# Modeling and Forecasting Electricity Demand and Prices: A Comparison of Alternative Approaches

Authors: Ismail Shah; Hasnain Iftikhar; Sajid Ali
Journal: Journal of Mathematics
 (2022)

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## Abstract

Electricity demand and price forecasting are key components for the market participants and system operators as precise forecasts are necessary to manage power systems effectively. However, forecasting electricity demand and prices are challenging due to their specific features, such as high frequency, volatility, long trend, nonconstant mean and variance, mean reversion, multiple seasonalities, calendar effects, and spikes/jumps. Thus, the main aim of this study is to propose models that can efficiently forecast electricity demand and prices. To this end, the time series (demand/price) is divided into two components. The first component is considered a deterministic component that includes a trend, yearly, seasonal, and weekly periodicities, calendar effects, and lagged exogenous information and is modeled by parametric and nonparametric approaches. The second component is known as a stochastic (residual) component that is estimated using univariate autoregressive (AR) and multivariate vector autoregressive (VAR) models. The estimation of these models is carried out by four different estimation methods, including ordinary least squares (O), Lasso (L), Ridge (R), and Elastic-net (E). The proposed modeling scheme is applied to Nordic electricity demand and price time series, and one-day-ahead out-of-sample forecasts are obtained for a whole year. Besides descriptive statistics, a statistical significance test is also used to evaluate the models’ forecasting accuracy. The results suggest that the proposed methodology effectively forecasts the price and demand for electricity. In addition, the choice of the estimation procedure used for both deterministic and stochastic components has a significant effect on the forecasting results. Furthermore, multivariate vector autoregressive gives superior performance compared to univariate autoregressive models.


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## Body

## 1. Introduction

In deregulated energy markets, day-ahead forecasting for the electricity demand and prices plays a fundamental role in formulating economic, reliable, and secure operation policies for the energy system. Electricity demand and price forecasts are vital components in developing any power planning model, especially in today’s reformed power system. Due to the global liberalization of energy markets, electricity can be bought and sold like any other commodity. However, unlike most commodities, electricity is not storable in large amount and need to deliver instantaneously when produced. In addition, electricity prices and demand generally have a complicated structure. To minimize risk and maximize profits, electricity market participants are highly interested in predicting future electricity prices and demand [1, 2].In today’s electricity markets, participants are generally interested in various forecasting horizons, including short, medium, and long term. Short-term forecasts commonly refer to the forecast obtained for a period of a few minutes to a week. These forecasts are generally used for reliable and efficient management of the electrical network, load management, and disposition of units [3]. Medium-term forecasts typically vary from a week to a few months and are widely used for unit maintenance planning, fuel purchase planning, and sales evaluation [4]. Finally, the long-term forecasts typically cover monthly and annual highs/lows that are frequently used for risk management and planning determinations. Short-term forecasts have received more attention in the literature as in many power markets, daily demand and prices are determined by auctions (semi-hourly or hourly) on the day before the physical delivery [5, 6].The time series of electricity demand and price has a complex structure. They exhibit specific features, including a long trend, multiple periodicities (daily, weekly, and annually), nonconstant mean and variance, spikes or jumps, calendar effects, and volatility (Figure1). For example, the linear and nonlinear trends, as well as the yearly periodicity, can be seen in Figure 1. The time series of demand and prices during working hours are more volatile than nonworking hours. On the other hand, the load curves are relatively different according to the days of the week; the weekdays have higher consumption than the weekends. In addition, the electricity demand is higher in winter and summer due to the increased use of heating and air conditioning, respectively, and is lower in spring and fall. The electricity demand and prices are also affected by the calendar effect (bridging or bank holidays); demand/prices are low during public holidays and bridging holidays. Moreover, the demand and price time series shows high volatility and unpredicted spikes/jumps caused by different factors.Figure 1
Specific features of electricity demand and price for the period January 1, 2013, to December 31, 2016 (a) prices series with superimposed linear (dark-green) and a nonlinear (red) trend, (b) demand annual seasonality, (c) average weekly and bank holidays curves, (d) average seasonal curves, (e) hourly demand, and (f) hourly prices box plots.
(a)(b)(c)(d)(e)(f)In the literature for electricity demand and price forecasting, various statistical, machine learning, econometric, and hybrid models have been proposed that vary in methodology, complexity, and performance [7–15]. They include various types of linear time series models (AR, ARMA, ARIMA, SARIMA, and ARIMAX) [16–21] and nonlinear time series models (NPAR, ARCH, GARCH, SGARCH, TGARCH, and EGARCH) [22–25]. Parametric and nonparametric regression-type (multiple, local polynomial, kernel, smoothing spline, quantile regression) models are easy to implement and are extensively used for electricity demand and price forecasting [15, 26–31]. For example, Ref. [32] combines wavelet, SARIMA, and GJR-GARCH models to forecast electricity prices. The wavelet transform is used to convert the price series into different components modeled and forecasted separately using the SARIMA and GJR-GARCH model. The electricity price time series of New South Wales is considered to assess the performance of the proposed model that shows superior forecasting accuracy. Using ARIMA and SARIMA, Ref. [33] forecasted hourly electricity demand using data from Assam, India. The models are fitted using a training set, and the forecasting results are validated on a testing set. The results suggested that SARIMA outperforms the nonseasonal counterpart. In addition, models based on exponential smoothing (simple, double, and triple Holt’s winters models) that account for various periodicities [34–40] are often used for forecasting purposes. For instance, Ref. [41] analyzed the parameters space of the multiple seasonal Holt-Winters models applied to electricity demand in Spain. The authors considered different time windows corresponding to varying periods of the year and different training set sizes. Artificial intelligence models have also been used to predict day-ahead electricity demand and price [42–46], as well as state-space models [47–50]. For example, Ref. [51] suggested a deep learning-based framework for forecasting electricity demand. Cluster analysis is used to convert electricity consumption data to obtain season-based segmented data. Based on the season, day, and interval data, long-short-term memory network multi-input multi-output models are trained to forecast electricity demand for the electricity consumption data of Union Territory Chandigarh, India. Other competing models such as support vector regression models, artificial neural network, and recurrent neural network are also used in the analysis. To forecast electricity demand or prices, various researchers combined the characteristics of two or more models and built a novel model generally known as a hybrid model [52–58]. For multistep electricity price forecasting, an adaptive deterministic and probabilistic interval forecasting system was proposed by Ref. [59]. The authors provided two cases and one comparative study to assess the performance of their proposal. The results suggested the proposal can be very helpful for the power market decision-makers. Some authors study the effect of different environmental and globalization trends [60, 61]. For instance, using panel estimation techniques, Ref. [61] investigated the effect of environmental-related technologies on energy demand and energy efficiency. The study results suggested that energy consumption decreases with increased environmental technologies. In addition, environmental technology plays a crucial role in improving energy efficiency by reducing energy intensity. In general, every model has its own functional and structural form, and the forecasting performance varies from market to market [62–64].This research compares various modeling approaches of parametric and nonparametric, univariate, and multivariate time series for the one-day-head forecast of electricity demand and prices. Accurate forecasting can be very helpful for the traders (buyers and suppliers) to optimize their bidding strategies to maximize their gains and to use the resources required for electricity generation more effectively. Consequently, this will also benefit the end-user in terms of reliable and economical electricity facilities. To this end, different estimation techniques are also discussed in this research work that suggests that forecasting accuracy can be improved by using these estimation techniques. Moreover, descriptive measures and statistical tests examine the overall accuracy of the day-ahead forecast.To this end, the main contribution of this work comprises the following:(i)
The current study provides a comprehensive and detailed comparison of different models within the approach, assuming that the dynamics of electricity demand and prices can be decomposed into deterministic and stochastic components.(ii)
The forecasting performance of different univariate and multivariate models within the framework is compared using parametric and nonparametric estimation techniques.(iii)
A parametric and nonparametric estimation technique is used for the deterministic component.(iv)
In the case of the stochastic component, four different estimation techniques are used to see their performance on the forecasting results.(v)
Finally, the significance analysis for the difference in prediction accuracy for different models is also considered in this study.The rest of the article is structured as follows: Section2 describes the proposed modeling methodology and the estimation procedure of each model. An application to the electricity demand and prices from the Nord Pool electricity market using the proposed modeling methodology is given in Section 3. The results are also discussed in the same section. Finally, the conclusions are shown in Section 4.


## 2. Methods

In this section, we introduce the general modeling framework, as well as the different univariate and multivariate models. The details about the estimation procedure are also provided in the following.


### 2.1. Modeling Framework

This study used different parametric and nonparametric univariate and multivariate time series models for one day-ahead electricity demand and price forecasts. The log demand and price series forhth hour and ith day are denoted by logDh,i and logPh,i and is modeled as(1)logDh,i=Gh,id+Şh,id,logPh,i=Gh,ip+Şh,ip,where Gh,id is a deterministic component and Şh,id is a residual or stochastic component. The superscripts d and p are used for demand and price time series, respectively. The Gh,id/Gh,ip (deterministic component) comprises a long-time trend, weekly, seasonal, and annual cycles, bank holidays days, and the lagged price/demand series and is modeled as(2)Gh,id=th,i+yh,i+sh,i+wh,i+bh,i+ph,i−1,(3)Gh,ip=th,i+yh,i+wh,i+bh,i+dh,i−1,where th,i represent the long trend component, yh,i is the yearly cycles, sh,i denotes the seasonal periodicity, wh,i is the weekly periodicity, bh,i represents the bank holidays for four main Nordic countries (Norway, Sweden, Denmark, and Finland), and ph,i−1 and dh,i−1 are the lagged prices and demand series, respectively. The intraday periodicity is addressed by analyzing each load hour separately. On the other hand, Sh,i is a stochastic component (residuals), which defines the short-term dynamics of the process. The deterministic component is estimated by parametric and nonparametric approaches, while univariate autoregressive model (AR) and multivariate vector autoregressive model (VAR) models are used for stochastic component.


### 2.2. Modeling Deterministic Component

This section describes the procedure for modeling and forecasting the deterministic component by parametric and nonparametric approaches.


#### 2.2.1. The Parametric Approach

The details about the estimation of deterministic components using the parametric regression method are provided in this section. The response variableDh,i is modeled parametrically by estimating equation (2). The long-time trend component th,i is modeled as a function of time i, yearly cycle as a function of the series 1,2,…,365,1,2,…,365,… using sinusoidal-type function such as yi=∑l=1qζ1,lsinωyi+ζ2,lcosωyi, where ω=2πl/365.5 and unknown constant parameters ζ1,andζ2 (l = 1, 2, …., q). Dummy variables are used for seasonal variation, weekly periodicity, and bank holidays, that is, si=∑j=14ρjIj,i, with Ij,i=1 if i refers to the jth season of the year and 0 otherwise, wi=∑j=17βjIi,t, with Ij,i=1 if i refers to the ith day of the week and 0 otherwise, and bi=∑j=12γjIj,i, with Ij,i=1 if i refers bank holiday or 0 otherwise. The lagged price and demand ph,i−1,dh,i−1 are used as exogenous information in each case. The coefficients of all components are estimated by the maximum-likelihood estimation (MLE) method. When all the regression coefficients are obtained, the estimated equation becomes(4)G^h,id=ζ0^th,i+∑j=1qζ^1,jsinωyi+ζ^2,jcosωyi+∑j=14ρj^Ij,i+∑j=17βj^Ij,i+∑j=12γj^Ij,i+δ^ph,i−1,G^h,ip=ζ0^th,i+∑j=1qζ^1,jsinωyi+ζ^2,jcosωyi+∑j=17βj^Ij,i+∑j=12γj^Ij,i+δ^dh,i−1.In the literature, several authors used this technique for the modeling of deterministic component [65, 66].


#### 2.2.2. The Nonparametric Approach

Nowadays, many complex processes are nonlinear, and hence, nonlinear modeling is a popular technique to account for this specific feature. In the case of electricity markets, many researchers capture the long trend and yearly and seasonal periodicity as smooth functions by using nonlinear smoothing techniques [67, 68]. In our case, using nonparametric estimation of these components, the following model is used:(5)Gh,id=f1th,i+f2yh,i+f3sh,i+f4ph,i−1+∑j=17βjIj,i+∑j=12γjIj,i,Gh,ip=f1th,i+f2yh,i+f3dh,i−1+∑j=17βjIj,i+∑j=12γjIj,i.The first four components, that is, the long-time trend, the yearly, seasonal, and the lagged prices/demand, are estimated using nonlinear smoothing functions, whereas dummy variables model the weekly periodicity and bank holidays. In this study, the cubic smoothing splines are used to estimate the smoothing functions. The regression coefficients are estimated using the MLE, and the final estimated equations are given by(6)G^h,id=f^1th,i+f^2yh,i+f^3sh,i+f^4ph,i+∑j=17β^jIj,i+∑j=12γ^jIj,i,G^h,ip=f^1th,i+f^2yh,i+f^3dh,i+∑j=17β^jIj,i+∑j=12γ^jIj,i.The out-of-sample forecast in both parametric and nonparametric cases is straightforward since they represent the time or calendar information. In addition, the lagged price/demand is also available and can be used for forecasting problems. Once the deterministic component is estimated, the residual part considered stochastic can be obtained as(7)S^h,id=Dh,i−G^h,id,(8)S^h,ip=Ph,i−G^h,ip.The estimation procedure of the stochastic component is given in the following. We will drop the superscriptd and P from now on as the two stochastic models are used for both residuals obtained by equations (7) and (8).


### 2.3. Modeling Stochastic Component

This section provides details about the modeling and forecasting of the stochastic component. To this end, we use univariate autoregressive and multivariate autoregressive models.


#### 2.3.1. Autoregressive Model

An autoregressive (AR) model is an illustration of a type of random process; that is, it is used to describe some time-varying processes. The AR models describe the response variable linearly dependent on its past values (lag) and a stochastic term. The general mathematical equation for an AR of ordern, that is, AR (n), is given as follows:(9)Sh,i=u+ζ1Sh,i−1+ζ2Sh,i−2+⋯+ζnSh,i−n+εh,i,where u indicates the intercept, ζii=1,2,…,n are unknown parameters of the model, and εh,i is a white noise random process that is normally distributed with zero mean and constant variance σε2. The main goal is to decide the number of lags that are to be used in the model. For this purpose, graphical analysis, that is, plotting the series, residual, autocorrelation function (ACF), and partial ACF, is used that indicated to use the lags 1, 2, 3, and 7 in the model for our data.


#### 2.3.2. Vector Autoregressive (VAR) Model

In the multivariate method, the response variable and predictors are not scalars but contain information about the entire daily load profile of a specified day. This makes it possible to take into account the possible interdependencies between price/demand in different load periods. This study uses a single standard multivariate technique known as the autoregressive vector model (VAR). In this method, each variable (in the current study, the load period vector) is a linear function ofr past lags of itself and an error term as(10)Si=H1Si−1+H2Si−2+⋯+HrSi−r+εi,where Si is the stochastic component, Hjj=1,2,…,r are matrices of coefficient, and εi is a disturbance term having zero mean and variance covariance matrix Σε.The unknown parameters of the AR and VAR models are estimated using four different estimation methods including ordinary least squares (O), Lasso (L), Ridge (R), and Elastic-net (E). The general mathematical equation for these models is given by(11)LEζ^=∑i=1dzi−yiζ^22d+α1−λ2∑j=1mζ^j2+λ∑j=1mζ^j.Equation (11) becomes ordinary least squares estimation when α=0. λ is the mixing parameter between Ridge λ=0 and Lasso λ=1. In the case of elastic net, there are two parameters to tune: α and λ, and their selection can be performed via cross-validation method.The estimation of the stochastic univariate model leads us to four parametric modelAPRO,APRL,APRR,APRE, and for nonparametric model ANPRO,ANPRL,ANPRR,ANPRE. Similarly, eight multivariate models are estimated and denoted by VPARO, VPARL, VPARR, VPARE, VNPARO, VNPARL, VNPARR, and VNPARE. After completing the estimation of both deterministic and stochastic components, the final forecast for electricity demand and prices is obtained as follows:(12)D^h,i+1=expG^h,i+1d+S^h,i+1,(13)P^h,i+1=expG^h,i+1p+S^h,i+1.A flowchart of the proposed modeling is given in Figure2.Figure 2
Flowchart of the proposed modeling framework.


## 2.1. Modeling Framework

This study used different parametric and nonparametric univariate and multivariate time series models for one day-ahead electricity demand and price forecasts. The log demand and price series forhth hour and ith day are denoted by logDh,i and logPh,i and is modeled as(1)logDh,i=Gh,id+Şh,id,logPh,i=Gh,ip+Şh,ip,where Gh,id is a deterministic component and Şh,id is a residual or stochastic component. The superscripts d and p are used for demand and price time series, respectively. The Gh,id/Gh,ip (deterministic component) comprises a long-time trend, weekly, seasonal, and annual cycles, bank holidays days, and the lagged price/demand series and is modeled as(2)Gh,id=th,i+yh,i+sh,i+wh,i+bh,i+ph,i−1,(3)Gh,ip=th,i+yh,i+wh,i+bh,i+dh,i−1,where th,i represent the long trend component, yh,i is the yearly cycles, sh,i denotes the seasonal periodicity, wh,i is the weekly periodicity, bh,i represents the bank holidays for four main Nordic countries (Norway, Sweden, Denmark, and Finland), and ph,i−1 and dh,i−1 are the lagged prices and demand series, respectively. The intraday periodicity is addressed by analyzing each load hour separately. On the other hand, Sh,i is a stochastic component (residuals), which defines the short-term dynamics of the process. The deterministic component is estimated by parametric and nonparametric approaches, while univariate autoregressive model (AR) and multivariate vector autoregressive model (VAR) models are used for stochastic component.


## 2.2. Modeling Deterministic Component

This section describes the procedure for modeling and forecasting the deterministic component by parametric and nonparametric approaches.


### 2.2.1. The Parametric Approach

The details about the estimation of deterministic components using the parametric regression method are provided in this section. The response variableDh,i is modeled parametrically by estimating equation (2). The long-time trend component th,i is modeled as a function of time i, yearly cycle as a function of the series 1,2,…,365,1,2,…,365,… using sinusoidal-type function such as yi=∑l=1qζ1,lsinωyi+ζ2,lcosωyi, where ω=2πl/365.5 and unknown constant parameters ζ1,andζ2 (l = 1, 2, …., q). Dummy variables are used for seasonal variation, weekly periodicity, and bank holidays, that is, si=∑j=14ρjIj,i, with Ij,i=1 if i refers to the jth season of the year and 0 otherwise, wi=∑j=17βjIi,t, with Ij,i=1 if i refers to the ith day of the week and 0 otherwise, and bi=∑j=12γjIj,i, with Ij,i=1 if i refers bank holiday or 0 otherwise. The lagged price and demand ph,i−1,dh,i−1 are used as exogenous information in each case. The coefficients of all components are estimated by the maximum-likelihood estimation (MLE) method. When all the regression coefficients are obtained, the estimated equation becomes(4)G^h,id=ζ0^th,i+∑j=1qζ^1,jsinωyi+ζ^2,jcosωyi+∑j=14ρj^Ij,i+∑j=17βj^Ij,i+∑j=12γj^Ij,i+δ^ph,i−1,G^h,ip=ζ0^th,i+∑j=1qζ^1,jsinωyi+ζ^2,jcosωyi+∑j=17βj^Ij,i+∑j=12γj^Ij,i+δ^dh,i−1.In the literature, several authors used this technique for the modeling of deterministic component [65, 66].


### 2.2.2. The Nonparametric Approach

Nowadays, many complex processes are nonlinear, and hence, nonlinear modeling is a popular technique to account for this specific feature. In the case of electricity markets, many researchers capture the long trend and yearly and seasonal periodicity as smooth functions by using nonlinear smoothing techniques [67, 68]. In our case, using nonparametric estimation of these components, the following model is used:(5)Gh,id=f1th,i+f2yh,i+f3sh,i+f4ph,i−1+∑j=17βjIj,i+∑j=12γjIj,i,Gh,ip=f1th,i+f2yh,i+f3dh,i−1+∑j=17βjIj,i+∑j=12γjIj,i.The first four components, that is, the long-time trend, the yearly, seasonal, and the lagged prices/demand, are estimated using nonlinear smoothing functions, whereas dummy variables model the weekly periodicity and bank holidays. In this study, the cubic smoothing splines are used to estimate the smoothing functions. The regression coefficients are estimated using the MLE, and the final estimated equations are given by(6)G^h,id=f^1th,i+f^2yh,i+f^3sh,i+f^4ph,i+∑j=17β^jIj,i+∑j=12γ^jIj,i,G^h,ip=f^1th,i+f^2yh,i+f^3dh,i+∑j=17β^jIj,i+∑j=12γ^jIj,i.The out-of-sample forecast in both parametric and nonparametric cases is straightforward since they represent the time or calendar information. In addition, the lagged price/demand is also available and can be used for forecasting problems. Once the deterministic component is estimated, the residual part considered stochastic can be obtained as(7)S^h,id=Dh,i−G^h,id,(8)S^h,ip=Ph,i−G^h,ip.The estimation procedure of the stochastic component is given in the following. We will drop the superscriptd and P from now on as the two stochastic models are used for both residuals obtained by equations (7) and (8).


## 2.2.1. The Parametric Approach

The details about the estimation of deterministic components using the parametric regression method are provided in this section. The response variableDh,i is modeled parametrically by estimating equation (2). The long-time trend component th,i is modeled as a function of time i, yearly cycle as a function of the series 1,2,…,365,1,2,…,365,… using sinusoidal-type function such as yi=∑l=1qζ1,lsinωyi+ζ2,lcosωyi, where ω=2πl/365.5 and unknown constant parameters ζ1,andζ2 (l = 1, 2, …., q). Dummy variables are used for seasonal variation, weekly periodicity, and bank holidays, that is, si=∑j=14ρjIj,i, with Ij,i=1 if i refers to the jth season of the year and 0 otherwise, wi=∑j=17βjIi,t, with Ij,i=1 if i refers to the ith day of the week and 0 otherwise, and bi=∑j=12γjIj,i, with Ij,i=1 if i refers bank holiday or 0 otherwise. The lagged price and demand ph,i−1,dh,i−1 are used as exogenous information in each case. The coefficients of all components are estimated by the maximum-likelihood estimation (MLE) method. When all the regression coefficients are obtained, the estimated equation becomes(4)G^h,id=ζ0^th,i+∑j=1qζ^1,jsinωyi+ζ^2,jcosωyi+∑j=14ρj^Ij,i+∑j=17βj^Ij,i+∑j=12γj^Ij,i+δ^ph,i−1,G^h,ip=ζ0^th,i+∑j=1qζ^1,jsinωyi+ζ^2,jcosωyi+∑j=17βj^Ij,i+∑j=12γj^Ij,i+δ^dh,i−1.In the literature, several authors used this technique for the modeling of deterministic component [65, 66].


## 2.2.2. The Nonparametric Approach

Nowadays, many complex processes are nonlinear, and hence, nonlinear modeling is a popular technique to account for this specific feature. In the case of electricity markets, many researchers capture the long trend and yearly and seasonal periodicity as smooth functions by using nonlinear smoothing techniques [67, 68]. In our case, using nonparametric estimation of these components, the following model is used:(5)Gh,id=f1th,i+f2yh,i+f3sh,i+f4ph,i−1+∑j=17βjIj,i+∑j=12γjIj,i,Gh,ip=f1th,i+f2yh,i+f3dh,i−1+∑j=17βjIj,i+∑j=12γjIj,i.The first four components, that is, the long-time trend, the yearly, seasonal, and the lagged prices/demand, are estimated using nonlinear smoothing functions, whereas dummy variables model the weekly periodicity and bank holidays. In this study, the cubic smoothing splines are used to estimate the smoothing functions. The regression coefficients are estimated using the MLE, and the final estimated equations are given by(6)G^h,id=f^1th,i+f^2yh,i+f^3sh,i+f^4ph,i+∑j=17β^jIj,i+∑j=12γ^jIj,i,G^h,ip=f^1th,i+f^2yh,i+f^3dh,i+∑j=17β^jIj,i+∑j=12γ^jIj,i.The out-of-sample forecast in both parametric and nonparametric cases is straightforward since they represent the time or calendar information. In addition, the lagged price/demand is also available and can be used for forecasting problems. Once the deterministic component is estimated, the residual part considered stochastic can be obtained as(7)S^h,id=Dh,i−G^h,id,(8)S^h,ip=Ph,i−G^h,ip.The estimation procedure of the stochastic component is given in the following. We will drop the superscriptd and P from now on as the two stochastic models are used for both residuals obtained by equations (7) and (8).


## 2.3. Modeling Stochastic Component

This section provides details about the modeling and forecasting of the stochastic component. To this end, we use univariate autoregressive and multivariate autoregressive models.


### 2.3.1. Autoregressive Model

An autoregressive (AR) model is an illustration of a type of random process; that is, it is used to describe some time-varying processes. The AR models describe the response variable linearly dependent on its past values (lag) and a stochastic term. The general mathematical equation for an AR of ordern, that is, AR (n), is given as follows:(9)Sh,i=u+ζ1Sh,i−1+ζ2Sh,i−2+⋯+ζnSh,i−n+εh,i,where u indicates the intercept, ζii=1,2,…,n are unknown parameters of the model, and εh,i is a white noise random process that is normally distributed with zero mean and constant variance σε2. The main goal is to decide the number of lags that are to be used in the model. For this purpose, graphical analysis, that is, plotting the series, residual, autocorrelation function (ACF), and partial ACF, is used that indicated to use the lags 1, 2, 3, and 7 in the model for our data.


### 2.3.2. Vector Autoregressive (VAR) Model

In the multivariate method, the response variable and predictors are not scalars but contain information about the entire daily load profile of a specified day. This makes it possible to take into account the possible interdependencies between price/demand in different load periods. This study uses a single standard multivariate technique known as the autoregressive vector model (VAR). In this method, each variable (in the current study, the load period vector) is a linear function ofr past lags of itself and an error term as(10)Si=H1Si−1+H2Si−2+⋯+HrSi−r+εi,where Si is the stochastic component, Hjj=1,2,…,r are matrices of coefficient, and εi is a disturbance term having zero mean and variance covariance matrix Σε.The unknown parameters of the AR and VAR models are estimated using four different estimation methods including ordinary least squares (O), Lasso (L), Ridge (R), and Elastic-net (E). The general mathematical equation for these models is given by(11)LEζ^=∑i=1dzi−yiζ^22d+α1−λ2∑j=1mζ^j2+λ∑j=1mζ^j.Equation (11) becomes ordinary least squares estimation when α=0. λ is the mixing parameter between Ridge λ=0 and Lasso λ=1. In the case of elastic net, there are two parameters to tune: α and λ, and their selection can be performed via cross-validation method.The estimation of the stochastic univariate model leads us to four parametric modelAPRO,APRL,APRR,APRE, and for nonparametric model ANPRO,ANPRL,ANPRR,ANPRE. Similarly, eight multivariate models are estimated and denoted by VPARO, VPARL, VPARR, VPARE, VNPARO, VNPARL, VNPARR, and VNPARE. After completing the estimation of both deterministic and stochastic components, the final forecast for electricity demand and prices is obtained as follows:(12)D^h,i+1=expG^h,i+1d+S^h,i+1,(13)P^h,i+1=expG^h,i+1p+S^h,i+1.A flowchart of the proposed modeling is given in Figure2.Figure 2
Flowchart of the proposed modeling framework.


## 2.3.1. Autoregressive Model

An autoregressive (AR) model is an illustration of a type of random process; that is, it is used to describe some time-varying processes. The AR models describe the response variable linearly dependent on its past values (lag) and a stochastic term. The general mathematical equation for an AR of ordern, that is, AR (n), is given as follows:(9)Sh,i=u+ζ1Sh,i−1+ζ2Sh,i−2+⋯+ζnSh,i−n+εh,i,where u indicates the intercept, ζii=1,2,…,n are unknown parameters of the model, and εh,i is a white noise random process that is normally distributed with zero mean and constant variance σε2. The main goal is to decide the number of lags that are to be used in the model. For this purpose, graphical analysis, that is, plotting the series, residual, autocorrelation function (ACF), and partial ACF, is used that indicated to use the lags 1, 2, 3, and 7 in the model for our data.


## 2.3.2. Vector Autoregressive (VAR) Model

In the multivariate method, the response variable and predictors are not scalars but contain information about the entire daily load profile of a specified day. This makes it possible to take into account the possible interdependencies between price/demand in different load periods. This study uses a single standard multivariate technique known as the autoregressive vector model (VAR). In this method, each variable (in the current study, the load period vector) is a linear function ofr past lags of itself and an error term as(10)Si=H1Si−1+H2Si−2+⋯+HrSi−r+εi,where Si is the stochastic component, Hjj=1,2,…,r are matrices of coefficient, and εi is a disturbance term having zero mean and variance covariance matrix Σε.The unknown parameters of the AR and VAR models are estimated using four different estimation methods including ordinary least squares (O), Lasso (L), Ridge (R), and Elastic-net (E). The general mathematical equation for these models is given by(11)LEζ^=∑i=1dzi−yiζ^22d+α1−λ2∑j=1mζ^j2+λ∑j=1mζ^j.Equation (11) becomes ordinary least squares estimation when α=0. λ is the mixing parameter between Ridge λ=0 and Lasso λ=1. In the case of elastic net, there are two parameters to tune: α and λ, and their selection can be performed via cross-validation method.The estimation of the stochastic univariate model leads us to four parametric modelAPRO,APRL,APRR,APRE, and for nonparametric model ANPRO,ANPRL,ANPRR,ANPRE. Similarly, eight multivariate models are estimated and denoted by VPARO, VPARL, VPARR, VPARE, VNPARO, VNPARL, VNPARR, and VNPARE. After completing the estimation of both deterministic and stochastic components, the final forecast for electricity demand and prices is obtained as follows:(12)D^h,i+1=expG^h,i+1d+S^h,i+1,(13)P^h,i+1=expG^h,i+1p+S^h,i+1.A flowchart of the proposed modeling is given in Figure2.Figure 2
Flowchart of the proposed modeling framework.


## 3. Out-of-Sample Demand and Price Forecasting

To examine the one-day-ahead out-of-sample forecast accuracy of the models described above, demand and prices series for the Nord Pool electricity market are considered. For both, demand and prices, datasets range from1stJanuary 2, 2013 to 31stDecember 31, 2017 (43824 data points, cover 1826 days), and some missing observations in the demand and price series were substituted by averaging the neighboring values. The entire dataset was divided into two segments: the first four years; that is, data from 1stJanuary 1, 2013 to 31thDecember 31, 2016 (35064 data points, covering 1461 days) are used for model estimation, whereas the fifth year; that is, data from 1stJanuary 1, 2017 to 31thDecember 31, 2017 (8784 data points, covering 365 days) for one day-ahead out-of-sample forecasting, depicted in Figure 3. The proposed methodology includes two main phases. In the first phase, the deterministic component is estimated distinctly for each load period as defined in Section 2.1, while in the second phase, the models earlier described for the stochastic component are applied to the Sh,i. In both phases, models are estimated and one-day-ahead forecasts are obtained for 365 days with the rolling window method. The final demand and price forecasts are obtained using equations (12) and (13).Figure 3
Nord Pool electricity data: demand (a) and prices (b) over the period of1stJanuary 1, 2013 to 31stDecember 31, 2017. The out-of-sample forecasting period is separated by two vertical lines.
(a)(b)To evaluate the model forecasting performance, we computed four descriptive accuracy measures of the errors and conducted a statistical test, namely, the Diebold and Mariano (DM) [69] test for determining the statistical significance of the differences in the models’ performance. The four descriptive accuracy measures are the mean absolute error (MAE), mean absolute percentage error (MAPE), root mean square error (RMSE), and day-specific MAPE DSMAPE, which is the MAPE for each day of the week. Suppose Eh,i=xh,i−x^h,i be the one-day-ahead forecast error for the demand/price time series where xh,i is the observed and x^h,i is the forecasted value. Then, the descriptive accuracy measures can be written as(14)MAE=MeanEh,i,MAPE=MeanEh,ixh,i∗100,MDSAPE=MeanEh,i∈Dayxh,i∗100,RMSE=MeanEh,i2.The descriptive statistics related to the out-of-sample demand and price forecast using all sixteen models are listed in Tables1 and 2. From Table 1, the descriptive results for demand forecasting suggest that within univariate models, ANPRE produced the lowest errors values for the MAPE, MAE, and RMSE of 2.312, 997.598, and 1358.278, respectively. In the case of multivariate modeling, VNPARE produced the smallest MAPE, MAE, and RMSE of 2.017, 872.241, and 1176.278, respectively. In general, VNPARE resulted in the lowest mean error values, while VPARO is the second best model. Furthermore, one can see that the univariate models perform relatively worse than their multivariate counterparts. For example, APRO produces a MAPE value of 2.861, whereas its multivariate counterpart VPARO resulted in a 2.529 MAPE value. In the nonparametric case, values of 2.313 and 2.027 are obtained for ANPRO and VNPARO, respectively. In addition, it can be seen from the results that the estimation procedure used for both deterministic and stochastic components has a significant effect on the forecasting results.Table 1
Nord Pool electricity demand: one-day-ahead out-of-sample forecasting errors.
ModelMAPEMAERMSEAPRO2.8611226.871667.86APRL2.8671228.71667.181APRR2.8611226.931665.924APRE2.8631227.551666.072ANPRO2.313997.511359.029ANPRL2.313998.0851358.657ANPRR2.312997.6041358.624ANPRE2.312997.5981358.278VPARO2.5291080.821446.78VPARL2.5221078.921448.603VPARR2.5261079.811445.773VPARE2.5211078.341446.981VNPARO2.027876.3441178.927VNPARL2.022874.4171179.981VNPARR2.025875.5711178.161VNPARE2.017872.2411176.078Table 2
Nord Pool electricity prices: one-day-ahead out-of-sample forecasting errors.
ModelMAPEMAERMSEAPRO5.6991.6742.833APRL5.7111.6782.835APRR5.7051.6752.833APRE5.7081.6762.834ANPRO5.3361.5542.636ANPRL5.3341.5542.633ANPRR5.3381.5552.635ANPRE5.3361.5542.634VPARO5.0231.4962.543VPARL4.9431.4732.524VPARR4.9721.482.522VPARE4.9531.4762.522VNPARO4.8231.4312.401VNPARL4.7461.4082.378VNPARR4.7911.4212.387VNPARE4.7621.4122.38The descriptive results for electricity prices in Table2 show that the multivariate models outperform the univariate counterparts. Although the estimation technique used for the stochastic component has a significant effect by reducing the forecasting errors in the case of univariate models, the errors are still higher than the multivariate models. This indicates that the intraday correlation between different load period prices is highly important for forecasting prices. Moreover, within the univariate models, ANPRE produced the lowest mean error values of 5.334, 1.554, and 2.633 for MAPE, MAE, and RMSE, respectively, whereas in the case of multivariate models, the lowest MAPE, MAE, and RMSE of 4.746, 1.408, and 2.378 are obtained by VNPARL. Among all models, VNPARL produced the least mean errors values, while VPARE is the second best model.The DM test is performed for each pair of models to evaluate the statistical significance of the differences in the model descriptive accuracy measures for demand and price listed in Tables1 and 2. The results (p-values) of the DM test are listed in Tables 3 and 4. In these tables, each entry is a p-value of a hypothesis system. The null hypothesis assumes no difference in the precision of the model in the column and row versus the alternative assumes that the model in the column is more accurate than the model in the row. From these tables, we observed that multivariate vector autoregressive models combined with both parametric and nonparametric estimation of the deterministic component are statistically significant compared to the univariate autoregressive models at forecasting one day-ahead electricity prices. In the case of electricity demand forecasting, nonparametric estimation of the deterministic component combined with univariate or multivariate model shows more accurate and efficient results than the parametric case.Table 3
Nord Pool electricity demand:p-values for the DM test for the null hypothesis of the same accuracy of two models in row and column versus the alternative hypothesis that the model in the column is more accurate than the row model (squared loss function is used).
Univariate modelsModelsAPROAPRLAPRRAPREANPROANPRLANPRIANPREAPRO—0.5960.9720.901>>0.001>>0.001>>0.001>>0.001APRL0.404—0.9130.89>>0.001>>0.001>>0.001>>0.001APRR0.0280.087—0.07>>0.001>>0.001>>0.001>>0.001APRE0.0990.110.931—>>0.001>>0.001>>0.001>>0.001ANPRO0.9990.9990.9990.999—0.8430.990.958ANPRL0.9990.9990.9990.9990.157—0.8080.784ANPRR0.9990.9990.9990.9990.010.192—0.19ANPRE<<0.999<<0.999<<0.999<<0.9990.0420.2160.81—Multivariate modelsModelsVPAROVPARLVPARRVPAREVNPAROVNPARLVNPARRVNPAREVPARO—0.5760.2330.598>>0.001>>0.001>>0.001>>0.001VPARL0.425—0.3610.484>>0.001>>0.001>>0.001>>0.001VPARR0.7670.64—0.697>>0.001>>0.001>>0.001>>0.001VPARE0.4020.5160.303—>>0.001>>0.001>>0.001>>0.001VNPARO0.9990.9990.9990.999—0.5690.1080.274VNPARL0.999<<0.999<<0.999<<0.9990.431—0.3710.032VNPARR<<0.999<<0.999<<0.999<<0.9990.8920.629—0.315VNPARE<<0.999<<0.999<<0.999<<0.9990.7260.9680.685—Univariate and multivariate modelsModelsAPROAPRLAPRRAPREANPROANPRLANPRRANPREVPAROVPARLVPARRVPAREVNPAROVNPARLVNPARRVNPAREAPRO—0.5960.9720.901>>0.001>>0.001>>0.001>>0.0010.001>>0.0010.0010.001>>0.001>>0.001>>0.001>>0.001APRL0.404—0.9130.89>>0.001>>0.001>>0.001>>0.0010.001>>0.0010.0010>>0.001>>0.001>>0.001>>0.001APRR0.0280.087—0.07>>0.001>>0.001>>0.001>>0.0010.001>>0.0010.0010.001>>0.001>>0.001>>0.001>>0.001APRE0.0990.110.931—>>0.001>>0.001>>0.001>>0.0010.001>>0.0010.0010.001>>0.001>>0.001>>0.001>>0.001ANPRO0.9990.9990.9990.999—0.8430.990.9580.9870.9890.9870.988>>0.001>>0.001>> 0.001>>0.001ANPRL0.9990.9990.9990.9990.157—0.8080.7840.9860.9880.9850.987>>0.001>>0.001>>0.001>>0.001ANPRR0.9990.9990.9990.9990.010.192—0.190.9830.9850.9830.984>>0.001>>0.001>>0.001>>0.001ANPRE0.9990.9990.9990.9990.0420.2160.81—0.9850.9870.9840.986>>0.001>>0.001>>0.001>>0.001VPARO0.9990.9990.9990.9990.0130.0140.0170.015—0.5760.2330.598>>0.001>>0.001>>0.001>>0.001VPARL0.9990.9990.9990.9990.0110.0120.0150.0130.425—0.3610.484>>0.001>>0.001>>0.001>>0.001VPARR0.9990.9990.9990.9990.0130.0150.0170.0160.7670.64—0.697>>0.001>>0.001>>0.001>>0.001VPARE0.9990.9990.9990.9990.0120.0130.0160.0140.4020.5160.303—>>0.001>>0.001>>0.001>>0.001VNPARO0.9990.9990.9990.9990.9990.9990.9990.9990.9990.9990.9990.999—0.5690.1080.274VNPARL0.9990.9990.9990.9990.9990.9990.9990.9990.9990.9990.9990.9990.431—0.3710.032VNPARR0.9990.9990.9990.9990.9990.9990.9990.9990.9990.9990.9990.9990.8920.629—0.315VNPARE0.9990.9990.9990.9990.9990.9990.9990.9990.9990.9990.9990.9990.7260.9680.685—Table 4
Nord Pool electricity prices:p-values for the DM test for the null hypothesis of the same accuracy of two models in row and column versus the alternative hypothesis that the model in the column is more accurate than the row model (squared loss function is used).
Univariate modelsModelsAPROAPRLAPRRAPREANPROANPRLANPRIANPREAPRO—0.99820.99540.9976>>0.001>>0.001>>0.001>>0.001APRL0.0018—0.72240.6812>>0.001>>0.001>>0.001>>0.001APRR0.00460.2776—0.2396>>0.001>>0.001>>0.001>>0.001APRE0.00240.31880.7604—>>0.001>>0.001>>0.001>>0.001ANPRO0.9990.9990.9990.999—0.63040.9630.9063ANPRL0.9990.9990.9990.9990.3696—0.95270.9557ANPRR0.9990.9990.9990.9990.0370.0473—0.0522ANPRE0.9990.9990.9990.9990.09370.04430.9478—Multivariate modelsModelsVPAROVPARLVPARRVPAREVNPAROVNPARLVNPARRVNPAREVPARO—0.52730.33840.45380.00360.00240.00120.0011VPARL0.4727—0.33160.29110.02540.00170.00950.0031VPARR0.66160.6684—0.62850.01360.00180.0040.0016VPARE0.54620.70890.3715—0.01990.00150.00650.0021VNPARO0.99640.97460.98640.9801—0.43440.27780.358VNPARL0.99760.99830.99820.99850.5656—0.47420.3622VNPARR0.99880.99050.9960.99350.72220.5258—0.4405VNPARE0.99890.99690.99840.99790.6420.63780.5595—Univariate and multivariate modelsModelsAPROAPRLAPRRAPREANPROANPRLANPRRANPREVPAROVPARLVPARRVPAREVNPAROVNPARLVNPARRVNPAREAPRO—0.9980.9950.998>>0.001>>0.001>>0.001>>0.0010.001>>0.0010.0010.001>>0.001>0.001>>0.001>>0.001APRL0.002—0.7220.681>>0.001>>0.001>>0.001>>0.0010.001>>0.0010.0010>>0.001>0.001>>0.001>>0.001APRR0.0050.278—0.24>>0.001>>0.001>>0.001>>0.0010.001>>0.0010.0010.001>>0.001>0.001>>0.001>>0.001APRE0.0020.3190.76—>>0.001>>0.001>>0.001>>0.0010.001>>0.0010.0010.001>>0.001>0.001>>0.001>>0.001ANPRO0.9990.9990.9990.999—0.630.9630.9060.1050.080.0810.0790.0070.0010.0030.002ANPRL0.9990.9990.9990.9990.37—0.9530.9560.1040.080.0810.0780.0060.0010.0030.002ANPRR0.9990.9990.9990.9990.0370.047—0.0520.1010.0770.0780.0750.0060.0010.0030.002ANPRE0.9990.9990.9990.9990.0940.0440.948—0.1030.0780.0790.0770.0060.0010.0030.002VPARO0.9990.9990.9990.9990.8950.8960.8990.898—0.5270.3380.4540.0040.0020.0010.001VPARL0.9990.9990.9990.9990.920.9210.9230.9220.473—0.3320.2910.0250.0020.010.003VPARR0.9990.9990.9990.9990.9190.9190.9220.9210.6620.668—0.6290.0140.0020.0040.002VPARE0.9990.9990.9990.9990.9210.9220.9250.9230.5460.7090.372—0.020.0020.0070.002VNPARO0.9990.9990.9990.9990.9940.9940.9940.9940.9960.9750.9860.98—0.4340.2780.358VNPARL0.9990.9990.9990.9990.9990.9990.9990.9990.9980.9980.9980.9990.566—0.4740.362VNPARR0.9990.9990.9990.9990.9970.9970.9970.9970.9990.9910.9960.9940.7220.526—0.441VNPARE0.9990.9990.9990.9990.9980.9980.9980.9980.9990.9970.9980.9980.6420.6380.56—The day-specific mean errors (MDSAPE, MDSAE, and RDSMSE) are listed in Tables 5 and 6 for electricity demand and prices, respectively. From Table 5, one can notice that day-specific mean errors are low on Friday, Saturday, and Sunday, while high on the rest of the weekdays. For electricity prices, Table 6, the day-specific mean errors are considerably low on Wednesday for all models compared to other weekdays. In both cases (demand and price), the multivariate models produced the lowest day-specific mean errors compared to other competitors. A graphical illustration of day-specific MAPE is given in Figure 4 for demand (a) and prices (b). From these plots, it is clear that multivariate models performed better than univariate models as their MDSAPE values are significantly lower than the univariate models.Table 5
Day-specific descriptive statistics for the electricity demand.
ErrorsModelsMondayTuesdayWednesdayThursdayFridaySaturdaySundayMAPEAPRR3.6793.0713.1532.8682.5172.2542.426ANPRE2.9772.5272.4522.4411.9741.8151.937VPARE3.0552.2252.6722.9732.6441.9662.022VNPARE2.5312.0802.0352.1501.8771.7171.635MAEAPRR2268.3122077.1141854.0981504.5321103.3551063.5331286.313ANPRE1873.9591755.6531364.3071301.025858.629827.8791042.292VPARE1927.8561477.3531558.3831586.8501180.265909.5491088.127VNPARE1588.4371401.0961141.7781195.991840.231738.303900.813RMSEAPRR1758.4381571.9501433.2851124.624875.044800.940997.363ANPRE1434.2821307.0411116.705969.972684.786638.853802.760VPARE1488.3881149.6581231.4441166.200939.004690.836840.726VNPARE1246.9461085.889931.889850.718658.074605.595682.408Table 6
Day-specific descriptive statistics for the electricity prices.
ErrorsModelsMondayTuesdayWednesdayThursdayFridaySaturdaySundayMAPEAPRO7.2615.2013.8795.9107.0005.1655.530ANPRL6.4465.3413.6675.4266.4124.6945.392VPARL6.5744.8503.4815.1455.3944.5054.670VNPARL5.5725.1453.5155.0035.0544.3954.557MAEAPRO4.4602.8891.6732.2212.1341.8451.983ANPRL4.0612.7471.5632.0751.9541.6941.928VPARL4.0932.7611.5481.9761.7391.6051.717VNPARL3.6622.6791.4901.9001.6281.5751.668RMSEAPRO2.7551.8201.1721.6521.5211.3591.454ANPRL2.4701.8331.1131.4981.3671.2191.390VPARL2.5001.6941.0591.4271.2321.1681.240VNPARL2.1541.7701.0671.3831.1441.1391.205Figure 4
Day-specific MAPE for the best univariate and multivariate models. (a) Demand and (b) prices.
(a)(b)The demand hourly MAPE for the best six combination of models, that is,ANPRL, ANPRE, VNPARO, VNPARL, VNPARR, and VNPARE, is plotted in the left panel of Figure 5. From this figure, note that the hourly MAPE is lower at starting load period and gradually increases with the first peak around 8 : 00 am while the second peak around 7 : 00 pm. The price hourly MAPE can be seen from Figure 5(b) for the six best combinations of models, namely, ANPRL, VPARE, VNPARO, VNPARL, VNPARR, and VNPARE. As it can be seen from the figure, errors are lower at initial hours and increase at later hours with multiple peaks throughout the day.Figure 5
Hourly MAPE for best univariate and multivariate models. (a) Demand and (b) prices.
(a)(b)Finally, the forecasted values for the four best models along with the observed values are presented in Figure6 for both demand (a) and prices (b). From the figure, we can see that our models’ forecast follows actual demand and prices very well.Figure 6
The observed and forecasted electricity demand (a) and prices (b) for four best models over a week period.
(a)(b)It is important to mention that our best MAPE values are comparatively lower than those mentioned in other research articles. For illustration, using different univariate and multivariate functional-based models to forecast Nord Pool electricity load and prices, Lisi and Shah [70] reported the minimum MAPE value of 5.55 for electricity price forecast, which is 14.23% higher than our minimum MAPE value of 4.76. In the case of the electricity demand forecasting, they obtained a minimum MAPE value of 2.10 that is 3.95% greater than our minimum MAPE value of 2.017. In another work, Lisi and Pelagatti [67] reported a minimum mean MAPE value of 6.30 and 2.46 for electricity prices and demand, respectively, which are significantly higher than our corresponding values of 4.74 and 2.017.


## 4. Conclusion

The main objective of this study is to compare various models and estimation methods when forecasting electricity demand and prices. To this end, the electricity demand (price) time series is divided into deterministic and stochastic components. The deterministic component includes the long trend, annual, seasonal, and weekly periodicities, calendar effects, and a lagged exogenous variable. Parametric and nonparametric approaches are used to estimate the deterministic component. For the stochastic (residual) component, univariate autoregressive (AR) and multivariate vector autoregressive (VAR) time series models are used. The ordinary least squares, Lasso, Ridge, and Elastic-net methods are used to estimate the stochastic models. The proposed modeling framework is applied to the Nordic electricity demand and prices data that range from1stJanuary 1, 2013 to 31stDecember 31, 2017. The first four years are used for model estimation, whereas the last complete year is used for the one-day-ahead out-of-sample forecast. Different descriptive statistics and a statistical significance test are used to evaluate the models’ forecasting accuracy.The results suggest that the proposed methodology efficiently forecasts electricity prices and demand. The multivariate vector autoregressive models combined with deterministic component estimated by both parametric and nonparametric approaches give superior forecast accuracy compared to the univariate autoregressive models. In the case of electricity demand, the estimation of the stochastic model made by Elastic-net leads to lower errors, whereas models estimated by Lasso give superior results for prices. In particular, the MAPE and MAE values for electricity demand in the case of nonparametric VAR estimated by Elastic-net are 2.02 and 872.24, respectively. On the other hand, when considering the electricity prices, the lowest MAPE and MAE are 4.75 and 1.41, respectively, estimated by nonparametric VAR using the Lasso estimation method. The results obtained in this study are compared with the results reported in the literature and are found better in terms of different errors.Accurate forecasting can be very helpful for the traders (buyers and suppliers) to optimize their bidding strategies to maximize their gains and to use the resources required for electricity generation more effectively. Consequently, this will also benefit the end-user in terms of reliable and economical electricity facilities. Nevertheless, the prediction of spikes or jumps can be considered a limitation of this study. In addition, other exogenous variables, that is, temperature and wind speed, that can be helpful for electricity demand and price forecasting are also not considered due to their availability. As this study considered data from one electricity market, in the future, the proposed framework can be applied to other electricity markets to evaluate its forecasting performance. In addition, the performance of the deep learning models, such as artificial neural networks, can also be assessed within the current modeling framework.



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