Analysis on the Reverted Mean of Log Price Dynamics Through Stochastic Modeling

: Expectation of the future trend and volatility of log prices has an impact on forest management decisions under stochastic environments. In this paper, we investigate the reverted mean level of log price dynamics using several variants of the mean-reverting process. Market-based log price data in Fukuoka Prefecture, along with nationwide averages, are used for the analysis. Target timber products are 3 and 4 m sugi ( Cryptomeria japonica ) and hinoki ( Chamercypress obtusa ) dimensional log. Parameter estimation is carried out by a quasi-maximum likelihood method. Our analysis shows that the reverted mean price differs significantly between the nationwide average and market-based price dynamics. Because the nationwide average price data smoothes market-based price dynamics, the underlying volatility is underestimated when compared to market-based price data. The estimated parameter values of all models from the nationwide average price are also much smaller than the estimated parameters from the market-based models. The reverted mean derived from the market-based data tends to show a decreasing trend over the time horizon, while the nationwide data does not.

mean of the price dynamics using four of the thirteen stochastic models with the mean-reverting property proposed by Yoshimoto and Shoji (2002). The paper is organized as follows. In the next section, the four stochastic models used here are described using the parameter estimation method. In the third section, the estimated parameters, and the reverted mean of each price dynamic, are provided as the results of our analysis. The final section presents some concluding remarks.

State Dependent Volatility Models and Parameter Estimation
In this paper, we use the state dependent volatility process with the meanreverting property for log price dynamics (denoted as SDVP-MR hereafter). Letting x t be a log price at time t, SDVP-MR is expressed by, [1] Case 1: where B t is a standard Brownian motion with the following characteristics: 1.
{ , 0} t B t ≥ has stationary and independent change 3. for all t (>0), t B follows the normal distribution with a variance of t and a mean of 0 The set of parameters ( , , ) α β σ are positive coefficients and thus strictly constrain equation [1] to be mean-reverting. Changing the value of γ to 1, ½, and 0, as in Yoshimoto and Shoji (2002), three other mean-reverting models can be considered: With the exception of model [4], parameter estimation is carried out by the pseudolikelihood approach based on discretization by the local linearization method. The idea of the local linearization method was first introduced by Ozaki (1985) (see also Ozaki, 1992Ozaki, , 1993, and has been utilized for parameter estimation of nonlinear stochastic models with a finite sample. Shoji and Ozaki (1997) and Shoji (1998) implemented an extension of this method. An original nonlinear stochastic differential equation is first converted into a stochastic differential equation with a constant diffusion term, and then a nonlinear drift term of the derived stochastic differential equation is locally approximated by a linear function of the state and time over a small time elapse. Because the resultant stochastic differential equation can be solved analytically, the corresponding likelihood function for parameter estimation is derived. A detailed description of this method can be also found in Yoshimoto and Shoji (2002).
Following Yoshimoto and Shoji (2002), for SDVP-MR the quasi log-likelihood function for the data set, { t x }, under 1, 0 γ ≠ is derived as Elements in the above two equations [7] and [8] are: For 0 γ = , we have an analytical solution, so that the log-likelihood function is Note that n t is the time of the n-th observation and n t y is the corresponding logtransformed data of n t x . In order to strictly constrain parameters ( , , ) α β σ to be positive, the following exponential transformation is applied where 1 2 3 ( , , ) θ θ θ are unrestricted in the range of ( , ) −∞ ∞ . [20]

Analysis of Estimated Parameters and Reverted Means
Analysis was performed for two types of the time series data, nationwide average and market-based log prices for sugi (Cryptomeria japonica) and hinoki (Chamaecypress obtusa). Nationwide price data was taken from monthly time series data of log prices between January 1975 and September 2006, along with annual data from 1970 to 1975, to develop 392 data points covering the duration from 1970 to 2006. This data was taken from the Japanese Forestry Agency's annual report on the supply and demand of timber (Rinyacho, 1975(Rinyacho, -2006. Logs included in the report were offered for sale in the Japanese timber market and had a diameter of 14-22cm and a length of 3.65-4.0m. The market-based data was taken from the log market at Fukuoka Prefecture (Ukiha Log Auction Market) and included 985 data points. Logs included in this data had a diameter of 14-18cm and lengths of 3 or 4m. Figure 1 shows price dynamics of these data sets from 1970 to 2006. Sugi3m, sugi4m, and sugi4mave represent sugi with 3 and 4m length marketbased data and 4m nationwide average data, respectively. Likewise, hinoki3m, hinoki4m, and hinoki4mave represent the same corresponding data points for hinoki logs. An increasing price trend can be observed for the first decade, with a peak in the late 1980s, followed by a decreasing trend to the present time (see Figure 1).
Based on the estimated parameter values, the reverted mean was calculated as follows: First, the target stochastic differential equation was converted into an equation with a constant volatility process With this conversion, we have the following constant volatility process: Setting the drift term equal to zero, we estimate the reverted mean from the following equation: Figure 4 depicts the reverted mean derived for different log products and different models over different periods of time, while Table 2 shows their statistics.
As observed in Figure 4, the nationwide average log data tended to result in a lower

Concluding Remarks
From an economic viewpoint on price movement, a mean-reverting stochastic model would be preferable because of the microeconomic reasons for the long-run marginal cost of production reflecting the mean price (Insley 2002). We do not expect price to increase infinitely or decrease to zero or lower over the time horizon.
In this paper, we applied a state dependent volatility process with a mean-reverting property to investigate how the reverted mean differs when price dynamics of different products and models are applied. Our analysis demonstrated that reverted mean price varies depending on the models and products used. There was a significant difference between the nationwide average and market-based price dynamics. Because the nationwide average price data smoothes the market-based price dynamics, the underlying volatility is underestimated when compared to the a) 1965 1970 1975 1980 1985 1990 1995   market-based price data. The estimated parameter values of all models from the nationwide average price are also much smaller than the estimated parameters from the market-based models. This implies that using the nationwide average data could result in the wrong reverted mean for forecasting the future trend of price dynamics for local forest owners. Forest owners must face market-based prices; thus, it may be important to focus analysis on local markets. There is an urgent need for the analysis of local market data because almost all research on log prices uses nationwide average data that is frequently more available. Forestry practices in local forest areas are increasingly diminished by unfavorable price changes.