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.
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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)
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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.
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Characteristics of Pulp Market
The Available Data
and Industry
Production costs tend to be lower in Here we deal with quoted prices (in US$) -
emerging countries, but technological individual contract prices may vary.
changes in plants have lowered Quoted prices serve as a guideline for
considerably the costs of production in contracts (usually an upper bound)
the USA, Canada and Scandinavian NORSCAN (North America + Scandinavian
Countries) inventories are available on a
countries, which were historically the
monthly basis and may serve as an
largest pulp producers. explanatory variable for pulp prices.
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3.
The Available Data The Available Data
Quarterly Prices: available since the 1st NORSCAN inventories available monthly
quarter of 1978. since January 1985.
The price series used refers to constant To match NORSCAN inventories, we will
prices (base = December 1996) whose use price data from January 1985 on.
values in US dollars were inflated using The idea of using an artificial monthly
the Consumer Price Index (CPI) in the price series is due to the fact that we
USA. intend to use monthly inventories as a
Monthly prices are obtained by leading indicator of price changes.
interpolation (via CPI) quarterly prices.
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NORSCAN Inventories versus
Pulp Prices
Pulp Prices
Low prices usually correspond to high
Pulp Prices (US$)
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.
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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
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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
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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?
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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.
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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.
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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.
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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.
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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”.
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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
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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:
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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?
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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.
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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 %.
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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%.
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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).
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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
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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.
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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.
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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.
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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.
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