- Email: [email protected]
Classical and variance bounds tests of the production smoothing hypothesis
international journal of
production economics ELSEVIER
Classical
Int. J. Production
and variance Sophia “Athens University ‘Western
Economics
35 (1994) 15-22
bounds tests of the production hypothesis P. Dimelisa’*,
Moheb
smoothing
A. Ghalib
qf Economics and Business, 76 Patission Str., 10434 Athens, Greece Washington Universit~~. Bellingham, WA 98225-9038, USA
Abstract In this paper we test the “controversial” production smoothing role of inventories by applying both classical econometric techniques and variance bounds tests. We use seasonally unadjusted data measured in physical units for five industries. The model used is a version of the familiar Holt et al. (1960) inventory model. We also use the data on one industry disaggregated at the district level to investigate potential bias which may arise from aggregation. Overall, the empirical evidence suggests production smoothing in the great majority of the industries examined.
1. Introduction
Research on inventories has not yet yielded definite conclusions with regard to the “controversial” production smoothing/buffer stock motive of holding inventories. The negative results obtained by empirical studies have raised several questions as to what might have caused the apparent rejection of this classical inventory mode1 [ 11. Recent investigations have focused on either the quality and appropriateness of the data used in testing empirically the production smoothing/buffer stock hypothesis, or on the validity of the inventory models employed for that purpose. Despite the growing concern about these issues, very few studies have addressed them adequately, partly due to the lack of appropriate data. In this paper we exploit the availability of a particular set of seasonally unadjusted monthly
* Corresponding
author
0925-5273/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0925-5273(93)E0098-G
data measured in physical units to test the production smoothing hypothesis of inventories. To test this hypothesis we follow two different approaches: starting from a version of the Holt et al. [2] inventory model, we first estimate a reduced form econometric equation, and second we perform variance bounds tests [3,4] using estimates of the Euler condition associated with the above model. In doing this, we can compare the results from the two different types of tests and draw conclusions regarding the reliability of the tests and the validity of the hypothesis. Finally, by using the disaggregated data on one industry, we also investigate the implications of any potential bias arising from aggregation.
2. The model For the purposes of this paper, we start a genera1 linear quadratic mode1 of inventories
with that
S.P. Dimelis,
16
M.A.
Ghali: Int. J. Production
embodies production smoothing, derive the cost minimizing relationship between production, inventories and sales, and then apply the different tests of the model. The cost structure we use in this is a simplified version of the Holt et al. [2] model. Introducing uncertainty in the model, the representative firm is minimizing its expected discounted future stream of real costs: min&
f. B’{c,Q: + c2(Qt- QtpI)’ r=o
subject
to
(2)
H _ 1 given,
(3)
where Q, and S, are production and sales in period t, respectively; H, is the firm’s stock of inventories at the end of period t; E,( .) denotes mathematical expectations conditional on information available at time 0; fi is a discount factor between zero and one; the Ci’S (i = 1, 2, 3, 4) are positive cost parameters; the term Q: represents production costs, which under the assumption of decreasing returns are convex. The term (Qr - QI _ I )’ reflects the costs of changing production, so that a penalty is imposed on fluctuations of the production level. Finally, the last term, (H,- c&+ 1)2, is a cost associated with the level of inventories; it implies increasing costs as inventories (H,)deviate from a desired level (which is assumed to relate linearly with expected next period sales (cqSI+ i)). Inventories in this model are held for two reasons: to smooth production in the face of fluctuating demand as implied by the first two types of costs and to avoid stocking-out as implied by the last type of cost. If this last cost is large relative to the two previous ones, then production smoothing will not occur. Sales here are taken to be exogenously determined. Differentiating (1) with respect to H, we obtain the first-order conditions for cost minimization of (1) subject to (2) and (3): 2(B2c2Ht+2
-
BCCI
+
2c2(1
+
B)lH,+r
+ [c,(l + /I) + cz(B* + 48 + 1) + c,]H,
35 IlYY4)
15-22
H,m1+ c2H1-2
- [cl + 2c2(1 + 8)-j +
P2”2&+2
-
CPCl
+
+ [c, + c2(1 + zp)]s,
Bc2V
+
PI
- C2Sr-1)
+
CAlS,+1
= 0.
(4)
This is the Euler equation associated with problems (l))(3). It is an expectational difference equation that can be solved to derive the optimal decision rule for inventory behavior. Following Blanchard [S] in normalizing by c2 = 1, we obtain
H, = (iI+ A,)H,_,
Q,= S,+ H,- H,mI,
E,
Economics
-
(i28Y”l
- 3.,j.2Hf_2
B(L)E,St+2+,,
(5)
where B(L) = - /PC2 + [!JCi + fiC2(2 + fi) + c3cJLmi - [Cl + c2(1 + 2fi)L-l
+ (.2Lm3
and ii, i2 are the smallest absolute value roots of the Euler eq. (4). According to (5), the optimal inventory stock depends on its values lagged once and twice, as well as on lagged, current and expected sales. Had we allowed for the existence of stochastic cost elements in (1) then inventory behavior would have also depended on the unobservable stochastic disturbances. Furthermore, the roots 1, and i2 can be expressed in terms of the structural cost parameters, ci, so that all coefficients in (5) are functions of these parameters. The first approach in testing this model is to estimate an unconstrained regression of a testable form of Eq. (5). This is equivalent to estimating a reduced form equation in standard econometrics and we will refer to that as “classical econometric testing”. This is a simple and useful approach, but it does not allow estimation of the structural parameters. However, the coefficient estimates obtained from such regressions provide important information as to the validity of the model. We also test the model using the variance bounds test developed originally by West [3] and generalized recently by Kollintzas [4]. The application of these tests requires estimation of the structural parameters. Before proceeding to the estimation
S.P. Dimelk,
M.A.
Ghalillnr.
J. Production
process, we describe in the following section the two tests as applied to the model presented in this paper.
3. The two tests and the data 3.1. Classical econometric
testing
In order to obtain an estimable form of (5) we should first derive its backward looking solution. For that purpose we assume an AR(4) process for S, and express B(L)S,+2 of (5) in a quasi-first-order form as in Blanchard [S], in which case we derive H, = m,H,_,
- m2H,_,
+ m3St
Economics
35 (1994)
15-22
17
these conditions, each factor by itself (production costs or adjustment costs) can be considered as necessary to result in production smoothing. Within a multifactorial framework each factor acts independently and therefore the absence of one of them is not evidence against production smoothing, while if both factors are present a synergistic effect can exist, producing more smoothing. Furthermore, with respect to sales, it can be inferred that, under the assumption of increasing marginal costs, firms would prefer to meet current sales out of current inventories. This implies a negative and less than one value for m3 in Eq. (7). As a result, a value exceeding unity of the coefficient on S, in (8) can be evidence against production smoothing.
3 +
1
m3iS,-i
+
c’,,
i=l
3.2. The generalized
where the parameters ml, m2, m3i, i= 1,2,3, are complicated functions of the structural parameters cl, c3, c4 and B. The term v, represents the residuals. Equivalently, we can write (6) in terms of production by using the identity Qt = H, H,_, + S,, so that
Q, = mQlml - (1 - ml + m2)H,-1 + (m3 + l)S, + 1
mziS,-i
+ f-4
(7)
variance bounds test
The model is also tested using the generalized variance bounds test (GVBT) developed by Kollintzas [4], according to which a weighted sum of autocovariances of the covariance stationary component of inventories should be nonnegative. Following the steps described in Kollintzas [4], it can be shown that for the model to be accepted, the GVBT requires that2,3 [(l + j3-‘)c,
+ (5 + pml)cZ + jj-lc3]
var(h)
i=l
Estimation of Eq. (7), or (6) yields consistent estimates of the reduced form coefficients. These coefficients can then be interpreted in association with the structural parameters of Eq. (1). As explained in the previous section, for the production smoothing hypothesis to hold, either c1 or c2 or both should be different from zero and large relative to cj. In terms of the reduced form coefficients, this implies that either (1 - ml + mz) or m2 or both be different from zero, respectively.’ Under
- 2(c, + 4c,)cov(hl, + 2c, cov(hlhm2)
h_ ,)
2 0,
(8)
where var(h) = E(h:), cov(hl,
cov(h,,
h- 1) = E(h,, h,- A,
h-2) = E(h,, h,-2)
and h, = H, - I?,, where l?, is the trend component.
’ Within a similar setup, Blanchard [S] provides explicitly the relationship between the structural parameters c,, c, and the reduced form estimates m, and rnz. However, these relations are subject to the particular normalization of c2 = I, while parameters, such as c4, not included in the lag polynomial B(L), cannot be expressed in terms of the m,‘s
’ Details for the derivation of this condition can be found in the appendix of Ref. [6]. 3 All variables are assumed to have zero unconditional means for expositional simplicity.
S. P. Dimeli.\, M. A. Ghuli: ItIt. J. Produc~rion Economic.s 3.5 i 1994) IS-22
IX
This inequality measures the cost savings the firm would expect by adjusting inventories optimally, i.e. in response to random sales fluctuations as suggested by the model. These cost savings were calculated by comparing the expected costs under this optimal policy and the necessarily higher costs that firms would face from a policy of letting inventories grow at a constant growth rate, which implies meeting random fluctuations by variations in production. A violation of inequality (8) would indicate that firms do not choose their inventories according to the optimal policy, which would mean a rejection of the model and the lack of production smoothing motive of inventories. The weights in (8) are functions of the structural parameters of the model which can be estimated from the Euler condition (4). By defining q, = /3Q, - QI_ 1 and under the arbitrary normalization c’~ + ~~(1 + p) = 1, Eq. (4) can be written as q,+, = c2(/@+2
+ 41) + (.3Ht - (.3~Jr+l
+ ~1. (9)
The parameters c2, c3 and c4 were estimated (up to a normalization factor) using a two-step two-stage procedure described by Cumby et al. [7] with a fixed fl = 0.995. The first step includes estimation of the Euler equation by instrumental variables. As instruments we used a constant, a linear trend and three lags each of inventories and sales. The disturbance term was allowed to follow an MA (2) process. In the second step, the parameter estimates were obtained from the optimal instrumental variable estimator: h = (X’z~“z’X)-‘x’z~-‘z’~~, where fi = (l/T)(Z’iZ) and i is the variancecovariance matrix of the first-step residuals, X is the matrix of the right-hand side variables of (9), Z is the matrix of instruments and y the left-hand side variable of (9). Variances and autocovariances were calculated from the estimates of a VAR system of inventories and sales and the Yule-Walker equations. The asymptotic standard error of the test statistic was derived from Hansen’s [S] asymptotic varianceecovariance matrix of the estimated parameters (see [3]).
3.3. The duta The data used are for six industries: asphalt, oil burners, glass containers, printing paper, beer and Portland cement. For each industry we use 120 monthly observations for the period 19.50-1960. In addition, for the Portland cement industry, disaggregated data for each production district were also available. This industry is divided into 19 production districts which are described in Ghali [9]. These data are not seasonally adjusted and are in physical units. For the variance bounds test the series were first deseasonalized using seasonal dummies.
4. The results 4.1. Unconstrained
regression estinmtes
In Table 1 we present the results of OLS estimation of Eq. (7) fitted to monthly data on the six industries4 The results indicate strong production smoothing in the industries of glass containers, and Portland cement, as judged by the high significance and magnitude of the lagged production coefficients, the significance of the lagged inventory coefficients and the fact that current sales coefficient estimates are significantly less than unity. Furthermore, following the analysis in Section 3.1, these results indicate that both adjustment and production costs are strongly significant in these industries. Considerable smoothing is also observed in the asphalt and oil burners industries. Although lagged production is not significant in these two industries, the coefficients of lagged inventories and current sales are significant, with the latter also being significantly less than unity. In the printing paper and beer industries the evidence is conflicting. Lagged inventories in these industries seem to play a statistically important role, while lagged production is insignificant. These
’ It should be mentioned that the presence of a lagged dependent variable in the OLS regressions may produce serial correlation in the errors, in which case OLS estimates are inconsistent. Using Durbin’s procedure 10 check for autocorrelation. it was found that only in the oil burners and the printing paper industries was it significant. Correcting for serial correlation did not change the conclusions drawn earlier about these two industries.
S.P. Dimelis.
Table 1 Unconstrained
regression
estimates
M.A.
Ghnliilnt.
of the production
J. Production
smoothing
Economics
35 (1994)
model for six industries
15-22
(r-values
19
in parentheses)
Industry - 0.06
Asphalt
( - 0.54) Oil burners Glass containers Printing
paper
Portland
- 0.12
- 0.10
( - 1.35)
( - 2.34)
0.40 (4.40)
( - 3.61)
0.20 (1.35)
( - 3.68)
- 0.04
Beer cement
- 0.28”
( - 4.39)
- 0.16 - 0.26” - 0.41”
( - 0.39)
( - 4.52)
0.36 (3.63)
( - 2.92)
- 0.12”
0.98
1.95
0.96
1.54
0.93
2.04
0.95
1.61
0.0 1 (0.33)
0.98
1.99
- 0.14”
- 0.13”
0.97
2.01
( - 2.39)
( - 2.71)
0.92 (26.76)
- 0.02
- 0.03
( - 0.13)
( - 0.93)
( - 4.02)
0.9 1a (15.59)
( - 1.02)
0.12 (1.74)
( - 0.95)
0.32” (7.06)
- 0.08
- 0.03
- 0.07
( - 1.32)
( - 0.60)
( - 1.56)
1.01” (25.05)
( - 1.27)
0.02 (0.36)
( - 0.31)
1.05” (38.11)
0.05 (0.50)
0.03 (1.31)
0.71” (13.59)
0.07 (1.24)
0.12
- 0.20
For each estimation, 117 observations were used as required by the lagged variables A set of monthly dummies and a time trend were included in each regression. Estimates obtained by OLS procedure. a The coefficient is statistically different from zero at the 5% level.
facts imply that only production costs matter whereas adjustment costs are not important in the production smoothing process. However, the sales coefficients both exceed unity and statistically so in the beer industry only. It could be argued therefore that “weak” production smoothing is evident in the printing paper industry, while no clear conclusion can be drawn in the case of beer without knowledge of the relative magnitude of all the cost parameters. In Table 2 we present the results of OLS estimation of Eq. (7) for the 19 production districts of the Portland cement industry. Lagged production is highly significant in all but five districts, lagged inventories are highly significant in all but three cases, while the current sales coefficient estimates are significant and smaller than unity in all cases. Lagged production and inventories are both insignificant only in district 10. These results give strong support to the production smoothing hypothesis and are in agreement with those derived from the aggregate estimates of this particular industry. It is interesting to note here that these results confirm the evidence implied by the relative variability of unadjusted production to sales reported by Ghali [9] for exactly the same industries. Comparing the results from the first two columns of
- 0.14” - 0.05
- 0.01
of the model.
Table 1 in Ghali ([9], p. 466), the following can be observed. In industries in which the variance of unadjusted production exceeds the variance of unadjusted sales and the correlation between sales and inventory change is positive or negative but weak (district 10 of Portland cement, printing paper and beer), the corresponding estimates from Tables 1 and 2 indicate “weak” or “lack of” production smoothing. It should also be noted that on the basis of the seasonally adjusted and detrended data of Table 1 in Ghali [9], only three out of the nine possibly “negative” cases of that table are confirmed by our regression estimates.
4.2.
Variance bounds test estimates
Tables 3 and 4 report the results of the GVBT given by Eq. (8). The evidence from Table 3 supports the production smoothing motive in all but one of the industries. The results are statistically significant5 in three industries: asphalt, glass containers and cement. In the case of printing paper the
’ By statistically significant we mean a test value which is more than two asymptotic standard errors.
20
S.P. Dinwlis.
Table 2 Unconstrained
regression
Portland cement production district
estimates
M.A.
Giud:‘Int.
of the production
J. Produc~rron Econotnics
smoothing
model for the Portland
Qr-1
H,-,
S,
~ ~ ~ -
II 12 I3 I4 I5 16 I7 I8 I9
0.44 0.15 0.41” 0.52” 0.44” 0.20 0.24” 0. I7 0.35” 0.05 0.20 0.33” 0.35” 0.20” 0.22” 0.13 0.42” 0.20” 0.13
0.18” 0.15” 0.22* 0.16” 0.19” 0.08” 0.24 0.29 0.31” 0.02 0.06 0.10” 0.13” 0.06 0.16 0.13” 0.24” 0.27” 0.29”
0.11” 0.77” 0.53” 0.49 0.57” 0.40” 0.85” 0.69” 0.57” 0.66 0.41” 0.39” 0.47 0.56” 0.50” 0.52” 0.56 0.55” 0.69”
- 0.1 I - 0.01 0.23” - 0.05 - 0.03 0.02 - 0.07 - 0.03 - 0.01 0.37” - 0.02 0.08 - 0.07 0.24” 0.23” 0.1 I - 0.09 - 0.04 0.06
Aggregate
0.36
- 0. I 2”
0.71”
0.07”
I 2 3 4 5 6 7 8 9
I0
See notes to Table
Table 3 Variance
bounds
Industry
Asphalt Oil burners Glass containers Portland cement Beer Printing paper ” Test
~ ~ ~ ~ ~ ~ -
35 (1994)
15-22
cement
production
5 ~2
S 1
~ -
~ -
~ ~
distracts
R2
DW
- 0.04 0.00 - 0.13 0.00 - 0.16” 0.01 - 0.05 - 0.02 - 0.07 ~ 0.16 0.0 I - 0.06 0.01 - 0.08 ~ 0.15” 0.03 0.02 0.04 - 0.02
0.87 0.93 0.93 0.86 0.96 0.82 0.93 0.84 0.86 0.90 0.9 I 0.92 0.83 0.94 0.93 0.96 0.92 0.93 0.91
2.15 2.03 2.14 2.04 2.16 I .94 2.02 1.99 2.12 I .9x 2.03 I .98 2.04 2.08 I .90 I .95
~ 0.13”
0.97
2.01
& 3
0.14” 0.04 0.21” 0.11 0.12 0.07 0.06 0.04 0.10 0.05 0.06 0.05 0.04 0.01 0.07 0.03 0.10 0.01 0.00
~ 0.14
1.84 2.04 I .98
I
tests on industry
Table 4 Euler equation
data”
Generalized variance bounds test (GVBT)b I23 67X’ - 966 533 931958‘ 2434791’ 39 577 d
normalization estimated under the (‘r + (I + pjcz = 1. ’ Kollintras [4] test. ’ Stgniticant at the 5% level. ’ A large positive and highly insignificant value.
(r-values)
estimates
of the cost parameters
Industry
(‘I
Asphalt
1.335 (0.1 I)
- 0.168 (0.053)
- 0.170 (0.10)
1.655 (2.77)
(2.26) (1.10) (3.94) (2.25) (1.34) (0.00)
Oil burners
1.634 (0.49)
- 0.318 (0.24)
- 0.178 (0.22)
- 5.95 (4.90)
Glass containers
0.578 (0.301)
0.21 I (0.15)
- 0.016 (0.02)
4.420 (25.34)
0.568 (0.156)
0.043 (0.08)
~ 3.015 (3.98)
restriction:
Beer
0.005 (0.5 17)
~ 0.361 (39.60)
value of the test was positive but extremely large, and so was its variance, rendering it statistically insignificant. These results can be compared with those obtained by Krane and Braun [lo], who also used
Portland
cement
Asymptotic standard Estimates obtained C’, + (I + B)C, = I.
- 0.133 (0.31) I.263 (0.12)
(‘2
~ 0.132 (0.06)
(‘3
errors in parentheses. under the normalization
(‘4
restriction:
disaggregated, monthly physical product data but from different sources and periods (mostly for the period 1977-1988). They found that in the asphalt
S.P. Dimelis,
Table 5 Tests of homogeneity tricts
across
the Portland
M.A.
GhalilInt.
production
J. Production
dis-
Model
Hypothesis
1 2 3
Common regressions in all 19 districts Different constants, common all other coefficients Different constants, different lagged production coefficients, common all other coefficients Different constants. different lagged production and inventory coefficients, common all other Different regressions in all 19 districts
4 5
testing
and glass containers industries the smoothing hypothesis is supported, while for the printing paper and beer it is rejected. With respect to the cement industry, the same evidence is provided by Fair [l I], who also used physical data from the Bureau of Mines for the period 1947-1964. No comparison with other studies, such as, for example, West’s [3] or Blinder’s [12] results can be made due to the high degree of aggregation in their data. Table 4 contains the cost parameter estimates we derived from the Euler equation (9). The marginal production cost parameter c1 is positive in four out of the five disaggregated industries we considered. However, the parameter c2 that measures marginal costs of changing production is mostly negative and so are most of the other cost parameters. It should be noted that the normalization used [ci + (1 + /?)cZ = l] constrains either ci or c2 to be positive in each equation. Therefore, no clear statement about the positive cost parameters can be made. Krane and Braun [lo] also estimated positive cost parameters in about three quarters of the disaggregated industries they examined.
Economics
35 (1994)
IS-22
21
the parameter estimates.6 Using the data on the Portland cement industry which were available disaggregated at the district level, we also tested for possible aggregation bias. This was achieved by pooling the sample of 117 x 19 = 2223 observations and defining a set of dummy variables for each district and each regressor of Eq. (7) separately. Then we performed various tests of homogeneity for the constants and the coefficients as a whole or sets of them. The hypotheses tested are described in Table 5. Each hypothesis was tested using the Ftest and the results are presented in a form of variance analysis in Table 6. The calculated values of the F-test indicate that no homogeneity across districts either of the intercepts or slope coefficients is existent in the data. As a result, one would expect that estimates obtained by using the aggregated data suffer from bias. It is not, however, possible to draw any conclusions as to the direction and magnitude of this bias.
5. Conclusions
4.3. Aggregation
In this paper the production smoothing property of inventories was tested using two different tests. It was found that both tests provide positive evidence for production smoothing in the asphalt, glass containers and Portland cement industries (Tables 1 and 3). Among the remaining three industries, oil burners, beer and printing paper, the “classical test” is less supportive of the production smoothing hypothesis only in the case of beer, while the “variance bounds test”, although insignificant in all three of them rejects it only in the case of oil burners. Finally, the lack of homogeneity across districts yields the conclusion that aggregation introduces bias in the estimates obtained from highly aggregated data. Therefore, the tests should be performed on disaggregated data where possible. However, in our case aggregation did not alter the conclusions supporting the production smoothing hypothesis.
It has been suggested that disaggregated data should be used in testing the production smoothing hypothesis since aggregation may introduce bias in
’ See, for example, others.
Blinder [ 131, Seitz [ 141 and Lai [ 151among
S.P.
22
Table 6 Analysis of variance
Residual
I
RSS, RSSz RS& RSS‘, RSS, RSS, RSS> RSS3 RSS, RSS, RSS, RSSl
versus versus versus versus I versus I versus 1 versus
M.A.
Ghali,‘Int.
J. Production
Ecormnics
35
i 19941
15 -22
and F-tests
Model
2 3 4 5 1 2 3 4
Dimelis,
2 3 4 5 3 4 5
The F-test was computed
= = = = = ~ ~ ~ ~
sum of squares 0.47655 0.445589 0.433779 0.426129 0.370851 RSSz = RSS, = RSS, = RSSS = RSS, = RSS, = RSSS =
x IO8 x IO” x lo8 x lo8 x lo8 0.031366 0.01181 0.07650 0.055278 0.043176 0.050826 0.106104
Degrees
of freedom
T-K
x IO” x IO” x lo8 x IO8 x 10’ x IO’ x IO8
= 2216 T-p-ktl=2198 T - 2p - k + 2 = 2180 T ~ 3p - k + 3 = 2162 T - kp = 2090 18 18 I8 72 36 54 126
F-test
FLH.219H = 8.60 FIX.2180 = 3.30 F 18.2,hz
= 2.16
F7 l,I”‘)fl = 4.33 F,,.,,,, = 6.03 F S4.2LhZ ~ - 4.77 F1Zh.21190 = 4.75
by
where RS& is the residual sum of squares in the restricted case, RSS,, is the residual sum of squares in the unrestricted case, L’, is the degrees of freedom of the numerator obtained by subtracting the degrees of freedom of the two cases and t12is the degrees of freedom of the denominator.
Acknowledgement We are grateful to Michael Love11 and the other participants of the 6th International Symposium on Inventories, Budapest 27-3 1 August 1990, for their suggestions. An earlier version of this paper was presented at the Annual Meeting of the Southern European Association for Economic Theory (ASSET), 21-23 November 1991, Athens. We also thank two anonymous referees for their valuable comments.
References Cl1 Blinder,
A.S., and Maccini, L.J., 1991. Taking stocks: A critical assessment of research on inventories. J. Econom. Perspectives, 5(l): 73-96. PI Holt, C.C., Modigliani, F., Muth, J.F. and Simon, H., 1960. Planning Production, Inventories, and Work Force. Prentice-Hall, Englewood Cliffs, NJ. 131 West, K.D., 1986. A Variance bounds test of the linear quadratic inventory model. J. Political Econom., 94: 374-401. c41 Kollintzas, T., 1988. A generalized variance bounds test. Staff Report, 113.Federal Reverse Bank of Minneapolis, Research Department, to appear in J. Econom. Dynamics and Control.
O.J., 1983. The production and inventory I51 Blanchard, behavior of the american automobile industry. J. Political Economy. 91: 365-400. [61 Dimelis, S., 1991. A Test of the linear rational expectations equilibrium inventory model in the EEC. Int. J. Prod. Econom., 26: 25-32. [71 Cumby, R.E., Huizinga. J. and Obstfeld, M., 1983. Twostep two-stage least squares estimation in models with rational expectations. J. Econometrics, 21: 333-355. PI Hansen, L.P.. 1982. Large sample properties of generalized method of moments estimators. Econometrica. 50: 1029-1054. aggregation. and the test191 Ghali, M.A., 1987. Seasonality. ing of the production smoothing hypothesis. Am. Econom. Rev., 71: 464-469. smoothing evidCl01 Krane, S. and S. Braun, 1991. Production ence from physical product data. J. Political Economy., 99: 558.-58 I. smoothing model is alive 1111 Fair, R., 1989. The production and well. J. Monetary Econom., 353-370. smoothing model Cl21 Blinder, A.S., 1986. Can the production of inventory behavior be saved’?. Quart. J. Econom., 101: 43 l-453. in invenCl31 Blinder, A.S., 1986. More on the speed ofadjustment tory models. J. Money, Credit and Banking, 18: 355- 365. in 1141 Seitz, H., 1993. Still more on the speed of adjustment inventory models: A lesson in aggregation. Empirical Econom. 1151 Lai, KS., 1991. Aggregation and testing of the production smoothing hypothesis. Int. Econom. Rev., 32: 391-403.
production economics ELSEVIER
Classical
Int. J. Production
and variance Sophia “Athens University ‘Western
Economics
35 (1994) 15-22
bounds tests of the production hypothesis P. Dimelisa’*,
Moheb
smoothing
A. Ghalib
qf Economics and Business, 76 Patission Str., 10434 Athens, Greece Washington Universit~~. Bellingham, WA 98225-9038, USA
Abstract In this paper we test the “controversial” production smoothing role of inventories by applying both classical econometric techniques and variance bounds tests. We use seasonally unadjusted data measured in physical units for five industries. The model used is a version of the familiar Holt et al. (1960) inventory model. We also use the data on one industry disaggregated at the district level to investigate potential bias which may arise from aggregation. Overall, the empirical evidence suggests production smoothing in the great majority of the industries examined.
1. Introduction
Research on inventories has not yet yielded definite conclusions with regard to the “controversial” production smoothing/buffer stock motive of holding inventories. The negative results obtained by empirical studies have raised several questions as to what might have caused the apparent rejection of this classical inventory mode1 [ 11. Recent investigations have focused on either the quality and appropriateness of the data used in testing empirically the production smoothing/buffer stock hypothesis, or on the validity of the inventory models employed for that purpose. Despite the growing concern about these issues, very few studies have addressed them adequately, partly due to the lack of appropriate data. In this paper we exploit the availability of a particular set of seasonally unadjusted monthly
* Corresponding
author
0925-5273/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0925-5273(93)E0098-G
data measured in physical units to test the production smoothing hypothesis of inventories. To test this hypothesis we follow two different approaches: starting from a version of the Holt et al. [2] inventory model, we first estimate a reduced form econometric equation, and second we perform variance bounds tests [3,4] using estimates of the Euler condition associated with the above model. In doing this, we can compare the results from the two different types of tests and draw conclusions regarding the reliability of the tests and the validity of the hypothesis. Finally, by using the disaggregated data on one industry, we also investigate the implications of any potential bias arising from aggregation.
2. The model For the purposes of this paper, we start a genera1 linear quadratic mode1 of inventories
with that
S.P. Dimelis,
16
M.A.
Ghali: Int. J. Production
embodies production smoothing, derive the cost minimizing relationship between production, inventories and sales, and then apply the different tests of the model. The cost structure we use in this is a simplified version of the Holt et al. [2] model. Introducing uncertainty in the model, the representative firm is minimizing its expected discounted future stream of real costs: min&
f. B’{c,Q: + c2(Qt- QtpI)’ r=o
subject
to
(2)
H _ 1 given,
(3)
where Q, and S, are production and sales in period t, respectively; H, is the firm’s stock of inventories at the end of period t; E,( .) denotes mathematical expectations conditional on information available at time 0; fi is a discount factor between zero and one; the Ci’S (i = 1, 2, 3, 4) are positive cost parameters; the term Q: represents production costs, which under the assumption of decreasing returns are convex. The term (Qr - QI _ I )’ reflects the costs of changing production, so that a penalty is imposed on fluctuations of the production level. Finally, the last term, (H,- c&+ 1)2, is a cost associated with the level of inventories; it implies increasing costs as inventories (H,)deviate from a desired level (which is assumed to relate linearly with expected next period sales (cqSI+ i)). Inventories in this model are held for two reasons: to smooth production in the face of fluctuating demand as implied by the first two types of costs and to avoid stocking-out as implied by the last type of cost. If this last cost is large relative to the two previous ones, then production smoothing will not occur. Sales here are taken to be exogenously determined. Differentiating (1) with respect to H, we obtain the first-order conditions for cost minimization of (1) subject to (2) and (3): 2(B2c2Ht+2
-
BCCI
+
2c2(1
+
B)lH,+r
+ [c,(l + /I) + cz(B* + 48 + 1) + c,]H,
35 IlYY4)
15-22
H,m1+ c2H1-2
- [cl + 2c2(1 + 8)-j +
P2”2&+2
-
CPCl
+
+ [c, + c2(1 + zp)]s,
Bc2V
+
PI
- C2Sr-1)
+
CAlS,+1
= 0.
(4)
This is the Euler equation associated with problems (l))(3). It is an expectational difference equation that can be solved to derive the optimal decision rule for inventory behavior. Following Blanchard [S] in normalizing by c2 = 1, we obtain
H, = (iI+ A,)H,_,
Q,= S,+ H,- H,mI,
E,
Economics
-
(i28Y”l
- 3.,j.2Hf_2
B(L)E,St+2+,,
(5)
where B(L) = - /PC2 + [!JCi + fiC2(2 + fi) + c3cJLmi - [Cl + c2(1 + 2fi)L-l
+ (.2Lm3
and ii, i2 are the smallest absolute value roots of the Euler eq. (4). According to (5), the optimal inventory stock depends on its values lagged once and twice, as well as on lagged, current and expected sales. Had we allowed for the existence of stochastic cost elements in (1) then inventory behavior would have also depended on the unobservable stochastic disturbances. Furthermore, the roots 1, and i2 can be expressed in terms of the structural cost parameters, ci, so that all coefficients in (5) are functions of these parameters. The first approach in testing this model is to estimate an unconstrained regression of a testable form of Eq. (5). This is equivalent to estimating a reduced form equation in standard econometrics and we will refer to that as “classical econometric testing”. This is a simple and useful approach, but it does not allow estimation of the structural parameters. However, the coefficient estimates obtained from such regressions provide important information as to the validity of the model. We also test the model using the variance bounds test developed originally by West [3] and generalized recently by Kollintzas [4]. The application of these tests requires estimation of the structural parameters. Before proceeding to the estimation
S.P. Dimelk,
M.A.
Ghalillnr.
J. Production
process, we describe in the following section the two tests as applied to the model presented in this paper.
3. The two tests and the data 3.1. Classical econometric
testing
In order to obtain an estimable form of (5) we should first derive its backward looking solution. For that purpose we assume an AR(4) process for S, and express B(L)S,+2 of (5) in a quasi-first-order form as in Blanchard [S], in which case we derive H, = m,H,_,
- m2H,_,
+ m3St
Economics
35 (1994)
15-22
17
these conditions, each factor by itself (production costs or adjustment costs) can be considered as necessary to result in production smoothing. Within a multifactorial framework each factor acts independently and therefore the absence of one of them is not evidence against production smoothing, while if both factors are present a synergistic effect can exist, producing more smoothing. Furthermore, with respect to sales, it can be inferred that, under the assumption of increasing marginal costs, firms would prefer to meet current sales out of current inventories. This implies a negative and less than one value for m3 in Eq. (7). As a result, a value exceeding unity of the coefficient on S, in (8) can be evidence against production smoothing.
3 +
1
m3iS,-i
+
c’,,
i=l
3.2. The generalized
where the parameters ml, m2, m3i, i= 1,2,3, are complicated functions of the structural parameters cl, c3, c4 and B. The term v, represents the residuals. Equivalently, we can write (6) in terms of production by using the identity Qt = H, H,_, + S,, so that
Q, = mQlml - (1 - ml + m2)H,-1 + (m3 + l)S, + 1
mziS,-i
+ f-4
(7)
variance bounds test
The model is also tested using the generalized variance bounds test (GVBT) developed by Kollintzas [4], according to which a weighted sum of autocovariances of the covariance stationary component of inventories should be nonnegative. Following the steps described in Kollintzas [4], it can be shown that for the model to be accepted, the GVBT requires that2,3 [(l + j3-‘)c,
+ (5 + pml)cZ + jj-lc3]
var(h)
i=l
Estimation of Eq. (7), or (6) yields consistent estimates of the reduced form coefficients. These coefficients can then be interpreted in association with the structural parameters of Eq. (1). As explained in the previous section, for the production smoothing hypothesis to hold, either c1 or c2 or both should be different from zero and large relative to cj. In terms of the reduced form coefficients, this implies that either (1 - ml + mz) or m2 or both be different from zero, respectively.’ Under
- 2(c, + 4c,)cov(hl, + 2c, cov(hlhm2)
h_ ,)
2 0,
(8)
where var(h) = E(h:), cov(hl,
cov(h,,
h- 1) = E(h,, h,- A,
h-2) = E(h,, h,-2)
and h, = H, - I?,, where l?, is the trend component.
’ Within a similar setup, Blanchard [S] provides explicitly the relationship between the structural parameters c,, c, and the reduced form estimates m, and rnz. However, these relations are subject to the particular normalization of c2 = I, while parameters, such as c4, not included in the lag polynomial B(L), cannot be expressed in terms of the m,‘s
’ Details for the derivation of this condition can be found in the appendix of Ref. [6]. 3 All variables are assumed to have zero unconditional means for expositional simplicity.
S. P. Dimeli.\, M. A. Ghuli: ItIt. J. Produc~rion Economic.s 3.5 i 1994) IS-22
IX
This inequality measures the cost savings the firm would expect by adjusting inventories optimally, i.e. in response to random sales fluctuations as suggested by the model. These cost savings were calculated by comparing the expected costs under this optimal policy and the necessarily higher costs that firms would face from a policy of letting inventories grow at a constant growth rate, which implies meeting random fluctuations by variations in production. A violation of inequality (8) would indicate that firms do not choose their inventories according to the optimal policy, which would mean a rejection of the model and the lack of production smoothing motive of inventories. The weights in (8) are functions of the structural parameters of the model which can be estimated from the Euler condition (4). By defining q, = /3Q, - QI_ 1 and under the arbitrary normalization c’~ + ~~(1 + p) = 1, Eq. (4) can be written as q,+, = c2(/@+2
+ 41) + (.3Ht - (.3~Jr+l
+ ~1. (9)
The parameters c2, c3 and c4 were estimated (up to a normalization factor) using a two-step two-stage procedure described by Cumby et al. [7] with a fixed fl = 0.995. The first step includes estimation of the Euler equation by instrumental variables. As instruments we used a constant, a linear trend and three lags each of inventories and sales. The disturbance term was allowed to follow an MA (2) process. In the second step, the parameter estimates were obtained from the optimal instrumental variable estimator: h = (X’z~“z’X)-‘x’z~-‘z’~~, where fi = (l/T)(Z’iZ) and i is the variancecovariance matrix of the first-step residuals, X is the matrix of the right-hand side variables of (9), Z is the matrix of instruments and y the left-hand side variable of (9). Variances and autocovariances were calculated from the estimates of a VAR system of inventories and sales and the Yule-Walker equations. The asymptotic standard error of the test statistic was derived from Hansen’s [S] asymptotic varianceecovariance matrix of the estimated parameters (see [3]).
3.3. The duta The data used are for six industries: asphalt, oil burners, glass containers, printing paper, beer and Portland cement. For each industry we use 120 monthly observations for the period 19.50-1960. In addition, for the Portland cement industry, disaggregated data for each production district were also available. This industry is divided into 19 production districts which are described in Ghali [9]. These data are not seasonally adjusted and are in physical units. For the variance bounds test the series were first deseasonalized using seasonal dummies.
4. The results 4.1. Unconstrained
regression estinmtes
In Table 1 we present the results of OLS estimation of Eq. (7) fitted to monthly data on the six industries4 The results indicate strong production smoothing in the industries of glass containers, and Portland cement, as judged by the high significance and magnitude of the lagged production coefficients, the significance of the lagged inventory coefficients and the fact that current sales coefficient estimates are significantly less than unity. Furthermore, following the analysis in Section 3.1, these results indicate that both adjustment and production costs are strongly significant in these industries. Considerable smoothing is also observed in the asphalt and oil burners industries. Although lagged production is not significant in these two industries, the coefficients of lagged inventories and current sales are significant, with the latter also being significantly less than unity. In the printing paper and beer industries the evidence is conflicting. Lagged inventories in these industries seem to play a statistically important role, while lagged production is insignificant. These
’ It should be mentioned that the presence of a lagged dependent variable in the OLS regressions may produce serial correlation in the errors, in which case OLS estimates are inconsistent. Using Durbin’s procedure 10 check for autocorrelation. it was found that only in the oil burners and the printing paper industries was it significant. Correcting for serial correlation did not change the conclusions drawn earlier about these two industries.
S.P. Dimelis.
Table 1 Unconstrained
regression
estimates
M.A.
Ghnliilnt.
of the production
J. Production
smoothing
Economics
35 (1994)
model for six industries
15-22
(r-values
19
in parentheses)
Industry - 0.06
Asphalt
( - 0.54) Oil burners Glass containers Printing
paper
Portland
- 0.12
- 0.10
( - 1.35)
( - 2.34)
0.40 (4.40)
( - 3.61)
0.20 (1.35)
( - 3.68)
- 0.04
Beer cement
- 0.28”
( - 4.39)
- 0.16 - 0.26” - 0.41”
( - 0.39)
( - 4.52)
0.36 (3.63)
( - 2.92)
- 0.12”
0.98
1.95
0.96
1.54
0.93
2.04
0.95
1.61
0.0 1 (0.33)
0.98
1.99
- 0.14”
- 0.13”
0.97
2.01
( - 2.39)
( - 2.71)
0.92 (26.76)
- 0.02
- 0.03
( - 0.13)
( - 0.93)
( - 4.02)
0.9 1a (15.59)
( - 1.02)
0.12 (1.74)
( - 0.95)
0.32” (7.06)
- 0.08
- 0.03
- 0.07
( - 1.32)
( - 0.60)
( - 1.56)
1.01” (25.05)
( - 1.27)
0.02 (0.36)
( - 0.31)
1.05” (38.11)
0.05 (0.50)
0.03 (1.31)
0.71” (13.59)
0.07 (1.24)
0.12
- 0.20
For each estimation, 117 observations were used as required by the lagged variables A set of monthly dummies and a time trend were included in each regression. Estimates obtained by OLS procedure. a The coefficient is statistically different from zero at the 5% level.
facts imply that only production costs matter whereas adjustment costs are not important in the production smoothing process. However, the sales coefficients both exceed unity and statistically so in the beer industry only. It could be argued therefore that “weak” production smoothing is evident in the printing paper industry, while no clear conclusion can be drawn in the case of beer without knowledge of the relative magnitude of all the cost parameters. In Table 2 we present the results of OLS estimation of Eq. (7) for the 19 production districts of the Portland cement industry. Lagged production is highly significant in all but five districts, lagged inventories are highly significant in all but three cases, while the current sales coefficient estimates are significant and smaller than unity in all cases. Lagged production and inventories are both insignificant only in district 10. These results give strong support to the production smoothing hypothesis and are in agreement with those derived from the aggregate estimates of this particular industry. It is interesting to note here that these results confirm the evidence implied by the relative variability of unadjusted production to sales reported by Ghali [9] for exactly the same industries. Comparing the results from the first two columns of
- 0.14” - 0.05
- 0.01
of the model.
Table 1 in Ghali ([9], p. 466), the following can be observed. In industries in which the variance of unadjusted production exceeds the variance of unadjusted sales and the correlation between sales and inventory change is positive or negative but weak (district 10 of Portland cement, printing paper and beer), the corresponding estimates from Tables 1 and 2 indicate “weak” or “lack of” production smoothing. It should also be noted that on the basis of the seasonally adjusted and detrended data of Table 1 in Ghali [9], only three out of the nine possibly “negative” cases of that table are confirmed by our regression estimates.
4.2.
Variance bounds test estimates
Tables 3 and 4 report the results of the GVBT given by Eq. (8). The evidence from Table 3 supports the production smoothing motive in all but one of the industries. The results are statistically significant5 in three industries: asphalt, glass containers and cement. In the case of printing paper the
’ By statistically significant we mean a test value which is more than two asymptotic standard errors.
20
S.P. Dinwlis.
Table 2 Unconstrained
regression
Portland cement production district
estimates
M.A.
Giud:‘Int.
of the production
J. Produc~rron Econotnics
smoothing
model for the Portland
Qr-1
H,-,
S,
~ ~ ~ -
II 12 I3 I4 I5 16 I7 I8 I9
0.44 0.15 0.41” 0.52” 0.44” 0.20 0.24” 0. I7 0.35” 0.05 0.20 0.33” 0.35” 0.20” 0.22” 0.13 0.42” 0.20” 0.13
0.18” 0.15” 0.22* 0.16” 0.19” 0.08” 0.24 0.29 0.31” 0.02 0.06 0.10” 0.13” 0.06 0.16 0.13” 0.24” 0.27” 0.29”
0.11” 0.77” 0.53” 0.49 0.57” 0.40” 0.85” 0.69” 0.57” 0.66 0.41” 0.39” 0.47 0.56” 0.50” 0.52” 0.56 0.55” 0.69”
- 0.1 I - 0.01 0.23” - 0.05 - 0.03 0.02 - 0.07 - 0.03 - 0.01 0.37” - 0.02 0.08 - 0.07 0.24” 0.23” 0.1 I - 0.09 - 0.04 0.06
Aggregate
0.36
- 0. I 2”
0.71”
0.07”
I 2 3 4 5 6 7 8 9
I0
See notes to Table
Table 3 Variance
bounds
Industry
Asphalt Oil burners Glass containers Portland cement Beer Printing paper ” Test
~ ~ ~ ~ ~ ~ -
35 (1994)
15-22
cement
production
5 ~2
S 1
~ -
~ -
~ ~
distracts
R2
DW
- 0.04 0.00 - 0.13 0.00 - 0.16” 0.01 - 0.05 - 0.02 - 0.07 ~ 0.16 0.0 I - 0.06 0.01 - 0.08 ~ 0.15” 0.03 0.02 0.04 - 0.02
0.87 0.93 0.93 0.86 0.96 0.82 0.93 0.84 0.86 0.90 0.9 I 0.92 0.83 0.94 0.93 0.96 0.92 0.93 0.91
2.15 2.03 2.14 2.04 2.16 I .94 2.02 1.99 2.12 I .9x 2.03 I .98 2.04 2.08 I .90 I .95
~ 0.13”
0.97
2.01
& 3
0.14” 0.04 0.21” 0.11 0.12 0.07 0.06 0.04 0.10 0.05 0.06 0.05 0.04 0.01 0.07 0.03 0.10 0.01 0.00
~ 0.14
1.84 2.04 I .98
I
tests on industry
Table 4 Euler equation
data”
Generalized variance bounds test (GVBT)b I23 67X’ - 966 533 931958‘ 2434791’ 39 577 d
normalization estimated under the (‘r + (I + pjcz = 1. ’ Kollintras [4] test. ’ Stgniticant at the 5% level. ’ A large positive and highly insignificant value.
(r-values)
estimates
of the cost parameters
Industry
(‘I
Asphalt
1.335 (0.1 I)
- 0.168 (0.053)
- 0.170 (0.10)
1.655 (2.77)
(2.26) (1.10) (3.94) (2.25) (1.34) (0.00)
Oil burners
1.634 (0.49)
- 0.318 (0.24)
- 0.178 (0.22)
- 5.95 (4.90)
Glass containers
0.578 (0.301)
0.21 I (0.15)
- 0.016 (0.02)
4.420 (25.34)
0.568 (0.156)
0.043 (0.08)
~ 3.015 (3.98)
restriction:
Beer
0.005 (0.5 17)
~ 0.361 (39.60)
value of the test was positive but extremely large, and so was its variance, rendering it statistically insignificant. These results can be compared with those obtained by Krane and Braun [lo], who also used
Portland
cement
Asymptotic standard Estimates obtained C’, + (I + B)C, = I.
- 0.133 (0.31) I.263 (0.12)
(‘2
~ 0.132 (0.06)
(‘3
errors in parentheses. under the normalization
(‘4
restriction:
disaggregated, monthly physical product data but from different sources and periods (mostly for the period 1977-1988). They found that in the asphalt
S.P. Dimelis,
Table 5 Tests of homogeneity tricts
across
the Portland
M.A.
GhalilInt.
production
J. Production
dis-
Model
Hypothesis
1 2 3
Common regressions in all 19 districts Different constants, common all other coefficients Different constants, different lagged production coefficients, common all other coefficients Different constants. different lagged production and inventory coefficients, common all other Different regressions in all 19 districts
4 5
testing
and glass containers industries the smoothing hypothesis is supported, while for the printing paper and beer it is rejected. With respect to the cement industry, the same evidence is provided by Fair [l I], who also used physical data from the Bureau of Mines for the period 1947-1964. No comparison with other studies, such as, for example, West’s [3] or Blinder’s [12] results can be made due to the high degree of aggregation in their data. Table 4 contains the cost parameter estimates we derived from the Euler equation (9). The marginal production cost parameter c1 is positive in four out of the five disaggregated industries we considered. However, the parameter c2 that measures marginal costs of changing production is mostly negative and so are most of the other cost parameters. It should be noted that the normalization used [ci + (1 + /?)cZ = l] constrains either ci or c2 to be positive in each equation. Therefore, no clear statement about the positive cost parameters can be made. Krane and Braun [lo] also estimated positive cost parameters in about three quarters of the disaggregated industries they examined.
Economics
35 (1994)
IS-22
21
the parameter estimates.6 Using the data on the Portland cement industry which were available disaggregated at the district level, we also tested for possible aggregation bias. This was achieved by pooling the sample of 117 x 19 = 2223 observations and defining a set of dummy variables for each district and each regressor of Eq. (7) separately. Then we performed various tests of homogeneity for the constants and the coefficients as a whole or sets of them. The hypotheses tested are described in Table 5. Each hypothesis was tested using the Ftest and the results are presented in a form of variance analysis in Table 6. The calculated values of the F-test indicate that no homogeneity across districts either of the intercepts or slope coefficients is existent in the data. As a result, one would expect that estimates obtained by using the aggregated data suffer from bias. It is not, however, possible to draw any conclusions as to the direction and magnitude of this bias.
5. Conclusions
4.3. Aggregation
In this paper the production smoothing property of inventories was tested using two different tests. It was found that both tests provide positive evidence for production smoothing in the asphalt, glass containers and Portland cement industries (Tables 1 and 3). Among the remaining three industries, oil burners, beer and printing paper, the “classical test” is less supportive of the production smoothing hypothesis only in the case of beer, while the “variance bounds test”, although insignificant in all three of them rejects it only in the case of oil burners. Finally, the lack of homogeneity across districts yields the conclusion that aggregation introduces bias in the estimates obtained from highly aggregated data. Therefore, the tests should be performed on disaggregated data where possible. However, in our case aggregation did not alter the conclusions supporting the production smoothing hypothesis.
It has been suggested that disaggregated data should be used in testing the production smoothing hypothesis since aggregation may introduce bias in
’ See, for example, others.
Blinder [ 131, Seitz [ 141 and Lai [ 151among
S.P.
22
Table 6 Analysis of variance
Residual
I
RSS, RSSz RS& RSS‘, RSS, RSS, RSS> RSS3 RSS, RSS, RSS, RSSl
versus versus versus versus I versus I versus 1 versus
M.A.
Ghali,‘Int.
J. Production
Ecormnics
35
i 19941
15 -22
and F-tests
Model
2 3 4 5 1 2 3 4
Dimelis,
2 3 4 5 3 4 5
The F-test was computed
= = = = = ~ ~ ~ ~
sum of squares 0.47655 0.445589 0.433779 0.426129 0.370851 RSSz = RSS, = RSS, = RSSS = RSS, = RSS, = RSSS =
x IO8 x IO” x lo8 x lo8 x lo8 0.031366 0.01181 0.07650 0.055278 0.043176 0.050826 0.106104
Degrees
of freedom
T-K
x IO” x IO” x lo8 x IO8 x 10’ x IO’ x IO8
= 2216 T-p-ktl=2198 T - 2p - k + 2 = 2180 T ~ 3p - k + 3 = 2162 T - kp = 2090 18 18 I8 72 36 54 126
F-test
FLH.219H = 8.60 FIX.2180 = 3.30 F 18.2,hz
= 2.16
F7 l,I”‘)fl = 4.33 F,,.,,,, = 6.03 F S4.2LhZ ~ - 4.77 F1Zh.21190 = 4.75
by
where RS& is the residual sum of squares in the restricted case, RSS,, is the residual sum of squares in the unrestricted case, L’, is the degrees of freedom of the numerator obtained by subtracting the degrees of freedom of the two cases and t12is the degrees of freedom of the denominator.
Acknowledgement We are grateful to Michael Love11 and the other participants of the 6th International Symposium on Inventories, Budapest 27-3 1 August 1990, for their suggestions. An earlier version of this paper was presented at the Annual Meeting of the Southern European Association for Economic Theory (ASSET), 21-23 November 1991, Athens. We also thank two anonymous referees for their valuable comments.
References Cl1 Blinder,
A.S., and Maccini, L.J., 1991. Taking stocks: A critical assessment of research on inventories. J. Econom. Perspectives, 5(l): 73-96. PI Holt, C.C., Modigliani, F., Muth, J.F. and Simon, H., 1960. Planning Production, Inventories, and Work Force. Prentice-Hall, Englewood Cliffs, NJ. 131 West, K.D., 1986. A Variance bounds test of the linear quadratic inventory model. J. Political Econom., 94: 374-401. c41 Kollintzas, T., 1988. A generalized variance bounds test. Staff Report, 113.Federal Reverse Bank of Minneapolis, Research Department, to appear in J. Econom. Dynamics and Control.
O.J., 1983. The production and inventory I51 Blanchard, behavior of the american automobile industry. J. Political Economy. 91: 365-400. [61 Dimelis, S., 1991. A Test of the linear rational expectations equilibrium inventory model in the EEC. Int. J. Prod. Econom., 26: 25-32. [71 Cumby, R.E., Huizinga. J. and Obstfeld, M., 1983. Twostep two-stage least squares estimation in models with rational expectations. J. Econometrics, 21: 333-355. PI Hansen, L.P.. 1982. Large sample properties of generalized method of moments estimators. Econometrica. 50: 1029-1054. aggregation. and the test191 Ghali, M.A., 1987. Seasonality. ing of the production smoothing hypothesis. Am. Econom. Rev., 71: 464-469. smoothing evidCl01 Krane, S. and S. Braun, 1991. Production ence from physical product data. J. Political Economy., 99: 558.-58 I. smoothing model is alive 1111 Fair, R., 1989. The production and well. J. Monetary Econom., 353-370. smoothing model Cl21 Blinder, A.S., 1986. Can the production of inventory behavior be saved’?. Quart. J. Econom., 101: 43 l-453. in invenCl31 Blinder, A.S., 1986. More on the speed ofadjustment tory models. J. Money, Credit and Banking, 18: 355- 365. in 1141 Seitz, H., 1993. Still more on the speed of adjustment inventory models: A lesson in aggregation. Empirical Econom. 1151 Lai, KS., 1991. Aggregation and testing of the production smoothing hypothesis. Int. Econom. Rev., 32: 391-403.