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Time series modeling of pulp prices

Time series modeling of pulp prices

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  • 1. Time Series Modelling of Pulp Objectives and Overview Prices Mônica Barros To analyze quarterly (and monthly) data on pulp prices. Reinaldo Castro Souza Pulp prices: highly volatile series - large drops in prices have occurred on the past DEE, PUC-RIO years. On recent years, pulp prices had suffered August 1997 from an erratic behavior, which can be related to global demand for paper and the level of inventories. info@mbarros.com 1 info@mbarros.com 2 We present 2 distinct Characteristics of Pulp Market approaches to the problem: and Industry 1- Dynamic regression model to forecast Capital intensive pulp prices and, International (over 80 % of Bleached Kraft 2- Simulation approach based on returns Pulp is exported from country of production and there are about 30 calculated from pulp prices. producing countries) Scattered production - even the largest supplier cannot control prices (has less than 6 % of global market share) info@mbarros.com 3 info@mbarros.com 4
  • 2. Characteristics of Pulp Market Characteristics of Pulp Market and Industry and Industry Highly Integrated - most companies are New pulp plants under construction in both pulp and paper producers. Asia and Latin America. Supply tends to increase substantially for Pulp Price increases are not necessarily a the next 5 years. major problem for paper producers, since they also increase their prices. Pulp prices directly affect profitability of paper mills. However, there is some upper bound for Thus, pulp and paper prices tend to move paper prices - above a certain level in the same direction. demand tends to decrease. info@mbarros.com 5 info@mbarros.com 6 Characteristics of Pulp Market The Available Data and Industry Production costs tend to be lower in Here we deal with quoted prices (in US$) - emerging countries, but technological individual contract prices may vary. changes in plants have lowered Quoted prices serve as a guideline for considerably the costs of production in contracts (usually an upper bound) the USA, Canada and Scandinavian NORSCAN (North America + Scandinavian Countries) inventories are available on a countries, which were historically the monthly basis and may serve as an largest pulp producers. explanatory variable for pulp prices. info@mbarros.com 7 info@mbarros.com 8
  • 3. The Available Data The Available Data Quarterly Prices: available since the 1st NORSCAN inventories available monthly quarter of 1978. since January 1985. The price series used refers to constant To match NORSCAN inventories, we will prices (base = December 1996) whose use price data from January 1985 on. values in US dollars were inflated using The idea of using an artificial monthly the Consumer Price Index (CPI) in the price series is due to the fact that we USA. intend to use monthly inventories as a Monthly prices are obtained by leading indicator of price changes. interpolation (via CPI) quarterly prices. info@mbarros.com 9 info@mbarros.com 10 NORSCAN Inventories versus Pulp Prices Pulp Prices Low prices usually correspond to high Pulp Prices (US$) inventories, the converse is also true !!! 1100.00 1000.00 900.00 NORS CAN Inv e nto rie s v e rs us Pric e 800.00 2500 2300 700.00 2100 600.00 1900 1700 500.00 1500 400.00 1300 1100 300.00 900 700 500 300.00 400.00 500.00 600.00 700.00 800.00 900.00 1000.00 1100.00 p ric e
  • 4. Basic Dynamic Regression Dynamic Regression Models Model 1- Basic Model Structure All variables are significant, no further pricet = b0 + b1.(trendt) + b2.(invt-1) + b3.(pricet-1) lags of Y or inv are necessary, residuals + b4.(pricet-3) + et do not show serial correlation BUT: where : The model is unable to identify radical pricet = pulp price at time t price changes! trendt = linear trend The usefulness of the model is restricted invt-1 = NORSCAN inventory at time t-1 to “normal” periods, that is, whenever et = error term substantial price increases or drops do not occur. info@mbarros.com 13 info@mbarros.com 14 Basic Dynamic Regression Basic Dynamic Regression Model Model For example, the 6 months ahead All forecasts seem reasonable and real forecasts obtained by fitting the model values fall within the confidence limits. until June 1995 are: A totally different situation occurs when P eriod 1995.07 L ower 2.5 % 897 F orecast 965 U p per 97.5 % 1033 R eal V alue 969 we consider 6 months ahead forecasts 1995.08 856. 939 1022 927 produced by the model fitted until 1995.09 845 933 1022 924 December 1995, as shown next: 1995.10 833 928 1023 1005 1995.11 787 887 987 956 1995.12 725 828 931 910 info@mbarros.com 16
  • 5. Basic Dynamic Regression Basic Dynamic Regression Model Model The 6 months ahead forecasts obtained by Now the forecasts generated by the model fitting the model until December 1995 are: are clearly inappropriate, since the model fails to capture the substantial price fall P erio d L o w er 2 .5 F o reca st U p p er 9 7 .5 R ea l V a lu e occurring on the second quarter of 1996. % % 1 9 9 6 -0 1 786 854 922 638 1 9 9 6 -0 2 704 787 870 634 1 9 9 6 -0 3 623 712 802 632 1 9 9 6 -0 4 543 639 734 433 1 9 9 6 -0 5 504 605 706 431 1 9 9 6 -0 6 506 611 716 430 info@mbarros.com 18 What to do then ????? What to do then ????? Inventories seem to be a reasonable Moreover, the inventory series is more leading indicator for prices. “predictable” than the price series, so more accurate and reasonable forecasts Also, NORSCAN inventories are available can be generated for NORSCAN monthly, and this might give us an idea, inventories. within one quarter, of the price movements on the next quarter. The important question is: when are the inventories “atypical”, and can this information be useful in the prediction of pulp prices? info@mbarros.com 19 info@mbarros.com 20
  • 6. What to do then ????? Dummy Variable Creation To access whether inventory levels are The Moving Averages computed with 6 “typical” or not we compare observed and 12 months of data are much NORSCAN inventories with forecasts “smoother” estimators of inventories generated by Moving Averages (MA) using than 2 and 3 months moving averages. 2, 3, 6 and 12 months. We would expect forecasting errors to be The discrepancies between each MA large when using the moving averages prediction and the actual inventory is with 6 and 12 points as predictors. measured, and we define dummy variables for the corresponding periods. info@mbarros.com 21 info@mbarros.com 22 Dummy Variable Creation Dummy Variable Creation The dummy variables constructed from Why use Moving Averages to construct the dummies? the forecasting errors of the moving average models indicate whether The basic answer is: simplicity! The inventories are within “reasonable” procedure can be easily implemented in a bounds. worksheet without any knowledge of Time Series methods. Any other automatic method would also work, but the client would need access to a statistical software. info@mbarros.com 23 info@mbarros.com 24
  • 7. Dummy Variable Creation Dummy Variable Creation Define the percent discrepancy between where MAt denotes the moving average the observed NORSCAN inventory at time prediction for the inventory at time t and t and its moving average prediction as: invt is the real value of NORSCAN inventory at t. DIFt = 100. ( MAt − invt ) If DIFt > 0, inventories were overestimated MAt by the moving averages, otherwise they were underestimated. info@mbarros.com 26 Dummy Variable Creation Dummy Variable Creation We next compute the DIF variables based Percentage errors (in absolute value) tend on 2, 3 , 6 and 12 months Moving are larger when we estimate inventories Averages in the period January/1986 to using the “smoother” estimators (6 and 12 February/1997 and look at their descriptive months moving averages). statistics. We observe that: For example, the largest ( in absolute value) forecast error is -64.1%, indicating that the inventory was severely underestimated by a moving average with 12 points. info@mbarros.com 27 info@mbarros.com 28
  • 8. Dummy Variable Creation Dummy Variable Creation Negative errors tend to be larger (in The 25% percentile for all forecast error series absolute value) than positive errors. (except that generated from the 12 point moving average) is around -9. Thus, moving averages estimators tend to The 75% percentile for all error series is larger grossly underestimate inventories. than 7. The moving average estimators are very We propose the use of the percentiles of the forecast error distributions as bounds used to bad when inventory level rises. indicate whether the inventory level at a given month is “atypical”. info@mbarros.com 29 info@mbarros.com 30 Dummy Variable Creation Dummy Variable Creation Let DIFMM2, DIFMM3, DIFMM6, DIFMM12 denote, respectively, the forecast errors Percentile DIFMM2 DIFMM3 DIFMM6 DIFMM12 generated by predictions based on 2, 3, 6 10 % 20 % -14.3 -11.4 -14.8 -10.8 -21.4 -11.5 -34.4 -16.3 and 12 months moving averages. 30 % -6.3 -6.9 -7.4 -9.0 40 % -3.7 -3.4 -3.0 -3.9 50 % (median) -0.95 +0.25 +0.40 +1.95 The next table shows the percentiles for 60 % 70 % 2.1 4.9 3.3 5.9 3.6 7.4 5.9 9.8 the forecast errors based on 2, 3, 6 and 12 80 % 8.7 8.3 11.4 17.1 months moving averages. 90 % 13.8 12.2 15.5 22.0 info@mbarros.com 31
  • 9. Dummy Variable Creation Dummy Variable Creation Based on these values we define as An “abnormal” inventory level is detected “abnormal” situations those when the when : forecast error is outside the [p30% , p70%] DIFMM2 [ -6, 5] or interval. DIFMM3 [ -7, 6] or DIFMM6 [ -7 , 7] or We round these percentiles when creating DIFMM12 [ -9, 10] the dummy variables, and we obtain the following decision rule: info@mbarros.com 33 info@mbarros.com 34 Two Fundamental Questions Dummy Variable Creation Now Arise We next create dummy variables based on 1- Do the dummy variables have any this decision rule. The dummies are explanatory power over pulp prices? constructed as: This can be verified through one way ANOVA models using the dummies as  −1 if DIFM belowthe lower limit of the interval M factors.  I =  0 if DIFM is inside the interval M 2- Do we improve forecasts of pulp prices  1 if DIFM is above the interval's upper limit M when dummies are included in the  dynamic regression model? info@mbarros.com 36
  • 10. Do Dummies have explanatory Regression Model including power over Pulp Prices ? dummies We fit one way ANOVA models for Pulp We add dummy variables constructed Prices using the dummies as factors. from 2 and 12 month MA inventory All ANOVA models are significant at 90% estimators to the basic model. level. The new model structure is: Thus, different levels of dummies affect pricet = b0 + b1.(trendt) + b2.(invt-1) + pulp prices. b3.(pricet-1) + b4.(pricet-3) + b5.(dummy2t) + We next insert dummies into the basic b6.(dummy12t) + et regression model to access whether any gains in forecasting are accomplished. info@mbarros.com 37 info@mbarros.com 38 Regression Model including Regression Model including dummies dummies Where dummy2 and dummy12 are Dummies based on 3 and 6 months MA constructed from the prediction errors were not included in the model, because using 2 and 12 month moving averages to they’re highly insignificant. predict inventories. These 2 variables are not “strictly” significant, their significance level is about 85 %. info@mbarros.com 39 info@mbarros.com 40
  • 11. Forecast Comparison Forecast Comparison The next table shows the forecasts produced from 2 models, that exclude and include dummy variables A B S O LU T E V ALU E S O F % E R R O R S m o d e l w it h m o d e l w it h o u t O N E S T E P AHE AD F O R E C AS T S d u m m ie s d u m m ie s J a n /9 6 3 3 .2 3 3 3 .8 6 a ctu a l fo r e c a s t % error fo r e c a s t % error F eb /9 6 5 .9 9 3 .6 3 p r ic e ( w it h d u m m ie s ) ( w it h o u t d u m m ie s ) % J a n /9 6 638 850 - 3 3 .2 3 854 - 3 3 .8 6 M a r/9 6 9 .8 1 7 .9 1 F eb /9 6 634 596 5 .9 9 611 3 .6 3 A p r /9 6 2 4 .2 5 2 1 .7 1 M a r/9 6 632 570 9 .8 1 582 7 .9 1 M a y/9 6 0 .0 0 1 .1 6 A p r /9 6 433 538 - 2 4 .2 5 527 - 2 1 .7 1 J u n /9 6 1 1 .6 3 1 1 .1 6 M a y/9 6 431 431 0 .0 0 426 1 .1 6 J u l/9 6 1 1 .0 9 1 2 .2 0 J u n /9 6 430 480 - 1 1 .6 3 478 - 1 1 .1 6 A u g /9 6 3 .9 0 2 .7 8 J u l/9 6 541 481 1 1 .0 9 475 1 2 .2 0 S ep /9 6 3 .9 0 4 .2 8 A u g /9 6 539 518 3 .9 0 524 2 .7 8 O c t/9 6 1 .8 1 2 .1 8 S ep /9 6 538 517 3 .9 0 515 4 .2 8 N o v /9 6 2 .0 0 0 .7 3 O c t/9 6 551 541 1 .8 1 539 2 .1 8 D ec/9 6 4 .7 5 4 .3 9 N o v /9 6 549 538 2 .0 0 545 0 .7 3 J a n /9 7 3 .2 0 5 .0 0 D ec/9 6 547 521 4 .7 5 523 4 .3 9 F eb /9 7 0 .6 4 1 .0 6 J a n /9 7 500 516 - 3 .2 0 525 - 5 .0 0 F eb /9 7 470 473 - 0 .6 4 475 - 1 .0 6 M a r/9 7 0 .7 9 1 .4 7 M a r/9 7 443 446 - 0 .7 9 449 - 1 .4 7 A p r /9 7 0 .8 8 2 .8 6 A p r /9 7 455 451 0 .8 8 442 2 .8 6 M a y/9 7 2 .0 4 2 .8 6 M a y/9 7 490 480 2 .0 4 476 2 .8 6 m ean : 7 .0 5 7 .0 1 m ean % error : - 1 .6 2 - 1 .7 2 m ax : 3 3 .2 3 3 3 .8 6 m in . % e r r o r : - 3 3 .2 3 - 3 3 .8 6 m in : 0 .0 0 0 .7 3 m ax. % error : 1 1 .0 9 1 2 .2 0 Forecast Comparison Forecast Comparison ABS OLUTE VALUES OF % ERRORS "QUARTERLY" FORECAS TS Some gain was obtained by including the model with model without dummies, but it is marginal!!! dummies dummies Jan/96 33.23 33.86 Apr/96 24.25 21.71 Jul/96 11.09 12.20 The comparison of forecasts was done in Oct/96 1.81 2.18 a “worst case” scenario. Jan/97 3.20 5.00 Apr/97 0.88 2.86 mean : 12.41 12.97 The price drop from 1996/Q1 to 1996/Q2 max : 33.23 33.86 min : 0.88 2.18 was roughly 32%. info@mbarros.com 44
  • 12. Forecast Comparison Forecasts until December 1997 One should attempt to include other Basic scenario uses automatic forecasts explanatory variables, but there aren’t for NORSCAN inventories. many available candidates! These projections correspond to dummies Shipment/Inventory ratio is one of the with value 0 until the end of 1997. candidates, but shipments series is short (since 1994). info@mbarros.com 45 info@mbarros.com 46 Alternative Approach: Forecasts until December 1997 Simulation We now adopt a totally different approach. m with odel m without odel m with odel dum ies m dum ies m dum ies m +ARCHerror Jun/97 495 492 497 We will work with returns constructed Jul/97 514 507 518 from quarterly prices, and attempt to Aug/97 510 504 513 Sep/97 498 493 501 model and forecast these returns. Oct/97 511 504 515 Nov/97 512 504 515 Dec/97 499 493 502 info@mbarros.com 48
  • 13. Alternative Approach: We propose the following Simulation model for the returns: Let Pt and Pt-1 denote pulp prices on  P  yt = log t  = µt + σ t Zt where Zt are iid N(0,1) quarters t and t-1.  Pt −1  The geometric returns are defined as: where both µt and σt2 are time varying according to the equations:  P  y t = log  t   Pt − 1  µ t = (1 − λ 1 ). y t + λ 1 . µ t − 1 and Where log denotes the natural logarithm. σ 2 t = (1 − λ 2 ). y t2 + λ 2 . σ 2 t −1 Rationale behind this structure Drawbacks Both µt and σt2 are time varying, and their Choice of smoothing constants λ1 and λ2 updating equations are easy to implement. If we perform a grid search we find out The equation for σt2 resembles that of an that the mimimum mean squared error integrated GARCH model. constants are obtained when λ1 and λ2 If the lambdas are large ( > 0.7 or 0.8), tend to zero. information decay is slow; This is due to the fact that we are Otherwise, the latest observations will attempting to optimize both constants for have a large weight, and the memory is the whole sample ! short. info@mbarros.com 51 info@mbarros.com 52
  • 14. Drawbacks Implementation We need to make an arbitrary choice. One step ahead prediction (corresponding Intuitively, the smoothing constant for σt2 to 1997/Q3 can be easily obtained by should be larger than that of µt . simulating a large Normal sample and applying the last estimated values for µt and σt2 . Besides the point estimates of pulp prices, we can use additional information from the generated probability distribution, that can lead, for example, to Value at Risk (VaR) estimates. info@mbarros.com 53 info@mbarros.com 54 Implementation Example k step ahead forecast could be obtained in We next show the result of one an analogous fashion, by incorporating at implementation where: each step the last prediction, re-estimating the smoothing constants are λ1 = 0.5 and µt and σt2 by the same procedure and λ2 = 0.7, generating another sample of iid Normal the last observed price is US$ 490 (2nd variables. quarter 1997), 5000 iid Normal variables are generated. info@mbarros.com 55 info@mbarros.com 56
  • 15. Example Example Example Example The last estimates of µt and σt are, respectively, 1.01% and 22.27% The simulated price distribution is: D i s t ri b u t i o n o f P re d i c t e d P ri c e s 3 rd Q u a rt e r 1 9 9 7 m e a n : 5 0 7 m in im u m : 2 3 7 m a x im u m : 1 1 9 0 s td . d e v . : 1 1 5 p e rc e n tile s v a lu e 5 % 3 4 2 1 0 % 3 7 3 2 5 % 4 2 4 5 0 % 4 9 4 7 5 % 5 7 7 9 0 % 6 5 6 9 5 % 7 1 4
  • 16. Comparison of Estimates Conclusion For comparison, the estimated values for The alternative approach is completely Prices in July 1997 are: heuristic, but may serve as an model with dummies : 514 approximation when the true data model without dummies : 507 generating process is hard to identify, or model with dummies and ARCH errors : 518 when dynamic regression models do not median of simulated prices : 494 seem to provide a completely satisfactory mean of simulated prices : 507 answer. info@mbarros.com 62

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