On the estimation of short- and long-run elasticities in U.S. petroleum consumption: comment.

I. Introduction

In a recent paper in this journal, Jones [13] has advocated for the use of the general-to-simple modelling strategy in estimating short- and long-run elasticities in energy consumption. In contrast to the simple-to-general modelling approach, which is most often used in energy demand studies, the general-to-simple strategy starts by estimating a deliberately overparameterized unrestricted autoregressive distributed lag model. Various parameter restrictions are then tested sequentially until a parsimonious representation of the underlying data-generating-process is obtained.(1)

In an empirical application using this approach, with U.S. time-series data for petroleum consumption, the real price of oil, and real GNP, Jones ends up with an apparently well-specified restricted model containing only seven parameters, and with economically plausible estimates of short- and long-run price and income elasticities.

In this note we argue that there is an important initial step missing in the analysis carried out by Jones, namely an investigation of the time-series properties in terms of integration and cointegration of the time series involved. The implicit assumption underlying the modelling approach followed by Jones is that the variables are stationary stochastic processes, which means that they have constant unconditional means and variances. However, recent research suggests that the variables involved in energy demand relationships are usually not stationary in levels, but have to be differenced in order to be stationary.(2) Such variables are said to be integrated of order one, I(1), which means that they have a unit root in their autoregressive representation, see Engle and Granger [3]. I(1) behavior of economic variables has profound implications for estimation and statistical inference. First, the use of traditional asymptotic inference (e.g., t-tests and F-tests of linear restrictions on the parameters) may become invalid in regressions containing the levels of the variables, like equation (1) in Jones [13]. Second, the calculation of long-run elasticities from such level-regressions presupposes that the variables are cointegrated in the sense of Engle and Granger [3], since cointegration among non-stationary variables is a necessary condition for the existence of a long run relationship between them.

Below we investigate the time-series properties of the data used by Jones, and find that petroleum consumption, the real price of oil, and real GNP all appear to be I(1) processes that do not cointegrate. This implies either that there does not exist a stable long run relationship between the levels of petroleum consumption, oil prices and income, or that there is one or more important variables missing in explaining energy consumption in the long run. Furthermore, it implies that the proper specification of the energy demand schedule is a first-difference specification without an error-correction term. When we estimate such a model using Jones's data, we end up with a structurally stable model, which is more parsimonious compared to his final model (in the sense that fewer parameters are estimated). In accordance with the time-series properties of the data, our model does not contain any long-run relationships between the levels of the variables. It only contains long-run relationships between their growth rates.

Table I. Tests for Unit Roots(*)

                     ADF {lags}     PP {3}

[q.sub.t]             -1.29 {1}     -1.06
[y.sub.t]             -2.33 {0}     -2.36
[p.sub.t]             -1.57 {0}     -1.84
[Delta][q.sub.t]      -4.20 {0}     -4.11
[Delta][y.sub.t]      -5.46 {0}     -5.39
[Delta][p.sub.t]      -5.70 {0}     -5.72

* ADF is Augmented Dickey-Fuller test, with augmentation lags shown
in { }. PP {3} is Phillips-Perron test with a non-parametric
correction for third order autocorrelation. A constant and linear
trend are included in the regressions. The 5% critical value in a
sample with 50 observations is: -3.50 (Table 8.5.2 in Fuller [6]).

[TABULAR DATA FOR TABLE II OMITTED]

II. Evidence(3)

The data are from the appendix in Jones [13] and covers the period 1947-1989, with [q.sub.t] denoting U.S. petroleum consumption, [p.sub.t] denoting the real price of oil, and [y.sub.t] denoting real GNP. All variables are measured in natural logarithms.

Table I reports Dickey and Fuller [2] tests and Phillips and Perron [15] tests for a unit root in [q.sub.t], [y.sub.t], and [p.sub.t]. As seen, the hypothesis of a unit root cannot be rejected at even the 10% level of significance for any of the variables. The table also depicts tests for non-stationarity of [Delta][q.sub.t], [Delta][y.sub.t], and [Delta][p.sub.t], and these tests strongly reject the unit root hypothesis. This means that the levels of the variables may be considered non-stationary I(1) processes which have to be first-differenced in order to be stationary.

The presence of a unit root in [q.sub.t], [y.sub.t], and [p.sub.t] implies that in order for them to be tied together in a stable long-run relationship, they have to be cointegrated. Engle and Granger [3] propose to test for cointegration by regressing the variables on one another in a static OLS regression, and then testing for stationarity of the residuals using the Dickey-Fuller test.

Table II contains the results of such a cointegration analysis, where all the variables in turn are used as the dependent variable. The parameter estimates have been normalized by the [q.sub.t] coefficient so that, in case that cointegration is found, they can be readily interpreted as long-run petroleum demand elasticities. Judging from the DW and ADF statistics, absolutely no evidence of cointegration is found, no matter what variable is chosen as the dependent variable. This implies that the regressions are spurious in the sense of Granger and Newbold [7] and Phillips [14], and hence that one should be very careful in interpreting the parameter estimates obtained as long-run petroleum demand elasticities.

Table III. Johansen Cointegration Analysis(*)

                              [L.sub.max]   [L.sub.trace]

r = 0                             15.6          24.6
r [less than or equal to] 1        7.8           8.9
r [less than or equal to] 2        1.2           1.2

* r is the cointegration rank. [L.sub.max] and [L.sub.trace] are
the
two tests for the value of r (the number of cointegration vectors).
For r = 0 the 5% critical values for [L.sub.max] and [L.sub.trace]
are, respectively: 20.8 and 29.5 (Table A1 in Johansen and Juselius
[12]). The tests are based on a VAR model with two lags, and a
constant term included.

This non-cointegration result is confirmed by applying the multivariate maximum likelihood approach proposed by Johansen [10; 11].(4) Table III reports the socalled Lambda-max test and Trace test from the Johansen analysis. None of the test statistics are significant at even the 10% level. This again indicates that [q.sub.t], [y.sub.t], and [p.sub.t] are not cointegrated, so that either there does not exist a stable long-run relationship between the levels of U.S. petroleum consumption, real oil prices, and real GNP, or that there is one or more important non-stationary variable missing in explaining petroleum demand in the long run.(5)

These results imply that the t-tests and F-tests carried out by Jones to reach his final model (equation (2) in Jones' paper) are invalid, and that the demand elasticity estimates reported should not be relied upon. In the following we will apply the general-to-simple strategy and estimate an alternative specification in which all variables are first-differenced. We start by including three lags of all variables and then we test sequentially the significance of the individual parameters using standard t-tests until a parsimonious model is obtained. This gives the following restricted model (t-values in parentheses)

[Mathematical Expression Omitted]

T = 39, SSR = 0.008976, [Mathematical Expression Omitted], DW = 2.18, LM(4) = 4.98, LMARCH(4) = 6.98.

In accordance with the cointegration tests reported earlier, the model given by (1) does not contain any long run relationships between [q.sub.t], [y.sub.t], and [p.sub.t]. It does, of course, contain long run relationships between [Delta][q.sub.t], [Delta][y.sub.t], and [Delta][p.sub.t]:

[Mathematical Expression Omitted] [Mathematical Expression Omitted].

An interesting hypothesis to be tested is that [Mathematical Expression Omitted]. This hypothesis implies that the [Delta]y-coefficients and the [Delta][q.sub.t - 1]-coefficient in (1) are summing to one. The F(1,34) test for this hypothesis gives a value of 0.765 which is highly insignificant. We therefore impose the restriction and reestimate the model. The result is:

[Mathematical Expression Omitted]

T = 39, SSR = 0.009178, [Mathematical Expression Omitted], DW = 2.13, LM(4) = 5.46, LMARCH(4) = 3.82, CHOW = 2.21.

We also report a number of misspecification tests in order to assess the statistical validity of the estimated model. LM(4) and LMARCH(4) are Lagrange Multiplier tests for fourth order residual autocorrelation and autoregressive conditional heteroscedasticity, respectively. These tests are [[Chi].sup.2](4) distributed. CHOW is an F-test for structural stability of the estimated relationship (breakpoint: 1974). This test has an F(4, 31) distribution. As seen, there are no signs of misspecification in terms of autocorrelation, ARCH, or structural instability.

Comparing our final model, equation (2), with Jones's final model we see that our model is more parsimonious in that it only contains four estimated parameters (Jones's model contains seven estimated parameters). Furthermore, in accordance with the time-series properties of [q.sub.t], [y.sub.t], and [p.sub.t], our model does not contain a long-run relationship between these variables.(6) Instead, it contains long run relationships between the growth rates of the variables. The short and long run elasticities in terms of growth rates are:

[Mathematical Expression Omitted] [Mathematical Expression Omitted]

[Mathematical Expression Omitted] [Mathematical Expression Omitted].

A possible explanation for the above findings is that in the long run we would expect the level of energy consumption to be just as closely related to energy efficiency (or technological progress) as to real income and real energy prices. However, the growth rates of energy consumption is much less likely to be strongly related to changes in energy efficiency, so that by focusing on growth rates instead of levels we do not have to include efficiency as an explanatory variable, as shown by the final model (2) where only growth rates of real income and real energy prices are included.

III. Conclusion

U.S. petroleum consumption, the real price of oil, and real GNP appear to be non-stationary I(1) processes that do not cointegrate. This implies that regressions involving only these three variables are spurious. A possible explanation for the non-cointegration result is that energy efficiency (or technological progress) is likely to exert a strong influence on the level of energy consumption. However, regressions involving the first-differences of the variables are non-spurious. In estimating such a first-difference specification, using the general-to-simple modelling approach, we end up with a well-specified and structurally stable model containing only four parameters. The implied short and long run income elasticities, in terms of growth rates, are, respectively: 0.558 and 1.000. The equivalent short and long run oil price elasticities are, respectively: -0.084 and -0.341.

Jan Bentzen Tom Engsted Aarhus School of Business Aarhus, Denmark

1. For a detailed description of the general-to-simple modelling strategy, see Hendry, Pagan, and Sargan [8].

2. See Bentzen and Engsted [1], Engsted and Bentzen [5], Hunt and Manning [9], and Yu and Jin [16].

3. No detailed description of the employed tests is given. The reader is referred to the literature cited.

4. See Bentzen and Engsted [1] and Engsted and Bentzen [5] for a more detailed description of the Johansen approach in the context of energy demand.

5. An obvious candidate for such missing variable is energy efficiency, see below.

6. That there really should not be such a relationship in the model is confirmed by including a so-called error-correction term (given by the residuals from any one of the cointegrating regressions reported in Table II). This term is in all three cases insignificant at a 5% level.

References

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