Contributions to Risk Assessment with Edgeworth–Sargan Density Expansions (I): Stability Testing

Abstract

This paper analytically derives a stability test for the probability distribution of a random variable that follows the Edgeworth–Sargan density, also called Gram–Charlier. The distribution of the test is a weighted sum of Chi-squared densities of increasing degrees of freedom, starting with the standard equivalent Chi-squared under the same conditions. The weights turn out to be linear combinations of the parameters of the distribution and the moments of a Gaussian density, and can be computed exactly. This is a convenient result, since then the probability intervals can be easily calculated from existing Chi-squared distribution tables. The test is applied to assess the weekly solar irradiance data stability for a twelve-year period. It shows that the density is acceptably stable overall, except for some eventual and localised dates. It is also shown that the usual probability intervals implemented in stability testing are larger than those of the equivalent Chi-squared distribution under comparable conditions. This implies that the common upper tail interval values for rejecting the null stability hypothesis are larger.

1. Introduction

This paper presents and derives an analytical expression for the stability distribution of a random variable that follows the general Edgeworth–Sargan (ES) type of probability distribution. The distribution possesses several interesting properties and is able to capture departures from Gaussianity of various kinds, notably asymmetry and thick tails, i.e., skewness and kurtosis. The paper shows that the final expression is a weighted sum of Chi-squared distributions with increasing degrees of freedom, starting with the standard Chi-squared if the variables followed a Gaussian distribution. Thus, it departs from the standard stability test, and the usual upper tail probability intervals implemented in stability testing will be less restrictive. This exact analytic result is convenient, since then probability intervals can be easily calculated from the probabilities for Chi-squared distribution tables, which are readily available.

The test is applied to the analysis of solar irradiance data provided by the PV-GIS European database [1]. First, a distribution of this type is fitted to the data. Second, the stability test is implemented to check eventual disruptions over the period considered—weekly data spanning the period 2005–2016.

The distribution was formally introduced by Edgeworth more than a century ago [2], but it was Sargan who introduced it into the wide econometrics and statistical theoretical and applied fields [3,4]. A further significant contribution was provided by the work of Gallant [5,6], who suggested a transformation to avoid some potential negativity problems. This transformation, nevertheless, although theoretically better and sufficiently general, is challenging to implement and has not shown a clear advantage over the first and simpler version.

An early empirical application of this distribution is [7]. Since then, it has slowly made its way into the applied financial field, with significant contributions being made by [8,9,10]. In particular, ref. [11] provides a generalisation to a multivariate setting, accounting for crossed moments beyond covariances, like co-skewness and co-kurtosis [12]. The distribution has been compared to alternatives [13,14], notably the Student’s t [15] and its asymmetric generalisation [16]. In general, and although from a theoretical point of view this distribution can also capture some non-normal anomalies, in practice it has failed to show its superiority, and it is far more challenging to obtain generalisations and derive convenient results given its high nonlinearity. Other popular alternatives include the multivariate skew-normal [17], although this distribution cannot account for any other non-standard feature beyond asymmetry. However, beyond the field of applied finance, the ES distribution is largely unknown. A secondary purpose of this research is, therefore, to show its applicability in the burgeoning field of solar irradiance statistical characterisation.

The following section presents the formal derivation in a simplified case. Use is made of a significant number of supporting and related results, discussed in several appendices in order to clarify the derivation. Section 3 presents the results of estimating the ES distribution for the data analysed and implements the test for several configurations. A discussion on possible alternative distributions and their empirical feasibility is conducted in Section 4, and Section 4 summarises the main contributions of the paper, suggesting immediate avenues for future research. Several appendices present and derive results required in the main derivation. In particular, a generalisation of the primary result of Section 2 is presented in Appendix C.1, and Appendix C.2 summarises a list of results required in several steps of the derivation for easy reference. Finally, Appendix D reports additional empirical results and provides a detailed description and reference for the data analysed.

2. A Simplified Case

This section is devoted to deriving the proposed test in a simplified case that, nevertheless, involves the main steps and helps clarify the derivation. A general case will be dealt with in Appendix C. Let us start by considering the distribution of a random variable, ${\epsilon }_i$, with probability density function (p.d.f.) given by:

$$f_{{\epsilon }_i}\left({\epsilon }_i\right)=\alpha \left({\epsilon }_i\right)\times \left[1+d_2 H_2\left({\epsilon }_i\right)+d_3 H_3\left({\epsilon }_i\right)\right]$$

(1)

where, $\alpha \left({\epsilon }_i\right)$ is the p.d.f. of a $N\left(0,1\right)$, the $d_s$ are some constants, and the $H_s$ are Hermite polynomials of orders 2 and 3, respectively—see Appendix B.3. This can be conveniently rewritten as follows:

$$f_{{\epsilon }_i}\left({\epsilon }_i\right)=\alpha \left({\epsilon }_i\right)\times \left({\theta }_0+{\theta }_1 {\epsilon }_i+{\theta }_2 {\epsilon }_i^2+{\theta }_3 {\epsilon }_i^3\right)$$

(2)

These transformed coefficients must fulfil the following properties: (1) $\left({\theta }_0+{\theta }_2=1\right)$, so that the probability integral is one, and (2) ${\theta }_1+{\theta }_3=0$, for the mean to be zero. It is straightforward to check that the p.d.f. in (1) meets both conditions—see Appendix B.3. Note, also, that ${\sigma }_u^2=1+2{\theta }_2\ne 1$, in general, although transforming to unitary variance is immediate. It will be assumed from now on that this correction has been implemented and denotes the new variable, $u_i$. It may be helpful before proceeding to gather the assumptions implied in what follows: (a) $\left(p_0+p_2=1\right)$ so that the probability integral is one; (b) $\left(p_0+3p_2=1\right)$ so that the variance is one; (c) $\left(p_1+3p_3=0\right)$ so that the mean is zero; (d) $\alpha \left(u_i\right)$ is the p.d.f. of a standard $N\left(0,1\right)$; (e) the $u_i\prime s$ are i.i.d., i.e., independently and identically distributed. These assumptions are implied in the specification of (1), (2).

Consider now the joint distribution of the vector of independent variates $u^{\prime }=\left(u_1,u_2\right)$ given by:

$$f_u\left(u\right)={\displaystyle \prod }_{i=1}^2\left[\alpha \left(u_i\right)\times \left(p_0+p_1 u_i+p_2 u_i^2+p_3 u_i^3\right)\right]$$

(3)

It is now convenient to implement the polar coordinate transform, i.e., $z=u^{\prime }u=u_1^2+u_2^2$ and $\eta =u/z^{1/2}$. The joint p.d.f. of the transformed variables will be trivially given by:

$$f_{z,{\eta }_2}\left(z,{\eta }_2\right)=f_u\left(z^{1/2} \eta \right) \Vert J\Vert$$

(4)

where$\Vert J\Vert$ is the absolute value of the relevant Jacobian—see Appendix A.2 for a complete derivation. Note that ${\eta }_1$in this context is just a shorthand for ${\eta }_1={\left(1-{\eta }_2^2\right)}^{1/2}$ so that it is not an independent variate. An explicit form for this p.d.f. is given by:

$$f_{z,{\eta }_2}\left(z,{\eta }_2\right)={\chi }_2^2\left(z\right)\times f_{{\eta }_2}\left({\eta }_2\right)\times \left[{\displaystyle \prod }_{i=1}^2\left(p_0+p_1 z^{1/2} {\eta }_i+p_2 z {\eta }_i^2+p_3 z^{3/2} {\eta }_i^3\right)\right]$$

(5)

where ${\chi }_2^2\left(z\right)$ is the p.d.f. of a Chi-squared distribution with two degrees of freedom and $f_{{\eta }_2}\left({\eta }_2\right)$ is the marginal density of ${\eta }_2$; see Appendix C for a detailed derivation of the general case. Let us suppose now that this p.d.f., i.e., the parameters $\left\{p_i,i=0,\dots ,3\right\}$, have been estimated over a given sample $\left\{u_t,t=-T,\dots ,0\right\}$. For forecasting purposes, a stability test is in order. Consider, then, the simple case where two additional observations are available to conduct the test, $\left\{u_t,t=1, 2\right\}$. Note that when $\left\{p_0=1,p_i=0,i=1,2,3\right\}$ in (3), $u_i$ is distributed as a standardised normal, i.e., $u_i~N\left(0,1\right)$ and therefore, the standard forecasting stability test follows a Chi-squared distribution with two degrees of freedom, i.e., $\left(u_1^2+u_2^2\right)~{\chi }_2^2$. By analogy to the standard normal case, a convenient statistic to conduct the test would also be $z=u^{\prime }u$. The p.d.f. of $z$ can be obtained immediately now as the marginal of the joint distribution of $\left(z,{\eta }_2\right)$, i.e.:

$$\begin{array}{ll}{f}_z\left(z\right)& ={{\displaystyle \int }}^{ }{f}_{z,{\eta }_2}\left(z,{\eta }_2\right)\partial {\eta }_2\\ & ={\chi }_2^2\left(z\right)\times {{\displaystyle \int }}^{ }\left\{f_{{\eta }_2}\left({\eta }_2\right)\times \left[{\displaystyle {\displaystyle \prod }_{i=1}^2}\left(p_0+p_1 z^{1/2} {\eta }_i+p_2 z {\eta }_i^2+p_3 z^{3/2} {\eta }_i^3\right)\right]\right\}\partial {\eta }_2\end{array}$$

(6)

$$={\chi }_2^2\left(z\right) \left( p_0^2+p_2^2 z^2 E\left({\eta }_1^2 {\eta }_2^2\right)p_0 p_2 z E\left({\eta }_1^2+{\eta }_2^2\right) \right)$$

(7)

where use is made of (A14)–(A16) whereby terms involving odd powers of ${\eta }_i$ are zero, and cross-products moments are equal to the product of the individual moments. Solving now for the moments of ${\eta }_i$ as given in (A16), yields:

$$f_z\left(z\right)={\chi }_2^2\left(z\right)\times \left(p_0^2+p_2^2 z^2{\mu }_2^2{\left[E\left({\delta }_2^4\right)\right]}^{-1}+p_0 p_2 z 2 {\mu }_2 {\left[E\left({\delta }_2^2\right)\right]}^{-1} \right)$$

(8)

Finally, applying the result in (A19) yields the explicit p.d.f. sought after as:

$$f_z\left(z\right)=q_0^2 {\chi }_2^2\left(z\right)+2 q_0 q_2 {\chi }_4^2\left(z\right)+q_2^2 {\chi }_6^2\left(z\right)$$

(9)

where $q_2=p_2 {\mu }_2=p_2$, because ${\mu }_2=1$, and $q_0=p_0$.

Using the operator defined in Appendix B.4, this last expression (9) can be written more compactly as:

$$\begin{array}{ll}{f}_z\left(z\right)& ={\chi }_2^2\left(z\right)\times {\left(q_0+q_2 z I\right)}^2{\delta }_2^2\\ & ={\chi }_2^2\left(z\right)\times {\left[Q\left(z I\right)\right]}^2{\delta }_2^2\end{array}$$

(10)

With:

$$Q\left(z I\right)={\displaystyle \sum }_{s=0}^1\left[q_{2s} {\left(z I\right)}^s\right]$$

(11)

which can also be written as ${\left[Q\left({\left(z I\right)}^{1/2}\right)\right]}^2$, with $q_1=0$. This is a slightly more general, but entirely equivalent notation, since odd terms in (10) vanish, given that all odd moments of ${\eta }_i$ are zero. It is of interest to note, as well, that the distribution can be written as:

$$f_z\left(z\right)={\displaystyle \sum }_{r=0}^2\left\{{\omega }_{2r} {\chi }_{2+2r}^2\left(z\right)\right\}$$

(12)

i.e., a weighted sum of Chi-squared distributions of degree 2 and above, where:

$${\displaystyle \sum }_{r=0}^2{\omega }_{2r}={\left(q_0+q_2\right)}^2=1^2=1$$

(13)

as it should, for (12) to integrate to one and therefore be a proper p.d.f. The cumulative probability function required to establish probability confidence intervals is immediately given finally as:

$$\begin{array}{ll}{F}_z\left(\overline{z}\right)& =Prob\left(z\le \overline{z}\right)={{\displaystyle \int }}_0^{\overline{z}}{f}_z\left(z\right) \partial z \\ & ={\displaystyle {\displaystyle \sum }_{s=0}^2}\left\{{\omega }_{2s}\times \left[{{\displaystyle \int }}_0^{\overline{z}}{\chi }_{2s}^2\left(z\right)\partial z\right]\right\}\end{array}$$

(14)

A generalisation of this result to more complex and realistic cases, along with some computational considerations and additional results, is left to Appendix C.

3. Empirical Results

The ES p.d.f. and related proposed distributions addressing an assorted array of issues have been implemented almost exclusively in applied financial analysis; see the introduction for a summary survey. However, it can also be applied in many other settings, including the study of meteorological and, in particular, solar radiation data. This is a promising field of research, given the urgency to tackle the climate change threat by deploying a whole array of renewable energy technologies, particularly solar photovoltaic (PV), given its impressive and sustained cost decreases since it was commercially introduced at the beginning of the 1980s. It is convenient to clarify at the outset that stability, or its lack thereof, is a property of a model. In order to check empirically whether a model is stable, appropriate statistical tests are required. In the present case, the focus is on the p.d.f. of the errors of a series, once the annual cycle has been removed. The test proposed here is completely general, but nevertheless can be applied to the residuals of any given model, e.g., a standard linear dynamic model, possibly estimated by an ordinary least-squares procedure. Note, also, that what is being tested is the stability of the underlying model.

The primary data set analysed has been the radiation database PVGIS-SARAH provided by the EU [1]. For technical details and other discussions related to its applicability, see [18,19]. The starting point was hourly data for the period 2005–2016 (both end-years inclusive), and PV power generation in central Spain; see Appendix D for details. Weekly observations were calculated from the hourly data, yielding a series with 624 data points—again, see Appendix D for details.

The main series considered, the weekly generation of PV power,$z_t$, exhibits a substantial cycle over the year. The first step is removing it and obtaining a ‘de-cycled’ series, as explained next. The weekly average over the years is denoted as:

$$z_t={\displaystyle \sum }_{i=2005}^{2016}{\left(\frac{z_{it}}{12}\right)}_{it}$$

(15)

where the subindices, i, t, refer respectively to a given year, $i=1,\dots 12$, and the week within that year, $t=1,\dots ,52.$ One straightforward and accurate way to define the cycle is given by:

$$z_t^c={\displaystyle \sum }_{n=-5}^{n=5}\left({\omega }_i z_{t+n}\right)$$

(16)

$${\omega }_i>0, {\displaystyle \sum }_{n=-5}^{n=5}{\omega }_n=1$$

(17)

i.e., a weighted sum given by the moving average using appropriately selected weights; see, e.g., [20] for a related discussion on alternative patterns and their optimality. Note that this weighted sum can be understood in the framework of a ‘circular’ time series, and hence, there are no gaps at both extremes: i.e., $z_{53}=z_1$, $z_0=z_{52}$, and similarly for other periods. The ‘de-cycled’ observation, ${\tilde{z}}_{it}$, is now straightforwardly given as:

$${\tilde{z}}_{it}=z_{it}-z_t^c$$

(18)

The cycle as it has been calculated and its fitting accuracy are analysed graphically in Figure 1—kWh is kilowatt hour, i.e., one thousand watts per hour of energy generated, electric power in this case; the specific details and references for the series analysed are further considered in Appendix D; appropriate literature references are [1,18,19].

Eventual remaining structures in the series have been considered by means of regression analysis, and no significant dynamic relationship has been detected. The residuals do not show clear signs of heteroskedasticity of any kind, although the normality hypothesis is strongly rejected: a relevant test yields a value of $34~{\chi }_2^2$, strongly rejecting the null, and thereby suggesting that a more general p.d.f. is warranted. Therefore, an ES p.d.f. of the type considered in this research has been estimated, yielding the following results for the coefficients associated with the Hermite polynomials of order 3 and 4, respectively:

$$d_3=-0.039646 \left(3.164\right)$$

$$d_4=0.017698 \left(2.236\right)$$

(19)

where the t-ratios are the figures in brackets, and no other polynomial is statistically significant. For these estimates, specific values for the confident intervals of the test presented in (14) in Section 2 can be derived: accordingly, stability tests for the following periods have been calculated, $\left(2, 5, 10, 20\right)$, and for the standard confidence intervals $\left(90\%, 95\%, 99\%\right)$. The values, jointly with the corresponding values of the relevant Chi-squared distribution, are reported in

Table 1. Chi-Squared and ES stability Probability intervals.

	Probability Intervals for the Stability Test (2, 5, 10 and 20 Weeks Ahead)
	ES and Chi-Squared Intervals (10%, 5%, 1% Upper Tail Probabilities)
Deg. freedom	2	5	10	20
	(A) 10%
Chi-squared	4.605	9.236	15.987	28.412
ES	4.650	9.627	16.758	29.175
	(B) 5%
Chi-squared	5.991	11.070	18.307	31.410
ES	6.341	11.793	19.203	32.678
	(C) 1%
Chi-squared	9.210	15.086	23.209	37.566
ES	10.448	16.559	25.034	39.683

Notes. Values for the ES test are derived with the estimates reported in (19).

.

It is immediately remarkable that the intervals for the ES case are higher in all cases, as the expression for the ES c.d.f. in (12–14) suggests. In the present case, however, the differences are not too large because the original variable does not depart strongly from the normality assumptions, as shown in the estimated coefficients for $\left(d_3,d_4\right)$ that imply skewness and excess kurtosis values of $\left(6d_3=-0.238, 18d_4=0.319\right)$, respectively—statistically significant, but not large in absolute value. Note that skewness and excess kurtosis, compared to the standard Gaussian, are respectively measured by $\left\{E\left(u^3\right),\left(E\left(u^4\right)-3\right)\right\}$. They are also the two most immediate departures from the Gaussianity assumption and what every p.d.f. should purport to capture adequately in empirical distributions. Note, also, that the Bera–Jarque test against non-normality is based precisely on the joint departure from zero of these two values—recent empirical applications of the Bera–Jarque test can be seen in, e.g., [21,22].

For the estimated distribution and the values in Table 1, the results for the five weeks’ stability test over the whole sample are displayed in Figure 2—note that 624/5 = 124 period tests, plus a remainder of 4 weeks. In this case, the general conclusion would be that the distribution is acceptably stable over the whole period analysed, save a few localised exceptions. Nevertheless, this does not imply that data at other frequencies, e.g., daily and even hourly, exhibit the same stability over time, in the same or different historical dates. The results for the remaining stability periods considered, 2, 10, and 20 weeks ahead, are reported in Appendix D.

In this context, it is also worth considering that equivalent results could be produced from computer-generated pseudo-random numbers for the ES distribution. A random number from the ES p.d.f., similarly to any other p.d.f. for that matter, can be derived from the following expression:

$$\begin{array}{ll}U\left(y\right)=y& =\underset{-\infty }{\overset{x}{{\displaystyle \int }}}\left\{\alpha \left(\epsilon \right)\times \left[1+d_3 H_3\left(\epsilon \right)+d_4 H_4\left(\epsilon \right)\right]\right\}\partial \epsilon \\ & =\Phi \left(x\right)-\left[d_3 H_2\left(x\right)+d_4 H_3\left(x\right)\right]\end{array}$$

(20)

where $\alpha \left(\epsilon \right)$,$\Phi \left(x\right),$ are respectively the p.d.f. and c.d.f. of a Gaussian (0,1) p.d.f., $U(.)$ is the c.d.f. of a uniform p.d.f. over the (0,1) interval, and ‘y’ a pseudo-random number generated with a suitable algorithm, like, e.g., [23]. Solving this expression for ‘x’ yields a random number that follows precisely that distribution, i.e., the ES with Hermite polynomials $\left(H_3,H_4\right)$ and their associated coefficients $\left(d_3,d_4\right)$; see, e.g., [24]. Solving this highly non-linear equation for a large number of random values is computationally demanding since it involves the inverse of $\Phi \left(x\right)$. There are available computational approximations [25], although there may be workarounds, e.g., generating and storing in a first step values for the ES c.d.f. Nevertheless, although generally, it is much easier and exact to derive the exact values of the test as given in (13) in Section 2, the random numbers derived using (20) may be helpful in specific cases, and even to provide an independent check for the analytical results.

4. Conclusions and Discussion

4.1. Results

This paper has derived an analytical expression for a stability test applicable when the underlying distribution is of the general ES type, capable of accounting for departures from the Gaussianity hypothesis of several kinds. Several related and relevant results have also been presented.

These non-standard anomalies have been studied in other fields with considerable length, notably in the applied financial literature; see, e.g., [21,22]. Here, the distribution has been implemented in a new area: the analysis of solar irradiance. This is all the more relevant, given the urgency to implement the energy transition to a low or zero-carbon system based on renewable energy sources, especially solar.

The solar irradiance data analysed show significant skewness and kurtosis, adequately accounted for by the ES distribution. The stability test has been applied to the solar data provided by the PV-GIS database, which shows some eventual localised instability. Otherwise, it displays a fairly stable behaviour over the period analysed. The theoretical probability intervals have been compared to the standard Chi-squared distribution for the equivalent cases. It is shown that even for cases where the departures from Gaussianity are moderate, the differences can be substantial. The general result is that the Chi-squared suggests a more stringent interval, i.e., lower, that will result in incorrect rejections of the null stability hypothesis in some cases. This can lead to multiple independent estimations in restricted samples, and is therefore less reliable.

4.2. Discussion and Implications

Besides the ES, other p.d.f. discussed and implemented in the empirical literature, mainly financial, have been considered. All can handle departures from Gaussianity to some extent. One that has been widely considered is the Student’s t and its asymmetric generalisation [16]. A multivariate generalisation is also possible, but limiting the co-moments of univariate distributions to the variance–covariance matrix. This p.d.f. has been implemented in the univariate case with some success, although the multivariate generalisation is somewhat limited. Besides, beyond fitting the observational data, it is difficult to work with it in other applications, like risk analysis or the stability test presented and discussed here.

Alternative ES specifications have been suggested in the literature, notably [5,6], that solve the problem of eventual negative values, but introduce new ones since they require complex non-linear corrections to produce a p.d.f. with zero mean and unitary variance, ready for empirical applications. Besides, beyond theoretical properties, the practical suitability of any p.d.f. has to be proven in practice, and this p.d.f. is challenging to implement computationally.

Non-standard Gaussian p.d.f.s have been estimated in several fields, notably finance. However, the issue of stability testing beyond the Gaussian framework has hardly ever been discussed in the literature, and the only available test statistics up to now assume that the underlying error distribution follows precisely a $N\left(0,1\right)$; see, e.g., [26]. The test derived in this research is a further motive to favour the ES distribution for empirical applications instead of eventual alternatives like [16], or [5,6]. Finally, it must be pointed out that testing the stability of sufficiently long periods may also be conducted in the framework of the likelihood analysis, i.e., implementing standard likelihood ratio tests.

The ES distribution and the test have been applied to weekly data. An immediate extension of the research would be to apply it to daily and even hourly frequencies, since the data are available. Extending the study to the multivariate framework is also a clear field for future research, as is solar radiation analysis at additional geographical locations. Finally, it must be stressed that a correct distribution is a requirement to establish proper forecasting intervals and conduct an accurate risk analysis.