0.2r

g 0.0

-0.1

-0.2

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Quarters

FIGURE 4.7 The effects of a 19 the wage and the interest rate

government-purchases shock on the paths of

nology shocks, there is no overshooting. Because technology is unchanged and the capital stock moves little, the movements in output are small and track the changes in employment fairly closely. Consumption declines at the time of the shock and then gradually returns to normal. The increase in employment and the fall in the capital stock cause the wage to fall and the interest rate to rise. The anticipated wage movements after the period of the shock are small and positive; thus, as before, the source of the increases in labor supply are the increases in the interest rate.

As with technology, the persistence of movements in government purchases has important effects on how the economy responds to shocks. If pc falls to 0.5, for example, acc falls from -0.13 to -0.03, olg falls from 0.15 to 0.03, and increases from -0.004 to -0.020: because movements in purchases are much shorter-lived, much more of the response takes the form of reductions in capital holdings. These values imply that output rises by about one-tenth of the increase in government purchases, that consumption falls by about one-tenth of the increase, and that investment falls by about four-fifths of the increase. In response to a 1% shock, for example, output increases by just 0.02% in the period of the shock and then faUs below normal, with a low of -0.004% after seven quarters.

Nelson and Plossers Test

The persistence of fluctuations was first addressed by Nelson and Plosser 11982), who consider the question of whether fluctuations have a permanent component (see also McCuUoch, 1975). The idea behind their test is conceptually simple, though it turns out to involve some econometric complications. If output movements are fluctuations around a deterministic trend, then output growth will tend to be less than normal when output is above its trend and more than normal when it is below its trend. That is, consider a regression of form

Alnyt = a+ b{lnyf-, - [a + /3(f - 1)]} + St, (4.54)

where Iny is log real GDP, a - /Sf is its trend path, and f f is a mean-zero disturbance uncorrelated with In i - [a -i- /3(f - 1)]. (The regression can also mclude other variables that may affect output growth.) The term lnyf i -[a + pit - 1)] is the difference between log output and the trend in period

4.8 Empirical Application: The

Persistence of Output Fluctuations

Introduction

Real-business-cycle models emphasize shifts in technology as a central source of output fluctuations. The specific model analyzed in this chapter assumes that technology fluctuates around a deterministic trend; as a result, the effects of a given technological shock eventually approach zero. But this assumption is made purely for convenience. It seems plausible that changes in technology have a significant permanent component. For example, an innovation today may have little impact on the likelihood of additional innovations in the future, and thus on the expected behavior of the growth of technology in the future. In this case, the innovation raises the expected path of the level of technology permanently. Thus real-business-cycle models are quite consistent with a large permanent component of output fluctuations. In traditional Keynesian models, in contrast, output movements are largely the result of monetary and other aggregate demand disturbances coupled with sluggish adjustment of nominal prices or wages. Since the models assume that prices and wages adjust eventually, under natural assumptions they imply that changes m aggregate demand have no long-run effects. For this reason, natural baseline versions of these models predict that output fluctuates around a deterministic trend path. These considerations have sparked a considerable literature on the persistence of output movements.

"" term trend-stationary means that the difference between actual output and a deterministic trend is not explosive. The term unit root arises from the lag-operator methodology (see n. 17, above, and Section 6.8). If output has a permanent component, it must be differenced to produce a stationary series. In lag-operator notation, 1 ( 1 is written as Z,lny„ and thus 1 , is written as (1 - I)Iny,. The polynomial 1 - L is equal to zero for 1=1; that is, it has a "unit root." For comparison, consider, for example, the stationary process Iny, = piny, i + p < 1. In lag-operator notation, this is (1 pi)Iny, = a,. The polynomial 1 -pi is equal to zero for L = 1 /p, which is greater than 1 in absolute value. More generally, stationary processes have roots outside the unit circle.

For a simple case, see Problem 4.16.

t -1. Thus if output tends to revert toward the trend, h is negative; if it does not, h is zero.

We can rewrite (4.54) as

AIny, = o--b/Sr-b 1-b ft, - (4.55)

where a = a ~ ba + bp and p = -bp. Thus to test for trend-reversion versus permanent shocks, we need only estimate (4.55) and test whether b = 0. Note that with this formulation, the null hypothesis is that output does not revert toward a trend. Formally, the null hypothesis is that output is nonstationary or has a unit root, the alternative is that it is trend-stationary.

There is, however, an important econometric complication in carrying out this test: under the null hypothesis, ordinary least squares (OLS) estimates of b are biased toward negative values, lo see why, consider the case of p = 0; thus (4.55) becomes

Alnyr = a + folnyt-i + ff (4.56)

Assume for simplicity that the fs are independent, identically distributed, mean-zero disturbances. The Iny, Is are combinations of the fs. Specifically, under the nuU hypothesis of fo = 0, lnyf i is Inyo + (t - 1)« - f i - f2 - • • • + ff 1. Since the fs are not correlated with one another, ff is uncorrelated with In yt 1. It might therefore appear that OLS is unbiased. But the requirement for OLS to be unbiased is not just that the disturbance term is uncorrelated with the contemporaneous value of the right-hand-side variable, but that it is uncorrelated with the right-hand-side variable at all leads and lags. The fact that the past fs enter positively into In yt 1 means that In yt 1 is positively correlated with past values of the error term. One can show that this causes the estimates of b from OLS to be biased toward negative values. That is, even when the null hypothesis that output has no tendency to revert toward a trend is true, OLS tends to suggest that output is trend-reverting.

This econometric complication is an example of a more general difficulty: the behavior of statistical estimators when variables are highly per-