## Estimating The Production Function

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For a theory of growth, if one does not want to wait 20 or 30 years for confirmation, the best hope is to explain past economic growth reasonably well for a very long period, such as a century. This is what we attempt here in this chapter. The starting point is to specify the form of a production function that fits historical data with as few independent parameters as possible, subject to certain statistical requirements. This was the aim of Chapter 6. The next step looks simple, at first glance: plug in the time series data and turn the crank.

Alas, things are not so simple. Important questions about the time series data themselves remain to be addressed. They have implications for the method to be used for parameter estimation. The five economic variables in question are capital K, labor L, energy (actually exergy) E, useful work U and output (GDP) Y. Questions that might be asked include: are the variables well-behaved? Do the variables exhibit a systematic trend or do they vary randomly? Is there evidence of transitory shocks or structural breaks? Is there evidence of a stable long-run relationship among the variables, an essential precondition for a production function to be meaningful? Can we say anything about the direction of causality between the factors of production and GDP?

Before we can have great confidence in the outcome of calculations with a production function, especially if after introducing a new and unfamiliar factor of production (U), it is desirable to conduct a number of statistical tests on the variables. To do statistical analysis on time series variables, they must be converted to logarithmic form, to eliminate any exponential time trend. The next step is to determine whether the time series (of logarithms) is 'covariance stationary', meaning that the year-to-year differences are finite, random and not dependent on previous values of the variable. In statistical language, the mean and covariances are normally distributed and do not increase or decrease systematically over time. It happens that many macroeconomic variables, including the ones of interest to us, are not covariant stationary. When this condition is not met, there is said to be a 'unit root'. The first statistical test for this situation is known as Dickey-Fuller (DF) (Dickey and Fuller 1981).

The first question is whether the unit root is 'real' (that is, due to a missing variable) or whether it is due to an external shock or structural break (discontinuity) in the time series. We have carried out extensive tests, not only using the Dickey-Fuller statistic but also several more recent variants, to determine whether our time series do, in fact, exhibit structural breaks (Phillips and Perron 1988; Zivot and Andrews 1992). The results, as is often the case, are somewhat ambiguous: unit root tests of the time series show some evidence of 'mini-breaks' in individual time series. But rarely do these mini-breaks occur in the same years in all series. These mini-breaks may be due to various possible causes, from external events to major changes in government policy, especially in Japan. Some of the years correspond to identifiable events (such as the onset of the Great Depression in 1930), but others do not.

However the unit root tests we have carried out all point to the existence of one major structural break for both the US and Japan closely corresponding to the dates of World War II. Thus we have carried out our model-fitting procedures for two cases, namely for the entire 100-plus year period (1900-2005) and separately for the prewar (1900-41) and postwar (1946-2005) periods.

The next step is to test specific model formulations, such as the Cobb-Douglas or LINEX forms discussed in the last chapter. The most familiar statistical fitting procedure is known as 'ordinary least squares' (OLS). The question is whether OLS is legitimate for testing a model. The answer is easily stated: it can be shown that, when the model variables are not covari-ance stationary in the above sense, OLS model fits are likely to be spurious, except in one very special case which we return to below. Because of this special case, we cannot reject the use of OLS just yet.

The Durbin-Watson (DW) statistic is another test frequently applied to the residuals of econometric models. It checks for serial auto-correlation, meaning that the residuals (errors) of a model are (or are not) correlated (Durbin and Watson 1950, 1951). N.B. the DF test, as applied to a model, has the same purpose, but the DW test does not apply to an individual time series. The DW test statistic is defined as a 1 et - et-122

where et is the model residual error at time t. The statistic takes values ranging from 0 to 4, where a value of 2 means that there is no statistical evidence of auto-correlation, positive or negative, meaning that the errors are truly random. A DW value less than 2 implies positive auto-correlation between successive error. A DW value greater than 2 implies negative autocorrelation, which is extremely unlikely. A value close to (but less than) 2 is regarded as very good. A value less than 1.5 is regarded as 'cause for alarm'. A very small positive DW value means that successive error terms are consistently very close to one another. This implies that the errors are systematic, probably due to a missing variable, and hence not randomly distributed. Thus the smaller the DW statistic, the more likely it is that some important factor has been omitted.

However, like many statistical tests, the DW test is very specialized. It is quite possible for a model fitted over a short period to have a better (that is, larger) DW statistic than a model fitted over one long period. This could happen, for example, in the case of a model characterized by several segments, each displaying serial correlation, where the errors in different segments have opposite signs. The DW statistic is also perverse, in the sense that it bears no relationship to the magnitudes of the errors. The errors could be very small and yet give rise to a small DW statistic.

Having established that we are dealing with variables that are not covari-ance stationary, the second issue of importance is multi-collinearity. This means that the variables, and their logarithms, tend to be highly correlated with each other, although their year-to-year differences may not be. In such a case, high values of the correlation coefficient (R2) are meaningless, and goodness of fit must be assessed in other ways. However, our variables are 'first-difference stationary'. This means that we could construct a model that explains past year-to-year differences very accurately but that has lost essential long-term information about the future. In fact, apart from the special case noted earlier, it has been shown that where the variables are first-difference stationary any OLS regression is likely to be spurious, meaning that no robust relationship can be detected between the variables (Granger and Newbold 1974).

The next step was to examine the residuals from OLS estimates, for both C-D and LINEX models both over the whole century and over the pre- and postwar periods taken separately. In the US case, the C-D model appeared to show breaks in 1927, 1942, 1957 and 1986. The implication is that the model should be re-calibrated for each period. This can be done by introducing dummy variables that modify the exponents and multipliers for each period, of which there are (5 x 3) -1 = 14 parameters in all. As it turns out, even with so many additional parameters, the fit is not particularly good (in fact, some of the fitted coefficients are negative). Hence, we decided to use the simpler two-period version of the model. To make a rather long story short (we have tested literally dozens of combinations), we found that the period of World War II (1942-5) is the only structural break that needs to be taken into account in both the US case and the Japanese case.1

To anticipate results shown in the following pages, it turns out that OLS regressions of the Cobb-Douglas model are indeed spurious, as expected, despite high values of R2, because of both the existence of unit roots in the model residuals and extremely small values of the DW statistic (strong serial correlation).

The LINEX model is not estimated by OLS, however, but by a method of constrained non-linear optimization. The constraints we imposed on the optimization are that the output elasticities be non-negative and add up to unity (constant returns). It happens that there are multiple solutions that satisfy the constraints, because of multiple collinearity. Ideally, the independent variables would each be positively correlated with the dependent variable, but not correlated with the other independent variables. However, we think that in our case the variables do not divide neatly into 'independent' and 'dependent' categories. Rather, they are all mutually dependent. In simple terms, the problem with multiple collinearity is that the variables are measuring the same phenomenon (economic growth) and are consequently - to some extent - redundant. This situation can theoretically result in over-fitting.

We must also acknowledge at the outset that a good fit of the output (GDP) to the input variables (capital, labor, exergy or useful work) - even though not arrived at by OLS - does not, by itself, constitute proof of a postulated model relationship. It is theoretically possible that the causality runs the other way, that is, that the changes in the input variables (factors of production) in the model were consequences of changes in the state of the economy. However, recalling Figure 1.1 from Chapter 1, we actually expect causality to run both ways, although not necessarily at the same time. In fact, we suspect that the business cycle may consist of two alternating 'regimes' in the sense of Hamilton (1989, 1996).

The last step is to determine whether the variables (K, L, U, Y) cointe-grate. In other words, we want to know if there is a stable long-term relationship among them. As pointed out earlier, the logarithms of most macroeconomic variables are not covariance stationary. However, in most cases, they are 'first-difference' stationary. Two such variables with non-stationary residuals (unit roots) are cointegrated if and only if there exists a linear combination of them, known as a vector auto-regressive (VAR) model, that has stationary residuals (that is, no unit root). A single integrating equation suffices in the bi-variate case. However, the general multivariate case is more complicated, because if there are N variables, there can be up to N - 1 cointegrating relationships. The challenge is not only to prove that such a linear combination exists - this is the special case, mentioned earlier in which the use of OLS is legitimate - but to find the best one. The form of the relationship is to express the rate of change of the target variable (say GDP) at time t in terms of a linear combination of the previous years' values of the variables (the cointegrating equation or error-correction term or ECT). Each variable in the ECT is weighted by a coefficient that describes the rate at which each variable adjusts to the long-term relationship. The cointegrating model also incorporates a numberp of lagged values of the differences (rates of change) of each of the variables at prior times t - 1 through t - p. The system (with p specified) can be expressed most conveniently as a matrix equation, called a vector error-correction model or VECM (for example, Engle and Granger 1987; Johansen 1988, Johansen 1995). The absence of constant returns and non-negativity constraints on elasticities means that the VECM cannot usually be interpreted as a conventional production function.2

Cointegration analysis is a prerequisite of testing for causality when the variables are not covariance stationary (that is, they exhibit unit roots). The first application of cointegration analysis to the specific case of GDP and energy consumption was by Yu and Jin, using a bi-variate model (Yu and Jin 1992). These authors concluded that there is no long-run cointegra-tion between energy consumption, industrial production or employment.

However, Stern (1993) used a multivariate VECM and reached the opposite conclusion, that is, that cointegration does occur among the variables and that energy consumption, adjusted for quality, does Granger-cause GDP growth (Stern 1993). He explains this contradiction of Yu and Jin's results as the consequence of the inclusion of two more variables, which allow for indirect substitution effects that are not possible when only two variables are considered. Stern's results were reconfirmed by a later study by himself (Stern 2000). A more recent application of the multivariate method, as applied to Canada, concluded that Granger-causation runs both ways (Ghali and El-Sakka 2004).3