F . Verschueren Economics Letters 69 2000 289 –297
293
The key difference between the latter specification and the long run one originally proposed by Jorgenson 1963 is that 18 links the transformed change of capital stock to the level of output,
and not the level of both variables. In fact the latter equation may be viewed as a particular case of the former, but expressed in an attractive way to investigate co-integration assumption. When DK 5 0 the
t
] usual user cost proxy relative to price investment appears since s 5r 1
d and dividing each side of
t
18 by this constant term leads to the model of Jorgenson. Turning to the more realist case that a . 0
adjustment costs, Eqs. 2, 11 and 14 suggest that investment has the following long run representation
1
K
]] I 5
P 1 S 19
]
t t
t
1 1r a
K
] with
P 5 p Y 2 w L 2rK .
t t
t t
t t 21
Consistently with economic theory, investment is inversely related to interest rate and to the importance of installation burden. But more interestingly, the long run equation identifies structural
parameter a, so that a dynamic equation linking successive levels of investment is not required as is
1
usual in the literature . Furthermore it is important to note that when a 5 0 the latter specification
K
becomes I 5 P 1 S , implying a unit long run parameter in 19. When a parametric Cobb Douglas
t t
t
production function 16 is introduced as before, it is possible to identify both structural parameters a
and u since the above procedure gives
] u
r ]]
]] I 5
p Y 2 K
1 S 20
] ]
t t
t t 21
t
1 1r a
1 1r a
When r is to be estimated together with a a slight modification of 20 allows for identification of
these parameters since 1
r ]]
]] I 5
p Y 2 w L 2 K
1 S 21
t t
t t
t t 21
t
1 1 r a
1 1 r a
4. Data, methodology and results
The data set from which the variables are constructed has two sources. OECD database for total economy has provided the Gross Domestic Product Y 2 C , the Gross Fixed Capital Formation I ,
t t
t I
both at 1990 prices and exchange rates, and deflator 1990 5 1 of GDP p and of GFCF p . Stock
t t
of capital K has been simulated with the accumulation Eq. 3 with d 5 0.10. The labour income
t I
w L comes from the Eurostat national account ESA. All prices are relative to p , and all quantities
t t
t
] are expressed in US at 1990 exchange rates. When interest rate is given, r 5 0.10. Countries under
investigation are Belgium, Denmark, Spain, USA, France, Ireland, Italy, Japan, the Netherlands, UK and Germany.
First of all we have to check that our modelled variables are I1. To determine the order of integration of the series we use the standard Dickey–Fuller procedure Dickey and Fuller, 1979.
1
Though in the financial Tobin’s q model see Hayashi, 1982 the structural parameter a also appears in the long run
equation explaining the rate of investment.
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. Verschueren Economics Letters 69 2000 289 –297
Under the null hypothesis of a random walk the distribution of the t-statistics associated with the coefficient
g in the augmented regression
k
Dx 5 m 1 zt 1 gx
1
O
d Dx 1 u
22
t t 21
j t 2j
t j 51
is non standard, with u a white noise, and has to be compared with simulated critical values
t
depending on the sample size and the inclusion of the constant term m or the trend term z .
K
˜ Implementation of the unit root test for the variables under investigation [K ,
P , P , I , I , p Y
t t
t t
t t
t
and p Y 2 w L ] clearly indicates that all the variables fit the random walk hypothesis. Therefore the
t t
t t
co-integration hypothesis from present value relation and neo-classical theory can be inferred by checking if residuals from Eqs. 15, 17, 19, 20 and 21 respectively are I0. For all
specifications the testing procedure is undertaken with help of the two-step Engle and Granger EG test Engle and Granger, 1987.
Tables 1–3 hereafter summarize the results. The second column gives the estimated long-run coefficients. The next column is the adjusted coefficient of determination, which is followed by the
Durbin–Watson statistics. The final column refers to the t-statistics associated to the Engle and Granger test, with augmented order between parentheses. From Table 1 comparing the EG test with
the critical value there is clear evidence that optimal value and profit are not co-integrated, so that
Table 1 Co-integration hypothesis for optimal value–profit model without convex adjustment costs Eq. 8, 1965–1991
2 a
¯ 1 r
R d.w.
EGk Germany
2.085 0.95
0.14 22.38 0
0.028 Belgium
1.812 0.89
0.12 21.78 0
0.029 Denmark
2.361 0.86
0.14 21.76 0
0.038 Spain
2.056 0.93
0.09 22.17 1
0.041 France
2.073 0.94
0.08 22.08 0
0.034 The Netherlands
2.102 0.97
0.16 22.83 0
0.026 Ireland
1.900 0.86
0.09 21.83 2
0.052 Italy
2.136 0.80
0.08 21.96 1
0.049 Japan
2.779 0.98
0.14 21.87 1
0.045 UK
1.689 0.82
0.11 22.28 0
0.035 USA
1.705 0.85
0.08 22.19 1
0.039
a
Critical value for EG test 5, N 527: 23.57.
F . Verschueren Economics Letters 69 2000 289 –297
295 Table 2
Co-integration hypothesis for optimal investment model without convex adjustment costs. Identification of structural parameter
u Eq. 8, 1965–1991
2 a
¯ u
R d.w.
EGk Germany
0.376 0.95
0.53 24.75 1
0.002 Belgium
0.338 0.90
0.20 21.45 1
0.004 Denmark
0.410 0.84
0.17 20.90 2
0.005 Spain
0.387 0.97
0.39 22.74 1
0.003 France
0.380 0.96
0.19 21.86 2
0.003 The Netherlands
0.379 0.97
0.60 22.04 2
0.002 Ireland
0.538 0.82
0.13 22.02 2
0.009 Italy
0.385 0.88
0.13 21.39 1
0.005 Japan
0.496 0.99
0.36 23.33 1
0.003 UK
0.320 0.91
0.39 23.07 1
0.003 USA
0.323 0.90
0.12 21.36 2
0.004
a
Critical value for EG test 5, N 527: 23.57.
with respect to our theoretical discussion we cannot relate the value for optimal path given by the economic model, and a present value model-like behaviour. Besides, our estimates of r are much
larger than what would be expected for interest rate. But it is worth stressing that all regressions exhibit a very good explanatory power. Tables 2 and 3 refer to investment specifications. Table 2
provides estimates for the elasticity of capital in the production function
u , which takes very reasonable values for this time-series approach. For Japan
u is high while it is low for US economy. ]
2
Moreover R is nearly perfect for most countries. Nevertheless I0 residuals are checked only for Germany, and the co-integration hypothesis is not rejected at a slightly higher significance level for
Japan and United Kingdom. The same conclusions are drawn about co-integration in the case of adjustment technology since from Table 3 this only occurs for Germany and United Kingdom besides
in a not too significant way. But all the estimated coefficients have the right sign, with values far from unity. Also the quality of the fit remains rather good except for Denmark and Ireland. When
structural parameters
a are retrieved they take abnormally high values, a common finding in the empirical literature of investment, from
a 5 16 for Japan up to a 5 36 for United Kingdom and US. So at an equilibrium rate of investment of 10, and retaining the installation function 13, a 1
additional amount of new investment involves adjustment costs for 1.6 and 3.6 in respective countries.
We also performed the co-integration tests for the extended Eqs. 20 and 21 but got very bad
296 F
. Verschueren Economics Letters 69 2000 289 –297 Table 3
Co-integration hypothesis for optimal investment. Model with convex adjustment costs. Identification of structural parameter a unit value under costless adjustment hypothesis Eq. 12, 1965–1991
2 a
¯ ¯
1 11r a
a R
d.w. EGk
Germany 0.256
29.03 0.74
0.34 23.49 1
0.003 Belgium
0.227 34.03
0.65 0.19
21.61 2 0.005
Denmark 0.282
25.36 0.28
0.18 21.04 1
0.007 Spain
0.274 26.42
0.88 0.29
22.27 1 0.004
France 0.263
27.96 0.88
0.13 21.54 2
0.004 The Netherlands
0.259 28.48
0.73 0.21
21.67 2 0.004
Ireland 0.246
30.53 0.57
0.16 21.62 2
0.009 Italy
0.264 27.74
0.82 0.15
20.95 1 0.004
Japan 0.379
16.38 0.97
0.31 22.71 1
0.004 UK
0.214 36.68
0.83 0.60
23.51 1 0.003
USA 0.216
36.24 0.83
0.28 21.06 1
0.003
a
Critical value for EG test 5, N 527: 23.57.
results not reported here. The major problem comes from the estimated coefficients associated to K
t 21
which are systematically positive while expected to be negative. Actually there seems to be a conflict between the specification and the economic model, since, by its scale effect in 13, capital stock
lowers the adjustment burden and therefore should be related positively to investment. To overcome this difficulty, we have followed a less rigourous treatment by dividing each relation 20 and 21 by
K
, so that the rate of investment is explained by the ratio output–capital and a constant for which
t 21
negative sign may have some interpretation. Of course since all error terms have also been weighted by this non stationary variable, co-integration inference has no meaning with respect to the present
value model 8 from which all the results of Section 3 have been deduced. We however mention that for some countries
u takes a reasonable value together with an a value much lower than the one found in Table 3, while interest rates between 0.08 and 0.10 are obtained together with lower values
of structural parameter a.
5. Concluding remarks
