A. Conceptual Framework

Assume a Cobb-Douglas production function that incorporates the public capital stock, G, as an input:⁶

where A denotes an index of economy-wide productivity, K is private capital, L denotes employment, and y is output. In this setup, an increase in public capital raises output directly (i.e., Y_G=β(Y/G)>0, where a subscript denotes a partial derivative), but also indirectly through its positive effect on the marginal productivity of private capital and labor (i.e., Y_KG>0 and Y_LG>0).

By taking natural logs on both sides of the equation, and denoting lowercase variables as the natural log of the respective uppercase variable, the following equation results:

The coefficients α, β, and γ are the output elasticities of the factor inputs. Inclusion of public capital in the production function raises the issue of returns to scale. Imposing the restriction of constant returns to scale across all inputs (i.e., α+β+γ=1), which is a common assumption in the literature, yields an expression featuring decreasing returns with respect to private inputs:

An alternative model assumes constant returns to scale in both private inputs, allowing for increasing returns to scale across all inputs:

This specification has been employed at times in the endogenous growth literature (see, for example, Barro, 1990).

In the econometric specification, all equations to be estimated include a capital utilization rate, cu, to capture the effects of the business cycle on factor use. Because the capital utilization rate enters the equation in an additive fashion, it does not affect the optimal capital-labor ratio.⁷ Many studies also include a constant and a time trend to capture Hicks-neutral technological progress. The current study will represent the technology variable, a, by a constant. This specification reflects the underlying hypothesis that economic growth in Portugal has been mainly driven by factor accumulation and not by increases in factor productivity. In addition, a dummy variable for the period 1975–85 will be introduced to capture the period in between the new regime after the 1974 revolution and EU accession in 1986.

B. Evidence for OECD Countries

Most time series studies employ a Cobb-Douglas production function to estimate the output effects of public capital. On average, these studies estimate a production elasticity of public capital (β) of 0.25 for various OECD countries when the production function is estimated in levels. Estimates of β vary considerably across countries but lie in the interval 0.20–0.30 at a 95 percent level of confidence (Table 3). If the model is estimated in first differences, estimates of β are on average higher and confidence intervals are wider. Panel studies—based on regional data for a single country—find in general much lower estimated coefficients, which could be ascribed to “leakages” reflecting the fact that, at the regional level, not all beneficial spillover effects of public investment can be internalized (see Munnell, 1992).

Table 3.

Summary Descriptive Statistics for Studies Estimating the Production Elasticity of Public Capital

Source: Based on the overview of studies presented in Table 2.

^1/Studies for selected OECD countries.

^2/The average production elasticity derived from studies employing first differences drops to 0.23 if the study of Ford and Poret (1991) is excluded.

Table 3.

Summary Descriptive Statistics for Studies Estimating the Production Elasticity of Public Capital

	Time Series: National Data				Cross-Section/Panel		All Studies
	U.S.		OECD 1/2/		U.S.	OECD 1/
	Levels	First difference	Levels	First difference	Levels	First difference	Levels
Average	0.25	0.22	0.25	0.39	0.14	0.17	0.22
Number of observations	11	3	17	9	7	9	26
Standard deviation	0.11	0.06	0.10	0.17	0.06	0.13	1.2
95 percent confidence interval	[0.19, 0.31]	[0.15, 0.30]	[0.20, 0.30]	[0.28, 0.50]	[0.09,0.19]	[0.08, 0.26]	[0.18,0.27]

Source: Based on the overview of studies presented in Table 2.

^1/Studies for selected OECD countries.

^2/The average production elasticity derived from studies employing first differences drops to 0.23 if the study of Ford and Poret (1991) is excluded.

View Table

Table 3.

Summary Descriptive Statistics for Studies Estimating the Production Elasticity of Public Capital

	Time Series: National Data				Cross-Section/Panel		All Studies
	U.S.		OECD 1/2/		U.S.	OECD 1/
	Levels	First difference	Levels	First difference	Levels	First difference	Levels
Average	0.25	0.22	0.25	0.39	0.14	0.17	0.22
Number of observations	11	3	17	9	7	9	26
Standard deviation	0.11	0.06	0.10	0.17	0.06	0.13	1.2
95 percent confidence interval	[0.19, 0.31]	[0.15, 0.30]	[0.20, 0.30]	[0.28, 0.50]	[0.09,0.19]	[0.08, 0.26]	[0.18,0.27]

Source: Based on the overview of studies presented in Table 2.

^1/Studies for selected OECD countries.

^2/The average production elasticity derived from studies employing first differences drops to 0.23 if the study of Ford and Poret (1991) is excluded.

C. The Data

The empirical analysis employs annual data for Portugal over the period 1965-95. Data on GDP, the number of employed persons, the private capital stock, and the public capital stock are obtained from the Historical Series for the Portuguese economy.⁸ All series are expressed in constant prices. Estimates of the private and public capital stock are constructed by employing the perpetual inventory method (OECD, 1993). This approach computes the value of the capital stock by summing over past investments, appropriately adjusted for the rate of depreciation. It is assumed that asset lifespans are the following: residential buildings, 70 years; investments in machinery and equipment, 16 years; and public works (roads, railways, etc.), 35 years. The perpetual inventory method has been widely applied in the literature but is not free of criticism. Some authors (e.g., Sturm and de Haan, 1995) have shown that assumptions concerning the lifespans of capital goods matter for the size of the production elasticity estimate for public capital.

Table 4 shows the composition of the estimated Portuguese capital stock. The ratio of public to private capital was 18 percent for Portugal in 1995 (27 percent if only structures and equipment are included), compared with, for example, a ratio of 31 percent for the United States (58 percent if only equipment and structures are counted) with a PPP-based per capita income more than twice that of Portugal. Nearly 60 percent of public capital consists of core infrastructure such as roads, railways, airports, and the like. Investment in equipment and transport material amounts to only 9 percent of the public capital stock.

Table 4.

Composition of the Portuguese Capital Stock, 1975 and 1995

Source: Historical Scries of the Bank of Portugal.

Table 4.

Composition of the Portuguese Capital Stock, 1975 and 1995

			Percent of Total 1975	Percent of Total 1995
Total capital stock			100.0	100.0
Total private capital stock			87.2	85.1
	Equipment and transport material		25.4	29.6
	Construction		61.8	55.5
		Residential	33.1	30.7
		Nonresidential	28.7	24.8
Total public capital stock			12.8	14.9
	Equipment and transport material		0.6	1.4
	Construction		12.3	13.5
		Buildings	3.0	4.9
		Other (including core infrastructure)	9.2	8.6
Public-private capital stock ratio			0.15	0.18

Source: Historical Scries of the Bank of Portugal.

View Table

Table 4.

Composition of the Portuguese Capital Stock, 1975 and 1995

			Percent of Total 1975	Percent of Total 1995
Total capital stock			100.0	100.0
Total private capital stock			87.2	85.1
	Equipment and transport material		25.4	29.6
	Construction		61.8	55.5
		Residential	33.1	30.7
		Nonresidential	28.7	24.8
Total public capital stock			12.8	14.9
	Equipment and transport material		0.6	1.4
	Construction		12.3	13.5
		Buildings	3.0	4.9
		Other (including core infrastructure)	9.2	8.6
Public-private capital stock ratio			0.15	0.18

Source: Historical Scries of the Bank of Portugal.

D. Empirical Results for Portugal

Ordinary least squares estimates

For purposes of comparison with the literature, equations (2)-(3) are first estimated in levels using ordinary least squares.⁹ Because no time series data on capital utilization over a sufficiently long time span are available, the estimated output gap is used as a proxy. Potential output is obtained by employing a Hodrick-Prescott (1997) filter to the actual real output series. An alternative measure, the unemployment rate, u, is used as an indicator of demand pressure in factor markets.¹⁰ The estimation results for equation (2) are presented in Table 5.¹¹ Fitting an unrestricted production function yields a production elasticity of public capital significantly different from zero (using standard inference criteria), amounting to 0.19, just at the lower bound of the confidence interval derived from the results of previous studies for other countries (Table 3). This implies that a 1 percent increase in the public capital stock raises GDP by 0.19 percent. The private capital elasticity amounts to 0.37, whereas the labor elasticity is on the order of 0.67. These values are closely in line with traditional assumptions on capital and labor shares in industrialized countries but higher than the Portuguese labor share of value added. Summing over the three input coefficients yields a value of 1.2, a little above unity. To test whether production can be characterized by constant returns to scale, a Wald test on the coefficients was conducted, which indicated that the restriction of constant returns to scale cannot be rejected. Accordingly, the focus below will be on the specification that assumes constant returns to scale.

Table 5.

Unrestricted Estimates of the Production Function in Levels, 1965–95

^1/Engle-Granger (E-G) procedure which tests whether the I(1) variables in the equation are cointegrated. The Davidson and MacKinnon (1993) asymptotic critical values for cointegration are -4.10 (-4.64) for the 5 percent (and 1 percent) level, respectively.

^2/Tests whether production can be characterized by a constant returns to scale specification.

Table 5.

Unrestricted Estimates of the Production Function in Levels, 1965–95

$\ln ({Y})=\underset{(1.07)}{1.43}+\underset{(16.06)**}{0.374\ln ({K})}+\underset{(5.10)**}{0.667\ln ({L})}+\underset{(3.15)**}{0.186\ln ({G})}+\underset{(6.97)**}{0.682\ln (\text{CU})}-\underset{(-3.96)**}{0.02*D_{7585}}$
R2 ajd. = 0.998, D-W = 0.74, E-G = -2.49 1/
	F-statistic	Probability
Wald test on the restriction: α + β+γ=12/	1.52	0.229
$\ln ({Y})=\underset{(3.70)**}{0.337\ln ({K})}+\underset{(7.93)**}{0.193\ln ({G})}+\underset{(32.11)**}{0.901\ln ({L})}$
R2 ajd. = 0.992, D-W = 0.62, E-G = -2.38

^1/Engle-Granger (E-G) procedure which tests whether the I(1) variables in the equation are cointegrated. The Davidson and MacKinnon (1993) asymptotic critical values for cointegration are -4.10 (-4.64) for the 5 percent (and 1 percent) level, respectively.

^2/Tests whether production can be characterized by a constant returns to scale specification.

View Table

Table 5.

Unrestricted Estimates of the Production Function in Levels, 1965–95

$\ln ({Y})=\underset{(1.07)}{1.43}+\underset{(16.06)**}{0.374\ln ({K})}+\underset{(5.10)**}{0.667\ln ({L})}+\underset{(3.15)**}{0.186\ln ({G})}+\underset{(6.97)**}{0.682\ln (\text{CU})}-\underset{(-3.96)**}{0.02*D_{7585}}$
R2 ajd. = 0.998, D-W = 0.74, E-G = -2.49 1/
	F-statistic	Probability
Wald test on the restriction: α + β+γ=12/	1.52	0.229
$\ln ({Y})=\underset{(3.70)**}{0.337\ln ({K})}+\underset{(7.93)**}{0.193\ln ({G})}+\underset{(32.11)**}{0.901\ln ({L})}$
R2 ajd. = 0.992, D-W = 0.62, E-G = -2.38

^1/Engle-Granger (E-G) procedure which tests whether the I(1) variables in the equation are cointegrated. The Davidson and MacKinnon (1993) asymptotic critical values for cointegration are -4.10 (-4.64) for the 5 percent (and 1 percent) level, respectively.

^2/Tests whether production can be characterized by a constant returns to scale specification.

Imposing the restriction of constant returns to scale—in line with Aschauer’s (1989) specification, which serves as a useful benchmark (see equation (3))—yields slightly higher values for the output elasticity of public capital, β, ranging from 0.22–0.27 (Table 6). The estimated private capital elasticity amounts to 0.33–0.48, which seems to be a plausible range of values both in terms of capital income shares and in terms of the results obtained for other countries. Given that the public capital stock amounted to 51 percent of GDP in 1995, an estimate of the production elasticity of public capital in the range of 0.22–0.27 implies a marginal productivity of public capital of 43 to 52 percent a year. This is roughly four to five times the implicit nominal interest rate on public debt in that year. Gramlich (1994) finds even larger rates of returns (on the order of 100 percent a year or more) for the United States. These returns on public capital are very large and should be interpreted with caution.¹²

Table 6.

Estimates of the Production Function in Levels, 1965–95 ^1/

^1/t-statistics in parentheses.

^2/Coefficient of the C-O estimation procedure.

^3/Engle-Granger (E-G) test for cointegration. The number in parentheses refers to the number of lags—determined by Akaike’s Information Criterion included in the unit root test on the estimated residuals.

Table 6.

Estimates of the Production Function in Levels, 1965–95 ^1/

	I		II		III
	OLS	C-0	OLS	C-0	OLS	c-o
Constant	3.061 (79.26)	3.075 (8.46)	2.482 (17.14)	2.337 (7.69)	3.043 (113.94)	3.211 (11.33)
Ln(L/K)	0.405 (59.31)	0.395 (11.48)	0.321 (15.46)	0.314 (8.72)	0.390 (71.50)	0.391 (16.58)
Ln(G/K)	0.215 (7.73)	0.273 (4.26)	0.256 (5.35)	0.205 (1.94)	0.282 (12.45)	0.371 (5.65)
Ln(CU)	0.775 (12.21)	0.807 (13.43)			0.820 (18.51)	0.883 (15.45)
Ln(U)			-0.07 (-5.16)	-0.07 (-3.88)
D7585	-0.03 (-5.48)	-0.01 (-1.94)			-0.688 (-5.78)	-0.546 (-3.26)
D7585 ln(G/K)					-0.355 (-5.50)	-0.288 (-3.18)
ρ2/		0.876 (7.48)		0.656 (4.40)		0.787 (5.071)
R2 adj.	0.997	0.998	0.986	0.992	0.998	0.999
D-W	0.63	1.34	0.68	1.42	1.14	1.88
E-G 3/	-2.14 (0)		-2.95 (4)		-3.30 (0)

^1/t-statistics in parentheses.

^2/Coefficient of the C-O estimation procedure.

^3/Engle-Granger (E-G) test for cointegration. The number in parentheses refers to the number of lags—determined by Akaike’s Information Criterion included in the unit root test on the estimated residuals.

View Table

Table 6.

Estimates of the Production Function in Levels, 1965–95 ^1/

	I		II		III
	OLS	C-0	OLS	C-0	OLS	c-o
Constant	3.061 (79.26)	3.075 (8.46)	2.482 (17.14)	2.337 (7.69)	3.043 (113.94)	3.211 (11.33)
Ln(L/K)	0.405 (59.31)	0.395 (11.48)	0.321 (15.46)	0.314 (8.72)	0.390 (71.50)	0.391 (16.58)
Ln(G/K)	0.215 (7.73)	0.273 (4.26)	0.256 (5.35)	0.205 (1.94)	0.282 (12.45)	0.371 (5.65)
Ln(CU)	0.775 (12.21)	0.807 (13.43)			0.820 (18.51)	0.883 (15.45)
Ln(U)			-0.07 (-5.16)	-0.07 (-3.88)
D7585	-0.03 (-5.48)	-0.01 (-1.94)			-0.688 (-5.78)	-0.546 (-3.26)
D7585 ln(G/K)					-0.355 (-5.50)	-0.288 (-3.18)
ρ2/		0.876 (7.48)		0.656 (4.40)		0.787 (5.071)
R2 adj.	0.997	0.998	0.986	0.992	0.998	0.999
D-W	0.63	1.34	0.68	1.42	1.14	1.88
E-G 3/	-2.14 (0)		-2.95 (4)		-3.30 (0)

^1/t-statistics in parentheses.

^2/Coefficient of the C-O estimation procedure.

^3/Engle-Granger (E-G) test for cointegration. The number in parentheses refers to the number of lags—determined by Akaike’s Information Criterion included in the unit root test on the estimated residuals.

Estimating an equation that includes the unemployment rate (rather than capital utilization) as an indicator of demand pressure in factor markets yields a slightly larger β coefficient (specification II in Table 6). Specification III includes an interaction dummy variable for the period 1975-85 to capture possible productivity effects on public capital during the period when the role of the state was greatly expanded. The negative interaction coefficient is large and suggests that public investment actually contributed to reducing output in these years.¹³ In addition, the negative value of the “intercept” dummy (as observed in all the specifications) indicates that during this time period of accelerated public investment, total factor productivity growth was also lower.¹⁴ Because of the presence of autocorrelation in the error terms, the equations are also estimated using the Cochrane-Orcutt (C-O) procedure, which changes the coefficient on public capital marginally.

Table 7 presents results for various components of public capital: core construction projects, gc, public buildings, gb, and transport equipment and machinery, ge. The equation including all three components shows that none of the coefficients on public capital are significant and two of them (core infrastructure and buildings) have the wrong sign. In addition, the coefficient on employment becomes implausibly large. That the coefficient on each component is insignificant while, as shown above and in Table 7, the coefficient on aggregate public capital is significant, suggests that there maybe important interrelationships between the different components of public capital. Statistically, the insignificant coefficients on public capital also reflect multicollinearity between the various components. Nevertheless, to obtain some idea about differences in productivity of various types of public capital, the equations are also estimated for each component separately. The coefficient on core infrastructure is now significant and close to the estimate found for the aggregate capital stock. The growth effect of transport material and equipment is also statistically significant, but substantially smaller than core infrastructure (i.e., β=0.10 compared with β=0.18 in the benchmark case). Government buildings and equipment do not appear to play a significant role in explaining capital productivity, which is confirmed by the statistically insignificant coefficient. These results should be interpreted with care, given the poor results that were obtained when all three components of public capital were included together.

Table 7.

Disaggregated Estimates of the Production Function ^1/

^1/t-statistics in parentheses.

^2/Asterisks denote significance at the 1 percent level.

Table 7.

Disaggregated Estimates of the Production Function ^1/

	All Three Components	Core Infrastructure	Buildings	Transport Material and Equipment
Constant	0.440 (0.11)	0.344 (1.25)	-1.070 (-0.39)	2.535 (1.60)
Ln(K)	0.506 (6.15)	0.403 (18.85)	0.367 (3.54)	0.374 (15.01)
Ln(GC)	-0.018 (-0.06)	0.185 (4.71)
Ln(GB)	-0.147 (-1.61)		0.078 (0.80)
Ln(GE)	0.139 (0.75)			0.104 (4.63)
Ln(L)	0.930 (3.76)	0.751 (3.53)	1.195 (3.61)	0.720 (3.28)
Ln(CU)	0.468 (2.92)	0.651 (6.55)	0.462 (2.77)	0.608 (6.41)
D7585	-0.03 (-3.62)	-0.032 (6.55)	-0.029 (-2.66)	-0.029 (-3.70)
R2 adj.	0.999	0.998	0.996	0.998
D-W	1.04	0.78	0.87	0.72
E-G 2/	-2.95	-2.65	-4.00**	-2.46

^1/t-statistics in parentheses.

^2/Asterisks denote significance at the 1 percent level.

View Table

Table 7.

Disaggregated Estimates of the Production Function ^1/

	All Three Components	Core Infrastructure	Buildings	Transport Material and Equipment
Constant	0.440 (0.11)	0.344 (1.25)	-1.070 (-0.39)	2.535 (1.60)
Ln(K)	0.506 (6.15)	0.403 (18.85)	0.367 (3.54)	0.374 (15.01)
Ln(GC)	-0.018 (-0.06)	0.185 (4.71)
Ln(GB)	-0.147 (-1.61)		0.078 (0.80)
Ln(GE)	0.139 (0.75)			0.104 (4.63)
Ln(L)	0.930 (3.76)	0.751 (3.53)	1.195 (3.61)	0.720 (3.28)
Ln(CU)	0.468 (2.92)	0.651 (6.55)	0.462 (2.77)	0.608 (6.41)
D7585	-0.03 (-3.62)	-0.032 (6.55)	-0.029 (-2.66)	-0.029 (-3.70)
R2 adj.	0.999	0.998	0.996	0.998
D-W	1.04	0.78	0.87	0.72
E-G 2/	-2.95	-2.65	-4.00**	-2.46

^1/t-statistics in parentheses.

^2/Asterisks denote significance at the 1 percent level.

Integration and cointegration issues

So far, the issue of stationarity of the variables has not been addressed. If variables are nonstationary, the usual test statistics have nonstandard distributions, implying that the use of standard inference tests may give rise to seriously misleading inferences. A related problem is the possibility of finding spurious relationships between variables. Only when variables are cointegrated—expressing the presence of a long-run equilibrium relationship between a group of nonstationary economic time series—can the equations be estimated in levels. The first step is to determine the order of integration of the variables. To this end, tests for unit roots are performed on the levels, first differences, and second differences of the variables. The results of the augmented Dickey-Fuller (1981) tests are presented in Table 8. The evidence suggests that the variables g, k, l, and y are all integrated of order one (i.e., they are I(1) variables) and thus nonstationary in levels, but stationary in first differences. The public capital stock appears to be 1(2) but its coefficient is very far from unity (i.e., -0.35), indicating that it could be an I(1) variable. Most of the variables expressed in terms of the private capital stock are stationary in levels, except the labor-capital ratio, which seems to be integrated of order two.

Table 8.

Augmented Dickey-Fuller Tests for Nonstationarity ^1/

^1/The tests are conducted with a constant, $, included in the following equation: $\Delta y_t=\varphi +\rho y_{t-1}+{\displaystyle \sum _{s=1}^n{{\Omega }}_s{\Delta }{y}_{t-s}+{\epsilon }_t}$, where y_t is the relevant time series, ε_t is an i.i.d. sequence of random variables.

^2/The number of autoregressive lags is chosen so as to minimize Akaikc’s (1969) Information Criterium. The null hypothesis is that the variable under investigation has a unit root (i.e., ρ= 1) against the alternative that it does not. A value of the augmented Dickey-Fuller (ADF) statistic exceeding the critical value for the specific lag length leads to a rejection of the null hypothesis.

* Significant at the 5 percent level.

^**Significant at the 1 percent level.

Table 8.

Augmented Dickey-Fuller Tests for Nonstationarity ^1/

variable	ADF	Lags 2/	variable	ADF	Lags	Variable	ADF	lags
ln(Y/K)	-4.68**	7	Δln(Y/K)			Δ2 ln(Y/K)
ln(L/K)	-2.59	1	Δln(L/K)	-1.70	0	Δ2 ln(L/K)	-7.28**	0
ln(G/K)	-3.97**	5	Δln(G/K)			Δ2 ln(G/K)
ln(GC/K)	-4.73**	6	Δln(GC/K)			Δ2 ln(GC/K)
ln(GB/K)	-0.25	7	Δln(GB/K)	-2.79	5	Δ2 ln(GB/K)	-7.03**	0
ln(GE/K)	-0.69	1	Δln(GE/K)	-2.15	1	Δ2 ln(GE/K)	-4.05**	0
ln(Y)	-1.96	1	Δln(Y)	-4.00**	2	Δ2 ln (Y)
ln(K)	-2.60	2	Δln(K)	-3.79**	7	Δ2 ln(K)
ln(L)	-0.74	1	Δln(L)	-3.72**	0	Δ2 ln(L)
Ln(G)	-0.47	2	Δln(G)	-2.66	1	Δ2 ln(G)	-4.39**	0
Ln(GC)	0.25	1	Δln(GC)	-2.09	0	Δ2 ln(GC)	-5.18**	0
Ln(GB)	-2.61	1	Δln(GB)	-1.93	4	Δ2 ln(GB)	-7.44**	0
Ln(GE)	-0.65	3	Δln(GE)	-3.84**	2	Δ2 ln(GE)
Ln(CU)	-1.37	1	Δln(CU)	-3.76**	0	Δ2 ln(CU)
Ln(u)	-5.51**	2	Δln(u)			Δ2 ln(u)

^1/The tests are conducted with a constant, $, included in the following equation: $\Delta y_t=\varphi +\rho y_{t-1}+{\displaystyle \sum _{s=1}^n{{\Omega }}_s{\Delta }{y}_{t-s}+{\epsilon }_t}$, where y_t is the relevant time series, ε_t is an i.i.d. sequence of random variables.

^2/The number of autoregressive lags is chosen so as to minimize Akaikc’s (1969) Information Criterium. The null hypothesis is that the variable under investigation has a unit root (i.e., ρ= 1) against the alternative that it does not. A value of the augmented Dickey-Fuller (ADF) statistic exceeding the critical value for the specific lag length leads to a rejection of the null hypothesis.

* Significant at the 5 percent level.

^**Significant at the 1 percent level.

View Table

Table 8.

Augmented Dickey-Fuller Tests for Nonstationarity ^1/

variable	ADF	Lags 2/	variable	ADF	Lags	Variable	ADF	lags
ln(Y/K)	-4.68**	7	Δln(Y/K)			Δ2 ln(Y/K)
ln(L/K)	-2.59	1	Δln(L/K)	-1.70	0	Δ2 ln(L/K)	-7.28**	0
ln(G/K)	-3.97**	5	Δln(G/K)			Δ2 ln(G/K)
ln(GC/K)	-4.73**	6	Δln(GC/K)			Δ2 ln(GC/K)
ln(GB/K)	-0.25	7	Δln(GB/K)	-2.79	5	Δ2 ln(GB/K)	-7.03**	0
ln(GE/K)	-0.69	1	Δln(GE/K)	-2.15	1	Δ2 ln(GE/K)	-4.05**	0
ln(Y)	-1.96	1	Δln(Y)	-4.00**	2	Δ2 ln (Y)
ln(K)	-2.60	2	Δln(K)	-3.79**	7	Δ2 ln(K)
ln(L)	-0.74	1	Δln(L)	-3.72**	0	Δ2 ln(L)
Ln(G)	-0.47	2	Δln(G)	-2.66	1	Δ2 ln(G)	-4.39**	0
Ln(GC)	0.25	1	Δln(GC)	-2.09	0	Δ2 ln(GC)	-5.18**	0
Ln(GB)	-2.61	1	Δln(GB)	-1.93	4	Δ2 ln(GB)	-7.44**	0
Ln(GE)	-0.65	3	Δln(GE)	-3.84**	2	Δ2 ln(GE)
Ln(CU)	-1.37	1	Δln(CU)	-3.76**	0	Δ2 ln(CU)
Ln(u)	-5.51**	2	Δln(u)			Δ2 ln(u)

^1/The tests are conducted with a constant, $, included in the following equation: $\Delta y_t=\varphi +\rho y_{t-1}+{\displaystyle \sum _{s=1}^n{{\Omega }}_s{\Delta }{y}_{t-s}+{\epsilon }_t}$, where y_t is the relevant time series, ε_t is an i.i.d. sequence of random variables.

^2/The number of autoregressive lags is chosen so as to minimize Akaikc’s (1969) Information Criterium. The null hypothesis is that the variable under investigation has a unit root (i.e., ρ= 1) against the alternative that it does not. A value of the augmented Dickey-Fuller (ADF) statistic exceeding the critical value for the specific lag length leads to a rejection of the null hypothesis.

* Significant at the 5 percent level.

^**Significant at the 1 percent level.

Table 9.

Estimates of the Production Function in First Differences, 1965–95 ^1/

^1/t-statistics in parentheses below the coefficients.

^2/Engle-Granger (E-G) test for cointegration. The number in parenthesis is the number of lags included in the unit root test on the estimated residuals.

Table 9.

Estimates of the Production Function in First Differences, 1965–95 ^1/

	Total Stock		Core Infrastructure		Buildings		Equipment and Transport Material
Constant	0.010 (1.03)	0.000 (0.13)	0.007 (1.68)	0.006 (1.34)	0.013 (3.43)	0.014 (3.41)	0.011 (1.70)	0.004 (0.72)
Δln(L/K)	0.481 (4.51)	0.347 (3.14)	0.489 (4.81)	0.449 (4.22)	0.673 (13.54)	0.687 (12.85)	0.639 (7.74)	0.537 (6.40)
Δln(G/K)	0.199 (2.03)	0.371 (3.30)
Δln(GC/K)			0.175 (2.05)	0.216 (2.35)
Δln(GB/K)					0.022 (0.36)	0.035 (0.56)
Δln(GE/K)							0.034 (0.54)	0.188 (2.33)
Interaction Dummy		-0.415 (-2.54)		-0.184 (-1.16)		-0.081 (-0.75)		-0.185 (-2.67)
Δln(CU)	0.825 (14.35)	0.893 (15.18)	0.829 (14.26)	0.854 (13.83)	0.770 (14.11)	0.773 (14.01)	0.762 (13.82)	0.770 (15.45)
R2 adj.	0.947	0.954	0.944	0.945	0.936	0.935	0.943	0.948
D-W	1.51	1.50	1.50	1.46	2.11	2.25	2.00	2.04
E-G 2/	-4.22 (0)		-4.19 (0)		-4.63 (1)		-5.43 (0)

^1/t-statistics in parentheses below the coefficients.

^2/Engle-Granger (E-G) test for cointegration. The number in parenthesis is the number of lags included in the unit root test on the estimated residuals.

View Table

Table 9.

Estimates of the Production Function in First Differences, 1965–95 ^1/

	Total Stock		Core Infrastructure		Buildings		Equipment and Transport Material
Constant	0.010 (1.03)	0.000 (0.13)	0.007 (1.68)	0.006 (1.34)	0.013 (3.43)	0.014 (3.41)	0.011 (1.70)	0.004 (0.72)
Δln(L/K)	0.481 (4.51)	0.347 (3.14)	0.489 (4.81)	0.449 (4.22)	0.673 (13.54)	0.687 (12.85)	0.639 (7.74)	0.537 (6.40)
Δln(G/K)	0.199 (2.03)	0.371 (3.30)
Δln(GC/K)			0.175 (2.05)	0.216 (2.35)
Δln(GB/K)					0.022 (0.36)	0.035 (0.56)
Δln(GE/K)							0.034 (0.54)	0.188 (2.33)
Interaction Dummy		-0.415 (-2.54)		-0.184 (-1.16)		-0.081 (-0.75)		-0.185 (-2.67)
Δln(CU)	0.825 (14.35)	0.893 (15.18)	0.829 (14.26)	0.854 (13.83)	0.770 (14.11)	0.773 (14.01)	0.762 (13.82)	0.770 (15.45)
R2 adj.	0.947	0.954	0.944	0.945	0.936	0.935	0.943	0.948
D-W	1.51	1.50	1.50	1.46	2.11	2.25	2.00	2.04
E-G 2/	-4.22 (0)		-4.19 (0)		-4.63 (1)		-5.43 (0)

^1/t-statistics in parentheses below the coefficients.

^2/Engle-Granger (E-G) test for cointegration. The number in parenthesis is the number of lags included in the unit root test on the estimated residuals.

The second step is to examine whether the series of the variables used in the estimation of the production function are cointegrated. To this end the Engle-Granger (1987) procedure can be employed, which consists of a unit root test on the residuals of the estimated equation.¹⁵ The Engle-Granger statistics (see the bottom row in Tables 5 and 6) indicate that the null hypothesis of no cointegration cannot be rejected at the 95 percent level of confidence.¹⁶ From the literature it is well known that the Engle-Granger procedure has low power in finite samples¹⁷ and may therefore be unable to detect cointegration when it is present in the data (see Kremers, Ericsson, and Dolado, 1992).

As an alternative to the single equation Engle-Granger test, the Johansen (1988) procedure—based on estimating a VAR—can be employed to test for cointegration. The following equation is estimated:

$\begin{array}{ccc}{\Delta }{x}_{{t}}={{\alpha }}^{*}{{\beta }}^{\prime }{x}_{t-1}+{\displaystyle \sum _{i=1}^{k-1}{{\Gamma }}_i{\Delta }{x}_{t-i}+{\in }_p}& & (5)\end{array}$

where x_t is a vector of variables, α^* is an adjustment coefficient,β′ is a long-run elasticity (or set of eigenvectors), and ε_t is an error term. The maximum likelihood test statistics (i.e., the maximum eigenvalue statistic, λ_max, and the trace statistic, λ_trace, both adjusted for the degrees of freedom) for the system of equations (5) are reported in Table 10. In contrast to the Engle-Granger results, both test statistics strongly reject the null hypothesis of no cointegration in favor of at least one cointegrating vector.¹⁸ There is little evidence of more than one cointegrating relationship.

Table 10.

Johansen’s Cointegration Analysis of Portuguese Production ^1/

^1/The VAR includes two lags on each variable and is estimated over the period 1967–95.

^2/The statistics λ_max and λ_trace are the maximum eigenvalue and trace eigenvalue statistics (${\lambda }_{\max }^a\text{and}{\lambda }_{trace}^a$ adjust for ihe degrees of freedom) for testing cointegration in the Johansen procedure. The null hypothesis is defined in terms of the cointegration rank, r. The critical values are taken from Osterwaid and Lenum (1992): (**) denotes significance at the 1 percent level whereas (*) indicates significance at the 5 percent level.

Table 10.

Johansen’s Cointegration Analysis of Portuguese Production ^1/

	Null hypothesis for test statistics
Statistic 2/	r = 0	r ≤ 1	r ≤ 2	r ≤ 3
Eigenvalue	0.727	0.477	0.362	0.124
λmax	37.6**	18.2	13.0	3.9*
${\lambda }_{\max }^a$	27.3*	13.6	9.4	2.8
95 percent critical value	27.1	21.0	14.1	3.8
λtrace	73.3**	35.7**	16.9*	3.9*
${\lambda }_{\mathrm{trace}}^a$	53.1*	25.8	12.2	2.8
95 percent critical value	47.2	29.7	15.4	3.8

^1/The VAR includes two lags on each variable and is estimated over the period 1967–95.

^2/The statistics λ_max and λ_trace are the maximum eigenvalue and trace eigenvalue statistics (${\lambda }_{\max }^a\text{and}{\lambda }_{trace}^a$ adjust for ihe degrees of freedom) for testing cointegration in the Johansen procedure. The null hypothesis is defined in terms of the cointegration rank, r. The critical values are taken from Osterwaid and Lenum (1992): (**) denotes significance at the 1 percent level whereas (*) indicates significance at the 5 percent level.

View Table

Table 10.

Johansen’s Cointegration Analysis of Portuguese Production ^1/

	Null hypothesis for test statistics
Statistic 2/	r = 0	r ≤ 1	r ≤ 2	r ≤ 3
Eigenvalue	0.727	0.477	0.362	0.124
λmax	37.6**	18.2	13.0	3.9*
${\lambda }_{\max }^a$	27.3*	13.6	9.4	2.8
95 percent critical value	27.1	21.0	14.1	3.8
λtrace	73.3**	35.7**	16.9*	3.9*
${\lambda }_{\mathrm{trace}}^a$	53.1*	25.8	12.2	2.8
95 percent critical value	47.2	29.7	15.4	3.8

^1/The VAR includes two lags on each variable and is estimated over the period 1967–95.

^2/The statistics λ_max and λ_trace are the maximum eigenvalue and trace eigenvalue statistics (${\lambda }_{\max }^a\text{and}{\lambda }_{trace}^a$ adjust for ihe degrees of freedom) for testing cointegration in the Johansen procedure. The null hypothesis is defined in terms of the cointegration rank, r. The critical values are taken from Osterwaid and Lenum (1992): (**) denotes significance at the 1 percent level whereas (*) indicates significance at the 5 percent level.

In sum, the evidence on cointegration is mixed. The Engle-Granger results point to a lack of cointegration, which would invalidate the earlier ordinary least squares results. However, the Johansen test (generally recognized to be superior to Engle-Granger tests) indicates cointegration. In light of these results, the ordinary least squares estimates should be interpreted with caution, and carefully compared with those obtained by other techniques.¹⁹