|mEIilIEM PI RI CAL Empirical Economics (1993)18 : 707-727 IIEIII/ECONOMICS The HUMP-Shaped Behavior of Macroeconomic Fluctuations PIERRE PERRON1 D6partement de Sciences Economiques et C.R.D.E. Universit+de Montr6aI C.P. 6128, Succ. A Montreal, Qu6bec, Canada, H3C-3J7 Abstract: We analyze the nature of persistence in macroeconomic fluctuations. The current view is that shocks to macroeconomic variables (in particular real GNP) have effects that endure over an indefinite horizon. This conclusion is drawn from the presence of a unit root in the univariate time series representation. Following Perron (1989), we challenge this assessment arguing that most macroeconomic variables are better construed as stationary fluctuations around a breaking trend function. The trend function is linear in time except for a sudden change in its intercept in 1929 (The Great Crash) and a change in slope after 1973 (followingthe oil price shock). Using a measure of persistence suggested by Cochrane (1988) we find that shocks have small permanent effects, if any, To analyze the effects of shocks at finite horizon, we select a member of the ARMA(p, q) class applied to the appropriately detrended series. For the majority of the variables analyzed the implied weights of the moving-average representation have the once familiar humped shape. Key Words: Measures of persistence, unit root, trend-stationarity, ARMA models, non-stationarity, structural change. JEL Classification System-Numbers: C22, E32 I Introduction The most widely used m o d e l i n g device in empirical economics characterizes a time series variable b y some d y n a m i c structure subject to shocks that are u n c o r r e l a t e d over time. There are m a n y ways to i n c o r p o r a t e this a p p r o a c h in a particular analysis. The a p p r o a c h popularized by Box a n d Jenkins (1970) analyzes the b e h a v i o r of a u n i v a r i a t e series by a n autoregressive m o v i n g average representation. Here, the d y n a m i c s of the system, or the p r o p a g a t i o n m e c h a n i s m , is represented by the effect of past values of the variable a n d of the u n d e r l y i n g shocks o n the present value of the variable. M o v e m e n t s of a given variable are essentially driven by the present a n d past realizations of the shocks. The n a t u r e of these shocks is generally left unspecified a n d can e n c o m p a s s various factors such as m o n e t a r y policy surprises, technological shocks, changes ] I wish to thank Christian Dea and Serena Ng for research assistance as well as Charles Nelson and John Campbell who kindly provided some of the data used in this study. Research support from the Social Sciencesand Humanities Research Council of Canada and the Fonds pour la Formation de Chercheurs et l'Aide/t la Recherche du Qu6bec is acknowledged. 0377- 7332/93/4/707 - 727 $2.50 9 1993 Physica-Verlag,Heidelberg 708 P. Perron in foreign conditions, exogenous governmental policy variations, etc. In general, the shocks may be viewed as a composite of factors. A problem that has received considerable attention in the literature concerns the effects of a shock on future values of the level of the variable. As an example of such concern, consider the opening of Blanchard's article (1981, p. 150): One of the few undisputed facts in maeroeconomics is that output is humped shaped, or more precisely that the distribution of weights of the moving average representation of the deviation of quarterly output from an exponential trend has a hump shape. Of course, this so-called 'undisputed fact' has been seriously challenged recently, to the extent that it is now perceived as counterfactual. A composite shock is now thought to have a lasting effect on the level of output and most other macroeconomic variables. The latter view contrasts sharply with the 'humpshaped' description, which defines a case where the effects of a shock are negligible after some period of time. The contrasting assessments result essentially from different methods of making a series stationary, i.e., different detrending procedures. According to the earlier wisdom used by Blanchard to obtain the hump-shaped result, the trend exhibited by many macroeconomic variables is essentially deterministic. The trend can be succinctly represented by a linear first-order polynomial in time applied (usually) to the logarithm of the series. This view has been challenged, most notably by Nelson and Plosser (1982), who argue that the trend is primarily stochastic due to the presence of a unit root (with drift) in the univariate representation of the series. The unit-root characterization implies that shocks have a permanent effect on the level of the series. Various studies have attempted to calculate the long-term percentage change in the level of a variable following a one percent innovation. These include Campbell and Mankiw (1987a, b, 1989), Cochrane (1988), Evans (1989), Clark (1987), Watson (1986) and Blanchard and Quah (1989). While the authors may disagree as to the precise long-term effect, and particularly whether it is greater or less than one percent, the consensus appears to be that it most likely is greater than zero. For example, Campbell and Mankiw (1987b, p. 1tl) argue that "Much disagreement remains over exactly how persistent are shocks to output. Nonetheless, among investigators using postwar quarterly data, there is almost unanimity that there is a substantial permanent effect." The purpose of the present study is to try to refloat the hypothesis that the effects of shocks on most macroeconomic variables are short-lived and are, in fact, hump-shaped. We extend a previous analysis (Perron (1989)), where we argued that the statistical non-rejection of the unit-root hypothesis for macroeconomic data was essentially due to the presence of the 1929 Great Crash and the post-1973 (post-oil-price shock) slowdown in growth. These extraordinary events correspond to a shift in the intercept of the trend function in 1929 and a sudden change in its slope in 1973. Regardless of whether these two events are The HUMP-Shaped Behavior of Macroeconomic Fluctuations 709 seen as large outliers or as exogenous (i.e., not realizations of the underlying probability structure of the shocks), the results in Perron (1989) suggest that the non-rejection of a unit root is due in large part to the occurrence of these two events. A way to isolate the events, i.e., to take them out of the noise function characterizing the variables, is to allow for non-linear trends. More specifically, if a sudden change in the intercept of the trend function in 1929 and a change in its slope in 1973 are permitted, the evidence for a unit-root hypothesis is considerably weakened. Here, shocks have no permanent effects except those associated with the 1929 crash and the slowdown in growth after 1973. The present paper deals with two issues. The first is a reassessment of the findings described above, using the method following Cochrane (1988) and Campbell and Mankiw (1987a), which calculates the long-term effect of shocks by estimating the spectral density of the first-differences of the series at the zero frequency. This approach is particularly useful since this measure of persistence will be shown to be (asymptotically) unaffected by the presence of a 'crash' (i.e., a sudden change in the intercept of the trend function). We investigate various series for real GNP, including a long span of annual data and a post-war quarterly series. We also analyze the series used in Nelson and Plosser (1982). Our findings tend to confirm our earlier results. The effects of shocks vanish over a long horizon. It must be noted, however, that the present analysis is not intended to provide a formal statistical test to discriminate between the classes of trend-stationary (possibly with breaks) and differenced-stationary processes. Rather, our intention is to provide qualitative results that support our earlier claim as well as offering a possible explanation of differing results found by Cochrane and Campbell and Mankiw. The second aim of the paper is to characterize the stochastic structure of the 'detrended' series using the non-linear trend described above. To this effect, we use the Akaike and Schwartz criteria to select members within the class of ARMA models. With very few exceptions, the appropriate stochastic structure is an AR(2) or ARMA(1, 1) with very similar dynamic properties. In most cases, the weights in the moving-average representation have a hump-shape. The paper is organized as follows. Section 2 introduces the models considered and discusses the non-parametric statistic used to measure the persistence effects. Section 3 analyses the behavior of this statistic under the hypothesis that the trend function contains a single break. Section 4 presents an empirical analysis of the Nelson-Plosser data set, and Section 5 considers various real GNP series. In each case, the impulse-response function corresponding to the selected stochastic models are described. Finally, Section 6 presents some concluding comments and a discussion of the implications of our results for macroeconomic modeling, forecasting and the issue of detrending to achieve stationarity. 710 2 ?. Perron The Models and the Statistics The *extraordinary' nature of the 1929 crash and the 1973 oil-price shock when compared to the modern historical experience motivates the present analysis. The 1929 crash witnessed a sudden and dramatic decline in aggregate economic activity, while the 1973 oil-price shock is associated with the beginning of the slowdown in growth experienced by most western industrialized countries. There are several possible statistical modeling strategies to incorporate these major events. On the one hand, one can view the 1929 crash as a realization of a shock issued from the same probability distribution as any other shock. This would imply a fat-tailed distribution, possibly one with infinite variance. The problem with this approach is that most statistical techniques are valid for finite variance distributions, and there is a clear possibility that such a single major event biases the statistics in favor of the unit-root hypothesis. This was shown formally in Perron (1989). Since the issue here is whether or not "regular" shocks have permanent effects, one may wish to remove the influence of a big outlier and see what the rest of the noise can tell us about the properties of the system. On the other hand, one can view such a large event, not as a realization issued from the same probability distribution as the other shocks, but as a quite separate random event. The idea is that the economy is regularly subjected to some shocks (call them "regular shocks," say) but that once in a while it is disrupted by a major event. This modeling strategy would allow two types of shocks issued from quite different probability distribution functions. For instance, one may wish to model the major events as realizations from a Poisson process. To be more precise, consider the following specification for a given variable y~: yt -- ~lt + Z t ; rh -- pt + ~tt (1) where A(L)Z~ = B(L)6; e~ ,~ i.i.d.(0, o=); #~ = #t-1 + V(L)vt and fl~ = ~-1 + W(L)w~. Here, the Z[s are (not necessarily stationary) deviations from the trend function ~/,. The intercept and the slope of the trend functions, #~ and ~, are themselves random variables modeled as integrated processes with W(L), V(L) stationary and invertible polynomials. However, the important distinction is that the timing of the occurrence of the shocks v, and w~ are rare relative to the sequence of innovations {6}; for example, Poisson processes with arriva! rates specified such that their occurrences are rare relative to the frequency of the realizations in the sequence {6}- The intuitive idea behind this type of modeling is that the coefficients of the trend function are determined by longterm economic fundamentals (e.g., the structure of the economic organization, population growth, etc.) and that these fundamentals are rarely changed. In our examples, v~is non-zero in 1929 (the great depression) and w~is non-zero in 1973 (the oil-price shock). In any event, since we are interested in the effects of "regular" shocks (the et in the notation of (1)) through time on various aggregate variables, a useful way The HUMP-Shaped Behaviorof MacroeconomieFluctuations 711 to approach the problem is to view these major events as part of the deterministic trend function. This follows the spirit of the Box-Tiao (1975) intervention analysis where we take "outlier" or "aberrant" events out of the noise function and analyze what the remaining noise can tell us about the properties of interest. This is the route we follow. Following Nelson and Plosser (1982), Campbell and Mankiw (1987a), Cochrane (1988) and others, we parameterize a model which possibly exhibits permanent effects of shocks by the imposition of an autoregressive unit root with a possibly non-zero drift. However, the approach is generalized to allow a one-time change in the structure occurring at a time T~ (I < Ts < T, where T is the sample size). Three different models are considered under this hypothesis: one that permits an exogenous change in the level of the series (a "crash"), one that permits an exogenous change in the rate of growth and one that allows both changes. The models are, under the null hypothesis of a unit root: Model (A): Yt = ,u + yD(TB)t + Yt-1 + et , Model (B): Yt = I2 + ODUt + Yt-~ + et , Model (C): Y t = 12 + 7D(TB), + ODU~ + Yt-~ + ez , where D ( T B ) , = 1 if t = Tn + 1, 0 otherwise, D U t = 1 if t > TB, 0 otherwise, and A ( L ) e t = B(L)v,, v, ,,~ i.i.d.(0, o-2) with A ( L ) and B(L), pth and q,h order polynomials, respectively, in the lag operator L. More general conditions on the errors {et} are possible but to ease the presentation we shall only consider these simpler ones. 2 The cases where the shocks have no permanent effect on the level of the series {y,} are represented by trend-stationary models parameterized as follows: M o d e l ( A ) : y , = 1 2 + 7 DU t + flt + w~ , Model (B): y, = (12 - OTs) + fit + ODT~ + w, , Model (C): Yt = (12 - OTB) + yDUt + fit + ODT, + w~ , where D Tt = t, if t > TB, 0 otherwise, and F ( L ) w t = D(L)vt; vt ,,, i.i.d.(0, a2), with F ( L ) and D(L) again finite order polynomials in L. Model (A) describes what we refer to as the crash model. The parameter ~ represents the magnitude of the change in the intercept of the trend function occurring at time T~. Model (B) is referred to as the "changing growth" model and 0 represents the magnitude of the change in the slope of the trend function. Model (C) allows for both effects to take place simultaneously, i.e., a sudden change in the level of the series followed by a different growth path. 2 Note also that we describe the models in terms of the "additive outlier version"as opposed to the "innovational outlier version,"a distinction which is discussed in Perron (1989).This does not affectany of the results that follow. 712 P. Perron In our earlier study, we considered testing the null hypothesis of a unit root in a variety of macroeconomic time series allowing for such changes in the trend function. The statistics derived were in the spirit of the Dickey-Fuller (1979) procedure and required the tabulation of a new set of critical values. An alternative is to consider the procedure suggested by Cochrane (1988) and Campbell and Mankiw (1987a) to provide a measure of the persistence effect of shocks~ It is based on the properties of the noise function of the first-differences of the data. Unit-root and trend-stationary models can be nested as: Model (A): Ay, = ~ + 7D(TB), + ur , (2) Model (B): Ay~ = tr + ODUr + ut , (3) (4) Model (C): Ay t = ~c + 7D(TB)t + ODU, + u, . Under the unit-root hypothesis, ~c = # (the drift) and u, = e, = A(L)-~B(L)vv Under the hypothesis that the series is trend-stationary, ~c = fl (the initial slope of the deterministic trend function) and u~ = (1 - L)w t = (1 - L)F(L)-~D(L)vr Hence, when the models are parameterized in first-differences, the only difference between each hypothesis is with respect to the nature of the errors {ut}. When the model is trend-stationary, there is a unit root in the moving-average representation of the errors, denoted u, = O(L)v,. The long-horizon effect of a i unit shock in vt on the level of Yt is given by 0(1) = l i m i ~ ~j=o 0j, where % are the coefficients in the polynomial 0(L). If the model is trend-stationary, 0(1) = 0 since ~b(L) = (1 - L)F(L)-~D(L)vr On the other hand, if the model contains an autoregressive unit root 0(1) > 0. Cochrane (1988) has proposed an alternative measure of persistence that can be estimated non-parametrically. The measure is based on the autocovariance function of the differenced process, which can be related to the moving average polynomial ~,(z) as C(z) @(z)0(z-1)o "2 where C(z) = ~ = _ ~ Cjz j, Cj is the jth autocovariance of the first-differenced process, and a 2 is the variance of the innovations {vt}. Denoting the variance of the first-differenced process as a2 ( = Co), the measure of persistence is given by V = C(1)/a 2, and is related to ~p(1) by V = (az~/a 2) [~,(1)] 2. The measure V is simply 2~ times the normalized spectral density of the process {u~} evaluated at frequency zero. V and 0(1) are not equivalent except in some special cases, in particular when ~(1) = 0 when there is no persistence. For us, this is the interesting case, and we shall work directly with the measure V and show that for most series its estimate turns out to be close to zero. -Ay t = Yt _ Y~-i _ ~ where fi = T - 1 '~'~T z.x (ty ~ - Yt-~) = T - t ( Y T . Yo) is the Let estimate of the slope of the trend function. A general class of estimators of the spectral density at the origin is given by: = k ~ k = 1 -t-2 Z w(k, j)[)j , j=l where (5) The HUMP-Shaped Behavior of Macroeconomic Fluctuations ~j = ( T / ( T - j)) ~ (Ay, Ay,_i) (~,)2 , 713 (6) t=j+l with k a truncation lag parameter and w(k, j) a lag w i n d o w ? M a n y possible choices of lag windows are available, and for ease of c o m p a r i s o n of our results with the previous literature, we choose the Bartlett triangular window defined by w(k, j) = 1 - j/(k + 1). I~k is a consistent estimate of V if k ~ oe and k / T --. 0 as T - , oc. Its asymptotic standard error is given by (see, e.g., Priestley (1981, p, 463)): s.e. [I~k] = v k / [ ( 3 / 4 ) ( T / ( k + 1))] 1/2 (7) As a measure of persistence, 17k has several advantages due mainly to the fact that it is non-parametric. However, a problem in finite samples concerns the appropriate choice of the truncation lag parameter k. Campbell and M a n k i w (1987b) conducted a small M o n t e Carlo experiment with 130 observations with a r a n d o m walk and a stationary AR(2) model. They found that a value k of at least 30, and preferably 40 or 50 (i.e., more than 1/3 of the total sample size) is needed to be able to distinguish the above-mentioned models. P e r r o n and N g (1992) found that the exact mean-squared error of I7k is minimized using a large value of k if V is small, and a small value of k if V is large (e.g., in the vicinity of 1). Given our earlier results (Perron (1989)) that most m a c r o e c o n o m i c time series appear to be best construed as "trend-stationary" if allowance is m a d e for a shift in the trend function, more weight should be given to estimates of 17k with k quite large, say between 1/3 and 1/2 of total sample size. This a p p r o a c h is taken in the empirical sections, t h o u g h we present estimates ~k for a wide range of values for k. 4 It is by now u n d e r s t o o d that statistics such as the variance ratio or any estimate of the spectral density at the origin have p o o r properties as formal tests to distinguish trend-stationary versus difference-stationary processes (see, in particular, Cecchetti and L a m (1991)). The basic problem is that the finite sample confidence intervals are very wide if the process has a unit root. We verified 3 1?k as defined by (5) and (6) does not necessarily lead to a non-negative estimate of V. It would do so if the biased estimates ofpj were used, i.e., without the correction factor (T/(T - j)). We report results using 17k,as defined in (5) and (6), to permit a comparison with previous studies, namely those of Cochrane (1988) and Campbell and Mankiw (1987a). Estimates of V using a variety of windows and truncation lags were also computed and are available upon request. The results are not very sensitive to other choices of windows and lead to the same conclusions. 4 The use of data-dependent methods to select the truncation lag parameter k, as in Andrews (1991) for example, is problematic in the present context. Such methods usually start with a preliminary estimation of a simple time series model, such as an AR(1). The optimal truncation lag is then a function of the estimated value of this autoregressive parameter. The problem with applying such a procedure is that, as shown in Perron (1989), the presence of a break in the trend function will bias the estimated first-order autocorrelation coefficient towards 1. In this context, the chosen truncation lag parameter may be inappropriate. We therefore prefer the presentation of a range of values to assess the robustness of the results. 714 ?. Perron this by doing the following experiment. We first assumed no break in the data and estimated, for each series, an AR process with a unit root. We then simulated the statistic 17k for various values of k. Except in some rare cases, the confidence intervals were so wide that no discrimination was possible. With this caveat in mind we nevertheless applied the following modification of an asymptotic test suggested by Phillips and Ouliaris (1988). The test is based on the fact that I~ is asymptotically normal with mean V and variance given by the square of (7). Their procedure is a bounds test based on the asymptotic distribution. Let H be the hypothesis that the process is trend-stationary. Atso let z, be the e-percentage point of the normal distribution. We say that we accept H if lTk + Z,{S.e.(l~k)} < C*, for some bound c*; correspondingly we say that we reject H if 12k - z~{s.e.(l~k) } > c*. If none of the inequalities are satisfied the test is viewed as inconclusive. Of course, the procedure is quite arbitrary in the sense that a choice of c* must be made. Moreover, for any such choice of e* the test will likely have some size distortions in finite sample for particular classes of data-generating processes. In any event, it nevertheless provides an asymptotically valid procedure and permits us to infer, in some sense, the most likely hypothesis (see the discussion in Phillips and Ouliaris (1988)). The particular choice of c* is discussed in the empirical sections, s 3 Properties of pk with a Breaking Trend Function To understand the empirical results described in the next sections, it is instructive to analyze the asymptotic behavior of the usual sample autocorrelations (with only a constant mean subtracted) when the trend function contains a break. We wish to derive the limit of ~j when in fact the true process is either given by equation (2) (a change in the intercept of the trend function) or equation (3) (a change in the slope of the trend function). As will be apparent from the results below, the behavior of kj when the true model is equation (4) (both changes allowed) is equivalent to that under equation (3). To perform the asymptotic analysis, we require Tn to increase at the same rate as T. For simplicity, we let TB/T = 2 for all values of T. We denote by pj the true autocorrelation coefficient at lag j of the process {ut} and C(j) the true autocovariance at lag j of the same process. We collect our results in the following Theorem, whose proof can be obtained by slight modifications of the proof of Theorem 1 in Perron (1990). s We view the bounds test of Phillips and Ouliaris (1988) not as a formal test per se bm rather as providing indirect evidence. We discuss in more detail, in the empirical sections, the strategy used to select the bound and how such a choice can help us draw some inference of interest. The HUMP-Shaped Behaviorof MacroeconomicFluctuations 715 Theorem I: Let ~j be defined by (6). Then as T ~ oo with T~/T = 2 for all T, we have: a) if Ay~ is 9enerated accordin9 to (2): ~)j ~ pj; b) if Ay~ is 9enerated accordin9 to either (3) or (4), ~j ~ [2(1 - 2)0 2 + C(j)]/[),(1 - ).)0 z + C(O)~. This Theorem shows that the standard sample autocorrelations are consistent estimates of their population values only for Model (A) where a break is present in the intercept of the trend function. This result is intuitively clear, since the mean of the first-differenced series is constant except for a single period. The effect of neglecting this change vanishes asymptotically. There may, however, remain a bias in finite sample. Things are different when there is a break in the slope of the trend function. Here, the sample autocorrelations are inconsistent estimates of the population autocorrelation coefficient. An interesting feature is that the limit is greater than the population value at any lag. The asymptotic bias is positive for all autocorrelations. The bias is greater the larger the relative magnitude of 2(1 - 2)0 2 compared to C(j) and C(0). In particular, the larger the break the greater is the bias. For a given value of 0, the bias is maximized when 2 --= 1/2, with a break at mid-sample. A consequence of this result is that one can expect tTk to be an inconsistent estimate of V if a break in the slope of the trend function is not taken into account. More importantly, l~k (using the standard autocorrelations ~3j) will tend to overestimate V, thereby suggesting a much greater degree of persistence than exists. In particular, l~k will suggest some persistence even if the shocks have no long-term effects. To obtain consistent estimators in the presence of a change in the slope of the trend function we need alternative estimates of the autocorrelation function of {u~}. To this effect, first denote by z137t the "demeaned" series Ay t where allowance is made for a possible change in mean at time TB. More specifically AYt is defined as the residuals in the following regression estimated by ordinary leastsquares: Ay, = ~ + ODu, + ~y, (8) with DUt = 1 if t > Tn (0 otherwise). 12k is computed using (5-6) with Ayt instead of Ay r 4 Analysis of the Nelson-Piosser Series We analyze the series considered by Nelson and Plosser (1982) for several reasons. First, it is a rich data set with a wide range of annual series covering a long historical period. More importantly, it allows us to make some interesting comparisons between the results in the original Nelson-Plosser study and Perron 716 P, Perron (1989). An interesting feature is that all series end in 1970 and cover the period of the 1929 crash. Hence, for each series only one break is likely to be present. For reasons discussed in Perron (1989), we consider model (A) (only a change in the intercept in 1929) for the series Real GNP, Nominal GNP, Real Per Capita GNP, Industrial Production, Employment, GNP Deflator, Consumer Prices, Nominal Wages, Money Stock, Velocity and Interest Rate. We consider model (C) (a change in both the intercept and the slope in 1929) for Real Wages and Common Stock Prices. We do not consider any change in the trend function for the unemployment series since there is agreement that it is stationary for the period under consideration. 6 For most of the series considered there is only a change in the intercept of the trend function. On the basis of Theorem 1, I~k is not influenced, asymptotically, by the presence of such a change. However, as discussed in Perron (1989), stano dard Dickey-Fuller tests for a unit root are biased toward non-rejection of the unit-root hypothesis even asymptotically. Hence, it can be viewed as an alternative to standard tests for a unit root which is robust to a sudden change in the intercept of the trend function and should, in principle, yield results similar to those obtained in our earlier study. The estimated measures of persistence (without any corrections for changes in the trend function) are presented in Table 1. We tabulate results for the following values of the truncation lag parameter k: 8, 16, 20, 30, 40 and 50. The results are quite striking. The estimates show very little evidence of persistence for alt series, with the exception of Consumer Prices, Velocity and Interest Rate, precisely those series for which the unit root could not be rejected in Perron (1989). The results are sometimes quite extreme. For example, consider the Industrial Production index, where l~k is 0.07 with k = 50 (less than half of total sample size). This result is quite contrary to the unit-root hypothesis found in NelsonPlosser but is in accord with our earlier results. The results are not so dramatic for all series but are indeed suggestive that most of them exhibit little persistence, if at all. Apart from the above-mentioned three exceptions, all series show an estimate less than 0.33 at k = 40 (except GNP Deflator with 0.50 and Common Stock Prices with 0.49). We also present in Table 1.a estimates for the Real Wages and Common Stock Price series when allowance is made for a change in slope in 1929. Following the results of Zivot and Andrews (1992) and Perron (1991) we also present the estimates with a change in slope in 1939 for Real Wages and 1936 for Common Stock Prices. These dates were selected as the outcome of tests for unit roots allowing the break point to be unknown. As expected from the theoretical result of Section 3, the estimates are noticeably smaller when allowance is made for a change in slope, For the Common Stock Prices series the choice of t936 as opposed to 1929 as the break date has a more significant impact. 6 Relatedresults, using the Nelson-Plosserdata set can be found in Zivot and Andrews (1992) and Raj (1993).Evidenceof rejectionsof the unit root with historical data from many countries can be found in Raj (1992), The HUMP-Shaped Behavior of Macroeconomic Fluctuations 717 Table I. Non-parametric estimates of persistence (l?k); Nelson-Plosser data Series T Real G N P 62 Nominal G N P 62 Real Per Capita G N P 62 industrial Production 111 Employment 81 Unemployment Rate 81 G N P Deflator 82 Consumer Prices ! 11 Nominal Wages 71 Real Wages 71 Money Stock 82 Velocity Interest Rate Common Stock Prices !02 71 !00 k= 8 k = 16 k = 20 k = 30 k = 40 k = 50 1.01 (.45) 1.59 r' (.71) 1.03 (.46) .45 (.15) 1.07" (.41) ,56 (.22) 2.16' (.83) 2.39 r (.79) 1.9W (.79) .82 (.34) 2.63 r (1.01) .76 (.26) 2.06 r (.85) .74 (.26) .41 (.25) 1.01 (.62) .43 (.26) .17 a (.08) .55 (.29) ~20~" (.11) 1.81 (.96) 2.23" (i.01) 1.i5 (.65) .49 (.28) 1.35 (.71) .72 (.34) 2.95 (1.68) .67 (,32) .16" (.11) .69 (.47) .45 (.30) .214" (.11) .49 (.30) .16" (.09) 1.38 (.81) 2.00 (1.01) .68 (.43) .49 (.31) .76 (.45) .81 (.42) 3.03 (1.91) .63 (.34) .37 (.30) .35 (.29) .34 (.28) .190" (.12) .29 (.21) .12 ~ (.09) .45 (.32) 1.65 (1.01) .32 (.25) .48 (.37) .40 (.29) .86 (.55) 2.82 (2.17) .57 (.37) .21 (.20) .23 (.22) .20 (.19) .15 a (.11) .11" (.09) .01" (.01) .50 (.41) 1.49 (1.05) .33 (.29) .31 (.27) .19 (.16) .86 (.63) 1.73 (1.53) .49 (.36) .11" (.12) .06" (.06) .12" (.13) .07" (.06) .17 (.16) .06" (.06) .68 (.62) 1.18 (.93) .17 (.17) .19 (.19) .28 (.26) .92 (.75) .76 (.75) .46 (.38) Table 1.a: Non-parametric estimates of persistence (~k) with break in trends at time Tb Series Tb k= 8 k = 16 k = 20 k = 30 k = 40 k = 50 Real Wages 1929 .65 (.27) .71 (.29) .70 (.24) .59 (.20) .27 (.15) .30 (.17) .58 (.28) ,40 (. 19) .24 (.15) .29 (.18) .52 (.28) .31 (. 16) .19 (.15) .27 (.21) .46 (.30) .19" (.08) .09" (.08) .14~ (.12) .44 (.33) .12" (.09) .05" (.05) .08" (.08) .41 (,34) .094 (.07) 1939 Common Stock Prices 1929 1936 Notes: i) Asymptotic standard errors computed according to (7) are in parentheses, ii)" and "denote a rejection of the trend-stationarity hypothesis, at the 10~ and 5~o level respectively, using the bounds test described in Section 2 with c* = 0.40. Similarly, " and " denote acceptance of this hypothesis at the I 0 ~ and 5~o level, respectively. 718 P. Perron Tables 1 and !.a also present the outcome of Phillips and Ouhmls (1988) bounds test. As noted in Section 2, this test depends on the arbitrary choice of the bound c*. We selected .40 as the value to be used in the test. The reason for such a choice is that it allows accepting the trend-stationarity hypothesis for the Unemployment series at the 5~ level for values of k greater than or equal to 20 and at the 10~ level for all values of k greater than or equal to 16. Since it is generally agreed that over this sample period the Unemployment Rate series is stationary, one would like the outcome of the test to reflect this feature, and this particular choice achieves this goat. Nevertheless, the tests presented should be viewed as rather suggestive and subject to the caveats discussed in Section 2. In many instances the tests yield an inconclusive outcome 9For values of k other than 8, only one series, the Consumer Price index, shows a test that suggests rejection of the trend-stationarity hypothesis 0br k = t6 only). The tests suggest accepting the trend-stationarity hypothesis with large values of k for Real GNP, Nominal GNP, Real per Capita GNP, Industrial Production, Employment as well as for Real Wages and Common Stock Price, when allowance is made for a change in slope in the latter two series. These results are, in an important sense, complementary to those obtained in Perron (1989). The issue is as follows. The measures presented in Table l suggest the absence of tong-term persistence effects of shocks. However, standard unitroot testing procedures fail to reject the unit root hypothesis. Furthermore, as argued by Nelson and Plosser (1982), univariate ARMA models of these series suggest a process close to an IMA(1, 1) whose first-differences are positively correlated. The latter would suggest a high degree of persistence. Hence, the two approaches yield quite different implications. One way to reconcile these results is by viewing the 1929 crash as a change in the intercept of the trend function. As argued in Section 3, the presence of such a change will have no effect (in large samples) on the non-parametric measure of persistenceu However, as argued in Perron (1989), it will bias the unit-root tests towards non-rejection of the unitroot hypothesis suggesting that standard univariate ARMA modeling will show a higher degree of persistence than is present. Given our earlier results of the many rejections of the unit root when allowance is made for a change in the intercept of the trend function in 1929, we incorporate such a change prior to estimating the ARMA models for the noise function. To analyze the effects of shocks at finite horizons we therefore adopt the following strategy. We detrend the data by allowing a change in the intercept between 1929 and 1930. The residuals, considered estimates of the noise of the series, are analyzed within the class of ARMA(p, q) models. We estimated all models with p and q less than or equal to 5. The optimal parameterization was chosen using either the Schwartz (1978) or the Akaike (1974, 1976) criteria. The Akaike criterion minimizes 2 In L + 2m where L is the likelihood and m = p + q is the number of parameters. The Schwartz criterion minimizes 2 In L + m In T where T is the total number of observations. For the sample sizes considered here, the Schwartz criterion penalizes extra parameters more heavily. We carried The H UM P-Shaped Behavior of Macroeconomic Fluctuations 7t9 o ~oo~w167I ~ ~ ~ o ~ o ~ I I I i I I I I t I I l I N.= i l o m o 0 ~~ z 9 .~ e- ~a n~ 9 ~2 u~ 720 P. Perron out this procedure for all Series except Consumer Prices, Velocity and Interest Rate, given that for these series the unit-root hypothesis is not rejected. For the majority of the series, the first and second choices, according to both criteria, are AR(2) and ARMA(1, 1), One notable exception is the Industrial Production series, where the chosen model is ARMA(O, 4). Table 2 presents the implied weights of the moving-average representation for each model selected. Not only are the impulse responses similar across models for a given series, they are also quite similar across series. The typical pattern is that of a hump-shape with most of the effects vanishing within 4 to 8 years. The evidence presented in this section appears inconsistent with the notion that shocks have permanent effects on a wide variety of macroeconornic variables. The fluctuations around the trend function are transitory, and the weights in the moving-average representation have the once-familiar humped-shape. Given that more attention is given to results pertaining to real GNP in the literature, the next section presents a more detailed analysis focusing on a number of available indices of Real GNP. 5 An Empirical Analysis of Real GNP We analyze three GNP series that are commonly used in the literature. The first one is Post-war Quarterly Real GNP 1947: 1-1985:4 (seasonally adjusted) from the National Income and Product Accounts. This is the same series analyzed in Campbell and Mankiw (1987a) except for some minor data revisions. In this case, we allow for a change in the slope of the trend function after 1973:1 (designated as detrending method B). The second series is Annual Real GNP 1869-1983 taken from Balke and Gordon (1986). Here, we allow two breaks: a change in the intercept after 1929 and a change in the slope after 1973 (detrending method D). The third series is Annual Real Per Capita GNP 1869-1975 taken from Friedman and Schwartz (1982). This series was analyzed by Perron and Phillips (1987) and Stock and Watson (1986). Here, we allow only for a change in the intercept after 1929 (detrending method A) since the bias due to the change in the slope of the trend function is likely to be small with only two data points after the change in slope in 1973. The estimated measures of persistence are presented in Table 3. Consider first the Post-war Quarterly Real GNP series. Row (1) reproduces the results of Campbell and Mankiw (1987a). Row (2) gives the estimates allowing for a shift in the slope of the trend function (Model B) after 1973: 1. It uses the residuals from regression (8) with TR set at 1973:1. The results clearly show much weaker evidence for persistence. For instance, at k = 50 (1/3 of total sample size), the estimate is reduced from 0.68 to 0.13, allowing for the break; at k = 100, it is reduced from 0.57 to 0.05. Furthermore, the outcome of Phillips and Ouliaris' The HUMP-Shaped Behavior of Macroeconomic Fluctuations 721 C~ C~ il II ~-~i ~I~ ~'~ I~ C~ II ~ F~ _~ C~ II . . . . . . . . ~ II .~ "~.~'~.~ ~.. ..~ ~ , . . ~ . . . . . ~ 9 ~.~_~ ~ ~ ~..~ ~ ~ ~ ~ ~'~.'~.Q ~..--~. ~-~.-~. ~ ~.; . . . . . . . . . . . . ~..~ ,_> ,._~ ,..; ..~ ~_~ ,..: ._> . . . . . . . . . . . . 0 0 te~ c~ . . . . c~ Z I ~.-. .-7. 0 722 P. Perron -- T T T 0 ~ ~ 0 =~ .'~ ~ ~ ~,~ " ~ ~ o_-~, T ~tl .~ ~ ~ , ~ ~ g ~ ~ 0 H "2. ~. H . .-2.. ~ . . . . . . . . . ~-= ~ ~.~ ~ "2. = H H ~ ~ ~'~ ~ ~ ~.~ .a 8 8 0 0 0 0 N~ g ..~ ~ ~: ~= "~ ~ - g ]i % The HUMP-Shaped Behaviorof MacroeconomicFluctuations 723 bounds test, with c * = .40 again, suggests accepting the trend-stationarity hypothesis when allowance is made for this change in slope in 1973. Rows (3) and (4) present results using split-sample estimates with" no correction for break, i.e., pre- and post-1973: 1. At k = 30, the post-1973 sample yields an estimate of - 0 . 0 6 while at k = 50, the pre-1973 sample estimate is 0.03] Hence, these results are consistent with the fact that the high value of the measure of persistence using the full sample with no break correction is simply due to a change in the slope of the trend function after 1973: 1. Rows (5) through (10) present results using annual real G N P from 1869 to 1983. Rows (5) and (6) show the estimates without and with corrections for shifts in the trend function respectively (the corrected estimates are constructed with the residuals from (8) setting TB = 1973). Again, the estimates are lower with corrections, though here, the differences are smaller. For both cases the outcome of the bounds test suggests accepting the trend-stationarity hypothesis for large values of k. The result, that the differences in the estimates with and without corrections for breaks are small, is because the proportion of the total sample where the change in the slope of the trend function occurs is much smaller with the sample of annual data (2 = 105/115) than with the sample of quarterly data ()~ = 105/156). From Theorem 1 we can expect the bias to be smaller with the former set of data if no correction is made. Campbell and Mankiw (1987a) argued that the small measure of persistence obtained by Cochrane (1988) was due partially to the presence of the pre-1929 data in the annual sample (compare rows (7) and (8)). However, if a correction is made for the shift in the trend function in the post-1929 sample (again allowing for a shift in the slope of the trend function in 1973), 1~k is smaller for the post-1929 than for the pre-1929 period for any value of k greater than 10 (compare row (7) and row (9)). This conclusion is supported from results for the period 1930-1972 without any corrections; for k greater than 10, the measure of persistence is indeed small. Hence, the differences between Cochrane's and Campbell and Mankiw's results appear not to be due to the inclusion or not of the pre-1929 sample but to the importance of the post-1973 period relative to the total sample. Rows (11) through (14) present the results for Friedman and Schwartz's Annual Real Per Capita G N P series. Here no correction is made in constructing the estimates l~k. Using either the full sample or any sub-samples, the estimates suggest that the shocks have little permanent effects. For example, for 18691975, ITk = 0.02 at k = 60, for 1869-1929 it is 0.07 at k = 30, for 1930-1975 it is 7 As noted in footnote 2, pk does not yield a non-negative estimate by construction. When a negative value is obtained, the asymptotic standard errors given by (7) are non-sensible,hence we do not report them. We interpret negative values of 17k as suggestiveof a true value close to zero. This interpretation is supported by other estimates of V (not reported) which use alternative lag windows that yield non-negativeestimates by construction. 724 P. Perron ~o ,9~ e. eq Ne-~ r a, rr t"q ,.A , . ~ ~'-1 ,,A o The HUMP-Shaped Behavior of Macroeconomic Fluctuations 725 0.20 at k = 20 and for 1909-1975 it is 0.21 at k = 30. In all cases, the bounds test suggests accepting the trend-stationarity hypothesis. To analyze the effects of shocks at finite horizons we adopted the following strategy. Since the estimates 17~, when appropriately corrected for the 1973 shift in the slope of the trend function, show little evidence of persistence, we detrended the data by allowing a change in the intercept after 1929 and a change in the slope of the linear time trend after 1973. The residuals are then considered as the noise of the series and are analyzed within the class of ARMA(p, q) models. We estimated all models with p and q less than or equal to 5 and the models were selected using either the Schwartz or Akaike criteria. For the Post-war Quarterly Real GNP series, the Akaike criterion selects the ARMA(1, 3) model, while the Schwartz criterion favors an AR(3) model. However, the AR(2) model is a close contender according to both criteria. Table 4 presents the estimated impulse response function for each model; i.e., the weights in the moving-average representation. The results are very similar across the three models and clearly show a hump-shaped pattern with a peak after 2 quarters. Almost all the effects vanish after 16 quarters. The results for the Annual Real GNP series are similar. According to both criteria the chosen model is an AR(2) with the ARMA(1, 1) a close second and ARMA(1, 2) a close third. The estimated impulse-response functions in Table 4 show again the hump-shaped pattern with a peak in the first year consistent with the evidence from the quarterly data. Most of the effects of the shocks vanish sometime between the 4th and 6th years. The evidence presented in this section is inconsistent with the notion that shocks have a permanent effect on real GNP. As with the Nelson-Plosser data, the fluctuations around the trend function are transitory and the weights in the moving-average representation have the once popular humped-shape. 6 Conclusions The present study has several implications for empirical macroeconomics. First, it suggests an alternative procedure for detrending to achieve stationarity. Some authors, using post-war quarterly data on economic aggregates, have detrended using a break in trend following the 1973 oil-price shock (e.g., Blinder (1981, 1986) and Blinder and Holtz-Eakin (1986)). Our results provide a formal justification for such a procedure. More interestingly, our results present an alternative picture of economic fluctuations somewhere in-between the current view that fluctuations have permanent effects and the view that fluctuations are stationary around a deterministic (strictly) linear time trend. This alternative view suggests that, most of the time, fluctuations have transitory effects and that the economy tends to revert 726 P. Perron to a stable-trend growth path. However, occasionally the economy is disrupted by a major event which has a lasting effect. The view suggested here is that such permanent shocks have been quite rare but are very important in magnitude. The non-rejection of tests of the unit root, when only a time invariant linear trend function is allowed, can be explained within the present context by noting that, even though the shocks which have permanent effects are rare, they are of such magnitude as to account for a substantial portion of the total variability of the series. Hence, it is not too surprising to find such wide evidence for unit roots. However, this evidence tells us only that something significant and permanent occurred. We cannot infer that most shocks have permanent effects. A third interesting issue concerns confidence intervals for long-range forecasts. The view of stationary fluctuations around a strictly linear time trend implies an accurate long-horizon forecast. This is thought to be quite implausible, thereby making the unit-root characterization more plausible, since it implies ltbrecasts with ever-increasing confidence intervals. The view suggested here implies that forecasts constructed using the estimated trend-stationary model are adequate for short-term forecasting, i.e., conditional on no major break in the trend function occurring in the near future. For long-horizon fbrecasting, it becomes implausible to suggest that no shift in the trend function witl occur. 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