Campbell and Mankiws Test

An obvious limitation of simply testing for the existence of a permanent component of fluctuations is that it cannot tell us anything about how big such a permanent component might be. The literature since Nelson and

See Stock and Watson (1988) and Campbell and Perron (1991) for introductions to the issues arising in time-series econometrics when series are highly persistent.

sistent is often complex and unintuitive. Care needs to be taken in such situations, and conventional econometric tests often carmot be used.

Because of the negative bias in estimates of b under the null hypothesis, one carmot use conventional t-tests of the significance of the OLS estimates of b from (4.55) or (4.56) to test whether output is trend-stationary. Nelson and Plosser therefore employ a Dickey-Fuller unit-root test (Dickey and Fuller, 1979). Dickey and Fuller use a Monte Carlo experiment to determine the distribution of the t-statistic on b from OLS estimates of equations like (4.55) and (4.56) when the true value of b is zero. That is, they use a random-number generator to choose fs; they then generate a time series for Iny using (4.55) or (4.56) with b set to zero; and then they estimate (4.55) or (4.56) by OLS and find the t-statistic on b. They repeat this procedure many times. The resulting distribution of the t-statistic, instead of being symmettic around zero, is considerably skewed toward negative values. For example. Nelson and Plosser report that for the case of 100 observations with true parameter values of a = 1 and fo = 0, the average value of the t-statistic on b is -2.22. The t-statistic is greater in absolute value than the standard 5% critical value of -1.96 65% of the time, and it is greater than -3.45 5% of the time. Thus an investigator who is unaware of the econometric complications and therefore uses standard critical values is more likely than not to reject the hypothesis of nonstationarity at the 5% level even if it is true. In a Dickey-Fuller test, however, one compares the t-statistic on b not with the standard t-disttibution, but with the distribution produced by the Monte Carlo experiment. Thus, for example, a t-statistic greater than -3.45 in absolute value is needed to reject the null hypothesis of fo = 0 at the 5% level.

With this lengthy econometric preface, we can now describe Nelson and Plossers results. They estimate equations shghtly more complex than (4.5 5) for U.S. real GNP, real GNP per capita, industrial production, and employment; they find that the OLS estimates of fo are between -0.1 and -0.2, with t-statistics ranging from -2.5 to -3.0. All of these are comfortably less than the correct 5% critical value of -3.45. Based on this and other evidence. Nelson and Plosser conclude that one cannot reject the null hypothesis that fluctuations have a permanent component.

Discussion

There are two major problems with the general idea of investigating the persistence of fluctuations, one statistical and one theoretical. The statistical problem is that it is difficult to learn about long-term characteristics of output movements from data from limited time spans. The existence of a permanent component to fluctuations and the asymptotic response of output to an innovation concern characteristics of the data at infinite horizons. As a result, no finite amount of data can shed any light on these issues. Suppose, for example, output movements are highly persistent in some sample. Although this is consistent with the presence of a permanent component to fluctuations, it is equally consistent with the view that output reverts extremely slowly to a deterministic trend. Alternatively, suppose we observe that output returns rapidly to some trend over a sample. Such a finding is completely consistent not only with trend stationarity, but also with the view that a smaU portion of output movements are not just perma-

"If £ is perturbed by 1 in a single period, (4.57) Implies that 1 is changed by 1 in that period, bi in the next period, bf + in the following period, and so on. Iny is therefore changed by 1 in the period of the shock, I + bi m the next period, I + bj + b( + bz in the following period, and so on.

Plosser has therefore focused on determining the extent of persistence in output movements. CampbeU and Mankiw (1987) propose a natural measure of persistence. They consider several specific processes for the change in log output. To take one example, they consider the third-order autoregressive (or AR-3) case:

Alnyf = a + foiAlnyf i + fo2Alnyf-2 + 1 - + ff. (4.57)

Campbell and Mankiw estimate (4.57) and compute the implied response of the level of In to a one-unit shock to e. Their measure of persistence is the value that this forecast converges to. Intuitively, this measure is the answer to the question: If output is 1 percent higher this period than expected, by what percent should I change my forecast of output in the distant future? If output is trend-stationary, the answer to this question is zero. If output is a random walk (so Alnyf is simply a - ff), the answer is 1 percent.

Campbell and Mankiws results are surprising: this measure of persistence generally exceeds 1. That is, shocks to output are generally followed by further output movements in the same direction. For the AR-3 case considered in (4.57), the estimated persistence measure is 1.57. Campbell and Mankiw consider a variety of other processes for the change in log output; for most of them (though not all), the persistence measure takes on similar values.

See Blough (1992a, 1992b) and CampbeU and Perron (1991).

See, for example, Cochrane (1988, 1994); Christiano and Eichenbaum (1990); Perron (1989); Watson (1986); and Beaudry and Koop (1993). Campbell and Mankiw (1989b) and Cogley (1990) present evidence for countries other than the United States.

nent, but explosive-so that the correct reaction to an output innovation is to drastically revise ones forecast of output in the distant future.

Thus at the very least, the appropriate questions are whether output fluctuations have a large, highly persistent component, and how output forecasts at moderately long horizons should be affected by output inno-V ations, and not questions about characteristics of the data at infinite horizons. Clearly, similar modifications are needed in any other situation where researchers claim to be providing evidence about the properties of series at infinite horizons.

Even if we shift the focus from infinite to moderately long horizons, the data are unlikely to be highly informative. Consider, for example, CampbeU and Mankiws procedure for the AR-3 case described above. Campbell and Mankiw are using the relationship between current output growth and its three most recent lagged values to make inferences about outputs long-run behavior. This is risky. Suppose, for example, that output growth is actually AR-20 instead of AR-3, and that the coefficients on the 17 additional lagged \alues of A In are all small, but all negative. In a sample of plausible size, it is difficult to distinguish this case from the AR-3 case. But the long-run effects of an output shock may be much smaller.

This difficulty arises from the brevity of the sample, not from the specifics of Campbell and Mankiws procedure. The basic problem is that samples of plausible length contain few independent, long subsamples. \s a result, no procedure is likely to provide decisive evidence about the long-term effects of shocks. Various approaches to studying persistence have been employed. The point estimates generally suggest considerable persistence (though probably somewhat less than Campbell and Mankiw found). At horizons of more than about five years, however, the estimates are not very precise. Thus the data are also consistent with the view that the effects of output shocks die out gradually at moderate horizons.

The theoretical difficulty with this literature is that there is only a weak case that the persistence of output movements, even if it could be measured precisely, provides much information about the driving forces of economic fluctuations. Since technology may have an important trend-reverting component, and since real-business-cycle models allow for shocks coming from sources other than technology, these models are consistent with low as well as high persistence. And Keynesian models do not require that persistence be low. To begin with, although they attribute the bulk of short-run fluctuations to aggregate demand disturbances, they do not assume that the processes that drive long-run growth follow a deterministic trend; thus they allow at least one part of output movements to be highly persistent. More