In order to detect and quantify asymmetry of two time series, a novel cross-correlation coefficient is proposed based on recent asymmetric detrended cross-correlation analysis (A-DXA), which we called A-DXA coefficient. The A-DXA coefficient, as an important extension of DXA coefficient $\rho DXA$, contains two directional asymmetric cross-correlated indexes, describing upwards and downwards asymmetric cross-correlations, respectively. By using the information of directional covariance function of two time series and directional variance function of each series itself instead of power-law between the covariance function and time scale, the proposed A-DXA coefficient can well detect asymmetry between the two series no matter whether the cross-correlation is significant or not. By means of the proposed A-DXA coefficient conducted over the asymmetry for California electricity market, we found that the asymmetry between the prices and loads is not significant for daily average data in 1999 yr market (before electricity crisis) but extremely significant for those in 2000 yr market (during the crisis). To further uncover the difference of asymmetry between the years 1999 and 2000, a modified *H* statistic (*MH*) and Δ*MH* statistic are proposed. One of the present contributions is that the high *MH* values calculated for hourly data exist in majority months in 2000 market. Another important conclusion is that the cross-correlation with downwards dominates over the whole 1999 yr in contrast to the cross-correlation with upwards dominates over the 2000 yr.

As a typical regional regulatory electricity market, California electricity market has attracted the interest of researchers due to its complexity and asymmetry of price and loads, especially after the power crisis in the year of 2000. To investigate the asymmetry between those two time series in the California before and during the crisis, namely, in the years of 1999 and 2000, we first propose a new coefficient based on asymmetric detrended cross-correlation analysis (A-DXA). The so-called A-DXA coefficient contains two directional asymmetric cross-correlated indexes, describing upwards and downwards asymmetric cross-correlations, respectively. Differing from the original A-DXA may produce pseudo correlation between the two studied series, our proposed coefficient is a comprehensive utilization of the information of the directional covariance functions and directional variance functions of each series itself, instead of using the power-law between the above functions and the scales in A-DXA. Therefore, it is able to quantify the asymmetry between two series with unknown correlations. And then, we use the proposed two directional coefficients (upwards and downwards) together with the classical DXA coefficient to investigate the asymmetry between the prices and loads data in different months and the whole years of 1999 and 2000. By means of significant analysis, it indicates that the asymmetry of trends between the prices and loads is not significant in 1999 yr but extremely significant in 2000 yr. Next, we define a modified *H* statistic (*MH*) and *ΔMH* statistic to further access the degree and direction of the asymmetry. Drawing on the idea of Ref. 6, we use equally weighted method to compare the degree of asymmetry in every month and the whole year. The results show that the asymmetries of 2000 yr are much significant than those of 1999 yr. Finally, by investigating *ΔMH* statistic with four cases, an interesting finding is found that the cross-correlation with downwards dominates over the whole 1999 yr in contrast to the cross-correlation with upwards prevails over the 2000 yr.

## I. INTRODUCTION

The fundamental objective of electric power industry deregulation is to provide a competitive business environment. The operation of the electricity market is far more involved than the traditional markets are, since the electric commodity must be generated, distributed, and consumed in real-time under strict physical laws and extremely high reliability requirements.^{1} As the power system is considered a non-stationary system of non-linearity and weak relevance, electricity prices are not a result of long-term but instant, usually on an hourly interval, balance of supply and demand. As the price evolution and the load generation form a complex coupled dynamic process, many researchers have pointed out that there is a certain correlation between the adjacent price signals and the load signals.^{2–5} Since California broke out the largest power crisis in the United States after World War II in 2000 yr, the phenomenon of asymmetric correlation between the above two signals had been exposed as an ordinary behavior as in financial systems, where many studies have conducted spontaneously to explore the universal existence of asymmetric correlations of stock returns.^{6–9} Of which, Ang and Chen^{6} proposed a model-based method to estimate asymmetries in the correlations between stock portfolios and the US market. Hong and Zhou^{7} proposed a model-free method to address whether the actual data are asymmetric in correlation.

Advancement of physical statistics has greatly contributed to the researches on asymmetric correlations. As an effective methodologies to explore the cross-correlations between two signals, the detrended cross-correlation analysis (DCCA, also DXA)^{10} and its multifractal version—MF-DXA^{11} were proposed to be applied into lots of fields.^{12–22} To further quantify the cross-correlated levels, a new cross-correlation index called DXA coefficient, denoted as $\rho DXA$, was proposed by Zebende.^{23} In addition, Podobnik *et al*.^{24} provided a statistical test using the $\rho DXA$ to quantify the significance of the cross-correlations between two time series. Although the DXA and its related researches provided approaches to investigate the cross-correlation of two series, asymmetries of the cross-correlation are not in attention. Recently, Alvarez-Ramirez *et al*.^{25} presented a new method called asymmetric detrended fluctuation analysis (A-DFA) to assess asymmetries in the scaling behavior for an individual time series as a straightforward extension of the DFA method,^{26} in which Alvarez-Ramirez also found the asymmetry phenomenon exited in price signals of Australia Victoria regional electricity market. The A-DFA method and its modified versions have well worked on the asymmetries of the time signals itself of stock market^{27} and electricity market.^{28} Moreover, to assess asymmetric cross-correlation for two series each other, combining methods of MF-DFA and A-DFA, Cao *et al*.^{29} and Yin and Shang^{30} proposed model-free methods of multifractal asymmetric detrended cross-correlation analysis (MF-ADXA) and asymmetric multiscale detrended cross-correlation analysis (MS-ADXA), respectively. In these methods, bivariate upwards and downwards together with overall Hurst exponents were calculated by the power-law relationship between the average fluctuation function and time scale, which were able to describe the asymmetric cross-correlation for the two series with significant cross-correlations. However, if there is no power-law between the average fluctuations function and time scale, say, no any bivariate Hurst exponents will be obtained. Thus, the methods are invalid.

Therefore, in this paper, the goal of the present contribution is to assess and quantify the asymmetry for the two series with whether strong or weak cross-correlations. To this end, combining the modified A-DXA and A-DFA, we propose a new cross-correlation coefficient, which we called A-DXA coefficient. The new coefficient is a generalization of DXA coefficient.^{23} Besides the overall case, it contains two directional asymmetries indexes, namely, upwards and downwards coefficients. Since the proposed A-DXA coefficient showed of using the information of directional covariance function and directional variance function instead of the power-law of the covariance function and time scale, it can quantify the asymmetries of the two series without any limitation, i.e., no matter whether the cross-correlations are genuine or not. We then apply the proposed A-DXA coefficient to investigate the asymmetric cross-correlation between the prices and loads in California electricity market. We find that there is huge difference of asymmetry of those data between 1999 yr and 2000 yr. To further explore the difference, the second of our contribution is to propose a modified *H* statistic to quantify the asymmetry between the electricity data in the two years. One of the main conclusions is the degree of asymmetries the 2000 yr has which is much larger than those of the 1999 yr. Another interesting finding is that the direction of asymmetry of 1999 yr is different from the 2000 yr.

The remainder of this paper is organized as follows. Sec. II presents asymmetric detrended cross-correlation alaysis. Subsec. II A gives a detailed description of the modified A-DXA; based on it, the A-DXA coefficient is proposed as well as its properties are discussed in Subsec. II B. Sec. III applies the proposed A-DXA coefficient to California electrcity market. Sec. IV studies the measurement of asymmetric cross-correlations between the prices and loads. Sec. V concludes.

## II. ASYMMETRIC DETRENDED CROSS-CORRELATION ANALYSIS

In this section, we first review the A-DXA reported in Ref. 25. Next, a coefficient is proposed based on the A-DXA, which is the so-called A-DXA coefficient.

### A. Description of the modified A-DXA

The modified A-DXA procedure consists of four steps, which is combined with the idea of DXA and A-DFA. Assume that two time-series {*x*^{(1)}(*t*)} and {*x*^{(2)}(*t*)} have the same lengths *N*, and these series are of compact support.

**Step 1**: Construct the profile

where $\u27e8x(i)\u27e9$ denotes the mean value of the {*x*^{(i)}(*t*)}.

**Step 2**: Divided the time series {*x*^{(i)}(*t*)} and its profile {*y*^{(i)}(*t*)} (*i* = 1, 2) into *N _{n}* = [

*T*/

*n*] non-overlapping segments of equal lengths

*n*, respectively. Since the length

*N*of the series is often not a multiple of the considered time scale

*n*, a short part at the end of the profile may remain. In order not to disregard this part of the series, the same procedure is repeated starting from the opposite end. Thereby, 2

*N*segments are obtained altogether. Let $Sj(i)={sj,k(i),k=1,...,n}$ be the

_{n}*j*th sub-time series with length

*n*and $Yj(i)={yj,k(i),k=1,...,n}$ denote the according profile series in the

*j*th time interval,

*j*= 1, 2,…, 2

*N*. In the

_{n}*j*th segment, we have $sj,k(i)=x(i)[(j\u22121)n+k],yj,k(i)=y(i)[(j\u22121)n+k]$ for

*j*= 1, 2,…,

*N*and $sj,k(i)=x(i)[N\u2212(j\u2212Nn)n+k],yj,k(i)=y(i)[N\u2212(j\u2212Nn)n+k]$ for

_{n}*j*=

*N*+ 1,…, 2

_{n}*N*, where

_{n}*k*= 1, 2,…,

*n*.

**Step 3**: For each subtime-series $Sj(i)$ and its profile time series $Yj(i)$, the local linear trends can be calculated as $LSj(i)(k)=aSj(i)+bSj(i)k$ and $LYj(i)(k)=aYj(i)+bYj(i)k$, respectively, where *k* refers to the horizontal coordinate and *i* = 1 and 2. The linear $LSj(i)(k)$ is used to discriminate via the slope $bSj(i)$ whether the trend of the subtime series $Sj(i)$ is positive or negative. The linear fit $LYj(i)(k)$ is used to detrend the integrated time series $Yj(i)$. Then, the detrended covariance in the *j*th segment is as follows, where *j* = 1, 2,…, 2*N _{n}*

**Step 4**: The average detrended variance functions and the detrended covariance functions for several cases when the time *x*^{(i)} have picewise positive and negative linear trends are considered to assess the asymmetric correlation scaling properties for the series *x*^{(i)} itself and cross-correlation scaling properties for the two series between the series *x*^{(1)} and *x*^{(2)}. This trend discrimination is made by using the sign of the slope $bSj(i)$, say, $bSj(i)>0$ (resp. $bSj(i)<0$) refers to the time series *x*^{(i)} with positive (resp. negative) trend in the subtime series $Sj(i)$.

In this token, we can access the asymmetric cross-correlation for the two series by different trends of one time series. Then, the directional 2nd-order average fluctuation functions of the *f*_{12}(*n*, *j*) over the all segments are calculated as

where $Mi+=\u2211j=12Nnsign(bSj(i))+12$ and $Mi\u2212=\u2211j=12Nn\u2013[sign(bSj(i))\u22121]2$ are the numbers of subtime series {*x*^{(i)}(*t*)} (*i* = 1 or 2) with positive and negative trends, respectively. If $bSj(i)\u22600$ for all *j* = 1, 2,…, 2*N _{n}*, then $Mi++Mi\u2212=2Nn$.

The average 2nd order fluctuation functions in traditional DXA^{8} are calculated by

Thereby, if the cross-correlations exist, there are power-law relationships present between the above average fluctuation functions and the scale *n*, and some exponents are obtained consequently, that is

The *λ*, *λ*^{+}, and *λ*^{−} are the so-called bivariate directional Hurst exponents. In addition, one can analyze different kinds of asymmetric correlation natures according to the relationships between the exponents.^{29,30}

However, in the present case, the $FDXA2(n)$ ($F122+(n)$ and $F122\u2212(n)$) can take both positive and negative values, which is shown around the zero, then the cross-correlation is absent.^{17} Thus, we can do nothing by the exponent *λ* (*λ*^{+} and *λ ^{−}*) determined by the Eq. (5) due to the complex root. To solve this problem, most literatures

^{16,18,29,30}modified the Eq. (2) as

Nevertheless, we should point out that there are lots of weak correlated or even un-correlated series that present perfect correlation by this modification (for example, see Sec. III Fig. 2(c), also see Fig. 9 in Ref. 17). This, of course, is a false detection as these series are not expected to be correlated. By this regards, we may make mistake using Eq. (6) when we access the asymmetric correlation between two time series. A valid way to access the asymmetry between two time series with unknown correlations is a comprehensive utilization of the information of $F122+(n),F122\u2212(n)$, and $FDXA2(n)$ based on *f*_{12}(*n*, *j*) shown in Eq. (2) together with the directional fluctuation functions for each series itself, instead of using the exponents determined by Eq. (5). In this spirt, we propose a new coefficient based on A-DXA, introduced in Subsection II B.

### B. A-DXA coefficient

As mention above, the potential false exponents determined by Eq. (5) may interfere with our judgment on asymmetry of two series. Only the fluctuation functions can be used. Recently, a DXA cross-correlation coefficient $\rho DXA$ is introduced in Ref. 23 to test the significance of the cross-correlation between two signals. In Eq. (2), if *x*^{(1)}(*t*) = *x*^{(2)}(*t*), then the detrended covariance *f*_{12}(*n*, *j*) degenerates into the detrended variance as

Thereby, the 2nd-order average fluctuation function of series *x*^{(1)}(*t*) and *x*^{(2)}(*t*) is computed by

The dimensionless coefficient $\rho DXA$ is dependent on the two time series of length *T* and the time scale *n*. In our work, we take *n* ranging from 6 to *T*/4 with an equal logarithmic increment. Here, by using the directional 2nd-order average of the covariance $F122+(n)$ and $F122\u2212(n)$ (determined by Eq. (3)) together with the directional variance (denoted as $Fi+(n)$ and $Fi\u2212(n)$, proposed in Ref. 21, are determined by Eq. (10)), two directional coefficients are defined by Eqs. (11) and (12), which we called A-DXA coefficient, denoted as $\rho ADXA+$ and $\rho ADXA\u2212$, respectively.

The dimensionless $\rho ADXA+$ and $\rho ADXA\u2212$ describe the long-term cross-correlation with upwards and downwards of time series {*x*^{(i)}(*t*)} (*i* = 1 or 2), respectively. Together with $\rho DXA$, which denotes overall long-term cross-correlation, the three coefficients can make us acquire the asymmetry between the time series {*x*^{(1)}(*t*)} and {*x*^{(2)}(*t*)}.

In practice, if $\rho ADXA+(n)=\rho ADXA\u2212(n)$ for every time-scale *n*, then the cross-correlations between the two time series are symmetric. By contrast, if $\rho ADXA+(n)\u2260\rho ADXA\u2212(n)$, then the cross-correlations between the two time series are asymmetric, which implies that the cross-correlations are different when the trending of the time series {*x*^{(i)}(*t*)} is positive than when it is negative.

## III. A-DXA FOR CALIFORNIA ELECTRICITY MARKET

We put our interest on California electricity market, which is a typical regional regulated market in the world. The power crisis broke out in 2000 yr made the prices skyrocket over $1200/(MWh), leading to severe electricity shortage and wide-area blackout. The relationship of the prices and loads had changed greatly before the crisis and during the crisis. We put the hourly data in Fig. 1. One notes that the loads are relatively stable during the two years compared with mutation of the prices. We believe that our research on asymmetric cross-correlation of them is helpful to explore market operation mechanism of California electricity market. To this end, we start our study from the cross-correlation of the prices and loads in the years of 1999 and 2000 by the method of A-DXA (the upwards and downwards cases are obtained based on the prices series, i.e., the *x*^{(i)}(t) in Eqs. (11) and (12) denotes prices series), respectively, and present our results in Fig. 2. As expected, in the California's 1999 yr electricity market, prior to the crisis, as shown in Fig. 2(a), the nearly perfect fitting lines suggest that there are cross-correlations between the prices and the loads. According to A-DXA using Eq. (2), as demonstrated in Fig. 2(b), the (directional) 2nd-order covariance of the prices and loads in California's 2000 yr market fluctuates around 0; as a result, no power-law exists between $FDXA2(n)$($F122+(n),F122\u2212(n)$) and *n*. Therefore, we conclude that there is no significant cross-correlation between the prices and loads during the crisis, which is consistent with our previous study.^{5,12,31} However, if we apply A-DXA with Eq. (6) to the same data, as shown in Fig. 2(c), false cross-correlations are detected. By this regard, when we further investigate the asymmetry of prices and loads in the California's 2000 yr market, the existed methods are invalid due to no power-law between $FDXA2(n)$($F122+(n),F122\u2212(n)$) and *n*.

Next, we apply the proposed A-DXA coefficient to analyze the asymmetry of the prices and loads between the California's 1999 and 2000 yrs markets, and report the results in Figs. 3–5. In particular, Fig. 3 shows the A-DXA coefficients with overall, upwards, and downwards calculated for the daily average data of the whole year. Figs. 4 and 5 show the A-DXA coefficients of the hourly data in the chosen four months. For the whole year data, the differences between upward and downward coefficients are not obvious for smaller scale intervals, while the two coefficients enlarge gaps from about 30 days and 16 days for the years 1999 and 2000, respectively. For the hourly data in 1999 yr market illustrated in Fig. 4, the nice linear relationship between the *F _{DXA}*(

*n*) ($F122+(n),F122\u2212(n)$) and

*n*in log-log plot explains that there are significant cross-correlation existed in prices and loads. By contrast, as shown in inset plots in Figs. 5(c) and 5(d),

*F*(

_{DXA}*n*) ($F122+(n),F122\u2212(n)$) around the zero implies that there is no power-law cross-correlation between the prices and loads of August and November in 2000 yr. Furthermore, we notice that the degree of asymmetry is different in different months though there are differences existed among the three A-DXA coefficients for most scale intervals. It demonstrates that the coexistence of symmetry and asymmetry may be found in both California's 1999 and 2000 yrs markets.

In order to test whether the asymmetric cross-correlation determined by the proposed A-DXA coefficient is significant or not, we conduct hypothesis tests for this purpose. In practice, for the A-DXA coefficient with overall, upwards, and downwards of the prices, three null hypothesis H_{0} are constructed, namely, H_{0}^{(1)}: *ρ*^{+}_{ADXA} = *ρ _{DXA}*; H

_{0}

^{(2)}:

*ρ*

^{−}

_{ADXA}=

*ρ*; H

_{DXA}_{0}

^{(3)}:

*ρ*

^{+}

_{ADXA}=

*ρ*

^{−}

_{ADXA}=

*ρ*. The

_{DXA}*p*-values are reported in Table I. One notes that there is a huge different significance in different time intervals between the two years. Specifically, for the 1999 yr market, the hourly prices and loads suffer severe asymmetry in February in contrast to contemporaneity in the 2000 yr, in which all of the three

*p*-values show that we cannot reject null hypothesis. The same situation also occurred in October. However, for the average daily prices and loads of the whole year, we have enough reasons to reject the null hypothesis that DXA coefficients calculated by the overall, upwards, and downward case are equal and believe that there is a severe trend asymmetry of average daily prices and loads existed in the 2000 yr market. Instead, we should concede that the trend of average daily prices and loads is symmetrical in 1999 market due to the

*p*-values close to 1.

Time interval . | Ca. 1999 electricity market . | Ca. 2000 electricity market . | ||||
---|---|---|---|---|---|---|

ρ^{+}_{ADXA} = ρ
. _{DXA} | ρ^{−}_{ADXA} = ρ
. _{DXA} | ρ^{+}_{ADXA} = ρ^{−}_{ADXA} = ρ
. _{DXA} | ρ^{+}_{ADXA} = ρ
. _{DXA} | ρ^{−}_{ADXA} = ρ
. _{DXA} | ρ^{+}_{ADXA} = ρ^{−}_{ADXA} = ρ
. _{DXA} | |

Jan (h) | 0.4144 | 0.3252 | 0.2329 | 0.7597 | 0.8767 | 0.9022 |

Feb (h) | 0.0131^{b} | 0.0280^{b} | 0.0003^{a} | 0.4800 | 0.6962 | 0.5476 |

Mar (h) | 0.7094 | 0.7152 | 0.7838 | 0.5068 | 0.2624 | 0.2164 |

Apr (h) | 0.2251 | 0.8003 | 0.4089 | 0.4242 | 0.0320^{b} | 0.0124^{b} |

May (h) | 0.0993 | 0.1376 | 0.0147^{b} | 0.3005 | 0.6986 | 0.5609 |

Jun (h) | 0.7228 | 0.3178 | 0.6172 | 0.8011 | 0.8687 | 0.9313 |

Jul (h) | 0.9973 | 0.4549 | 0.7330 | 0.8644 | 0.9254 | 0.9652 |

Aug (h) | 0.5417 | 0.5149 | 0.7766 | 0.1644 | 0.0061^{a} | 0.0138^{b} |

Sep (h) | 0.5328 | 0.0156^{b} | 0.0718 | 0.4107 | 0.5066 | 0.3442 |

Oct (h) | 0.0089^{a} | 0.2692 | 0.0021^{a} | 0.2266 | 0.6530 | 0.5055 |

Nov (h) | 0.7477 | 0.3158 | 0.6110 | 0.7488 | 0.8472 | 0.8741 |

Dec (h) | 0.6229 | 0.9702 | 0.8984 | 0.2287 | 0.3899 | 0.1416 |

Whole (d) | 0.9306 | 0.9785 | 0.9933 | 0.0081^{a} | 0.1078 | 0.0005^{a} |

Time interval . | Ca. 1999 electricity market . | Ca. 2000 electricity market . | ||||
---|---|---|---|---|---|---|

ρ^{+}_{ADXA} = ρ
. _{DXA} | ρ^{−}_{ADXA} = ρ
. _{DXA} | ρ^{+}_{ADXA} = ρ^{−}_{ADXA} = ρ
. _{DXA} | ρ^{+}_{ADXA} = ρ
. _{DXA} | ρ^{−}_{ADXA} = ρ
. _{DXA} | ρ^{+}_{ADXA} = ρ^{−}_{ADXA} = ρ
. _{DXA} | |

Jan (h) | 0.4144 | 0.3252 | 0.2329 | 0.7597 | 0.8767 | 0.9022 |

Feb (h) | 0.0131^{b} | 0.0280^{b} | 0.0003^{a} | 0.4800 | 0.6962 | 0.5476 |

Mar (h) | 0.7094 | 0.7152 | 0.7838 | 0.5068 | 0.2624 | 0.2164 |

Apr (h) | 0.2251 | 0.8003 | 0.4089 | 0.4242 | 0.0320^{b} | 0.0124^{b} |

May (h) | 0.0993 | 0.1376 | 0.0147^{b} | 0.3005 | 0.6986 | 0.5609 |

Jun (h) | 0.7228 | 0.3178 | 0.6172 | 0.8011 | 0.8687 | 0.9313 |

Jul (h) | 0.9973 | 0.4549 | 0.7330 | 0.8644 | 0.9254 | 0.9652 |

Aug (h) | 0.5417 | 0.5149 | 0.7766 | 0.1644 | 0.0061^{a} | 0.0138^{b} |

Sep (h) | 0.5328 | 0.0156^{b} | 0.0718 | 0.4107 | 0.5066 | 0.3442 |

Oct (h) | 0.0089^{a} | 0.2692 | 0.0021^{a} | 0.2266 | 0.6530 | 0.5055 |

Nov (h) | 0.7477 | 0.3158 | 0.6110 | 0.7488 | 0.8472 | 0.8741 |

Dec (h) | 0.6229 | 0.9702 | 0.8984 | 0.2287 | 0.3899 | 0.1416 |

Whole (d) | 0.9306 | 0.9785 | 0.9933 | 0.0081^{a} | 0.1078 | 0.0005^{a} |

^{a}

Denotes 0.01 significant level.

^{b}

Denotes 0.05 significant level.

## IV. MEASUREMENT OF ASYMMETRIC CROSS-CORRELATIONS

To quantify the degree of asymmetric cross-correlations, an *H* statistic was proposed by Ang and Chen.^{6} The statistic has several advantages over graphical approaches, such as, which can formally summarize the magnitudes of correlation asymmetries by providing a succinct numerical measure. However, the statistic is proposed according to a given null distribution, such as normal distribution. Hence, one can only measure the degree of the exceedance asymmetric correlations by comparing with those implied by the null distribution. Here, we propose a new statistic inspired by *H* statistic, which we called modified *H* statistic (*MH*), defined as

for which the weights *W*(*n _{i}*) ≥ 0 satisfy: $\u2211i=1NW(ni)=1$.

*ρ*(

*n*) denotes standard correlation coefficient calculated by the DXA method with overall case, determined by Eq. (9). $\rho \u2322(ni)$ is the asymmetric correlation coefficient calculated by A-DXA with upwards and downwards case, shown in Eqs. (11) and (12). By this means, two kinds of

_{i}*MH*statistics are employed to describe the asymmetry of two series, i.e., upwards and downwards of

*MH*, denoted as $MH+$ and $MH\u2212$, respectively, determined by

In virtue of the proposed statistics defined from two directions, the $MH+$ measures the degree of asymmetry between the upwards $\rho ADXA+$ and the standard $\rho DXA$; meanwhile, the $MH\u2212$ quantifies the degree of asymmetry between the downwards $\rho ADXA\u2212$ and the standard $\rho DXA$. To synthesize the impact of $MH+$ and $MH\u2212$ on the asymmetry, *MH* is set to $(MH+)2+(MH\u2212)2$, which represents a non-linear average of $MH+$ and $MH\u2212$. The larger of the *MH*, for which the asymmetric degree is stronger.

For the proposed statistic, a key point is how to determine the weights of the *i*th square deviation between the two correlation coefficients of the up(down)-ward case and overall case. According to Ang and Chen,^{6} three kinds of weights are used. Here, we also employ the three notions:

where *N* is the number of time scales *n*, in our work, we take *N* = 20. As shown in Eq. (15), the first kind of weight *W*_{1}(*n _{i}*) is determined by the notion of using weights inversely proportional to the sample variance of $\rho \u2322(ni)$, $\sigma 2(\rho \u2322(ni))$ using a standardized measure of the inverse of $\sigma 2(\rho \u2322(ni))$. The larger the sampling variance of $\rho \u2322(ni)$, the smaller the weight placed on asymmetry. It indicates that the $\rho \u2322(ni)$ with small deviation dominates the

*MH*. Because increasing the number of samples increases the accuracy of the covariance estimation, the second set of weight

*W*

_{2}(

*n*) is proposed to depend on the sample size, which is calculated by the weights proportional to the number of observations. In Eq. (16),

_{i}*N*is the number of observations used in calculating $\rho \u2322(ni)$. This choice of weights places more emphasis on exceedance correlations for which more data are available. The third weight

_{i}*W*

_{3}(

*n*) shown in Eq. (17) is equally weighted by the number of $\rho \u2322(ni)$. The simple notion places greater weight on observations in the extreme tails of the distribution than the previous choice of weights. Due to the enormous influence of some extreme

_{i}*n*

_{i}on the correlation, the W

_{3}is the most reasonable weight for calculating the

*MH*statistics and we prefer this weight in our following analysis. However, in Tables II and III, we show all of our results of calculated $MH+,MH\u2212$, and

*MH*statistics for five chosen time intervals. One notes that the results to be robust to different choices of weights for 1999 yr but huge fluctuations of them for 2000 yr. By using W

_{3}, the

*MH*statistics of 12 months and whole year between the two years are compared in Fig. 6. As expected, the degree of asymmetry between the prices and loads during the crisis (2000 yr) is larger than the normal year.

Ca.1999 market . | W_{1}: Weighted by σ^{2}(ρ)
. _{i} | W_{2}: Weighted by number of observations
. | W_{3}: Equally weighted
. | ||||||
---|---|---|---|---|---|---|---|---|---|

Time interval . | MH ^{+}
. | MH ^{−}
. | MH
. | MH ^{+}
. | MH ^{−}
. | MH
. | MH ^{+}
. | MH ^{−}
. | MH
. |

February (h) | 0.0419 | 0.0671 | 0.0791 | 0.0462 | 0.0602 | 0.0759 | 0.0414 | 0.0560 | 0.0696 |

April (h) | 0.0798 | 0.0861 | 0.1174 | 0.1179 | 0.1222 | 0.1698 | 0.0884 | 0.0959 | 0.1304 |

August (h) | 0.0743 | 0.0771 | 0.1071 | 0.0644 | 0.1429 | 0.1568 | 0.0670 | 0.1056 | 0.1251 |

October (h) | 0.1847 | 0.1165 | 0.2184 | 0.2499 | 0.1439 | 0.2883 | 0.2001 | 0.1262 | 0.2366 |

Whole year (d) | 0.0808 | 0.1065 | 0.1337 | 0.0970 | 0.1387 | 0.1693 | 0.0914 | 0.1196 | 0.1505 |

Ca.1999 market . | W_{1}: Weighted by σ^{2}(ρ)
. _{i} | W_{2}: Weighted by number of observations
. | W_{3}: Equally weighted
. | ||||||
---|---|---|---|---|---|---|---|---|---|

Time interval . | MH ^{+}
. | MH ^{−}
. | MH
. | MH ^{+}
. | MH ^{−}
. | MH
. | MH ^{+}
. | MH ^{−}
. | MH
. |

February (h) | 0.0419 | 0.0671 | 0.0791 | 0.0462 | 0.0602 | 0.0759 | 0.0414 | 0.0560 | 0.0696 |

April (h) | 0.0798 | 0.0861 | 0.1174 | 0.1179 | 0.1222 | 0.1698 | 0.0884 | 0.0959 | 0.1304 |

August (h) | 0.0743 | 0.0771 | 0.1071 | 0.0644 | 0.1429 | 0.1568 | 0.0670 | 0.1056 | 0.1251 |

October (h) | 0.1847 | 0.1165 | 0.2184 | 0.2499 | 0.1439 | 0.2883 | 0.2001 | 0.1262 | 0.2366 |

Whole year (d) | 0.0808 | 0.1065 | 0.1337 | 0.0970 | 0.1387 | 0.1693 | 0.0914 | 0.1196 | 0.1505 |

Ca.2000 market . | W_{1}: Weighted by σ^{2}(ρ)
. _{i} | W_{2}: Weighted by number of observations
. | W_{3}: Equally weighted
. | ||||||
---|---|---|---|---|---|---|---|---|---|

Time interval . | MH^{+}
. | MH^{−}
. | MH
. | MH^{+}
. | MH^{−}
. | MH
. | MH^{+}
. | MH^{−}
. | MH
. |

February (h) | 0.1029 | 0.1160 | 0.1551 | 0.2389 | 0.1237 | 0.2690 | 0.1509 | 0.1009 | 0.1815 |

April (h) | 0.0428 | 0.0852 | 0.0953 | 0.0577 | 0.2452 | 0.2519 | 0.0491 | 0.1632 | 0.1704 |

August (h) | 0.1578 | 0.0938 | 0.1836 | 0.5645 | 0.1644 | 0.5879 | 0.3476 | 0.1542 | 0.3803 |

October (h) | 0.0955 | 0.0739 | 0.1208 | 0.1046 | 0.0735 | 0.1279 | 0.0859 | 0.0731 | 0.1128 |

Whole year (d) | 0.3086 | 0.3712 | 0.4827 | 0.5193 | 0.3024 | 0.6009 | 0.3543 | 0.3360 | 0.4883 |

Ca.2000 market . | W_{1}: Weighted by σ^{2}(ρ)
. _{i} | W_{2}: Weighted by number of observations
. | W_{3}: Equally weighted
. | ||||||
---|---|---|---|---|---|---|---|---|---|

Time interval . | MH^{+}
. | MH^{−}
. | MH
. | MH^{+}
. | MH^{−}
. | MH
. | MH^{+}
. | MH^{−}
. | MH
. |

February (h) | 0.1029 | 0.1160 | 0.1551 | 0.2389 | 0.1237 | 0.2690 | 0.1509 | 0.1009 | 0.1815 |

April (h) | 0.0428 | 0.0852 | 0.0953 | 0.0577 | 0.2452 | 0.2519 | 0.0491 | 0.1632 | 0.1704 |

August (h) | 0.1578 | 0.0938 | 0.1836 | 0.5645 | 0.1644 | 0.5879 | 0.3476 | 0.1542 | 0.3803 |

October (h) | 0.0955 | 0.0739 | 0.1208 | 0.1046 | 0.0735 | 0.1279 | 0.0859 | 0.0731 | 0.1128 |

Whole year (d) | 0.3086 | 0.3712 | 0.4827 | 0.5193 | 0.3024 | 0.6009 | 0.3543 | 0.3360 | 0.4883 |

The last topic we care about is the direction of asymmetry between the prices and loads series for the different time intervals in the two years. To do so, $\Delta MH$ statistic is difened as follows:

$\Delta MH>0$ refers positive asymmetry and $\Delta MH<0$ refers negative asymmetry. We will prove that there are four cases about the sign of $\Delta MH$ and the relationship of the A-DXA coefficients with overall, upwards, and downwards trends of prices series as below.

Case 1: $\Delta MH>0\u2009\u2009and\u2009\u2009\rho +>max(\rho ,\rho \u2212)$; Case 2: $\Delta MH>0\u2009\u2009and\u2009\u2009\u2009\rho +<min(\rho ,\rho \u2212)$;

Case 3: $\Delta MH<0\u2009\u2009and\u2009\u2009\rho \u2212>max(\rho ,\rho +)$; Case 4: $\Delta MH<0\u2009\u2009and\u2009\u2009\rho \u2212<min(\rho ,\rho +)$. where *ρ*, *ρ*^{+}, and *ρ*^{−} denote weighted average of $\rho DXA$, $\rho ADXA+$, and $\rho ADXA\u2212$ over the time scale *n*, respectively. To this end

When $\Delta MH>0$, if $\rho +>\rho \u2212$, then $2\rho <\rho ++\rho \u2212<2\rho +$, hence $\rho +>\rho $; if $\rho +<\rho \u2212$ then $2\rho >\rho ++\rho \u2212>2\rho +$, hence $\rho +<\rho $, the cases 1 and 2 hold. Similarly, when $\Delta MH<0$, the cases 3 and 4 hold. The case 1 and case 4 mean that $\rho ADXA+$ dominates the asymmetric correlation, which implies that the cross-correlation with simultaneous rising of the prices and loads is stronger than falling of them. On the contrary, the cases 2 and 3 purport that $\rho DXA\u2212$ prevails the asymmetric correlation, which indicates that the cross-correlation with simultaneous falling is stronger than rising of them. Following, to investigate the four cases in different time intevals of the California 1999 and 2000 yrs markets, our preferred form of the *ΔMH* statistic uses the weight W_{3} presented in Eq. (13), as listed in Table IV.

Markets . | Jan . | Feb . | Mar . | Apr . | May . | Jun . | Jul . | Aug . | Sep . | Oct . | Nov . | Dec . | Whole year . | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Ca. 1999 | △MH | − | − | + | − | − | + | − | − | − | + | + | − | − |

ρ^{+}-ρ^{-} | − | + | − | + | + | − | − | − | − | + | − | + | − | |

Case | 3 | 4 | 2 | 4 | 4 | 2 | 3 | 3 | 3 | 1 | 2 | 4 | 3 | |

Ca. 2000 | △MH | − | + | − | − | + | − | − | + | + | + | + | + | + |

Ρ^{+}-ρ^{-} | + | − | − | − | + | + | + | − | + | + | − | + | + | |

Case | 4 | 2 | 3 | 3 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 1 | 1 |

Markets . | Jan . | Feb . | Mar . | Apr . | May . | Jun . | Jul . | Aug . | Sep . | Oct . | Nov . | Dec . | Whole year . | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Ca. 1999 | △MH | − | − | + | − | − | + | − | − | − | + | + | − | − |

ρ^{+}-ρ^{-} | − | + | − | + | + | − | − | − | − | + | − | + | − | |

Case | 3 | 4 | 2 | 4 | 4 | 2 | 3 | 3 | 3 | 1 | 2 | 4 | 3 | |

Ca. 2000 | △MH | − | + | − | − | + | − | − | + | + | + | + | + | + |

Ρ^{+}-ρ^{-} | + | − | − | − | + | + | + | − | + | + | − | + | + | |

Case | 4 | 2 | 3 | 3 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 1 | 1 |

One notes in Table IV indicate that most months in 1999 market possess stronger cross-correlation when the price series has a negative trending than when it has a positive one, which leads to the whole year of 1999 market belongs to the case 3. Instead, most months in 2000 yr market possess stronger cross-correlation when the price series has a positive trending than when it has a negative one, which brings about the case 1 for the whole year of 2000 market.

As mentioned above, the cases 1 and 4 refer the cross-correlations with positive trends of the prices dominated, and cases 2 and 3 refer those with negative trends of the prices dominated. The trend change process in the 12 successive months in each year can be considered as a Markov chain, and we can calculate transition probabilities about it. In that spirit, the one-step transition probability matrix of the 1999 (*P*_{1}) and 2000 (*P*_{2}) markets is obtained as follows:

In each matrix, “+” denotes positive trends and “−” denotes negative trends, of which, the first row in *P*_{1} shows in California 1999 yr market, there are 1/4 probability from the cross-correlations with positive trends in current month to those with positive trends in next month and 3/4 probability from positive trends in current month to negative trends in next month. We can conclude that the transition process in both of the two years is ergodic, which means the cross-correlations with positive or negative trends in the current month free transfers to the next month of cross-correlations with positive or negative trends. An interesting finding is that the “sum” of two probability (here, we define $14\u229547=1+44+7=511$, as 11 is the total number of the transition) in the first column of *P*_{1} is less than those in the second column ($34\u229537=611$) in contrast to $36\u229535>36\u229525$ in *P*_{2}. The former relationship suggests that the cross-correlation with downwards (refers to negative trends of price series) dominates over the whole year, coincided with the fact of case 3 for the year of average daily data in 1999 yr market. The latter relationship demonstrates that the the cross-correlation with upwards (refers to positive trends of price series) dominates over the whole year, which is consistent with the case 1 for the 2000 yr average daily data. In fact, one can get the same conclusion from the limiting distribution about *P*_{1} and *P*_{2}, which are denoted as *π*_{1} and *π*_{2}, respectively, say, $\pi 1=(1637,2137)$ and $\pi 2=(611,511)$, in which $1637<2137$ and $611>511$.

## V. CONCLUSIONS

We developed a DXA coefficient extension to explore the existence of asymmetric cross-correlation between two time series. The so-called A-DXA coefficient was proposed using the information of directional covariance function of two time series together with the directional variance function of each series itself comprehensively. The new coefficient is able to detect the asymmetry in two time series, no matter whether the cross-correlation is significant or not. The empirical results on the California electricity markets indicate that the asymmetry of trends between the prices and loads is not significant in 1999 yr daily market but extremely significant in 2000 yr market. In order to quantify the degree of asymmetry, a modified *H* statistic (*MH*) was defined. Synthesizing the impact of upward *MH*^{+} and downward *MH*^{−}, the *MH* shows that the degree of the asymmetries in the 2000 yr is much bigger than those in the 1999 yr. In addition, by investigating the direction of the asymmetry, the proposed *ΔMH* statistic suggests that the cross-correlation with downwards dominates over the whole 1999 yr in contrast to the cross-correlation with upwards dominates over the 2000 yr. The interesting finding uncovers that the relationship between the prices and loads has changed essentially before and during the electricity crisis.

## ACKNOWLEDGMENTS

The author wishes to thank the anonymous reviewers and the handling editor Dr. David Campbell for their comments and suggestions, which led to a great improvement to the presentation of this work.

This work was supported by National Natural Science Foundation of China (Grant No. 31501227), Social Science Foundation of Hunan Province (Grant No. 15YBA204), the Key R & D Project Funds of Hunan Province, China (Grant No. 2015JC3098), and Young Scholar Project Funds of the Department of Education of Hunan Province, China (Grant No. 14B087).