2.1 The Svedberg critique, unobserved, country-specific heterogeneity, and random measurement error

2.2 Sensitivity to data structure

The recent empirical literature on the determinants of growth is replete with new and original explanations that seek to explain observed differentials in growth performance. Unfortunately, much of this literature is far from being robust, as has been made clear by Levine and Renelt (1992), Sala-i-Martin (1996), and Arcand and Dagenais (1999). Thus, it is important to be cautious regarding results such as those presented in Tables 1 and 2 until it has been established that they are robust to a certain number of standard concerns.

2.1 The Svedberg critique, unobserved, country-specific heterogeneity, and random measurement error

In a recent article, Svedberg (1999) has criticized the procedure followed by the FAO in constructing both measures of the DES per caput and the PFI. Svedberg points out that there are three potential sources of error. First, the DES per caput itself may be measured with error. Second, the parameters of the lognormal distribution that food consumption is assumed to follow (in essence, the coefficient of variation of the distribution, given that its mean is given by the average DES per caput) is assumed to be measured with substantial error (as noted by Svedberg, the lognormal distribution may be an inappropriate assumption in and of itself). Third, the calorie cut-off-point, which defines the threshold below which individuals are assumed to be undernourished may be country specific and it may therefore be inappropriate to assume a common cutoff point of 1.55 or 1.56 the Basic Metabolic Rate (BMR).

In econometric terms there may therefore be two sources of error that plague the nutritional variables included in the regressions presented in Tables 1 and 2. Let the "true" nutritional variable be denoted by X_it, where i indexes the country and t indexes the time period (decade). Then the points raised by Svedberg imply that the observed value of the nutritional variable, denoted by , is equal to :

(13) ,

where is a composite error term. If the only source of measurement error is given by the country-specific term , then use of the within estimator (country-specific fixed effects) will potentially eliminate all sources of bias and the coefficient associated with the nutritional variable may be estimated consistently. It is likely that this source of error is the dominant one in the PFI estimates given that the parameters of the lognormal distribution used to compute the proportion of the population that is undernourished is assumed to be the same over the three decades for which the figure is reported in the World Food Survey (1996). Moreover, the calorie cutoff point, alluded to above, may also be taken to be a country-specific datum.

If, in addition to the country-specific effect, there is also a country-time specific random shock , then the parameter estimates presented in Tables 1 and 2 are subject to a standard error in variables problem and instrumental variable estimation is called for in order to purge the observed value of the nutritional variable from stochastic elements that will be correlated with the growth regression's disturbance term.

As noted previously, solving the first problem is relatively straightforward, and results corresponding to panel estimation with country-specific fixed effects are presented in the first six columns of Table 3 (see Islam, 1996 for the standard results using panel data in the context of the neo-classical model). As can be seen from the estimates, the coefficients associated with the nutritional variables in columns 3 and 5 (the linear specifications) are robust to the "within" procedure, although the estimated standard error in the case of the PFI is slightly larger than in the benchmark results presented in Table 1. For the DES per caput, on the other hand, the coefficient increases substantially, from 1.38 x10^-5 in column (7) of Table 1 to 2.20 x10^-5 in column (5) of Table 3. The associated t-statistic remains approximately the same. In terms of country-specific errors in the measurement of the nutritional variables, our results regarding the impact of nutritional concerns are therefore robust to the Svedberg critique.

Several other remarks are in order concerning the "within" estimator results. First, the absolute value of the coefficient associated with the logarithm of initial GDP per caput are much larger (by a factor of ten) once one controls for country-specific fixed effects, and the point estimates are much more precise. This implies that the estimated annual rate of convergence is much greater once country-specific fixed effects are taken into account. Second, the specification in which the PFI enters as a polynomial of degree three is not robust to the inclusion of country-specific fixed effects. The same can be said of the quadratic specification in terms of the DES per caput. Third, a specification involving country-specific random effects is strongly rejected in favour of fixed effects on the basis of the usual Hausman test; in all cases the p-value associated with this test statistic is extremely small.

In contrast to unobserved country-specific heterogeneity, dealing with the second source of bias -random measurement error-is notoriously difficult in the context of growth regressions. This is because, generally speaking, it is extremely problematic to come by admissible exogenous instruments for variables included in the growth regression that should not, in and of themselves, already be included in the equation. One option that is frequently pursued in the literature is to use the initial level of GDP per caput, lagged two periods (two decades, here, see Baltagi, 1996, for the standard treatment). The procedure that is pursued here, in order to eliminate both the country-specific error term and random measurement error involves

1. first-differencing (in order to eliminate ) followed by
2. application of a Generalized Method of Moments (GMM) estimator (in order to eliminate the correlation between the explanatory variables and the equation's disturbance term because of the presence of ).⁸

More formally, write the basic growth regression as

(14)

where Y_it is the growth rate of GDP per caput of country i at time t,

Z_it is the vector of control variables,
is the nutritional variable included in the regression (which is assumed to be measured with error),
is the constant term. and
is a vector of parameters associated with the control variables.

The parameter that one wishes to estimate consistently is, of course, . In the present context, there are three time periods, corresponding to the three decades for which data are available (1960s, 1970s, 1980s). First differencing the 1980s and the 1970s yields:

(15) ,

where it should be obvious that has been eliminated. There remains the correlation between and . In order to correct for this standard error in variables problem, GMM estimation is applied to equation (15), where the instruments are the variables themselves lagged two periods. That is, the instruments are given by the matrix .

TABLE 4
PFI, DES per caput, and economic growth
Sensitivity to data structure : GMM estimation with first-differencing
(t-statistics below coefficients)

Note : first difference of observations for the 1980s (t) minus the observations for the 1970s (t - 1); columns (1) to (4) are estimated by non-linear least squares in order to recover a direct estimate of the annual rate of convergence ; in the GMM estimates presented in columns (5) to (8), instruments are the variables themselves for the 1960s (t - 2), plus the growth rate of GDP per caput for the 1960s; hence, the GMM equations are all overidentified with degree of freedom equal to one.
Data source : same as Table 1.

Of course, a crucial identifying assumption is also that there is no serial correlation in the error term v_it . The consistency of the resulting GMM estimator is ensured by the standard result due to Hansen (1982). This yields a system that is overidentified with degree of freedom equal to one. The overidentifying restriction can then be tested, using the usual Sargan test in order to assess the validity of the proposed set of instruments.

Results are presented in columns (5) to (8) of Table 4. Keeping in mind that one decade of observations (the 1960s) has been lost due to their use as instruments, it is noteworthy that the estimated impact of the DES per caput (column (6)) increases with respect to the within estimator results presented in column (5) of Table 3 by a factor of seven. The same is true when one considers the results in terms of the PFI (column (8) of Table 4; c.f. column (3) of Table 3), although the coefficient "only" increases by a factor of five. As one would expect, these results suggest that an error in variables problem biases the coefficients presented in Table 3 downwards. In order to isolate the impact of the random error term from that of , in terms that are strictly comparable, the first 4 columns of Table 4 present results corresponding to first-differencing without resort to GMM estimation. For the linear specification in terms of the DES, accounting for the presence of through the GMM procedure increases the marginal impact of the nutritional variable by a factor of 4 compared with first-differencing alone (column (2) versus column (6)). For the linear specification in terms of the PFI (columns (4) and (8), the corresponding figure is 6. None of the specifications including nutritional variables (columns (6), (7) and (8)) are rejected by the test of the overidentifying restriction. Note also that the quadratic specification in terms of the DES per caput is not robust to first-differencing (column (3)) or, for that matter, to first differencing followed by the GMM procedure (column (7)).

Another result of interest involves the estimated annual rate of convergence. Here, it is found that the estimated value of (see equation (12) above) is extremely low when compared to the results obtained in the case of the specification based on the Solow model (Table 2) or our benchmark specification (Table 1). This is in sharp contrast to the results presented by Caselli, Esquivel and Lefort (1996) who find that the econometric procedure based on first-differencing and GMM estimation yields estimated annual convergence rates that are greater by a factor of ten than those usually reported in the literature (the commonly accepted figure being in the order of 2 percent per year). This difference in results despite the similarity in estimation procedure may be due to the different time-frame considered, since Caselli, Esquivel and Lefort base their results on data spanning five-year periods and, in the context of the neo-classical model, it is often argued that such a time-span is inappropriate.

The upshot is that while Svedberg's (1999) critique of FAO data may indeed be well taken when it comes to assessing the prevalence of food inadequacy or dietary energy supply, his concerns do not translate into the disappearance of the statistical significance of the impact of the DES per caput or the PFI on growth. On the contrary, as one would expect on the basis of received econometric theory, controlling for country-specific fixed effects through first-differencing and potential errors in variables through GMM estimation strengthens and amplifies the impact of nutrition on economic growth.

2.2 Sensitivity to data structure

It is possible that the effect of nutritional variables on growth is purely a medium-term phenomenon that is discernible only when high frequency data ("high" in terms of a growth regressions, that is) is used. That this is probably not the case is already suggested by the results presented in Table 2 in terms of the Solow model with data covering a 30 year period. A further manner of assessing this is to compute the "between" estimates, that is, to apply OLS to country averages using our pooled country-decade dataset.

In Table 5, are presented results of re-estimating the basic specification using the between estimator. In this context, the coefficient associated with the PFI (column (1)) remains statistically significant and of the same order of magnitude as in the pooling results presented in Table 1. Inter-decade variations in the PFI are therefore not the source of the statistically significant coefficient associated with the PFI in the growth regressions. The cubic specification (column (2)) is not robust to the "between" procedure, and is therefore present only in our pooling results. Much the same obtains in terms of the DES per caput, although the quadratic specification, which was not robust to the inclusion of country-specific fixed effects, remains robust to the "between" procedure. This points to the conclusion that the threshold effect of the DES per caput on growth is a phenomenon that arises because of country-specific characteristics that are not reflected in our explanatory variables. The Africa dummy variable, for its part, also remains significant, as with the results presented in Table 1.

A second manner of assessing the robustness of our results is to change dataset entirely. It has already been shown that our results hold up in the context of panel estimation where each observation corresponds to a country-decade, as well as when one considers a long-run relationship implied by the Solow model (based on the Levine and Renelt, 1992, dataset). An additional test of robustness is provided by examining the impact of nutritional concerns using the empirical framework proposed by Sachs and Warner (1997a, 1997b) who claim to have identified all of those factors that are fundamental in explaining long run growth. The reader is referred to the original contributions by Sachs and Warner for the justifications behind the inclusion of the various explanatory variables.

TABLE 5
PFI, DES per caput, and economic growth
Sensitivity to data structure (panel data estimation: "between estimator")
(t-statistics below coefficients)

Dependent variable : growth rate of per caput GDP (1960-1990)	(1)	(2)	(3)	(4)
Intercept	-0.090	-1.998	-0.017	-0.178
	-2.690	-0.763	-0.752	-3.681
Africa dummy	-0.013	-0.010	-0.014	-0.014
	-2.706	-1.784	-3.143	-3.156
Latin America dummy	-0.016	-0.011	-0.014	-0.016
	-3.902	-2.296	-3.020	-3.736
Log of initial GDP (by decade)	-0.003	-0.007	-0.001	-0.003
	-1.131	-2.065	-0.345	-0.797
Estimated annual rate of convergence	3.3E-04	7.0E-04	1.6E-04	3.1E-04
	1.266	2.132	0.487	0.952
Nutritional variables
100 - PFI (%)	0.001	0.075
	3.441	0.809
(100 - PFI (%))2		-0.0009
		-0.853
(100 - PFI (%))3		3.81E-06
		0.910
DES per caput (kcal/day)			1.17E-05	1.44E-04
			1.802	3.985
DES per caput, squared				-2.40E-08
				-3.7143
Mean of dependent variable	0.019	0.019	0.018	0.018
Adjusted R2	0.362	0.373	0.332	0.407
	0.016	0.016	0.016	0.015
Number of observations	110	110	108	108

Note : standard errors are White heteroskedasticity-consistent; threshold value of DES per caput in quadratic specification (column (4)) : 3000 kcal/day. Data source : same as Table 1.

Results are presented in Table 6, and they confirm our previous findings. First, the impact of the DES per caput (see column (2) of Table 6) on the growth rate of GDP per caput is of the same order of magnitude as in our benchmark results presented in Table 1 (note that, in Table 1, the dependent variable is expressed in decimal terms, i.e., a growth rate of 2 percent is expressed as 0.02, whereas, in the Sachs-Warner dataset, the same growth rate appears as 2), although it is only marginally significant. Second, the PFI, when entered in logistic form, is highly significant (corresponding t-statistic of 4.592). If denotes the coefficient associated with the PFI, the logistic specification implies that the marginal impact of an increase of the PFI on the growth rate of per caput GDP is equal to:

(16) .

Estimated at the sample mean (PFI = 5.72%), this implies that a decrease of 1 percent in the PFI increases the growth rate of GDP per caput by 0.014 percentage points. For countries with a PFI of one percent, the corresponding figure for the total elimination of food inadequacy is 0.85 percentage points of additional growth. Needless to say, the latter is an exceedingly large number, especially since the mean value of the growth rate of GDP per caput in the dataset is equal to 1.81 percent, and it reflects the functional assumption that the marginal gains in terms of growth stemming from reductions in the PFI increase as the PFI decreases.

TABLE 6
PFI, DES per caput, and economic growth
Sensitivity to data structure : Sachs-Warner (1997) dataset
(t-statistics below coefficients)

Dependent variable : growth rate of per caput GDP (1960-1990)	(1)	(2)	(3)
Intercept	-68.470	-77.135	-101.058
	-1.748	-1.937	-2.840
Log of real GDP per economically active inhabitant in 1965	-1.523	-1.565	-1.518
	-6.421	-6.491	-7.222
Openness times log GDP per ea. in 1965	-1.105	-1.070	-1.086
	-3.095	-2.949	-3.421
Openness to international trade (share of years open 1965-90)	11.102	10.767	10.553
	3.771	3.607	4.032
Landlocked dummy variable	-0.607	-0.579	-0.506
	-2.405	-2.290	-2.262
Log life expectancy circa 1970	37.986	42.303	55.413
	1.900	2.082	3.047
Square of log life expectancy	-4.419	-5.041	-6.780
	-1.729	-1.938	-2.907
Central government savings, 1970-90	0.114	0.113	0.115
	5.124	5.021	5.854
Dummy for tropical climate	-0.875	-0.665	-0.180
	-2.996	-1.994	-0.581
Institutional quality index	0.319	0.306	0.241
	3.837	3.506	3.075
Natural resource exports / GDP 1970	-4.022	-4.000	-3.454
	-4.040	-3.989	-3.886
Growth in ea. pop - pop growth	0.945	0.977	1.018
	2.621	2.707	3.196
Nutritional variables
DES per caput (kcal/day)		0.001
		1.717
1/(1+exp{PFI})			4.370
			4.592
Mean of dep. var.	1.798	1.791	1.810
Adjusted R2	0.837	0.842	0.874
	0.774	0.773	0.683
Number of observations	82	79	81

Note: Sachs and Warner (1997) dataset currently available at http://www.ksg.harvard.edu/cid/cid_public_data.htm.
See aforementioned http for details concerning the sources of the data and the definition of the variables. Method of estimation : OLS.

An interesting corollary to these results is that the dummy variable associated with a tropical climate becomes statistically insignificant. This suggests that the tropical climate dummy, in Sachs and Warner's results, is simply proxying for climatically-determined differences in nutritional requirements. Moreover, the coefficients associated with the life expectancy variables become both larger and more significant, indicating potentially interesting interaction effects between nutrition and life expectancy which will be explored at greater length below.

While the empirical analysis presented in Tables 1 to 6 highlights that nutritional status does appear to affect the growth rate of GDP per caput and the simple theoretical models proposed show that it is relatively straightforward to include malnutrition within the context of the usual theoretical constructs, it does not tell us how this impact obtains. In other words, the interesting question is the following : what is the mechanism through which nutritional status affects economic growth ? In what follows, those mechanisms are examined that provide the most likely explanations for the correlations presented above and subject them to formal econometric analysis motivated by solid theoretical foundations.

8 This estimation strategy was first proposed by Arellano and Bond (1991). See Caselli, Esquivel and Lefort (1996) for an application to the growth regressions first presented by Barro and Lee (1994).