The Distributed Lag Between Capital Appropriations and Expenditures Author(s): Shirley Almon Source: Econometrica, Vol. 33, No. 1 (Jan., 1965), pp. 178-196 Published by: The Econometric Society Stable URL: http://www.jstor.org/stable/1911894 . Accessed: 05/01/2015 11:12 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp
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Econometrica, Vol. 33, No. 1 (January, 1965)
THE DISTRIBUTED LAG BETWEEN CAPITAL APPROPRIATIONS AND EXPENDITURES1 BY SHIRLEY ALMON
THISPAPER PRESENTS a new distributed lag, very flexible and easy to estimate, and
its applicationto the problem of predictingquarterlycapital expendituresin manufacturingindustriesfrom present and past appropriations.2One would expect, encouragedby a few simple correlationcoefficients,that computingthe weightedsum of past appropriationswhichcomprisescurrentexpenditureswould requirea distributionof weightswhich rise for a time, then decline. Here it is assumedonlythat successiveweightslie on a polynomial.Theplanis to estimateas regressioncoefficientsa few points on the curve, taking account of the fact that polynomialinterpolationwill be used to interpolatebetweenthemfor the remaining points.Thesameequationmayincludeothervariableswithdifferentdistributed lags, or unlaggedvariables.Sincethe length of the distributedlag is generallynot knownin advance,it is necessaryto estimatethe distributionusing varyingnumbersof periods,thenchoosethe bestamongthem. The quarterlydata on appropriationsand expenditurescome from the survey conducted by the National Industrial ConferenceBoard (NICB) among the thousandlargestmanufacturingcompanies.Estimatesin this paperare based on the nine years, 1953 through 1961. The distributedlag was estimatedfor all manufacturing,for durablesand nondurablesseparately,and for their fifteen constituent industries. The distributedlag on appropriations,plus estimated seasonalcoefficients,givesa good explanationof the variancein expenditures,with the best lags centeringaround8 and 9 quartersin length. The distributionsare quite stable,that is, among the severaldistributionsestimatedfor each industry, the weightsfor a given periodvary little after a sufficientlylong distributionhas beenreached. Thedistributedlag will be describedfirst,thenthe resultsof its applicationto the data will be reported.Finally,becauseone is usually appropriations-expenditures interestedin total capitalexpendituresfor the manufacturingsector, ratherthan the 60 per cent or so accountedfor by the largecompaniesreportingto the NICB, we shall look brieflyat the relationshipbetweenthe expendituresreportedin the 1 This paperwas read at the September,1963, EconometricSociety meetings.I am gratefulto James S. Duesenberry,under whose directionthe work was done as a doctoral dissertation,for helpfulcomments.The computingwas done at the MIT ComputationCenter. 2 Attempts have also been made to estimate (non-statistically) the rate of spending from applicationsfor acceleratedamortizationand from another questionnairedesigned specifically for this purpose [7, 8]. Kareken and Solow [1] have made statisticalestimatesof the distributed lag between new orders for nonelectricalmachineryand the Federal Reserve Board index of productionof new businessequipment. 178
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179
DISTRIBUTEDLAG
andExchange NICBsurveyandthosein theOfficeof BusinessEconomics-Securities Commission(OBE-SEC)surveys. 1. THE DISTRIBUTEDLAG
All distributedlag equationsstate that a dependentvariable,Y, is determined by a weightedsum of pastvaluesof an independentvariable,X: n-i (1)
Yt=
Y,W(i)
xt
-i.
i=O
If n, the numberof relevantvaluesof X is small,as may well be the case for some problemsif annualdata are involved,and if these successivepast observationsare not collinear,then the w(i), the weightswith which the severalpresentand past valuesare combined,can be estimateddirectlyby least squares.3Whenn is large, however,or whensuccessiveobservationsaretoo collinearfor this straightforward treatment,as will frequentlybe the case with quarterlydata, it becomesnecessary to makesome reasonable,restrictiveassumptionaboutthe patternof the weights. The point is, of course,to choose an assumptionwhich makesthe individuallag coefficientsdependon a few parameters,whichin turn can be estimatedin some reasonablysimple way.4 The "interpolationdistribution"assumesthat the w(i) are values at x=0, . . n -1 of a polynomialw(x) of degreeq+ 1, q< n, wheren is the numberof periods overwhichthe distributedlag extends.Its estimationis basedon the factthat once q+2
points on the curve are known-w(x0)=b0,
w(xl)=bl,.
.
., w(xq+i)=
bq+ -all the w(i) can be calculatedas linear combinationsof these known valuesby q+1
(2)
w(i)= , qj(i) bj j=0
(i =O~... ,n-1).
Herethe Oj(i) are the valuesat x= i of the Lagrangianinterpolationpolynomials
3For an exampleof a distributedlag estimatedin this way see [4]. 4 Two assumptionsfor which estimatingprocedureshave been workedout are that the weights decline geometricallyand, a more generalform of the same thing, that they are points on the Pascaldistribution[6, 11]. The first-mentionedseemednot so well suitedfor the presentproblem since it was expected that the true distributionsmight be more or less symmetric.Numerous data for attemptswere made to estimatethe Pascaldistributionwith appropriations-expenditures the chemicalindustry,but estimatesof one of the parameters,A, in the notation of [11], did not fall within the requiredrange, 0 to 1, possibly becausethe estimationmethodsare rathersensitive to specificationerror. The distributiondescribedbelow developed more directly from a third distributedlag, which assumesthat the patternof the weights is an invertedV [3].
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180
SHIRLEY ALMON (X-Xl)(X-X2)
P0(X)
...
...
~(X -XO) (X -Xi)
(x)
1
1
'
(Xo-Xq+1)
... (X-Xq+l) (X(X-X )(X-X2) X2) .... * (X1-XO(Xl) -Xq+
0P1(x
*
(X-Xq+l)
...
(XO-X2)
(XO-X1)
x1)
(X-x) (x
1)
q)
Xq) x0) ... Xq+iXo)(Xq+iXi).I. (xq-+ (xq-+ Noting that these polynomials have the property that (j=O, . . ., q + 1),~ (jOk; j =O . , q Jr1; k=O, . q.q+1)
o(XJ.)= 1 0AXk) =0
it is easily seen that q+1 w(x) =
E j=O
j(x)b
is indeed a polynomial of degree q +1 having the values bj at the points xj as required. Hence equation (2) is justified. Since we shall always want w(-1) = w(n) = 0, i.e., zero weights before time 0 and after time n -1, we may take xo =-1,xq+ 1= n, and bo =bq+ =1 0. Then equation (2) simplifies to q (2a) w(i) = E (i)bj. j=1
0
=-1 x0
1
/\
b2
bl.j
2
3
4
5
6
7 X3
x2
XI
I
b
8
9 X4
FIGURE1.-Example of LagrangianInterpolation.
Figure 1 shows with dashed lines an example of the Pj(x) for q =3, n =9, and xO= -1, xi = 1, x2 = 4, X3 = 7, X4 = 9, assuming that b, = b2= b3. The interpolated polynomial w(x), computed from (2a), is drawn with a solid line. Substituting (2a) into (1) gives n-1
(3)
YtZ=
i
q
(E
q 1j=l
bj
=)Xt-i
n-1
b= i=0
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DISTRIBUTEDLAG
181
The bj can now be estimatedby simplyregressingthe Y, on the q variableszj = r-
Ij(i)Xt
,
j=
1, ..
.,
q. The distributed lag weights are then calculated from
(2a). Anotherdistributedlag or unlaggedvariablescould of coursebe includedin (3). The computationalsteps are as follows: (a) Pick n and q. One will frequentlyhave some idea about how long a lag is relevantfor a particularproblem,but if furtherindicationsare desired,the method problem can be recommended.It used with the appropriations-expenditures was simplyto calculatefor one industrythe simplecorrelationcoefficientsbetween the dependentvariableand successivelaggedvalues of the independentvariable. That quarterin which appropriationsexplainedexpendituresabout as well as appropriationswith no lag at all was chosen as the centerof a rangeof n's. The planis to estimatea numberof lags of differentlength,thenchoosethe best among them. The numberof parameterpoints, q, mightbe determinedto some extentby the numberof observationsavailable.In additionto the points estimated,it must be rememberedthat the two points (t+ 1) and (t -n) are specifiedin advanceto be zero, so that the polynomialbeingfittedwill alwayspass through(q+ 2) points. (b) Choose the location of the q points xj. It makesno differencewherein the interval[0, n] these parameterpoints are located.The intuitiveexplanationis that within this intervalthere can be only one polynomial of degree (q+ 1) which minimizesthe sum of squares.The parameterpoints do not need to be integers, and sinceit is most convenientto choose themin standardform, they usuallywill not be. As a rule, therefore,all the distributedlag weightswill be calculatedas linearcombinationsof the q estimatedones. (c) Calculatethe LagrangianinterpolationcoefficientsOj(i),as indicatedabove. for all t n and j=1,. . ., q. This transfor(d) ComputeZtj= 12j'1ij(i)Xt-i mationof the independentvariableexpandsit into q variables.In timeseries,it also reducesthe numberof observationsby n-1. (e) Use multipleregressionto estimatethe bj for q
(4)
Yt=
j=1
bjztj+ut .
(f) Use equation(2a) to computew(i), i= 0, 1,..., n - 1. The standarderrorsof the distributedlag weights, combiningboth variancesand covariancesof the estimatedregressioncoefficients,are calculatedby (5)
St =
i 2(Zl
Z)- 0,
matrixof bj. whereX = (0P1(i),. . ., Iq(l)) andC2(Z'Z)- I is the variance-covariance (g) Varyn overthe rangealreadychosenandrepeatsteps(a) through(f). Waysto choose the best lag distributionfrom amongthe severalestimatedwill be discussedin connectionwith the use of the modelbelow.
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182
SHIRLEYALMON 2. THE DATA
The NICB's quarterly survey among the thousand largest manufacturing companiescovers appropriationsmade during the quarter,the appropriations committed(i.e., orders backlogat beginningandend of the quarter,appropriations placed)duringthe quarter,cancellationsof previouslymade appropriations,and capital expenditures.The informationrequestedon the last mentioneditem is identicalwith that collectedin the OBE-SECquarterlysurvey.Appropriationsare thoughtof as the finalstageof approvalfor capitalexpenditures,a confirmationof plans representedin the annual budget.5They are data routinelytabulatedby companiesusingformalcapitalbudgetingprocedures.The NICB publishes(in the ConferenceBoard Business Record) resultsof the surveyby 2-digitSIC industries. Since 1956,for all manufacturingand for total durablesand nondurables,it has also blown up the repliesof the reportingcompaniesto the level of the thousand largestcompanies,using total assets in three differentsize groupsas the basis of inflation. Whenthe surveywas begunin 1956about 500 companiesrepliedwith data for 1955 and 1956. These respondentsrepresented69 per cent of total assets in the survey'suniverse,and 55 per cent of employment.The responserate by industry varied from 95 per cent of assets, in primaryiron and steel, to 40 per cent, in transportationequipment.Of the first 500 respondents,353 were able to provide data for 1953and 1954, and these werelinked by the NICB to the 500-company series,whichran from 1955through1959. Sincethe beginningof 1958respondentshave included602 companies,owning 80 per cent of total assets among the thousandlargestcompanies.I used the 8 quartersoverlapin 1958-59to link the earlierseriesto the 602-companylevel. For appropriations(A), for example,each quarter'sappropriationspriorto 1958were multipliedby the ratio 8
E A, (602 companies)
t=l 8
E At (500 companies)
t=l
During 1958,the differencesbetweenthe 500-companyseriesthus inflatedto the 602-companylevel and the actual602-companyseriesweredivided,with gradually increasingweight to the latter. Since the first quarterof 1959 (1959-I) the 602companyseries has been used as reported.Expendituresand cancellationswere linkedin the sameway. 5 For further discussion of the relevant aspects of capital budgeting and the NICB survey see [9] and [10].
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DISTRIBUTEDLAG
3.
183
THE MODEL
The basic assumptionunderlyingthe following analysisof the appropriationsexpendituresdata is that expendituresaccrueentirelyfrom previousappropriations, that no capitalexpenditureis made without an appropriation,and that all appropriationsare eventuallyspent. In other words, the economic decision to investis madeat the appropriationstage,and thereafterspendingis determinedby technologicaland institutionalfactors.6Spendingof an appropriationis assumed to beginin the same quarterin whichit is made,and the problemnow is to determine what part of it is spent in this and each succeedingquarter.Cancellations must be mentioned,since companiesreportedthat they cancelledappropriations, in the period 1953through1961,amountingto 6.7 per cent of total appropriations made. Not to adjustin some way for cancellationswill cause the distributedlags to subtractthe averagecancellationin everyquarter,whereasin fact cancellations have been quite erratic.An attemptto take account of cancellationswill be described,but the conclusionwas that it is usuallynot worththe effort. It was early decided to use a three-parameterdistribution,i.e., to estimate fourth-degreepolynomialsto describethe patternof spendingof appropriations. In some sense, n, the numberof periodsin the distribution,must be considereda fourth parameter,since its optimalvalue is routinelydeterminedby tryingalternativevalues.The rangeof n's to be triedwas arrivedat, as noted above,by computing the simple correlationcoefficientsbetween expendituresand successive laggedvalues of appropriationsin the chemicalindustry,both for levels of these two seriesand for first differences.The correlationcoefficientsfor levels ran .08, .45, .68, .82, .86, .77, .56, .36, and .09. Firstdifferencesyieldedpositivecoefficients of moreerraticsizes until the ninth quarterback, whichwas slightlynegative.On the basis of these numbersit was expectedthat the optimallength of lag for this industrywas aboutnine quarters.AccordinglyI decidedto try n = 6, 7, . . ., 12 for all industries. Both appropriationsand expendituresfor manufacturingas a whole have a seasonalpattern,and, althoughsome industries'appropriationsdo not, it was clearlynecessaryto make some seasonaladjustments.Ratherthan correctingthe seriesbefore estimatingthe distributedlag betweenthem, three seasonaldummy variableswere includedin the estimatingequation,the fourth substitutedfor by definingit to be the numbernecessaryto makethe sum of the four equalto zero, in accordancewith the assumptionof a zero interceptfor the year as a whole. The six-variableequationto be estimatedby multipleregressionthenis 3
(6)
Et=a1s1+a2s2+a3S3+
E j=l
n-I
bj E j(i)At_i+ut i=O
6
This approachshifts the more interestingproblem,so far as economictheoryis concerned,to the study of appropriations.Work using appropriationsas the dependentvariableis underway and will be reportedlater.
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184
SHIRLEY ALMON
for n = 6, 7, . . ., 12, whereE is expenditures;s1, S2, S3 are seasonaldummies;A, appropriations;and the remainderof the terminologyis as definedin the previous discussionof the distributedlag. The parameterpoints x were located at -.6, 0, +.6, in the standardizedinterval,with -1 and 1 definedin advanceto be zero.7 for durableandnondurable Equation(6) thenwas estimatedfor all manufacturing, industries. fifteen constituent goods industriesseparately,andfor their 4. EMPIRICALRESULTS
Fromamongthe 7 distributedlagsestimatedfor eachindustry,one has beenchosen as the best, usingtwo rulesof thumband a modicumof judgment.Thesedistributions explainappropriationsratherwell. The coefficientsof multipledetermination adjustedfor numberof observations(A2)in 12 of the 15industriesaregreaterthan .8, and for the aggregates,all manufacturing,durables,and nondurables,they are above .9. We firstpresentsome observationson the sets of distributedlags estimatedfor each industry,although,in general,the empiricalresultswill be presentedfirstfor all manufacturingand then for the constituentindustries.The first but not the final criterionto be used in choosingthe best distributedlag is A2. In this respect therefrequentlyis not muchdifferencebetweenthe worstandthe bestdistributions. In the 18 industriesand aggregatesthereof,the gap runs from .02 to .32. For all it is .04; for most industries,less than .1. In general,the correlation manufacturing, coefficientsdo not bounce about from one distributedlag to the next, but move slowly up to a peak, and then decline. Withtheseusuallyslight differencesin degreeof explanationof the dependent variableofferedby successivedistributedlags, it is not surprisingthat consecutive distributionsdo not changemucheither,especiallyaftera sufficientlylong one has beenreached.In most industries,the shorterdistributionsarethe least satisfactory, givingthe lowest correlationcoefficients,and frequentlyhavingkinks and jagged peaks. Usuallyby the time 8 or 9 quartersare included,both these characteristics disappearand a smooth curve takes shape. Thereafter,most of the weights of succeedinglag distributionsfall within one standarderrorof the best one for the correspondingperiod.A concomitantof this behaviorin the early,"relevant"part of the longer-than-optimaldistributionsis that the later weights frequently become negative,since the polynomialcannot, of course, continueat zero. All manufacturingis a typical,if subdued,exampleof the changesthat occur as the 7 I didn'trealizeuntil I had finishedcomputingthat the location of the parameterpoints was a matterof indifference.The ones I used werechosen accordingto the theoremin numericalanalysis which says that the way to minimizethe maximumpossible deviation of an interpolatedpolynomial from the true function, providing the latter has n continuous derivatives and the nth derivativeis differentiable,is to pick as the parameterpoints the roots of the Chebyshev polynomial.
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185
DISTRIBUTED LAG
number of periods in the distributed lag moves from 6 to 12, and its 7 lag distributions are shown in Figure 2. The highest R' is reachedwith the 8-period distribution, and this is also the one where relative stability in the weights appears. Of the first 8 weights of distributions with 9 through 12 periods, all (except one in the longest distribution) fall within one standard error of the corresponding weights in the optimal 8-period distribution. In the 9-period distribution, all the weights are still positive, but beginning with 10 periods, the last one and then two are negative by 1 or 2 per cent. In view of the slight differences sometimes found in the correlation coefficients, close similarity of the weights among distributed lags of optimal and longer length may be considered a second criterion for choosing the optimal distribution. 25 -
20/ b1
15-
0
2
4-
6
Quarter
8
~ '
'
FIGURE2.-Distributed Lags on Appropriations,All Manufacturing.
It should be noted here that the sum of the distributed lag weights is substantially the same from the shortest to the longest distributions. The individual weights of the shorter distributions are simply larger, as seen in Figure 2. For all manufacturing, for example, the weights always add to .92. An additional 7 per cent of appropriations is accounted for by cancellations, and the remaining 1 per cent may possibly be due to lack of reporting on cancellations. The weights of the optimal 8-period distribution for all manufacturing are also shown in the first column of Table I. They are roughly symmetric, 15 per cent of appropriations being spent within the first half year, 45 per cent in a year, 77 per cent during the first 6 quarters and, as noted above, 92 per cent by the end of 2 years. This distribution yields a coefficient of multiple determination of .920. Expenditurespredicted using it and the estimated seasonal coefficients are shown in Figure 3 against actual expenditures. Appropriations are also shown. All turning points are correctly predicted. The residuals range from virtually zero to a little
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186
SHIRLEY ALMON TABLE I ALLMANUFACTURING, BESTDISTRIBUTED LAGS 1
2
3
Explanatoryvariablesincluding Appropria- Appropriations AppropriationsBacktions Only withBacklog log. Cancellations Seasonalintercepts, for quarters (million $)
Distributedlag weights,for quarters
I II III IV
-283. 13.
-311. 6.
-318. 34.
-
-
-
320.
347.
330.
0
.048 (.023) .099 (.016) .141 (.013) .165 (.023) .167 (.023) .146 (.013) .105 (.016) .053 (.024)
.068 (.023) .122 (.017) .156 (.013) .168 (.021) .157 (.022) .127 (.014) .084 (.017) .037 (.022)
.066 (.018) .113 (.016) .141 (.010) .153 (.014) .151 (.018) .136 (.016) .111 (.011) .078 (.014) .04(0
1 2 3 4 5 6 7
50.
42.
8
47.
(.016)
RegressionCoefficientfor backlogdummy Sum of weights Cancellations/Appropriations R?2 Durbin-WatsonStatistic
.922 .067 .920 .890
-113. (48.) .919 .067 .933 1.409
-
76. (47.) .990 .067 .940 1.314
over 1 per cent of actualexpenditures,averagingabout 4 per cent. Smallas they are on the whole, it is of someinterestto examinethemfurther. From 1954-IVthrough 1955-II expendituresare a little largerthan predicted. The investmentboom of 1956-57 is characterizedfirst by a period when actual expendituresfell shortof predicted,from 1955-IVthrough1957-1I,followedby a burstof overspending,in the last 2 quartersof 1957.(Butthe underspending in the first7 quartersof this periodtotaledmorethantwicethe sumwhichwas eventually ".madeup" in the last half of 1957.)Overspending was cut shortat the beginningof
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187
DISTRIBUTED LAG
Millions of
$
4000 Ak Appropriations
3000
A~~~ 2000
\-
iooc
*j4
+ihilVJ<
Calculated
expenditures
Millions of $
300cancellations:
200-
Billions of$ l l Unfilled orders for fabricated metal products and nonelectrical machinery
10.
1953
1955
1957
1959
1961
FIGURE3.-Appropriations, Expenditures,and Related Series for All Manufacturing.
1958, when in every quarter spending fell short of predicted. From 1959 through 1961 spending was larger than predicted from appropriations, by amounts ranging from virtually zero to nearly 7 per cent of spending. This summary suggests that the residuals from equation (6) are autocorrelated. That they are is confirmed by the Durbin-Watson statistic, which is .9, indicating definite autocorrelation. Autocorrelation here is not terribly serious, since it does not mean biased estimates, and the residuals are quite small. It only indicates that the standard errors of regression coefficients (and indirectly of the distributed lag weights) are understated. The regression coefficients, however, are substantially more than twice their standard errors (for all manufacturing the ratios are 4.9, 6.7, and 5.1), and at any rate no decisions about the usefulness of variables have been made on those grounds. Nevertheless, since two of the most important runs of
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188
SHIRLEYALMON TABLE II BEST DISTRIBUTED
Seasonal intercepts, for quarters (million $)
Distributed lag weights, for quarters
I
II III IV 0 1 2 3 4 5 6 7
LAGS FOR INDUSTRIES
1
2
3
4
5
6
7
8
All Manufacturing
All Durable Goods
Iron and Steel
Nonferrous Metals
Electrical Machinery
Machinery Except Electrical
Transportation Equipment
Stone, Clay and Glass
-158. 2. - 28. 188.
-45.4 8.8 -22.9 59.6
-10.7 - 3.3 - 2.5 16.5
-36.9 - 5.9 3.2 39.5
-10.9 - 3.0 - 2.5 16.4
-32.5 2.2 3.9 26.5
-15.3 2.6 1.0 13.7
.057 (.021) .101 (.016) .132 (.014) .151
.013 (.030) .047 (.024) .092 (.021) .138 (.029) .175 (.028) .190 (.019) .174 (.026) .114 (.033)
.076 (.023) .112
.072 (.019) .110
.058 (.016) .119
(.023) .123
(.018) .124
(.014) .119 (.009) .109
(.010) .123 (.012) .113 (.019) .099 (.020) .084 (.014) .068 (.010) .052 (.015) .031 (.017)
.064 (.023) .127 (.018) .169 (.014) .181 (.020) .161 (.021) .114 (.015) .056 (.018) .007 (.021)
.061 (.021) .114 (.017) .150 (.010) .166 (.017) .160 (.022) .136 (.017) .097 (.010) .053 (.017) .016 (.020)
-
-12.7 (13.9)
-311. 6. - 42. 347. .068 (.023) .122 (.017) .156 (.013) .168 (.021) .157 (.022) .127 (.014) .084 (.017) .038 (.023)
(.021) .156 (.020) .145 (.013) .118 (.019) .071 (.024)
8 9
Regressioncoefficients for backlog dummy
BLC
-113. (48.)
Sum of weights Cancellations/Approp.
SW C/A R2
.919 .067 .933 1.396
R2
Durbin-Watson statistic DW 1
-
55. (35.) .931 .057 .922 1.487
(.016) .097 (.020) .086 (.018) .076 (.013) .063 (.013) .040 (.013)
-30.3 -24.5 (8.5) (16.0) .944 .017 .844 1.631
.903 .073 .915 1.015
14.6 (5.4) .875 .121 .863 .900
2.6 (3.1)
.879 .089 .839 1.557
(.014) .166 (.016) .186 (.020) .177 (.016) .140 (.011) .086 (.020) .032 (.024)
.964 .067 .906 .903
Includes lumber products, furniture and fixtures, and miscellaneous manufactures.
autocorrelated residuals correspond to the behavior of variables whose influence had been suspected earlier, these variables were now added, one at a time, to equation (6). The obvious explanation for expenditures falling short of predicted during 1956-57, and the compensatory overspending in the last half of 1957, is the temn-
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.952 .052 .886 1.631
189
DISTRIBUTEDLAG
I TI
III IV 0 1 2 3 4 5 6 7
9
10
11
12
13
14
15
16
17
18
Fabricated Metal Products
Other' Durables
All Nondurable Goods
Food and Beverages
Textiles
Paper
Chemicals
Petroleum
Rubber
Other Nondur.2
-7.38 - .08 -1.77 9.24
-5.08 -2.00 -1.21 8.29
-176. 6. 5. 175.
-11.4 6.7 1.1 3.6
-3.62 .67 1.77 2.18
12.7 -13.5 .3 1.1
-39.9 .5 - 2.0 41.4
-68. -22. -27. 117.
-6.59 .41 -1.14 7.32
-1.44 .75 -1.36 2.05
.085 (.042) .153 (.038) .191 (.035) .195 (.041) .166 (.037) .113 (.030) .051 (.046) .004 (.052)
.081 (.042) .137 (.028) .169 (.026) .176 (.037) .161 (.031) .124 (.027) .070 (.036)
.126 (.021) .174 (.017) .172 (.009) .144 (.016) .108 (.021) .075 (.018) .051 (.011) .035 (.014) .022 (.018)
.011 (.052) .105 (.041) .200 (.037) .245 (.045) .221 (.039) .139 (.043) .041 (.054)
.022 (.033) .155 (.021) .266 (.031) .285 (.029) .201 (.022) .068 (.039)
.102 (.022) .148 (.021) .159 (.014) .148 (.017) .128 (.020) .105 (.018) .083 (.015) .062 (.020) .037 (.021)
.063 (.022) .103 (.018) .127 (.011) .138 (.019) .138 (.024) .130 (.019) .113 (.012) .087 (.019) .051 (.023)
.184 (.044) .270 (.027) .247 (.032) .144 (.030) .027 (.040)
.103 (.018) .136 (.016) .130 (.011) .111 (.015) .093 (.018) .084 (.015) .084 (.012) .083 (.017) .064 (.018)
.120 (.060) .130 (.043) .115 (.045) .121 (.057) .157 (.044) .197 (.047) .177 (.064)
-
-
-3.82 (1.11)
-10.0 (4.3)
.998 .047 .853 1.554
.972 .046 .870 1.300
.888 .056 .819 1.096
1.017 .010 .596 .718
.
8 9
BLC
-5.45 (3.25)
SW C/A
.958 .037 .647 1.640
f? 2
DW
3.03 (1.72) .918 .048 .855 1.582
49. (17.)
.908 .074 .945 1.556
1.68 (2.69) .962 .043 .782 .978
-
.950 .064 .850 .701
7.2 (8.2)
.872 .091 .883 1.636
2 Includes tobacco, apparel, printing and publishing, and leather products.
porarylimitationin supplyof capitalgoods, causedby heavydemandand aggravated by the steel strikein 1956. It is reflectedin the end-of-quarterbacklog of unfilledordersfor fabricatedmetalproductsand nonelectricalmachinery,whichis shownat the bottom of Figure3. It was decidedto take accountof this supplyconstraintby includinga variable
This content downloaded from 191.98.191.8 on Mon, 5 Jan 2015 11:12:39 AM All use subject to JSTOR Terms and Conditions
190
SHIRLEYALMON
whichwouldbe + 1 duringthe 5 quartersbeginningin 1955-IV, -1 in the following 5 quarters,andzeroelsewhere.Equation(6) thenbecomes 3
(7)
Et=a1s1+a2s2+a3S3+a4S+
n-I
bj E 0j(i)At_i+ut, j=1 i=o ,
whereS is the dummyvariablerepresentingthe supply constraint.Expenditures were then regressed on these 7 variables for n = 6, . . ., 12 in the same 18 industries andaggregatesas before. The effect of the additionalvariableon the best distributedlag for all manufacturingcan be seen in Column2 of TableI. The estimatedregressioncoefficient appearswith the expectednegativesign, it makesa significantcontributionby the usualstandards,and, as hoped,the autocorrelationin the residualshas decreased. The Durbin-Watsonstatistichas risen from .9 to 1.4. The best distributedlag is again 8 quarterslong, the weightsstill add to about .92, and individuallyare little changed,the maximumdifferencebeingabout 2 per cent. All the new weightsfall within2 standarderrorsof the originalones, and 4 of the 8 arewithinone standard error. Most important,the generaleffect of the backlog variableis to shift the distributionforward,i.e., it raisesthe first4 weightsandlowersthe last 4, indicating that the new variabledoes indeedcompensatefor the delayedspending.Incidentally, R2 risesslightly,from .920 to .933. As for the effecton the residuals1955-IV through 1958-I, the overestimateof spendingis reducedin each of the first 5 quarters,though not until 1956-IVis the small overestimateswitchedto a larger In the last half of the 10-quarterperiodinvolved,whenthe backlog underestimate. of orderswas being workeddown, the original predictionswere underestimates only in the last 2 quartersof 1957.Takingaccountof the backlogof ordersdoes reducethese2 errorsto abouthalftheirformersize, but does not changethe sign of any of the 5. In short,the new variableproduceschangesin the expecteddirection, but does not completelyerasethe spell of autocorrelatedresiduals. Besides 1956-57, the other period when there is an obvious explanationfor deviationof predictedfrom actualspendingis in 1958,when expenditureswerea good dealless thanpredicted.Herethe underspendingof a predictedlowerlevel of investmentappearsto be voluntary.Cancellationsshot up in 1957-IIandremained high in 3 of the 4 quartersin 1958.Theytoo are shownin Chart2. Comparisonof these cancellationswith the residualsfrom equation (7) reveals no generalcomovement,exceptin 1958.Nevertheless,since some adjustmentfor cancellations had been seriouslyconsideredbeforeany computingwas done, but delayedover the matter of which appropriationsare cancelled, an attempt to answer that questionwas now made. Expenditurespredictedby equation(7) were multiplied by the reciprocalof the sum of weightsshown in Column2 of Table 1, and the residualswere recalculated.The new residuals,almost all of them positive,were now used as a dependentvariableto be explainedby cancellations,laggedsucces-
This content downloaded from 191.98.191.8 on Mon, 5 Jan 2015 11:12:39 AM All use subject to JSTOR Terms and Conditions
191
DISTRIBUTED LAG
sively from zero to threequarters.The four simple correlationcoefficientsthus obtainedwere, respectively,.264, .122, .197, and -.022. None of these is significantlydifferentfrom zero at the .95 confidencelevel, and the considerationof cancellationsmighthavebeendroppedat thispoint. Havingcome this far, however,the decisionwas made to re-estimateequation (7) after adjustingthe data for cancellations.Cancellationsin each quarterwere dividedby three,and one-thirdaddedto expendituresin the currentand each of the following 2 quarters.Equation(7) was re-estimatedwith this data, but as a 20
All manufacturing
10-
Other durable goods 10
C 20
0 All durable goods
10
All nondurable goods 10 O .
20
Primary iron an steel
11,
Food and beverages 10
O .
\
O
a H\ X Primary nonferrous metals
0 Textile mill products
. ba ?10
..,
0
X
0
Paper and allied
Electrical machinery
~~~io.
?
