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Effects of alternative seasonal adjustment procedures on monetary policy
EFFECTS OF ALTERNATIVE SEASONAL ADJUSTMENT PROCEDURES ON MONETARY POLICY* Agustin
MARAVALL
Bunk of Spain, Madrid
14, Sptrirl
1. Introduction The seasonal adjustment method presently used at the Board for adjusting M, is a blend of X-11 and judgmental corrections applied to the monthly series of demand deposits and currency. Since the monthly (unadjusted) series are averages of daily values, an alternative procedure for obtaining the monthly adjusted figure would be’ to adjust the daily series and then aggregate them into months. In other words, temporal aggregation can precede or succeed seasonal adjustment. The use of a daily method was recommended by the Bach Committee on Monetary Statistics.’ In this paper we compare the official Board method with the daily one developed at the Board, which is an extension of the one actually used by the Bach Committee. It is argued that there are reasons for preferring a daily approach and reasons for preferring the monthly one. It seems safe to conclude that, to the extent that neither of the two approaches is unquestionably superior, they provide two alternative methods for adjusting the series. Insofar as the two methods do not yield identical results, the difference between them reflects an ambiguity in the computation of the series of interest (monthly seasonally adjusted M,). Logically this ambiguity sets a lower limit to the precision with which the series can be measured and therefore has the character of a measurement error. Naturally, if in most practical situations the difference between the two methods happens to be very small, the ambiguity could be easily ignored. On the other hand it may be that on occasion the judgment as to whether growth of M, has been adequate or not and therefore the *This work was primarily done while the author was an economist on the staff of the Division of Research and Statistics of the Federal Reserve Board of Governors. Thanks are due to D.A. Pierce. K.J. Singleton, A. Zellner, H.H. Stokes, R. Meese and R.D. Porter for their comments. The opinions expressed are those of the author and do not necessarily represent those of the Federal Reserve Board of Governors. ‘See Advisory Commtttee on Monetary Statistics (1976, pp. 4 and 39-40).
direction of monetary policy depends crucially on the method used for seasonal adjustment. Of course, economic time series are used by decision makers when setting policy. These series, we know, have errors, yet this knowledge is usually ignored. One can wonder what part of economic policy is simply the result of measurement errors. This paper can be viewed as a crude attempt at providing some insights into the problem for a specific error of measurement in the context of monetary policy.’ In the next section we apply the two methods of seasonal adjustment to the M, series for the period 1971-1977. This yields a series of differences between the two adjusted values which we proceed to analyze. Then we provide an estimate of how likely it is that the direction of monetary policy depends on the method used for seasonal adjustment and how is translates into an equivalent imprecision in terms of interest rate movements. It is also seen how the effect of the ambiguity can be decreased.
2. Daily versus monthly seasonal adjustment of M, The first series we consider is the seasonally adjusted monthly money supply (M,) series published by the Federal Reserve. This series is obtained by applying X-11, and then judgmentally modifying the factors obtained, to the two components of M,: currency and private demand deposits. The procedure is discussed in Fry (1976) and in Pierce et al. (1978). The second series we consider is the monthly average of seasonally adjusted daily values of M,, obtained also as the sum of adjusted currency and demand deposit components. The daily seasonal adjustment procedure we consider was developed at the Federal Reserve as an extension of the one used by the Bach Committee. The procedure is described in Pierce and Fries (1978).3 Perhaps some general comments on the two methods are in order: Aside from the stochastic characteristics of the series, optimality of a seasonal adjustment method depends on the use to be made of the series4 In our application, this use is the setting of monetary policy by the Federal Open Market Committee (FOMC). Since the growth of M, (seasonally adjusted) has been for some time a crucial variable in monetary control, optimal seasonal adjustment of M, would require separate estimation of the seasonal which is the result of past policy action and the one which is due to ?Attempts at quantifying measurement errors are infrequent m economics. According to Morgenstern (1963, p. 7): ‘there is much less occupation with errors than in other fields. This is undoubtedly one of the reasons why the social sciences have had a rather uncertain development.’ 3The method is an extension of the daily method described in Pierce et al. (1978), along the lines developed in Pierce (197X). “See Grether and Nerlove (1970).
A. Murtrtall,
Seusor~al adjustment
and monetary
policy
117
‘exogenous’ causes. Otherwise, monetary policy would be contaminated by undesirable variables, such as for example, past errors in the estimation of seasonal factors.’ Thus, strictly speaking, optimality of the seasonal adjustment of the monetary aggregates in the context we consider possibly requires the use of an appropriate econometric model. Although some progress in that direction has been made by Plosser (1978, 1979), practical enforcement of such methods in ongoing monetary policy is still very distant6 Meanwhile, what can be said of the relative merits of the two methods we consider? A theoretical reason for preferring a daily approach is provided in Geweke every conceivable time (1978), where it is shown that ‘for virtually series . . . and a reasonably inclusive class of potential adjustment procedures, minimum mean-square-error adjustment implies that seasonal adjustment should always precede temporal or sectorial aggregation.’ Yet his result cannot be applied blindly. First, it is hard to assess whether the combination of X-11 and judgment falls into his ‘reasonably inclusive class’ of adjustment procedures [see Love11 (1978)]. Second. as shown in Taylor (1978): ‘It is not the preaggregation adjustment that is crucial for optimality, but rather the utilization of all observable components simultaneously.’ Insofar as the judgment corrections may use daily information, Geweke’s result would not apply. Other relative advantages of the method based on the daily procedure are that it allows for a more complete treatment of special-day events, and that it makes possible the construction of perfectly consistent seasonally adjusted figures expressed in different time intervals. On the other hand, there are advantages associated with the aggregate monthly method. First, the computational complications of the daily approach are not trivial. Second, monthly data tend to have smaller measurement errors.’ Finally, misspecification may be more of a problem in the daily approach.8 Thus it seems safe to conclude that neither of the two approaches can be seen as unquestionably superior. Since presently one of the methods is being used, and the other is being considered for use by the Federal Reserve, they provide a meaningful comparison for answering the question of how sensitive monetary policy is to alternative seasonal adjustment methods. Let m, stand for the seasonally adjusted monthly figure of M, at time t. ‘This point is discussed in the report of the Bach Committee [see Advisory Committee on Monetary Statistics (1978, pp. 37.39)]. ‘See Lombra (1978). Possrbly this distance is reinforced by the additional (optimality) requirement of specifying the correct aggregate loss function of the 12 FOMC members. ‘See Porter et al. (1978b). sThe preference for the daily approach expressed by several of this paper’s referees is in contrast with that of the members of the Federal Reserve Advisory Committee on Seasonal Adjustment and with that of Shiskin (1978).
Since monetary policy utilizes annualized periods, we shall operate with the variable
rates
of growth
over two-month
rr = 6[ (in, + z - rn, )/WI,]. When m is the official Board estimate, the variable rt shall be denoted rp. When III is computed through the daily approach, we shall write rp. The two seasonally adjusted series are displayed in table 1. Table 2 displays the autocorrelation function (ACF) of ry and r:. For both series the functions are extremely close. No seasonality seems to remain, and only low-order autocorrelation is significantly different from zero. Also both series are seen to be stationary. Possibly the most noticeable difference between the two is the difference in variances, which indicates that more seasonality is removed from the series when the daily procedure is used. Let is denote the difference between the two series: 6, = ry - ra f’
Tables 1, 2 and fig. 1 display the series 6, and its ACF. Despite that no seasonality can be detected in the r-series, their difference strong semi-annual and annual patterns.
3. The effect of alternative measurements versus non-tolerable growth of M,
3.1. Alternutive
mea.surements
on monetary
of’ M, and monetary
control:
the fact exhibits
Tolerable
control
At each monthly meeting, the Federal Open Market Committee sets a range for the rate of growth of the monetary aggregates (seasonally adjusted) which is considered tolerable over a two-month period. Policy action will depend among other things -- on whether or not the growth of the aggregates falls within this range.” In the case we are analysing, there are two alternative ways of measuring the rates of growth. Consider two monetary authorities, A and B, both of which follow identical behaviour. They set a range of tolerance for the rate of growth of M, seasonally adjusted, and rates outside the interval trigger some kind of intervention. Assume A and B are identical in every respect, with the only difference being the way they adjust for seasonality. A adjusts the monthly aggregates, while B adjusts the daily data, then aggregates them into months. Specifically, A operates with PO,while B operates with r“. “For a more complete
description,
see Wallich
and Kerr (1978) or Lombra
and Torte
(lY75).
235732 235867 0.3584
257737 257839 0.3051
271761 271841 -0.5376
282637 283017 - 0.7697
295918 29628 I - 0.7604
314919 314407 1.1202
1972
1973
1974
1975
1976
1977
316336 316309 1.105
297936 297853 2.2393
282787 283133 0.7626
273107 273371 0.3342
258162 258304 ~ 1.7674
237721 238041 - 0.6723
222746 223177
Feb.
228302 227711 1.7824
225828 225685 1.5568 241407 241288 1.1172 259012 259023 0.3046
224362 224437 I .3426
240025 240040 0.3127
258179 258069 0.493 1
322382 322221 - 0.3220
301462 301692 0.6327 322008 321865 0.2183
299328 299047 1.313
318313 3 17986 ~ 0.3627
303347 303029 0.0664
284756 285042 0.1330
284975 29494 1 0.8821
278967 278746 - 1.0946 292405 292418 _ I .9856
278316 278026 PO.2050 291068 290609 0.0088 304134 304469 - 1.2944
277832 277109 1.6786 291205 29026 1 2.6025 303537 303006 1.5166
(My).
327463 327611 -0.5812
265186 264532 - 0.7233
264677 263852 0.3642
264043 263076 2.2659
324306 323797 0.6823
247265 246646 0.4949
245099 244654 0.0463
242860 242448 0.7280
329231 329279 - I.0488
305840 306002 _ I .3797
231854 231011 0.3988
230723 229806 0.8439
229436 228753 1.4312
Aug.
July
I
June
(MF). from the daily method
287635 287185 0.8766
275389 275435 0.4848
274839 274768 0.3354
276228 275847 0.6765
261896 261236 I .2745
242205 241783 1.0909
May
April
March
“Seasonally adjusted monthly series of M, Published %easonally adjusted monthly series of M, Obtained ‘Monthly series of 6,( = rp - rp ).
220797” 22135@’ _c
1971
Jan.
Table
331649 331798 0.0032
306756 307224 - 0.2563
334730 333811 1.765
310133 309746 1.0815
292923 292813 0.2525
280704 280864 -0.8212
279615 279599 -0.5948 293442 293217 -0.4918
266168 266174 - 1.502X
265075 265065 - 1.8573
251050 25 1079 - 1.601
232714 233073 -3.1281
232270 232306 - 2.5047 249237 249367 _ 1.4306
Oct.
Sept.
B 2’
282822 283528 - 1.167
334941 334657 0.7854
310563 3 I0630 0.7944
295430 295418 - 0.4403
282311 282328 -0.0715
337225 336610 -0.5581
312576 313119 - 1.8077
294509 295518 -2.2941
g g g
270474 270885 -0.9145 268629 26847 1 0.3366
$ 5
? P 2 ‘2
: ‘4
g _ s ;
g 255276 254675 1.5087
252179 252404 - 0.2249
.~
234012 234069 0.7840
Dec.
233099 233370 - 0.6082
No\.
1
0.50 -0.23
0.49 - 0.21
0.34 0.13
0.37 -0.10
Lags
l-12 13-24
l-12 13-24
1-12 13-24
l-12 13 24
-0.18 0.10
- 0.24 -0.20
- 0.02 - 0.08
0.02 - 0.07
2
-0.18 0.04
-0.12 - 0.07
- 0.05 0.09
- 0.06 0.07
3
- 0.05 0.2 I
0.01 0.03
-0.01 0.24
-0.12 0.22
4
0.08 0.17
-0.18 -0.14
0. I8 0.21
0.10 0.15
5
Table
7
<
- 0.02 -0.10
-0.17 - 0.09
e,
-0.15 - 0.07
(d) Variable
- 0.49 - 0.29
(c) Vuriable 6,
0.00 -0.11
(b) Variabk 0.17 - 0.01
8
- 0.06 - 0.08
0.09 - 0.02
0.04 - 0.09
0.04 -0.15
functions.
2
(a) Variable r: 0.17 -0.01 - 0.05 0.06
6
Autocorrelation
0.01 -0.16
-0.09 -0.11
0.11 - 0.06
0.12 -0.14
9
0.02 0.00
-0.12 0.01
0.15 0.02
0.14 0.04
10
0.10 0.03
0.31 0.28
0.15 - 0.05
0.16 0.01
11
0.02 0.07
0.59 0.33
- 0.09 -0.14
- 0.08 - 0.08
12
0.12 0.15
0.11 0.18
0.11 0.15
0.11 0.15
Error
2 S 2 % 0 z c
2 9. 3
& 0 B a $ :: 3 2
2
$ _%
A. Marcrcall,
Seasonal
adjustment
and monetary
policy
Fig. 1. Series [b,J
Let A and tolerance for the previous =0) so that, for both. For that A and B
B each hold a meeting at time t in which they set a range of the growth of M, over a two-month period. Assume that for period, both measures of r yielded the same figure (thus a,_, everything being equal, the range of tolerance will be the same a specific growth of M, over the period, what are the chances would disagree as to whether or not that growth is tolerable?
The probability
of disagreement
(1) The probability inside (P,). (2) The probability
that Y: falls outside that
not (p,,). (3) The probability that opposite sides (P,,,). Assume
that
is the sum of three components:
rp falls inside both
the forecast
rf and
the interval the interval
of tolerance of tolerance
rp fall outside
of rp at the time
the
the meeting
and VPfalls and rf does
interval
is held
but
on
is the
122
____-
I
St
_-___
,
,
/’
/’ 1’
Fig. 2
midpoint of the interval of tolerance (0 in fig. 2).” The forecast error et is then the distance between rp and 0. Let x denote the midrange of the interval of tolerance. It is easily seen that (deleting the subscript t for notational simplicity): (1) When r” is outside the tolerance interval, then 1.“ will be inside if and only if 16- ~1< CI (see fig. 2). (2) When Y’ is inside the interval, then P will be outside whenever 16- rj > x. (3) When Y* is outside, for P to be also outside but on the opposite side it has to be that 6>r+cc if P>O, and 6
“A typical ‘Record of Policy Actions’ of FOMC (published in the Federcd Resrrw Bulktir~) contains the following statement: ‘The Committee agreed that if growth rates of the aggregates over the 2-month period appeared to be deviating significantly from the midpoints of the indicated ranges, the operational objective in the weekly-average Federal funds rate should be modified _’Also, Lombra and Torte (1975, p. 11) state that ‘from the viewpoint of the staff the ranges _._ represent a standard error around a point estimate at the midpoint of the range.’ Thus the assumption that the mldpomt of the range can be seen as an approximation to a staff forecast seems reasonable. Of course. this will not always be the case.
A. Mururull,
and the total probability
Seusonul udjustmrnt und monrtur~ polic,!
of disagreement
123
is equal to
P = P, + P,, + P,,,. In order to compute the probabilities P,, P,, and P,,,, we need the joint distribution of 6, and e,. In the appendix we conclude that this distribution can be assumed to be bivariate normal, with zero mean vector and covariance matrix R, as given in the appendix. The value of c(, the midrange of the the interval of tolerance is published in the Federal Reserve Bulletin. For the past years it has oscillated between 2 and 3. We set tl=2.5.” Before performing the integrations, a minor modification has to be made. In order to simplify the comparison between the two monetary authorities we set before cS_~=O. Thus in our computation, instead of the marginal distributions of 6,, we should employ the distribution of 6, conditional on 6,_, EO.12 From the expression derived in the appendix, it is obtained that r&,,_2=o=1.22. Numerical computation yields the following results: P,=O.O76,
P,,=O.O53,
P,,,=O.O28
(10-5),
P=O.13.
P, is the probability that, for the same growth of M, in period t, B considers the growth tolerable and A does not; P,, is the probability that A considers the growth tolerable and B does not. Obviously I’,,, can be neglected.13 Thus, for a given growth of M, in period t, there would be a 7.67; probability that A would think monetary intervention was granted and B that A would think monetary would thrnk the opposite, a 5.3 ‘;‘, p robability intervention was not granted and B would think the opposite, and in general, a 13 y0 probability that the two monetary authorities would disagree. Approximately, one out of eight times disagreement would occur. The estimated probabilities of disagreement are, so to speak, unconditional probabilities in that they do not take account of economic or policy conditions which may affect the joint p.d.f. For example, it could happen that deviations of ~4, from range midpoints are positively correlated with the rate of expansion of economic activity. In general, it would not be correct to “One implicit assumption in our formulation is that the p.d.f. of r is independent of n. This can only be seen as a simplification, since it could be that a wider tolerance range indtcates a greater wrllingness by the FOMC to tolerate wider departure of M, from the range midpoint. I am grateful to a referee for pointing out this qualification. ‘2Naturally. using u,“,,,_ *= 0 in the computatton of P biases downwards the estimntcd probability of disagreement. “A necessary (though by no means sufficient) condition for r0 and r’ to fall outside the tolerance range and on opposite sides, is (6/> 22. Since P([6[ > 2~) = ZP(r > 4.3), where z - N(0, I j, it follows that P,,, is smaller than (0.04)x lo-‘. Therefore, for the rest of the analysis. the P,,, type probability of disagreement can be safely ignored.
124
A. Morarall,
Seasonal
adjustment
and monetury
policy
assume that the probability of disagreement is invariant nomic conditions and monetary policy directives.14
3.2. SensiticitJ?
qf‘ the
result
to purumeter
changes:
The
to different
eff;ct
eco-
qf forecasting
improvement
The numerical result of the previous paragraph was computed for values of c1 and r~, typical of recent years. Naturally, the smaller the value of CI, the larger would be the probability of disagreement. Since over the last years, a has oscillated between 2 and 3, the probability of disagreement would have ranged between 13.8 and 11.5 percent. More interesting is the relationship between the probability of disagreement and G<, the standard error of the forecast of the rate of growth of M, over the two-month period. We shall assume that c( remains constant, although it would seem plausible that improvements in the ability to forecast M, could lead to a reduction of the interval of tolerance. However, the oscillations of u over the past years seem to have been motivated by other considerations. Also, we shall assume that crBremains constant. It is easily seen that the function that relates P to ge tends to a minimum for g,-+O and O,+KI. This means that, for a positive value of oe, a maximum for P has to exist (since the function is bound). Thus in some region, improvements in forecasting accuracy increase the probability of disagreement. Fig. 3 plots P as a function
of (T,. Two things can be noticed:
(a) The result obtained (disagreement would occur one out of eight times) is robust with respect to present (and at least short-run expectations of) forecasting accuracy. For the range 2.5
to a referee for pointing
out this qualification
A. Maruoall.
Srasonal
adjustment
and mon~ur)
policy
125
Thus P, and P,, can be seen as the probabilities of type I and type I1 errors, respectively. If non-tolerable growth implies intervention and viceversa. then P, represents the probability of intervention when none should have been granted, and P,, represents the probability of no intervention when there should have been some. From the values computed, it is seen that, over the relevant range of (TV,the probability of a type I error is slightly larger than the probability of a type II error. This may reflect an implicit FOMC belief that it is more dangerous not to intervene when intervention is granted than to intervene when it is not needed.
-----a_-____ i 0.05 0.00
I
I 1.67
I
I 3.33
5.00
I 6.67
I 8.33
10.00
Fig. 3.
4. The effect of alternative measurements
on interest rates
In the previous section we provided a quantitative measure to a qualitative question. We estimated what would be the probability that two alternative seasonal adjustment methods would lead to disagreement in terms of whether or not the monetary authority should act to modify the funds rate. In this section we provide an answer to a different question: what would be the difference in interest rates when the two alternative seasonal adjustment methods are used in the setting of monetary policy? Since the difference in funds rate changes (according to which method is used) can be interpreted as an imprecision in detecting federal funds movements, it sets a lower limit to the ‘finesse’_ with which monetary policy can be instrumented given present operating procedures.
Consider a monetary authority with a reaction function which specifies that changes in the funds rate are related to a set of variables, some of which are related to the seasonally adjusted rate of growth of M ,,while others are independent of M, measurements. Assume that monthly changes in the Federal funds rate, J,, are expressed as a function of forecast errors of the two-month rate of growth, of M, (seasonally adjusted), e,. plus a set of variables such as changes in the unemployment rate, the rate of inflation, etc., which we assume unaffected by errors of measurement in M,.’ 5 Let z denote the vector of these variables so that
where t(B) denotes a polynomial lag. If rp were to be used, the forecast error would become c,, where (;I=P, -0,. The difference in x, from using one rate or the other is, therefore.‘”
In Farr (1978) several reaction
functions
are estimated.
The filter
with parameters equal to to =0.4, t, =0.03, c2 =O.Ol, seems to provide a formulation in agreement with all the results obtained. Applying t(B) to the series ii,, a series do, is obtained. Expressed in terms of basis points,” the standard deviation of the difference in interest rates is equal to 6.2. Since the standard deviation of month-to-month changes in the funds rate (over the sample period) is, approximately, 50 basis points, these results imply that 13 YiOof it could be accounted for simply by the difference in the two seasonal adjustment methods we considered (when applied to the series). Also, since 203,,= 12.4, a change in the funds rate of l/8 of a percent point could be attributed to that difference. To obtain an additional estimate, we used the reaction function presented in De Rosa and Stern (1977) for the period 197&74. In their formulation. month-to-month percentage change in the funds rate is related to a three‘“This last assumption is, again, a simplification. Since, for example. measurementa of M, influence policy and policy Influences inflation, IndIrect relationships could exist. lhWe are assumin_g that the lag structure c(B) is the same whether we use official adjustment or adJustmerIt based on the daily method. Since the ACF of rp and r; are extremely close, possibly the difference m t(B) would be small, and the results presented below would not be much affected. “A basis point represents a change of I Oo.
month
moving
average
percentage
change
in M, through
the filter
as 14.64, 0.304, -0.282 and 0.016, where cc,, cr, c1 and c3 are estimated respectively. If SF denotes the difference in the three-month moving average percentage change in M, when the two seasonal adjustment methods are used, the standard deviation of the series, AI‘: = i’*(B)h:, is equal to 7.43, where the unit again is basis points. Therefore, the imprecision in detecting month-to-month funds changes is of similar magnitude for the two formulations of the reaction function. 5. Some implications
5.1. Ambiguities
in seasonal adjustment
In the previous two sections we have analysed the effects of using the seasonally adjusted monthly average of daily values of A4, versus the monthly average of seasonally adjusted daily values of M,. We found that, depending on which one is used, the monetary authority would disagree one out of eight times as to whether or not intervention is warranted and that a difference of l/8 of a percent point in the Federal funds rate could separate policies using the two alternative methods. So despite the fact that no criterion seems to be available that would establish which of the two methods is the right one to use, the implications of choosing one versus the other do not appear to be negligible. From a general point of view, the problem we have analysed reinforces the idea that seasonal adjustment may possibly affect the data considerably more than is intended, and that this is not only due to the shortcomings of the seasonal adjustment methods presently available, or to the existence of revision errors. In fact, there are problems due to ambiguities which arise from the fact that we do not know exactly what we want to measure.” Despite all its limitations, there is little doubt that, for the time being, seasonal adjustment along the lines we have analysed will continue to be used in the setting of monetary policy. Then, the undesirable effects of seasonal adjustment errors can be smoothed out with simple modifications in operating procedures. ‘“Another example of this type of ambiquity is provided of using seasonally adjusted rates versus rates of seasonally and found to be non-trivial.
in Maravall (1978), where the effect adjusted levels of M, are dwussed
One such modification would be to extract the irregular component from the seasonally adjusted figures. Since the official adjustment does not provide final estimates of the irregular component, we cannot compute how the 6series of errors would change if such a replacement were to be enforced. Another modification could be to replace the monthly rate of growth of MI seasonally adjusted over a two-month period with some other time series related to the growth of M,. For example, if the monthly annualized rate of growth of consecutive nonoverlapping three-month moving averages is used, the variance of the difference between the two seasonally adjusted rates of growth is one third of 6,.
As we already mentioned. of the two alternative seasonal adjustment methods considered in this paper, neither can be seen as unquestionably superior; more reasonably, the two methods could complement each other. Therefore, one can think of combining them to improve upon the results that each one provides when taken individually. It is well known that forecasts obtained with different models can be combined into a weighted average, where the weights are chosen so as to minimize the MSE of the forecast. However. the true seasonal factor. s,, is unobservable, no matter how large the time series becomes. and the MSE of the estimate S, is therefore never known. As an alternative criterion, we could compute the weights of the linear combination of seasonal factors so as to minimize the variance of the revision error, for some chosen period of time. Denote by 1-f the seasonally adjusted rate of growth of M, given by
where O
the variance
dll - P ~rno~rna p, =-p dx, + do - 2P OGzi, where 11is the (contemporaneous)
of (I$ is equal to
*
cross-correlation
between
,I$ and v$‘.‘~
As a consequence, if rf is used instead of either I$ or rp, a decrease in the magnitude of the revision errors can be expected. The comparison between the revision errors corresponding to the published seasonal factors and the ones obtained with the daily method are given in Pierce and Fries (1978). They conclude that the daily method produces considerably smaller revision errors. We have computed the of these revision errors corresponding to s,,‘J for /? =flo. The comparison errors with the ones obtained through the daily method. in terms of the mean square error, absolute average error and average error. is presented in table 3.20 It is seen that there is an overall decrease in the magnitude of the revision errors when the linear combination of the two methods is used. However, the weight given to the method based on the daily procedure is very large, and the improvement is therefore relatively small. In fact, only values inside the interval O
1974 1975 1976 1977
0.92 1.25 2.11 3.66
0.91 1.41 2.30 3.89
-2.45 2.82 -0.58 -4.81
“,WSE= x 10h: AE= x10-‘;
-2.58 3.00 0.88 -4.97
0.77 0.92 1.29 1.73
0.76 0.96 1.27 1.72
A.AE= x lo-”
Although a method with smaller revision errors is not necessarily better. it is nevertheless an important advantage, particularly when the series are used in policy making. One is tempted to conclude that, when this empirical superiority of the daily method is put together with the theoretical result provided by Geweke, more attention should be placed on the daily seasonal adjustment of the monetary aggregate series.
6. Summary Monthly seasonally adjusted rates of growth of the monetary aggregates have become the key elements in the design of US monetary policy. Since the monthly series are aggregates of daily values, two possibilities are open, depending on the order in which the two operations aggregation and seasonal adjustment ~- are performed. Although the currently published ““The revisions
are computed
after a new year of data.
1977. becomes
available
figures are based on a direct monthly adjustment method, an alternative method based on daily adjustment has been developed at the Board for possible consideration. This paper compares the difference between the results obtained with the two procedures from the point of view of monetary policy. The first part of the paper estimates the difference between the two methods for the period 1971-77. In order to assess the relevance of this difference in terms of monetary policy, we provide two measures. First, section 3 deals with the following question: Our simplified policy decision-making process is based on a range of tolerance for the rate of growth of M,, so that growth outside this range triggers intervention. Within this framework. we consider two monetary authorities. identical in all respects except in the fact that, when computing the rates of growth. one uses the monthly adjustment method, while the other uses the method based on the daily data. What is then the probability that. starting with the same initial conditions, a given growth of M, over one period would be judged tolerable by one authority and non-tolerable by the other’? Under sotne simplifying assumptions it is found that, under present operating procedures. disagreement could be expected to happen one out of eight times, approximately. Several additional results are obtained. First, the estimated probability of disagreement is quite robust with respect to reasonable assumptions for the width of the range of tolerance and for the accuracy of short-run forecasts. Second, it is found that short-run improvements in this accuracy may lead to increases in the probability of disagreement. Third, under the assumption that the appropriate method is the one where adjusttnent precedes aggregation, probabilities of type 1 and type 11 errors for present operating procedures are computed, reflecting the probability of intervention when none is warranted and the probability of non-intervention when some is deserved. Probabilities of the first type are seen to be somewhat larger. Section 4 tries to assess the relevance of the difference between the two methods by looking at its effect in terms of interest rates. It is seen that a change in the federal funds rate of approximately one-eighth of a percent point could be accounted for simply by the difference between the two methods. Naturally, the analysis is conditional on present operating procedures. The last part of the paper discusses how (minor) changes in the procedures may attenuate undesirable effects of present seasonal adjustment methods. Also. combinations of the alternative methods of computing the series could decrease the magnitide of the revision errors implied by seasonal adjustment. lncidentially, analysis of the revision errors indicates a marked superiority (in that respect) of the method based on daily adjustment.
131
Appendix A.I.
The distribution
Prewhitening
of 6
of the b-series is achieved
through
the filter
-1
~(1--718”-B2B”) 6,,
a,=
1
[
(A.11
where 4= -0.27, 0= -0.88, yi = -0.27, yZ =0.47. The ACF of the residuals LI, is displayed in table 4, and no lagged autocorrelation is significantly different from zero. The variance of a, is 42% of the variance of is and its mean can be assumed zero. Let ti, = N,/cI,. We wish to test the hypothesis that the series ‘I, is normally distributed. Let p3 and n.4 denote the coefficients of skewness and kurtosis, respectively. In samples from a normal process, the distribution of ,nLJ is symmetric about zero, with variance approximately given by var(p3)=6/T, where T is the sample size. Similarly, as Tbecomes larger, the distribution of ~1~ approaches normality with a mean of 3 and a variance given by var(,n,) 2 24/T” Since our sample yields the values & = 0.123, I4 = 2.830,
o(n3)=0.27, 0(n4) =0.54,
both coefficients, p3 and n4, can be assumed to be generated by a sample from a normal distribution. This conclusion is further reinforced by performing KolmogorovvSmirnov and Chi-Square tests. In the first case, the sample quantity D, = max IF,(x)-
F(x)/,
where F(x) is N(0, 1) and FT(x) the empirical c.d.f. of the sample, value D,=O.O48. Since at test levels of ~=0.10, 0.05, 0.01 and for hypothesis is rejected when D >0.09, 0.099, 0.113, respectively, we that FT(x) is N(0, 1). 22 When the Chi-Square goodness of performed, with the observations grouped in 10 classes, the quantity C2= TX
yields the T=84, the can accept fit test is
(Pi-Pi)2/pi,
where pi and Pi are the empirical
and theoretical
frequencies
“See Durand (1971, pp. 346 349). lZWe use critical values of D corrected for the case in which the mean distribution have been estimated from the sample [see Lilliefors (1967)].
in the ith class,
and variance
of the
Lags
1 12 13-24
lm 12 t 3-24
I
- 0.07 - 0.06
- 0.05 - 0.06 __~
Lags
l-10 1 I-20
1 10 I I-20
2
- 0.03 -0.12
-0.11 - 0.08
1
0.13 0.14
- 0.02 0.15
3
-0.14 -0.11
0.09 0.02
2
- 0.09 0.16
0.02 0.03
4
5
0.06 0.07
(a) Vrrriuhte -0.13 0.09
functions.
Table 4 Autocorrelation
7 of h,)
6 u, (innovations
0.00 0.02
0.01 -0.10
-0.16 -0.12
of c,)
- 0.05 0.10
- 0. I? 0.07
5
6
0.05 - 0.07
0.04 - 0.08
0.05 -0.10
8
0.08 0.08
-0.1 I -0.10
9
-0.13 0.09
- 0.04 -0.01
10
0.15 -0.03
0.09 0.14
11
- 0.02 0.04
0.05 0.04
I2
0.12 0.18
0.11 0.15
Error
- 0.02 0.00
~ 0.08 -0.13
~ 0.09 -0.0x
- 0.05 0.00
0.06 0.20
9
0. I 5 - 0.15
0.01 0.07
10 8
-0.13 - 0.05
7 -.___
- 0.05 0.01
of h, wifh fuIccrr rtdw.5 of 0,
-0.10 0.09
- 0.00 - 0.02
(‘rlrid~lrioll r1fh, wirtt pl\t l~l//LWuf
-0.11 0.09
Cnrrclutiorf
4
Cross-correlation function between II, and h, Lag-zero cross-correlation = 0.21.
0.08 0.05
(b) Vtrrirrhl~ h, (innovation\ - 0.05 0.14
3
0.19 -0.21
~ 0. I 2 -0.01
yields the value c2 =3.22. Since pothesis for (I, is not rejected. A.2. The effect
xt,Os (9)= 16.9, again
the
normality
hy-
ofrevision errors
Since the series we analyse are the ones available as of July 1978 and the sample spans the period 1971-77, the seasonal factors for the last years of this -period will be affected by future revisions. This raises the possibility that the distribution of 6 could be different for recent data, still subject to revision errors. To test for this possibility we divide the sample period into two subperiods, one covering the first five years, the other the last two years. The statistic f= 2
f
(,,,,-U,)2’
y
1-l
(~,,-ti,)~=1.32,
t=59
where m= 58, n=24, and ti, and U2 denote the sample means of the two subperiods, is smaller than F,,,,, (0.975)=2.20. Hence we do not reject the assumption of equal variances, Since the statistic
Cul-~,-(cll-~2)l(~+n-2)f
t=
(l/m+l/n)+
(T
(ul,-a,)2+
1=1
+y (a2t--2)2 t=59
f 1 ,
under the assumption p1 = p2, takes the values t = 1.15, we do not reject the assumption that the means are equal (at the 5 “; level). A.3. The variance
of 6,/h, _2= 0
In the computation of the probability an estimate of var (6,/S, _ 2 = 0). From (A.l), (1-@)6,=
of disagreement
(1 -BB)v,,
where y(B)v, = a, and y(B) = 1 - y,B6 - y2B”.
Thus,
S,=~261_2+v,+(~-e)u,_1-~eU,_,. Since c’,, v,_ 1 and v, _ z are uncorrelated, var(6,/6,_,=0)=ot[l+(f$-8)2+$202],
l-Y2
1
~:=Ifg,(1_-71)2_y~4~
of section
3 we need
A.4. The distribution
qf the
e-series
In section 3 we argued that, to a first approximation, the deviation of I$’ from the midpoint of the tolerance interval could be interpreted as a staff forecast error corresponding to the forecast done at the time of the FOMC meeting. The series of such forecast errors has been analysed in Porter et al. (1978a, pp. 5-19) for the period 1972-1977, where it is shown that the error distribution can be assumed to be Normal. Since the forecasts of VP are month-to-month forecasts of a rate of growth over a two-month period, e, cannot be considered a one-step ahead forecast error. Instead, two consecutive forecast errors will span two two-month periods with a month of overlap. Hence we should expect e, to follow a firstorder moving average process. This is in complete agreement with the sample autocorrelation of e,, displayed in table 2. Estimation of the MA(l) model e,=(l
-M)b,
yields I.= -0.60, (TV=2.53. Table 4 also exhibits the ACF of the residuals where again no lagged autocorrelation is significantly different from zero.
AS.
The joint distrihutiorl
h,,
of’6, an<1 e,
Although the marginal distributions of 6, and eI are Normal, this does not imply that the joint distribution has to be Normal. Yet two uncorrelated random variables with marginal Normal distributions will be jointly normally distributed if and only if the variables are mutually independent.‘” Thus, as we find that the difference between the two seasonally adjusted estimates and the forecast errors are independent, we can conclude that their joint distribution is Normal. The tests for independence will be carried out on the innovations of the (5 and e processes, that is, on the u- and h-series. On the bivariate sample [u,, h,] we perform first a contingency table test for independence. The statistic Tl’, where I2 is given by
when a and b are independent, converges in probability to x2 with (r- 1) (s - 1) d.f. Setting r= 3. s = 3, we obtain T12 = 1.92. Since xi.05 (4)=9.5, we do not reject the hypothesis of independence. Second, we perform a Spearman’s rank correlation test for independence. Let uk be the A,th largest value of (I in the sample, and b, be the B,th largest ‘“See Anderson
(1958, p. 38)
value of h (k = 1,. ., 7’). If u and b are statistically statistic
R=l-
T(T:_l)
; k
independent,
then the test
(Ak--k)Z I
is asymptotically Normal with zero mean Since, for a two-tailed test,
and
variance
equal
to l/(T-
1).
p, - 0.05/2 (T_ 1 )+ =0.24, and, for our sample, R=0.21, we conclude that the hypothesis of independence cannot be rejected (at the 0.05 level of significance). Finally, table 4 presents the cross-correlation function between the innovations clI,and b,. Summarizing, the joint distribution of (S,,r,) can be assumed to be bivariate Normal, with zero mean vector and covariance matrix given by
where C$ = 1.34 and 0: = 3.25.
References2” Advisory Committee on Monetary Statistics, 1976. Improving the monetary aggregates (Board of Governors of the Federal Reserve System, Washington, DC). Anderson. T.W., 1958, An introduction to multivarlate statistical analysis (Wiley. New York). Bradley, J.W.. 1968. Distribution-free statistical tests (Prentice-Hall, Englewood Cliffs, NJ). De Rosa. P. and G.H. Stern, 1977, Monetary control and the federal funds rate, Journal of Monetary Economics 3, 217-230. Durand, D., 1971, Stable chaos: An introduction to statistical control (General Learning Press, New York). Farr, H.T.. 1978, Attempts to estimate a reaction function for the federal reserve. Internal memorandum (Board of Governors of the Federal Reserve System, Washington, DC). Fry, E.R.. 1976, Seasonal adjustment of M,. Currently published and alternative methods. Staff economic studies (Board of Governors of the Federal Reserve. Washington, DC). Geweke, J.. 1978, The temporal and sectoral aggregation of seasonally adjusted time series.* Grether, D.M. and M. Nerlove, 1970, Some properties of ‘optimai’ seasonal adjustment. Econometrica 38. no. 5, 682 703. Lilliefots, H.W., 1967, On the Kolmogorov -Smirnov test for normality with mean and variance unknown, Journal of the American Statistical Association 62, 399 402. Lombra, R.E., 1978. Comment.*
24The references Series
marked
* can be found in the volume Serrso~rul Anu/~si.s of‘ Ecurromic Time Washington, DC, Sept. 9-10, 1976) edited by A. Zellner, of Commerce. Bureau of the Census. Dec. 1978).
(Proceedings of a conference,
(U.S. Department
Lombra. R.E. and R.Ci. Torte. 1975. The atratcgy of monetary policy, Economrc Rc\xw. Sept. Oct. 3 14. Lovell, M.C., 197X. Comment.* MaravaIl, A., 1978. Seasonally adjusted rates of growth versus rates of growth of seasonally adjusted levels: Some implrcntions for monetary control. Special studies paper 111). 120 (Federal Reser\c Board of Governors. Washington. DC). Morgenstern. 0.. 1963. On the accuracy of economrc observations (Princeton Uni\,crsity I’rc<\. Prtnceton. NJ). Prcrcc. D.A.. 197X. Seasonal adJuatmcnt when both determmistic and stochastic seasonality are pr-escnt.” Pierce. D.A. and G. Fries. 197X. Seasonal adJustment t>f daily data with special reference to the U.S. money supply. Special studies paper no. I29 (Board of Governors of the Federal Reserve Sy\tem.‘W&hington. DC). . Pierce. D.A.. N Vanpeski and E.R. Fry. 1978. Seasonal adjustment of the monetary aggregates. in: Improxmg the monetary aggrcgatcs: Staff papers (Board of Goternora of the Federal Reserve System. Washington. DC). Plosser. C.I., 1979. The analysrs of seasonal rconomrc models, Journal of Econometrics 10. 147 163. Plosser. C.I., 1978. A time series analysts of seasonality in econometric models.* Porter. R.D.. H.T. Farr and J. Peres. 197%. An evaluation of Board Staff forecasts of M,, internal memorandum (Board of Governors of the Federal Reserve System, Washington, DC). Porter. R.D., A. MaravaIl, D.W. Parke and D.A. Pierce, 197Xb. Transitory variations in the monetary aggregates. in: Improving the monetary aggregates: Staff papers (Board of Governors of the Federal Reserve System. Washington. DC) I 33. Shiskin, J.. 1078. Seasonal adjustment of sensittve mdicators.* Taylor, J.B.. 1978. Comment.* Wallich, H.C. and P.M. Kcir. 1978. The role of operating guides in U.S. monetary policy: A histortcal review, in: Krcdit und Kapital. Jan II (Berlin).
MARAVALL
Bunk of Spain, Madrid
14, Sptrirl
1. Introduction The seasonal adjustment method presently used at the Board for adjusting M, is a blend of X-11 and judgmental corrections applied to the monthly series of demand deposits and currency. Since the monthly (unadjusted) series are averages of daily values, an alternative procedure for obtaining the monthly adjusted figure would be’ to adjust the daily series and then aggregate them into months. In other words, temporal aggregation can precede or succeed seasonal adjustment. The use of a daily method was recommended by the Bach Committee on Monetary Statistics.’ In this paper we compare the official Board method with the daily one developed at the Board, which is an extension of the one actually used by the Bach Committee. It is argued that there are reasons for preferring a daily approach and reasons for preferring the monthly one. It seems safe to conclude that, to the extent that neither of the two approaches is unquestionably superior, they provide two alternative methods for adjusting the series. Insofar as the two methods do not yield identical results, the difference between them reflects an ambiguity in the computation of the series of interest (monthly seasonally adjusted M,). Logically this ambiguity sets a lower limit to the precision with which the series can be measured and therefore has the character of a measurement error. Naturally, if in most practical situations the difference between the two methods happens to be very small, the ambiguity could be easily ignored. On the other hand it may be that on occasion the judgment as to whether growth of M, has been adequate or not and therefore the *This work was primarily done while the author was an economist on the staff of the Division of Research and Statistics of the Federal Reserve Board of Governors. Thanks are due to D.A. Pierce. K.J. Singleton, A. Zellner, H.H. Stokes, R. Meese and R.D. Porter for their comments. The opinions expressed are those of the author and do not necessarily represent those of the Federal Reserve Board of Governors. ‘See Advisory Commtttee on Monetary Statistics (1976, pp. 4 and 39-40).
direction of monetary policy depends crucially on the method used for seasonal adjustment. Of course, economic time series are used by decision makers when setting policy. These series, we know, have errors, yet this knowledge is usually ignored. One can wonder what part of economic policy is simply the result of measurement errors. This paper can be viewed as a crude attempt at providing some insights into the problem for a specific error of measurement in the context of monetary policy.’ In the next section we apply the two methods of seasonal adjustment to the M, series for the period 1971-1977. This yields a series of differences between the two adjusted values which we proceed to analyze. Then we provide an estimate of how likely it is that the direction of monetary policy depends on the method used for seasonal adjustment and how is translates into an equivalent imprecision in terms of interest rate movements. It is also seen how the effect of the ambiguity can be decreased.
2. Daily versus monthly seasonal adjustment of M, The first series we consider is the seasonally adjusted monthly money supply (M,) series published by the Federal Reserve. This series is obtained by applying X-11, and then judgmentally modifying the factors obtained, to the two components of M,: currency and private demand deposits. The procedure is discussed in Fry (1976) and in Pierce et al. (1978). The second series we consider is the monthly average of seasonally adjusted daily values of M,, obtained also as the sum of adjusted currency and demand deposit components. The daily seasonal adjustment procedure we consider was developed at the Federal Reserve as an extension of the one used by the Bach Committee. The procedure is described in Pierce and Fries (1978).3 Perhaps some general comments on the two methods are in order: Aside from the stochastic characteristics of the series, optimality of a seasonal adjustment method depends on the use to be made of the series4 In our application, this use is the setting of monetary policy by the Federal Open Market Committee (FOMC). Since the growth of M, (seasonally adjusted) has been for some time a crucial variable in monetary control, optimal seasonal adjustment of M, would require separate estimation of the seasonal which is the result of past policy action and the one which is due to ?Attempts at quantifying measurement errors are infrequent m economics. According to Morgenstern (1963, p. 7): ‘there is much less occupation with errors than in other fields. This is undoubtedly one of the reasons why the social sciences have had a rather uncertain development.’ 3The method is an extension of the daily method described in Pierce et al. (1978), along the lines developed in Pierce (197X). “See Grether and Nerlove (1970).
A. Murtrtall,
Seusor~al adjustment
and monetary
policy
117
‘exogenous’ causes. Otherwise, monetary policy would be contaminated by undesirable variables, such as for example, past errors in the estimation of seasonal factors.’ Thus, strictly speaking, optimality of the seasonal adjustment of the monetary aggregates in the context we consider possibly requires the use of an appropriate econometric model. Although some progress in that direction has been made by Plosser (1978, 1979), practical enforcement of such methods in ongoing monetary policy is still very distant6 Meanwhile, what can be said of the relative merits of the two methods we consider? A theoretical reason for preferring a daily approach is provided in Geweke every conceivable time (1978), where it is shown that ‘for virtually series . . . and a reasonably inclusive class of potential adjustment procedures, minimum mean-square-error adjustment implies that seasonal adjustment should always precede temporal or sectorial aggregation.’ Yet his result cannot be applied blindly. First, it is hard to assess whether the combination of X-11 and judgment falls into his ‘reasonably inclusive class’ of adjustment procedures [see Love11 (1978)]. Second. as shown in Taylor (1978): ‘It is not the preaggregation adjustment that is crucial for optimality, but rather the utilization of all observable components simultaneously.’ Insofar as the judgment corrections may use daily information, Geweke’s result would not apply. Other relative advantages of the method based on the daily procedure are that it allows for a more complete treatment of special-day events, and that it makes possible the construction of perfectly consistent seasonally adjusted figures expressed in different time intervals. On the other hand, there are advantages associated with the aggregate monthly method. First, the computational complications of the daily approach are not trivial. Second, monthly data tend to have smaller measurement errors.’ Finally, misspecification may be more of a problem in the daily approach.8 Thus it seems safe to conclude that neither of the two approaches can be seen as unquestionably superior. Since presently one of the methods is being used, and the other is being considered for use by the Federal Reserve, they provide a meaningful comparison for answering the question of how sensitive monetary policy is to alternative seasonal adjustment methods. Let m, stand for the seasonally adjusted monthly figure of M, at time t. ‘This point is discussed in the report of the Bach Committee [see Advisory Committee on Monetary Statistics (1978, pp. 37.39)]. ‘See Lombra (1978). Possrbly this distance is reinforced by the additional (optimality) requirement of specifying the correct aggregate loss function of the 12 FOMC members. ‘See Porter et al. (1978b). sThe preference for the daily approach expressed by several of this paper’s referees is in contrast with that of the members of the Federal Reserve Advisory Committee on Seasonal Adjustment and with that of Shiskin (1978).
Since monetary policy utilizes annualized periods, we shall operate with the variable
rates
of growth
over two-month
rr = 6[ (in, + z - rn, )/WI,]. When m is the official Board estimate, the variable rt shall be denoted rp. When III is computed through the daily approach, we shall write rp. The two seasonally adjusted series are displayed in table 1. Table 2 displays the autocorrelation function (ACF) of ry and r:. For both series the functions are extremely close. No seasonality seems to remain, and only low-order autocorrelation is significantly different from zero. Also both series are seen to be stationary. Possibly the most noticeable difference between the two is the difference in variances, which indicates that more seasonality is removed from the series when the daily procedure is used. Let is denote the difference between the two series: 6, = ry - ra f’
Tables 1, 2 and fig. 1 display the series 6, and its ACF. Despite that no seasonality can be detected in the r-series, their difference strong semi-annual and annual patterns.
3. The effect of alternative measurements versus non-tolerable growth of M,
3.1. Alternutive
mea.surements
on monetary
of’ M, and monetary
control:
the fact exhibits
Tolerable
control
At each monthly meeting, the Federal Open Market Committee sets a range for the rate of growth of the monetary aggregates (seasonally adjusted) which is considered tolerable over a two-month period. Policy action will depend among other things -- on whether or not the growth of the aggregates falls within this range.” In the case we are analysing, there are two alternative ways of measuring the rates of growth. Consider two monetary authorities, A and B, both of which follow identical behaviour. They set a range of tolerance for the rate of growth of M, seasonally adjusted, and rates outside the interval trigger some kind of intervention. Assume A and B are identical in every respect, with the only difference being the way they adjust for seasonality. A adjusts the monthly aggregates, while B adjusts the daily data, then aggregates them into months. Specifically, A operates with PO,while B operates with r“. “For a more complete
description,
see Wallich
and Kerr (1978) or Lombra
and Torte
(lY75).
235732 235867 0.3584
257737 257839 0.3051
271761 271841 -0.5376
282637 283017 - 0.7697
295918 29628 I - 0.7604
314919 314407 1.1202
1972
1973
1974
1975
1976
1977
316336 316309 1.105
297936 297853 2.2393
282787 283133 0.7626
273107 273371 0.3342
258162 258304 ~ 1.7674
237721 238041 - 0.6723
222746 223177
Feb.
228302 227711 1.7824
225828 225685 1.5568 241407 241288 1.1172 259012 259023 0.3046
224362 224437 I .3426
240025 240040 0.3127
258179 258069 0.493 1
322382 322221 - 0.3220
301462 301692 0.6327 322008 321865 0.2183
299328 299047 1.313
318313 3 17986 ~ 0.3627
303347 303029 0.0664
284756 285042 0.1330
284975 29494 1 0.8821
278967 278746 - 1.0946 292405 292418 _ I .9856
278316 278026 PO.2050 291068 290609 0.0088 304134 304469 - 1.2944
277832 277109 1.6786 291205 29026 1 2.6025 303537 303006 1.5166
(My).
327463 327611 -0.5812
265186 264532 - 0.7233
264677 263852 0.3642
264043 263076 2.2659
324306 323797 0.6823
247265 246646 0.4949
245099 244654 0.0463
242860 242448 0.7280
329231 329279 - I.0488
305840 306002 _ I .3797
231854 231011 0.3988
230723 229806 0.8439
229436 228753 1.4312
Aug.
July
I
June
(MF). from the daily method
287635 287185 0.8766
275389 275435 0.4848
274839 274768 0.3354
276228 275847 0.6765
261896 261236 I .2745
242205 241783 1.0909
May
April
March
“Seasonally adjusted monthly series of M, Published %easonally adjusted monthly series of M, Obtained ‘Monthly series of 6,( = rp - rp ).
220797” 22135@’ _c
1971
Jan.
Table
331649 331798 0.0032
306756 307224 - 0.2563
334730 333811 1.765
310133 309746 1.0815
292923 292813 0.2525
280704 280864 -0.8212
279615 279599 -0.5948 293442 293217 -0.4918
266168 266174 - 1.502X
265075 265065 - 1.8573
251050 25 1079 - 1.601
232714 233073 -3.1281
232270 232306 - 2.5047 249237 249367 _ 1.4306
Oct.
Sept.
B 2’
282822 283528 - 1.167
334941 334657 0.7854
310563 3 I0630 0.7944
295430 295418 - 0.4403
282311 282328 -0.0715
337225 336610 -0.5581
312576 313119 - 1.8077
294509 295518 -2.2941
g g g
270474 270885 -0.9145 268629 26847 1 0.3366
$ 5
? P 2 ‘2
: ‘4
g _ s ;
g 255276 254675 1.5087
252179 252404 - 0.2249
.~
234012 234069 0.7840
Dec.
233099 233370 - 0.6082
No\.
1
0.50 -0.23
0.49 - 0.21
0.34 0.13
0.37 -0.10
Lags
l-12 13-24
l-12 13-24
1-12 13-24
l-12 13 24
-0.18 0.10
- 0.24 -0.20
- 0.02 - 0.08
0.02 - 0.07
2
-0.18 0.04
-0.12 - 0.07
- 0.05 0.09
- 0.06 0.07
3
- 0.05 0.2 I
0.01 0.03
-0.01 0.24
-0.12 0.22
4
0.08 0.17
-0.18 -0.14
0. I8 0.21
0.10 0.15
5
Table
7
<
- 0.02 -0.10
-0.17 - 0.09
e,
-0.15 - 0.07
(d) Variable
- 0.49 - 0.29
(c) Vuriable 6,
0.00 -0.11
(b) Variabk 0.17 - 0.01
8
- 0.06 - 0.08
0.09 - 0.02
0.04 - 0.09
0.04 -0.15
functions.
2
(a) Variable r: 0.17 -0.01 - 0.05 0.06
6
Autocorrelation
0.01 -0.16
-0.09 -0.11
0.11 - 0.06
0.12 -0.14
9
0.02 0.00
-0.12 0.01
0.15 0.02
0.14 0.04
10
0.10 0.03
0.31 0.28
0.15 - 0.05
0.16 0.01
11
0.02 0.07
0.59 0.33
- 0.09 -0.14
- 0.08 - 0.08
12
0.12 0.15
0.11 0.18
0.11 0.15
0.11 0.15
Error
2 S 2 % 0 z c
2 9. 3
& 0 B a $ :: 3 2
2
$ _%
A. Marcrcall,
Seasonal
adjustment
and monetary
policy
Fig. 1. Series [b,J
Let A and tolerance for the previous =0) so that, for both. For that A and B
B each hold a meeting at time t in which they set a range of the growth of M, over a two-month period. Assume that for period, both measures of r yielded the same figure (thus a,_, everything being equal, the range of tolerance will be the same a specific growth of M, over the period, what are the chances would disagree as to whether or not that growth is tolerable?
The probability
of disagreement
(1) The probability inside (P,). (2) The probability
that Y: falls outside that
not (p,,). (3) The probability that opposite sides (P,,,). Assume
that
is the sum of three components:
rp falls inside both
the forecast
rf and
the interval the interval
of tolerance of tolerance
rp fall outside
of rp at the time
the
the meeting
and VPfalls and rf does
interval
is held
but
on
is the
122
____-
I
St
_-___
,
,
/’
/’ 1’
Fig. 2
midpoint of the interval of tolerance (0 in fig. 2).” The forecast error et is then the distance between rp and 0. Let x denote the midrange of the interval of tolerance. It is easily seen that (deleting the subscript t for notational simplicity): (1) When r” is outside the tolerance interval, then 1.“ will be inside if and only if 16- ~1< CI (see fig. 2). (2) When Y’ is inside the interval, then P will be outside whenever 16- rj > x. (3) When Y* is outside, for P to be also outside but on the opposite side it has to be that 6>r+cc if P>O, and 6
“A typical ‘Record of Policy Actions’ of FOMC (published in the Federcd Resrrw Bulktir~) contains the following statement: ‘The Committee agreed that if growth rates of the aggregates over the 2-month period appeared to be deviating significantly from the midpoints of the indicated ranges, the operational objective in the weekly-average Federal funds rate should be modified _’Also, Lombra and Torte (1975, p. 11) state that ‘from the viewpoint of the staff the ranges _._ represent a standard error around a point estimate at the midpoint of the range.’ Thus the assumption that the mldpomt of the range can be seen as an approximation to a staff forecast seems reasonable. Of course. this will not always be the case.
A. Mururull,
and the total probability
Seusonul udjustmrnt und monrtur~ polic,!
of disagreement
123
is equal to
P = P, + P,, + P,,,. In order to compute the probabilities P,, P,, and P,,,, we need the joint distribution of 6, and e,. In the appendix we conclude that this distribution can be assumed to be bivariate normal, with zero mean vector and covariance matrix R, as given in the appendix. The value of c(, the midrange of the the interval of tolerance is published in the Federal Reserve Bulletin. For the past years it has oscillated between 2 and 3. We set tl=2.5.” Before performing the integrations, a minor modification has to be made. In order to simplify the comparison between the two monetary authorities we set before cS_~=O. Thus in our computation, instead of the marginal distributions of 6,, we should employ the distribution of 6, conditional on 6,_, EO.12 From the expression derived in the appendix, it is obtained that r&,,_2=o=1.22. Numerical computation yields the following results: P,=O.O76,
P,,=O.O53,
P,,,=O.O28
(10-5),
P=O.13.
P, is the probability that, for the same growth of M, in period t, B considers the growth tolerable and A does not; P,, is the probability that A considers the growth tolerable and B does not. Obviously I’,,, can be neglected.13 Thus, for a given growth of M, in period t, there would be a 7.67; probability that A would think monetary intervention was granted and B that A would think monetary would thrnk the opposite, a 5.3 ‘;‘, p robability intervention was not granted and B would think the opposite, and in general, a 13 y0 probability that the two monetary authorities would disagree. Approximately, one out of eight times disagreement would occur. The estimated probabilities of disagreement are, so to speak, unconditional probabilities in that they do not take account of economic or policy conditions which may affect the joint p.d.f. For example, it could happen that deviations of ~4, from range midpoints are positively correlated with the rate of expansion of economic activity. In general, it would not be correct to “One implicit assumption in our formulation is that the p.d.f. of r is independent of n. This can only be seen as a simplification, since it could be that a wider tolerance range indtcates a greater wrllingness by the FOMC to tolerate wider departure of M, from the range midpoint. I am grateful to a referee for pointing out this qualification. ‘2Naturally. using u,“,,,_ *= 0 in the computatton of P biases downwards the estimntcd probability of disagreement. “A necessary (though by no means sufficient) condition for r0 and r’ to fall outside the tolerance range and on opposite sides, is (6/> 22. Since P([6[ > 2~) = ZP(r > 4.3), where z - N(0, I j, it follows that P,,, is smaller than (0.04)x lo-‘. Therefore, for the rest of the analysis. the P,,, type probability of disagreement can be safely ignored.
124
A. Morarall,
Seasonal
adjustment
and monetury
policy
assume that the probability of disagreement is invariant nomic conditions and monetary policy directives.14
3.2. SensiticitJ?
qf‘ the
result
to purumeter
changes:
The
to different
eff;ct
eco-
qf forecasting
improvement
The numerical result of the previous paragraph was computed for values of c1 and r~, typical of recent years. Naturally, the smaller the value of CI, the larger would be the probability of disagreement. Since over the last years, a has oscillated between 2 and 3, the probability of disagreement would have ranged between 13.8 and 11.5 percent. More interesting is the relationship between the probability of disagreement and G<, the standard error of the forecast of the rate of growth of M, over the two-month period. We shall assume that c( remains constant, although it would seem plausible that improvements in the ability to forecast M, could lead to a reduction of the interval of tolerance. However, the oscillations of u over the past years seem to have been motivated by other considerations. Also, we shall assume that crBremains constant. It is easily seen that the function that relates P to ge tends to a minimum for g,-+O and O,+KI. This means that, for a positive value of oe, a maximum for P has to exist (since the function is bound). Thus in some region, improvements in forecasting accuracy increase the probability of disagreement. Fig. 3 plots P as a function
of (T,. Two things can be noticed:
(a) The result obtained (disagreement would occur one out of eight times) is robust with respect to present (and at least short-run expectations of) forecasting accuracy. For the range 2.5
to a referee for pointing
out this qualification
A. Maruoall.
Srasonal
adjustment
and mon~ur)
policy
125
Thus P, and P,, can be seen as the probabilities of type I and type I1 errors, respectively. If non-tolerable growth implies intervention and viceversa. then P, represents the probability of intervention when none should have been granted, and P,, represents the probability of no intervention when there should have been some. From the values computed, it is seen that, over the relevant range of (TV,the probability of a type I error is slightly larger than the probability of a type II error. This may reflect an implicit FOMC belief that it is more dangerous not to intervene when intervention is granted than to intervene when it is not needed.
-----a_-____ i 0.05 0.00
I
I 1.67
I
I 3.33
5.00
I 6.67
I 8.33
10.00
Fig. 3.
4. The effect of alternative measurements
on interest rates
In the previous section we provided a quantitative measure to a qualitative question. We estimated what would be the probability that two alternative seasonal adjustment methods would lead to disagreement in terms of whether or not the monetary authority should act to modify the funds rate. In this section we provide an answer to a different question: what would be the difference in interest rates when the two alternative seasonal adjustment methods are used in the setting of monetary policy? Since the difference in funds rate changes (according to which method is used) can be interpreted as an imprecision in detecting federal funds movements, it sets a lower limit to the ‘finesse’_ with which monetary policy can be instrumented given present operating procedures.
Consider a monetary authority with a reaction function which specifies that changes in the funds rate are related to a set of variables, some of which are related to the seasonally adjusted rate of growth of M ,,while others are independent of M, measurements. Assume that monthly changes in the Federal funds rate, J,, are expressed as a function of forecast errors of the two-month rate of growth, of M, (seasonally adjusted), e,. plus a set of variables such as changes in the unemployment rate, the rate of inflation, etc., which we assume unaffected by errors of measurement in M,.’ 5 Let z denote the vector of these variables so that
where t(B) denotes a polynomial lag. If rp were to be used, the forecast error would become c,, where (;I=P, -0,. The difference in x, from using one rate or the other is, therefore.‘”
In Farr (1978) several reaction
functions
are estimated.
The filter
with parameters equal to to =0.4, t, =0.03, c2 =O.Ol, seems to provide a formulation in agreement with all the results obtained. Applying t(B) to the series ii,, a series do, is obtained. Expressed in terms of basis points,” the standard deviation of the difference in interest rates is equal to 6.2. Since the standard deviation of month-to-month changes in the funds rate (over the sample period) is, approximately, 50 basis points, these results imply that 13 YiOof it could be accounted for simply by the difference in the two seasonal adjustment methods we considered (when applied to the series). Also, since 203,,= 12.4, a change in the funds rate of l/8 of a percent point could be attributed to that difference. To obtain an additional estimate, we used the reaction function presented in De Rosa and Stern (1977) for the period 197&74. In their formulation. month-to-month percentage change in the funds rate is related to a three‘“This last assumption is, again, a simplification. Since, for example. measurementa of M, influence policy and policy Influences inflation, IndIrect relationships could exist. lhWe are assumin_g that the lag structure c(B) is the same whether we use official adjustment or adJustmerIt based on the daily method. Since the ACF of rp and r; are extremely close, possibly the difference m t(B) would be small, and the results presented below would not be much affected. “A basis point represents a change of I Oo.
month
moving
average
percentage
change
in M, through
the filter
as 14.64, 0.304, -0.282 and 0.016, where cc,, cr, c1 and c3 are estimated respectively. If SF denotes the difference in the three-month moving average percentage change in M, when the two seasonal adjustment methods are used, the standard deviation of the series, AI‘: = i’*(B)h:, is equal to 7.43, where the unit again is basis points. Therefore, the imprecision in detecting month-to-month funds changes is of similar magnitude for the two formulations of the reaction function. 5. Some implications
5.1. Ambiguities
in seasonal adjustment
In the previous two sections we have analysed the effects of using the seasonally adjusted monthly average of daily values of A4, versus the monthly average of seasonally adjusted daily values of M,. We found that, depending on which one is used, the monetary authority would disagree one out of eight times as to whether or not intervention is warranted and that a difference of l/8 of a percent point in the Federal funds rate could separate policies using the two alternative methods. So despite the fact that no criterion seems to be available that would establish which of the two methods is the right one to use, the implications of choosing one versus the other do not appear to be negligible. From a general point of view, the problem we have analysed reinforces the idea that seasonal adjustment may possibly affect the data considerably more than is intended, and that this is not only due to the shortcomings of the seasonal adjustment methods presently available, or to the existence of revision errors. In fact, there are problems due to ambiguities which arise from the fact that we do not know exactly what we want to measure.” Despite all its limitations, there is little doubt that, for the time being, seasonal adjustment along the lines we have analysed will continue to be used in the setting of monetary policy. Then, the undesirable effects of seasonal adjustment errors can be smoothed out with simple modifications in operating procedures. ‘“Another example of this type of ambiquity is provided of using seasonally adjusted rates versus rates of seasonally and found to be non-trivial.
in Maravall (1978), where the effect adjusted levels of M, are dwussed
One such modification would be to extract the irregular component from the seasonally adjusted figures. Since the official adjustment does not provide final estimates of the irregular component, we cannot compute how the 6series of errors would change if such a replacement were to be enforced. Another modification could be to replace the monthly rate of growth of MI seasonally adjusted over a two-month period with some other time series related to the growth of M,. For example, if the monthly annualized rate of growth of consecutive nonoverlapping three-month moving averages is used, the variance of the difference between the two seasonally adjusted rates of growth is one third of 6,.
As we already mentioned. of the two alternative seasonal adjustment methods considered in this paper, neither can be seen as unquestionably superior; more reasonably, the two methods could complement each other. Therefore, one can think of combining them to improve upon the results that each one provides when taken individually. It is well known that forecasts obtained with different models can be combined into a weighted average, where the weights are chosen so as to minimize the MSE of the forecast. However. the true seasonal factor. s,, is unobservable, no matter how large the time series becomes. and the MSE of the estimate S, is therefore never known. As an alternative criterion, we could compute the weights of the linear combination of seasonal factors so as to minimize the variance of the revision error, for some chosen period of time. Denote by 1-f the seasonally adjusted rate of growth of M, given by
where O
the variance
dll - P ~rno~rna p, =-p dx, + do - 2P OGzi, where 11is the (contemporaneous)
of (I$ is equal to
*
cross-correlation
between
,I$ and v$‘.‘~
As a consequence, if rf is used instead of either I$ or rp, a decrease in the magnitude of the revision errors can be expected. The comparison between the revision errors corresponding to the published seasonal factors and the ones obtained with the daily method are given in Pierce and Fries (1978). They conclude that the daily method produces considerably smaller revision errors. We have computed the of these revision errors corresponding to s,,‘J for /? =flo. The comparison errors with the ones obtained through the daily method. in terms of the mean square error, absolute average error and average error. is presented in table 3.20 It is seen that there is an overall decrease in the magnitude of the revision errors when the linear combination of the two methods is used. However, the weight given to the method based on the daily procedure is very large, and the improvement is therefore relatively small. In fact, only values inside the interval O
1974 1975 1976 1977
0.92 1.25 2.11 3.66
0.91 1.41 2.30 3.89
-2.45 2.82 -0.58 -4.81
“,WSE= x 10h: AE= x10-‘;
-2.58 3.00 0.88 -4.97
0.77 0.92 1.29 1.73
0.76 0.96 1.27 1.72
A.AE= x lo-”
Although a method with smaller revision errors is not necessarily better. it is nevertheless an important advantage, particularly when the series are used in policy making. One is tempted to conclude that, when this empirical superiority of the daily method is put together with the theoretical result provided by Geweke, more attention should be placed on the daily seasonal adjustment of the monetary aggregate series.
6. Summary Monthly seasonally adjusted rates of growth of the monetary aggregates have become the key elements in the design of US monetary policy. Since the monthly series are aggregates of daily values, two possibilities are open, depending on the order in which the two operations aggregation and seasonal adjustment ~- are performed. Although the currently published ““The revisions
are computed
after a new year of data.
1977. becomes
available
figures are based on a direct monthly adjustment method, an alternative method based on daily adjustment has been developed at the Board for possible consideration. This paper compares the difference between the results obtained with the two procedures from the point of view of monetary policy. The first part of the paper estimates the difference between the two methods for the period 1971-77. In order to assess the relevance of this difference in terms of monetary policy, we provide two measures. First, section 3 deals with the following question: Our simplified policy decision-making process is based on a range of tolerance for the rate of growth of M,, so that growth outside this range triggers intervention. Within this framework. we consider two monetary authorities. identical in all respects except in the fact that, when computing the rates of growth. one uses the monthly adjustment method, while the other uses the method based on the daily data. What is then the probability that. starting with the same initial conditions, a given growth of M, over one period would be judged tolerable by one authority and non-tolerable by the other’? Under sotne simplifying assumptions it is found that, under present operating procedures. disagreement could be expected to happen one out of eight times, approximately. Several additional results are obtained. First, the estimated probability of disagreement is quite robust with respect to reasonable assumptions for the width of the range of tolerance and for the accuracy of short-run forecasts. Second, it is found that short-run improvements in this accuracy may lead to increases in the probability of disagreement. Third, under the assumption that the appropriate method is the one where adjusttnent precedes aggregation, probabilities of type 1 and type 11 errors for present operating procedures are computed, reflecting the probability of intervention when none is warranted and the probability of non-intervention when some is deserved. Probabilities of the first type are seen to be somewhat larger. Section 4 tries to assess the relevance of the difference between the two methods by looking at its effect in terms of interest rates. It is seen that a change in the federal funds rate of approximately one-eighth of a percent point could be accounted for simply by the difference between the two methods. Naturally, the analysis is conditional on present operating procedures. The last part of the paper discusses how (minor) changes in the procedures may attenuate undesirable effects of present seasonal adjustment methods. Also. combinations of the alternative methods of computing the series could decrease the magnitide of the revision errors implied by seasonal adjustment. lncidentially, analysis of the revision errors indicates a marked superiority (in that respect) of the method based on daily adjustment.
131
Appendix A.I.
The distribution
Prewhitening
of 6
of the b-series is achieved
through
the filter
-1
~(1--718”-B2B”) 6,,
a,=
1
[
(A.11
where 4= -0.27, 0= -0.88, yi = -0.27, yZ =0.47. The ACF of the residuals LI, is displayed in table 4, and no lagged autocorrelation is significantly different from zero. The variance of a, is 42% of the variance of is and its mean can be assumed zero. Let ti, = N,/cI,. We wish to test the hypothesis that the series ‘I, is normally distributed. Let p3 and n.4 denote the coefficients of skewness and kurtosis, respectively. In samples from a normal process, the distribution of ,nLJ is symmetric about zero, with variance approximately given by var(p3)=6/T, where T is the sample size. Similarly, as Tbecomes larger, the distribution of ~1~ approaches normality with a mean of 3 and a variance given by var(,n,) 2 24/T” Since our sample yields the values & = 0.123, I4 = 2.830,
o(n3)=0.27, 0(n4) =0.54,
both coefficients, p3 and n4, can be assumed to be generated by a sample from a normal distribution. This conclusion is further reinforced by performing KolmogorovvSmirnov and Chi-Square tests. In the first case, the sample quantity D, = max IF,(x)-
F(x)/,
where F(x) is N(0, 1) and FT(x) the empirical c.d.f. of the sample, value D,=O.O48. Since at test levels of ~=0.10, 0.05, 0.01 and for hypothesis is rejected when D >0.09, 0.099, 0.113, respectively, we that FT(x) is N(0, 1). 22 When the Chi-Square goodness of performed, with the observations grouped in 10 classes, the quantity C2= TX
yields the T=84, the can accept fit test is
(Pi-Pi)2/pi,
where pi and Pi are the empirical
and theoretical
frequencies
“See Durand (1971, pp. 346 349). lZWe use critical values of D corrected for the case in which the mean distribution have been estimated from the sample [see Lilliefors (1967)].
in the ith class,
and variance
of the
Lags
1 12 13-24
lm 12 t 3-24
I
- 0.07 - 0.06
- 0.05 - 0.06 __~
Lags
l-10 1 I-20
1 10 I I-20
2
- 0.03 -0.12
-0.11 - 0.08
1
0.13 0.14
- 0.02 0.15
3
-0.14 -0.11
0.09 0.02
2
- 0.09 0.16
0.02 0.03
4
5
0.06 0.07
(a) Vrrriuhte -0.13 0.09
functions.
Table 4 Autocorrelation
7 of h,)
6 u, (innovations
0.00 0.02
0.01 -0.10
-0.16 -0.12
of c,)
- 0.05 0.10
- 0. I? 0.07
5
6
0.05 - 0.07
0.04 - 0.08
0.05 -0.10
8
0.08 0.08
-0.1 I -0.10
9
-0.13 0.09
- 0.04 -0.01
10
0.15 -0.03
0.09 0.14
11
- 0.02 0.04
0.05 0.04
I2
0.12 0.18
0.11 0.15
Error
- 0.02 0.00
~ 0.08 -0.13
~ 0.09 -0.0x
- 0.05 0.00
0.06 0.20
9
0. I 5 - 0.15
0.01 0.07
10 8
-0.13 - 0.05
7 -.___
- 0.05 0.01
of h, wifh fuIccrr rtdw.5 of 0,
-0.10 0.09
- 0.00 - 0.02
(‘rlrid~lrioll r1fh, wirtt pl\t l~l//LWuf
-0.11 0.09
Cnrrclutiorf
4
Cross-correlation function between II, and h, Lag-zero cross-correlation = 0.21.
0.08 0.05
(b) Vtrrirrhl~ h, (innovation\ - 0.05 0.14
3
0.19 -0.21
~ 0. I 2 -0.01
yields the value c2 =3.22. Since pothesis for (I, is not rejected. A.2. The effect
xt,Os (9)= 16.9, again
the
normality
hy-
ofrevision errors
Since the series we analyse are the ones available as of July 1978 and the sample spans the period 1971-77, the seasonal factors for the last years of this -period will be affected by future revisions. This raises the possibility that the distribution of 6 could be different for recent data, still subject to revision errors. To test for this possibility we divide the sample period into two subperiods, one covering the first five years, the other the last two years. The statistic f= 2
f
(,,,,-U,)2’
y
1-l
(~,,-ti,)~=1.32,
t=59
where m= 58, n=24, and ti, and U2 denote the sample means of the two subperiods, is smaller than F,,,,, (0.975)=2.20. Hence we do not reject the assumption of equal variances, Since the statistic
Cul-~,-(cll-~2)l(~+n-2)f
t=
(l/m+l/n)+
(T
(ul,-a,)2+
1=1
+y (a2t--2)2 t=59
f 1 ,
under the assumption p1 = p2, takes the values t = 1.15, we do not reject the assumption that the means are equal (at the 5 “; level). A.3. The variance
of 6,/h, _2= 0
In the computation of the probability an estimate of var (6,/S, _ 2 = 0). From (A.l), (1-@)6,=
of disagreement
(1 -BB)v,,
where y(B)v, = a, and y(B) = 1 - y,B6 - y2B”.
Thus,
S,=~261_2+v,+(~-e)u,_1-~eU,_,. Since c’,, v,_ 1 and v, _ z are uncorrelated, var(6,/6,_,=0)=ot[l+(f$-8)2+$202],
l-Y2
1
~:=Ifg,(1_-71)2_y~4~
of section
3 we need
A.4. The distribution
qf the
e-series
In section 3 we argued that, to a first approximation, the deviation of I$’ from the midpoint of the tolerance interval could be interpreted as a staff forecast error corresponding to the forecast done at the time of the FOMC meeting. The series of such forecast errors has been analysed in Porter et al. (1978a, pp. 5-19) for the period 1972-1977, where it is shown that the error distribution can be assumed to be Normal. Since the forecasts of VP are month-to-month forecasts of a rate of growth over a two-month period, e, cannot be considered a one-step ahead forecast error. Instead, two consecutive forecast errors will span two two-month periods with a month of overlap. Hence we should expect e, to follow a firstorder moving average process. This is in complete agreement with the sample autocorrelation of e,, displayed in table 2. Estimation of the MA(l) model e,=(l
-M)b,
yields I.= -0.60, (TV=2.53. Table 4 also exhibits the ACF of the residuals where again no lagged autocorrelation is significantly different from zero.
AS.
The joint distrihutiorl
h,,
of’6, an<1 e,
Although the marginal distributions of 6, and eI are Normal, this does not imply that the joint distribution has to be Normal. Yet two uncorrelated random variables with marginal Normal distributions will be jointly normally distributed if and only if the variables are mutually independent.‘” Thus, as we find that the difference between the two seasonally adjusted estimates and the forecast errors are independent, we can conclude that their joint distribution is Normal. The tests for independence will be carried out on the innovations of the (5 and e processes, that is, on the u- and h-series. On the bivariate sample [u,, h,] we perform first a contingency table test for independence. The statistic Tl’, where I2 is given by
when a and b are independent, converges in probability to x2 with (r- 1) (s - 1) d.f. Setting r= 3. s = 3, we obtain T12 = 1.92. Since xi.05 (4)=9.5, we do not reject the hypothesis of independence. Second, we perform a Spearman’s rank correlation test for independence. Let uk be the A,th largest value of (I in the sample, and b, be the B,th largest ‘“See Anderson
(1958, p. 38)
value of h (k = 1,. ., 7’). If u and b are statistically statistic
R=l-
T(T:_l)
; k
independent,
then the test
(Ak--k)Z I
is asymptotically Normal with zero mean Since, for a two-tailed test,
and
variance
equal
to l/(T-
1).
p, - 0.05/2 (T_ 1 )+ =0.24, and, for our sample, R=0.21, we conclude that the hypothesis of independence cannot be rejected (at the 0.05 level of significance). Finally, table 4 presents the cross-correlation function between the innovations clI,and b,. Summarizing, the joint distribution of (S,,r,) can be assumed to be bivariate Normal, with zero mean vector and covariance matrix given by
where C$ = 1.34 and 0: = 3.25.
References2” Advisory Committee on Monetary Statistics, 1976. Improving the monetary aggregates (Board of Governors of the Federal Reserve System, Washington, DC). Anderson. T.W., 1958, An introduction to multivarlate statistical analysis (Wiley. New York). Bradley, J.W.. 1968. Distribution-free statistical tests (Prentice-Hall, Englewood Cliffs, NJ). De Rosa. P. and G.H. Stern, 1977, Monetary control and the federal funds rate, Journal of Monetary Economics 3, 217-230. Durand, D., 1971, Stable chaos: An introduction to statistical control (General Learning Press, New York). Farr, H.T.. 1978, Attempts to estimate a reaction function for the federal reserve. Internal memorandum (Board of Governors of the Federal Reserve System, Washington, DC). Fry, E.R.. 1976, Seasonal adjustment of M,. Currently published and alternative methods. Staff economic studies (Board of Governors of the Federal Reserve. Washington, DC). Geweke, J.. 1978, The temporal and sectoral aggregation of seasonally adjusted time series.* Grether, D.M. and M. Nerlove, 1970, Some properties of ‘optimai’ seasonal adjustment. Econometrica 38. no. 5, 682 703. Lilliefots, H.W., 1967, On the Kolmogorov -Smirnov test for normality with mean and variance unknown, Journal of the American Statistical Association 62, 399 402. Lombra, R.E., 1978. Comment.*
24The references Series
marked
* can be found in the volume Serrso~rul Anu/~si.s of‘ Ecurromic Time Washington, DC, Sept. 9-10, 1976) edited by A. Zellner, of Commerce. Bureau of the Census. Dec. 1978).
(Proceedings of a conference,
(U.S. Department
Lombra. R.E. and R.Ci. Torte. 1975. The atratcgy of monetary policy, Economrc Rc\xw. Sept. Oct. 3 14. Lovell, M.C., 197X. Comment.* MaravaIl, A., 1978. Seasonally adjusted rates of growth versus rates of growth of seasonally adjusted levels: Some implrcntions for monetary control. Special studies paper 111). 120 (Federal Reser\c Board of Governors. Washington. DC). Morgenstern. 0.. 1963. On the accuracy of economrc observations (Princeton Uni\,crsity I’rc<\. Prtnceton. NJ). Prcrcc. D.A.. 197X. Seasonal adJuatmcnt when both determmistic and stochastic seasonality are pr-escnt.” Pierce. D.A. and G. Fries. 197X. Seasonal adJustment t>f daily data with special reference to the U.S. money supply. Special studies paper no. I29 (Board of Governors of the Federal Reserve Sy\tem.‘W&hington. DC). . Pierce. D.A.. N Vanpeski and E.R. Fry. 1978. Seasonal adjustment of the monetary aggregates. in: Improxmg the monetary aggrcgatcs: Staff papers (Board of Goternora of the Federal Reserve System. Washington. DC). Plosser. C.I., 1979. The analysrs of seasonal rconomrc models, Journal of Econometrics 10. 147 163. Plosser. C.I., 1978. A time series analysts of seasonality in econometric models.* Porter. R.D.. H.T. Farr and J. Peres. 197%. An evaluation of Board Staff forecasts of M,, internal memorandum (Board of Governors of the Federal Reserve System, Washington, DC). Porter. R.D., A. MaravaIl, D.W. Parke and D.A. Pierce, 197Xb. Transitory variations in the monetary aggregates. in: Improving the monetary aggregates: Staff papers (Board of Governors of the Federal Reserve System. Washington. DC) I 33. Shiskin, J.. 1078. Seasonal adjustment of sensittve mdicators.* Taylor, J.B.. 1978. Comment.* Wallich, H.C. and P.M. Kcir. 1978. The role of operating guides in U.S. monetary policy: A histortcal review, in: Krcdit und Kapital. Jan II (Berlin).