OverviewThe motivation for this work derives from (i) prior experience in the application of structural time series models for the purpose of modelling underlying demand trends (Broadstock and Hunt, 2010 inter alia) and dynamic energy demand functions (Broadstock and Chen, 2010) and energy production functions with dynamic components (Broadstock, 2010) and (ii) the recent emergence of an econometrically simpler, and intuitively appealing, procedure grounded on the tenets of rolling regression methods, which can approximate dynamic model components (Yalta et al 2015, Altinay and Yalta 2016).

Admittedly rolling regressions are not the only alternative to structural time series models. One may for instance consider nonparametric procedures such as Yatchew and Dimitropoulos (2017), or other methods which capture variation through the frequency domain rather than the time domain. The rolling regression is however of special interest for at least two reasons. First is that it has had direct application to energy demand problems, making the comparison with structural time series methods more salient. The second is that rolling-regressions are also widely utilised in econometric studies in for example financial econometric applications but also across a range of testing procedures. Thus the analysis in our simple horse race, though applied to the context of energy demand functions, has the potential to inform a much wider body of applied econometric literature.

Faced with two alternative approaches to achieve a similar outcome, the question is whether they are equivalent, and if not under which circumstances one procedure may be suited over the other. The a-priori expectation was that rolling regression should have some ability to capture reasonable approximations of the underlying dynamics, but probably that the structural time series model (a tool designed precisely for this purpose) would outperform it.

MethodsThe length of the abstract prohibits any extensive detail of the competing methodologies, and the full monte-carlo experiment (the horse race). Here only a brief sketch is offered.

We establish a synthetic world. In this world, we draw two (initially stationary, for simplicity) random variables-i.e. one for price and the other for income-from normal distributions with zero mean and unit variance. The relevance of having two variables stretches beyond simply baring a similarity to the demand function context, but permits a more thorough experimental design through which complex dynamics (coefficients with differing time-varying structures) but also missing variable effects can be explored.

We next construct the coefficients. We consider a range of alternative structures for the coefficients, but for brevity highlight only two core cases here. First is the case where the intercept, and price and income elasticities are constant. In principle rolling regression should fare well in such a scenario, since the model is correctly specified insofar as parameters are constant by design, compared against the forced time-varying parameter structure in the structural time series model. In the other extreme, the intercept, price elasticity and income elasticity are each assumed to follow a random walk, directly reflecting the standard assumption in the structural time series model, and thus delivering an environment where it should perform better. Using the variables and coefficients we calculate demand, and add on a further idiosyncratic error. We then use the two competing methods to attempt to recover the true coefficients, by regressing the ‘observed’ demand (i.e. the true demand plus the idiosyncratic error) against the observed price and income series. Thus the two horses (methods) are pitched in a race against each other.

DYNAMIC DEMAND FUNCTIONS & UNDERLYING DEMAND TRENDS: A HORSE RACE BETWEEN ROLLING REGRESSION & STRUCTURAL TIME SERIES MODELS

David C. Broadstock, Hong Kong Polytechnic University, Hong Kong, david.broadstock@polyu.edu.hk

XiaoQi CHEN, Hong Kong Polytechnic University, Hong Kong, 18380490324@163.com

Dong WANG, Southwestern University of Finance & Economics, China, xphoenixx@163.com

The key types of questions (though not only questions) are firstly can it be confirmed that OLS based rolling regressions approximate constant parameters much better than a structural time series model would, and secondly can it be confirmed that they approximate time-varying parameters with some reasonable degree of accuracy.

The methodology so far is rather statistical in nature, which fails to capture the ‘so-what’ nature of the problem. Are the differences between the models economically important? To evaluate this, we revisit the results in Altinay and Yalta (2016) who uses rolling regression to capture time-varying coefficients in modelling the demand for natural gas in Turkey, and re-run the analysis using a structural time series model. As highlighted below, the differences are economically meaningful (refer to the income coefficients in particular), in fact they are strikingly different in some cases. Though in other cases the results are qualitatively similar.

Figure 1: Black lines are rolling regression estimates of selected model parameters; blue lines are structural time series models (Kalman filter/KF) estimates of the same parameters. These results are generated using the same data as in Altinay and Yalta (2016).

ResultsWe illustrate several key points:

- The rolling regression framework never really outperforms the structural time series model, even in the scenario where the true coefficients are in fact time invariant

o This seems initially counter-intuitive, but is in fact quite straightforward to rationalize as being a consequence of imposed statistical inefficiency, coupled with the fact that a structural time series model can mimic constant parameters very effectively when it needs to

- On the contrary the structural time series model has surprisingly good performance under all model evaluation criteria considered (though not discussed here for brevity), suggesting a high degree of relevance of previous results reported using these methods.

- The above conclusions are invariant to sample size, estimation in the presence of missing variables, the presence of moving average errors, complex dynamic processes (i.e. that would not follow a definable functional form) and mixtures of stationary and non-stationary variables.

- From the empirical application, it seems that rolling regression can substantially compromise the lessons-learned and policy implications. We illustrate that (i) time variation is imposed where it may in fact appear not to really exist, and also that (ii) elasticity estimates are susceptible to taking values that are inconsistent with any reasonable understanding of economic theory.

ConclusionsThrough this work we highlight the accuracy of the structural time series model as a tool for modelling energy demand and obtaining accurate price and income elasticities to help inform energy policy. A statistical horse-race is used in conjunction with an empirical replication study to illustrate, emphatically, that our conclusions have considerable statistical and economic significance.