Modeling Latent Carbon Emission Prices for Japan: Theory and Practice

Climate change and global warming are significantly affected by carbon emissions that arise from the burning of fossil fuels, specifically coal, oil, and gas. Accurate prices are essential for the purposes of measuring, capturing, storing, and trading in carbon emissions at regional, national, and international levels, especially as carbon emissions can be taxed appropriately when the price is known and widely accepted. This paper uses a novel Capital (K), Labor (L), Energy (E) and Materials (M) (or KLEM) production function approach to calculate the latent carbon emission prices, where carbon emission is the output and capital (K), labor (L), energy (E) (or electricity), and materials (M) are the inputs for the production process. The variables K, L, and M are essentially fixed on a daily or monthly basis, whereas E can be changed more frequently, such as daily or monthly, so that changes in carbon emissions depend on changes in E. If prices are assumed to depend on the average cost pricing, the prices of carbon emissions and energy may be approximated by an energy production model with a constant factor of proportionality, so that carbon emission prices are a function of energy prices. Using this novel modeling approach, this paper estimates the carbon emission prices for Japan using seasonally adjusted and unadjusted monthly data on the volumes of carbon emissions and energy, as well as energy prices, from December 2008 to April 2018. The econometric models show that, as sources of electricity, the logarithms of coal and oil, though not Liquefied Natural Gas (LNG,) are statistically significant in explaining the logarithm of carbon emissions, with oil being more significant than coal. The models generally displayed a high power in predicting the latent prices of carbon emissions. The usefulness of the empirical findings suggest that the methodology can also be applied for other countries where carbon emission prices are latent.


Introduction
Climate change and global warming are two of the most important environmental issues presently facing the international community. Global warming is typically defined as the observed century-scale rise in the average temperature of the Earth's climate system and its related effects, while climate

Literature Review
Much of the research literature on pricing carbon emissions in recent years has concentrated on Europe and China. The European Union Emissions Trading Scheme established the world's first greenhouse gas emissions trading scheme in 2005, and remains the largest and most influential organization mitigating the effects of global warming and climate change.
In the sparse literature for Europe, most of which has been based on the European Union Emissions Trading Scheme (EUETS), [2] Daskalakis and Mrkellos (2009) developed a framework for future pricing and hedging, and examined data from EEX, Nord Pool, and Powernext, to analyze whether electricity risk premia were affected by the prices of emission allowances. [3]   provided evidence from modeling the prices of CO 2 emissions allowances and derivatives using European Trading Scheme data.
More recently, [4] Bushnell et al. (2013) examined profiting from regulation, based on European carbon market data. [5] Oestreich and Tsiakas (2015) analyzed carbon emissions and stock returns using EUETS data. Additionally, [6] Martin et al. (2016) examined the impacts of the EUETS on regulated firms for ten years from inception.
China's market for national carbon emissions was established at a later stage than in Europe, and has been progressing steadily. The development of such markets since their inception in China in 2013 has not necessarily been based on competitive market principles. The National Government in Beijing has made it clear that the regional markets are a first step in the development of a national carbon emission pricing scheme. Just as the market for carbon emissions trading is presently at a formative stage in China, the academic theoretical and empirical research on the emerging regional and national carbon emissions pricing markets is also at a development stage.
A review of the admittedly sparse literature on the prices of carbon emissions in Europe and China, as well as a comprehensive analysis regarding the availability of price data at the domestic regional level, is analyzed in [7,8]  . Following earlier work, the authors discuss a number of developments in China since 2005. [9] Chen (2005) evaluated the costs associated with mitigating carbon emissions. [10] Gregg et al. (2008) analyzed the carbon emissions patterns associated with fossil fuel consumption and cement production. Moreover, [11] Li and Colombier (2009) evaluated carbon emissions and energy efficiency. Using a multivariate modeling approach, [12] Chang (2010) developed a multivariate causality test of various combinations of carbon dioxide emissions, energy consumption, and economic growth.
In more recent years, there have been analyses of interactions between China and one or more other regions. For example, [ There seems to be even less research on pricing carbon emissions at an international level. [20] Zhang and Sun (2016) applied a Full BEKK multivariate conditional volatility model to analyze the time-varying covariances and the associated dynamic spillovers between the respective markets for Euro carbon and fossil fuel (for caveats regarding the Full BEKK model, particularly regarding the lack of mathematical regularity conditions, invertibility, the absence of a likelihood function, and hence the lack of valid asymptotic statistical properties, see [21] Chang and McAleer (2019)) [22]. Chang et al. (2017) and [21] Chang and McAleer (2019) analyzed the novel issues of volatility spillovers and Granger causality on the basis of daily data for the futures prices of EU carbon emissions, spot prices of US carbon emissions, and spot and future prices of oil and coal. Interesting recent research has been undertaken on analyses of price and volatility from the perspective of derivative markets and alternative volatility models, such as the price impact, trading volume and volatilities associated with mission permits and the announcement of realized emissions in [23] Hitzemann, Uhrig-Homburg, and Ehrhart (2015); a Bayesian approach to the stochastic volatility of future prices of emission allowances in [24] Kim, Park, and Ryu (2017); and the empirical performance of reduced form models for emission permit prices in [25] Hitzemann and Uhrig-Homburg (2019).
As stated above, there seems to have been no published research on pricing carbon emissions in Japan. The existing published research seems to have primarily focused on Europe and, very recently, China. For these reasons, this paper makes a novel contribution to the pricing of carbon emissions for Japan. The usefulness of the empirical findings suggests that the methodology can also be applied for other countries where carbon emission prices are latent.

KLEMS Production Function for Carbon Emissions and Energy
Inputs are required to produce outputs, and the relationship is captured in a production function. A relatively broad specification is the KLEMS production function ( [26] OECD, 2001), which uses capital (K), labor (L), energy (E), materials (M), and services (S) as the inputs for the production process. The function captures the most widely-used inputs for producing the output. Typically, the most widely observable input is energy, which is based on the use of electricity, whereas the services associated with a specific output are typically difficult to observe.
As a special case of KLEMS, consider the KLEM production function (see, for example, [27] Lecca et al., 2011) that is given as where CE denotes the production of the output of carbon emissions; capital (K), labor (L), energy or electricity (E), and materials (M) are the inputs; A is a constant; for simplicity, the partial production parameters are assumed to satisfy constant returns to scale, that is, a1 + a2 + a3 + a4 = 1 (as in the Cobb-Douglas production function); and the random error term is assumed to be u ∼ iid(0, 1). In practice, the output CE and input E may be observed at daily or monthly data frequencies, whereas the inputs K, L, and M are generally fixed at much lower data frequencies, such as one year. Consequently, when the data are observed at monthly frequencies, as in this paper, CE and E may be assumed to be proportional, with the random factor of proportionality given by k = AK a1 L a2 M a4 exp(u). ( It is worth noting that k is not restricted in its range, such as (0, 1). As k has a unit of measurement, k can take any value, depending on the units of measure of the variables, such as where a3 is a scalar. If it is assumed that output prices are set according to an average cost pricing rule (see [28] Chang et al. (2018)), the prices of CE and E may be approximated by a model with a constant factor of proportionality. Let this relationship be given as where P CE is the price of CE, and P E is the price of energy (such as electricity), such that k in Equation (4) is the constant factor of proportionality. In practice, k may be known in advance or may be determined empirically using appropriate modeling techniques. k through the use of Equations (3b) and (4). The estimation of P CE through the use of econometric models, and acknowledging the presence of measurement errors, will be developed in the next section.

Modeling Latent Carbon Emission Prices
The definitional relationship for a random variable, Y, can be seen in Equation (5): where Y is the measured value of the random variable, which is subject to measurement error, and Y* is the true value of the random variable, with no measurement error, for which the following four relationships are possible: Here, P CE and P * CE denote the (possibly) observed prices of carbon emissions and their underlying true prices, respectively, and P E and P * E denote the observed and underlying true prices of energy, respectively.
In cases (i) and (ii), the price of carbon emissions is not available, together with the true underlying prices of carbon emissions and energy, respectively. case (iii) is not of particular interest as it relates to the observed and true underlying prices of energy, respectively. In case (iv), the price of energy is available, but the true underlying price of carbon emissions is not. Therefore, this paper is concerned with case (iv).
The remainder of the paper uses a novel econometric model to calculate the latent prices of carbon emissions, as will be discussed below. The approach has been used in the literature to calculate latent variables, and this paper is the first to use the methodological approach to calculate latent carbon emission prices.
For the sake of argument, in case (iv), if the price of energy is stationary, P E can be regressed on a set of observable variables using Ordinary Least Squares (OLS), as given in Equation (6): to yield estimated fitted values, as given in Equation (7): where b i denotes the OLS estimates of the unknown parameters b i (i = 0, 1, 2, 3), and P E denotes the estimated fitted values of energy prices from Equation (6), which are equivalent to the estimated fitted values of carbon emissions, ( P * CE ), adjusted by the factor (1/k) (see Equation (4)), where k may be known in advance or may be determined empirically.
If energy prices are nonstationary rather than stationary, the price model for energy in Equation (6) can be estimated using a cointegrating relationship or a vector autoregressive distributed lag model to accommodate the nonstationarity. Alternatively, prices can be transformed into log differences in prices, which are equivalent to the rate of growth in prices, and then estimated by OLS (in empirical finance, log differences in prices are referred to as rates of return), which would be based on stationary processes.

Estimating Latent Carbon Emission Prices
This section describes a novel method of estimating monthly latent carbon emissions prices for Japan that are based on the observed monthly prices of alternative forms of energy and the models presented in the previous section. In what follows, the models presented in the previous section will be estimated using the available price and volume data to obtain estimated fitted latent prices of carbon emissions. It is assumed that there is a known expected relationship between carbon emissions and the generation of electricity using the data in cases (2)- (12) above. This is based on applied engineering practice and structural form modeling, as presented in the previous section, as follows: or equivalently as log(p 0 q 0 ) = log(k) + log(p 1 q 1 ) + u, where k is assumed to be a known factor of proportionality, and u is assumed to be a random measurement error, possibly independently and identically distributed, with a mean of 0 and constant variance. Equation (8) relates the unknown price and known quantity of carbon emissions to the known price and known quantity of energy (or electricity), which confirms that the resulting price equation directly refers to carbon emissions. Equation (9) is the logarithmic equivalent of Equation (8).
It follows that as E(u) = 0, such that the estimated fitted value of Equation (10) is given as log(e 0 ) = log(e 1 ).
This novel method would seem to be the first to calculate carbon emissions prices in theory, as well as in practice, using monthly data for Japan.
Further to the above discussion, if data were available for p 0 , the price of energy (or electricity), as well as prices on some renewable (Z 1 , say) and nonrenewable (Z 2 , say) energy inputs, we could compare whether Z 1 jointly was more or less significant than Z 2 in explaining p 0 .
The Japanese Government imposes energy taxes on industry, in particular, as well as on households. The use of government data is crucial for consumers and businesses that use energy, as they are expected to pay carbon taxes as users of energy. This is not a financial product, but an application of optimal carbon tax policies.

Monthly Data and Diagnostic Checks
The empirical analysis uses a monthly sample period from December 2008 to April 2018 for Japan, where carbon emission prices have not yet been calculated by any government agency or research institute. The definitions of the variables are given in Table 1 as the volume of seasonally unadjusted and seasonally adjusted (using X12) values of coal, LNG, petroleum, electricity, carbon emissions, the spot electricity price, the total transaction volume of electricity, and the price of electricity that represents the price of energy. The sources of the data are the Trade Statistics of Japan, Ministry of Finance, Government of Japan; the Japan Metrological Agency; and the Japan Electricity Power eXchange. Prior to econometric estimation, the seasonally unadjusted and adjusted volumes of the logarithms of electricity, coal, oil, and LNG were tested for unit roots, for both levels and first differences of the variables, using the non-parametric Phillips-Perron (PP) test of stationarity to ensure that asymptotic theory was valid for the purposes of drawing statistical inferences (see [29] Phillips and Perron (1979)). The test of the null hypothesis that a time series is integrated by an order of 1 is robust to unspecified autocorrelation and heteroscedasticity in the errors of the auxiliary test equation.
The PP tests of the logarithms of the levels and first differences of the seasonally unadjusted and adjusted volumes of electricity, coal, oil, and LNG in Table 2 were conducted using a constant term with a deterministic trend in order to provide alternative simulated critical values of the non-standard test. OLS was used to obtain the estimator of the residual spectrum at a frequency of zero. The PP test detects unit roots of the logarithmic levels of the variables at all reasonable levels of significance, but rejects the null hypothesis of unit roots in logarithmic first differences, so that the log differences of the data are stationary. The correlations in the logarithmic levels of the seasonally unadjusted and adjusted data are shown in Table 3(a), with the corresponding correlations in the logarithmic first differences of the seasonally unadjusted and adjusted data shown in Table 3(b). The highest correlations for the logarithmic levels of the seasonally unadjusted and adjusted data are between oil and LNG, with surprisingly close values of 0.73 and 0.737, respectively. The highest correlations for the logarithmic first differences of the seasonally unadjusted and adjusted data are quite different, with the highest correlation for the former being between oil and LNG at 0.445, and at a much lower 0.267 for coal and oil, as well as for coal and LNG.  Table 4. The null hypothesis of no cointegration (that is, R = 0) can be rejected at the 5% level of significance using both tests for the combination of logarithms of the seasonally unadjusted and adjusted volumes of coal, LNG, oil, and electricity.

Empirical Estimates and Analysis
The results of the PP unit root tests and Johansen tests of the number of cointegrating vectors in the previous section led to fully-modified OLS estimation ( [31] Phillips and Hansen, 1990) of the cointegrating regressions, presented in Table 5.  Two cases are considered, as follows: Case 1: Seasonally unadjusted data, with a deterministic trend and dummy variable; Case 2: Seasonally adjusted data, with a deterministic trend and dummy variable.
The dummy variable, which was introduced to accommodate the influence of increases in electricity prices on May 2017 arising from tax changes in Japan, equals 0 before September 2017, and 1 after September 2017.
For both seasonally unadjusted and seasonally adjusted data, that is, Cases 1 and 2, the logarithms of coal and oil, as well as the deterministic time trend and the tax change dummy variable, are statistically significant in explaining the logarithm of carbon emissions, with oil being more significant than coal. The logarithm of LNG is not statistically significant at any conventional level of significance in explaining the logarithm of carbon emissions in either Case 1 or Case 2. The adjusted R 2 values are reasonably high, at 0.943 and 0.958 for Cases 1 and 2, respectively, which suggests that the models are quite successful in explaining the logarithm of carbon emissions using both seasonally unadjusted and seasonally adjusted data.
The forecasts of log(E) for seasonally unadjusted and seasonally adjusted data are shown in Figure 1a,b, respectively, where two standard errors in each figure provide the 95% confidence intervals. In each case, there were noticeable falls in the forecasts of the logarithm of electricity from mid-2015 through to early-2016, and substantial rises toward the end of 2017. the models are quite successful in explaining the logarithm of carbon emissions using both seasonally unadjusted and seasonally adjusted data.
The forecasts of log( ) for seasonally unadjusted and seasonally adjusted data are shown in Figures 1a and 1b, respectively, where two standard errors in each figure provide the 95% confidence intervals. In each case, there were noticeable falls in the forecasts of the logarithm of electricity from mid-2015 through to early-2016, and substantial rises toward the end of 2017. The predicted latent carbon emission prices for seasonally unadjusted and adjusted data are depicted in Figure 2a  The predicted latent carbon emission prices for seasonally unadjusted and adjusted data are depicted in Figure 2a The predicted latent carbon emission prices for seasonally unadjusted and adjusted data are depicted in Figure 2a  The predicted values are based on the formulae for the logarithmic values presented in Equation (16), with the predicted carbon emissions prices having been obtained using the exponential function for the seasonally unadjusted data, as follows: and for the seasonally adjusted data, as follows: The patterns of the predicted carbon emissions prices in Figure 2a,b emulate the patterns described in Figure 1a,b with significant upward movements in both the seasonally unadjusted and seasonally adjusted data after mid-2017.
The correlation between log( ) and log( ) of 0.997 is given in Table 6, while the correlation between the seasonally unadjusted and seasonally adjusted predicted carbon emissions prices of 0.992 is given in Table 7. It is clear that the forecasts of carbon emission prices are very similar, regardless of whether seasonally unadjusted or seasonally adjusted data are used.  Table 7. Correlation between predicted carbon emissions price and predicted carbon emissions price (SA).
The predicted values are based on the formulae for the logarithmic values presented in Equation (16), with the predicted carbon emissions prices having been obtained using the exponential function for the seasonally unadjusted data, as follows: and for the seasonally adjusted data, as follows: The patterns of the predicted carbon emissions prices in Figure 2a,b emulate the patterns described in Figure 1a,b with significant upward movements in both the seasonally unadjusted and seasonally adjusted data after mid-2017.
The correlation between log(E) and log(E) SA of 0.997 is given in Table 6, while the correlation between the seasonally unadjusted and seasonally adjusted predicted carbon emissions prices of 0.992 is given in Table 7. It is clear that the forecasts of carbon emission prices are very similar, regardless of whether seasonally unadjusted or seasonally adjusted data are used.

Concluding Remarks
It is widely accepted in the international scientific and global communities that climate change and global warming are significantly affected by carbon emissions that arise from the burning of fossil fuels, specifically coal, oil, and gas. Although prices of fossil fuels are readily available, the prices of carbon emissions arising from competitive markets for the product are not yet commercially available.
As carbon emissions can be taxed appropriately when accurate commercial market prices are known and accepted at an international level, carbon emissions can be taxed appropriately for the purposes of measuring, capturing, storing, and trading in carbon emissions at regional, national, and international levels.
This paper used a novel KLEM production function approach to calculate the latent carbon emission prices, where carbon emissions are regarded as the output and capital (K), labor (L), energy (E) (or electricity), and materials (M), are the inputs for the production process. As the inputs K, L, and M are essentially fixed on a monthly basis, whereas E can be changed daily, it follows that changes in carbon emissions are functions of changes in E.
On the condition that prices depend on the average cost pricing, the prices of carbon emissions and energy may be approximated by an energy production model with a constant factor of proportionality, so that carbon emission prices will be a function of energy prices. Using this novel modeling approach, this paper estimated carbon emission prices for Japan using seasonally adjusted and unadjusted monthly data on the volumes of carbon emissions and energy, as well as energy prices, from December 2008 to April 2018.
The novel methods presented in the paper are straightforward to understand and implement, based on simple methods of estimation. The usefulness in establishing carbon emission prices goes to the heart of tackling global warming and climate change.
The econometric models showed that, as sources of electricity, the logarithms of coal and oil, though not LNG, are statistically significant in explaining the logarithm of carbon emissions, with oil being more significant than coal. The models generally displayed a high power in predicting the latent prices of carbon emissions. In particular, the forecasts of carbon emission prices were very similar, regardless of whether seasonally unadjusted or seasonally adjusted data were used. The usefulness of the empirical findings suggests that the methodology can also be applied for other countries where carbon emission prices are latent.
The theoretical approach and empirical application developed in the paper should be useful for purposes of public policy debate and decision making. Accurate predictions of the latent prices of carbon emissions, and the imposition of appropriate environmental taxes, are essential in order to mitigate the effect of carbon emissions on global warming and climate change.
(ii) OLS is the simplest technique that can be employed to obtain optimal weights through estimation, but it is possible to use Logistic regression to obtain optimal weights and the inherent associated probabilities; (iii) In the context of Cognitive Computing, it is widely argued that computers, computing facilities, machine hardware, mathematical algorithms, and computer software should be perceived as aids to learn dynamically, to reach managerial decisions, and to achieve strategic aims; (iv) As advanced machines can be programmed to learn through feedback, an important implication is that the outputs obtained from inputs and processing systems are not the same if learning is allowed because the outputs can differ. Consequently, outputs from such processing systems should be modeled and analyzed in a probabilistic context which, in turn, helps to make managerial decisions.
The cognitive approach is precisely related to realized latent rankings models, in which the probabilistic outputs, namely realized rankings, can change dynamically according to the generated (or realized) regressors and dynamic learning. Therefore, the probabilities would assist in strategic management decision making.
Conditional on the sets of features (such as the latent endogenous variable, and measurable or latent exogenous variables), the model can provide different realized latencies, such as latent carbon emission prices, which can assist management in making optimal decisions. It is assumed in latent probabilistic models that the exogenous regressors are measurable, such as where y* in Equation (A15) is latent, but we observe a binary variable y that takes the value of 1 or 0, according to whether or not y* crosses a threshold, that is, y = 1 if y* > 0, and 0 otherwise. The probabilistic model is given by P(y = 1 x) = P(y * > 0) = P(−e < X β) = F(X β) where F in Equation (A16) is typically a normal or logistic cumulative distribution function (CDF). Then, y* is given in Equation (A17) by y * = F X β .
The model and associated analysis change drastically if the basic latent regression model is changed to incorporate X*, as in Equation (A18), rather than X in Equation (A15): as an appropriate model would be required for X* to obtain generate regressors for X* and realized latencies for y*.
Such an extension of logistic probabilistic models to incorporate latent exogenous variables to estimate and forecast latent endogenous variables is a novel extension of the classical latent probabilistic model, and can be analyzed as follows: (v) Defining and measuring a wide range of latent variables, such as unknown carbon emission prices and academic quality; ranking individuals, departments, faculties, and institutions based on the new measure; and establishing the theoretical properties of the new measures, as well as the associated parametric estimators; (vi) Ranking individuals, departments, faculties and institutions based on non-academic measures, and establishing the theoretical properties of the new measures, as well as of the associated parametric estimators; (vii) Applying the approach based on "realized latent rankings" commercially to any decision making strategies in business, using structural models; multiple decision making based on recursive models, that is, sequential decision making; and strategic decision making using probabilistic models, among others.
It is clear that sequential models will be necessary to obtain "sequential generated regressors", and subsequent "sequential realized latencies", which would present challenging and novel theoretical and practical developments. The approach given above could also be extended to probabilistic models, which would be an entirely new and innovative development.
Each of the points stated above is novel from theoretical and practical perspectives, namely applications of generated regressors and realized latents to obtain measurable and optimal rankings of any latent endogenous variables, and extensions to a wide range of more complicated models and decision strategies. The techniques of the basic model presented above can be extended, and are portable to different problems and disciplines, such as latent carbon emission prices, rankings of important factors, and associated management decisions.