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On the adjustment matrix in error correction models
Journal of Monetary Economics 42 (1998) 427—444
On the adjustment matrix in error correction models Robert J. Rossana* Department of Economics, Wayne State University, Detroit, MI 48202, USA Received 24 March 1997; received in revised form 26 August 1997; accepted 1 October 1997
Abstract This paper explores the determinants of the adjustment matrix in error correction models within two intertemporal models of the firm. In a production smoothing model of inventories, it is shown that the adjustment matrix contains the speed of adjustment of inventories as conjectured in previous work but this parameter matrix also contains the parameters from the autoregressive polynomials associated with the stochastic, unobservable shocks in the model. Two empirical examples are provided suggesting that these unobservable shocks cannot be assumed to be serially uncorrelated. In a flexible wage model of a labor market, the impact upon the adjustment matrix of normalizing the cointegrating matrix is studied. In contrast to one-state variable problems, normalization of the cointegrating matrix is arbitrary on economic and statistical grounds in this model and it is shown that, even with serially uncorrelated shocks, estimating the elements of the adjustment matrix may not provide an estimate of the speed of adjustment under some normalizations of the cointegrating matrix. The implication of the analysis is that without guidance from an economic model about the interpretation of elements of the adjustment matrix under alternative normalizations of the cointegrating matrix, or when unobservable shocks are serially correlated, economic interpretation of estimates of the adjustment matrix will be hazardous. ( 1998 Elsevier Science B.V. All rights reserved. JEL classification: C17; E00; E32 Keywords: Cointegration; Vector autoregression
* Corresponding author. E-mail: [email protected] 0304-3932/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 9 3 2 ( 9 8 ) 0 0 0 2 6 - 9
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1. Introduction Vector autoregressive time series systems comprised of cointegrated time series are known to obey the error correction representation k~1 *X " + P *X !PX #m , t i t~i t~k t i/1 where X is a vector of time series (and may also include the unit vector), m refers to a vector of iid disturbances, and the coefficient matrix P has reduced rank so that P"ab@. The a matrix is called the adjustment matrix and b is known as the cointegrating matrix where the latter contains the parameters from the long-run equilibrium maintained by the time series in the system. Johansen and Juselius (1990) and Johansen (1991), among others, have stated that the adjustment matrix measures the speed at which time series approach their equilibrium levels, suggesting that this matrix contains the adjustment speeds which arise in intertemporal models of the firm or household. In this paper, this claim is examined and, for two reasons, it is shown that it may be inappropriate to regard the adjustment matrix as containing these adjustment speeds. One reason is that unobservable shocks in the economic environment may be serially correlated stochastic processes. Ignoring seasonality, vector autoregressions estimated with macroeconomic data typically reveal lag lengths considerably longer than one. Economic theory generally cannot explain lag lengths beyond one; for example, costs of adjustment are the main theoretical explanation for serial persistence in economic time series such as inventories and capital goods and this theory implies that there will be only one lag of a quasi-fixed time series in vector autoregressions.1 One way to justify long lags in estimated VARs is to assume that unobservable stochastic shocks are serially correlated. In a production smoothing model of the firm, the traditional model used to study inventories (see, e.g., Eichenbaum, 1989; Lai, 1991; Durlauf and Maccini, 1995), it will be shown that the speed of adjustment is contained by the adjustment vector but, if the unobservable shocks in the model obey univariate autoregressive stochastic processes, the adjustment vector will also contain the parameters in the autoregressive polynomial appearing in the stochastic processes obeyed by these shocks. The cointegrating vector is shown to be free of these nuisance
1 It could also be true that costs of adjustment affect the firm’s cash flow for more than one period which is an additional manner in which lags beyond one for a state variable might arise in these VARs. This would be a plausible assumption with high frequency data but this is an assumption which has never, to my knowledge, been a prominent assumption in the adjustment cost or inventory literature. The time to build (Kydland and Prescott, 1982) and delivery lags (Maccini, 1973; Nickell, 1977) are additional mechanisms which could be used in some settings to motivate long lags in vector autoregressions.
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parameters, so elements of this vector may be estimated without concern for the degree of serial persistence manifested by the data. A second reason why the adjustment matrix may not only contain adjustment speeds is that alternative normalizations of the cointegrating matrix affect the elements of the adjustment matrix. Normalizing transformations of the cointegrating matrix are statistically arbitrary and may also be arbitrary on economic grounds. There is no reason to believe that alternative normalizations of the cointegrating matrix are irrelevant to the interpretation of estimates of the adjustment matrix, just as one would expect the interpretation of the parameters in ordinary regression to depend upon the choice of dependent variable. The impact upon the adjustment matrix of alternative normalizations of the cointegrating matrix will be studied in a variant of a labor market model with flexible wages adapted from Sargent (1987), a model similar to those developed in Lucas and Rapping (1969) and Kennan (1988). Only one normalization of the cointegrating vector makes economic sense in one-state variable optimization problems. The cointegrating vector in such models represents the long-run equilibrium of the state variable. But the labor market model in this paper will produce a trivariate VAR in the flow of labor services, real wages, and a unit root exogenous factor input price, say for raw materials, which permits the derivation of the adjustment matrix under alternative normalizations of the cointegrating matrix. It will be seen that the adjustment matrix contains not just the speed of adjustment of the state variable; the adjustment matrix contains parameters arising from the optimization problems solved on both sides of the labor market. Further, it will be observed that individual elements of the adjustment matrix are considerably different under two alternative normalizations which are examined (adjustment parameters are defined in dynamic optimization problems independently of any normalizations taken in an economic model); normalization of the cointegrating matrix is arbitrary on statistical and economic grounds in this model. The implication is that the interpretation of estimates of the adjustment matrix will be of little use, unless there is an explicit analysis of an economic model providing guidance about the relationship between normalization of the cointegrating matrix and its effects upon the adjustment matrix.2
2 In Rossana (1995) it is shown that the adjustment matrix will contain the speed of adjustment in a one-state variable model of the firm. However, that paper ignores the effects upon the adjustment matrix of serially correlated shocks and normalization of the cointegrating matrix. This earlier paper is concerned with providing one possible rationale for cointegrating matrices having rank greater than one, showing that the rank of the cointegrating matrix will be identical to the number of quasifixed state variables contained in an optimizing model of a firm.
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2. The effects of serially correlated unobservable shocks This section provides an analysis of a production smoothing model of inventories. It will be shown that lags greater than one will arise in the VAR arising from the model when there are serially correlated unobservable cost shocks, and the autoregressive parameters appearing in the stochastic process obeyed by these shocks will be contained in the adjustment vector arising from the model. These autoregressive parameters do not appear in the cointegrating vector. Empirical evidence is provided, in estimated bivariate VARs for selected nondurable goods industries, that these serially correlated shocks seem to be present since there are lags beyond one which are statistically significant in all of these industries. In this section, and throughout this paper, models are simplified to deliver the results of interest with minimal economic detail. So, for example, constants are omitted wherever possible. 2.1. A model of a firm holding finished goods The firm produces a storable output with the inventory cost function c "(a /2)(i !a s )2#(a /2)i2, a '0, it 0 t 1t 2 t i
(1)
where i denotes the stock of finished goods, and s is the level of shipments or sales. The first term in Eq. (1) implies that the firm has a target or desired level of inventories determined by its planned level of sales and the firm incurs costs if the level of inventories deviates from its planned level. This captures the stockout costs arising if the firm is unable to service an incoming new order. Inventory holding costs are assumed to rise at the margin. The flow of sales to the firm obeys a univariate stochastic process specified below. The firm has a cost function for the production of output which is given by c "t q #(b/2)q2, b'0, t qt ct t
(2)
where t is a stochastic shock to the firm’s cost function, and q denotes the flow c of output produced. This shock may be thought of as a factor price (Eichenbaum, 1989) or it may alternatively be a transitory shock to the firm’s technology. Either interpretation is suitable for present purposes. The shock to the cost function also obeys a univariate time series process. The firm’s total costs are given by c "c #c . t qt it The univariate time series processes obeyed by sales and the shock to the firm’s costs are (1!h ¸)(1!¸)s "ms, Dh D(1, t s s t
(1!h ¸)t "mt, c ct t
Dh D(1, c
(3)
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where ¸ is the lag operator. The random shocks, ms and mt, are iid with mean t t zero and are mutually uncorrelated. The sales of the firm are assumed to be ARI(1,1), a nonstationary, unit root stochastic process whereas the cost shock is AR(1) and thus stationary. Normalizing output prices to one, the firm wishes to solve the intertemporal problem = max E + /t[s !c ], 0 t t t/0 where E is the expectation operator conditional on an information set available to the firm and / is the discount rate (0(/(1). Using the accounting constraint q "s #*i , the optimization problem above will be solved by t t t choosing the path of finished goods to obey the Euler equation
C
A CA
E i ! t t`1 "E
t
B
D
a #a #(1#/)b 0 2 i #/~1i t t~1 /b
B
A
BD
b!a a t !/t 0 1 s !s # ct ct`1 t t`1 /b /b
.
(4)
Familiar methods may be used to derive the optimal investment equation for inventories given by
C A A B
a a i "k i # 1!k 1! 0 1 t 1 t~1 1 b
BD
= + (k /)iE s !s 1 t t`i t t/0
1!k = 1 + (k /)iE t !b~1t (5) 1 t ct`i ct b t/0 which displays the well-known property that all future values of the exogenous variables (sales and the technology shock) determine current inventory investment. The inventory investment Eq. (5) displays partial adjustment since 0(k (1. 1 The Weiner—Kolmogorov prediction formula (Sargent, 1987, p. 304) can be used if the firm uses linear least squares projections in forming its expectations. For sales, this formula is #
= h(k /)!k /¸~1h(¸) 1 + (k /)iE s " 1 s (6) 1 t t`i h(k /)(1!k /¸~1) t 1 1 i/0 for an arbitrary polynomial in the lag operator h(L). Evaluating Eq. (6) using Eq. (3) yields = 1!h k /¸ s 1 + (k /)iE s " s. 1 t t`i (1!k /)(1!h k /) t 1 s 1 i/0
(7)
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For the technology shock, the same derivation gives = 1 + (k /)iE t " t . (8) 1 t ct`i (1!h k /) ct c 1 i/0 Substituting the results in Eqs. (7) and (8) into Eq. (5), the VAR for the production smoothing model may be derived. This bivariate VAR describing the time path of inventories and shipments is
CD C
DC D D C D C
i h #k X (1#h !h )!X i t " c 1 1 s c 2 t~1 s 0 1#h s t s t~1 !k h X h !X h i X ms#X mt 3 t 1 c 2 c 1 s # t~2 # 1 t # ms 0 !h s t s t~2 a a 1 c" 1!k 1! 0 1 , K" , 1 b (1!h /k )(1!/k ) c 1 1 1!h / k ch /k c . X "cK!1, X " s 1, X " 1 1 2 3 K b 1!h /k c 1 The error correction form of Eq. (9) is
C
.
C
A
BD
C
C D C C
D
DC D DC D C
D (9)
*i h #k !1 X (1#h !h )X *i t " c 1 1 s c 2 t~1 *s 0 h *s t s t~1 (1!h )(1!k ) !(1!h )(X !X ) i X ms#X mt 3 t c 1 c 1 2 t~2 # 1 t ! ms 0 0 s t t~2
D
(10) Since there are costs of adjusting inventories, lagged inventories appear in the inventory demand function as shown in Eq. (9). Notice that this VAR contains the lag two level of inventories due to the serial persistence in the technology shock. If the series in Eq. (10) are cointegrated, the matrix of coefficients preceding the lagged levels of the time series must be of reduced rank. Inspection of Eq. (10) reveals that this is true (since k O1, it has rank one). This coefficient 1 matrix may be decomposed as P"ab@"(1!h ) c a"(1!h ) c
C
C
1!k 1 0
D
1!k 1 , 0
DC
D
1!/k !c 1 , (1!/k )(1!k ) 1 1 1 b" 1!/k !c . 1 (1!/k )(1!k ) 1 1 1
C
D
(11)
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A number of observations can be made about the adjustment and cointegrating vectors. The reduced rank coefficient matrix can be written as the product of a scalar and two vectors. This may at first appear to produce an indeterminacy in interpreting parameter estimates of coefficients in the cointegrating and adjustment vectors for this autoregressive parameter could be taken to appear in either the adjustment or cointegrating vector. But the cointegrating vector contains the parameters of the long-run equilibrium of the firm and this equilibrium is independent of the transient effects of serially correlated shocks. This is easily established by inspection of Eq. (9) and is entirely plausible.3 Thus the parameters of the cointegrating vector are identical to the case where shocks in the model are not serially correlated. The adjustment vector must be regarded as containing this autoregressive parameter. The adjustment vector contains the speed of adjustment of inventories as suggested above (this is discussed further below). In addition, generalization to the case where the technology shock obeys an AR(p) stochastic process is immediate. If the lag operator polynomial appearing in the stochastic process obeyed by the cost shock is given by h (¸)"1!h ¸!h ¸2!2!h ¸p, the reduced rank coefficient matrix in c 1c 2c pc the error correction model of the firm will now be
C
P"h (1) c
DC
1!k 1 0
1
D
1!/k !c 1 . (1!/k )(1!k ) 1 1
This extension merely adds additional autoregressive parameters to the adjustment matrix. Second, note that the analysis quite naturally leads to a unit coefficient attached to the stock of inventories so that the issue of how to normalize the cointegrating vector is irrelevant. Since the cointegrating vector contains the parameters of the long-run equilibrium of the firm, it makes little economic sense to attach a unit coefficient to anything other than the state variable held by the firm. This normalization would be appropriate in any one state variable problem. Third, note that the autoregressive parameter associated with the sales time series process does not affect the lag length of inventories in the VAR. It is only the autoregressive parameter associated with the technology shock appearing in the adjustment vector. Sales are cointegrated with the level of inventories so the
3 Set *i"0 and observe that the bivariate system reduces to the long-run equilibrium of the firm, an equilibrium which does not depend upon any of the autoregressive parameters in the stochastic process for the cost shock.
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autoregressive parameter contained in the stochastic process for sales is irrelevant to the adjustment vector. Only those parameters associated with the unobservable stochastic processes outside of the cointegrating relationship appear in the adjustment vector. Since there are no serial correlation parameters appearing in the cointegrating vector, estimation of this vector reveals information about the underlying structural model of the firm; it is not contaminated by nuisance parameters associated with any stochastic process. This analysis shows that, contrary to statements in Johansen and Juselius (1990) and Johansen (1991), the adjustment vector contains the speed of adjustment (1!k ) only for the case where the technology or cost shock is not serially 1 correlated. Generally, if it is necessary to match higher order serial correlation in the state variables contained in economic models, often evident in macroeconomic data, it will be necessary to assume serially correlated shocks to permit a VAR to match the data. The implication of this serial persistence is that the adjustment matrix will contain nuisance parameters associated with the stochastic shocks in the model, making it impossible to recover an estimate of the speed of adjustment from the estimated parameters of the adjustment matrix.4 2.2. Empirical results This section presents estimates of bivariate VAR systems for selected twodigit nondurable goods industries to see if serial correlation beyond lag one in the estimated VAR is evident in the data. If this is so, then we have compelling evidence that the adjustment matrix will not provide estimates of structural parameters which might be of interest in these industries since there will be serial correlation parameters appearing in the adjustment matrix. Two sets of VARs will be estimated. Since the production smoothing model of inventories was discussed above, a system with finished goods and sales will be examined. Since the statement was made earlier that the implications of the inventory model would arise in any intertemporal optimization framework, a second set of VARs will be estimated containing a measure of labor services in production and the real wage, series related to the model of the next section. The data series are all available from public sources. Data on inventories are provided by the Department of Commerce in seasonally adjusted constant
4 The results in this section are quite analogous to the ordinary regression problem with a lagged dependent variable and a disturbance which is serially correlated. Once the original equation is quasi-differenced with the serial correlation parameter from the disturbance process, the resulting equation will have coefficients on the lagged dependent variables which contain both structural and serial correlation parameters. This is essentially what happens to the adjustment matrix in an error correction model when shocks are serially correlated.
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Table 1 Bivariate error correction models in selected two-digit industries — inventories of finished goods and sales SIC 20
SIC 21
SIC 23
SIC 28
SIC 29
SIC 30
Dependent variable: *i t *i 0.076 t~1 (0.047) *s 0.110 t~1 (0.043)* i 0.033 t~2 (0.012)* s !0.026 t~2 (0.011)*
!0.25 (0.050)* !0.019 (0.038) 0.159 (0.029)* !0.008 (0.028)
0.065 (0.047) !0.039 (0.040) 0.044 (0.012)* !0.077 (0.023)*
0.107 (0.047)* !0.003 (0.008) 0.017 (0.006)* !0.014 (0.007)
0.110 (0.048)* 0.092 (0.052) 0.072 (0.016)* !0.040 (0.012)*
0.021 (0.048) !0.008 (0.024) 0.036 (0.010)* !0.020 (0.006)*
Dependent variable: *s t *i 0.171 t~1 (0.052)* *s !0.257 t~1 (0.046)* i !0.022 t~2 (0.013) s 0.022 t~2 (0.047)
0.239 (0.060)* !0.521 (0.045)* !0.013 (0.035) 0.160 (0.033)*
0.131 (0.056)* !0.250 (0.048)* !0.065 (0.014)* 0.138 (0.027)*
!0.109 (0.258) !0.630 (0.045)* !0.228 (0.035)* 0.281 (0.041)*
0.061 (0.044) !0.154 (0.048)* !0.042 (0.015)* 0.038 (0.011)*
0.143 (0.094) !0.137 (0.047)* !0.024 (0.020) 0.017 (0.012)
Note: In the table, i refers to constant-dollar inventories of finished goods, and s is constant-dollar sales. Standard errors are given in parentheses beneath each estimated coefficient. A constant was included in all regressions but the constant is not reported in the table. All data are in natural logarithms and the lag length is k"2. An asterisk (*) indicates significance at the 5% significance level.
dollars. The same organization provides constant dollar shipments, also seasonally adjusted. These series are used along with data on current dollar sales provided by the Census Bureau to produce a price index used in the construction of the real wage rate. Labor will be measured by manhours, the product of average weekly hours for production workers and the number of production workers, both adjusted for seasonality. The wage rate is measured by average hourly earnings. Both series are available from the Bureau of Labor Statistics and are published in Employment and Earnings. Data are measured at monthly frequency over the period 1959:1 to 1995:12, and are in natural logarithms. Unit root tests are available elsewhere which indicate that a unit root null hypothesis cannot be rejected for the data contained in the estimated VARs.5 Tables 1 and 2 contain the estimates of bivariate error correction VARs for six two digit nondurable goods industries. These industries, regularly used in
5 See Rossana (1993) and Rossana and Seater (1995) for Dickey—Fuller tests on the times series used here.
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Table 2 Bivariate error correction models for selected two-digit industries — labor services and real wages SIC 20
SIC 21
SIC 23
SIC 28
SIC 29
SIC 30
Dependent variable: *n t *n !0.312 t~1 (0.046)* *w 0.030 t~1 (0.030) n 0.020 t~2 (0.011) w !0.004 t~2 (0.004)
!0.282 (0.046)* 0.043 (0.032) 0.004 (0.006) !0.005 (0.010)
!0.156 (0.047)* !0.114 (0.078) 0.003 (0.007) 0.009 (0.006)
!0.003 (0.048) 0.002 (0.004) 0.020 (0.007)* !0.004 (0.003)
!0.097 (0.048)* !0.048 (0.056) 0.091 (0.022)* 0.012 (0.007)
0.037 (0.050) 0.029 (0.088) 0.033 (0.009)* !0.046 (0.015)*
Dependent variable: *w t *n 0.040 t~1 (0.071) *w 0.152 t~1 (0.047)* n 0.003 t~2 (0.018) w 0.010 t~2 (0.006)
!0.123 (0.066) !0.365 (0.045)* !0.006 (0.009) 0.035 (0.014)*
!0.039 (0.029) !0.127 (0.048)* !0.001 (0.005) 0.005 (0.004)
0.495 (0.512) !0.563 (0.044)* !0.212 (0.074)* 0.155 (0.032)*
!0.040 (0.038) 0.356 (0.045)* 0.046 (0.017)* 0.016 (0.006)*
0.028 (0.029) !0.097 (0.050) 0.010 (0.005) !0.011 (0.009)
Note: In the table, n refers to man-hours of labor, and w refers to real wages. Standard errors are given in parentheses beneath each estimated coefficient. A constant was included in all regressions but the constant is not reported in the table. All data are in natural logarithms and the lag length is k"2. An asterisk (*) indicates significance at the 5% significance level.
applied inventory studies, are classified as pure production to stock industries where no unfilled order data exist. The same industries are used in the labor and real wage rate error correction VARs. Constants are included in all regressions but they are not reported in the table for the sake of brevity. A lag length of two was used in all estimated VARs since this is the lag length arising in the inventory model examined earlier in this section. The results show that there are a large number of estimated equations where lagged differences and levels achieve statistical significance at conventional significance levels. Indeed it is quite likely that the actual lag lengths which might be chosen to test for cointegration in these data sets would be much longer than two as a brief examination of regressions with longer lag lengths suggests that lags longer than two would be appropriate in nearly every industry. Thus it seems reasonable to conclude that a lag length of at least two is a correct choice for all of these industries since, in no case, is there an industry where lagged differences are always statistically insignificant. In the context of the production smoothing model of this section, serially correlated shocks appear to be present in every industry.
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An interesting exercise to conduct is to estimate the adjustment matrix to see how badly mislead one might be by interpreting estimated parameters as estimates of an adjustment speed, rather than a nonlinear function of the adjustment speed and serial correlation parameters. To carry out this exercise, the maximum likelihood estimator of Johansen (1991) was used to estimate the adjustment vector in SIC 20 for inventories and sales. With a lag length of two, the column one, row one element of the adjustment vector is estimated to be 0.234. If we were to ignore the serial correlation parameters which may be present in the adjustment matrix, this parameter estimate would be regarded as a direct estimate of (1!k ), the speed of adjustment of inventories. This value of 1 the adjustment speed is not as low as it has often been estimated to be in other studies but it is also not as large as many economists might expect it to be, given the perceptions held by economists working in the inventory literature about the costs of adjustment attached to inventories and the size of sales expectations errors. But, if what is being estimated is (1!h )(1!k ), then given a parameter c 1 estimate of 0.234, the speed of adjustment could be quite rapid if the stochastic process for the cost shock is highly persistent, say with a positive autoregressive parameter of at least 0.5. Thus the presence of these serial correlation parameters can have a substantial influence upon one’s perception of the speed at which state variables appear to reach their long-run equilibrium.
3. The effects of normalization The model in the preceding section contains a cointegrating vector for which there is only one normalization which makes economic sense. Thus the effects of normalization upon the adjustment matrix must be studied in a framework where more than one choice of normalizing transformation of the cointegrating matrix may be used. The labor market model in this section has the property that more than one choice of normalizing transformation is available. 3.1. A model of the firm The firm is assumed to maximize = E + /t[(md !w )n !(d /2)n2!(d /2)*n2!d n m t 2 t t t 1 nt t t 0 0 t/0 (12) #(m !p )m !(d /2)m2], t mt t t 3 where the discount rate is again denoted by /, n refers to labor services, m is an intermediate input in production, say raw materials, with an associated factor price given by p, w is the real wage (the firm’s output price is normalized to one), and m refers to iid shocks. There are shocks specific to each input, shifting the
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marginal product of each input, and the factor price p will be assumed to be I(1). t It is assumed that d '0. i For an unspecified information set, the Euler equations describing the optimal choices of labor and materials are
C
A
B
A BD C
D
d w !md d #d nt , (13) 1 n #/~1n ! 2 m "E t E n ! 1# 0 t t`1 t t~1 t t /d /d /d 1 1 1 E [m !p !d n !d m ]"0. t mt t 2 t 3 t
(14)
Eq. (13) may be solved to show that the discounted marginal product of labor equals the real wage net of the shock to labor’s productivity. Eq. (14) is a simple marginal productivity condition for the input m . This is a static condition t because the stock of materials is a variable factor input. These decision rules, along with an appropriate transversality condition, characterize optimal behavior by the firm. 3.2. A model of a household The household problem is very simple. It is a static model so that the dynamics displayed by the labor market VAR come entirely from the demand side of the labor market.6 The household wishes to solve the problem. = (15) max E + /t[p c !ms n !(p /2)n2]. t nt t 1 0 0 t t/0 Consumption is given by c "w n #D , where D is a dividend stream received t t t t t by the household which may come from the firm studied above or it may be received from another unspecified source. Inserting the definition of consumption into Eq. (15), the Euler equation which describes the optimal choice of labor supply is E [p w !ms !p n ]"0. nt 1 t t 0 t
(16)
This expression, along with Eqs. (13) and (14), will be used to obtain the labor market VAR.
6 Lagged labor supply could also be assumed to affect current utility which would add additional lagged labor parameters from the supply side of the labor market to the VAR. This assumption does not offer any additional insights beyond those arising in a simpler model.
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3.3. The var for the labor market Using Eq. (16) to eliminate the wage rate from Eq. (13) and using Eq. (14) to eliminate m from Eq. (13), the resulting expression for labor is t (1!j ¸)(1!j )n "(/d )~1[p~1ms !md #d d~1(m !p )], t nt 2 3 mt 0 nt 1 2 t`1 1
(17)
where the expectation operator is suppressed and 0(k (1 and j '1. If p is 1 2 t a driftless random walk with iid shock m , one may solve forward with the pt unstable root and use Eq. (6) to obtain n "j n #R p #K , t 1 t~1 1 t~1 1t j d 1 2 R " , 1 d d (1!/j ) 1 3 1
C
A
d j ms m pt K "! 1 nt!md # 2 m ! nt 1t mt d d p 1!/j 3 1 0 1
BD
.
(18)
The time path of the second endogenous variable to be determined by the labor market VAR, real wages, can be derived in a similar way. This is R n #w "d n #R p #K , 0 t t 1 t~1 2 t~1 2t
C A
d #d !d2d~1 2 3 1 R "!/d j ! 1# 0 0 1 1 /d 1
C
D
BD
d 2 , R " , 2 d (1!/d ) 3 1
C
D
d ms d 2 K "(1#/d )md # m !/d nt# 2m . 2t 1 nt pt 1 d (1!/d ) p d mt 3 1 0 3
(19)
The VAR for labor and the two factor prices may be written as
C DC D C DC D C D 1
0 0
R 1 0 0 0 0 1
n j 0 R t 1 1 w " d 0 R t 1 2 p 0 0 1 t
K t~1 1t w # K . t~1 2t p m t~1 pt n
(20)
The error correction form of Eq. (20) is
C D C D C D C DC D C DCD *n 1 0 0 1 0 0 ~1 j 0 R t 1 1 *w "! 0 1 0 ! R 1 0 d 0 R 0 t 1 2 0 0 1 0 0 1 *p 0 0 1 t 0 0 ~1 K
1
# R 0
0
1t
1 0
K
0 1
m pt
2t
.
n t~1 w t~1 p t~1
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The matrix of coefficients preceding the lagged levels of time series is
C
D
1!j
0 !R 1 1 P" R j !d 1 R R !R . 0 1 1 0 1 2 0 0 0
(21)
This coefficient matrix has rank two as long as j O1 which is the case in 1 this model. With this matrix, we can now observe how alternative normalizations of the cointegrating matrix will affect the elements of the adjustment matrix. Before carrying out this exercise, it is instructive to obtain the non-normalized adjustment and cointegrating matrices which may be found by using the equilibrium version of Eq. (20). These matrices are
C D C 1 0
a" 0 1 ,
1!j 1 b" 0
D
R j !d 0 1 1 . 1
(22)
!R R R !R 1 0 1 2 Notice that the adjustment matrix, prior to normalization, does not correspond well with the definitions of adjustment parameters developed in the neoclassical adjustment cost literature. In intertemporal models under static expectations, adjustment parameters were defined as eigenvalues arising from the transition equations in an economic model or, in multiple state variable problems, adjustment parameters are made up of coefficients, evaluated in equilibrium, arising from the transition equations and eigenvalues from a dynamic model. These adjustment coefficients were generally found, in multiple state variable problems, to be nonzero on theoretical grounds (indeed a critical part of the analysis was precisely to show that adjustment parameters were all nonzero) and adjustment parameters were generally not equal to one. These parameters are nonlinear functions of parameters exogenous to economic agents and are defined independently of any normalizing transformations of the long-run decisions rules obeyed by economic agents. However, as will be seen below, the adjustment matrix in Eq. (22) can be made more comparable to the adjustment matrix developed in the neoclassical investment literature once a normalizing transformation of the cointegrating matrix has been carried out. Now choose two normalizations of b in Eq. (22) in order to observe the resulting adjustment matrices. First, set the first row of coefficients in the cointegrating matrix, corresponding to labor, equal to one. The resulting normalized matrices are 0 0
C
D
1!j 0 1 a" 0 R j !d , 0 1 1 0 0
R.J. Rossana / Journal of Monetary Economics 42 (1998) 427–444
C
1
441
D
1
(R j !d )~1 . (23) 0 1 1 !R (1!j )~1 (R R !R )(R j !d )~1 1 1 0 1 2 0 1 1 Under this normalization, estimates of the elements of the adjustment matrix would yield a direct estimate of the speed of adjustment of labor, residing in the row one, column one element of the adjustment matrix.7 This normalization corresponds to the traditional view in the literature. As a second normalization, set the third row of coefficients in the cointegrating matrix equal to one. With this choice of normalization, the coefficients associated with the materials price p are being set equal to one. In this case, the t adjustment and cointegrating matrices are b"
C C
!R
a"
0 0
0
1
0
D
R R !R , 0 1 2 0
D
!R~1(1!j ) (R j !d )(R R !R )~1 1 1 0 1 1 0 1 2 0 (R R !R )~1 b" . 0 1 2 1 1
(24)
To see the effects of normalization, compare the row one, column one element of the adjustment matrix under each normalization. Rewriting the definition of R for convenience, we have 1 j d 1 2 R " . 1 d d (1!/j ) 1 3 1 Now note that when the labor coefficients in the cointegrating matrix are normalized to one, Eq. (23) shows that estimation of the elements of the adjustment matrix will give a direct estimate of the speed of adjustment of labor. But if the second normalization on the price coefficients were chosen, this same element now contains other underlying structural parameters from the optimization problem solved by the firm, as well as j . Estimation of this parameter of 1 the adjustment matrix would reveal very different information as a consequence of alternative normalizations of the cointegrating matrix. Therefore, empirical estimates of the parameters of the adjustment matrix provide estimates of
7 The presence of the adjustment speed in the cointegrating matrix in Eq. (23) suggests that, in some economic problems, an estimate of the speed of adjustment could also be achieved by estimation of the cointegrating matrix.
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adjustment speeds and structural parameters that are conditioned upon the particular normalization of the cointegrating matrix chosen by the analyst, a normalization which itself is an arbitrary choice. Without explicitly deriving the adjustment matrix arising as a consequence of normalization applied to the cointegrating matrix in a model, one is likely to draw incorrect inferences from empirical estimates of the elements of the adjustment matrix.
4. Concluding remarks Vector autoregressions applied to economic time series will often display high order lag lengths, particularly when the data used for estimation are measured at monthly or quarterly frequency. Economic theories have been devised to rationalize this serial persistence in economic time series but these theories typically do not explain lag lengths beyond one. One way to explain these high order lags is to assume that unobservable shocks to the preferences of households or the technology of firms are serially correlated. In this paper, the implications of these serially correlated shocks are studied for the adjustment matrix in error correction VARs. The second issue examined here concerns the effect of normalization of the cointegrating matrix upon the elements of the adjustment matrix. The traditional view in the literature is that the adjustment matrix will contain adjustment speeds associated with the time series in the VAR. However there is nothing in previous research which suggests how the choice of normalizing transformation will affect the parameters of interest in the adjustment matrix and so, in this paper, a model is specified which permits an explicit derivation of the adjustment matrix under alternative normalizations of the cointegrating matrix. When unobservable shocks obey autoregressive stochastic processes, it is shown in this paper that the parameters of the autoregressive polynomial appearing in the stochastic process for these shocks will be contained in the adjustment matrix in the error correction model. The presence of these nuisance parameters makes it generally impossible to obtain estimates of structural parameters of interest in an economic model without the use of nonlinear estimation methods which would make it possible to separate structural parameters from the parameters appearing in the stochastic processes for unobservable shocks. Empirical evidence is provided that high order lags (i.e., lags beyond one) evident in estimated VARs are readily found at monthly (and presumably quarterly) data frequencies so that estimates of the adjustment matrix should not be interpreted as providing estimates of the speed at which times series reach their long-run equilibrium. However it is also shown that the parameters of the cointegrating matrix will be free of these nuisance parameters so that estimates of the cointegrating matrix can provide information about the underlying structure of an economic model.
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A model of a labor market with flexible wages is studied here in order to observe how normalization of the cointegrating matrix affects the elements of the adjustment matrix. Optimization problems which contain one state variable cannot address this issue because there is only one normalization which can be used in such models. Since the cointegrating vector represents the long-run equilibrium in the model, it only makes economic sense to attach a unit coefficient to the state variable. However, in a market context, there is no obvious normalization which must be used and it is found that the ability to estimate adjustment speeds and other structural parameters of interest is conditional upon the normalization of the cointegrating matrix which is used. If an economic model is unavailable which traces out the effects upon the adjustment matrix of normalizing the cointegrating matrix, estimation of the parameters of the adjustment matrix is unlikely to provide useful information since one cannot know what is being estimated without an explicit model framework. The results in this paper clearly illustrate the (perhaps obvious) point that empirical work which is to be both convincing and interpretable should be based upon an explicit analysis of an economic model because, without such a framework, the interpretation of parameter estimates is likely to be hazardous.
Acknowledgements I am indebted to Mehmet Balcilar, John J. Seater and Anke Sindermann for helpful comments on earlier drafts of this paper. Thanks are due to an anonymous referee who provided a number of very valuable suggestions. The usual disclaimer applies regarding responsibility for errors and omissions.
References Eichenbaum, M., 1989. Some empirical evidence on the production level and production cost smoothing models of inventory investment. American Economic Review 79, 853—864. Durlauf, S.N., Maccini, L.J., 1995. Measuring noise in inventory models. Journal of Monetary Economics 36, 65—89. Johansen, S., Juselius, K., 1990. Maximum likelihood estimation and inference on cointegration — with applications to the demand for money. Oxford Bulletin of Economics and Statistics 52, 169—210. Johansen, S., 1991. Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 1551—1580. Kennan, J., 1988. An econometric analysis of fluctuations in aggregate labor supply and demand. Econometrica 56, 313—333. Kydland, F.E., Prescott, E.C., 1982. Time to build and aggregate fluctuations. Econometrica 50, 1345—1370. Lai, K.S., 1991. Aggregation and testing of the production smoothing hypothesis. International Economic Review 32, 391—403.
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Lucas, R.E., Rapping, L.A., 1969. Real wages, employment, and inflation. Journal of Political Economy 77, 721—754. Maccini, L.J., 1973. Delivery lags and the demand for investment. Review of Economic Studies 40, 269—281. Nickell, S.J., 1977. Uncertainty and lags in the investment decisions of firms. Review of Economic Studies 44, 249—263. Rossana, R.J., 1993. Testing the long-run implications of the production smoothing model of inventories. Journal of Applied Econometrics 8, 295—306. Rossana, R.J., 1995. Technology shocks and cointegration in quadratic models of the firm. International Economic Review 56, 325—353. Rossana, R.J., Seater, J.J., 1995. Temporal aggregation and economic time series. Journal of Business and Economic Statistics 4, 441—451. Sargent, T., 1987. Macroeconomic Theory (2nd ed.), Academic Press, Orlando.
On the adjustment matrix in error correction models Robert J. Rossana* Department of Economics, Wayne State University, Detroit, MI 48202, USA Received 24 March 1997; received in revised form 26 August 1997; accepted 1 October 1997
Abstract This paper explores the determinants of the adjustment matrix in error correction models within two intertemporal models of the firm. In a production smoothing model of inventories, it is shown that the adjustment matrix contains the speed of adjustment of inventories as conjectured in previous work but this parameter matrix also contains the parameters from the autoregressive polynomials associated with the stochastic, unobservable shocks in the model. Two empirical examples are provided suggesting that these unobservable shocks cannot be assumed to be serially uncorrelated. In a flexible wage model of a labor market, the impact upon the adjustment matrix of normalizing the cointegrating matrix is studied. In contrast to one-state variable problems, normalization of the cointegrating matrix is arbitrary on economic and statistical grounds in this model and it is shown that, even with serially uncorrelated shocks, estimating the elements of the adjustment matrix may not provide an estimate of the speed of adjustment under some normalizations of the cointegrating matrix. The implication of the analysis is that without guidance from an economic model about the interpretation of elements of the adjustment matrix under alternative normalizations of the cointegrating matrix, or when unobservable shocks are serially correlated, economic interpretation of estimates of the adjustment matrix will be hazardous. ( 1998 Elsevier Science B.V. All rights reserved. JEL classification: C17; E00; E32 Keywords: Cointegration; Vector autoregression
* Corresponding author. E-mail: [email protected] 0304-3932/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 9 3 2 ( 9 8 ) 0 0 0 2 6 - 9
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1. Introduction Vector autoregressive time series systems comprised of cointegrated time series are known to obey the error correction representation k~1 *X " + P *X !PX #m , t i t~i t~k t i/1 where X is a vector of time series (and may also include the unit vector), m refers to a vector of iid disturbances, and the coefficient matrix P has reduced rank so that P"ab@. The a matrix is called the adjustment matrix and b is known as the cointegrating matrix where the latter contains the parameters from the long-run equilibrium maintained by the time series in the system. Johansen and Juselius (1990) and Johansen (1991), among others, have stated that the adjustment matrix measures the speed at which time series approach their equilibrium levels, suggesting that this matrix contains the adjustment speeds which arise in intertemporal models of the firm or household. In this paper, this claim is examined and, for two reasons, it is shown that it may be inappropriate to regard the adjustment matrix as containing these adjustment speeds. One reason is that unobservable shocks in the economic environment may be serially correlated stochastic processes. Ignoring seasonality, vector autoregressions estimated with macroeconomic data typically reveal lag lengths considerably longer than one. Economic theory generally cannot explain lag lengths beyond one; for example, costs of adjustment are the main theoretical explanation for serial persistence in economic time series such as inventories and capital goods and this theory implies that there will be only one lag of a quasi-fixed time series in vector autoregressions.1 One way to justify long lags in estimated VARs is to assume that unobservable stochastic shocks are serially correlated. In a production smoothing model of the firm, the traditional model used to study inventories (see, e.g., Eichenbaum, 1989; Lai, 1991; Durlauf and Maccini, 1995), it will be shown that the speed of adjustment is contained by the adjustment vector but, if the unobservable shocks in the model obey univariate autoregressive stochastic processes, the adjustment vector will also contain the parameters in the autoregressive polynomial appearing in the stochastic processes obeyed by these shocks. The cointegrating vector is shown to be free of these nuisance
1 It could also be true that costs of adjustment affect the firm’s cash flow for more than one period which is an additional manner in which lags beyond one for a state variable might arise in these VARs. This would be a plausible assumption with high frequency data but this is an assumption which has never, to my knowledge, been a prominent assumption in the adjustment cost or inventory literature. The time to build (Kydland and Prescott, 1982) and delivery lags (Maccini, 1973; Nickell, 1977) are additional mechanisms which could be used in some settings to motivate long lags in vector autoregressions.
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parameters, so elements of this vector may be estimated without concern for the degree of serial persistence manifested by the data. A second reason why the adjustment matrix may not only contain adjustment speeds is that alternative normalizations of the cointegrating matrix affect the elements of the adjustment matrix. Normalizing transformations of the cointegrating matrix are statistically arbitrary and may also be arbitrary on economic grounds. There is no reason to believe that alternative normalizations of the cointegrating matrix are irrelevant to the interpretation of estimates of the adjustment matrix, just as one would expect the interpretation of the parameters in ordinary regression to depend upon the choice of dependent variable. The impact upon the adjustment matrix of alternative normalizations of the cointegrating matrix will be studied in a variant of a labor market model with flexible wages adapted from Sargent (1987), a model similar to those developed in Lucas and Rapping (1969) and Kennan (1988). Only one normalization of the cointegrating vector makes economic sense in one-state variable optimization problems. The cointegrating vector in such models represents the long-run equilibrium of the state variable. But the labor market model in this paper will produce a trivariate VAR in the flow of labor services, real wages, and a unit root exogenous factor input price, say for raw materials, which permits the derivation of the adjustment matrix under alternative normalizations of the cointegrating matrix. It will be seen that the adjustment matrix contains not just the speed of adjustment of the state variable; the adjustment matrix contains parameters arising from the optimization problems solved on both sides of the labor market. Further, it will be observed that individual elements of the adjustment matrix are considerably different under two alternative normalizations which are examined (adjustment parameters are defined in dynamic optimization problems independently of any normalizations taken in an economic model); normalization of the cointegrating matrix is arbitrary on statistical and economic grounds in this model. The implication is that the interpretation of estimates of the adjustment matrix will be of little use, unless there is an explicit analysis of an economic model providing guidance about the relationship between normalization of the cointegrating matrix and its effects upon the adjustment matrix.2
2 In Rossana (1995) it is shown that the adjustment matrix will contain the speed of adjustment in a one-state variable model of the firm. However, that paper ignores the effects upon the adjustment matrix of serially correlated shocks and normalization of the cointegrating matrix. This earlier paper is concerned with providing one possible rationale for cointegrating matrices having rank greater than one, showing that the rank of the cointegrating matrix will be identical to the number of quasifixed state variables contained in an optimizing model of a firm.
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2. The effects of serially correlated unobservable shocks This section provides an analysis of a production smoothing model of inventories. It will be shown that lags greater than one will arise in the VAR arising from the model when there are serially correlated unobservable cost shocks, and the autoregressive parameters appearing in the stochastic process obeyed by these shocks will be contained in the adjustment vector arising from the model. These autoregressive parameters do not appear in the cointegrating vector. Empirical evidence is provided, in estimated bivariate VARs for selected nondurable goods industries, that these serially correlated shocks seem to be present since there are lags beyond one which are statistically significant in all of these industries. In this section, and throughout this paper, models are simplified to deliver the results of interest with minimal economic detail. So, for example, constants are omitted wherever possible. 2.1. A model of a firm holding finished goods The firm produces a storable output with the inventory cost function c "(a /2)(i !a s )2#(a /2)i2, a '0, it 0 t 1t 2 t i
(1)
where i denotes the stock of finished goods, and s is the level of shipments or sales. The first term in Eq. (1) implies that the firm has a target or desired level of inventories determined by its planned level of sales and the firm incurs costs if the level of inventories deviates from its planned level. This captures the stockout costs arising if the firm is unable to service an incoming new order. Inventory holding costs are assumed to rise at the margin. The flow of sales to the firm obeys a univariate stochastic process specified below. The firm has a cost function for the production of output which is given by c "t q #(b/2)q2, b'0, t qt ct t
(2)
where t is a stochastic shock to the firm’s cost function, and q denotes the flow c of output produced. This shock may be thought of as a factor price (Eichenbaum, 1989) or it may alternatively be a transitory shock to the firm’s technology. Either interpretation is suitable for present purposes. The shock to the cost function also obeys a univariate time series process. The firm’s total costs are given by c "c #c . t qt it The univariate time series processes obeyed by sales and the shock to the firm’s costs are (1!h ¸)(1!¸)s "ms, Dh D(1, t s s t
(1!h ¸)t "mt, c ct t
Dh D(1, c
(3)
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where ¸ is the lag operator. The random shocks, ms and mt, are iid with mean t t zero and are mutually uncorrelated. The sales of the firm are assumed to be ARI(1,1), a nonstationary, unit root stochastic process whereas the cost shock is AR(1) and thus stationary. Normalizing output prices to one, the firm wishes to solve the intertemporal problem = max E + /t[s !c ], 0 t t t/0 where E is the expectation operator conditional on an information set available to the firm and / is the discount rate (0(/(1). Using the accounting constraint q "s #*i , the optimization problem above will be solved by t t t choosing the path of finished goods to obey the Euler equation
C
A CA
E i ! t t`1 "E
t
B
D
a #a #(1#/)b 0 2 i #/~1i t t~1 /b
B
A
BD
b!a a t !/t 0 1 s !s # ct ct`1 t t`1 /b /b
.
(4)
Familiar methods may be used to derive the optimal investment equation for inventories given by
C A A B
a a i "k i # 1!k 1! 0 1 t 1 t~1 1 b
BD
= + (k /)iE s !s 1 t t`i t t/0
1!k = 1 + (k /)iE t !b~1t (5) 1 t ct`i ct b t/0 which displays the well-known property that all future values of the exogenous variables (sales and the technology shock) determine current inventory investment. The inventory investment Eq. (5) displays partial adjustment since 0(k (1. 1 The Weiner—Kolmogorov prediction formula (Sargent, 1987, p. 304) can be used if the firm uses linear least squares projections in forming its expectations. For sales, this formula is #
= h(k /)!k /¸~1h(¸) 1 + (k /)iE s " 1 s (6) 1 t t`i h(k /)(1!k /¸~1) t 1 1 i/0 for an arbitrary polynomial in the lag operator h(L). Evaluating Eq. (6) using Eq. (3) yields = 1!h k /¸ s 1 + (k /)iE s " s. 1 t t`i (1!k /)(1!h k /) t 1 s 1 i/0
(7)
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For the technology shock, the same derivation gives = 1 + (k /)iE t " t . (8) 1 t ct`i (1!h k /) ct c 1 i/0 Substituting the results in Eqs. (7) and (8) into Eq. (5), the VAR for the production smoothing model may be derived. This bivariate VAR describing the time path of inventories and shipments is
CD C
DC D D C D C
i h #k X (1#h !h )!X i t " c 1 1 s c 2 t~1 s 0 1#h s t s t~1 !k h X h !X h i X ms#X mt 3 t 1 c 2 c 1 s # t~2 # 1 t # ms 0 !h s t s t~2 a a 1 c" 1!k 1! 0 1 , K" , 1 b (1!h /k )(1!/k ) c 1 1 1!h / k ch /k c . X "cK!1, X " s 1, X " 1 1 2 3 K b 1!h /k c 1 The error correction form of Eq. (9) is
C
.
C
A
BD
C
C D C C
D
DC D DC D C
D (9)
*i h #k !1 X (1#h !h )X *i t " c 1 1 s c 2 t~1 *s 0 h *s t s t~1 (1!h )(1!k ) !(1!h )(X !X ) i X ms#X mt 3 t c 1 c 1 2 t~2 # 1 t ! ms 0 0 s t t~2
D
(10) Since there are costs of adjusting inventories, lagged inventories appear in the inventory demand function as shown in Eq. (9). Notice that this VAR contains the lag two level of inventories due to the serial persistence in the technology shock. If the series in Eq. (10) are cointegrated, the matrix of coefficients preceding the lagged levels of the time series must be of reduced rank. Inspection of Eq. (10) reveals that this is true (since k O1, it has rank one). This coefficient 1 matrix may be decomposed as P"ab@"(1!h ) c a"(1!h ) c
C
C
1!k 1 0
D
1!k 1 , 0
DC
D
1!/k !c 1 , (1!/k )(1!k ) 1 1 1 b" 1!/k !c . 1 (1!/k )(1!k ) 1 1 1
C
D
(11)
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A number of observations can be made about the adjustment and cointegrating vectors. The reduced rank coefficient matrix can be written as the product of a scalar and two vectors. This may at first appear to produce an indeterminacy in interpreting parameter estimates of coefficients in the cointegrating and adjustment vectors for this autoregressive parameter could be taken to appear in either the adjustment or cointegrating vector. But the cointegrating vector contains the parameters of the long-run equilibrium of the firm and this equilibrium is independent of the transient effects of serially correlated shocks. This is easily established by inspection of Eq. (9) and is entirely plausible.3 Thus the parameters of the cointegrating vector are identical to the case where shocks in the model are not serially correlated. The adjustment vector must be regarded as containing this autoregressive parameter. The adjustment vector contains the speed of adjustment of inventories as suggested above (this is discussed further below). In addition, generalization to the case where the technology shock obeys an AR(p) stochastic process is immediate. If the lag operator polynomial appearing in the stochastic process obeyed by the cost shock is given by h (¸)"1!h ¸!h ¸2!2!h ¸p, the reduced rank coefficient matrix in c 1c 2c pc the error correction model of the firm will now be
C
P"h (1) c
DC
1!k 1 0
1
D
1!/k !c 1 . (1!/k )(1!k ) 1 1
This extension merely adds additional autoregressive parameters to the adjustment matrix. Second, note that the analysis quite naturally leads to a unit coefficient attached to the stock of inventories so that the issue of how to normalize the cointegrating vector is irrelevant. Since the cointegrating vector contains the parameters of the long-run equilibrium of the firm, it makes little economic sense to attach a unit coefficient to anything other than the state variable held by the firm. This normalization would be appropriate in any one state variable problem. Third, note that the autoregressive parameter associated with the sales time series process does not affect the lag length of inventories in the VAR. It is only the autoregressive parameter associated with the technology shock appearing in the adjustment vector. Sales are cointegrated with the level of inventories so the
3 Set *i"0 and observe that the bivariate system reduces to the long-run equilibrium of the firm, an equilibrium which does not depend upon any of the autoregressive parameters in the stochastic process for the cost shock.
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autoregressive parameter contained in the stochastic process for sales is irrelevant to the adjustment vector. Only those parameters associated with the unobservable stochastic processes outside of the cointegrating relationship appear in the adjustment vector. Since there are no serial correlation parameters appearing in the cointegrating vector, estimation of this vector reveals information about the underlying structural model of the firm; it is not contaminated by nuisance parameters associated with any stochastic process. This analysis shows that, contrary to statements in Johansen and Juselius (1990) and Johansen (1991), the adjustment vector contains the speed of adjustment (1!k ) only for the case where the technology or cost shock is not serially 1 correlated. Generally, if it is necessary to match higher order serial correlation in the state variables contained in economic models, often evident in macroeconomic data, it will be necessary to assume serially correlated shocks to permit a VAR to match the data. The implication of this serial persistence is that the adjustment matrix will contain nuisance parameters associated with the stochastic shocks in the model, making it impossible to recover an estimate of the speed of adjustment from the estimated parameters of the adjustment matrix.4 2.2. Empirical results This section presents estimates of bivariate VAR systems for selected twodigit nondurable goods industries to see if serial correlation beyond lag one in the estimated VAR is evident in the data. If this is so, then we have compelling evidence that the adjustment matrix will not provide estimates of structural parameters which might be of interest in these industries since there will be serial correlation parameters appearing in the adjustment matrix. Two sets of VARs will be estimated. Since the production smoothing model of inventories was discussed above, a system with finished goods and sales will be examined. Since the statement was made earlier that the implications of the inventory model would arise in any intertemporal optimization framework, a second set of VARs will be estimated containing a measure of labor services in production and the real wage, series related to the model of the next section. The data series are all available from public sources. Data on inventories are provided by the Department of Commerce in seasonally adjusted constant
4 The results in this section are quite analogous to the ordinary regression problem with a lagged dependent variable and a disturbance which is serially correlated. Once the original equation is quasi-differenced with the serial correlation parameter from the disturbance process, the resulting equation will have coefficients on the lagged dependent variables which contain both structural and serial correlation parameters. This is essentially what happens to the adjustment matrix in an error correction model when shocks are serially correlated.
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Table 1 Bivariate error correction models in selected two-digit industries — inventories of finished goods and sales SIC 20
SIC 21
SIC 23
SIC 28
SIC 29
SIC 30
Dependent variable: *i t *i 0.076 t~1 (0.047) *s 0.110 t~1 (0.043)* i 0.033 t~2 (0.012)* s !0.026 t~2 (0.011)*
!0.25 (0.050)* !0.019 (0.038) 0.159 (0.029)* !0.008 (0.028)
0.065 (0.047) !0.039 (0.040) 0.044 (0.012)* !0.077 (0.023)*
0.107 (0.047)* !0.003 (0.008) 0.017 (0.006)* !0.014 (0.007)
0.110 (0.048)* 0.092 (0.052) 0.072 (0.016)* !0.040 (0.012)*
0.021 (0.048) !0.008 (0.024) 0.036 (0.010)* !0.020 (0.006)*
Dependent variable: *s t *i 0.171 t~1 (0.052)* *s !0.257 t~1 (0.046)* i !0.022 t~2 (0.013) s 0.022 t~2 (0.047)
0.239 (0.060)* !0.521 (0.045)* !0.013 (0.035) 0.160 (0.033)*
0.131 (0.056)* !0.250 (0.048)* !0.065 (0.014)* 0.138 (0.027)*
!0.109 (0.258) !0.630 (0.045)* !0.228 (0.035)* 0.281 (0.041)*
0.061 (0.044) !0.154 (0.048)* !0.042 (0.015)* 0.038 (0.011)*
0.143 (0.094) !0.137 (0.047)* !0.024 (0.020) 0.017 (0.012)
Note: In the table, i refers to constant-dollar inventories of finished goods, and s is constant-dollar sales. Standard errors are given in parentheses beneath each estimated coefficient. A constant was included in all regressions but the constant is not reported in the table. All data are in natural logarithms and the lag length is k"2. An asterisk (*) indicates significance at the 5% significance level.
dollars. The same organization provides constant dollar shipments, also seasonally adjusted. These series are used along with data on current dollar sales provided by the Census Bureau to produce a price index used in the construction of the real wage rate. Labor will be measured by manhours, the product of average weekly hours for production workers and the number of production workers, both adjusted for seasonality. The wage rate is measured by average hourly earnings. Both series are available from the Bureau of Labor Statistics and are published in Employment and Earnings. Data are measured at monthly frequency over the period 1959:1 to 1995:12, and are in natural logarithms. Unit root tests are available elsewhere which indicate that a unit root null hypothesis cannot be rejected for the data contained in the estimated VARs.5 Tables 1 and 2 contain the estimates of bivariate error correction VARs for six two digit nondurable goods industries. These industries, regularly used in
5 See Rossana (1993) and Rossana and Seater (1995) for Dickey—Fuller tests on the times series used here.
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Table 2 Bivariate error correction models for selected two-digit industries — labor services and real wages SIC 20
SIC 21
SIC 23
SIC 28
SIC 29
SIC 30
Dependent variable: *n t *n !0.312 t~1 (0.046)* *w 0.030 t~1 (0.030) n 0.020 t~2 (0.011) w !0.004 t~2 (0.004)
!0.282 (0.046)* 0.043 (0.032) 0.004 (0.006) !0.005 (0.010)
!0.156 (0.047)* !0.114 (0.078) 0.003 (0.007) 0.009 (0.006)
!0.003 (0.048) 0.002 (0.004) 0.020 (0.007)* !0.004 (0.003)
!0.097 (0.048)* !0.048 (0.056) 0.091 (0.022)* 0.012 (0.007)
0.037 (0.050) 0.029 (0.088) 0.033 (0.009)* !0.046 (0.015)*
Dependent variable: *w t *n 0.040 t~1 (0.071) *w 0.152 t~1 (0.047)* n 0.003 t~2 (0.018) w 0.010 t~2 (0.006)
!0.123 (0.066) !0.365 (0.045)* !0.006 (0.009) 0.035 (0.014)*
!0.039 (0.029) !0.127 (0.048)* !0.001 (0.005) 0.005 (0.004)
0.495 (0.512) !0.563 (0.044)* !0.212 (0.074)* 0.155 (0.032)*
!0.040 (0.038) 0.356 (0.045)* 0.046 (0.017)* 0.016 (0.006)*
0.028 (0.029) !0.097 (0.050) 0.010 (0.005) !0.011 (0.009)
Note: In the table, n refers to man-hours of labor, and w refers to real wages. Standard errors are given in parentheses beneath each estimated coefficient. A constant was included in all regressions but the constant is not reported in the table. All data are in natural logarithms and the lag length is k"2. An asterisk (*) indicates significance at the 5% significance level.
applied inventory studies, are classified as pure production to stock industries where no unfilled order data exist. The same industries are used in the labor and real wage rate error correction VARs. Constants are included in all regressions but they are not reported in the table for the sake of brevity. A lag length of two was used in all estimated VARs since this is the lag length arising in the inventory model examined earlier in this section. The results show that there are a large number of estimated equations where lagged differences and levels achieve statistical significance at conventional significance levels. Indeed it is quite likely that the actual lag lengths which might be chosen to test for cointegration in these data sets would be much longer than two as a brief examination of regressions with longer lag lengths suggests that lags longer than two would be appropriate in nearly every industry. Thus it seems reasonable to conclude that a lag length of at least two is a correct choice for all of these industries since, in no case, is there an industry where lagged differences are always statistically insignificant. In the context of the production smoothing model of this section, serially correlated shocks appear to be present in every industry.
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437
An interesting exercise to conduct is to estimate the adjustment matrix to see how badly mislead one might be by interpreting estimated parameters as estimates of an adjustment speed, rather than a nonlinear function of the adjustment speed and serial correlation parameters. To carry out this exercise, the maximum likelihood estimator of Johansen (1991) was used to estimate the adjustment vector in SIC 20 for inventories and sales. With a lag length of two, the column one, row one element of the adjustment vector is estimated to be 0.234. If we were to ignore the serial correlation parameters which may be present in the adjustment matrix, this parameter estimate would be regarded as a direct estimate of (1!k ), the speed of adjustment of inventories. This value of 1 the adjustment speed is not as low as it has often been estimated to be in other studies but it is also not as large as many economists might expect it to be, given the perceptions held by economists working in the inventory literature about the costs of adjustment attached to inventories and the size of sales expectations errors. But, if what is being estimated is (1!h )(1!k ), then given a parameter c 1 estimate of 0.234, the speed of adjustment could be quite rapid if the stochastic process for the cost shock is highly persistent, say with a positive autoregressive parameter of at least 0.5. Thus the presence of these serial correlation parameters can have a substantial influence upon one’s perception of the speed at which state variables appear to reach their long-run equilibrium.
3. The effects of normalization The model in the preceding section contains a cointegrating vector for which there is only one normalization which makes economic sense. Thus the effects of normalization upon the adjustment matrix must be studied in a framework where more than one choice of normalizing transformation of the cointegrating matrix may be used. The labor market model in this section has the property that more than one choice of normalizing transformation is available. 3.1. A model of the firm The firm is assumed to maximize = E + /t[(md !w )n !(d /2)n2!(d /2)*n2!d n m t 2 t t t 1 nt t t 0 0 t/0 (12) #(m !p )m !(d /2)m2], t mt t t 3 where the discount rate is again denoted by /, n refers to labor services, m is an intermediate input in production, say raw materials, with an associated factor price given by p, w is the real wage (the firm’s output price is normalized to one), and m refers to iid shocks. There are shocks specific to each input, shifting the
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marginal product of each input, and the factor price p will be assumed to be I(1). t It is assumed that d '0. i For an unspecified information set, the Euler equations describing the optimal choices of labor and materials are
C
A
B
A BD C
D
d w !md d #d nt , (13) 1 n #/~1n ! 2 m "E t E n ! 1# 0 t t`1 t t~1 t t /d /d /d 1 1 1 E [m !p !d n !d m ]"0. t mt t 2 t 3 t
(14)
Eq. (13) may be solved to show that the discounted marginal product of labor equals the real wage net of the shock to labor’s productivity. Eq. (14) is a simple marginal productivity condition for the input m . This is a static condition t because the stock of materials is a variable factor input. These decision rules, along with an appropriate transversality condition, characterize optimal behavior by the firm. 3.2. A model of a household The household problem is very simple. It is a static model so that the dynamics displayed by the labor market VAR come entirely from the demand side of the labor market.6 The household wishes to solve the problem. = (15) max E + /t[p c !ms n !(p /2)n2]. t nt t 1 0 0 t t/0 Consumption is given by c "w n #D , where D is a dividend stream received t t t t t by the household which may come from the firm studied above or it may be received from another unspecified source. Inserting the definition of consumption into Eq. (15), the Euler equation which describes the optimal choice of labor supply is E [p w !ms !p n ]"0. nt 1 t t 0 t
(16)
This expression, along with Eqs. (13) and (14), will be used to obtain the labor market VAR.
6 Lagged labor supply could also be assumed to affect current utility which would add additional lagged labor parameters from the supply side of the labor market to the VAR. This assumption does not offer any additional insights beyond those arising in a simpler model.
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439
3.3. The var for the labor market Using Eq. (16) to eliminate the wage rate from Eq. (13) and using Eq. (14) to eliminate m from Eq. (13), the resulting expression for labor is t (1!j ¸)(1!j )n "(/d )~1[p~1ms !md #d d~1(m !p )], t nt 2 3 mt 0 nt 1 2 t`1 1
(17)
where the expectation operator is suppressed and 0(k (1 and j '1. If p is 1 2 t a driftless random walk with iid shock m , one may solve forward with the pt unstable root and use Eq. (6) to obtain n "j n #R p #K , t 1 t~1 1 t~1 1t j d 1 2 R " , 1 d d (1!/j ) 1 3 1
C
A
d j ms m pt K "! 1 nt!md # 2 m ! nt 1t mt d d p 1!/j 3 1 0 1
BD
.
(18)
The time path of the second endogenous variable to be determined by the labor market VAR, real wages, can be derived in a similar way. This is R n #w "d n #R p #K , 0 t t 1 t~1 2 t~1 2t
C A
d #d !d2d~1 2 3 1 R "!/d j ! 1# 0 0 1 1 /d 1
C
D
BD
d 2 , R " , 2 d (1!/d ) 3 1
C
D
d ms d 2 K "(1#/d )md # m !/d nt# 2m . 2t 1 nt pt 1 d (1!/d ) p d mt 3 1 0 3
(19)
The VAR for labor and the two factor prices may be written as
C DC D C DC D C D 1
0 0
R 1 0 0 0 0 1
n j 0 R t 1 1 w " d 0 R t 1 2 p 0 0 1 t
K t~1 1t w # K . t~1 2t p m t~1 pt n
(20)
The error correction form of Eq. (20) is
C D C D C D C DC D C DCD *n 1 0 0 1 0 0 ~1 j 0 R t 1 1 *w "! 0 1 0 ! R 1 0 d 0 R 0 t 1 2 0 0 1 0 0 1 *p 0 0 1 t 0 0 ~1 K
1
# R 0
0
1t
1 0
K
0 1
m pt
2t
.
n t~1 w t~1 p t~1
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The matrix of coefficients preceding the lagged levels of time series is
C
D
1!j
0 !R 1 1 P" R j !d 1 R R !R . 0 1 1 0 1 2 0 0 0
(21)
This coefficient matrix has rank two as long as j O1 which is the case in 1 this model. With this matrix, we can now observe how alternative normalizations of the cointegrating matrix will affect the elements of the adjustment matrix. Before carrying out this exercise, it is instructive to obtain the non-normalized adjustment and cointegrating matrices which may be found by using the equilibrium version of Eq. (20). These matrices are
C D C 1 0
a" 0 1 ,
1!j 1 b" 0
D
R j !d 0 1 1 . 1
(22)
!R R R !R 1 0 1 2 Notice that the adjustment matrix, prior to normalization, does not correspond well with the definitions of adjustment parameters developed in the neoclassical adjustment cost literature. In intertemporal models under static expectations, adjustment parameters were defined as eigenvalues arising from the transition equations in an economic model or, in multiple state variable problems, adjustment parameters are made up of coefficients, evaluated in equilibrium, arising from the transition equations and eigenvalues from a dynamic model. These adjustment coefficients were generally found, in multiple state variable problems, to be nonzero on theoretical grounds (indeed a critical part of the analysis was precisely to show that adjustment parameters were all nonzero) and adjustment parameters were generally not equal to one. These parameters are nonlinear functions of parameters exogenous to economic agents and are defined independently of any normalizing transformations of the long-run decisions rules obeyed by economic agents. However, as will be seen below, the adjustment matrix in Eq. (22) can be made more comparable to the adjustment matrix developed in the neoclassical investment literature once a normalizing transformation of the cointegrating matrix has been carried out. Now choose two normalizations of b in Eq. (22) in order to observe the resulting adjustment matrices. First, set the first row of coefficients in the cointegrating matrix, corresponding to labor, equal to one. The resulting normalized matrices are 0 0
C
D
1!j 0 1 a" 0 R j !d , 0 1 1 0 0
R.J. Rossana / Journal of Monetary Economics 42 (1998) 427–444
C
1
441
D
1
(R j !d )~1 . (23) 0 1 1 !R (1!j )~1 (R R !R )(R j !d )~1 1 1 0 1 2 0 1 1 Under this normalization, estimates of the elements of the adjustment matrix would yield a direct estimate of the speed of adjustment of labor, residing in the row one, column one element of the adjustment matrix.7 This normalization corresponds to the traditional view in the literature. As a second normalization, set the third row of coefficients in the cointegrating matrix equal to one. With this choice of normalization, the coefficients associated with the materials price p are being set equal to one. In this case, the t adjustment and cointegrating matrices are b"
C C
!R
a"
0 0
0
1
0
D
R R !R , 0 1 2 0
D
!R~1(1!j ) (R j !d )(R R !R )~1 1 1 0 1 1 0 1 2 0 (R R !R )~1 b" . 0 1 2 1 1
(24)
To see the effects of normalization, compare the row one, column one element of the adjustment matrix under each normalization. Rewriting the definition of R for convenience, we have 1 j d 1 2 R " . 1 d d (1!/j ) 1 3 1 Now note that when the labor coefficients in the cointegrating matrix are normalized to one, Eq. (23) shows that estimation of the elements of the adjustment matrix will give a direct estimate of the speed of adjustment of labor. But if the second normalization on the price coefficients were chosen, this same element now contains other underlying structural parameters from the optimization problem solved by the firm, as well as j . Estimation of this parameter of 1 the adjustment matrix would reveal very different information as a consequence of alternative normalizations of the cointegrating matrix. Therefore, empirical estimates of the parameters of the adjustment matrix provide estimates of
7 The presence of the adjustment speed in the cointegrating matrix in Eq. (23) suggests that, in some economic problems, an estimate of the speed of adjustment could also be achieved by estimation of the cointegrating matrix.
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adjustment speeds and structural parameters that are conditioned upon the particular normalization of the cointegrating matrix chosen by the analyst, a normalization which itself is an arbitrary choice. Without explicitly deriving the adjustment matrix arising as a consequence of normalization applied to the cointegrating matrix in a model, one is likely to draw incorrect inferences from empirical estimates of the elements of the adjustment matrix.
4. Concluding remarks Vector autoregressions applied to economic time series will often display high order lag lengths, particularly when the data used for estimation are measured at monthly or quarterly frequency. Economic theories have been devised to rationalize this serial persistence in economic time series but these theories typically do not explain lag lengths beyond one. One way to explain these high order lags is to assume that unobservable shocks to the preferences of households or the technology of firms are serially correlated. In this paper, the implications of these serially correlated shocks are studied for the adjustment matrix in error correction VARs. The second issue examined here concerns the effect of normalization of the cointegrating matrix upon the elements of the adjustment matrix. The traditional view in the literature is that the adjustment matrix will contain adjustment speeds associated with the time series in the VAR. However there is nothing in previous research which suggests how the choice of normalizing transformation will affect the parameters of interest in the adjustment matrix and so, in this paper, a model is specified which permits an explicit derivation of the adjustment matrix under alternative normalizations of the cointegrating matrix. When unobservable shocks obey autoregressive stochastic processes, it is shown in this paper that the parameters of the autoregressive polynomial appearing in the stochastic process for these shocks will be contained in the adjustment matrix in the error correction model. The presence of these nuisance parameters makes it generally impossible to obtain estimates of structural parameters of interest in an economic model without the use of nonlinear estimation methods which would make it possible to separate structural parameters from the parameters appearing in the stochastic processes for unobservable shocks. Empirical evidence is provided that high order lags (i.e., lags beyond one) evident in estimated VARs are readily found at monthly (and presumably quarterly) data frequencies so that estimates of the adjustment matrix should not be interpreted as providing estimates of the speed at which times series reach their long-run equilibrium. However it is also shown that the parameters of the cointegrating matrix will be free of these nuisance parameters so that estimates of the cointegrating matrix can provide information about the underlying structure of an economic model.
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A model of a labor market with flexible wages is studied here in order to observe how normalization of the cointegrating matrix affects the elements of the adjustment matrix. Optimization problems which contain one state variable cannot address this issue because there is only one normalization which can be used in such models. Since the cointegrating vector represents the long-run equilibrium in the model, it only makes economic sense to attach a unit coefficient to the state variable. However, in a market context, there is no obvious normalization which must be used and it is found that the ability to estimate adjustment speeds and other structural parameters of interest is conditional upon the normalization of the cointegrating matrix which is used. If an economic model is unavailable which traces out the effects upon the adjustment matrix of normalizing the cointegrating matrix, estimation of the parameters of the adjustment matrix is unlikely to provide useful information since one cannot know what is being estimated without an explicit model framework. The results in this paper clearly illustrate the (perhaps obvious) point that empirical work which is to be both convincing and interpretable should be based upon an explicit analysis of an economic model because, without such a framework, the interpretation of parameter estimates is likely to be hazardous.
Acknowledgements I am indebted to Mehmet Balcilar, John J. Seater and Anke Sindermann for helpful comments on earlier drafts of this paper. Thanks are due to an anonymous referee who provided a number of very valuable suggestions. The usual disclaimer applies regarding responsibility for errors and omissions.
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