# The growth trend

Anticipating the contribution of Nelson and Plosser (1982), discussed below, Blatt (1980) points out that the Frisch-Slutsky hypothesis implies that business cycles are caused by shocks and that any attempt to include business cycle-like oscillations as part of the trend curve produced by the model runs directly counter to the Frischian view. Nevertheless, he notes that the trend does not have to be strictly linear to be consistent with the Frisch-Slutsky hypothesis. If nonlinear, however, it should be stable with smooth curvature. It should not itself display fluctuations in the business cycle time-scale because otherwise detrending would remove part of the business cycle phenomena to be explained. In the light of the discussion of section 4.3.1, it should be added that the trend should not display regular cycles of longer duration either, because these would instead be consistent with the various long swing hypotheses.

In section 1.2 it was noted that there is a growing tendency, even at the NBER, to analyse cycles using detrended data.16 In many analyses (log) linear trends are assumed and elsewhere moving average trends are estimated. The presumption in favour of (log) linear trends is a natural extension of the linearity hypothesis, but the data should at least be examined to see if it is appropriate; otherwise detrending will distort the series. It was noted in section 4.3.1, however, that the more sophisticated approach of estimating a moving average trend introduces its own distortions which often show up as spurious cycles of longer duration than the business cycle, as defined by the NBER. Detrended data used for business cycle analysis is, therefore, highly likely to be distorted and in the absence of better methods of trend estimation, which take account perhaps of the structural changes that occur during long runs of data, it is hard to escape Burns and Mitchell's (1946) conclusion that it is better to work with non-detrended data.

Nelson and Plosser (1982) challenge the Frischian view, as expressed by Blatt (1980), that macroeconomic time series are best characterised as stationary fluctuations around a deterministic trend. They argue instead that they should be viewed as nonstationary processes that have no tendency to return to a deterministic (trend) path. They see no reason why the secular movement in economic time series should not itself be stochastic and observe that if it is, then models based on deterministic time trend residuals will be misspecified. They illustrate the types of misspecification that can arise from inappropriate detrending by considering the properties of residuals from a random walk on time, which are known to display drift, similar to secular movements, which is stochastic rather than deterministic in nature (see section 3.4). They show that the autocorrelation function of the deterministic time trend residuals is a statistical artifact which is determined entirely by sample size. The autocorrelation function displays strong autocorrelation at low lags and pseudo-periodic behaviour at long lags. Empirical investigations that ignore the possibility that a stochastic trend is the source of the autocorrelation might, therefore, be led to overestimate both the persistence and variance of the business cycle. Further, to the extent that the stochastic nature of the trend can be associated with real shocks, the use of a deterministic trend will underestimate the influence of real shocks. Since the basic statistical issue is the appropriate representation of nonstationary economic time series, Nelson and Plosser (1982) consider two fundamentally different classes of nonstationary processes as alternative hypotheses. One class consists of deterministic function of time plus a stationary stochastic process with zero mean, referred to as the trend-stationary (TS) process. It is judged that such processes are most appropriately applied to the natural logs of economic time series, and deviations from trend, the so-called cyclical components of the series, are represented as invertible ARIMA processes.

The second class of nonstationary process considered is that in which first, or higher order differences, are a stationary and invertible ARIMA process. This is referred to as a difference-stationary (DS) process and the first order case is used to explain the natural logs of the economic time series examined. The DS class is purely stochastic and the TS class is fundamentally deterministic.

Various historical time series from the United States, including measures of output, spending, money, prices and interest rates, are examined and the relationship of the analysis to McCulloch's test of the Monte Carlo hypothesis is noted (see section 1.2). In particular McCulloch (1975) finds some evidence of periodicity in the logs of real income, investment and consumption after fitting a linear trend but finds no periodicity in their first differences, a finding consistent with their results. The sample autocorrelation structures for the series are found to be consistent with those expected from a random walk.17 The exception is the unemployment rate series, which exhibits autocorrelation properties consistent with a stationary series. The autocorrelation structures of real, nominal and per capita GNP, real and nominal wages and common stock prices display positive autocorrelation at lag one only. This is characteristic of first order MA processes and inconsistent with the TS model. The GNP deflator, consumer prices, the money stock and the bond yield exhibit more persistent autocorrelation in first differences but do not show evidence of having been generated by a differenced TS process. In sum, Nelson and Plosser find their evidence to be consistent with the DS representation of nonstationary economic time series. They do, however, recognise that their tests have little power against the alternative hypothesis of a TS process with an AR root close to unity. This alternative implies little tendency to return to trend and could be indicative of a stable limit cycle produced by a nonlinear model.

Their results, therefore, suggest that economic time series contain stochastic trends of the DS type rather than deterministic time trends. In this case, if (the log of) output is viewed as the sum of a secular or growth component and a cyclical component, and the latter is assumed to be transitory (stationary), then any underlying nonstationarity must be attributed to the secular component. Thus if actual output is in the DS class then so too must be the secular component. The separation of the secular component from the observed data can, they note, be thought of as a problem of signal extraction when only the information in observed series itself is used. Using, as an example, Friedman's permanent income model they show that it is not always possible to identify the cyclical and secular components. However, if the cyclical component is stationary and, as they discover, the autocorrelations in the first differences of output are positive at lag one and zero elsewhere, then they demonstrate that the variation in actual output changes will be dominated by changes in the secular, rather than the cyclical, component. They acknowledge that they cannot prove empirically that cyclical fluctuations are stationary or transitory but feel that their evidence is strongly supportive of the hypothesis that the business cycle is a stochastic process of the DS class.

The hypothesis that the cycle is stationary is implicit in the Frisch 1 hypothesis (see section 1.4), which assumes that cyclical fluctuations dissipate over time and the cycle is the result of hitting the propagation model with repeated random shocks. Long-run or permanent movements (nonstationarities) are attributed to the secular (trend) component and are the result of real factors. Nelson and Plosser believe most economists accept both the Frisch I hypothesis and this view of the trend and, therefore, that the cyclical component is stationary. Finally, they observe that assigning a major portion of the variance in output to innovations in the nonstationary component gives an important role to real factors in output fluctuations and places limits on theories of the business cycle that stress the importance of unanticipated monetary disturbances, such as the Lucas (1975) model. The debate between proponents of real and monetary causes of the business cycle is reviewed in section 2.2.

Nelson and Plosser (1982) therefore provide another strong warning against using residuals from fitted deterministic trend lines for the empirical analysis of business cycles. If the trend follows a nonstationary stochastic process, a possibility that cannot be discounted, then the residuals will contain both cyclical and stochastic trend variation and the magnitude and duration of the supposedly cyclical component will be overstated. They also warn that first differencing will not remove the stochastic growth component but may render the time series stationary, and the problem of inferring the behaviour of each unobserved component from the observed sum, the signal extraction problem, will remain.