Directory UMM :Data Elmu:jurnal:E:Economics Letters:Vol69.Issue3.Dec2000:

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Co-integration inference in the value–profit relation and

investment models

a,b ,

_*

´ ´

Frederic Verschueren

a

´ ´

ARPEGE, Facultes Universitaires Catholiques de Mons, 151 Chaussee de Binche, B-7000 Mons, Belgium b

´

GREMARS, Universite Charles de Gaulle Lille III, BP 149, F-59653 Villeneuve d’Ascq Cedex, France Received 11 August 1999; received in revised form 25 April 2000; accepted 25 April 2000

Abstract

In this paper we make use of the co-integration property in a present value model to obtain long run specifications for investment within a neoclassical framework with adjustment cost technology. These attractive theoretical models are implemented for eleven OECD countries.  2000 Elsevier Science S.A. All rights reserved.

Keywords: Present value model; Co-integration; Investment JEL classification: C22; E22

1. Introduction

Since the beginning of this century much research work has been dedicated to understanding and modelling investment behaviour, producing an abundant set of literature, both theoretical and empirical (see Chirinko, 1993, for an interesting and critical synthesis). On the other hand, time-series based econometrics has developed drastically since the 70s and it is now common to deal with non-stationarity (see Dickey and Fuller, 1979) and to test co-integration or stable relations between variables belonging to the model under analysis (see Engle and Granger, 1987). Such an inference has been intensively implemented in many contexts (e.g. consumption, money demand or foreign trade). Nevertheless few studies have been proposed to apply the above temporal procedures to the analysis of long run fixed investment. In a disequilibrium context, de la Croix and Licandro (1993) have tested co-integration between rate of investment, Tobin’s q and the degree of utilisation of capacity for the Belgian economy. In a neoclassical model Cuthbertson and Gasparro (1995) performed the same

*Corresponding author. Tel.: 132-65-323-395; fax: 132-65-323-223. E-mail address: verschue@fucam.ac.be (F. Verschueren).

0165-1765 / 00 / $ – see front matter  2000 Elsevier Science S.A. All rights reserved. P I I : S 0 1 6 5 - 1 7 6 5 ( 0 0 ) 0 0 2 8 9 - 5

exercise between investment, output, Tobin’s q and debt using United Kingdom data. An argument for this lack of interest is that theory of optimal stock of capital gathers variables with flow and with stock profiles. Recognizing that investment has a rather volatile profile with respect to smoother economic variables, the search for balanced specifications in a time-series point of view is hence made more difficult. The direction proposed in the paper is based on co-integration inference in a present value model linking the value and the profit of the firm when these two variables are integrated of order 1, or I(1). The main result is that acceptance of stable long-run residuals (i.e.

co-integration hypothesis)in our specifications,which are constructed from a neoclassical model with convex adjustment costs, implies that the factors are chosen optimally with respect to the economic model. The paper is organised as follows. Section 2 gives details on the economic model. The general framework involving the co-integration hypothesis, and from which are deduced long run relations and investment specifications, is set up in Section 3. Empirical results with OECD data are presented in Section 4 and Section 5 summarizes the paper.

2. The economic model

In a neoclassical behaviour with adjustment costs, the value of the firm is defined as the maximum of the discounted sum of current and expected future flows of profits

` i

Vt5max

O

b EtfPt1ig (1)

I,L,K i50

21

with V the value,_t P_t the profit andb the discount factor, b5(11r) , r the interest rate assumed to be constant, and E . the expectation of . based on information available at time t.tf g f g

The profit variable has the usual definition

Pt5p Yt t2w Lt t2It2Gt (2)

Y is the output from the production function F(K_{t t}21, L ); K is the end-period capital stock and L thet t t

labour force; I is the amount of investment; p and w are the price of output and the wage cost_{t t t}

I

respectively, both expressed in terms of p the price of investment; G(K_{t t}21, I ) is the adjustmentt

function reflecting the costs of installing new capital goods. The installation technology is a convex function in I (the more is invested, the more the associated costs) and is inversely related to current_t capital stock (scale effects). Both production and adjustment technologies are assumed to be homogeneous of degree 1 in respective arguments. The dynamic constraint on K is_t

K_t2K_t215It2dKt21 (3)

To solve the optimization problem (1) with (2) and (3), the current-value Hamiltonian is constructed as H[l_t]5p Y_{t t}2w L_{t t}2I_t2G_t1l_thI_t2dK_t21j so that the necessary first-order conditions for a maximum are given by

≠Gt

]]

H_I5 2 _≠ 211l_t50 (4)

≠Y_t ]

H_L5p_t≠ 2w_t50 (5)

L_t

≠Yt

]]

l_t2l_t215rl_t212

H

p_t≠ 2dl_t

J

(6)

K_t21

with the transversality condition

i

limb l_tK_t50 (7)

t→`

3. The empirical specifications

3.1. Present value model and co-integration

We now give up for the moment the optimization context of Section 2 and assume that a particular variable V and a particular profit_t P_t are linked in a present value model, so that V is generated by_t

` i

Vt5

O

b EtfPt1ig (8)

i50

with b and E . having the same definition as above. Campbell and Shiller (1987) have shown thattf g

(8) can be attractively transformed into the following three equations

21

C_t5V_t2b _hV_t212Pt21_j (9)

Ct5Vt2Et21f gVt (10)

21

St5Vt2(12b) Pt (11)

In (9) C links contemporaneous and lagged observations of V and_{t t} P_t. By combining (9) and (8) a crucial economic interpretation of this variable C is obtained in (10), since it is revealed to be the_t expectation error from predicting V at time t_t 21. When expectations are naive, C_t5 DV while in the_t perfect expectation case C_t50. Alternatively (8) can be expressed as (11), showing how the value and the profit are linked in the long run. A crucial property of the variable S emerges when it is_t related to C . Indeed using (9), Eq. (11) can be reformulated in a dynamic way as_t

21 21 21

S_t5(12b) DV_t2(12b) DP_t2b(12b) C_t (12)

The latter expression indicates that (11) is the candidate co-integration equation, with S the long runt

residual. Indeed when both P_t and V are I(1) then from (10), C is a martingale difference which is_{t t} I(0), and which implies from (12) that the residual S is also I(0), and therefore that value and profitt

21 21

3.2. Implication for optimal factors demand

Now we study the implications of the co-integration hypothesis of (11) when this equation is

*

related to the theoretical model discussed in section 2. We denote V_t as the value associated to an

*

*

*

optimal choice of factors (I , Kt t21, L ) andt Pt as the profit constructed as in (2). When these two variables are both I(1) and when they are co-integrated of order (1, 1), then with respect to results

i i

*

*

*

*

*

*

from Section 3.1 we have thatPt5Pt 5P(I , Kt t21, L ) and Vt t 5maxob EtfPt1ig5ob EtfPt1ig. In other words, co-integration implies that factors are chosen optimally at each period, since constructed profit fits its optimal level given by the economic theory.

In the neo-classical point of view with adjustment technology as described above, using (4), (5), (6) and (3), summing up from 0 to infinity, discounting and using (7), the value of the firm is checked

*

to be V_t 5(11r)l_tK_t21. To give an observable equivalence to the multiplier l_t we select the simplest quadratic form for the installation function (defining hence ≠G /≠I )

2

I

a t

] ]]

G_t5 (13)

2 K_t21

with a.0. Now using (13) and (4) we are able to rewrite the value at optimum path as

*

V_t 5(11r)(K_t211aI )t (14)

*

The idea is thus to put V_t andP_ttogether. When there are no adjustment costs (a50), (2), (11) and (14) quickly leads to

1 ]

K_t215_rPt1St (15)

So constant interest rate plays the role of long run coefficient. However whena.0 equations such as (15) cannot be deduced since the adjustment parameter enters the left side of the specification.

3.3. Implications for long run investment

]

In a first stage interest rate is assumed to be given and is denoted r. The sensitivity of investment to its determinants is expected to depend on the structural parameters which characterize the economy in which firms behave. In that line when adjusting the stock of capital is costless (a50) we have to rely on a parametric production technology. We chose the Cobb–Douglas functional

u 12u

Yt5AKt21Lt (16)

withu the capital–output elasticity and A a constant. Using (2), (5) and (16), when factors are chosen

*

optimally, profit is commonly defined as P_t 5up Y_{t t}2I , and the value–profit relation (11) can be_t reformulated to obtain

˜It5up Yt t1St (17)

] ˜

with I_t5I_t1rK_t21, or using (3) ]

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. WhenDK_t50 the

]

usual user cost proxy (relative to price investment) appears since st5r1d and dividing each side of (18) by this constant term leads to the model of Jorgenson. Turning to the more realist case thata.0 (adjustment costs), Eqs. (2), (11) and (14) suggest that investment has the following long run representation

1 K

]]

It5₁₁]_raPt 1St (19)

K ]

with P_t 5p Y_{t t}2w L_{t t}2rK_t21.

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 a50 the latter specification

K

becomes I_t5P_t 1S , implying a unit long run parameter in (19). When a parametric Cobb Douglas_t production function (16) is introduced as before, it is possible to identify both structural parametersa

andu since the above procedure gives ]

u r

]] ]]

It5_11r]_ap Yt t2₁₁]_raKt211St (20)

When r is to be estimated together with a a slight modification of (20) allows for identification of these parameters since

1 r

]] ]]

It5_11ra( p Yt t2w L )t t 2_11raKt211St (21)

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 (Yt2C ), the Gross Fixed Capital Formation (I ),t t

I

both at 1990 prices and exchange rates, and deflator (199051) of GDP ( p ) and of GFCF ( p ). Stock_{t t} of capital (K ) has been simulated with the accumulation Eq. (3) witht d50.10. The labour income

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, r50.10. 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 I(1). 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 parametera also appears in the long run equation explaining the rate of investment.

Under the null hypothesis of a random walk the distribution of the t-statistics associated with the coefficientg in the augmented regression

k

Dx_t5m1zt1gx_t211

O

d_jDx_t2j1u_t (22)

j51

is non standard, with u a white noise, and has to be compared with simulated critical valuest

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 ,t Pt, Pt, I , I , p Yt t t t

and ( p Y_{t t}2w L )] clearly indicates that all the variables fit the random walk hypothesis. Therefore the_{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 I(0). 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 coefficient(s). 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. EG(k)

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

Table 2

Co-integration hypothesis for optimal investment model without convex adjustment costs. Identification of structural parameteru (Eq. (8), 1965–1991)

2 a

¯

u R d.w. EG(k)

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%, N527): 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 I(0) 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 a516 for Japan up toa536 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.

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 /(11ra) a R d.w. EG(k)

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%, N527): 23.57.

results not reported here. The major problem comes from the estimated coefficients associated to Kt21

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_t21, so that the rate of investment is explained by the ratio output–capital and a constant for which 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

This paper was concerned with providing long run specifications for which economic theory and non stationarity econometrics have room to co-exist. Based on a present value model, candidate

co-integrating relations have been obtained both characterizing a value–profit relation along the optimal path and explaining long run investment. When applied to our OECD aggregate data set empirical results are disappointing since the null of no co-integration can not be rejected in most cases. But extensions such as (i) the investigation of a present value model with profit weighted by the stock of capital variable, and (ii) the collection of more disaggregated data, are likely to yield more attractive empirical evidence using the long run specifications discussed in the paper.

Acknowledgements

Financial support from the Belgian Fonds National de la Recherche Scientifique under grant D.4507.93 is gratefully acknowledged. I would like to thank for helpful comments on an earlier version of the paper, my thesis advisers Laurence Broze and Marcel Gerard, and the participants to the ESEM’98 congress at Berlin and to the CEPREMAP seminar in Paris. Any error remains mine.

References

Campbell, J.Y., Shiller, R.J., 1987. Cointegration and tests of present value models. Journal of Political Economy 95, 1062–1088.

Chirinko, R.S., 1993. Business fixed investment spending modeling strategies, empirical results, and policy implications. Journal of Economic Literature 31, 1875–1911.

de la Croix, D., Licandro, O., 1993. The q theory of investment under unit root tests. Cahiers Economiques de Bruxelles 139, 329–339.

Cuthbertson, K., Gasparro, D., 1995. Fixed investment decisions in UK manufacturing: the importance of Tobin’s Q, output and debt. European Economic Review 39, 919–941.

Dickey, D.A., Fuller, W.A., 1979. Distribution of the estimators for autoregressive time series with a unit root. Journal of American and Statistical Association 74, 427–431.

Engle, R.F., Granger, C.W.J., 1987. Co-integration and error correction: representation, estimation and testing. Econometrica 55, 254–276.

Hayashi, F., 1982. Tobin’s q and average q: a neoclassical interpretation. Econometrica 50, 213–224. Jorgenson, D.W., 1963. Capital theory and investment behaviour. American Economic Review 53, 247–259.

3.2. Implication for optimal factors demand

Now we study the implications of the co-integration hypothesis of (11) when this equation is

*

related to the theoretical model discussed in section 2. We denote V_t as the value associated to an

*

*

*

optimal choice of factors (I , Kt t21, L ) andt Pt as the profit constructed as in (2). When these two variables are both I(1) and when they are co-integrated of order (1, 1), then with respect to results

i i

*

*

*

*

*

*

from Section 3.1 we have thatPt5Pt 5P(I , Kt t21, L ) and Vt t 5maxob EtfPt1ig5ob EtfPt1ig. In other words, co-integration implies that factors are chosen optimally at each period, since constructed profit fits its optimal level given by the economic theory.

In the neo-classical point of view with adjustment technology as described above, using (4), (5), (6) and (3), summing up from 0 to infinity, discounting and using (7), the value of the firm is checked

*

to be V_t 5(11r)l_tK_t21. To give an observable equivalence to the multiplier l_t we select the simplest quadratic form for the installation function (defining hence ≠G /≠I )

2 I

a t

] ]]

G_t5 (13)

2 K_t21

with a.0. Now using (13) and (4) we are able to rewrite the value at optimum path as

*

V_t 5(11r)(K_t211aI )t (14)

*

The idea is thus to put V_t andP_ttogether. When there are no adjustment costs (a50), (2), (11) and (14) quickly leads to

1

]

K_t215_rPt1St (15)

So constant interest rate plays the role of long run coefficient. However whena.0 equations such as (15) cannot be deduced since the adjustment parameter enters the left side of the specification.

3.3. Implications for long run investment

]

In a first stage interest rate is assumed to be given and is denoted r. The sensitivity of investment to its determinants is expected to depend on the structural parameters which characterize the economy in which firms behave. In that line when adjusting the stock of capital is costless (a50) we have to rely on a parametric production technology. We chose the Cobb–Douglas functional

u 12u

Yt5AKt21Lt (16)

withu the capital–output elasticity and A a constant. Using (2), (5) and (16), when factors are chosen

*

optimally, profit is commonly defined as P_t 5up Y_{t t}2I , and the value–profit relation (11) can be_t

reformulated to obtain

˜It5up Yt t1St (17)

] ˜

with I_t5I_t1rK_t21, or using (3)

]

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. WhenDK_t50 the

]

usual user cost proxy (relative to price investment) appears since st5r1d and dividing each side of (18) by this constant term leads to the model of Jorgenson. Turning to the more realist case thata.0 (adjustment costs), Eqs. (2), (11) and (14) suggest that investment has the following long run representation

1 K

]]

It5_11r]_aPt 1St (19)

K ]

with P_t 5p Y_{t t}2w L_{t t}2rK_t21.

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 a50 the latter specification K

becomes I_t5P_t 1S , implying a unit long run parameter in (19). When a parametric Cobb Douglas_t

production function (16) is introduced as before, it is possible to identify both structural parametersa

andu since the above procedure gives

]

u r

]] ]]

It5_11r]_ap Yt t2_11r]_aKt211St (20)

When r is to be estimated together with a a slight modification of (20) allows for identification of these parameters since

1 r

]] ]]

It5_11ra( p Yt t2w L )t t 2_11raKt211St (21)

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 (Yt2C ), the Gross Fixed Capital Formation (I ),t t

I

both at 1990 prices and exchange rates, and deflator (199051) of GDP ( p ) and of GFCF ( p ). Stock_{t t} of capital (K ) has been simulated with the accumulation Eq. (3) witht d50.10. The labour income

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, r50.10. 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 I(1). 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 parametera also appears in the long run equation explaining the rate of investment.

Under the null hypothesis of a random walk the distribution of the t-statistics associated with the coefficientg in the augmented regression

k

Dx_t5m1zt1gx_t211

O

d_jDx_t2j1u_t (22) j51

is non standard, with u a white noise, and has to be compared with simulated critical valuest 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 ,t Pt, Pt, I , I , p Yt t t t and ( p Y_{t t}2w L )] clearly indicates that all the variables fit the random walk hypothesis. Therefore the_{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 I(0). 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 coefficient(s). 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. EG(k)

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

Table 2

Co-integration hypothesis for optimal investment model without convex adjustment costs. Identification of structural parameteru (Eq. (8), 1965–1991)

2 a

¯

u R d.w. EG(k)

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%, N527): 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 I(0) 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 a516 for Japan up toa536 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.

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 /(11ra) a R d.w. EG(k)

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%, N527): 23.57.

results not reported here. The major problem comes from the estimated coefficients associated to Kt21

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_t21, so that the rate of investment is explained by the ratio output–capital and a constant for which

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

This paper was concerned with providing long run specifications for which economic theory and non stationarity econometrics have room to co-exist. Based on a present value model, candidate

co-integrating relations have been obtained both characterizing a value–profit relation along the optimal path and explaining long run investment. When applied to our OECD aggregate data set empirical results are disappointing since the null of no co-integration can not be rejected in most cases. But extensions such as (i) the investigation of a present value model with profit weighted by the

stock of capital variable, and (ii) the collection of more disaggregated data, are likely to yield more

attractive empirical evidence using the long run specifications discussed in the paper.

Acknowledgements

Financial support from the Belgian Fonds National de la Recherche Scientifique under grant D.4507.93 is gratefully acknowledged. I would like to thank for helpful comments on an earlier version of the paper, my thesis advisers Laurence Broze and Marcel Gerard, and the participants to the ESEM’98 congress at Berlin and to the CEPREMAP seminar in Paris. Any error remains mine.

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