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Optimal prediction of cyclical downturns
Journal of Economic Dynamics and Control 4 (1982) 225-241. North-Holland
OPTIMAL
PREDICTION
OF CYCLICAL
DOWNTURNS
Salih N. NEFTCI* City
University
of New
York,
NY
10036,
USA
Received October 1980, final version received October 1981 In the conventional way of visualizing economic time series, the prediction problem does not dichotomize into predicting turning points and predicting conditional means. Yet, the prediction of turning points has always been considered a separate issue in Business Cycle literature. The paper first discusses these issues and then describes a model where the prediction of turning points does separate from the prediction of conditional means. Using the model an ‘optimal way of predicting turning points is derived and applied to U.S. data. The paper uses time series methodology and the theory of optimal stepping times.
1. Introduction A common theme of the work on business cycles has been the special treatment of cyclical downturns and their prediction. This seems to be partly motivated by the belief that the stochastic behavior of major aggregates changes suddenly as soon as the economy enters a downturn regime. For example, according to Hicks, ‘one of the most striking characteristics of actual cycles is that output, once it has passed the peak, falls off rather rapidly. (It) seems likely that a rapid downswing ought to be a feature of the theoretical cycles’.’ Further, ‘the substitution of a downward for an upward tendency often takes place suddenly and violently, whereas there is, as a rule, no sharp turning point when an upswing is substituted for a downward tendency’ [Keynes (1936)].2 Some evidence supporting these observations was reported in Neftci (1979). Yet, so far most existing work dealing with the issue, with the exception of Wecker (1978), seems to lack a precise statistical basis - i.e., optimality in some sense [e.g., Zarnowitz and Boschan (197511. In this paper we study the ‘optimal’ prediction of cyclical downturns in a model in which the need to forecast turning points results from switches in stochastic behavior of economic time series at (random) points. *I am grateful to Thomas Sargent, Robert Shiller, David Pierce, Bob Goldfarb, and especially to Christopher Sims for comments and criticisms. ‘The emphasis is by Hicks (1950). See also Hicks (1949). The study by Mitchell (1927) is an earlier example. ‘Some more recent examples of this literature are Schinasi (1979), Rose (1967) and Varian (1979).
0165-1889/82/0000-0000/!$02.75
0 1982 North-Holland
226
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of cyclical
downturns
The main contribution of the paper is to recognize turning point prediction as an optimal stopping time problem. ‘Stopping time’3 because what is being predicted is the time of occurrence of an event and not the value of an economic variable. ‘Optimal’ because the forecaster has to balance the benefit of calling an early signal against the cost of giving a false alarm, thus requiring a solution similar to that of dynamic programming. To summarize the model briefly, we assume that the forecaster is observing a process {X,} whose probability structure changes abruptly at some random time period. The popular notion of a ‘downturn’ is assumed to occur during this switch. The decision-maker’s objective is to predict in some ‘optimal’ sense when this switch will occur. The change in probability structure is not directly observed; as a result observations on {X,} will have to be used to make inferences on whether the economy has entered a new regime or not. Thus, the problem is to obtain a prediction rule which signals the switch in distribution as soon as possible, given that the number of false alarms are kept at a minimum. Based upon the work of Shiryayev (1978), we derive this optimal rule and apply it to the leading indicator series of the U.S. economy. Although this paper deals with the issue in a framework where there is an ‘indicator’ and an ‘actual’ series, the approach is actually more general than that and can be interpreted as an alternative way of dealing with the ‘switching regression’ problem for economic time series. It turns out that when regressions are subject to i.i.d. disturbances, the estimation of the switch points becomes difficult. The model presented in this paper shows how sequential analysis can be used to determine these switch points.4 The paper is organized as follows. In the next section we discuss the background of the issue. The model is then described and the ‘optimal’ prediction rule is derived. Finally, some applications are presented. 2. The background
It is not clear at the outset why the prediction of downturn periods should constitute a problem which differs significantly from estimating conditional means. Consider a typical case where a decision-maker observes a (possibly non-stationary) process {X,} and desires to minimize the expected value of a (quadratic) objective function depending on the current and future values of X,. Solutions of such optimization problems in general require forecasts of future X,‘s. These forecasts are then obtained by projecting X, +~, z > 0, on H,, a space spanned by {X,-,, s=O, 1,. . .} and complete under quadratic mean. If 3Let {F,} be a sequence of sub-sigma lields with respect to which {X,} is measurable. Then c(o) is said to be a stopping time (Markov time) if {o:r(o)s t} EF,, i.e., iI {F,} is regarded as a sequence of information sets given to a forecaster, T(O) must be a random variable whose value depends only on the information observed up to that time. ‘See Shiryayev (1978) for examples.
S.N. Neftci,
Optimal
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of cyclical
downturns
221
the process {X,} is known to be covariance stationary, then the projection operator will be time invariant. This would be the case of the standard least squares techniques. On the other hand, if the process is non-stationary, and the decision-maker has some a priori notion of how this non-stationarity evolves over time, then, the Kalman Filter techniques will apply. However, in either case one would only be interested in the projection itself and the issue of when X, would turn down will never arise. So, the prediction problem does not dichotomize into predicting the periods of downturns and predicting the conditional means at other times. Neither would the prediction of downturn periods be of much value to economic agents maximizing the expected value of (quadratic) inter-temporal objective functions, since these problems usually result in linear decision rules depending on the expected r>O only. [For example, see Sargent (1978) and Lucas and Prescott $Yi)., Thus, it appears that, in order to justify a separate effort directed towards predicting the periods of downturns, some additional structure needs to be imposed on the {X,} process. A clue to what this structure can be is provided by the pre-Keynesian business cycle literature, which suggests two possible characteristics that a stochastic process {X,}, representing major cyclical variables, may possess. First, the stochastic behavior of X, could be different before and after the downturns - i.e., a switch may occur at the turning point. And second, this switch may occur in a sudden and rather unexpected way. In terms of the discussion above, this may be paraphrased by saying that the equation describing the evolution of the projection operators over time, 9, =A$,- i, breaks down at some random time period due to a sudden change in the probability structure of {X,}. Under these conditions, dynamic optimization problems of economic agents would lead to a separate prediction of the periods during which this breakdown occurs. To see this, note that a decision-maker facing adjustment costs in inputs and maximizing the expected value of discounted future profits would find it ‘useful’ to predict the time period during which the probability structure of the observed series changes, as well as the future values of unobserved variables. From the point of view of such a decisionmaker there would be a genuine advantage in separating the prediction problem into one of obtaining the EX,,,, r >O, on the one hand and of forecasting the time period during which downturns occur on the other. Such optimization problems are discussed in a recent book by Kushner who also shows that the two prediction problems separate under some plausible conditions [Kushner (1977, p. 51)].5 ‘Note that iI economic agents possess an asymmetric loss function on the domain of prediction errors, this may be a sufficient reason to introduce some asymmetric behavior in economic time series. This is true, since an asymmetric loss function means that these prod,u~rs will change their behavior suddenly, under some conditions. See Wecker (1978). ’
S.N. NejQi, Optimal prediction of cyclical
228
downturns
In the next section a model incorporating unexpected switches in distribution function of the observed process is outlined. Although limited some respects, the model will be based upon explicit assumptions about behavior of the observed stochastic processes, and will be consistent with business cycle literature of the pre-Keynesian area.
the in the the
3. The model and the prediction rule 3.1. The framework
There are two stochastic processes {x} and {X,}. Y, represents observations on major macro-aggregates such as employment or production. The ultimate aim of the forecaster is to predict ‘cyclical turning points’ of {Y;} in some optimal sense. Heuristically, this prediction is based on the idea that the events starting a downturn in x will be present in {X,} before they show up in {Y;}. In more precise terms, we assume that turning points are characterized by sudden switches ih the distribution function of {X,}. Let 2, an integer valued random variable represent the period of downturn. This random variable has the property that when {Z= k} the probability distribution of {Xk+j, i=O, 1,2,. . .} will be different and independent of the distribution of {X,-j, j = 1,2,. . .}. Accordingly, for {Z = k, kc t} we will have P(XSx, ,...) X,IXk ,..., X,~xx,)=FO(x, ,...) Xk-J.F’(Xk )...) x,), (1) where F’(v) and F’(e) are two distribution functions which represent the probabilities associated with X, during normal and downturn regimes respectively. The value of Z - i.e., the period during which distributions switch - is not directly observed. Instead, inferences based on the X, observed up to time t have to be used in order to see whether a downturn has started (Z 5 t) or not (Z > t). This characterization of the {X,} process is assumed to reflect the two properties that major macro-aggregates are believed by the pre-Keynesian business cycle literature to possess. The switch in the distribution function being able to deal with a more rapid decline in the economy and the independence between {Xi,. . .,X,-r} and {X,, . . .,X,} representing the suddenness of the downturns. The last component of the model is the existence of an a priori probability on when downturns are expected to occur. It is assumed that economic agents who have observed past cyclical downturns have developed an a
S.N. Neftci,
Optimal
prediction
of cyclical
downturns
229
priori probability, P(Z = k) = Pk,
(2)
representing the a priori beliefs on the period of the next downturn. Given this framework, the problem of predicting. the cyclical downturn can now be summarized. The objective is to forecast the value of Z after successive observations on X,. Let this forecast be denoted by z. Then {r = t} will mean that the forecaster is signalling a downturn at time t, and z-Z would represent the prediction error. Every time period the decision-maker faces two choices. Based upon the available information he can either ‘signal’ a downturn or instead wait and take another observation on the process {X,}. The first choice implies that the decision-maker is predicting at time t, that {ZS t}; the second implies that he is instead opting for {Z> t}. The optimal decision will be made after comparing the cost of sounding a ‘false alarm’ represented by {z
To determine the optimal prediction rule, we will, as is often done, limit our attention to a particular class of estimators. Definition I.
0)
An estimate r is said to belong to class M(F,, a) if
{w:z=t} EF,, P{z
(3) O$aIl,
where F, is the information set available at time t and WEB are ‘events’ causing T to assume the value t. Here, the condition (i) implies that the events determining the value of r will be part of the information set available at that time - i.e., r will be an F,-measurable random variable. In probability theory integer-valued random variables having this property are called stopping times. Thus, obtaining the best estimator r* E M(F,, a) is equivalent to finding the optimal stopping time. Condition (ii), on the other hand, says that the probability of giving false alarms of having {r < Z} is constant at a.
S.N. Neftci, Optimal prediction of cyclical downturns
230
The ‘best’ estimate from this class is to be selected according following criterion: The estimate of Z denoted by the class M(F,, a) if for all 7 E M(F,, a),
Definition 2.
E[max
(7*
-Z, 0)] sE[max(r-Z,
7*
to the
is said to be optimal within
0)],
(5)
where the E( *) is the conditional expectation operator given the information set F,. According to this definition, 7* will be the optimal estimate, if for a given false alarm probability, ‘the average delay’ in reporting the downturn is minimized. Given Definition 1 and Definition 2 we are now ready to state the main result which is basically due to Shiryayev (1978, p. 195): Theorem 1. Let the F,-measurable statistic 7~~be defined by ~~=P(Zskj
Fk).
(6)
Then, the optimal prediction rule in the sense of Definition 2 will be given by
7*=inf{k:Xk~~*},
(7)
k
where 0
The next lemma shows an important will be mainly useful in applications:6
property
of the statistic 7ck,which
% should be noted that an accurate estimation of pz+, is crucial method. This is true since, in general a downturn is signalled if a ‘decline’ the range of historical experience of normal-upturn periods. Also denominator of (10) so that small changes in it can cause large changes in
for the success of the is observed well out of p,“,, appears in the Q+ ,.
231
S.N. Nefcgi, Optimal prediction of cyclical downturns
Lemma 1. Let pi and p: denote the value of the conditional during ‘normaP and ‘downswing’ regimes, respectively, P:+ 1 =dPOh+
11 xo,. . -,Mdxk+
densities of X,
13
(8)
Then xk + 1 can be calculated recursively from nk by using the relation ,
a,+,=[a,+P(Z=k+l~Z~k)(l-23lp:+,
/{CBk+P(Z=k+lIZ>k)(l-Kk)lPkl+l +(l-rr,)p,O+l[l-P(Z=k+l~Z>k)]}. Proof
(10)
Available from the author upon request.
According to this lemma rrk+ i can be calculated sequentially by using the current observation of {X,} and the Q. To do this one needs the conditional probabilities pi+ 1 and pi+ I and the a priori probability P(Z = k + 1 I Z> k). The expression in (10) can also be used to study what makes the statistic rrL+ I move towards or away from A *. For example, if the currently observed X,, 1 is an ‘unusual’ observation for a downturn period, then pi+ 1+O and the formula in (10) suggests that rck+l will move towards zero and away from A*. At the other extreme, if Xk+l is unusual for an upswing period, then pE+ I +O, and according to the expression in (lo), rrk+ i will move towards one, and thus will most likely exceed A*. Finally, suppose the value of Xk+i is such that it has equal (conditional) probabilities of coming from a downturn or from an upswing regime, i.e., suppose pi+1 zpi+ i. Then, a simple computation shows that (10) reduces to rc~+l~rr~+(l-rrJP(Z=k+l
IZ>k).
(11)
Thus, an ambivalent observation will lead to an increase in nk+r if the a priori probability P(Z = k + 1 I Z> k) is non-zero. If the n, had been less than A* and if one has a rather ‘flat’ prior on Z, then the rrnk+i will generally not exceed A* and one would not signal a downturn in period k + 1. At this point, it may be instructive to compare the procedure suggested by Theorem 1 with some other rules commonly used to predict downturn periods. One of these is ‘three consecutive declines in Leading Indicators’, a rule traditionally associated with NBER. According to this, the estimate of the downturn period will be given by ?=inf{k:AX,
AX,-,-CO,
AX,-,
(12)
232
S.N. Neftci,
Optimal
prediction
of cyclical
downturns
where 4X, is the change in leading indicator series. Obviously, the value of f will not in general be the same as the estimate given by Theorem 1, z* =inf{k:n,
zA*}.
k
Also note that the determination of ? is not explicit about the false alarm probability, although one may argue that the class of rules such as ?,=inf{k:dX,-j
j=O, l,.. .,m}
(13)
k
has an implicit way of taking into account the false alarm probability by using the value of m. For example, with m=2, as in the NBER approach, one would implicitly be settling for some specific false alarm probability, a smaller m leading to a higher false alarm probability. 3.3. Verification of the signals
The last point that needs to be discussed is the verification process of the signals. Suppose using Theorem 1 and Lemma 1, one obtains {z* = k} - i.e., signals at time k, that a downturn is ‘imminent’. How many periods is it going to be allowed to pass before this signal is labeled a ‘false alarm’? In other words, how is one going to distinguish between a ‘false’ and a ‘too early’ signal? This issue is closely rel,ated to the timing relationship between the ‘indicator’ {X,} and the ‘actual’ series {x} is discussed in NeftCi (1979). The usual way of dealing with this problem is to decide a priori, on a time interval A, which one would allow to elapse, before a signal is labeled as a false alarm. According to this, with {z* = k}, one would expect the {x} series to enter a downturn regime between k and k + A. If this doesn’t happen, then the signal {z*= k} would be called a false alarm. Obviously, this procedure assumes that there would be no difficulty in observing the turning points of {Y;}, an assumption which is reasonable in reality, where a recesssion is usually defined as two consecutive declines in the Gross National Product. Accordingly, {z*= k} would be a correct signal if inf{j:(&+jyk+j-l)
S.N. Neftci, Optimal prediction of cyclical downturns
233
one can select A by letting A =+(I+),
&>O.
This would then be consistent with the definition of a leading time series [see NeftCi (197911. Obviously, this way of dealing with signal verifi&tion is somewhat ad-hoc. For example, tile parameter A and the correlation between {X,} and {x} is not taken into account during the derivation of the ‘optimal’ prediction rule. However, the explicit determination of stopping rules in cases incorporating such characteristics is still an unresolved question and is beyond the scope of this paper. 4. Applications
The main characteristic of the business cycle model introduced in the previous section is the unobserved and sudden switch in the distribution of the economic process at some random time period. In this section we will present applications of the procedure introduced above. The index of Leading Indicators series from the U.S. is selected and used to compute estimates of {nEk} for the period 1970-79 to see whether the procedure predicts the 1974 recession, and the slowdown of the economy in 1979. The application involves a simple version of the framework outlined above. It assumes that increments, AX,, of an observed economic time series are generated by a switching model AX, = a0 + E;, =d+&:,
lS;t
(14)
where 2 is the unknown random time of downturn. The {$} are an i.i.d. sequence with a common mean 0. The {E:} are an i.i.d. sequence (independent of the Ep) with common mean 0.’ By observing the sequence X, we wish to ‘detect’ whether the process AX, has switched from regime 0 (normal) to regime 1 (downturn). Also, we need to start t from 1 rather than zero and interpret the break in distributions as if it applies to AX, rather than X,. Obviously this process has all the characteristics of the {X,} series discussed in earlier sections. The applications presented here will assume that the leading indicator series can be reasonably approximated with this model. ‘It is worth emphasizing that the sudden change in the distribution of X, need not necessarily be due to different variances of E,‘s.The distributions may, in fact, belong to different classes.’
234
S.N. Nejtci, Optimal prediction of cyclical
downturns
An application of the prediction rule introduced in Theorem 1 to the model above requires a number of intermediate steps. First among these is the need to estimate the probabilities p”=P(dX,=x,IZ>t)
and
p’=P(dX,=x,~Z~t),
and determine an a priori probability on Z. Second, one has to select an estimate for the initial value no and using (10) calculate successive values of R, recursively. After doing this one decides on the false alarm probability a, and compares each a, with ,4*(a). A downturn will be signalled at the first occurrence of the event {7r,zA*}. The estimates of {p”} and {p’} are shown in fig. 1 and table 1. To obtain these estimates the data for leading indicators belonging to 1948-1970 were split into periods of downturn and upturn regimes,8 the observations belonging to upturn periods were grouped and {p”} were estimated using the frequency distribution of {AX,}. The {p’} were estimated similarly using the frequency distribution of {AX,} belonging to downturn periods. The data for post-1970 were not used and instead were saved for out-of-sample predictions. The a priori probabilities P(Z= k 1Z> k- 1) were approximated using a symmetric density with a mean of 50 months (see table 2). According to this density, an economic agent with no information on {X,} would predict the next downturn to occur 50 months after the last downturn [see Taylor (1978)]. This seemed to be a reasonable approximation to the average .distance between two turning points in the U.S. economy. It turns out that the results are not sensitive to any reasonable variation in this density. Finally, the starting point of the prediction process, go, was selected as zero. This was believed to be a good guess for P(Z=O 1F,), the probability of having a downturn immediately following the upturn. Using this procedure, the detection rule given in Theorem 1 was applied to two periods of interest: 1971-1974 and 1975-1979. The first case involves the 1974 recession. This seemed a natural period to select. The second is the subsequent slowdown of the economy and appeared to be another test of the procedure. For both cases the false alarm probability was selected as 10% (i.e., a=O.l) and A* was approximated by (1 -a).’ sTo estimate pt and p:, the period was split in the following way: Overall period: 1948-1970. Downturn regimes: January 1948 through June 1949, April 1953 through November 1953, October 1955 through January 1958, May 1959 through December 1960, February 1969 through March 1970. Upturn regimes: The remaining intervals of the period. 91t is easy to prove that .A*6 1 -a. Shiryayev (1978, p. 200, remark 4)].
So that if n, >(I -a)
then of necessity xI> A*
[see
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236
S.N. Neftgi,
Optimal
prediction
o/cyclical
downturns
Table 1 Estimated distribution functionsa Estimated F”, the ‘upturn’ density
Estimated F’, the ‘downturn’ density
Value
Probability
Value
Probability
- 1.5 -1.4 -1.3 - 1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
0.002 0.002 0.004 0.006 0.006 0.011 0.015 0.020 0.017 0.017 0.013 0.020 0.020 0.019 0.025 0.035 0.050 0.060 0.063 0.070 0.066 0.070 0.057 0.053 0.050 0.047 0.032 0.032 0.032 0.033 0.023 0.015 0.015 0.005 0.003 0.003
-2.1 -2.0 -1.9 -1.8 - 1.7 -1.6 -1.5 -1.4 -1.3 - 1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.026 0.026 0.037 0.020 0.020 0.020 0.030 0.043 0.090 0.100 0.090 0.043 0.020 0.083 0.090 0.060 0.027 0.030 0.030 0.023 0.003 0.000 0.003 0.003 0.003 0.003
’
“This table is obtained by taking a centered 3-term moving average of the frequencies in Rg. 1. “Table 2 was obtained in the following manner: First a probability density was generated by taking a centered 38-term moving average of a Daniel window with 36 elements. The resulting numbers were averaged again in a similar manner. Then, using this density we calculated the conditional probabilities given above. Fig. 3 was calculated by starting from k= 13 (this is the case since the ny’s for 1975 are shown in fig. 3). One can also let these conditional probabilities approach a limit E>O as k gets large. This can easily be done by applying a Daniel window with more elements.
S.N. Nejtqi,
Optimal
A priori probabilities k 0 1 2 3
4 5 6 7
a 9 10 11 12 13 14 15 16 17 18
19 20 21 22 23 24
29 30 31 32 33 34 35 36 31 38
39 40 41 42 43
44 45 46 47 48
49
of cyclical
downturns
Table 2 (periods of downturn k = 1,. . ., 99).
P(Z=klZ>k-1)
k
P(Z=klZ>k-1)
0.000 0.000 0.000
50 51 52 53 54 55 56 51 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 14 75 76 77 78 79 80 81 82 83
0.048 0.049 0.050 0.051 0.052 0.053 0.054 0.055 0.056 0.058 0.060 0.062 0.065 0.066 0.067 0.068 0.071 0.075 0.078 0.081 0.085 0.089 0.095 0.099 0.103
Kt 0:002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018
0.019 0.020 0.021 0.023
25 26 27 28
prediction
0.024 0.025 0.026 0.027 0.028 0.029
0.031 0.032 0.033 0.034 0.035 0.036 0.037 0.038
84 85 86
0.039 0.040 0.041
91 92
0.042 0.043
i: 95 96
0.044
ii 89 90
0.045 0.047 0.047
“See previous page.
ii; 99
0.109 0.111 0.119 0.120 0.140 0.160 0.180 0.213 0.241 0.390 0.418 0.567 0.604 0.699 0.749 0.805 0.840 0.880 0.899 0.907 0.996 0.999 0.999 0.999 0.999
237
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N&i,
Optimal
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of cyclical
downturns
4.1. Case I: The 1974 downturn
Fig. 2 displays the results. The upper part of the table shows the behavior of the {AX,} process during 1971-1973. The lower part of the table displays the estimated (72,; k= 1971; l,...; 1973; 8). According to these results, Wk remains close to zero and does not approach the threshold (1 -a) during the years 1971 and 1972. But the situation changes in 1973. The & becomes 0.4 during April and eventually exceeds the critical level 1 -a during August 1973. Thus, in this particular case the rule gives a clear signal of the 1974 recession. This is a rather early detection of the 1974 recession. Moreover, this prediction is accompanied by an upper bound for the false alarm probability: 1974 recession is signalled at most with a 10% room for error. It is also interesting to note that the prediction of the 1974 downturn occurs before the oil shock, although it appears that oil shock has affected the severity of the recession. This suggests .that the 1974 recession was in some sense similar to previous downturns in the economy and was not a ‘unique’ product of exogenous shocks. To see this, note that the transition of Sk from k- 1 to k depends essentially on the values assumed by the ratio p”(dX,=x,)/pl(dX,=xJ [see (lo)]. But these weights, PO(.) and p’(e), were estimated by partitioning the data using the turning points experienced before the 1970s. Thus, one would expect the ratio p”/pl to decrease as the behavior of {AX,} becomes similar to that experienced during previous downturns. The prediction rule of Theorem 1 signals such a ‘resemblance’ during August 1973. This ‘signal’ is then followed by the 1974 recession. 4.2. Case II: The period 1975-1979
Fig. 3 displays the behavior of the B,‘s since 1974. Two interesting results emerge. First, the procedure does not signal any downturns for 1976, 1977 or 1978. In contrast, the popular rule of ‘three consecutive declines in X,’ does give a false alarm in August 1977. During the same months, fi, is around 0.5 and therefore far from signalling a downturn. The second interesting result is the prediction in mid-1979 that a downturn will occur ‘soon’. However, unlike the case of 1974, this signal is not given suddenly. The rrk go up and down for a number of periods before a downturn is signalled. 5. Conclusions
This paper presents a Business Cycle model which assumes that underlying economic processes are subject to shifts in their distributions at random time points. Further, these shifts occur suddenly, in the sense that once the economy enters a downturn regime dependence on the history before the turning point becomes negligible.
, .oo I’J
F
M
A
M
J
J
A
S
0
N DIJ
M
A
M
J
Fig. 2. Application
F
VALUES OF tnrl
A
S
0
N
DIJ
to the 1974 recession.
J
FOR 197l-1973
CHANGE IN LEADING INDICATORS
F
M
A
M
I J
1 J
I A
I S
I 0
I N
I DI
240
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Optimal
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of cyclical
downturns
S.N. Neftci, Optimal prediction of cyclical
downturns
241
It is shown that if economic time series possess these characteristics, an explicit optimal prediction rule can be derived to forecast cyclical downturns. This rule is optimal in the sense of signalling the downturn as early as possible for a given probability of false alarms. The application presented in the paper suggests that the optimal rule may indeed be a useful procedure for predicting cyclical downturns of the U.S. economy. References Box, George and Gwilym Jenkins, 1976, Tie series analysis (Holden-Day, San Francisco, CA). Frisch, Ragnar, 1965, Propagation problems and impulse problems in economics, in: Robert Gordon and Lawrence Klein, eds., Reading in business cycles (Irwin, Homewood, IL). Gordon, Robert, 1979, The end-of-expansion phenomenon in short-run productivity behavior, Brookings Papers on Economic Activity 2,447-461. Hicks, John, 1949, Mr. Harrod’s dynamic theory, Economica 16,32-47. Hicks, John, 1950, A contribution to the theory of trade cycle (Clarendon Press, Oxford). Keynes, John M., 1936, The general theory of employment, interest and money (Macmillan, London). Kushner, Harold, 1977, Probabilistic ,methods in approximation of elliptic equations and stochastic control (Academic Press, New York). Mitchell, Wesley, 1927, Business cycles (NBER, New York). NeftCi, Salih, 1979, Lead-lag relations, e.xogeneity and prediction of economic time series, Econometrica 47, 101-113. NeftG, Salih, 1979, The asymmetric behavior of economic time series, Manuscript. Rose, Henry, 1967, On the nonlinear theory of the employment cycle, Review of Economic Studies 34, 153-173. Sargent, Thomas, 1978, Estimation of dynamic labor demand schedules under rational expectations, Journal of Political Economy 86, 1009-1045. Schinasi, Gary, 1979, A non-linear disequilibrium dynamic macro-economic model, Ph.D. dissertation (Columbia University, New York). Shiryayev, Alexi N., 1973, Statistical sequential analysis: On optimal stopping rules, translations of mathematical monographs (American Mathematical Society, Providence, RI). Shiryayev, Alexi N., 1978, Optimal stopping rules, Application of mathematics (Springer-Verlag, Berlin). Sims, Christopher, 1975, Labor and production in manufacturing, Brookings Papers on Economic Activity 2, 357-401. Taylor, John, forthcoming, Output and price stability: An international comparison, Journal of Economic Dynamics and Control. Varian, Hal, 1979, Catastrophe theory and the business cycle, Economic Inquiry XVII, 14-28. Wecker, William, 1978, Asymmetric time series, Manuscript. Wecker, William, 1979, Prediction of turning points, Journal of Business 52, 35-50. Zarnowitz, Victor and Charlotte Boschan, 1975, Cyclical indicators: An evaluation and new indexes, Business Conditions Digest 15, v-xix.
OPTIMAL
PREDICTION
OF CYCLICAL
DOWNTURNS
Salih N. NEFTCI* City
University
of New
York,
NY
10036,
USA
Received October 1980, final version received October 1981 In the conventional way of visualizing economic time series, the prediction problem does not dichotomize into predicting turning points and predicting conditional means. Yet, the prediction of turning points has always been considered a separate issue in Business Cycle literature. The paper first discusses these issues and then describes a model where the prediction of turning points does separate from the prediction of conditional means. Using the model an ‘optimal way of predicting turning points is derived and applied to U.S. data. The paper uses time series methodology and the theory of optimal stepping times.
1. Introduction A common theme of the work on business cycles has been the special treatment of cyclical downturns and their prediction. This seems to be partly motivated by the belief that the stochastic behavior of major aggregates changes suddenly as soon as the economy enters a downturn regime. For example, according to Hicks, ‘one of the most striking characteristics of actual cycles is that output, once it has passed the peak, falls off rather rapidly. (It) seems likely that a rapid downswing ought to be a feature of the theoretical cycles’.’ Further, ‘the substitution of a downward for an upward tendency often takes place suddenly and violently, whereas there is, as a rule, no sharp turning point when an upswing is substituted for a downward tendency’ [Keynes (1936)].2 Some evidence supporting these observations was reported in Neftci (1979). Yet, so far most existing work dealing with the issue, with the exception of Wecker (1978), seems to lack a precise statistical basis - i.e., optimality in some sense [e.g., Zarnowitz and Boschan (197511. In this paper we study the ‘optimal’ prediction of cyclical downturns in a model in which the need to forecast turning points results from switches in stochastic behavior of economic time series at (random) points. *I am grateful to Thomas Sargent, Robert Shiller, David Pierce, Bob Goldfarb, and especially to Christopher Sims for comments and criticisms. ‘The emphasis is by Hicks (1950). See also Hicks (1949). The study by Mitchell (1927) is an earlier example. ‘Some more recent examples of this literature are Schinasi (1979), Rose (1967) and Varian (1979).
0165-1889/82/0000-0000/!$02.75
0 1982 North-Holland
226
S.N. Neftci, Optimal prediction
of cyclical
downturns
The main contribution of the paper is to recognize turning point prediction as an optimal stopping time problem. ‘Stopping time’3 because what is being predicted is the time of occurrence of an event and not the value of an economic variable. ‘Optimal’ because the forecaster has to balance the benefit of calling an early signal against the cost of giving a false alarm, thus requiring a solution similar to that of dynamic programming. To summarize the model briefly, we assume that the forecaster is observing a process {X,} whose probability structure changes abruptly at some random time period. The popular notion of a ‘downturn’ is assumed to occur during this switch. The decision-maker’s objective is to predict in some ‘optimal’ sense when this switch will occur. The change in probability structure is not directly observed; as a result observations on {X,} will have to be used to make inferences on whether the economy has entered a new regime or not. Thus, the problem is to obtain a prediction rule which signals the switch in distribution as soon as possible, given that the number of false alarms are kept at a minimum. Based upon the work of Shiryayev (1978), we derive this optimal rule and apply it to the leading indicator series of the U.S. economy. Although this paper deals with the issue in a framework where there is an ‘indicator’ and an ‘actual’ series, the approach is actually more general than that and can be interpreted as an alternative way of dealing with the ‘switching regression’ problem for economic time series. It turns out that when regressions are subject to i.i.d. disturbances, the estimation of the switch points becomes difficult. The model presented in this paper shows how sequential analysis can be used to determine these switch points.4 The paper is organized as follows. In the next section we discuss the background of the issue. The model is then described and the ‘optimal’ prediction rule is derived. Finally, some applications are presented. 2. The background
It is not clear at the outset why the prediction of downturn periods should constitute a problem which differs significantly from estimating conditional means. Consider a typical case where a decision-maker observes a (possibly non-stationary) process {X,} and desires to minimize the expected value of a (quadratic) objective function depending on the current and future values of X,. Solutions of such optimization problems in general require forecasts of future X,‘s. These forecasts are then obtained by projecting X, +~, z > 0, on H,, a space spanned by {X,-,, s=O, 1,. . .} and complete under quadratic mean. If 3Let {F,} be a sequence of sub-sigma lields with respect to which {X,} is measurable. Then c(o) is said to be a stopping time (Markov time) if {o:r(o)s t} EF,, i.e., iI {F,} is regarded as a sequence of information sets given to a forecaster, T(O) must be a random variable whose value depends only on the information observed up to that time. ‘See Shiryayev (1978) for examples.
S.N. Neftci,
Optimal
prediction
of cyclical
downturns
221
the process {X,} is known to be covariance stationary, then the projection operator will be time invariant. This would be the case of the standard least squares techniques. On the other hand, if the process is non-stationary, and the decision-maker has some a priori notion of how this non-stationarity evolves over time, then, the Kalman Filter techniques will apply. However, in either case one would only be interested in the projection itself and the issue of when X, would turn down will never arise. So, the prediction problem does not dichotomize into predicting the periods of downturns and predicting the conditional means at other times. Neither would the prediction of downturn periods be of much value to economic agents maximizing the expected value of (quadratic) inter-temporal objective functions, since these problems usually result in linear decision rules depending on the expected r>O only. [For example, see Sargent (1978) and Lucas and Prescott $Yi)., Thus, it appears that, in order to justify a separate effort directed towards predicting the periods of downturns, some additional structure needs to be imposed on the {X,} process. A clue to what this structure can be is provided by the pre-Keynesian business cycle literature, which suggests two possible characteristics that a stochastic process {X,}, representing major cyclical variables, may possess. First, the stochastic behavior of X, could be different before and after the downturns - i.e., a switch may occur at the turning point. And second, this switch may occur in a sudden and rather unexpected way. In terms of the discussion above, this may be paraphrased by saying that the equation describing the evolution of the projection operators over time, 9, =A$,- i, breaks down at some random time period due to a sudden change in the probability structure of {X,}. Under these conditions, dynamic optimization problems of economic agents would lead to a separate prediction of the periods during which this breakdown occurs. To see this, note that a decision-maker facing adjustment costs in inputs and maximizing the expected value of discounted future profits would find it ‘useful’ to predict the time period during which the probability structure of the observed series changes, as well as the future values of unobserved variables. From the point of view of such a decisionmaker there would be a genuine advantage in separating the prediction problem into one of obtaining the EX,,,, r >O, on the one hand and of forecasting the time period during which downturns occur on the other. Such optimization problems are discussed in a recent book by Kushner who also shows that the two prediction problems separate under some plausible conditions [Kushner (1977, p. 51)].5 ‘Note that iI economic agents possess an asymmetric loss function on the domain of prediction errors, this may be a sufficient reason to introduce some asymmetric behavior in economic time series. This is true, since an asymmetric loss function means that these prod,u~rs will change their behavior suddenly, under some conditions. See Wecker (1978). ’
S.N. NejQi, Optimal prediction of cyclical
228
downturns
In the next section a model incorporating unexpected switches in distribution function of the observed process is outlined. Although limited some respects, the model will be based upon explicit assumptions about behavior of the observed stochastic processes, and will be consistent with business cycle literature of the pre-Keynesian area.
the in the the
3. The model and the prediction rule 3.1. The framework
There are two stochastic processes {x} and {X,}. Y, represents observations on major macro-aggregates such as employment or production. The ultimate aim of the forecaster is to predict ‘cyclical turning points’ of {Y;} in some optimal sense. Heuristically, this prediction is based on the idea that the events starting a downturn in x will be present in {X,} before they show up in {Y;}. In more precise terms, we assume that turning points are characterized by sudden switches ih the distribution function of {X,}. Let 2, an integer valued random variable represent the period of downturn. This random variable has the property that when {Z= k} the probability distribution of {Xk+j, i=O, 1,2,. . .} will be different and independent of the distribution of {X,-j, j = 1,2,. . .}. Accordingly, for {Z = k, kc t} we will have P(XSx, ,...) X,IXk ,..., X,~xx,)=FO(x, ,...) Xk-J.F’(Xk )...) x,), (1) where F’(v) and F’(e) are two distribution functions which represent the probabilities associated with X, during normal and downturn regimes respectively. The value of Z - i.e., the period during which distributions switch - is not directly observed. Instead, inferences based on the X, observed up to time t have to be used in order to see whether a downturn has started (Z 5 t) or not (Z > t). This characterization of the {X,} process is assumed to reflect the two properties that major macro-aggregates are believed by the pre-Keynesian business cycle literature to possess. The switch in the distribution function being able to deal with a more rapid decline in the economy and the independence between {Xi,. . .,X,-r} and {X,, . . .,X,} representing the suddenness of the downturns. The last component of the model is the existence of an a priori probability on when downturns are expected to occur. It is assumed that economic agents who have observed past cyclical downturns have developed an a
S.N. Neftci,
Optimal
prediction
of cyclical
downturns
229
priori probability, P(Z = k) = Pk,
(2)
representing the a priori beliefs on the period of the next downturn. Given this framework, the problem of predicting. the cyclical downturn can now be summarized. The objective is to forecast the value of Z after successive observations on X,. Let this forecast be denoted by z. Then {r = t} will mean that the forecaster is signalling a downturn at time t, and z-Z would represent the prediction error. Every time period the decision-maker faces two choices. Based upon the available information he can either ‘signal’ a downturn or instead wait and take another observation on the process {X,}. The first choice implies that the decision-maker is predicting at time t, that {ZS t}; the second implies that he is instead opting for {Z> t}. The optimal decision will be made after comparing the cost of sounding a ‘false alarm’ represented by {z
To determine the optimal prediction rule, we will, as is often done, limit our attention to a particular class of estimators. Definition I.
0)
An estimate r is said to belong to class M(F,, a) if
{w:z=t} EF,, P{z
(3) O$aIl,
where F, is the information set available at time t and WEB are ‘events’ causing T to assume the value t. Here, the condition (i) implies that the events determining the value of r will be part of the information set available at that time - i.e., r will be an F,-measurable random variable. In probability theory integer-valued random variables having this property are called stopping times. Thus, obtaining the best estimator r* E M(F,, a) is equivalent to finding the optimal stopping time. Condition (ii), on the other hand, says that the probability of giving false alarms of having {r < Z} is constant at a.
S.N. Neftci, Optimal prediction of cyclical downturns
230
The ‘best’ estimate from this class is to be selected according following criterion: The estimate of Z denoted by the class M(F,, a) if for all 7 E M(F,, a),
Definition 2.
E[max
(7*
-Z, 0)] sE[max(r-Z,
7*
to the
is said to be optimal within
0)],
(5)
where the E( *) is the conditional expectation operator given the information set F,. According to this definition, 7* will be the optimal estimate, if for a given false alarm probability, ‘the average delay’ in reporting the downturn is minimized. Given Definition 1 and Definition 2 we are now ready to state the main result which is basically due to Shiryayev (1978, p. 195): Theorem 1. Let the F,-measurable statistic 7~~be defined by ~~=P(Zskj
Fk).
(6)
Then, the optimal prediction rule in the sense of Definition 2 will be given by
7*=inf{k:Xk~~*},
(7)
k
where 0
The next lemma shows an important will be mainly useful in applications:6
property
of the statistic 7ck,which
% should be noted that an accurate estimation of pz+, is crucial method. This is true since, in general a downturn is signalled if a ‘decline’ the range of historical experience of normal-upturn periods. Also denominator of (10) so that small changes in it can cause large changes in
for the success of the is observed well out of p,“,, appears in the Q+ ,.
231
S.N. Nefcgi, Optimal prediction of cyclical downturns
Lemma 1. Let pi and p: denote the value of the conditional during ‘normaP and ‘downswing’ regimes, respectively, P:+ 1 =dPOh+
11 xo,. . -,Mdxk+
densities of X,
13
(8)
Then xk + 1 can be calculated recursively from nk by using the relation ,
a,+,=[a,+P(Z=k+l~Z~k)(l-23lp:+,
/{CBk+P(Z=k+lIZ>k)(l-Kk)lPkl+l +(l-rr,)p,O+l[l-P(Z=k+l~Z>k)]}. Proof
(10)
Available from the author upon request.
According to this lemma rrk+ i can be calculated sequentially by using the current observation of {X,} and the Q. To do this one needs the conditional probabilities pi+ 1 and pi+ I and the a priori probability P(Z = k + 1 I Z> k). The expression in (10) can also be used to study what makes the statistic rrL+ I move towards or away from A *. For example, if the currently observed X,, 1 is an ‘unusual’ observation for a downturn period, then pi+ 1+O and the formula in (10) suggests that rck+l will move towards zero and away from A*. At the other extreme, if Xk+l is unusual for an upswing period, then pE+ I +O, and according to the expression in (lo), rrk+ i will move towards one, and thus will most likely exceed A*. Finally, suppose the value of Xk+i is such that it has equal (conditional) probabilities of coming from a downturn or from an upswing regime, i.e., suppose pi+1 zpi+ i. Then, a simple computation shows that (10) reduces to rc~+l~rr~+(l-rrJP(Z=k+l
IZ>k).
(11)
Thus, an ambivalent observation will lead to an increase in nk+r if the a priori probability P(Z = k + 1 I Z> k) is non-zero. If the n, had been less than A* and if one has a rather ‘flat’ prior on Z, then the rrnk+i will generally not exceed A* and one would not signal a downturn in period k + 1. At this point, it may be instructive to compare the procedure suggested by Theorem 1 with some other rules commonly used to predict downturn periods. One of these is ‘three consecutive declines in Leading Indicators’, a rule traditionally associated with NBER. According to this, the estimate of the downturn period will be given by ?=inf{k:AX,
AX,-,-CO,
AX,-,
(12)
232
S.N. Neftci,
Optimal
prediction
of cyclical
downturns
where 4X, is the change in leading indicator series. Obviously, the value of f will not in general be the same as the estimate given by Theorem 1, z* =inf{k:n,
zA*}.
k
Also note that the determination of ? is not explicit about the false alarm probability, although one may argue that the class of rules such as ?,=inf{k:dX,-j
j=O, l,.. .,m}
(13)
k
has an implicit way of taking into account the false alarm probability by using the value of m. For example, with m=2, as in the NBER approach, one would implicitly be settling for some specific false alarm probability, a smaller m leading to a higher false alarm probability. 3.3. Verification of the signals
The last point that needs to be discussed is the verification process of the signals. Suppose using Theorem 1 and Lemma 1, one obtains {z* = k} - i.e., signals at time k, that a downturn is ‘imminent’. How many periods is it going to be allowed to pass before this signal is labeled a ‘false alarm’? In other words, how is one going to distinguish between a ‘false’ and a ‘too early’ signal? This issue is closely rel,ated to the timing relationship between the ‘indicator’ {X,} and the ‘actual’ series {x} is discussed in NeftCi (1979). The usual way of dealing with this problem is to decide a priori, on a time interval A, which one would allow to elapse, before a signal is labeled as a false alarm. According to this, with {z* = k}, one would expect the {x} series to enter a downturn regime between k and k + A. If this doesn’t happen, then the signal {z*= k} would be called a false alarm. Obviously, this procedure assumes that there would be no difficulty in observing the turning points of {Y;}, an assumption which is reasonable in reality, where a recesssion is usually defined as two consecutive declines in the Gross National Product. Accordingly, {z*= k} would be a correct signal if inf{j:(&+jyk+j-l)
S.N. Neftci, Optimal prediction of cyclical downturns
233
one can select A by letting A =+(I+),
&>O.
This would then be consistent with the definition of a leading time series [see NeftCi (197911. Obviously, this way of dealing with signal verifi&tion is somewhat ad-hoc. For example, tile parameter A and the correlation between {X,} and {x} is not taken into account during the derivation of the ‘optimal’ prediction rule. However, the explicit determination of stopping rules in cases incorporating such characteristics is still an unresolved question and is beyond the scope of this paper. 4. Applications
The main characteristic of the business cycle model introduced in the previous section is the unobserved and sudden switch in the distribution of the economic process at some random time period. In this section we will present applications of the procedure introduced above. The index of Leading Indicators series from the U.S. is selected and used to compute estimates of {nEk} for the period 1970-79 to see whether the procedure predicts the 1974 recession, and the slowdown of the economy in 1979. The application involves a simple version of the framework outlined above. It assumes that increments, AX,, of an observed economic time series are generated by a switching model AX, = a0 + E;, =d+&:,
lS;t
(14)
where 2 is the unknown random time of downturn. The {$} are an i.i.d. sequence with a common mean 0. The {E:} are an i.i.d. sequence (independent of the Ep) with common mean 0.’ By observing the sequence X, we wish to ‘detect’ whether the process AX, has switched from regime 0 (normal) to regime 1 (downturn). Also, we need to start t from 1 rather than zero and interpret the break in distributions as if it applies to AX, rather than X,. Obviously this process has all the characteristics of the {X,} series discussed in earlier sections. The applications presented here will assume that the leading indicator series can be reasonably approximated with this model. ‘It is worth emphasizing that the sudden change in the distribution of X, need not necessarily be due to different variances of E,‘s.The distributions may, in fact, belong to different classes.’
234
S.N. Nejtci, Optimal prediction of cyclical
downturns
An application of the prediction rule introduced in Theorem 1 to the model above requires a number of intermediate steps. First among these is the need to estimate the probabilities p”=P(dX,=x,IZ>t)
and
p’=P(dX,=x,~Z~t),
and determine an a priori probability on Z. Second, one has to select an estimate for the initial value no and using (10) calculate successive values of R, recursively. After doing this one decides on the false alarm probability a, and compares each a, with ,4*(a). A downturn will be signalled at the first occurrence of the event {7r,zA*}. The estimates of {p”} and {p’} are shown in fig. 1 and table 1. To obtain these estimates the data for leading indicators belonging to 1948-1970 were split into periods of downturn and upturn regimes,8 the observations belonging to upturn periods were grouped and {p”} were estimated using the frequency distribution of {AX,}. The {p’} were estimated similarly using the frequency distribution of {AX,} belonging to downturn periods. The data for post-1970 were not used and instead were saved for out-of-sample predictions. The a priori probabilities P(Z= k 1Z> k- 1) were approximated using a symmetric density with a mean of 50 months (see table 2). According to this density, an economic agent with no information on {X,} would predict the next downturn to occur 50 months after the last downturn [see Taylor (1978)]. This seemed to be a reasonable approximation to the average .distance between two turning points in the U.S. economy. It turns out that the results are not sensitive to any reasonable variation in this density. Finally, the starting point of the prediction process, go, was selected as zero. This was believed to be a good guess for P(Z=O 1F,), the probability of having a downturn immediately following the upturn. Using this procedure, the detection rule given in Theorem 1 was applied to two periods of interest: 1971-1974 and 1975-1979. The first case involves the 1974 recession. This seemed a natural period to select. The second is the subsequent slowdown of the economy and appeared to be another test of the procedure. For both cases the false alarm probability was selected as 10% (i.e., a=O.l) and A* was approximated by (1 -a).’ sTo estimate pt and p:, the period was split in the following way: Overall period: 1948-1970. Downturn regimes: January 1948 through June 1949, April 1953 through November 1953, October 1955 through January 1958, May 1959 through December 1960, February 1969 through March 1970. Upturn regimes: The remaining intervals of the period. 91t is easy to prove that .A*6 1 -a. Shiryayev (1978, p. 200, remark 4)].
So that if n, >(I -a)
then of necessity xI> A*
[see
. . .‘C .*.... . .
5
. . . . . . . . . . . . . . .
2
. . . . . .
>
. . . . . . . . . . . . . . . . . .
q
. . . . . . . -,
I
. . . . l .*..**..
ry
5 i
. . . . . . . . . .
1 !?I PC ?
. . . . . . . . . . . . . .
. . . . . . . . . .
. . .
y
I
........... ......... ‘: ........ .1 0-Y
. . . . ........T ..1:y
.$I
. . . . . . . . . . . . . ..*......
u)
-i
.......y il 09
. . . .
..I1 . .. 2 .
-II
. . . . .
2 I
. . . . . T . P --i -.
236
S.N. Neftgi,
Optimal
prediction
o/cyclical
downturns
Table 1 Estimated distribution functionsa Estimated F”, the ‘upturn’ density
Estimated F’, the ‘downturn’ density
Value
Probability
Value
Probability
- 1.5 -1.4 -1.3 - 1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
0.002 0.002 0.004 0.006 0.006 0.011 0.015 0.020 0.017 0.017 0.013 0.020 0.020 0.019 0.025 0.035 0.050 0.060 0.063 0.070 0.066 0.070 0.057 0.053 0.050 0.047 0.032 0.032 0.032 0.033 0.023 0.015 0.015 0.005 0.003 0.003
-2.1 -2.0 -1.9 -1.8 - 1.7 -1.6 -1.5 -1.4 -1.3 - 1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.026 0.026 0.037 0.020 0.020 0.020 0.030 0.043 0.090 0.100 0.090 0.043 0.020 0.083 0.090 0.060 0.027 0.030 0.030 0.023 0.003 0.000 0.003 0.003 0.003 0.003
’
“This table is obtained by taking a centered 3-term moving average of the frequencies in Rg. 1. “Table 2 was obtained in the following manner: First a probability density was generated by taking a centered 38-term moving average of a Daniel window with 36 elements. The resulting numbers were averaged again in a similar manner. Then, using this density we calculated the conditional probabilities given above. Fig. 3 was calculated by starting from k= 13 (this is the case since the ny’s for 1975 are shown in fig. 3). One can also let these conditional probabilities approach a limit E>O as k gets large. This can easily be done by applying a Daniel window with more elements.
S.N. Nejtqi,
Optimal
A priori probabilities k 0 1 2 3
4 5 6 7
a 9 10 11 12 13 14 15 16 17 18
19 20 21 22 23 24
29 30 31 32 33 34 35 36 31 38
39 40 41 42 43
44 45 46 47 48
49
of cyclical
downturns
Table 2 (periods of downturn k = 1,. . ., 99).
P(Z=klZ>k-1)
k
P(Z=klZ>k-1)
0.000 0.000 0.000
50 51 52 53 54 55 56 51 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 14 75 76 77 78 79 80 81 82 83
0.048 0.049 0.050 0.051 0.052 0.053 0.054 0.055 0.056 0.058 0.060 0.062 0.065 0.066 0.067 0.068 0.071 0.075 0.078 0.081 0.085 0.089 0.095 0.099 0.103
Kt 0:002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018
0.019 0.020 0.021 0.023
25 26 27 28
prediction
0.024 0.025 0.026 0.027 0.028 0.029
0.031 0.032 0.033 0.034 0.035 0.036 0.037 0.038
84 85 86
0.039 0.040 0.041
91 92
0.042 0.043
i: 95 96
0.044
ii 89 90
0.045 0.047 0.047
“See previous page.
ii; 99
0.109 0.111 0.119 0.120 0.140 0.160 0.180 0.213 0.241 0.390 0.418 0.567 0.604 0.699 0.749 0.805 0.840 0.880 0.899 0.907 0.996 0.999 0.999 0.999 0.999
237
238
S.N.
N&i,
Optimal
prediction
of cyclical
downturns
4.1. Case I: The 1974 downturn
Fig. 2 displays the results. The upper part of the table shows the behavior of the {AX,} process during 1971-1973. The lower part of the table displays the estimated (72,; k= 1971; l,...; 1973; 8). According to these results, Wk remains close to zero and does not approach the threshold (1 -a) during the years 1971 and 1972. But the situation changes in 1973. The & becomes 0.4 during April and eventually exceeds the critical level 1 -a during August 1973. Thus, in this particular case the rule gives a clear signal of the 1974 recession. This is a rather early detection of the 1974 recession. Moreover, this prediction is accompanied by an upper bound for the false alarm probability: 1974 recession is signalled at most with a 10% room for error. It is also interesting to note that the prediction of the 1974 downturn occurs before the oil shock, although it appears that oil shock has affected the severity of the recession. This suggests .that the 1974 recession was in some sense similar to previous downturns in the economy and was not a ‘unique’ product of exogenous shocks. To see this, note that the transition of Sk from k- 1 to k depends essentially on the values assumed by the ratio p”(dX,=x,)/pl(dX,=xJ [see (lo)]. But these weights, PO(.) and p’(e), were estimated by partitioning the data using the turning points experienced before the 1970s. Thus, one would expect the ratio p”/pl to decrease as the behavior of {AX,} becomes similar to that experienced during previous downturns. The prediction rule of Theorem 1 signals such a ‘resemblance’ during August 1973. This ‘signal’ is then followed by the 1974 recession. 4.2. Case II: The period 1975-1979
Fig. 3 displays the behavior of the B,‘s since 1974. Two interesting results emerge. First, the procedure does not signal any downturns for 1976, 1977 or 1978. In contrast, the popular rule of ‘three consecutive declines in X,’ does give a false alarm in August 1977. During the same months, fi, is around 0.5 and therefore far from signalling a downturn. The second interesting result is the prediction in mid-1979 that a downturn will occur ‘soon’. However, unlike the case of 1974, this signal is not given suddenly. The rrk go up and down for a number of periods before a downturn is signalled. 5. Conclusions
This paper presents a Business Cycle model which assumes that underlying economic processes are subject to shifts in their distributions at random time points. Further, these shifts occur suddenly, in the sense that once the economy enters a downturn regime dependence on the history before the turning point becomes negligible.
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S.N. Neftci,
Optimal
prediction
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S.N. Neftci, Optimal prediction of cyclical
downturns
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It is shown that if economic time series possess these characteristics, an explicit optimal prediction rule can be derived to forecast cyclical downturns. This rule is optimal in the sense of signalling the downturn as early as possible for a given probability of false alarms. The application presented in the paper suggests that the optimal rule may indeed be a useful procedure for predicting cyclical downturns of the U.S. economy. References Box, George and Gwilym Jenkins, 1976, Tie series analysis (Holden-Day, San Francisco, CA). Frisch, Ragnar, 1965, Propagation problems and impulse problems in economics, in: Robert Gordon and Lawrence Klein, eds., Reading in business cycles (Irwin, Homewood, IL). Gordon, Robert, 1979, The end-of-expansion phenomenon in short-run productivity behavior, Brookings Papers on Economic Activity 2,447-461. Hicks, John, 1949, Mr. Harrod’s dynamic theory, Economica 16,32-47. Hicks, John, 1950, A contribution to the theory of trade cycle (Clarendon Press, Oxford). Keynes, John M., 1936, The general theory of employment, interest and money (Macmillan, London). Kushner, Harold, 1977, Probabilistic ,methods in approximation of elliptic equations and stochastic control (Academic Press, New York). Mitchell, Wesley, 1927, Business cycles (NBER, New York). NeftCi, Salih, 1979, Lead-lag relations, e.xogeneity and prediction of economic time series, Econometrica 47, 101-113. NeftG, Salih, 1979, The asymmetric behavior of economic time series, Manuscript. Rose, Henry, 1967, On the nonlinear theory of the employment cycle, Review of Economic Studies 34, 153-173. Sargent, Thomas, 1978, Estimation of dynamic labor demand schedules under rational expectations, Journal of Political Economy 86, 1009-1045. Schinasi, Gary, 1979, A non-linear disequilibrium dynamic macro-economic model, Ph.D. dissertation (Columbia University, New York). Shiryayev, Alexi N., 1973, Statistical sequential analysis: On optimal stopping rules, translations of mathematical monographs (American Mathematical Society, Providence, RI). Shiryayev, Alexi N., 1978, Optimal stopping rules, Application of mathematics (Springer-Verlag, Berlin). Sims, Christopher, 1975, Labor and production in manufacturing, Brookings Papers on Economic Activity 2, 357-401. Taylor, John, forthcoming, Output and price stability: An international comparison, Journal of Economic Dynamics and Control. Varian, Hal, 1979, Catastrophe theory and the business cycle, Economic Inquiry XVII, 14-28. Wecker, William, 1978, Asymmetric time series, Manuscript. Wecker, William, 1979, Prediction of turning points, Journal of Business 52, 35-50. Zarnowitz, Victor and Charlotte Boschan, 1975, Cyclical indicators: An evaluation and new indexes, Business Conditions Digest 15, v-xix.