Annex I: The Johansen Procedure
Johansen and Juselius (1990) consider the following general model:
where Xt is a vector of ρ variables, e1,..., et are independent normal errors with mean zero and covariance matrix Λ, X_k+1,....Xo are fixed, and μ is an intercept vector. Economic time series are often nonstationary and systems such as the above vector autoregressive representation (VAR) can be written in the conventional first difference form:
Γi = -(I-Π1-. . .-Πi) (i=1,..., k-1), and Π = -(I-Π1-. . .-Πk)
The only level term in (2) is IIXt-k. Thus, only the matrix Π contains information about the long-run relationships between the variables in the data vector. There are three cases:
i. If the matrix Π has rank zero, then all the variables in Xt are integrated of order one or higher and the VAR has no long-run properties;
ii. If Π has rank p, i.e., it is of full rank, the variables in Xt are stationary; and
iii. The interesting case when Π has rank r, 0<r<p, in which case Π can be decomposed into two distinct (p × r) matrices α and β such that Π=αβ’.
The third case implies that there are r cointegrating vectors. The parameters of the cointegrating vectors are contained in the β matrix. Therefore, βXt is stationary even though Xt itself is non-stationary. The α matrix gives the weights with which the cointegrating vectors enter each equation of the system.
To determine the number of cointegrating vectors, r, Johansen and Juśelius describe two likelihood ratio tests. In the first test, which is based on the maximal eigenvalue, the null hypothesis is that there are at most r cointegrating vectors against the alternative of r+1 cointegrating vectors. In the second test, which is based on the trace of the stochastic matrix, the null hypothesis is that there are at most r cointegrating vectors against the alternative hypothesis that there are r or more cointegrating vectors. The first test is generally considered to be more powerful because the alternative hypothesis is an equality.
Annex II: Data Sources and Definitions
Real value added, hours worked, and employment in the nonfarm business sector are from the INSEE data tape, in each case subtracting the farm sector (secteur agriculture, sylviculture, pêche) from the total for the business sector (secteurs marchands). The relevant INSEE codes for real value added are PN1_V008 and PN1_U018; for hours worked, ACM_V001, ACM_U011; and for employment, EFM_V001, EFM_U011.
The stock of business sector capital, the stock of residential capital, labor force, and population are from the OECD Analytical Data Bank. The stock of infrastructure capital is taken from the annual estimates of Ford and Poret (1991), reported by them as INF.N (p. 80), interpolated to a quarterly frequency. The share of the labor force aged 15-24 is calculated from OECD, Labour Force Statistics.
The stock of R&D capital is calculated analogously to the stock of physical capital with an assumed obsolescence rate of 5 percent: kr&d -kr&d (1-0.05) + (real R&D expenditures). A benchmark for kr&d was calculated using the procedure suggested by Griliches (1980). Real R&D expenditures are gross domestic expenditure on R&D deflated by an average of the GDP deflator and an index of business sector wages. R&D expenditures are from OECD, Main Science and Technology Indicators, 1991:1, p. 16. R&D expenditure for 1991 was estimated using the same procedure reported in the text to project R&D expenditures for 1992-97. Annual data for the stock of R&D capital were interpolated to a quarterly frequency.
Data to construct the EC variable are from IMF, Direction of Trade. EC is constructed with data for all 12 current members of the EC, even though not all 12 countries were members during the full 1971-91 period. World trade as a percent of world output was constructed from the IMF World Economic Outlook database. Annual data were interpolated to a quarterly frequency.
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The authors are in the Research Department and European I Department, respectively, and thank Robert Ford, Manmohan Kumar, Flemming Larsen, Paul Masson, Mark Taylor, and Hari Vittas for helpful comments and suggestions, and Toh Kuan for research assistance.
This slowdown has rekindled interest in the determinants of growth and in growth theory, as evidenced by the recent emergence of a large and expanding literature on endogenous growth. See Lucas (1988), Sala-i-Martin (1990), and Helpman (1992).
The intuition behind the super-consistency result is that, for values of the parameters which do not cointegrate, the residual series will itself be nonstationary and therefore have a very large estimated variance. When the estimated parameters are close to the true cointegrating parameters, the residual becomes stationary and its variance shrinks. Since least squares and maximum likelihood methods essentially minimize the residual variance, they will be extremely good at picking out the cointegrating parameters if they exist.
All of the variables used below are, in fact, integrated of order 1 based on the Dickey-Fuller and augmented Dickey-Fuller tests.
An increase in the stock of housing, for example, may increase labor mobility.
See, for example, Romer (1987) and Sala-i-Martin (1990). There are a number of practical problems with imposing factor shares, one of which is that they are not constant over the sample period. In addition, there are a variety of ways to calculate factor shares, depending, for example, on the way that self-employment income is allocated to capital or labor.
The Johansen procedure involves the simultaneous estimation of dynamic vector autoregressive (VAR) equations, for which fourth order lags were included. Estimation has been done on MICROFIT 3.0, see Pesaran and Pesaran (1991).
Recent studies at the aggregate level for the United States, Japan, and Germany find an elasticity of about 0.13; see Adams and Coe (1990), Citrin (forthcoming), and Coe and Krueger (1991), respectively. Griliches (1988) reports that estimated elasticities from firm and industry level data tend to lie between 0.06 and 0.1.
In general, the standard error of any equation including non-stationary variables is biased, irrespective of how it is estimated.
De Long and Summers (1991), based on cross-country data for industrial and developing countries, find equipment investment has a stronger association with growth than other components of investment. Estimates using only the stock of business equipment capital were not possible since this variable is not readily available for France.
If the coefficients on the first differences of each variable are constrained to the estimates reported in Table 1, a cointegrating vector is obtained when either the level of EC or kr&d are added. Although these results suggest that the level of EC and kr&d have an additional positive impact on growth, the impact is quantitatively very small (reflecting the parameter restrictions). Without the parameter restrictions on the differenced variables, the results were not interesting.
The concept of potential output is central to many economic policy issues. In the short run, the relationship between actual and potential output indicates the extent to which demand may be exerting either upward or downward pressures on inflation. In the medium to long run, the path of potential output determines the sustainable pace of noninflationary output growth or—alternatively—the scope for increases in real standards of living. See the references cited in the box on potential output in IMF (1991), p. 43; and Martin and Torres (1990).
Using a “*” to indicate the variable has been cyclically adjusted or that the unemployment rate (U) is at its “natural” level, the adjustment for hours worked is h-h* ≃ (U*-U)/100, which is obtained by substituting e/lf = log(l-U/100) ≃ -U/100 into the equation used to decompose h, and making a similar substitution for (e/lf)* into an equation for h*.
The prime-age male unemployment rate is often used in estimated wage equations instead of the aggregate unemployment rate; see Cotis and Loufir (1990). The natural rate of unemployment is estimated as a quadratic trend on the 24-50 year old male unemployment rate plus the differential between the aggregate and the prime-age-male unemployment rates in the early 1970s.
Even if there had been no increase in the natural rate of unemployment from 1980 to 1986, the average growth of potential would have fallen from 2.7 percent in 1976-80 to 2.0 percent in 1981-86.
The medium-term projections are summarized in Annex II of the October 1992 World Economic Outlook, pp. 69-76.
The estimated regression is log (R&D expenditures) = 1.1 log (output) + constant, R2 = 0.8, annual data 1970-89. The stock of R&D capital was then calculated for 1992-97 by cumulating R&D expenditures with an assumed obsolescence rate of 5 percent.