS, (1975). Velocity: Friedman and Schwartz (1963) with revisions kindly provided by Anna Schwartz. Employment 189@-1928: Lebergott (1964). Data files available from the authors upon request..
CR.
N&m
and Cl.
Phsm,
Trends aJ
run&m
walks in macroeconomic
timer series
147
Table 2 Sample autocorrelations of the natural logs of annul -- Sample autocorrelations data. -___
Period
Randomwlrlkb Ti
100
b
0.95
0.?6
0.90 0.91 090 0.89 0.88 0.94 0.91 0.47 0.93 0.92 0.91 0.92 009 0.92 0.72 0.90
0.85 0.86 0.84 0.83 0.81 0.90 0.86 0.32 0.89 0.87 0.86 0.88 0.89 0.88 0.60 0.85
0.81 0.82 0.79 0.77 0.75 0.87 0.81 0.17 0.84 0.84 0.82 0.84 0.85 0.85 0.52 0.79
l
0.76 0.77 0.74 0.72 0.70 0.84 0.76 0.04 0.80 0.81 0.77 0.80 0.81 0.81 0.46 0.75
0.70 0.73 0.69 0.67 0.65 0.81 0.71 -0.01 0.76 0.77 0.73 0.75 0.77 0.79 0.40 0.71
Money stock vtkcity Bond yield Common stock pric4zs
1909-1970 1909-1970 f909-fW0 t 86@-19'90 1890-1970 l&9@-1970 1889-1970 MO-197Q MB-1970 f9W-1970 1889-1970 1869-1970 19oQ-1970 1871-1970
100 62 62 62 111 81 81 82 111 :1 82 102 ?I 100
6) 3s 0.95 0.95 0.97 0.96 0.75 0.96 0.96 0.96 0.96 0.96 0.96 0.84 0.96
`The natural logs of ati the data are used except for the bond yield. T is the sample size md ri is the itb order autocorrefation coefficient. The large sample standard error under th: null hypothesis of no autocorrelation is T-+ or roughly 0.11 for series of the length considered Ilere. Computed by the authors from the ap,woximation due to Wichern (1973).
series. Satrdple autocorrelations of first differences are presented in tattle 3 and in each instance are positive and significant at lag one, but in r.lany cases are not significant at longer lags. One explanation of positive autocorrelation at lag one only is that the annual series are constructed by averaging shorter interval observations which themselves are generated by a DS process. Working (1960) demonstrates this e%xt of time gation on a random walk and shows that positive autocorrelation at e approaches -1-0.25as the number of egatcd becomes large. Tiao (1972) shows ~~rv~tio~s hein neralizes to the temporal aggregation of any DS that the Working of serial dependence in the underlying ~U.-XMis ion, in this case one year. real GNP, nominal GNP, real per capita GNP, ~I~~~~yrn~nt, nominal and real wages, and common stock prices display positive autocorrelation at lag ore only which is characteristic of ~rst~~~der MA processes. This representation of the data is inconsiymt with the TS model, The only TS process that gives rise to autocorrelation only at lag one is the case of serially random deviations around the trend. The value e a~~t~c~~r~~ation, however, would be -0.50. To salvage a TS
148
CR.
N&on
and Cl. Plosser, *,Vends and random
walks in macroeconomic
timp series
C.R. Nelson mui Cl. Plosser, Tknds and random w~&.sin macroeconomic
time series
149
representation for these series we would steed to hypothesize the presence of an autoregretive component in the deviations from trend that has a root close enough to unity to obscure the effect of diffcrencing on the autocorrelation structure. For example, suppose the deviations from trend were generated by the ARMA process
so that the Arst differen= oft, have the representation
(11)
The ratio (I- L)l(l - t#tL) has the expansion [ 1- (1 - +)L - @(1- $I)@- . . . 1, which may be difficult to distinguish empirically from unity if t#~is close to one, leaving the ap+arance of a first order MA process (1 - BL) for ((1
-L&J.
The GNP deflator, consumer prices, the money stock, and the bond yield, exhibit more persistent autocorrelation in first differences. None, however, shows evidence of being generated from a process containing MA terms with a unit root or AR terms arising from inversion of such an MA term as one would expect to find in a TS prooess that has been differenced. The presence of strong positive autocorrelation in deviations from trend (or from a. fixed mean) may again be the explanation. The conclusion we are pointed toward is that if these series do belong to the TS class, then the deviations from trend must be sufficiently autocorrelated to make it diffcult to distinguish them from the DS class on the basis of sample autocorrdations. The evidence against the TS represent? tion from levels and difWences is reinfor& by the sample autocorrelations ?f the deviations from fitted trend lines printed in table 4, The pattern i.; strikingly similar across series (except fat the unemployment rate) startir$ at about 0.9 at lag one and ly exponentially, The first two lines in table 4 give the ted s~nple autocorrelations for deviations of random walks of 61 and obmrvarions from a fitted trend line [l?elson and Kang (1931): and the data wlfh a simple form of the DS alss show that these results are rather lation in first t tiffetencos, s,uch as would be present in a tinre a rqated DS pl ocess.
"Tk ~p~~~x~~&~ tcci sample autocorrelations are based on the ratios of expeczed t Simulation cxphmnts by Nels m and Kang (1981) fur 100 sample aWmvariances. ohmWim~ suapst that tl,c cxoct expected sample autocorrel hns are smaller. At lag me the mean sample autmorrelatht was -3.88 instead of 0.91 and at Ia& 6 it was 0 43 instead of QSI.
3.50
C.R. N; ~sowt Cl. Plosser, Eznds and random walks in Woeconomic timeseries and
Table 4 _---Sample autocorrelatr,lns of the deviations from the time trend. Sample autocorrelations -RXiOd T r1 T3 "4 r2 61 `, 101 62 62 62 111 81 81 82 111 71 71 82 102 71 100
.-
Series
r5
?,
De&ended nandom walkb Real GNP Nominal GNP Real per capita GNP Industrial production Employment LJnemployment rate GNP dcftator Consumer prices Wag& Real wages Money stock Velocity Bond yield Common stock prices
_------
1909-1970
1909-1970 1909-1970 1860-1970 1890-1970 1890-1970 1889-1970 1860-1970 19o(F1970 1900-1970 1889-1970 1869-1970 MO-1970 1871-1970
0.85 0.91 0.87 0.93 0.87 0.84 0.89 0.75 0.92 0.97 0.93 0.87 0.95 0.91 0.85 0.90
0.71 0.82 0.66 0.79 0.65 0.67 0.71 0.46 0.81 0.91 0.81 0.69 0.83 0.81 0.73 0.76
0.58 0.74 0.46 0.65 0.43 0.53 0.55 0.30 (3.67 0.84 0.67 0.52 0.69 0.72 0.62 0.64
t.I.47 0.66 0.26 0.52 0.24 0.40 0.39 0.15 0.54 0.78 0.54 0.38 0.53 0.65 0.55 0.53
0.36 0.27 0.58 0.51 0.19 0.07 0.43 0.05 0.11 0.04 0.30 0.28 0.25 0.17 0.03 -0.01 0.4,2 0.30 0.71 0.63 0.42 0.31 0.26 0.19 0.37 0.21 0.59 0.56 0.49 0.43 3.46 0.43
"The data are residuis from linear least squares regression of the logs of the se+s (except the bond yield) on time. See footnote for tiole 3. bApproximate expected sample autocorrelations based UIINelson and Kang (1981).
To carry out the formal tests of Dickey and Fuller we must estimate regressions of the iorm o: eq. (9) which may be rewritten as
To s@y the marSmum lag k we consider both the values that would be suggested by the .:autocorrelations of first diffcences ard by the partial zutccorrela.tions of the deviations from trend, ,In cases where MA models fol Yist differences seem appropriate we fit AR approximations. In general the latter procedure indicates higher order Gtnregressions and our rule is to utilize the higher order models on the grounds that leaving out relevant terms might bias our results but inclusion of irrelevant onc;s would onfy reduce efficiency. 2 l The results of th#zseregressions are reported in table 5. Recall that we are interested in testin;g whether pi differs from unity. The values of fi range
`*We do not report. the sample partial autocorrelations of the deviations from trend. However, the pattern is very similar across almost .all of the se&s; a sharp cut-off &et lag two where there is negatiie and gezkrglly si~&itiiit' ,p8Ftial.`aiitdcu~lation, ,TItiyse eh@racteristics suggest second-order AR representations with comfjlex rootri, %d th&ef&e I'pseudoperiiidic behavior in the trend deviatious, again a property of de&ended randum walks melson and Yang (1981)-j.
Table 5 Tests for autoregressive unit roots'
2,=~-~~?+~,2,_,~~,(2,-,-2,_,)+~~~+~~(2,_~+*-2,_~+~,.
62 62 0.825 0.899 0.818 0.835 0.861 0.706 0.915 0.986 0.910 0.831 0.916 0.941 1.03 0.913 - 2.05 0.158 - 3.55. -2.52 - 1.97 -2.09 -3.04 - 3.08 - 1.66 0.686 0.407 0.046 0.042 0.060 ,0.034 0.047 0.067 0.283 0.02 - 0.03 -0.06 0.00 - 0.01 0.03 0.11 - 0.02 0.20 - 2.53 -2.66 0.03 0.10 0.097 0.035 - 3.04 0.059 - GO2 - 0.02 0.03 3.05 4.32 2.68 2.81 2.55 1.76 2.30 3.10 3.52 0.99 -0.95
-0.00 0.004
2 2 3.01 244 2.54
0.819 1.06
3.03 237
OANM 0.006
3.03 2.34
-2.99 - 2.32
0.058 0.087
Real GNP Nominal G?w Real per capita GNP Industriat. 0.007 0.002 0.002 0.001 0.004 0.004 0.005 -o.@Jo 0.003 0.003 2.37 - 0.23 2.65 2.84 2.30 3-14 3.03 - 0.65 1.75
62
2
1.28
productioIl
-pkrymant Unemproarcnt
111 81
6 3
0.103 1.42
81 82 111 71 71 82 102 71 2.02
4 2 4 3 2 2 1 3
0.513 0.2Ml 0.090 0.466 0.437 c1.133 0.052 -0.186
rate GNP deflator (IzcmUnter prices Wages R&V Money stock velocity interest rate common stock pl+ceS
100
3
0.481
=z, represents the natural logs of annual data except for the bond yield. t(G) and t(f) are the ratios of the OLS estm.ates of jr and 7 tu their respective standard errors. I&) is the ratio of I, - 1 to its standard error. &I) is the standard error of the regression and r, is the first-order autocorrelation coefficient of the &duals. The values of IQ?,) denoted by an (*) are smaller than the 0.05 one tail critical value of the distribution ofT($,) ald &G\arij [or ji Ir shoutd also be noted that r(fi) and t(j) are not distributed as normal random variables.
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C.,R. Nelson and C.I. Pksser, Trends and random walks in mtzcroecu;romic time series
from a low of 0.706 for the unemployment rate to a high of 1.03 for the bond yield. The majority of the estimates fall in the range 0.85 to 0.95 which is quite consistent with the mean of 0.900 and standard deviation of 0.054 reported in table 1 for realizations of a random walk. Also of interest is that all but two of the t-statistics for the hypothesis p1 = 1 [i.e., @)] are significant by conven'tional standards. However, we know from the sampling experiments in table 1 that z(fiJ has a of mean about - 2.2 ~=ier the null hypothesis that pz iis unity. Using the distributions tabuated by Fuller (1976), only the uuemgloyment rate exhibits ~"r value of z(b) below tkre 0.05 critical value of - 3.45 for samples sizes of 100. In this case, fi is also smaller than the 0.05 critical value given by Fuller. Moreover, there is no evidence from this regression that the slope is non-zero and we conclude that the series is well described as a stationary process. Ta sum up, the evidence we have presented is consistent with the DS representation of non-stationaruty in economic time series, l 3 We recognize that none of the tests presented, formal and informal, can have power against a TS alternative with an _ARroot arbitrarily close to unity. However, if we are observing stationary deviations from linear trends in these series then the tendency to return to the trend line must be so weak as to avoid detection Izvenin samples as long as sixty years to over a century. 4. Stodstic representationsof the secular component
Our tests in section 3 suggest that economic time series do no: %"I a'nim _ - ,r.r&a.r detemlinistic time trends but contain stochastic trends characteristic of the DS class of processes. To investigate the implications of this finding it is useful to focus on the behavior of output. Pursuing the decomposition discussed earlier we assume that actual jutput (presumably logged) can be viewed as the sum of a srcular or growth component, Jli, and a cyclical component, c,. If the cyclical tzomponent is assumed to be transitory {stationary), then any underlying non-stationarity in output must be attributable to the secula;* component. Thus, if actual output can be viewed
'3Tbe contrasting implications of the TS and IX models for long-run uncertainty can illustrated by real per capita GNP. &,der the TS hypothesis, uncertainty about future V&.W 1s founded by the marginal standard deviation of fluctuations around a linear trend which is estimated to be 0.133 (in natural logs) over the sample period. According to the DS model for this series, however, the standard deviation of forecast errors is given by SD[e,(k)]= O.O62[1+ -1)1.7ilJ*, where e#) denotes the forecast error for k years in tke future. This standar deviation exceeds 0.133 for any more than four years in the future and obviously grows without bound but at .a decreasing rate. Taking intt, accotni an estimated mean growth rate of the lower point of a 95:/, confidence intetvs.1 reaches its minimum when the forecast hori 24 yearb. At a horizon of 24 years the standard deviation is 0.39 compared with amam mean qowth of 0.38. Thus the possibility that ,ictuar real per ~a~t~~ ~v~ul~~ only show no not increase 24 years hence but decline by abo 3: 33:`sfrom its torrent lever is act exclu interval.
C.R. Nelson ad C.I. f'losser, De;::% and random walks in macroeconomic time .series
IS3
as belonging to the DS class, then so must the secular component. This decomposition of output can be expressed as
(33) where Y;= (1 -. L)- `@(Z&I, (I- L)$ = f$!Ju,, and c, = I,&!&,. The (possibly or infinite order) polynomials e(L) and I&L) are assumed to satisfy the conditions for statior.arity and invertibility and ur and U, are mean zero serially uncorrelated random variables. Eq. (13) assigns the non-stationarity of y to y' through the factor (t --Q-l. (Also note that we are ignoring any drift in y for convenience.) Separation of the secular component from observed data may be thought of as a problem in signal extraction when only information in the otJ;erved series itself is used, or it may be cast as a regression problem w&n determinants of the growth process are regarded as known and observable. 4.1. Regression srrat&es Perhaps the ideaf nethod of dealing with non-stationarity in output (i.e., growth or secular n I ,ements) is to include the variables in a regression that would account fb&such behavior. For example, Perloff and Wachter ( 1979) among many others, fir real GNP to labor, capital, and energy as inputs in a translog production function and represent technological change as a time trend. The first-order autocorrelation coefficient of the residuals, however, is reported to be 0.881, roughly the value expected from regression of a random walk on time given the number of observations in question. It would seem then that measured input and time trend variables may not adequately account for the growth component in real GNP.r4 Another regression strategy is to work in per capita values under the assumption that population is the primary source of non-stationarity. We can reject this strat based on our results in section 3 that indicate per capita real GNP s to the DS class. Thus, using observable variables to account r'or also ms unsatisfactory since neither factor inputs nor growth components p~pul~t~~n seem to 8 and direct measures of teshnology are not readily available. Ls
ert (IWO) ~~~~~~~this
issue and others that mult be cmsidered when N&I 89 those cstimttxl by PerloB lrnd Wachter (1YPl).
the iX nature of technological change is evident in the c:arly empirical of technolagy calculated by Solow (1957) for the period 1909-1949 (19Sg The Ilog of Solow's A(t) variable exhibits Me a~tocorrelation in 8.8) and $ and it8 r-ratio for regression (12) are k$ th close to their expected v&es. under the: F~~i~~~~~~~~~ ciistribution, s~l~e~tin~ that the stock 0"'tecbnoiogy is well ch~r~cte~~~~das a r~~~~r
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and random walks in macroecmmnic
tim
series
4.2. Signal extraction and an unobseruable components model of output
Signal extraction procedures imply, or are implied by, some model of the underlying compozent structure of the series. Therefore, it seems that prior to adoption of an unobserved components model, it should be investigated for consistency with the data, or, perhaps preferably, an attempt should be made to identify a class of models from the observed sample autocorrelations. The classic example of signal extraction in economics is the permanent I6 income model of Friedman (195'7). One version measures permanent income as an exponentially weighted average of past observed incomes. Muth (1968) shows that an optimal estimate of permanent income has that form if permanent income follows a random walk and transitory income is serially random and independent of changes in permanent income. The Friedman/Muth permanent income model may be written as a special case of ii23 with B(L)=$(L)= 1 Yt=fi+% %=fi-, +o,, (14)
where Y; is now the permanent component, generated by a random walk with innovations u,, and u, is the transitory component, a purely random series that is independent of 0,. The first differences are the stationary process
(.1-~)Y,=~,+w-~,-l,
W)
illustrating the general fact that differencing does not `remove the trend' since ;he innovation ;:I the permanent component, u,, is part of ;he first diflfcrenee. The first differences are autocorrelated at lag one only with coefficient + Pl = - a,2/(a,2 2c+), W)
which is confined to the range -0.05 r;pl SO and depends on the relative variances of u and U. Apparently, this model cannot account for tie positive autocorrelation at lag one only observed in the first difference of the historical series studied in section 3. In general, if an unobserved components version of (13) is restricted I( priori by assuming that (i) y; is a random walk [i.e., 8(L)= 13 and (ii) 11, and u, are independent, then the parameters of tire unobserved components model will be identified. This is clearly the case for the: permanant income model
`?For a genera1 discirssion signalextraction wonamictime setiias Piano 11978) of in SC* and Me&we, @ether, and Carvalho j1979). SOWreeerltexamples of ignat cxtrwtio y techniques applied to unobserved componects models are Beveridge and Nelson (1S1) rtardWedrick rend Prescott (1980).
LA. Nelson and C.1. Plosser, llends
and random w&s
in macroeconomic
time serif3
155
since at is computable from the autocovariance of the first differences at lag one [the numerator of (1611 and CF~ from the variance of the first differences [the denominator of (16)] and the computed value of at. If the cyclical or stationary component of (13) has the MA representation $@)u, and is of order q, then the first diBerence will be
witb non-zero autocovariances through lag (q c 1). The value of (q + I) can in principle be %ferred from a realization of y. There are then (q-t 2) parameters to solved For from the (q-f-2) autocovariance relations implied by (17), using values for the autocovariances computed from the data." Ii is clear from our discussion, however, that a decomposition satisfying both the above restrictions is not always feasible. The simplest example is a process with positive autocorrelation in first differences at lag one only. Eq. (17) implies that the Friedman/Muth model is the only linear model that satisfies both restrictions and leads to non-zero autocorrelation at lag one only. However, it is unable to account for positive autocorrelation at lag one only. To do so we must relax either the assumption that jj is a random walk (i.e., containing no transitory, only permanent movements) or the assumption that tb and u are independent. In general, if either of these assumptions is relaxed the parameters of the unobserved compnPc;?ts model are not identified.
Nevertheless, the assumption that the cyclical component is stationary combined with the observation that autocorrelations in the first difirences of output are positive at lag one and zero elsewhere are sufficient to `rnpl/ that the variation in actual output changes is dominated by changes 111 secular component j& rather than the cyclical component c,.
The above proposition can be demonstrated by considering first differences of (13)
of first-order autocorrelation only in (I- L)y, implies (barrin: relations) that 8(L) is first-order and e(L) is zero. order so that we can wrile
17
~~~~rn~n~ i
to the caw where the stationary comyorcent includes
AR
of the first difference;; for !a$s greater than (q-+ 1) using ~ul~~~W~l~e~equations. We note, however, that we do not have a formal proof that the nun-linear autcxovariarnce ec;uations will always have a solution or unique soMicra in tekms of inviable values af the +`s_
tams sixw they may
the autacsvariances
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C.R. Nelson and C.I. Plosser, Trendsand random walksin macroeconomic time series
with ItI1 1 being required by invertibility. While ut and vt may be < contemporaneously correlated, lagged cross-correlations would imply higher than first-order autocorrelation in (1 -LJy, and therefore are ruled out. The au,ltocovariance of output changes at lag one is therefore
y*=ea,2-(i--e)u
;-u;,
where oUV the contemporaneous covariance between u and D, Note that y1 is consists of the autocovariance of the change in the secular component, j, at lag Q39, I%:, the sum of the cross-covariances at iag one, --(l -6)~,, , and the zctncovariance of the char,,:: is the cyclical component, c, at lag one, - 0,2, which is necessarily negative. The faGlors that would account for the positive value of yt we observe ~xt:therefore (1) a positive value of 8 (positive autocorrelation in first differences of the secular component) combined with a suffrcientky large value of ai, and/or (2) a sufficiently large negative value of the covariiance 6," which also puts a lower bound on a: due to the farnihar inequality oUgU2 la,,l. We now prove that if y1>O then 0: >c:. Since the value of 0," is unknown, consider first the case @,,&O. For y1 ~0 and g,,z 0, eq. (20) implies that 6 > 0, i.e., thz secular component must be positively autocorrelated. Given this, eq. (20) also implies
using the fact t.hat 0x8 < 1. The other possible case is rrUv Using the fact ~0. that c,c,~ ~u,,~I, have we
Factoring the first expression yields
and hence both factors must be positrve or negative. If they are both posittve then the second factor gives us cr,> CT,. Note that there is nothing in this case to prevent Q from being negative since if the first factor is positive we have only that 18~ -(a&J. If both factors are negative then the first factor would imply 84 but also that g,> -Q%, > g, (again using the fact that 0~: fk I), however, the second facictorwould imply tr, CC,` thus leading to ;% contradiction that rules out negative factors. We conclude therefore that the standard deviation of innovations in the secular or growth component is larger than the standard deviation of innovations in the cyclical component. `Wecari now use these results to obtain a plausible range of values of ~,,/pr, ~Cuderalternative assumptions. Consider first the case q,,=O, so that the growth and cyclical components are uncorklated. From (119)and (20) it is
CR. Nelson and Cl. Plocser, Tknds and random w&s in macroeconomic time s&es
157
easy to show that
Our enqMcal results give us a reIevant range of values for p, and we know that 0414 when Q, =O* Computed values of the ratio of the standard deviations are given in the following table:
Values of cr,,b, for various values of p, and B when Q, = 0.
Pl
0.3 2s -
0.5 1.8 3.6 3u
0.6 1.6 z9 5.7
0.8 -::: 3.5
The blanks in the table are due to tl ?: fact that p1 cannot be larger than 0/( 1+ 0') regardless of how large we ma!;e a,"/&. The values in the table suggest that the standard deviation of innovatons in the non-stationary component may be several times larger than the stimdard deviation of innovations in the cyclical compone&. * Now consider the case where the secular or non-stationary component is a strict random walk, so that 0= 0. The value of p1 is then given bY
where p,,,, is the contemporaneous correlation between u and v. To account for positive vialues of pr, pro must be negative, in fact pyu< --(a,/aJ- l-c 0. random walk assumption on y" implies either strong n u and u, or a large variance ratio, or both. This the values of puv and (CT&,) consistent with observed vaiues in the followin
P!
- 0.2
- 0.6 1.7 4.4 --
-0.9 1.1 1.2 1.0
0.0 0.1 0.3
5.0
-
__."
15s
C.R. Nelson and Ci. r"l~~v, Wends and random walks in macroeconomic time series
The Manks, in&ate values of pI and put,that are inconsistent with any Q,/c,. It is nnteresting to note that the magnitude of c,,/c, implied by assuming 0~ C,.`==O fp=O are similar. `P above results are dependent on the stochastic structure of output he
being a first-order MA process with positive autlocorrelation at lag one. As I-nentioned in section 3, the positive autocorrelation could be attributed entirely to temporal aggregation. If this is the case then our inferences about a,/~, are distorted since it is well-known that time aggregatian amplifies low frequency (i.e., long-run) movements relative to high frequency (i.e., short-run) movements. However, we are somewhat. reluctant to accept this interpretation of the results since it implies that the short-run or c;fclical variability we are reducing through aggregation is variation that is dissipated within the aggregation interval of a year [see Tiao (f9'72)]. Another way of making this point is to say that by looking at annual data, we can make no inference regarding the variance of components whose memory (or life) is less than a year. We do not believe, however, this is a significant disadvantage of the annual time interval since most economists probably identify business cycles (transitory components) with periods that are longer than a year.la It is instructive to contrast our analysis to the signal extraction strategy proposed by Hodrick and Prescott (1980). Hodrick and Prescott decompose observed variables into growth and cyclical components under the maintained hypothesis that the growth component moves smoothly through time. The standard deviation of innovations in the growth component is assumed to be very small relative to the standard deviation of innovations in the cyclical component (specifically 1/4Oth). Optimal estimates are chosen through a criterion function that penalizes variance in the second differences of the growth component ar: well as variance in the cycle. A linear time trend emerges as a limiting case. The Hodrick and Prescott strategy implicitly imposes a components model on the data without investigating what restrictions are implied (a difficult task in their model) and whether those restrictions are consistent with the data. Our strategy, on the other hand., is to use the data as an aid in identifying certtin characteristics of an appropriate components model. Our results suggest that the ratio of the standard deviations of growth to cyclical innovations has a minimum in the neijghborhood of one with likely values up 110 five or six rather than the value of 1/4Oth assumed by Hodrick and Prescott.
`*Although havenot ~;arritiout In analysis we usingquarterly data, our experience with such data suggests tl1if.tour conclusions are not likely to be sensitive to the interval of observation. In other words, the autocoSret3tion structure of the quarterly data are not much different from that observed in the z.nnual <`at 1.
C.R.
Nelson
and Cl.
Plosser,
%nds
aud random
walks
in macroeconomic
rime series
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5. Some implicatiorrs for busi-
cy&
theorizing
The analysis of unobserved components models leads us to the inferegce that if (a) output is the sum of a non-stationary component of the DS class and a stationary (transitory) component, and (b) we observe non-negative autocorrelation at lag one only in the first differences of output then (i) the
variance of the innovations in the non-stationary component must be as large or larger than that of the purely stationa* y or transitory component, and (ii) either the non-stationary component contains significant transitory comnonents (i.e., it is nut a random walk) or, if the non-stationary component is assumed to be a random walk, the innovations in the random walk are correlated with the transitr.;y componznt. These inferences have potentially important implications for business cycle research. For example, most of the recent developments in business cycle theo, y stress the importance of monetary disturbances as a source of output fiuctuations." However, the disturbances are generally assumed to have only transitory impact (i.e., monetary disturbances have no permanent real e...