What Can Be Deduced From The Goodness Of The Fits

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A professional statistician, seeing our results, is likely to respond with some skepticism, at least at first. We have acknowledged already that the fact that the OLS correlation coefficients are extremely high does not mean that the underlying model is 'correct'. In the first place, it is well-known that with enough free parameters one can model nearly anything, including the traditional elephant. However, our close fits over quite long periods are achieved with very few parameters. In fact, our most lavish use of parameters was to create the rather ad hoc 'bridge' between the two historical periods (pre- and post-World War II). Most economists will probably agree that the second World War constituted a major shock or 'break' that justifies re-calibration of the production function. The real question might be whether other significant breaks may have occurred, such as the Korean War, the Viet Nam War, the 'oil shock' in 1973-4 and so on.

Granted we have not utilized many free parameters, the skeptical statistician will note that a very close correlation between two (or more) variables need not mean that there is a causal connection between them. The close correlation between any two variables, such as exergy consumption and GDP, might be attributed, in principle, to some third independent variable driving them both. However, it is difficult to imagine what such a meta-driver might be (population? migration? education?). This difficulty becomes more acute when there are four variables to be explained simultaneously by a fifth variable that we cannot identify a priori. We think it is much more likely that the four variables linked in our production function drive (and explain)

each other, in the sense of a positive feedback relationship. There is statistical evidence to support this hypothesis of mutual causation. What can be concluded from the cointegration analysis we have carried out thus far is that in both the US and Japanese cases the four variables (K, L, U, Y) do cointegrate, except during the 1942-5 break (and, for Japan, subject to a caveat below). This means that a stable long-term relationship really does exist among these variables for both countries. It is tempting to think of this hypothetical relationship as an expression of the rather elusive dynamic equilibrium that most growth theorists have always postulated.

In Japan, the situation is more complicated than for the US case, because of a long period between the 1950s and the 1980s during which our unit-root tests indicate a significant departure from equilibrium occurred for the capital stock variable, in particular. That departure from equilibrium is almost certainly attributable to the Japanese postwar 'economic miracle'. The unprecedented growth rate from 1952 on into the 1980s was brought about by a series of government-industry policies that strongly favored savings and investment over current consumption. Gross private investment increased from a solid 17.2 percent of GDP in 1952-4 to a high of 30.5 percent in 1970-71, when annual growth rates in excess of 10 percent were being achieved. Private savings followed a parallel path, rising from 16.5 percent in 1952-4 to 31.9 percent during 1970-71. By contrast, private savings in the US between 1961 and 1971 averaged only 15.8 percent of GDP (Henderson 2002).4

Having established that the variables are cointegrated, it is possible to carry out Granger-type causality tests regarding the question as to whether energy (exergy) and/or useful work drive growth or vice versa.5 Because of the extreme complexity of the procedure, we have done this in detail only for the US. The results are summarized in Table 7.3.

We have tested Granger-causal relations for both exergy (Model A) and useful work (Model B), and considered short-run and long-run causality separately. For Model A we find evidence of both short-run and long-run causality from exergy to GDP, but no evidence for the reverse. However, in the case of Model B, where useful work replaces exergy as an input, we find no evidence of short-run causality from useful work to GDP, but strong evidence of long-run causality from useful work to GDP. We find this result very plausible, for the simple reason that aggregate exergy efficiency changes very slowly and therefore cannot explain short-term changes in GDP growth. On the other hand, we also found evidence that capital and labor Granger-cause useful work consumption in the short run. These results taken together are reasonably consistent with Stern's work (Stern 2000) and they refute the so-called neutrality hypothesis (that growth is independent of energy consumption/production).

Table 7.3 Causality tests

Independent variable (source of causation)

Dependent variable Model A

Independent

Dependent variable Model B

AGDP

AExergy

ACapital

ALabor

variable (source of causation)

AGDP

AExergy

ACapital

ALabor

Short run

Short run

AGDP

0.66

2.1

0.54

AGDP

0.51

2.07

0.82

AExergy

20.46**

*

6.17**

12.55

AWork

0.34

8.35**

1.51

ACapital

4.58

2.34

0.03

ACapital

1.06

4.65*

0.98

ALabor

0.68

0.53

4.63*

ALabor

1.67

5.06*

0.75

Long run

Long run

ECT /AGDP

ECT /AGDP

ECT /AExergy

5.63

3.89

18.25***

ECT /AWork

1.99

2.19

6.18

ECT /Capital

56.78**

*

11.03***

25.96***

ECT /Capital

17.71**

*

12.89**

12.80***

In brief, we have found that there is reasonably good evidence of cointe-gration and mutual causality among the four variables, for both the US and Japan. Admittedly, while the evidence for causality is strong, it is not absolutely conclusive. This is because the residual errors for both models are not quite normally (Gaussian) distributed, as one would like. Hence, the statistical tests might still be spurious. We also find, as already noted, that there are significant structural breaks in the 100 plus year time series, for some variables, for both countries. This implies that the models being tested (Cobb-Douglas and LINEX) should be re-calibrated at the break points, or (better) that appropriate dummy variables should be introduced into the fitting equations. As previously noted, refitting after re-calibration might well result in better DW statistics for the segments.

Moving on, and regardless of the caveats above, the extraordinarily good fit to past GDP data exhibited by our LINEX model strongly suggests that it can be useful as a forecasting tool. The argument, in brief, is that if the model 'explains' the past with so few free parameters, there is no reason to suppose that the relationship will become invalid overnight, or in the course of a few years. In other words, other things remaining equal, the model should also provide strong clues as to what can be expected over the next few decades, even though some departures from historical trends can be expected. We develop this idea in Chapter 8.

As regards the past, our results clearly reflect the substitution, during the past century, of 'useful work', mostly by fossil-fuel-powered machines, for muscle work by humans and animals. In fact, the calculated output elasticity of energy, as useful work, is up to ten times higher than earlier estimates based on the factor cost-share theorem (Appendix A). Although the factor of ten may well turn out to be somewhat too high (because our model is still too simplistic), the fact that the difference is large is hard to ignore. While the calculated values of the elasticities are not absolutely trustworthy, having been obtained from non-linear fits, the results are still qualitatively consistent with the idea that 'pure' (unskilled) labor, in the absence of machines and sources of power, is now nearly unproductive at the margin. This result holds for both the US and Japan. In effect, our results suggest that labor is no longer a scarce resource. One more unskilled worker, without tools and mechanical or electrical power, adds almost nothing to economic output.

This result, tentative though it may be, has important implications for the future. Among them is that it contradicts the assertions by many politicians and pundits in Europe that a declining birth-rate needs to be reversed. On the contrary, it is getting harder to keep everybody who wants a job productively employed. The declining birth-rate in Europe and Japan may be more positive than negative.

Luckily there is some other evidence to support our qualitative results.

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