Forecasting Monthly Prices of Japanese Logs

Forecasts of prices can help industries in their risk management. This is especially true for Japanese logs, which experience sharp fluctuations in price. In this research, the authors used an exponential smoothing method (ETS) and autoregressive integrated moving average (ARIMA) models to forecast the monthly prices of domestic logs of three of the most important species in Japan: sugi (Japanese cedar, Cryptomeria japonica D. Don), hinoki (Japanese cypress, Chamaecyparis obtusa (Sieb. et Zucc.) Endl.), and karamatsu (Japanese larch, Larix kaempferi (Lamb.) Carr.). For the 12-month forecasting periods, forecasting intervals of 80% and 95% were given. By measuring the accuracy of forecasts of 12and 6-month forecasting periods, it was found that ARIMA gave better results than did the ETS in the majority of cases. However, the combined method of averaging ETS and ARIMA forecasts gave the best results for hinoki in several cases.


Introduction
Fluctuations and low levels of log prices in Japan are a challenge for forest owners in managing forests.As a result, processing mills experience difficulties in ensuring a stable supply of suitable logs.Additionally, the mismatch between the supply of and demand for logs leads to sharp fluctuations in log prices, such as that observed in the former half of 2012 [1].In other words, log prices not only affect the profitability of forest owners, logging companies and sawmills, but might also affect the daily operations in sawmills.Therefore, having information on future log prices can be useful for the aforementioned parties in their risk management.For example, information about potential price fluctuations will perhaps allow suppliers and users of logs to adjust their supply and demand.When private suppliers of logs feel it is difficult to adjust their supplies due to small size of their operations and the need to cover daily maintenance costs, state-owned forests managers might play an important role by adjusting their supplies of logs.However, forecasts of log prices in Japan have rarely been provided, though analyses on fluctuations in the prices of Japanese logs started long ago.
The impacts of various supply and demand factors on log prices have been studied and the existence of seasonal fluctuations confirmed as early as in the late 1920s [2].Log prices are found to be at their lowest in June and July and to reach their peak in October and November due to the seasonal nature of construction (spring and autumn were peak times), but, in 1910-1920's, seasonal fluctuations have become less pronounced, about 1% of the annual average prices [2].Recently, a monthly seasonal price index for logs of different origins (e.g., Mainland Japan, North America, Hokkaido, South Asia, and Taiwan) in Tokyo and Osaka markets has been calculated [3], but, in response to this research, it was noted that the seasonal increase in autumn is rather limited and should not be expected, because the general commodity market is thought to be more important [4].Since the 1960s, not only the seasonal fluctuations, but the trend movement in log prices with the changes in supply and demand has also been analyzed [5][6][7][8].The cyclical fluctuations and their relationship with the diffusion index, apart from demand and supply, have also attracted the attention of researchers [9].The number of months from one valley to the next valley were calculated to show the cycles of timber prices [9].In the 1980s, a decomposition method, which was developed by Economic Planning Agency, Japan, originating from the Census Method II approach (US Census Bureau), was used in analyzing the trend, cycle, and seasonal movements for log price time series along with their influencing factors [10].In the 2000s, the X-12-ARIMA approach, developed by U.S. Census Bureau and mainly used for seasonal adjustment, was adopted to decompose the price time series into trend components, seasonal components, and irregular components for nine forest products in two private auction markets in the Kyushu region [11].A recent study in the field analyzed the relationship between monthly prices and log inventory in sawmills and stated that pest damage is the reason for low prices from June to August [12].Although the aforementioned research studies paid attention to price fluctuations, few forecasts could be found among them.In the 1970s, a two-step foresting approach was once adopted in which the price time series was decomposed into the following components: secular trend variation, which is determined by stock supply; seasonal variation, which is caused by the seasonal change in activities in construction; cyclical fluctuation, which is largely caused by business cycle; and random variation, which is fitted into an AR mode.These components were then combined together using the models established in the above stages for forecasting.As for the forecasting period, two years of monthly forecasts were provided, as were the lower and upper limits of forecasts at the 70% level, but the methods used to calculate these lower and upper limits were not explained [7].Moving average (MA), autoregressive moving average (ARMA), and seasonal autoregressive integrated moving average (ARIMA) models were once used for fitting price time series of some sawnwood and logs, and it was concluded that the most reasonable forecast results were obtained by using the seasonal ARIMA [11], even though only a simple form of the ARIMA model was considered.
Short-term forecasting of a time series is possible because each time series has its own pattern of movements.To do a forecast well, a good grasp of the situation is important, and forecasters need to make subjective judgments at times.Therefore, statistical forecasting can be described as "the blend of art and science" [13], and the objective of time series forecasting is "to discover the pattern in the historical data series and extrapolate that pattern into the future" [14].More complicated models are not considered in this research because model simplicity is preferred, though exponential smoothing method (ETS) and ARIMA have evolved into complicated forms already.
ETS and ARIMA have made great progress since the 1990s [14,15].The free software R [16] and the package forecast [17] make specifying parameters and comparing models much easier.ETS's model framework has made progress by introducing the state space model as well as other developments, such as stochastic models, likelihood calculation, prediction intervals, and procedures for model selection [15].
In this research, by applying the latest model specification and selection instruments and the algorithm in software R (R Core Team, Vienna, Austria), we analyze the movements in prices of domestic Japanese sawlogs-sugi, hinoki, and karamatsu.We then forecast log prices 12 months ahead by using ETS and ARIMA.In addition to point forecasts, we give forecast intervals at the 80% and 95% levels.Additionally, we apply valuations for forecast accuracy to forecast results given by ETS and ARIMA and concluded that, in most cases, ARIMA obtains better forecasts than ETS does.

Study Objects and Their Data
Forests in Japan cover about 67% of the total area and, due to aggressive planting since 1950s, the area of planted forests in Japan extends to over 10 million ha, which is 40% of the total forest area.Due to the low level of harvest volume compared to the growth level, the total forest stock keeps increasing, and the average growing stock per hectare comes to over 190 m 3 [18].In Japan, most forest owners and sawmills are small in scale.It is difficult for small owners to provide a steady supply of logs, and it is difficult for small mills to produce kiln-dried sawnwood to compete against foreign sawnwood.Increasing costs, which are partly driven by increasing wages, are another factor for the low competitiveness of domestic logs and sawnwood.The production of logs declined to 15 million m 3 by 2002 from 51 million m 3 in 1967, when Japan opened the door to foreign timber products.After the efforts of the Japanese government and wood industry to promote domestic wood, the self-sufficiency rate of wood recovered from its lowest point of 18.2% in 2002 to 31.2% in 2014 [19].
The top three species for log production in Japan-sugi, hinoki, and karamatsu-are mainly harvested from the planted forests.In 2014, the production volumes for these three species are 11.19 million m 3 (56% of the total log production), 2.40 million m 3 (12%), and 2.37 million m 3 (12%), respectively [20].The main prefectures for sugi production are Miyazaki, Akita, Oita, and Kumamoto; the top producers for hinoki are Okayama, Kochi, Ehime, and Kumamoto; and for karamatsu, they are Hokkaido, Iwate, and Nagano (see Figure 1).Sugi and hinoki sawnwood are mainly used in housing construction, and karamatsu sawnwood is mainly used as packaging materials for the storage and transportation of commodities.
Forests 2016, 7, 94 3 of 20 sawnwood.Increasing costs, which are partly driven by increasing wages, are another factor for the low competitiveness of domestic logs and sawnwood.The production of logs declined to 15 million m 3 by 2002 from 51 million m 3 in 1967, when Japan opened the door to foreign timber products.After the efforts of the Japanese government and wood industry to promote domestic wood, the selfsufficiency rate of wood recovered from its lowest point of 18.2% in 2002 to 31.2% in 2014 [19].
The top three species for log production in Japan-sugi, hinoki, and karamatsu-are mainly harvested from the planted forests.In 2014, the production volumes for these three species are 11.19 million m 3 (56% of the total log production), 2.40 million m 3 (12%), and 2.37 million m 3 (12%), respectively [20].The main prefectures for sugi production are Miyazaki, Akita, Oita, and Kumamoto; the top producers for hinoki are Okayama, Kochi, Ehime, and Kumamoto; and for karamatsu, they are Hokkaido, Iwate, and Nagano (see Figure 1).Sugi and hinoki sawnwood are mainly used in housing construction, and karamatsu sawnwood is mainly used as packaging materials for the storage and transportation of commodities.Logs of different diameters and lengths might have different usages, which translates into different prices.In this research, we analyze sugi, hinoki, and karamatsu logs in their most common diameters and lengths for sawnwood processing.For sugi and hinoki, the diameter is 14-22 cm, for karamatsu, it is 14-28 cm; the length considered for the logs of the three species is 3.65-4.00m [20].The prices are monthly volume-weighted average prices for all grades of logs that are used in processing sawnwood under the above-stated diameters and lengths.Our objective is to provide short-term forecasts; thus, considering the data availability, we think that monthly data are suitable.
Given that 2002 was the year when domestic log production decreased to its lowest point since 1960 and subsequently increased, monthly data from January 2002 to September 2015 are used in this research.The current value for monthly prices for logs was sourced from the Ministry of Agriculture, Logs of different diameters and lengths might have different usages, which translates into different prices.In this research, we analyze sugi, hinoki, and karamatsu logs in their most common diameters and lengths for sawnwood processing.For sugi and hinoki, the diameter is 14-22 cm, for karamatsu, it is 14-28 cm; the length considered for the logs of the three species is 3.65-4.00m [20].The prices are monthly volume-weighted average prices for all grades of logs that are used in processing sawnwood under the above-stated diameters and lengths.Our objective is to provide short-term forecasts; thus, considering the data availability, we think that monthly data are suitable.
Given that 2002 was the year when domestic log production decreased to its lowest point since 1960 and subsequently increased, monthly data from January 2002 to September 2015 are used in this research.The current value for monthly prices for logs was sourced from the Ministry of Agriculture, Forestry and Fisheries (MAFF) [20].Japan experienced deflation after the mid-1990s.However, the situation has changed since 2012, when Abenomics policies began to be implemented.Considering the possible impacts of general price level changes on the movements in log prices, we introduced the Corporate Goods Price Index (CGP) to adjust the monthly log prices to a constant value as that in 2010.

Methods
In the short-term forecasting of monthly prices of logs, we applied ETS and ARIMA, two typical forecasting approaches, though many methods have been proposed and applied in the field [14,[21][22][23][24][25][26][27], and there are "as many forecasting methods as there are forecasters" [28].Further, the naïve (or seasonal naïve, shortened as Snaïve) method was also introduced as a reference for measuring the accuracy of forecasting.Because we only used the time series data of monthly log prices and no other variables were included (e.g., housing starts on the demand side, forest resources on the supply side), our research is a univariate time series analysis.Root mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE) are calculated in measuring the accuracy of forecasting results.

Naïve Method and Snaïve Method
Company managers or business people occasionally use simple methods to forecast, and these methods can be useful.Using the average value (the mean) of historical data might be useful for time series that fluctuate around some constant value.For a random walk time series, without any trend and seasonality, a naïve method might be useful in which the most recent observation is taken as a forecast for the next period or periods.For a seasonal time series, a Snaïve method might be useful; by which the actual value in the same period of the previous year is taken as a forecast for that period in this year.In measuring the accuracy of forecasting, the naïve or seasonal Snaïve method was used as a reference for that of ETS and ARIMA, because we think that ETS and ARIMA at least should make forecasts with errors as small as naïve or Snaïve method.

ETS
The ETS forecasting approach was proposed in the 1950s and used in inventory control [29][30][31].The simple ETS has the following form: ŷt`1|t " ŷt|t´1 `αpy t ´ŷ t|t´1 q, or ŷt`1|t " αy t `p1 ´αq ŷt|t´1 (1) where y t is a time series, ŷt|t´1 is the forecast value for y t by taking account of all previous values, y 1 , y 2 , . . ., y t´1 , and α is a smoothing parameter between 0 and 1.For longer-range forecasting by the simple ETS, the forecast formula could be written as ŷt`h|t " ŷt`1|t , h = 2, 3, . . ., where h means h periods ahead [15,32].The most suitable method for a specific time series varies with trend and seasonality.The trend component includes five possibilities: None (N), Additive (A), Additive damped (Ad), Multiplicative (M), and Multiplicative damped (Md).The seasonal component includes three possibilities: None (N), Additive (A), and Multiplicative (M).By combining the trend and seasonal components, in total, this results in 15 methods.If considering additive and multiplicative error terms, there will be 30 methods.The formulae for recursive calculations and point forecasts have been well summarized and can be accessed openly [32].The package forest in R was used in applying exponential smoothing to the three time series, and AIC (Akaike's Information Criterion) corrected, AICc, which is appropriate for small sample bias, was adopted for model selection [32,33].Finally, the best method was chosen by the lowest AICc value.

ARIMA
A time series is weakly stationary if neither the mean nor the autocovariances depend on the time t [34].Economic time series, such as log price, are usually not stationary because the mean and autocovariances sometimes vary with time t.When differencing the time series, the resulting time series, which represents the changes in the series, is usually stationary.The original time series is called a unit root process in this case.ARIMA can be used to describe these types of time series.By adding the seasonality, a general form for ARIMA can be described as ARIMA (p, d, q) (P, D, Q) m , where (p, d, q) is the non-seasonal part of the model; (P, D, Q) is the seasonal part; p and P are orders of AR (autoregressive part); d and D are the degrees of first differencing; q and Q are orders of MA (moving average part); and m is the number of periods in a year.For monthly prices, m is 12 [14,32].
In this research, we first confirmed the situations of autocorrelation and partial autocorrelation in the time series, which was helpful in choosing the order of AR and MA.Then, we tested stationarity and determined the meaningful degrees of first differencing.We tried the possible orders of AR and MA and degree of integration.The best ARIMA models were selected according to their AICc statistics.However, we stopped at one degree of differencing the time series.Differencing twice makes it difficult to explain the economic meaning of the time series and leads to wider forecasting intervals.The unit root test was implemented to examine the stationarity of both the original time series and the differenced time series.Finally, a diagnostic check of the residuals in ARIMA models was conducted to justify model estimations.

Forecasting Intervals
In addition to point forecasts, we need to present forecasting or prediction intervals to show the range of values within which we believe the actual values to fall with some level of probability.Forecasting intervals show the extent of variability and uncertainty, which can be calculated from variances.Under the Gaussian model assumption, the errors are Gaussian and 100 ˆ(1 ´α)% forecasting intervals can be calculated by: where µ is the forecast mean or point forecast at h periods ahead; z α{2 is the α{2 significant point in a Gaussian distribution; and var is variance [15].As shown in statistics textbooks, when the probability level is 0.95, z would be 1.960; when the probability level is 0.80, z would be 1.282.Forecasting intervals become wider when the forecast periods increase because uncertainty increases with time.Error term will not affect the point forecasts because its expectation value is zero, but it plays an important role in calculating variances and forecasting intervals.

Measures of Forecasting Accuracy
Forecast error is generally defined as e t " y t ´ŷ t , or the difference between the observation and the forecast at time t.This definition is good for one-step forecast.In the case of h-periods-ahead forecasts, ŷt`1 , ŷt`2 , . . ., ŷt`h , it is meaningless if e t is summed because positive and negative errors cancel each other out.Thus, a proper summation of the errors is needed.In our research, three accuracy measures were calculated: RMSE, MAE, and MAPE.The former two are scale-dependent measures, and the last is a percentage point error.Of course, smaller measurement values show more accurate forecasts.

Seasonal Characteristics
Seasonality in a time series reflects a pattern of ups and downs over a fixed period of time.In Japan, different regions have different climate characteristics, and each region could have its own pattern.The pattern would also affect logging, sawmilling, and the demand for sawnwood, such as demand due to seasonal wood house construction.In the summer season in southern Japan, it is hot and heavy rain can make logging difficult, can damage the road for transportation, and affect the harvest volume.In Northern Japan, rain can be a problem in summer, but in winter, heavy snow leads to lower logging productivity.This pattern of impacts stemming from changes in weather conditions might lead to a seasonal movement in prices for logs.According to industry experts, however, seasonal price fluctuations are most affected by the presence of pests, which damage the quality of logs.Therefore, sawmills try to adjust their log stock and control their acquisition of new logs in spring and summer to avoid or lessen pest damage.Therefore, log prices are low in summer but increase starting from autumn.In contrast, abnormal weather conditions such as extremely heavy snow or rainfall or typhoons affect supply and, thus, prices.When abnormal weather conditions occur, prices will also change due to the related changes impacting supply and demand.These types of price changes should be considered as irregular movements.Now, we discuss the actual seasonal changes in price for the three species of logs.According to the annual changes against individual months shown in Figures 2a and 3a, a pattern of decreasing prices in spring to summer and increasing prices from autumn to winter was found for sugi and hinoki across the majority of years examined.For both species, the prices usually reach their lowest point in June and July and start to increase in August.In Figures 2b, 3b and 4b, the changes in mean price values were shown by averaging the subseries of the same month for the years 2002-2015.This figure does not show their magnitude in terms of seasonality because irregular movements were also included, but a general overview can be obtained by their changing mean levels.The means of the prices for sugi and hinoki decreased from February to July and increased from August to October or November; therefore, the prices for both sugi and hinoki are seasonal.As for karamatsu, as shown in Figure 4a,b, no obvious pattern of price fluctuations can be found, though August witnessed the lowest level of mean price, in contrast to the findings for sugi and hinoki.This difference among the three species may be caused by the differences in usage and production areas.As aforementioned, hinoki and sugi sawnwood are mainly used in wood house constructions, but karamatsu sawnwood is mainly used in packaging materials for the transportation of commodities.Wood house construction is seasonal, but other industrial production experiences far less seasonality.Another reason might be the different degree of pest damage.The majority of karamatsu logs are harvested in Hokkaido, which is located in Northern Japan, where summer is short and not intensely hot, and, as such, pest damage is not a serious problem for karamatsu logs.harvest volume.In Northern Japan, rain can be a problem in summer, but in winter, heavy snow leads to lower logging productivity.This pattern of impacts stemming from changes in weather conditions might lead to a seasonal movement in prices for logs.According to industry experts, however, seasonal price fluctuations are most affected by the presence of pests, which damage the quality of logs.Therefore, sawmills try to adjust their log stock and control their acquisition of new logs in spring and summer to avoid or lessen pest damage.Therefore, log prices are low in summer but increase starting from autumn.In contrast, abnormal weather conditions such as extremely heavy snow or rainfall or typhoons affect supply and, thus, prices.When abnormal weather conditions occur, prices will also change due to the related changes impacting supply and demand.These types of price changes should be considered as irregular movements.Now, we discuss the actual seasonal changes in price for the three species of logs.According to the annual changes against individual months shown in Figures 2a and 3a, a pattern of decreasing prices in spring to summer and increasing prices from autumn to winter was found for sugi and hinoki across the majority of years examined.For both species, the prices usually reach their lowest point in June and July and start to increase in August.In Figures 2b, 3b, and 4b, the changes in mean price values were shown by averaging the subseries of the same month for the years 2002-2015.This figure does not show their magnitude in terms of seasonality because irregular movements were also included, but a general overview can be obtained by their changing mean levels.The means of the prices for sugi and hinoki decreased from February to July and increased from August to October or November; therefore, the prices for both sugi and hinoki are seasonal.As for karamatsu, as shown in Figure 4a,b, no obvious pattern of price fluctuations can be found, though August witnessed the lowest level of mean price, in contrast to the findings for sugi and hinoki.This difference among the three species may be caused by the differences in usage and production areas.As aforementioned, hinoki and sugi sawnwood are mainly used in wood house constructions, but karamatsu sawnwood is mainly used in packaging materials for the transportation of commodities.Wood house construction is seasonal, but other industrial production experiences far less seasonality.Another reason might be the different degree of pest damage.The majority of karamatsu logs are harvested in Hokkaido, which is located in Northern Japan, where summer is short and not intensely hot, and, as such, pest damage is not a serious problem for karamatsu logs.

Trend and Cycles
The movements for a time series are usually said to be made up of four components-i.e., trend, cyclical, seasonal, and irregular components-but for convenience in short-term forecasting, trend and cycle are usually combined to make trend.Trend is usually defined as a long-term movement (e.g., [35]).As shown in Figures 5-7, sugi, hinoki, and karamatsu each display a different trend.The price for sugi logs experienced a decrease and reached its lowest level during the world financial crisis of 2008-2009.However, its prices started to recover after 2010.In October 2013, the national plan to raise consumption tax from 5% to 8% in April 2014 was communicated to the public.Together with the impacts of the devalued yen, which started several months earlier, a rush demand for domestic wood occurred.Thereafter, sugi prices spiked temporarily and then sat at a higher level than those in the years since 2008.In comparison, the price for hinoki logs is still experiencing pressure

Trend and Cycles
The movements for a time series are usually said to be made up of four components-i.e., trend, cyclical, seasonal, and irregular components-but for convenience in short-term forecasting, trend and cycle are usually combined to make trend.Trend is usually defined as a long-term movement (e.g., [35]).As shown in Figures 5-7, sugi, hinoki, and karamatsu each display a different trend.The price for sugi logs experienced a decrease and reached its lowest level during the world financial crisis of 2008-2009.However, its prices started to recover after 2010.In October 2013, the national plan to raise consumption tax from 5% to 8% in April 2014 was communicated to the public.Together with the impacts of the devalued yen, which started several months earlier, a rush demand for domestic wood occurred.Thereafter, sugi prices spiked temporarily and then sat at a higher level than those in the years since 2008.In comparison, the price for hinoki logs is still experiencing pressure

Trend and Cycles
The movements for a time series are usually said to be made up of four components-i.e., trend, cyclical, seasonal, and irregular components-but for convenience in short-term forecasting, trend and cycle are usually combined to make trend.Trend is usually defined as a long-term movement (e.g., [35]).As shown in Figures 5-7 sugi, hinoki, and karamatsu each display a different trend.The price for sugi logs experienced a decrease and reached its lowest level during the world financial crisis of 2008-2009.However, its prices started to recover after 2010.In October 2013, the national plan to raise consumption tax from 5% to 8% in April 2014 was communicated to the public.Together with the impacts of the devalued yen, which started several months earlier, a rush demand for domestic wood occurred.Thereafter, sugi prices spiked temporarily and then sat at a higher level than those in the years since 2008.In comparison, the price for hinoki logs is still experiencing pressure to decrease.
There is a decreasing demand for homes having the esthetic appearance of traditional Japanese-style houses in which hinoki wood is required.An increasing amount of laminated wood made of other species is now used in housing construction: this has impacted the price of hinoki logs.In the 1960s, the gap between hinoki and sugi prices grew due to the increasing demand for hinoki.Since 1990, the gap has shrunk.Figure 6 shows a decreasing trend in hinoki log prices.Similar to that of sugi, the hinoki log price rose sharply during the period from October 2013 to January 2014.However, the high prices did not last long.Therefore, the spikes during this time should be considered as an irregular movement.The time series as in Figure 7 showed the karamatsu log price fluctuating with a different pattern.The karamatsu log price also declined after 2002, but it fell to its lowest point in 2006.After 2006, it experienced sharp fluctuations but a recovery could be seen.By September 2015, it had recovered to its price of 2000.
to decrease.There is a decreasing demand for homes having the esthetic appearance of traditional Japanese-style houses in which hinoki wood is required.An increasing amount of laminated wood made of other species is now used in housing construction: this has impacted the price of hinoki logs.In the 1960s, the gap between hinoki and sugi prices grew due to the increasing demand for hinoki.Since 1990, the gap has shrunk.Figure 6 shows a decreasing trend in hinoki log prices.Similar to that of sugi, the hinoki log price rose sharply during the period from October 2013 to January 2014.However, the high prices did not last long.Therefore, the spikes during this time should be considered as an irregular movement.The time series as in Figure 7 showed the karamatsu log price fluctuating with a different pattern.The karamatsu log price also declined after 2002, but it fell to its lowest point in 2006.After 2006, it experienced sharp fluctuations but a recovery could be seen.By September 2015, it had recovered to its price of 2000.to decrease.There is a decreasing demand for homes having the esthetic appearance of traditional Japanese-style houses in which hinoki wood is required.An increasing amount of laminated wood made of other species is now used in housing construction: this has impacted the price of hinoki logs.In the 1960s, the gap between hinoki and sugi prices grew due to the increasing demand for hinoki.Since 1990, the gap has shrunk.Figure 6 shows a decreasing trend in hinoki log prices.Similar to that of sugi, the hinoki log price rose sharply during the period from October 2013 to January 2014.However, the high prices did not last long.Therefore, the spikes during this time should be considered as an irregular movement.The time series as in Figure 7 showed the karamatsu log price fluctuating with a different pattern.The karamatsu log price also declined after 2002, but it fell to its lowest point in 2006.After 2006, it experienced sharp fluctuations but a recovery could be seen.By September 2015, it had recovered to its price of 2000.Overall, in the long term, hinoki appears to show a decreasing trend, but it is difficult to say that sugi or karamatsu is experiencing a declining, increasing, or converging trend to some constant value.However, if we shorten the period to only recent years, we may find that the mean price is increasing for karamatsu since 2006 and that the mean price in the period since October 2013 is higher than that from 2008 to 2012, implying a recovering trend for sugi.These findings can be obtained by loess-smoothing the trend obtained from decomposing the time series.Overall, in the long term, hinoki appears to show a decreasing trend, but it is difficult to say that sugi or karamatsu is experiencing a declining, increasing, or converging trend to some constant value.However, if we shorten the period to only recent years, we may find that the mean price is increasing for karamatsu since 2006 and that the mean price in the period since October 2013 is higher than that from 2008 to 2012, implying a recovering trend for sugi.These findings can be obtained by loesssmoothing the trend obtained from decomposing the time series.

ETS Forecast Results
In using ets function in the package forecast under the R software environment, the results were shown in the form of (E, T, S), representing the error, trend and season components, respectively.The results for sugi, hinoki and karamatsu log prices are, respectively, (M, N, A), (M, Ad, A) and (M, N, N).All of these time series have multiplicative errors.Hinoki log price shows a damped trend, whereas sugi and karamatsu display no trend.As for the seasonality, the results show that sugi and hinoki experience additive seasonal movements in prices, but seasonal changes cannot be found in the karamatsu log price, which supports the aforementioned argument regarding seasonal characteristics.The smoothing parameters for sugi are α = 0.9999 and γ = 0.0001; for hinoki, they are α = 0.9999, β = 0.1412, γ = 0.0001, and ∅ = 0.8007; and for karamatsu, α = 0.9999.A high α value shows that time series values are highly affected by the previous value.A small γ value shows that the seasonal component does not change much over the years.Given that the expectation of error is zero, both the additive and multiplicative error methods gave the same forecasts but different forecast intervals.
The above estimated parameters can be substituted into the corresponding formulae in the component form [32] to obtain the final models; for sugi, its result for ETS (M, N, A) is as follows: For hinoki, ETS (M, Ad, A) is as follows: For karamatsu, ETS (M, N, N) is as follows: In the formulae, lt stands for the level or smoothed value of the time series at time t, b for slope, s for seasonal component, m for number of periods in a year (here, m = 12), α, β, γ and ∅ for smoothing parameters, st−m for the seasonal component for the same month of the previous year.Figures 5a, 6a, and 7a show the results by ETS for sugi, hinoki, and karamatsu, respectively, while Figures 5b, 6b, and 7b show the results by ARIMA.The right parts in both sets of figures show the results for the forecasts and the forecast intervals at probability levels of 95% (wider and brighter shadow area) and 80%

ETS Forecast Results
In using ets function in the package forecast under the R software environment, the results were shown in the form of (E, T, S), representing the error, trend and season components, respectively.The results for sugi, hinoki and karamatsu log prices are, respectively , (M, N, A), (M, Ad, A) and (M, N, N).All of these time series have multiplicative errors.Hinoki log price shows a damped trend, whereas sugi and karamatsu display no trend.As for the seasonality, the results show that sugi and hinoki experience additive seasonal movements in prices, but seasonal changes cannot be found in the karamatsu log price, which supports the aforementioned argument regarding seasonal characteristics.The smoothing parameters for sugi are α = 0.9999 and γ = 0.0001; for hinoki, they are α = 0.9999, β = 0.1412, γ = 0.0001, and H = 0.8007; and for karamatsu, α = 0.9999.A high α value shows that time series values are highly affected by the previous value.A small γ value shows that the seasonal component does not change much over the years.Given that the expectation of error is zero, both the additive and multiplicative error methods gave the same forecasts but different forecast intervals.
The above estimated parameters can be substituted into the corresponding formulae in the component form [32] to obtain the final models; for sugi, its result for ETS (M, N, A) is as follows: y t " pl t´1 `st´m qp1 `εt q, l t " l t´1 `αpl t´1 `st´m qε t , s t " s t´m `γpl t´1 `st´m qε t For hinoki, ETS (M, Ad, A) is as follows: y t " pl t´1 `∅b t´1 `st´m qp1 `εt q, l t " l t´1 `∅b t´1 `αpl t´1 `∅b t´1 `st´m qε t , b t " ∅b t´1 `βpl t´1 `∅b t´1 `st´m qε t , s t " s t´m `γpl t´1 `∅b t´1 `st´m qε t For karamatsu, ETS (M, N, N) is as follows: In the formulae, l t stands for the level or smoothed value of the time series at time t, b for slope, s for seasonal component, m for number of periods in a year (here, m = 12), α, β, γ and H for smoothing parameters, s t´m for the seasonal component for the same month of the previous year.Figures 5a, 6a and 7a show the results by ETS for sugi, hinoki and karamatsu, respectively, while Figures 5b, 6b and 7b show the results by ARIMA.The right parts in both sets of figures show the results for the forecasts and the forecast intervals at probability levels of 95% (wider and brighter shadow area) and 80% (narrower and darker shadow area), respectively, for the 12 months ahead (see Table 1 for 12-months-ahead forecast and forecast interval values).

ARIMA Forecast Results
These three time series are not stationary, as shown by their actual values in their left parts in either both sets of figures; their levels changed and did not converge to some constant value.The Augmented Dickey-Fuller (ADF) test, a unit root test, was implemented [36,37].Package tseries in R was used in which the null hypothesis is that the time series has a unit root and is not stationary, and an alternative hypothesis we adopted in this case study is that it is "stationary".For sugi, Dickey-Fuller = ´1.860(p-value = 0.635); after differencing, Dickey-Fuller = ´7.142(p-value < 0.01).For hinoki, Dickey-Fuller = ´2.655(p-value = 0.303); after differencing, Dickey-Fuller =´6.918 (p-value < 0.01).For karamatsu, Dickey-Fuller = ´3.252(p-value = 0.082); after differencing, Dickey-Fuller = ´6.307(p-value < 0.01).p-values above 0.05 in the ADF tests for the original time series show that the null hypothesis that original time series is not stationary and cannot be rejected by a 5% significant level, i.e., providing no evidence against the need for differencing, but that after differencing, p-values become less than 0.01, showing that further integration of a 2nd order can be rejected, and the 1st differences become stationary at a 1% significance level.That is, a degree of one is suitable for the integrated part of the ARIMA models.
By implementing the Arima function in the package forecast, we obtained the following result for sugi as the best model due to its lowest AICc: ARIMA (2, 1, 0) (2, 1, 1), representing two non-seasonal autoregressive terms, two seasonal autoregressive terms, and one seasonal moving averages term.Their coefficients and standard errors are, respectively, 0.457 (0.079), ´0.245 (0.085), ´0.595 (0.145), ´0.441 (0.118), and ´0.367 (0.165).All of them are significant at a 1% significance level, though in forecasting, it is not important to pursue significance parameters.When summarizing this result, the following model can be obtained.

Diagnostic Check of Residuals in ARIMA Models
Finally, we need to verify the adequacy of our ARIMA models by checking their residuals.By fitting a model, we can obtain fitted values and residuals.For a forecasting model, we have observed values and forecasts based on previous observed values, and the differences are residuals: e t " y t ´ŷ t .Residuals should have two properties: uncorrelated and zero mean [32].Correlations between residuals mean that the model is not fitted well because patterns remained in the residuals and should be included in the model.In addition, if the mean of residuals is not zero, then the forecasts are biased.It would be better also to check two other properties for ideal residuals: constant variance and normal distribution [32].
For the three ARIMA models, we first checked residuals' correlations.Figure 8 shows residuals and their autocorrelations and partial autocorrelations with lags of up to 36 for the sugi log price.No pattern can be found in Figure 8a, though some irregular residuals existed.Figure 8b,c show that no significant correlations were confirmed.The situations for hinoki were similar.Figure 9 showed the analysis of the prices for karamatsu logs.Similarly, some irregular residuals were found in Figure 9a, but no pattern was confirmed.Two significant autocorrelations and partial correlations, respectively, were found by checking Figure 9b,c but most of them with a lag over 20.This result can be ignored by considering it an accidental result.We then implemented an ADF test, and the Dickey-Fuller results were ´5.992, ´5.360, and ´6.283, respectively, for sugi, hinoki, and karamatsu.For all of the cases, p-value <0.01, which shows that they are stationary.Finally, we implemented the Box-Ljung test [38].The results for sugi, hinoki, and karamatsu are χ 2 = 22.660 (p-value = 0.540), 15.874 (p-value = 0.893), and 32.014 (p-value = 0.127), which means that these residuals were not distinguishable from a white noise series.
Forests 2016, 7, 94 12 of 20 that no significant correlations were confirmed.The situations for hinoki were similar.Figure 9 showed the analysis of the prices for karamatsu logs.Similarly, some irregular residuals were found in Figure 9a, but no pattern was confirmed.Two significant autocorrelations and partial correlations, respectively, were found by checking Figures 9b and c but most of them with a lag over 20.This result can be ignored by considering it an accidental result.We then implemented an ADF test, and the Dickey-Fuller results were −5.992, −5.360, and −6.283, respectively, for sugi, hinoki, and karamatsu.For all of the cases, p-value <0.01, which shows that they are stationary.Finally, we implemented the Box-Ljung test [38].The results for sugi, hinoki, and karamatsu are χ 2 = 22.660 (p-value = 0.540), 15.874 (pvalue = 0.893), and 32.014 (p-value = 0.127), which means that these residuals were not distinguishable from a white noise series.that no significant correlations were confirmed.The situations for hinoki were similar.Figure 9 showed the analysis of the prices for karamatsu logs.Similarly, some irregular residuals were found in Figure 9a, but no pattern was confirmed.Two significant autocorrelations and partial correlations, respectively, were found by checking Figures 9b and c but most of them with a lag over 20.This result can be ignored by considering it an accidental result.We then implemented an ADF test, and the Dickey-Fuller results were −5.992, −5.360, and −6.283, respectively, for sugi, hinoki, and karamatsu.For all of the cases, p-value <0.01, which shows that they are stationary.Finally, we implemented the Box-Ljung test [38].The results for sugi, hinoki, and karamatsu are χ 2 = 22.660 (p-value = 0.540), 15.874 (pvalue = 0.893), and 32.014 (p-value = 0.127), which means that these residuals were not distinguishable from a white noise series.

Measuring the Accuracy of Forecasts
To measure the forecast accuracy, we divided our data set into two parts: sample data and outof-sample data.We dealt with two forecast periods here: 12 months and 6 months.We established the objective of the research as being to forecast monthly prices and acknowledged the forecasting as being short-term; therefore, being able to forecast one year ahead or some months (less than 12 months) ahead accurately is important.The errors for forecasting sugi, hinoki, and karamatsu log prices for 12 months ahead by ETS and ARIMA were shown in Table 2.As for other lengths, from one to five months, from seven to 11 months, the results for sugi and hinoki were shown in supplementary materials (Tables S1-S10).
Table 2 shows 10 valuations of forecast errors of 12-months-ahead forecasts.Firstly, data from October 2014 to September 2015 were taken as out-of-sample data, while the data from January 2002 to September 2014 were taken as sample data.In the next nine valuations, by keeping 12 months as the forecast period and deleting the last datum in the series every time and moving backward, we obtained nine sets of sample data and out-of-sample data.Thus, the second sample data were from January 2002 to August 2014, and out-of-sample data were from September 2014 to August 2015.Snaïve was used for sugi and hinoki accuracy valuations, but a naïve method was used for karamatsu because the karamatsu log price did not show obvious seasonality.
As shown in Table 2, MAPE, MAE, and RMSE have similar results in comparing the accuracy of forecasts among ETS, ARIMA, and Snaïve method to sugi and hinoki or naïve method to karamatsu; i.e., when MAPE value is the smallest for a method among the three methods, MAE and RMSE values are also smallest for this method.Valuation 1 was the case in which the most recent sample data and out-of-sample data were used.For the sugi log price, ETS had a smaller error than ARIMA, but both ETS and ARIMA had larger errors than the Snaïve method.The results showed the error ranged from 3.96% to 5.57% by MAPE, from 504 Yen to 685 Yen by MAE, and from 670 Yen to 748 Yen by RMSE.For the hinoki log price in Valuation 1, its forecast accuracy was much better.The smallest forecast errors were from ARIMA, which were 1.64% by MAPE, 282 Yen by MAE, and 332 Yen by RMSE.The ETS results were not as good as those of ARIMA but were better than those of the Snaïve method.As for the karamatsu log price in Valuation 1, ETS and ARIMA resulted in the same errors as the naïve method.
By comparing the errors for sugi and hinoki over 10 valuations, Valuations 9 and 10 gave the largest errors due to the impact of the irregular level of prices at the end of 2013.Among the 10 valuations, it can be found that for sugi, ETS had smaller errors twice (i.e., for Valuations 1 and 8); for hinoki, ETS had a better error only once (i.e., for Valuation 4); and in all other cases ARIMA showed smaller errors for sugi and hinoki.As for karamatsu, ETS and ARIMA had the same errors as those by naïve method, except in the two valuations due to the rounding errors (i.e., Valuations 7 and 9).

Measuring the Accuracy of Forecasts
To measure the forecast accuracy, we divided our data set into two parts: sample data and out-of-sample data.We dealt with two forecast periods here: 12 months and 6 months.We established the objective of the research as being to forecast monthly prices and acknowledged the forecasting as being short-term; therefore, being able to forecast one year ahead or some months (less than 12 months) ahead accurately is important.The errors for forecasting sugi, hinoki, and karamatsu log prices for 12 months ahead by ETS and ARIMA were shown in Table 2.As for other lengths, from one to five months, from seven to 11 months, the results for sugi and hinoki were shown in supplementary materials (Tables S1-S10).
Table 2 shows 10 valuations of forecast errors of 12-months-ahead forecasts.Firstly, data from October 2014 to September 2015 were taken as out-of-sample data, while the data from January 2002 to September 2014 were taken as sample data.In the next nine valuations, by keeping 12 months as the forecast period and deleting the last datum in the series every time and moving backward, we obtained nine sets of sample data and out-of-sample data.Thus, the second sample data were from January 2002 to August 2014, and out-of-sample data were from September 2014 to August 2015.Snaïve was used for sugi and hinoki accuracy valuations, but a naïve method was used for karamatsu because the karamatsu log price did not show obvious seasonality.
As shown in Table 2, MAPE, MAE, and RMSE have similar results in comparing the accuracy of forecasts among ETS, ARIMA, and Snaïve method to sugi and hinoki or naïve method to karamatsu; i.e., when MAPE value is the smallest for a method among the three methods, MAE and RMSE values are also smallest for this method.Valuation 1 was the case in which the most recent sample data and out-of-sample data were used.For the sugi log price, ETS had a smaller error than ARIMA, but both ETS and ARIMA had larger errors than the Snaïve method.The results showed the error ranged from 3.96% to 5.57% by MAPE, from 504 Yen to 685 Yen by MAE, and from 670 Yen to 748 Yen by RMSE.For the hinoki log price in Valuation 1, its forecast accuracy was much better.The smallest forecast errors were from ARIMA, which were 1.64% by MAPE, 282 Yen by MAE, and 332 Yen by RMSE.The ETS results were not as good as those of ARIMA but were better than those of the Snaïve method.As for the karamatsu log price in Valuation 1, ETS and ARIMA resulted in the same errors as the naïve method.
By comparing the errors for sugi and hinoki over 10 valuations, Valuations 9 and 10 gave the largest errors due to the impact of the irregular level of prices at the end of 2013.Among the 10 valuations, it can be found that for sugi, ETS had smaller errors twice (i.e., for Valuations 1 and 8); for hinoki, ETS had a better error only once (i.e., for Valuation 4); and in all other cases ARIMA showed smaller errors for sugi and hinoki.As for karamatsu, ETS and ARIMA had the same errors as those by naïve method, except in the two valuations due to the rounding errors (i.e., Valuations 7 and 9).factors into account.For example, Organization of the Petroleum Exporting Countries (OPEC) quota, OPEC production, and industrial stock level of oil can be used to forecast short-term oil price [39].However, there are not many empirical studies on price forecasting by using structural time series models, perhaps due to the difficulties in quantifying the related factors and those factors must be forecasted prior to forecasting prices.With univariate forecasting approaches, the only variable is log price in this research, which makes forecasting simple and sometimes useful.Of course, the ETS method simply decomposes a time series into trend, seasonal, and error components without taking other elements into account, such as cyclical movements.As for ARIMA models, it might be a good idea to add exogenous variables to the model [40,41].Furthermore, it is also worthwhile to apply equilibrium, structural, and reduced form models in forecasting the prices of Japanese logs.
Abenomics, a policy that went into effect in Japan in December 2012, ensured that corrections to the excessive yen appreciation were made.In addition, when the plan to raise consumption tax from 5% to 8% in April 2014 was communicated to the public in October 2013, a spike in demand for wood house construction led to sharp increases in sugi and hinoki log prices from October 2013 to January 2014.After this period, hinoki log prices dropped to their earlier level, whereas sugi log prices dropped and fluctuated at a higher level than previously.These contextual changes made short-term forecasts from this specific period uncertain.Fortunately, concerns that increased tax would case an economic recession have not become a reality.These price fluctuations in sugi and hinoki can be taken as irregular movements.That is, the structural changes in the logs market in Japan have not occurred.
However, any factors that affect demand and supply, including any changes in the international and domestic economic environment, might affect prices.The impact of the increase in consumption tax (from 8% to 10%) in the near future and fluctuations in exchange rates were not discussed in the research.Another important issue is the impact of the general price level.Constant price data were used in the research.Given the low inflation value, it did not make a difference in comparison to using current value log prices.Using a constant value mitigated the impact of general price changes.However, it will be necessary to recalculate the forecasts during periods of higher inflation.In addition, impacts of changes in the international market and policy are also not dealt with in the research.Incorporating influencing factors into forecasts of log prices should also be a focus of further research.Supplementary Materials: Supplementary materials are available online at http://www.mdpi.com/1999-4907/7/5/94/s1.

Figure 1 .
Figure 1.Top production regions of sugi, hinoki, and karamatsu logs.The darkest color shows the production regions for karamatsu; the brightest color for production regions of sugi; while others for hinoki.

Figure 1 .
Figure 1.Top production regions of sugi, hinoki, and karamatsu logs.The darkest color shows the production regions for karamatsu; the brightest color for production regions of sugi; while others for hinoki.

Figure 2 .
Figure 2. Annual and monthly sugi log prices: (a) Annual changes of sugi log prices against months; (b) monthly subseries of sugi log prices.In Figure 2b, the horizontal bars show the mean prices for the monthly subseries, and polygonal lines show the changes over the years from 2002 to 2015.

Figure 2 .
Figure 2. Annual and monthly sugi log prices: (a) Annual changes of sugi log prices against months; (b) monthly subseries of sugi log prices.In Figure 2b, the horizontal bars show the mean prices for the monthly subseries, and polygonal lines show the changes over the years from 2002 to 2015.

Figure 3 .
Figure 3. Annual and monthly hinoki log prices: (a) Annual changes of hinoki log prices against months; (b) Monthly subseries of hinoki log prices.In Figure 3b, the horizontal bars show the mean prices for the monthly subseries, and polygonal lines show the changes over the years from 2002 to 2015.

Figure 4 .
Figure 4. Annual and monthly karamatsu log prices: (a) Annual changes of karamatsu log prices against months; (b) monthly subseries of karamatsu log prices.The horizontal bars show the mean price for the monthly subseries, and polygonal lines show the changes over the years from 2002 to 2015.

Figure 3 .
Figure 3. Annual and monthly hinoki log prices: (a) Annual changes of hinoki log prices against months; (b) Monthly subseries of hinoki log prices.In Figure 3b, the horizontal bars show the mean prices for the monthly subseries, and polygonal lines show the changes over the years from 2002 to 2015.

Figure 3 .
Figure 3. Annual and monthly hinoki log prices: (a) Annual changes of hinoki log prices against months; (b) Monthly subseries of hinoki log prices.In Figure 3b, the horizontal bars show the mean prices for the monthly subseries, and polygonal lines show the changes over the years from 2002 to 2015.

Figure 4 .
Figure 4. Annual and monthly karamatsu log prices: (a) Annual changes of karamatsu log prices against months; (b) monthly subseries of karamatsu log prices.The horizontal bars show the mean price for the monthly subseries, and polygonal lines show the changes over the years from 2002 to 2015.

Figure 4 .
Figure 4. Annual and monthly karamatsu log prices: (a) Annual changes of karamatsu log prices against months; (b) monthly subseries of karamatsu log prices.The horizontal bars show the mean price for the monthly subseries, and polygonal lines show the changes over the years from 2002 to 2015.

Figure 5 .
Figure 5. Forecasts of the sugi log prices: (a) Point forecasts and forecast intervals at 80% and 95% levels by exponential smoothing method (ETS); (b) point forecasts and forecast intervals at 80% and 95% levels by autoregressive integrated moving average (ARIMA).

Figure 6 .
Figure 6.Forecasts of the hinoki log prices: (a) Point forecasts and forecast intervals at 80% and 95% levels by ETS; (b) point forecasts and forecast intervals at 80% and 95% levels by ARIMA.

Figure 5 .
Figure 5. Forecasts of the sugi log prices: (a) Point forecasts and forecast intervals at 80% and 95% levels by exponential smoothing method (ETS); (b) point forecasts and forecast intervals at 80% and 95% levels by autoregressive integrated moving average (ARIMA).

Figure 5 .
Figure 5. Forecasts of the sugi log prices: (a) Point forecasts and forecast intervals at 80% and 95% levels by exponential smoothing method (ETS); (b) point forecasts and forecast intervals at 80% and 95% levels by autoregressive integrated moving average (ARIMA).

Figure 6 .
Figure 6.Forecasts of the hinoki log prices: (a) Point forecasts and forecast intervals at 80% and 95% levels by ETS; (b) point forecasts and forecast intervals at 80% and 95% levels by ARIMA.

Figure 6 .
Figure 6.Forecasts of the hinoki log prices: (a) Point forecasts and forecast intervals at 80% and 95% levels by ETS; (b) point forecasts and forecast intervals at 80% and 95% levels by ARIMA.

Figure 7 .
Figure 7. Forecasts of the karamatsu log prices: (a) Point forecasts and forecast intervals at 80% and 95% levels by ETS; (b) point forecasts and forecast intervals at 80% and 95% levels by ARIMA.

Figure 7 .
Figure 7. Forecasts of the karamatsu log prices: (a) Point forecasts and forecast intervals at 80% and 95% levels by ETS; (b) point forecasts and forecast intervals at 80% and 95% levels by ARIMA.

Figure 8 .
Figure 8. Residuals and their autocorrelations in sugi ARIMA model: (a) Plot of residuals; (b) autocorrelations with lags of up to 36 for the sugi log prices; (c) partial autocorrelations with lags of up to 36 for the sugi log prices.

Figure 8 .
Figure 8. Residuals and their autocorrelations in sugi ARIMA model: (a) Plot of residuals; (b) autocorrelations with lags of up to 36 for the sugi log prices; (c) partial autocorrelations with lags of up to 36 for the sugi log prices.

Figure 8 .
Figure 8. Residuals and their autocorrelations in sugi ARIMA model: (a) Plot of residuals; (b) autocorrelations with lags of up to 36 for the sugi log prices; (c) partial autocorrelations with lags of up to 36 for the sugi log prices.

Figure 9 .
Figure 9. Residuals and their autocorrelations in karamatsu ARIMA model: (a) Plot of residuals; (b) autocorrelations with lags of up to 36 for the sugi log prices; (c) partial autocorrelations with lags of up to 36 for the sugi log prices.

Figure 9 .
Figure 9. Residuals and their autocorrelations in karamatsu ARIMA model: (a) Plot of residuals; (b) autocorrelations with lags of up to 36 for the log prices; (c) partial autocorrelations with lags of up to 36 for the sugi log prices.