Inventory and Price Forecasting: Evidence from US Containerboard
Industry
A Thesis
Presented to
The Academic Faculty
by
Lidia S. Marko
In Partial Fulfillment
of the Requirements for the Degree
Master of Science in Economics
Georgia Institute of Technology
May 2003
Inventory and Price Forecasting: Evidence from US containerboard Industry
Approved by:
______________________________
Dr. Haizheng Li, Advisor
______________________________
Dr. Patrick McCarthy
______________________________
Dr. Jim McNutt
Date Approved _______________
ii
DEDICATION
I would like to dedicate this work to my friends Natalia Bidos, Julia Liubchenko,
and Dorottya Pap for their love, support, patience, and encouragement.
Without them this paper would have never been written.
iii
ACKNOWLEDGEMENT
I would like to begin by thanking my project groupmates - Jifeng Luo, Aselia
Urmanbetova, and Pallavi Damani - for their help and cooperation. I take this
opportunity to express my profound gratitude to my advisor Dr. Haizheng Li for his
exemplary guidance and constant encouragement. I have yet to see the limits
of his patience, and his selfless concern for his students. I wish to offer particular
thanks to Dr. Jim McNutt for his invaluable insights into the workings of the
industry. I am grateful to Dr. Patrick McCarthy for his support in this work and for
serving as a member on the Committee. Any mistakes in the results are my own.
iv
TABLE OF CONTENTS
DEDICATION
iii
ACKNOWLEDGEMENT
iv
LIST OF TABLES
vii
LIST OF FIGURES
viii
LIST OF FIGURES
viii
LIST OF ABBREVIATIONS
x
SUMMARY
xi
CHAPTER 1. The Behavior of Linerboard Price
12
1.1 Introduction
12
1.2 Factors Influencing Linerboard Price: Industry Experts Point of View
14
1.3 Literature Review
16
1.4 Data
18
1.5 Trend and Seasonality
19
CHAPTER 2. Univariate Forecasting Methods
25
2.1 Naïve Forecast
25
2.2 Exponential Smoothing
27
2.3 Forecasting Using ARIMA Models
32
CHAPTER 3. Forecasting Using Vector Autoregressive Model
39
3.1 Vector Autoregressive Framework
39
3.2 VAR Model Forecast
40
CHAPTER 4. Forecast Comparison
42
4.1 Forecasting Methods Comparison
42
v
4.2 Evaluation of the Existing Linerboard Price Forecast
44
4.3 Comparison with the Published Forecasts
50
4.4 Forecast for 2003-2005
54
CHAPTER 5. Inventory and Price Changes
56
5.1 Introduction
56
5.2 Literature Review
57
5.2 Relationship between Price, Production, and Inventory
64
5.2.1 Data
65
5.2.2 Granger Causality Testing
66
5.3 Econometric Model and Estimation Results
68
CONCLUSION
74
APPENDIX A. Variables Description
76
REFERENCES
78
vi
LIST OF TABLES
Table 1. Trend and Seasonality in Linerboard Price Time series
21
Table 2. Unit Root Test Results
23
Table 3. Linerboard price forecast for year 2000
26
Table 4. Parameters of Exponential Smoothing
30
Table 5. Breusch-Godfrey Serial Correlation LM Test for fitted ARIMA (3,1,0)
36
Table 6. Different methods MAPE and MRSE.
44
Table 7. RMSE and MAPE of containerboard price forecast.
47
Table 8. RMSE and MAPE of unbleached linerboard price forecast.
47
Table 9. RMSE and MAPE of corrugating medium price forecast
47
Table 10. RMSE and MAPE of export liner price forecast.
47
Table 11. Different Forecasting Methods Comparison (unbleached linerboard) 51
Table 12. Forecast for 2003-2005
55
Table 13. Logit Model Estimation Results (not seasonally adjusted)
69
Table 14. Logit Model Estimation Results (not seasonally adjusted)
69
Table 15. Logit Model Estimation Results (seasonally adjusted)
70
Table 16. Logit Model Estimation Results (not seasonally adjusted)
70
Table 17. Logit Model Estimation Results (seasonally adjusted)
70
Table 19. Fixed Effect Logit Model for Panel Data (NSA)
71
Table 20. Fixed Effect Logit Model Estimation Results for Panel Data (SA)
72
Table 21. Fixed Effect Logit Model Estimation Results for Panel Data (SA)
72
vii
LIST OF FIGURES
Figure 1. Linerboard Price (1980-2002)
14
Figure 2. Linerboard Price and Mill to Box Plant Inventory Ratio
16
Figure 3. Linerboard Price Descriptive Statistics
19
Figure 4. Naïve Forecast of Linerboard Price
26
Figure 5. Nonseasonal Exponential Smoothing Forecast of Linerboard Price
31
Figure 6. Seasonal Exponential Smoothing Forecast of Linerboard Price
31
Figure 7. AC and PAC of Linerboard Price Time series
34
Figure 8. AC and PAC of Differenced Linerboard Price Time series
35
Figure 10. ARIMA Forecast of Linerboard Price
37
Figure 9. AC, PAC and Q-statistic Probabilities for Fitted ARIMA (3,1,0)
38
Figure 11. Different Methods Price Forecasts
41
Figure 12. Actual Containerboard Price and Forecasts from the "Forecaster"
45
Figure 13. Actual Linerboard Price and Forecasts from the “Forecaster”
45
Figure 14. Actual Medium Price and Forecasts from the “Forecaster”
46
Figure 15. Actual Export Liner Price and Forecasts from the “Forecaster”
46
Figure 16. Linerboard and Corrugating Medium Monthly Prices (1996-2000)
49
Figure 17. Linerboard Price and ARIMA Model Forecasts
51
Figure 18. Linerboard Price and VAR Model Forecasts.
52
Figure 19. Linerboard Price and One-step Ahead Forecasts.
52
Figure 20. Linerboard Price and Two-step Ahead Forecasts
53
Figure 21. Linerboard Price and Three-step Ahead Forecasts
53
Figure 22. Linerboard Price Forecast for 2003-2005
55
viii
Figure 23. Price and Inventories at Mills and Box Plants
65
Figure 24. Production and Inventories at Mills and Box Plants
66
ix
LIST OF ABBREVIATIONS
AC
Autocorrelations
ADF
Augmented Dickey-Fuller Unit Root Test
AR
Autoregressive
ARIMA
Autoreggressive Integrated Moving Average Model
MA
Moving Average
MAPE
Mean Absolute Percentage Error
NSA
Not Seasonally Adjusted
PAC
Partial Autocorrelations
PP
Phillips-Perron Unit Root Test
PPW
Pulp and Paper Week
RMSE
Root Mean Squared Error
SA
Seasonally Adjusted
VAR
Vector Autoregressive Model
x
SUMMARY
The goal of this study is to identify economic factors that have influenced price
movement in containerboard industry. Various econometric techniques, simple
and advanced ones, have been employed to develop linerboard price
behavior models and produce forecast. The performance of each model has
been evaluated according to their out-of-sample forecast performance. Both
RMSE an MAPE forecast error measures point out that exponential smoothing
method renders the most accurate forecast for 2000.
When compared with existing industry forecasts, VAR outperforms all other
techniques. The fact that different methods of forecasting turn out to be
efficient (when applied to particular time intervals) could be explained by the
complicated pattern of price movements in which abrupt fluctuations are
followed by prolonged periods of almost no change.
For further analysis, the role of inventories in price/output adjustment in
short-run is investigated.
To this purpose, we estimate the effect of previous
month’s inventory level on the probability of price increase or decrease.
According to our results, inventories render significant influence on the
probability of price changes. It turns out that containerboard prices are more
responsive to changes in demand than to changes in output. Such results may
indicate that prices have been kept at planned levels and that containerboard
companies choose to adjust output on a short run basis. Finally, prices seem to
demonstrate
upward
stickiness,
but
xi
not
a
downward
one.
CHAPTER 1. The Behavior of Linerboard Price
1.1 Introduction
Linerboard price behavior was hardly predictable especially over the last
decade. The industry-wide linerboard price could increase more than 60% over
one year as in happened in 1994. Even though during the following years the
amplitude of price fluctuations decreased, it still had quite a haphazard pattern.
Such price behavior led to a number of serious consequences for the
containerboard industry, such as excess capacity and difficulties in long-term
financial planning. According to industry experts, there have been many factors
that could influence linerboard prices.
Among these are the build-up of
inventories, decreases in exports and fiber box shipments.
Despite the
importance of these issues to the industry, there has been little research focused
on linerboard prices, and as far as we know, there are no studies devoted to
linerboard price forecasting. The goal of this study is to employ advanced time
series techniques in order to analyze linerboard price movements, model its
behavior, and produce efficient price forecasts.
We have pursued the following approach to model building. First, the
linerboard price behavior and factors influencing it are analyzed.
Then
univariate time series methods—naïve forecast, exponential smoothing, and BoxJenkins approach—are utilized to build the most fitting model and to produce its
forecast. Further, we use Granger causality test to examine the industry expert
opinion, supported by many economic studies, that inventory is one of the most
12
important factors influencing price movements. As a result, a multivariate Vector
Autoregressive Model (VAR), incorporating the price – inventory relationship, is
developed.
The quality of each model is evaluated according to its out of
sample forecast performance.
Finally, to conclude our price forecasting
analysis, we present the VAR model and compare it with the previously
published forecasts.
Consequently,
to
investigate
the
role
of
inventories
in
short-run
price/output adjustments, we estimate the effect of previous month’s inventories
on the probability of price increases or decreases. The results suggest possible
asymmetry of price adjustments. In addition, we evaluate the response in output
levels to previous inventory. By comparing the relative probability of price vs.
output changes, we are able to gain some insight to the degree of price and
output flexibility in response to short-run inventory fluctuations.
The thesis is structured as follows. Chapter 1 reveals recent linerboard
price behaviors and provides short literature review of the existing price
forecasting studies as well as the description of the data used in this analysis.
Chapter 2 gives a brief theoretical review of forecasting methods mentioned
above and presents the results obtained utilizing these methods.
Chapter 3
addresses multivariate forecasting technique of the Vector Autoregressive (VAR)
mode.
Chapter 4 discusses forecast evaluation and compares the forecast
generated in this paper with the other published forecasts. Further, in Chapter 5
we continue with the investigation of behavioral model of linerboard price –
inventory relationship presenting detailed review of methodology and existing
studies in the area. Chapter 5 discusses the method of estimation and empirical
13
results. Finally in Conclusion, we summarize and indicate the directions for future
research.
1.2 Factors Influencing Linerboard Price: Industry Experts Point of View
US linerboard prices have historically been highly cyclical, rising during the
middle and late stages of economic recovery and falling down under
weakening demand. In Figure 1 we can see the 1988 and 1995 cyclical peaks in
the linerboard prices that are followed by quite dramatic decreases.
600
600
500
500
400
400
300
300
200
200
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
linerboard price
Figure 1. Linerboard Price (1980-2002)
Presently, the US linerboard price movement is in the middle of the current
cycle. Prices have reached the peak level almost a year ago and have already
begun to slowly move down.
Experts predict that the current cycle can be
different from both 1988 and 1995. This time price landing might be softer than
previously given that producers manage to master a number of challenges while
sustaining current price levels and decreasing inventory by adjusting operating
rates.
14
The relationship between price, production, and inventory levels in
containerboard industry is of paramount importance. Changes in one of the
variables can produce respective decreases or increases in the other two
parameters. Among the three, inventories seem to be the leading indicator,
which causes price and production to change in response to its own
movements.
Linerboard manufacturers produce their product to meet the needs of
box plants. The demand for fiber boxes is strongly dependent on the general
economic conditions.
Changes in demand cause fluctuations in inventories,
and as a result, instability in price and output levels. Therefore, unpredictable
inventory movements lead to such serious concerns for the industry as excess
capacity and difficulties in long-term financial planning.
According to industry experts 1 , there are several other factors that can
affect the inventory-price-output relationship. First, the US fiber box shipments
growth has been quite low compared to the growth in the rest of the economy.
At the end of 2000, box shipments have grown only 0.5% compared with the
GDP growth of 5.2%. This discrepancy can be explained by the fact that in that
year growth in consumer expenditures has been met by significant increases in
net imports. Second, the US exports of linerboard in 2000 have been 17% and
34% below 1999 and 1998 levels respectively.
The two factors have had a
negative impact on linerboard demand and have lead to steadily increasing
inventories in 1999-2000. Producers could not reduce containerboard inventories
below the three million ton for most of year 2000. Traditionally, the linerboard
1
Torma (2001)
15
market is considered to be firm when the containerboard inventory settles near
2.5 million tons. The fact that prices have been increasing in 1999 and have
remained at the same level in 2000 is explained by aggressive downtime policy
and capacity closures. Decreases in capital utilization have allowed balancing
decreases in demand.
The other factor that has kept linerboard pricing from collapsing is a low
mill to box plant inventory ratio.
According to Figure 2, there is an obvious
negative correlation between linerboard price level and inventory ratio.
550
0.30
500
0.25
450
0.20
400
0.15
350
0.10
300
0.05
250
200
0.00
80
82
84
linerboard price
86
88
90
92
94
96
98
mill to boxplant inventory ratio
Figure 2. Linerboard Price and Mill to Box Plant Inventory Ratio
A ratio less than 15% typically represents strong market condition and in 19992000 the ratio has maintained at 10-12%.
1.3 Literature Review
There have been a number of studies on the process of price formation in forest
industries.
Most of them concentrated on econometric models of price
formation.
16
Buongiorno and Gilles (1980), Buongiorno and Lu (1989), Chas-Amil and
Buongiorno (1999), Buongiorno et al. (1982) attempted to explain prices as the
function of input cost and technological change. With regards to the US paper
industry, the results show that: (i) product price is not directly related to capacity
utilization rate or level of production; (ii) capital cost has a dominant influence
on price-setting; (iii) technological changes, other than labor-saving ones, do not
influence pulp and paper prices significantly.
Stier (1985) uses a translog function to investigate the implications of
factor substitution, returns to scale, and technological progress for the cost of
production in the aggregate US pulp and paper industry.
The results are
consistent with the cost minimizing behavior on the part of firms, and indicate
that in short run increases in the price of capital, labor, or wood inputs would
drive up the commodity price.
Booth et al. (1991) examines the process of price formation by means of
applying the market structure methodology.
Targeting North-American
newsprint industry, they construct a dynamic model that consists of demand,
price, and regional capacity. Their results confirm existence of barometric price
leadership in the industry with oligopolistic coordination. The model estimation
showed that mark-ups are function of the utilization rate, and that higher
concentration levels lead to reduced levels of capacity expansion.
Some other studies used cointegration method for studying the
relationship between the exchange rate and paper product price movements.
Alavalapati et al. (1997) apply cointegration analysis to investigate the effect of
Canada-US exchange rate and US pulp price on the Canadian price of pulp.
17
Naininen and Toppinen (1999) estimate log-run price effects of exchange rate
changes in Finnish pulp and paper exports.
Brannlund et al. (1999) is the only study, as far as we know, that focuses
directly on forecasting of prices of paper products. Additionally, it introduces
new forecasting technique – Maximum Autocorrelation Factors (MAF).
This
method is based, like the vector autoregressive model, on the idea that time
series of prices and quantities from different sectors of the forest industry are
mutually correlated over time. They compare the results of the MAF estimation
with the results from the univariate method (ARIMA) estimation and naïve
forecast. It turns out that for the Swedish forest industry it is difficult to forecast
prices significantly better than naïve forecast.
The MAF technique is also
outperformed by the ARIMA model.
Such results are consistent with another body of literature focused on
examining relative merits of various forecasting approaches and measures of
forecasting performance. The analysis of large sets of time series by Newbold
and Granger (1974), Makridakis and Hibon (1979), Fildes et al. (1998) has led to a
number of important conclusions.
One such conclusion is that statistically
sophisticated or complex methods do not necessary forecast better than simpler
ones.
1.4 Data
The data for univariate forecasting comes from “Pulp and Paper Week” edition.
It includes 240 monthly observations from January 1980 to December 1999.
Figure 3 contains the descriptive statistics for the linerboard price time series.
18
25
Series: PRICEL
Sample 1980:01 1999:12
Observations 240
20
15
10
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
347.4792
347.5000
530.0000
250.0000
63.08201
0.699095
3.472866
Jarque-Bera
Probability
21.78535
0.000019
5
0
280
320
360
400
440
480
520
Figure 3. Linerboard Price Descriptive Statistics
Appendix A presents the description of the linerboard price time series.
Additionally, data covering the interval from January 2000 to December 2000 is
utilized to evaluate forecast performance.
1.5 Trend and Seasonality
In order to conduct forecasting utilizing such methods as VAR and ARIMA it is
necessary to determine whether time series is stationary or conduct a unit root
test.
A time series is said to be stationary if the mean and autocovariances of
the series do not depend on time. Standard estimation procedure cannot be
applied to the model that contains nonstationary variables. Hence, we should
check whether the data is stationary before using it in modeling.
In order to perform the unit root test we need to find out whether price,
inventory and production contain any trend and seasonal components. It is
19
necessary since including too many of the deterministic regressors results in lost
power, whereas not including enough of them biases the test in favor of the unit
root null. Since in the next chapter we also use inventory and production time
series in order build a behavioral model of price movement, we determine
whether all three time series contain trend and seasonality.
Visual analysis shows that linear trend and quadratic trend probably exist
in the given time series, but it is not clear whether there is any seasonal pattern in
the data. In order to verify our visual conclusion we employ regression analysis.
The following equations are estimated in order to check for trend in the data:
X = β 0 + β 1 * Trend + ε
(1)
X = β 0 + β 1 * Trend + β 2 * Trend 2 + ε
(2)
To check for seasonality we add dummy variables for every month excluding
January that serves as the base. The results are summarized in Table 1.
X = β 0 + β 1 Trend + β 2 D2 + β 3 D3 + β 4 D4 + β 5 D5 + β 6 D6 + β 7 D7 +
+ β 8 D8 + β 9 D9 + β 10 D10 + + β 11 D11 + + β 12 D12 + ε
(3)
X = β 0 + β 1 Trend + β 2 D 2 + β 3 D3 + β 4 D 4 + β 5 D5 + β 6 D6 + β 7 D7 +
(4)
+ β 8 D8 + β 9 D9 + β 10 D10 + + β 11 D11 + + β 12 D12 + β 13 Trend 2 + ε
As it turns out price time series contains both linear and quadratic trends
(see
Table
1).
Since
autocorrelation
testing
indicates
that
there
is
autocorrelation in the residuals in all time series, Newey-West autocorrelation
robust standard errors are calculated. Linear and quadratic trend parameter
20
estimates coefficients are significant at least at 5% significance level. Regression
analysis shows that linerboard prices do not have any seasonality. This finding is
surprising since industry experts often mention seasonal fluctuations in prices.
Inventory and production also seem to comprise linear and quadratic
trend. The p-values of the most seasonal dummy coefficients are very large and
therefore insignificant except for D10 and D11 (October and November
dummies), which are significant at 2% level for the inventory. February, April,
September, November and December dummy variables are significant at 5%
significance level for production.
Based such results, we need to include intercept and trend into price,
inventory, and production unit root test equations. Additionally, in the case of
inventory and production, seasonality should be taken into account when
performing the tests.
Table 1. Trend and Seasonality in Linerboard Price Time series
Variable
Trend
Quadratic trend
Seasonality
Price
Inventory
Production
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes (October, November)
Yes (February, April, September,
November, and December)
21
1.6 Unit Root Testing
The formal method of testing for stationarity of a series is the unit root test. The
first step in unit root testing is to select an appropriate number of lags included
into regression.
On one hand, too many lags decrease the number of
observations. On the other hand too few lags can lead to biased results of the
unit root test. One of the approaches to lag length selection is to start with a
quite long lag and reduce the model using t or F-tests. Since we work with a
monthly data, the maximum number of lags we should include into the unit root
test is 12. The following equation for the ADF test was estimated:
ΔX = γX t −1 + αt + ∑i =1 βΔX t − i +1 + ε ,
p
(5)
where Xt is the value of the tested variable and t is the time trend. If the t-statistic
of the last lag has been insignificant, we have proceeded with the estimation of
the regression with lag length p-1. This process has been repeated until the last
lag has been significantly different from zero. Further, the LR test for the joint
significance of the redundant variables has been performed to insure that jointly
excluded lags do not improve the regression performance. After the lag length
has been determined we have conducted diagnostic checking of the residuals
by the Ljung-Box Q-statistics. If there is no strong evidence of serial correlation in
the residuals and they appear to represent white noise then an appropriate lag
length is chosen.
In this manner, for the price time series lag length of order 6 is chosen. The
lags 7-12 appear to be jointly insignificant.
redundancy at any significant level.
22
We could not reject the null of
With regard to the inventory and
production the maximum 12th lag is appropriate. The LR test showed that it is
significant at less than 1% level. For all variables a serial autocorrelation test for
the chosen lags showed that the residuals are white noise. Table 2 below shows
the results of the unit root testing. The t-statistics are computed for coefficients
and referred to the Dickey-Fuller table. 2 If absolute computed value exceeds
the critical value, the null hypothesis that time series is non-stationary is rejected.
As we can see price and inventory variables turned out to be stationary. But for
the production variable unit root tests 3 produce contradictory results. PhillipsPerron test rejects the null of a unit root at 1% significance level. The ADF fail to
reject it even at 20% level.
Table 2. Unit Root Test Results
Variables
Lag
Test statistics
P-value
Price
6
-4.089371(ADF)
0.0075
Inventory
12
-5.430392 (ADF)
0.0000
Production
12
-15.11692 (ADF)
0.0000
Production
12
-2.600536 (PP)
> 0.2000
Suspecting that the discrepancy in the tests might be caused by the
seasonal component in the production data, we modify the ADF procedure for
production and inventory to account for seasonality.
Both variables are
Since critical values are calculated for the model that includes intercept or
intercept and trend only, we could not include quadratic trend into the
equation.
2
23
regressed on the monthly dummies. The residuals from these regressions can be
viewed as the deseasonalized values of the inventory and operating rate
respectively. We use the residuals to estimate the test equations. Dickey, Bell,
and Miller (1986) have shown that the limiting distribution for γ is not affected by
the removal of the deterministic seasonal component.
Therefore, the test is
valid. The same procedure is performed to determine the lag length.
For the inventory unit root testing t-statistic equals –5.41 when 12 legs are
included. Thus we can reject the null of a unit root at 1% significance level
(critical value of the Dickey empirical distribution at 1% level is –3.46). For the
production data, only 7 lags are necessary. The number of lags decreases in
comparison with the nonseasonal testing.
rejected at 1% level (t-statistics –3.66).
The unit root hypothesis is also
Hence, we can conclude that both
variables are seasonally and nonseasonally stationary and can be utilized in VAR
modeling.
We report only ADF test statistics but for price and inventory variables PhillipsPerron test results also strongly reject the null of a unit root.
3
24
CHAPTER 2. Univariate Forecasting Methods
2.1 Naïve Forecast
Naïve forecasting is a quantitative tool that uses only historical data of the
variable being forecasted in the analysis.
It provides a convenient way to
generate quick and easy forecasts for short time horizon; i.e. a month, a quarter,
at most a year ahead. This method has minimal data requirements and is easy
to implement since generally it requires only simple arithmetic to generate the
forecast.
The drawback of this technique is that its forecast will miss turning points.
The naïve forecast is based only on recent actual values of the variable. Hence,
the forecast will not change direction (up or down) until after the actual data
has shown this change. The naïve method generally expects the data to have
no trend and if a trend is present in the data it will usually treat the trend as a
linear one.
The forecasts for the time series containing trend are generated
according to following formula:
∧
X t +1 = X t + (X t − X t −1 )
(6)
The results of the one-step ahead price forecast for the next twelve
months can be found in Table 3. As we have mentioned before, naïve forecast
is based only on the recent price values.
Since the second half of 1999,
linerboard prices have been fixed at $425 per short ton, producing the forecast
for the whole year equaling the same value.
25
Table 3. Linerboard price forecast for year 2000
Date
Naïve
Exp. smoothing
nonseasonal
Exp. smoothing
seasonal
ARIMA
(3,1,0)
VAR
Jan-00 425,00
428,24
426,85
425,37
427,62
Feb-00 425,00
431,44
429,31
425,79
429,60
Mar-00 425,00
434,64
433,89
426,24
431,11
Apr-00 425,00
437,84
437,09
426,79
432,30
May-00 425,00
441,04
436,55
427,36
433,23
Jun-00 425,00
444,24
436,50
427,95
433,96
Jul-00 425,00
447,44
443,33
428,56
434,52
Aug-00 425,00
450,64
446,66
429,19
434,95
Sep-00 425,00
453,84
450,12
429,82
435,27
Oct-00 425,00
457,04
459,70
430,47
435,51
Nov-00 425,00
460,23
461,65
431,11
435,70
Dec-00 425,00
463,43
461,36
431,76
435,85
In Figure 4 you can see the actual price values from January 1995 and
forecasted linerboard price for year 2000.
600
500
400
300
200
95:01
95:07
96:01
96:07
97:01
97:07
98:01
linerboard price
Figure 4. Naïve Forecast of Linerboard Price
26
98:07
99:01
naive forecast
99:07
00:01
00:07
2.2 Exponential Smoothing
Another simple and commonly used method of adaptive forecasting is the
exponential smoothing technique. In this method the forecasted values are the
weighted averages of past observations with heavier weights given to recent
values and exponentially decreasing weights to earlier values. This technique
has been successfully employed in practice to predict the future values of many
types of time series, such as price, sales, or inventory data.
Exponential
smoothing methods under some circumstances may be more feasible, accurate,
cheaper, and easier to use than more complicated forecasting techniques. At
the same time, on average it produces more reliable results than the naïve
forecast. Since there are several exponential smoothing method modifications in
the following short theoretical review, we describe the algorithm utilized by
Eviews 4.0 software, which we use for modeling and obtaining forecasts.
The most basic method of exponential smoothing is the single exponential
smoothing with one parameter. According to Eviews technical documentation,
this method is appropriate for series that move randomly above and below a
∧
constant mean with no trend or seasonal patterns. The smoothed series Yt of Yt
is computed recursively by evaluating:
∧
y = αy t + (1 − α )y t −1 ,
(7)
27
where 0 ≤ α ≤ 1 is the damping (or smoothing) factor. The smaller is the α, the
∧
smoother is the y t series. By repeated substitution, the recursion can be rewritten
as:
∧
t −1
y t = α ∑ (1 − α )s y t − s ,
(8)
s=0
This shows why this method is called exponential smoothing—the forecast
of y is a weighted average of the past values of y t , where weights decline
exponentially with time. The forecasts from single smoothing are constant for all
∧
∧
future observations. This constant is given by: y T + k = y t for all k>0 and where t is
the end of the estimation sample.
∧
To start the recursion, we need an initial value for y t and a value for α.
EViews uses the mean of the initial observations of y t to start the recursion or it
can also estimate α minimizing the sum of squares of one-step forecast errors.
The next type of exponential smoothing is Holt-Winters non-seasonal
algorithm with two parameters. This method is appropriate for series with a linear
time trend and no seasonal variation. The smoothed series is given by:
∧
y T + k = a + bk
(9)
where a and b are the permanent component and trend.
coefficients are defined by the following recursions:
28
These two
a t = α y t + (1 − α )(a t −1 + b t −1 )
(10)
bt = β (at − at −1 ) + 1 − βb t −1
(11)
where 0<α, β, γ<1 are the damping factors. This is an exponential smoothing
method with two parameters. Forecasts are computed by:
∧
y T + k = aT + bT k
(12)
These forecasts lie on a linear trend with intercept a(T) and slope b(T).
Finally, Holt-Winters additive algorithm with three parameters is often being
utilized for series with a linear time trend and additive seasonal variation. The
smoothed series is given by:
∧
y t + k = a + bk + ct + k ,
(13)
where a is a permanent component (intercept), b is a trend, and ct is additive
seasonal factor. These three coefficients are defined by the following recursions:
at = α(y t − ct (t − s) + (1 − α )(at −1 + bt −1 )
(14)
bt = β (at − at −1 ) + 1 − βbt −1
(15)
ct (t) = γ (y t − at −1 ) − γct (t − s)
(16)
where 0<α, β, γ<1 are the damping factors and s is the seasonal frequency.
Forecasts are computed by:
29
∧
y T + k = aT + bT k + cT + k − s ,
(17)
where the seasonal factors are used from last s estimates and T is a total number
of observations in the sample.
We have already discovered that the price data contains linear and
quadratic trends, hence simple exponential smoothing cannot be applied to the
data.
To make sure that the price data does not comprise a seasonal
component we use both non-seasonal and seasonal approaches utilizing HoltWinters algorithm. The α, β, and γ parameters are determined by minimizing of
sum of squared errors. They are shown in Table 4.
Table 4. Parameters of Exponential Smoothing
α (general)
β (trend)
γ (seasonal)
Nonseasonal HW
.99
.2
N/A
Seasonal HW
0.97
0.22
0
The results of parameter estimation show that γ parameter is equal to zero.
Therefore, we have one more proof that the price data do not contain
seasonality.
The forecasts produced from both models are very close as Figures 5 and
6 indicate. Table 3 contains numerical value of forecasts. In contrast to the
Naïve forecast, exponential smoothing method manage to capture the upward
movement of the price.
30
600
500
400
300
200
95:01
95:07
96:01
96:07
97:01
97:07
98:01
98:07
99:01
99:07
00:01
00:07
linerboard price
nonseasonal exponential smoothing forecast
Figure 5. Nonseasonal Exponential Smoothing Forecast of Linerboard Price
600
500
400
300
200
95:01
95:07
96:01
96:07
97:01
97:07
98:01
98:07
99:01
99:07
00:01
linerboard price
seasonal exponential smoothing forecast
Figure 6. Seasonal Exponential Smoothing Forecast of Linerboard Price
31
00:07
2.3 Forecasting Using ARIMA Models
According to Makridakis (1997), autoregressive (AR) models were first
introduced by Yule in 1926 and subsequently supplemented by Slutsky, who in
1937 presented moving average (MA) process. Wold in 1938 combined both AR
and MA and showed that ARMA processes can be used to model a large class
of stationary time series. In other words, a time series X can be modeled as a
combination of past Xt values and/or past et:
X t = θ 1 X t −1 + θ 2 X t − 2 + ... + θ p X t − p + et + φ1 et −1 − φ 2 et − 2 − ... − φ q et − q
(18)
The process of fitting a model to a real lifetime series requires four steps.
First, the original time series must be transformed to become stationary around its
mean and variance. Second, appropriate order of p and q must be specified.
Third, the value of the parameters θ and φ must be estimated using some nonlinear optimization procedure that minimizes the sum of square errors or some
other appropriate loss function.
The implementation of the theoretical framework introduced by Wold
became possible only in the late 1960s when the first computers, capable of
performing all necessary calculations, appeared.
In 1970, Box and Jenkins
published a landmark book on time series analysis and forecasting and
popularized the use of the ARIMA method. They introduced the guidelines for
making
time
series
stationary;
suggested
autocorrelation
and
partial
autocorrelation as a tool for determining the appropriate values of p and q;
proposed to check the residuals for white noise to determine whether the model
32
is adequate or not. This methodology became known as Box-Jenkins or ARIMA
approach, where ‘I’ stands for ‘integrated,’ signifying that time series might need
to be differenced to become stationary. The last stage of fitting ARIMA model is
actually performing and evaluating the forecast from the chosen model using
such error measures as the Mean Root Square Error (MRSE) and Mean Absolute
Percentage Error (MAPE).
In Box-Jenkins methodology of ARIMA modeling, it must be first established
that a given time series is stationary before trying to identify the orders of AR and
MA processes. There are two basic tools to be used in this procedure: unit root
testing and visual analysis of the sample’s autocorrelations (AC) and partial
autocorrelations (PAC). Since unit root testing is often considered to have weak
power we will check its results by analyzing the sample’s AC pattern. Figure 7
shows the AC and PAC of the linerboard price. The sample autocorrelations of
the undifferenced series exhibit smooth patterns at high lags, so it is likely that the
price series is not stationary. Hence, differencing may be necessary.
The next step is to examine the sample AC and PAC of the first
differenced series.
They are shown together with two standard error limits in
Figure 8. The pattern of the sample AC at high lags is not at all smooth. The first
three sample AC are moderately large, while the next two values are quite small,
suggesting that a MA(3) model for the first-differenced series is possible.
However, similar pattern is also present in the sample PAC, so that an AR(3)
model might be appropriate. We can limit the choice of the model to ARIMA
with difference of order one, with up to AR(3) orders of the autoregressive term
and up to MA(3) orders of the moving average term.
33
Autocorrelation
.|*******|
.|*******|
.|*******|
.|*******|
.|****** |
.|****** |
.|***** |
.|***** |
.|**** |
.|**** |
.|**** |
.|*** |
.|*** |
.|** |
.|** |
.|** |
.|** |
.|*
|
.|*
|
.|*
|
.|*
|
.|*
|
.|*
|
.|*
|
Partial
Correlation
.|*******|
*|.
|
*|.
|
**|.
|
.|.
|
.|.
|
.|.
|
.|.
|
.|.
|
.|.
|
.|.
|
.|*
|
.|.
|
*|.
|
.|.
|
.|.
|
.|*
|
.|.
|
.|.
|
.|.
|
.|.
|
.|.
|
.|*
|
|.
|
AC
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
0.974
0.941
0.903
0.855
0.803
0.748
0.690
0.630
0.574
0.517
0.461
0.412
0.369
0.324
0.285
0.249
0.218
0.194
0.171
0.152
0.139
0.128
0.123
0.121
PAC
Q-Stat
Prob
0.974
-0.150
-0.106
-0.206
-0.048
-0.054
-0.047
-0.039
0.054
-0.051
-0.001
0.078
0.051
-0.102
0.018
-0.009
0.073
0.036
-0.035
0.017
0.053
-0.037
0.082
0.016
230.64
446.94
646.92
826.76
986.08
1124.9
1243.5
1342.9
1425.7
1493.1
1547.0
1590.3
1625.2
1652.2
1673.1
1689.2
1701.6
1711.4
1719.0
1725.2
1730.3
1734.7
1738.7
1742.7
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Figure 7. AC and PAC of Linerboard Price Time series
To facilitate the process of choosing the best model, we use Eviews
macro, which we wrote specifically for automatic ARIMA modeling of a nonseasonal time series. The macro estimates all possible models up to order 4 for
both MA and AR parameters using Eviews. Below we can see the result of the
estimation (Table 4) sorted by the SBC criterion in the ascending order. The SBC
points out that ARIMA (1,1,1), ARIMA (3,1,0), and ARIMA(0,1,3) might be
adequate models for producing forecast.
34
Autocorrelation
Partial Correlation
.|*
.|*
.|**
.|*
.|*
.|*
.|.
*|.
.|.
.|.
*|.
*|.
.|.
*|.
*|.
*|.
*|.
*|.
*|.
*|.
*|.
*|.
*|.
*|.
.|*
.|*
.|**
.|.
.|.
.|*
.|.
*|.
.|.
.|.
*|.
*|.
.|*
*|.
*|.
*|.
*|.
.|.
.|.
*|.
.|.
*|.
.|.
*|.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
AC
PAC
0.176
0.135
0.264
0.087
0.076
0.156
0.035
-0.085
0.019
-0.018
-0.103
-0.071
0.013
-0.148
-0.119
-0.140
-0.176
-0.072
-0.103
-0.149
-0.058
-0.144
-0.097
-0.118
0.176
0.108
0.234
0.002
0.015
0.084
-0.025
-0.139
-0.010
-0.012
-0.066
-0.060
0.069
-0.093
-0.070
-0.128
-0.065
0.031
-0.045
-0.063
0.039
-0.104
-0.025
-0.112
Q-Stat
7.5161
11.975
29.006
30.878
32.296
38.274
38.577
40.366
40.456
40.539
43.237
44.500
44.544
50.173
53.799
58.867
66.941
68.307
71.077
76.934
77.811
83.309
85.813
89.527
Figure 8. AC and PAC of Differenced Linerboard Price Time series
35
Prob
0.006
0.003
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Table 4. All Estimated Models Sorted by SBC Criterion
AR order I order
1
1
1
0
1
2
1
1
1
0
1
1
1
1
1
2
1
3
1
0
1
4
1
2
1
0
1
4
1
1
1
2
1
3
1
4
1
3
1
2
1
4
1
3
1
4
1
MA
order
1
0
3
2
0
1
2
3
1
1
4
0
0
2
4
4
3
2
1
3
4
2
4
3
SBC
7.881782
7.884503
7.888521
7.891333
7.897337
7.897573
7.900429
7.90403
7.906532
7.90765
7.910259
7.912295
7.913093
7.916782
7.924627
7.925225
7.928647
7.930564
7.935295
7.938573
7.951653
7.95817
7.960135
7.965903
ARIMA (1,1,1) cannot be used for forecasting since the residual test does
not show white noise. The next model, ARIMA (3,1,0), seems to be adequate.
We could not reject the null hypothesis of white noise even at 10% significance
level (see Table 5) when Breush-Pagan test for autocorrelation in residuals is used.
Table 5. Breusch-Godfrey Serial Correlation LM Test for fitted ARIMA (3,1,0)
F-statistic
Obs*R-squared
0.800913
9.878386
Probability
Probability
36
0.649358
0.626629
As we can see in Figure 9, this model’s autocorrelations are inside the
interval limited by the dotted lines.
The dotted lines in the plots are the
approximate two standard error boundaries computed as ± 2 / T , where T is
total number of observations in the sample. If the autocorrelations are within
these bounds, they are not significantly different from zero at (approximately) 5%
significance level and therefore represent white noise. The last two columns in
Figure 8 are the Ljung-Box Q-statistics and their p-values. The Q-statistic at lag k is
a test statistic for the null hypothesis that there is no autocorrelation up to order k.
This statistic also strongly rejects autocorrelation of the residuals. Hence, we will
perform forecasting using ARIMA (3,1,0) model.
The produced forecast shows moderate increasing in price values for the
next twelve months but this increase is smaller than the one predicted when
exponential smoothing methods are utilized (see Figure 10 and Table 3).
550
500
450
400
350
300
250
1995
1996
1997
1998
linerboard price
1999
ARIMA (3,1,0) forecast
Figure 10. ARIMA Forecast of Linerboard Price
37
2000
Autocorrelation
Partial
Correlation
.|.
.|.
.|.
.|.
.|.
.|*
.|.
*|.
.|.
.|.
.|.
.|.
.|*
*|.
*|.
*|.
*|.
.|.
.|.
*|.
.|.
*|.
.|.
*|.
.|.
.|.
.|.
.|.
.|.
.|*
.|.
*|.
.|.
.|.
*|.
.|.
.|*
*|.
*|.
*|.
*|.
.|.
.|.
*|.
.|*
*|.
.|.
*|.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
AC
PAC
Q-Stat
Prob
0.000
-0.004
-0.021
-0.003
0.036
0.099
0.011
-0.106
0.016
-0.017
-0.054
-0.041
0.092
-0.083
-0.063
-0.099
-0.098
0.005
-0.013
-0.079
0.023
-0.094
-0.020
-0.102
0.000
-0.004
-0.021
-0.003
0.035
0.098
0.012
-0.105
0.020
-0.017
-0.066
-0.053
0.100
-0.067
-0.070
-0.102
-0.084
-0.001
-0.044
-0.080
0.066
-0.098
-0.026
-0.123
5.E-05
0.0034
0.1129
0.1154
0.4226
2.7933
2.8228
5.5993
5.6649
5.7358
6.4559
6.8722
8.9948
10.720
11.736
14.243
16.720
16.726
16.768
18.372
18.514
20.830
20.931
23.677
0.734
0.810
0.425
0.588
0.347
0.462
0.571
0.596
0.650
0.533
0.467
0.467
0.357
0.271
0.335
0.401
0.366
0.422
0.346
0.401
0.309
Figure 9. AC, PAC and Q-statistic Probabilities for Fitted ARIMA (3,1,0)
38
CHAPTER 3. Forecasting Using Vector Autoregressive Model
3.1 Vector Autoregressive Framework
Two decades ago, Christopher Sims (1980) provided a new framework
that held a great promise – Vector Autoregressions (VAR). As Stock and Watson
(2001) have put it, a univariate autoregression is a single-equation, singlevariable linear model, in which the current value of a variable is explained by its
own lagged values. A VAR is an n-equation; n-variable linear model, in which
each variable is in turn explained by its own lagged values, plus current and past
values of the remaining n-1 variables. This simple framework provides a
systematic way to capture rich dynamics in multiple time series.
VAR models are commonly used for forecasting systems of interrelated
time series and for analyzing dynamic impact of random disturbances on
systems of variables.
The VAR approach sidesteps the need for structural
modeling by treating every endogenous variable in the system as a function of
the lagged values of all of the endogenous variables in the system.
The
mathematical representation of a VAR model including linerboard price and
inventories level is:
yt = A1 yt −1 + .... + Apyt − p + Bxt + ε t
where yt is a
(19)
vector of endogenous variables, xt is a
vector of exogenous
variables, A1 ..... A p and B are matrices of coefficients to be estimated, and ε t is
a vector of innovations that may be contemporaneously correlated but are
39
uncorrelated with their own lagged values and uncorrelated with all of the righthand side variables.
Since only lagged values of the endogenous variables appear on the
right-hand side of the equations, simultaneity is not an issue and an OLS yields
consistent estimates.
Moreover, even though the innovations may be
contemporaneously correlated, the OLS is efficient and equivalent to a GLS
since all equations have identical regressors.
As an example, suppose that the linerboard price (P) and total inventory
(Inv) are jointly determined by a VAR process and let a constant be the only
exogenous variable. Assuming that the VAR contains two lagged values of the
endogenous variables, it may be written as:
Pt = a11 Pt −1 + b12 Inv t −1 +b11 Pt − 2 + b12 Inv t − 2 + c1 + ε 1t
(19)
Inv t = a 21 Pt −1 + a 22 Inv t −1 + b 21 Pt − 2 +b 22 Inv t − 2 + c 2 + ε 2t
(20)
where aij ,b ij , ci are the parameters to be estimated.
3.2 VAR Model Forecast
According to the industry expert opinion, one of the important factors
influencing linerboard price movements in short run is a change in total inventory
levels at mills and box plants. Following the principle of parsimony we include in
the VAR model only two variables – linerboard price and inventory. A crucial
element in a VAR specification is the determination of its lag length.
The
importance of the lag length has been demonstrated by Hafer and Sheemen
40
(1989). They show that forecast accuracy from VAR models varies substantially
depending on alternative lag length. Ivanov and Kilian (2000) argue that the lag
length determination procedure, based on the AIC criterion, performs well in
most cases for monthly data. According to this criterion, the VAR model that
includes linear trend two lags of both variables should be utilized in forecasting.
The resultant forecast is represented in Table 3. The values of this forecast are
close to those of the ARIMA model. Figure 11 shows the actual linerboard price
from 1980 to 1999, and 2000 forecasts obtained from all the above-mentioned
models.
480
460
440
420
400
380
360
340
320
98:01
98:07
99:01
99:07
00:01
liner boar s pr ice
naive f or ecast
exponent ial sm oot hing
Figure 11. Different Methods Price Forecasts
41
00:07
ar im a
var
CHAPTER 4. Forecast Comparison
Despite the fact that the linerboard price movements have been quite
haphazard during the recent decades, advanced techniques can be employed
to analyze such behavior and generate price forecast. In this chapter we will
assess the accuracy of different forecast methods that we have utilized and
compare them with the existing industry forecasts.
4.1 Forecasting Methods Comparison
The accuracy of different forecasting methods is a topic of continuing interest
and research. There is a number of error measurement methods, which allow
comparing forecast performance across different grades and forecasting
horizons.
The two most commonly used error measures are—the Root Mean
Squared Error (RMSE) and Mean Absolute Percentage Error (MAPE).
For a long time the RMSE has been the preferred error measure used by
practitioners and academicians.
It can be calculated using the following
formula:
∑
n
RMSE=
(Xi − Fi )2
i =1
n
,
(21)
where Xi is the actual price value; Fi is the forecasted value; and n is the number
of forecasts for a specific forecast horizon. The disadvantages of the RMSE are:
(1) it is very sensitive to outliers (extreme forecast errors), and (2) it is not scale
42
free. To avoid the problem of scaling, many use the most popular and scale free
error measurement, the MAPE:
n
MAPE =
∑ 100 * (X
i =1
i
n
− Fi ) / X i )
,
(22)
The MAPE calculates the forecast error as a percentage of the actual value. The
drawback of the MAPE measure is that it puts a heavier penalty on the forecasts
that exceed the actual value than on those that fall behind the actual value.
Given that both the RMSE and MAPE have advantages and disadvantages, they
should be equally utilized in evaluating the existing containerboard price
forecasts. Additionally, it is important to keep in mind that in general smaller
horizon forecasts are expected to be more accurate because they face less
uncertainty than longer horizon forecasts.
We have calculated the MRSEs and MAPEs for twelve months horizon
forecasts for naïve, nonseasonal exponential smoothing, ARIMA, and VAR
methods.
As Table 6 shows both forecast error measures point out that
exponential smoothing method renders the most accurate forecast. This result is
consistent with the previous findings by Newbold and Granger (1974), Makridakis
and Hibon (1979),
Fildes et al. (1998) that simple forecasting methods often
render better forecasts than statistically sophisticated ones. However, the fact
that exponential smoothing over-performed ARIMA and VAR methods may be
explained by the fact that during the period from July to December 1999
linerboard price values have been fixed at $425 per short ton. In this case simple
forecasting technique such as exponential smoothing that takes into account
43
mainly recent values of the price and linear trend element may produce better
results tan those which incorporate more complex algorithms.
Table 6. Different methods MAPE and MRSE.
Methos
MAPE
MRSE
Naive
9.06
45.85
Exp. Smoothing
4.78
25.28
ARIMA
8.36
42.29
VAR
7.40
37.36
4.2 Evaluation of the Existing Linerboard Price Forecast
Pulp and Paper Forecaster by Miller Freeman Inc. publishes regular forecasts of
prices for major grades of containerboard.
Unfortunately, no explanation is
given in regards to the techniques they use for their forecasts. Still the forecast
performance and accuracy can be evaluated by using existing error
measurement criteria.
Available quarterly forecasts cover the period from January 1996 to April
2000 and include the following categories: containerboard, corrugated medium,
unbleached linerboard 4 , and export liner. 5 For the export liner, the forecast data
is available only up to December of 1999. The data is collected manually and
then regrouped in order to obtain one-, two-, and three-step forecasts, where
one-step equals to one quarter ahead forecast, etc. Further, the RMSE and PE
are calculated for each grade and forecast horizon. In Figures 12-15 the graphs
of the actual price and one-, two-, and three-step ahead forecasts are
represented.
4
5
standard kraft linerboard (42-lb/1,000 ft2 )
kraft linerboard (175+ g/m2)
44
850
800
USD
750
700
650
600
550
500
96/1 96/3 97/1 97/3 98/1 98/3 99/1 99/3 00/1 00/3
forecaster(actual)
1-Step
2-Step
3-Step
USD
Figure 12. Actual Containerboard Price and Forecasts from the "Forecaster"
620
570
520
470
420
370
320
270
220
170
96/1
96/3
97/1
PPW
97/3
98/1
1-Step
98/3
99/1
2-Step
99/3
00/1
00/3
3-Step
Figure 13. Actual Linerboard Price and Forecasts from the “Forecaster”
45
620
570
520
USD
470
420
370
320
270
220
170
96/1
96/3
97/1
PPW
97/3
98/1
1-Step
98/3
date
2-Step
99/1
99/3
00/1
00/3
3-Step
Figure 14. Actual Medium Price and Forecasts from the “Forecaster”
670
620
570
520
USD
470
420
370
320
270
220
170
96/1
96/3
97/1
Forecaster(actual)
97/3
98/1
1-Step
date
98/3
99/1
99/3
2-Step
3-Step
Figure 15. Actual Export Liner Price and Forecasts from the “Forecaster”
As we can see in the graphs, in general the forecast performance for all
three horizons is relatively adequate. For the years 1996-1997 the forecasts are
less accurate and consistently exceed the actual prices. It means that the price
46
is considerably over-forecasted.
But forecasted prices in 1999-2000 are quite
close to the actual price values.
As expected, one-step ahead forecasts appear to be more accurate
than two-, or three-step forecasts since the shorter horizon implies that the
forecaster faces less uncertainty in the future. Both the RMSE and MAPE show
that one-step ahead containerboard, unbleached linerboard, corrugated
medium, and export liner price forecasts are the most accurate (see Tables 710).
Table 7. RMSE and MAPE of containerboard price forecast.
Error
measurement
RMSE
MAPE
1 step ahead
2 step ahead
3 step ahead
38.91594
4.3702
54.43666
6.938355
65.81337
8.310461
Table 8. RMSE and MAPE of unbleached linerboard price forecast.
Error
measurement
RMSE
MAPE
1 step ahead
2 step ahead
3 step ahead
48.63147
10.65847
64.08824
13.00216
73.67204
14.4374
Table 9. RMSE and MAPE of corrugating medium price forecast
Error
measurement
RMSE
MAPE
1 step ahead
2 step ahead
3 step ahead
63.87064
16.14924
87.55552
22.22057
100.5182
25.39659
Table 10. RMSE and MAPE of export liner price forecast.
Error
measurement
RMSE
MAPE
1 step ahead
2 step ahead
3 step ahead
61.89507
12.35728
86.11547
18.49232
95.96614
19.82277
47
According to the MAPE, forecast error comprise only 4.37% of actual
value for containerboard one-step ahead forecast, 10.66 %, 16.45%, and 12.35%
for unbleached linerboard, corrugated medium, and export liner respectively.
For the second and third forecasting horizons the accuracy of the forecasts
decrease significantly. The MAPE range of the three-step ahead forecasts varies
from 8.31% for the containerboard forecasts to 25.4% for the corrugated medium
price predictions.
In other words, the three-quarter ahead price of the
corrugated medium is on average over- or under-predicted by 25 percent. Even
with the nine-month horizon, the forecasting error appears to be too big.
To minimize the influence of outliers, the MAPE is used in the evaluation of
forecast performance between grades.
Forecast assessment shows that the
most accurate forecasts for all horizons are generated for containerboard prices,
then for unbleached linerboard, export liner, and corrugated medium.
The fact that containerboard forecast appears to look quite satisfactory is
not surprising. Containerboard price has experienced less fluctuation than the
prices of the other grades (see Figure 12). We expect that export liner would be
the most difficult grade to forecast. Price movements depend not only on the
internal but also on many external or exogenous factors such as global demand
for paperboard, exchange rate, and competition on the global paperboard
market. Export liner prices are affected by these factors to a greater extent than
other grades.
Corrugated medium prices usually follow quite closely the
linerboard prices; therefore one would expect that the accuracy of the
linerboard and corrugated medium forecasts would not differ a lot (see Figure
16). But in the first quarter of 1996 the price of corrugated medium is significantly
48
over-forecasted as well as in the first and second quarter of 1997. During these
periods of time corrugated and linerboard prices plummet down and forecasts
of both grades fail to predict such an abrupt decrease in the prices. For some
reason, corrugated medium forecast is much more inaccurate – its MAPE values
for these quarters exceed 45%.
520
470
USD
420
370
320
270
220
170
linerboard
corrugating medium
Figure 16. Linerboard and Corrugating Medium Monthly Prices (1996-2000)
Overall, the performance of the published forecasts appears to be quite
accurate for the containerboard and acceptable for unbleached linerboard
prices. For nine months ahead forecast the MAPE did not exceed 9% and 15 %
for containerboard and unbleached linerboard prices respectively. For export
liner and corrugated medium forecasts cannot be accessed as reasonably
reliable.
49
When percentages are converted to dollar values the MAPE of the third
quarter ahead forecast on average constitutes $80 6 for the price of export liner
and $90 for corrugated medium. Such deviations are too significant to rely on in
long-term planning. Therefore, the goal of our forecasting project is to improve
the forecast accuracy by employing advanced forecasting techniques.
4.3 Comparison with the Published Forecasts
We use some common forecasting techniques to predict price movements. For
the comparability of results, the forecasting analysis in this part is performed for
the same time span and forecasting horizons as in the published forecasts.
For the purpose of such an exercise we limit the scope of our examination
to unbleached linerboard. Simple forecasting techniques - nonseasonal HoltWinter method - as well as more advanced ones, such as the univariate ARIMA
approach and multivariate VAR model, are applied to the log value of the
existing data. 7
Based on these models, the sequence of the out-of-sample forecasts is
produced for all quarters within the period from 1996 to 2000. The results of the
published and produced forecasts are represented in Table 11. MRSE and MAPE
are calculated for each forecasting horizon. Often these forecast error produce
contradictory results measurements Often In this case both forecast error
produce contradictory results. In our case different models forecast evaluations
produced by MRSE and MAPE coincide.
Average export liner and corrugated medium prices over the period 1996 to
1999 are multiplied by the value of the average MAPE over the same period.
7 Price of delivered unbleached kraft linerboard (42-lb), published in Pulp and
Paper Week (PPW) edition, is used as the actual price in forecasting.
6
50
Table 11. Different Forecasting Methods Comparison (unbleached linerboard)
Method
MRSE
MAPE
1-Step 2-Step 3-Step 1-Step
2-Step
3-Step
48.63
64.09
73.67
10.66
13.00
14.44
Published
forecasts
Holt-Winter
ARIMA
VAR
32.78
27.22
30.08
69.15
48.36
49.94
115.11
62.75
58.33
7.71
6.28
5.98
15.09
10.67
10.36
23.07
13.67
12.68
The results show that simple exponential smoothing (Holt-Winter) method
outperforms the published one-step ahead forecast but fails to improve the
accuracy for the longer horizons. More complicated techniques render more
USD
accurate results in all horizons (see Charts 17-21).
670
620
570
520
470
420
370
320
270
220
170
96/1
96/3
97/1
PPW
97/3
98/1
1-Step
98/3
99/1
2-Step
99/3
00/1
3-Step
Figure 17. Linerboard Price and ARIMA Model Forecasts
51
00/3
670
620
570
520
USD
470
420
370
320
270
220
170
96/1
96/3
97/1 97/3 98/1 98/3 99/1
PPW
1-Step date 2-Step
99/3
00/1
3-Step
00/3
Figure 18. Linerboard Price and VAR Model Forecasts.
USD
500
400
300
200
96/1
96/3 97/1 97/3 98/1 98/3
PPW
date
1-Step Holt-Winter
1-Step ARIMA
99/1
99/3 00/1 00/3
1-Step Forecaster
1-Step VAR
Figure 19. Linerboard Price and One-step Ahead Forecasts.
52
600
USD
500
400
300
200
96/1
96/3
97/1
97/3
PPW
2-Step Holt-Winter
2-Step ARIMA
98/1
98/3
99/1
99/3
00/1
00/3
2-Step Forecaster
2-Step VAR
date
USD
Figure 20. Linerboard Price and Two-step Ahead Forecasts
600
550
500
450
400
350
300
250
200
150
100
96/1
96/3 97/1 97/3
PPW
3-Step Holt-Winter
3-Step ARIMA
98/1
98/3
date
99/1 99/3 00/1 00/3
3-Step Forecaster
3-Step VAR
Figure 21. Linerboard Price and Three-step Ahead Forecasts
The difference between the ARIMA and VAR forecasts is insignificant. On
average, the MAPE is lower by 4.5, 2.5, and 1.25 percentage points for the one-,
two-, and three-step ahead forecasts respectively than for the published
forecasts. When the MAPE error percentages are converted into dollar values,
53
they constitute as much as $22 per short ton improvement of the one stepahead forecast in the fourth quarter of 2000.
All three methods can be employed in order to predict price movements.
If there is a need to produce simple and cheap short-term forecast, especially in
case when prices have not been changing significantly during several months
prior to the starting point of the forecast, exponential smoothing method can be
used. The VAR and ARIMA approaches require specialized computer software
and knowledge of the methodology, and as a result they tend to produce more
accurate forecasts in most cases.
4.4 Forecast for 2003-2005
Our previous investigation of forecast models performance points out that both
exponential smoothing and VAR models are likely to produce reliable price
forecasts.
Unfortunately, monthly data on inventory is available only up to
December 1999. Hence, in order to produce forecast for next up to year 2005
we can not the VAR model. Instead of VAR, we will use the ARIMA technique.
Figure 22 presents the ARIMA and exponential smoothing forecasts from April
2003 to December 2005.
Interestingly, the two models predict quite different directions or price
behavior (see Table 12).
Exponential smoothing nonseasonal Holt-Winters
incorporates recent downward trend and points out at a gradual decrease in
price during next two years. In contrast, the ARIMA takes into account historical
price behavior and points at possible increases in linerboard prices.
54
575
525
USD
475
425
375
325
275
1995 1996 1997 1998
linerboard price
1999 2000 2001 2002
2003 2004 2005
exp. smoothing forecast
ARIMA forecast
Figure 22. Linerboard Price Forecast for 2003-2005
Table 12. Forecast for 2003-2005
Exp. smoothing
ARIMA
2003:02
422,55
422,64
2003:03
418,88
422,63
2003:04
415,20
423,65
2004:01
411,53
425,07
2004:02
407,86
426,63
2004:03
404,18
428,24
2004:04
400,51
429,87
2005:01
396,83
431,50
2005:02
393,16
433,13
2005:03
389,49
434,77
2005:04
385,81
436,40
55
CHAPTER 5. Inventory and Price Changes
5.1 Introduction
Despite the amount of attention devoted to the study of short-run price and
output adjustments in response to demand and cost shocks, the role of inventory
in this process is not well understood. Understanding the effect of inventories on
price and production movements can help both producers and consumers to
predict price changes, and facilitate production planning and inventory
management.
Theory implies that the smaller is industry inventory stock the higher is the
aggregate price/output level and vice versa. A number of short-run economic
models assert that short-term quantitative output adjustments are more
apparent than short-term price adjustments. However, empirical studies have
produced mixed results with regard to inventory-price/output relationship.
The purpose of this research is to study the role of inventory in price/output
adjustment in short-run, based on the discrete choice methodology, time series,
and panel data on the US containerboard industry. In particular, we estimate
the effect of previous month’s inventories on the probability of price increase or
decrease. The results will provide information on the possible asymmetry of price
adjustment.
In addition, we also estimate response in output to previous
inventory level. By comparing the relative probability of price change vs. output
change, we will be able to gain some insight on the relative flexibility of prices
and output in response to inventory changes in short-run.
56
The Chapter is organized as follows. In Section 5.2 we provide a detailed
review of methodology and existing studies in this area.
In Section 5.3, we
describe the data and define direction of Granger causality. Further, we discuss
empirical results. And finally, we summarize and indicate the directions for future
research.
5.2 Literature Review
Prices versus output flexibility argument originated from the fact that neither
Marshalian (assuming fixed quantities in a short run) nor Keynesian (fixed prices in
the short run) did not seem to be realistic. A number of short-run economic
models have been developed in the 1970’s assuming that short-term
quantitative adjustments are much more apparent than short-term price
adjustments (Barro and Grossman (1971), Malinvaud (1977)).
Most of the discussion on the role of inventory in the price and output
adjustment process has been focused around the model of the monopolistic
competitive firm holding inventory of finished goods and facing convex
production and inventory carrying costs.
Significant amount of research
addresses the price-inventories relation ships at aggregate level.
Ekstein and Fromm (1968) suggest one of the first insights on the role of
inventories in price formation. The derived price equation for an oligopolistic
industry incorporates both cost and demand variables, such as unit labor cost,
material input prices, operating rate and inventory disequilibrium variable.
Utilizing the US data aggregated at industry level they discover that price and
output levels change in response to changes in cost and demand factors.
57
Lagged inventory disequilibrium variable is significant and has an expected sign
in most of the equations. Tests for asymmetries in pricing show some evidence in
support of the proposition that prices have a greater tendency to increase than
decrease.
Hay (1970) takes different approach and makes an attempt to present
“integrated” model of firm behavior, in which decisions on all relevant variables –
price, inventory, and production- are assumed to result from a single optimization
process 8 . The major assumption is that decisions on these variables should be
treated as simultaneous and interdependent. The system of linear equations for
price, inventory, and production variables is derived and estimated using data
for the US lumber and paper industries. In both price and output equations the
lagged inventory variable has the expected negative sign and appears to be
highly significant except for the output equation for the lumber industry 9 . Hay’s
study also addresses price and output reaction to the impact of unit increase in
demand. His finding is that for both industries price change plays a small role in
absorbing a temporary increase in demand in comparison to output.
Following Hay (1970), Venieris (1973) investigates concurrent changes in
price and inventory behavior. He points out that most of the models that have
been utilized in previous studies have not allowed for a feedback process
between prices and quantities in case of the demand change. Venieris (1973)
estimates three equations system depicting demand, supply, and price
adjustment relations for durable and non-durable industries. Contrary to Hay
To our knowledge Baudin (2001) is the only study that follows and further
develops Hay’s (1970) framework using VAR technique.
9 In this equation inventory coefficient is significant at 15% level.
8
58
(1970), he estimates all equations simultaneously using the 2SLS procedure. The
results show that if the desirable level of inventories is larger than the actual level,
it leads to positive increases in price and output levels and vise versa. The results
also clearly indicate that the sensitivity of prices in the case of positive excess
demand is larger than in the case of negative excess demand for both industries.
McFetridge (1973) utilizes a simple industry mark-up model incorporating
cost and demand variables to study the determinants of pricing behavior of the
Canadian textile industry. Excessive demand variable in case of the industry
producing purely to stock is introduced as the difference between actual and
desired inventories. The coefficient of this variable has an expected negative
sign and is significant at 1% level. Testing of an asymmetric price response to
changes in demand factors results in rejection of the asymmetry of pricing
behavior hypothesis. It means that the rate of price change is proportionate to
the rates of changes that lead to excess or deficient demand.
Contrary to the above-mentioned studies, which do not suggest any
treatment of possible aggregation bias, Maccini (1976) addresses aggregation
issue directly.
He comes up with a theoretical model that can be used to
analyze the short-run dynamics of an average price level and total output and
interpret the aggregate behavior of prices and output.
Solving firm’s
optimization problem and further constructing an aggregate model Maccini
(1976) shows that the optimal aggregate behavioral relationships for price and
output can be expressed as a function of aggregate stock of inventories,
estimated or expected average price level, expected level of aggregate
demand (sales), and expected money wage rate. The model implies that the
59
smaller is the industry inventory stock, the higher is the aggregate price level. In
addition, an increase in the expected level of aggregate sales will induce price
rises. The same is applicable to the aggregate output level.
Maccini (1976) mentions that previous empirical studies on price behavior
have been dominated by the use of the markup models, which essentially assert
that prices are set as some markups over unit labor costs, where the markup is
affected by a variety of demand-pressure factors. The drawback of the markup
approach is that it fails to explain how demand factors should affect prices. The
advantage of the model developed in this paper is that it not only tracks the
influence of cost-push factors on prices but also incorporates the demand
factors that stem from a sound theoretical framework. In effect such demand
factors are indicative of how optimizing firms respond to changes in inventory
levels and growth in anticipated sales. In his two consecutive papers on price
behavior published in 1977 and 1978 Maccini further develops the theory
introduced in 1976 and introduces separate models for elastic and inelastic
prices. The models are fitted to the data from the manufacturing sector of the
US economy. The empirical testing shows that inventories appear to have some
influence on prices. The estimates of these parameters generally have the right
sign but unlike sale variable coefficients they achieve statistical significance only
in few cases.
All above-mentioned studies have used the data aggregated at least to
the industry level. These studies have produced mixed results with regard to the
inventory influence on price formation, price-output flexibility, and price
asymmetry testing. Some authors tend to explain these controversial findings by
60
the fact that the data aggregated at an industry level could lead to biased
results since individual firms’ price changes might cancel each other.
The
literature we review in the next part of the paper deals with the micro-level data
obtained from different business surveys.
The following studies analyze firms’
price and quantity decisions using qualitative data.
Kawasaki et al (1982) examine two main hypotheses. The first one is that
in short run, quantities are more flexible than prices; the second hypothesis is that
prices are more sticky downwards than upwards. The short run responsiveness of
firm prices and outputs to inventories changes is estimated by utilizing monthly
German industry data, which includes firms’ own appraisal of prices, production,
and lagged inventory levels.
All variables are qualitative.
Conditional
multivariate logit model is employed to explain the effect of the independent
variable on the probability of the price/production change. Even though it is
possible to estimate separate models for price and output the authors suggest
utilizing a joint model. They argue that this approach allows them to take into
account the correlation between price and output due to some omitted
variables. As a result of two-stage estimation procedure gamma coefficients are
calculated which summarize the ceteris paribus bivariate interaction between
independent variable and each dependent variable. The results show that firms
react to inventory disequilibrium within one month with both price and output
changes.
Each firm seems to change its output whenever there is inventory
disequilibrium, but changes its price only when disequilibrium persists long
enough that changes in demand or cost seem to be permanent. Therefore, we
can conclude that persistence in price levels is not necessarily caused by
61
collusion between producers. With regard to the asymmetric price adjustment
hypothesis, no evidence is found that prices are more flexible downwards than
upwards.
Kawasaki et al (1983) further investigate price versus output flexibility issue
without utilizing inventory variable.
A testable corollary of the Kirman-Sobel
theorem 10 for industries producing differentiated products is developed and
applied to the German industry survey data.
The data contains qualitative
information on price, production, and expected sales levels as well as the
information on the expected changes in the business conditions that ends up
being used as a proxy for changes in long-run demand. This paper also provides
empirical evidence supporting the proposition that firms tend to change both
price and output in response to permanent changes in demand, but only output
in response to a transitory change in the demand.
Carlson and Dunkelberg (1989) utilize the data on US Quarterly Economic
Survey of Small Businesses and estimate the ordered probit model for the
probability of the price and employment change depending on inventory, cost,
and sales variables.
They determine that no systematic relationship emerge
between price changes and inventory investment. The demand and inventory
investment variables are more consistently positive and significant in the
employment regressions than in the price regressions. Therefore, the authors
This theorem basically implies the following testable hypothesis: 1) A firm will
change the price it charges in response to a demand change only if it perceives
the demand change to be permanent; and 2). A firm will change its output to
any change in demand.
10
62
state that this result is supportive of the Keynesian notion that output and
employment respond more quickly to demand change than do prices. Carlson
and Dunkelberg (1989) also discover that price changes are positively and
significantly correlated with the previous period price changes. They consider
this finding as supportive of the notion of sticky prices in the sense that initial price
responses to new demand and cost information are slow and once set in one
period, the price motions tend to linger through the next one.
McIntosh et al (1993) is, to our knowledge, the most recent study
investigating the role of inventories in price/output changes. Similar to Carlson
and Dunkelberg (1989) they include cost and demand variable as well as
lagged inventory variable in the model.
The authors utilize qualitative
information on changes in production, prices, inventories, and on changes in
expected and actual demand and cost contained in six business surveys
collected by the Confederation of the British Industry. Theoretically the model
describes monopolistically competitive firm that holds inventory of its finished
goods and is subject to cost and demand shocks. Two different methods are
employed for estimation: bivariate probit model for prices and output and semiparametric regression method. McIntosh et al (1993) point out that the reason
for doing this is to make sure that the obtained results are robust and are not an
artifact of the specific estimation method chosen. The results of both regression
approaches indicate that the past level of inventories is not an important
determinant of production and price changes.
This finding is consistent with
Maccini’s (1977, 1978) and Kawasaki’s (1982) results. The authors also suggest
that anticipated demand shocks constitute the driving variable for output
63
changes, while both cost and demand shocks affect pricing decisions.
The
general results show that firm’s appraisals of the adequacy of inventories appear
to be significant determinants of output plans and less often of price plans.
Our study differs from the above-mentioned studies in several important
ways. First, the discrete variables that we use in our estimation are based on real
values of price and output levels that have been observed on the market,
whereas in previous papers they reflected firms’ qualitative appraisals of their
future price and output plans.
Second, our independent variables are not
categorical. Therefore, we avoid a bias that may exist in the estimation of the
models where both dependent and independent variables are qualitative (see
Ronning and Kukuk (1996)). And finally, we utilize standard logit procedure as
well as fixed effect logit for panel data. The latter method has not been used in
previous studies and it allows us to deal with possible heterogeneity and
autocorrelation problems and therefore correct for possible bias in the estimation
procedure.
5.2 Relationship between Price, Production, and Inventory
The main goal of our study is to model the relationship between price,
production, and inventory using discrete choice model approach. One of the
requirements for the estimation to be valid is the stationarity of the data. Hence,
before conducting the estimation, we need to check if the data are stationary.
If the data appear to be stationary we can further conduct the Granger
causality test which shows the direction of Granger causality between variables
64
of interest and determines whether the past values of one variable help explain
the current value of another.
5.2.1 Data
Monthly data cover the period from January 1980 to December 1999 (total 240
observations). For the analysis we use monthly data on price, production, and
inventory for linerboard, corrugating medium, and recycled containerboard (see
Appendix A for the full list of the variables). Figures 23-24 allow us to visually
examine the relationship between the variables of interest. According to the
charts, there is a negative correlation between inventories and price
(production) variables.
Increase in inventories stock is likely to cause the
decrease in price and production rate. Price and production adjustments do
not take place instantly. There is a time lag between these two variables and
inventories changes. The lag length varies depending on the point of the time
interval.
550
2400
500
2200
450
2000
400
1800
350
1600
300
1400
250
200
80
1200
82
84
86
88
90
k iner boar d pr ice
92
94
96
98
t ot al inv ent or y
Figure 23. Price and Inventories at Mills and Box Plants
65
40
2400
2200
35
2000
30
1800
25
1600
20
15
80
1400
1200
82
84
86
88
90
lin e r b o a r d p r o d u c tio n
92
94
96
98
to ta l in v e n to r ie s
Figure 24. Production and Inventories at Mills and Box Plants
5.2.2 Granger Causality Testing
According to Granger (1969), the question of whether one time series causes
another can be answered as follows - X1 is said to be Granger caused by X2, if
the past values of X2 help in the prediction of X1, or equivalently, if the
coefficients on the lagged values of X2 are statistically significant.
Using this
definition, an econometric implementation of the Granger-causality test can be
conducted in the following way. First, we estimate:
X1t = α 0 + α 1 X1t −1 + α 2 X1t − 2 + ... + α p X t − p + β 1 X 2t −1 + β 2 X 2t − 2 + ... + β p X 2t − p + ε (23)
by OLS. Then, we conduct an F-test of the null hypothesis H0: β0=β1=β2….=βp. If the
null is rejected, X2 does not Granger cause X1. A few issues should be noted in
carrying out the Granger causality test. First, it is a bivariate test and thus must
be used between two time series. Second, the test is normally interpreted as a
test whether one variable helps forecast another, rather than a test of whether
66
one
variable
causes
another.
Therefore,
Granger
causality
measures
precedence and information content but does not itself indicate causality. Also,
bidirectional or two-way causality case is quite frequent. Finally, since the results
of the causality tests can be sensitive to the choice of the lag length (p) we
need to determine statistically and practically sensible number of lags included
in the model. The number might be different from the number of lags used in the
unit root testing since in this case not one but two variables are included in the
test equation.
To identify the number of the lags included in the model usually a number
of different criteria can be used - likelihood ratio statistics, finite prediction error,
Akaike, Schwartz, and Hannan-Quinn criterion. The problem is that these criteria
often point out at different lag length and consequently lead to contradictory
results. We choose to utilize the SIC criterion since it tends to point out the most
parsimonious models. In our case, the SIC criterion indicates that one lag should
be included in price/inventory Granger and product/inventory test equation
should contain two lags.
The results of Granger test for the level data neither reject nor strongly
support the hypothesis that inventory causes changes in other variables levels. It
turns out that there is a bi-directional causality between lagged inventory/price
and lagged inventory/production variables 11 .
But when we utilize price and
lagged inventory change (growth) variable, instead of the level values, lagged
inventories are found to significantly help explain changes in current price and
production level fluctuations.
11
Contrary to that, neither price, nor production
Results are the same for not-seasonally adjusted and deseasonalized data.
67
growth variables Granger cause inventory movements. The results of the tests
are consistent with Toppinen et al (1996) results.
Their study focused on the
relationship between Finnish pulp export prices and international pulp
inventories. They found that Granger causality existed from inventory to price
but not vice versa.
5.3 Econometric Model and Estimation Results
Since dependent variable in this analysis is dichotomous, the use of the OLS
regression in order to analyze the probability of price/output change would be
inappropriate as it may predict values outside the [0;1] interval. Hence, first we
utilize logit model for this purpose.
We start with the examination of linerboard. The dependent variables,
“price change” and “production change,” are dichotomous having value “1”
when the current price, or production level, increases from the previous month
and “0” it decreases 12 . For both dependent variables two models are estimated.
The first model contains only previous month’s inventory level as the independent
variable.
The subsequent models also include demand and cost variables –
previous month’s sales level and pulp producer price index.
Since we
discovered seasonality in inventories and production time series, the data are
deseasonalized by using the X12 method. The logit model is estimated for both
seasonally adjusted and non-adjusted data. In the latter case seasonal dummy
variables are added in the model. The results are reported in Tables 13-17 13 .
The discrete price and production variables were created using level data on
linerboard price and production.
13 The results so far are presented only for the first panel.
12
68
The linerboard data proves that inventory plays an important role in short
run price adjustment. As expected, in almost all of the estimated models signs of
the coefficients show that there is negative correlation between price level and
inventories.
prices.
Increases in lagged sales, on the contrary, leads to increases in
The sales’ coefficients are also statistically significant at least at 5%
significance level. Finally, no evidence is found that linerboard prices are more
flexible downwards than upwards. The magnitude and statistical significance of
the inventory and sales variables are similar for the probability of both price
increase and decrease.
Table 13. Logit Model Estimation Results (not seasonally adjusted)
Variable
Lagged inventories
Price increase
-0.0007521
Price increase
-0.0016824
Price increase
-0.0011877
-3.4775461
-3.7667111
0.0012112
2.4031684
-2.5049803
0.0017722
3.2607252
Lagged sales
Table 14. Logit Model Estimation Results (not seasonally adjusted)
Variable
Lagged inventories
Price decrease
8.771E-05
0.5953584
Lagged sales
69
Price decrease
0.0010606
2.9648202
-0.0013562
-3.0182089
Price decrease
0.0010720
2.9107580
-0.0013332
-2.7606404
Table 15. Logit Model Estimation Results (seasonally adjusted)
Variable
Lagged inventories
Price increase
5.94E-05
1.391756
Lagged sales
Price increase
-0.00053
-1.9473
0.00082
2.19598
Price increase
-0.0003406
-1.1719061
0.0010832
2.7230387
Table 16. Logit Model Estimation Results (not seasonally adjusted)
Variable
Lagged inventories
Prod. increase
0.000534
3.004954
Lagged sales
Prod. increase
0.000313
0.824612
0.000301
0.656354
Prod. increase
0.0004088
1.0178723
0.0004264
0.8710973
Table 17. Logit Model Estimation Results (seasonally adjusted)
Variable
Lagged inventories
Prod. increase
4.94E-05
1.156287
Lagged sales
Prod. increase
-0.0003
-1.09733
0.000477
1.305304
Prod. increase
-0.0002576
-0.8996579
0.0005226
1.3599457
However, these results seem to be only partly satisfactory. Not one of the
models estimated with production change as the dependent variable renders
statistically significant results. It might indicate that inventory, demand, and cost
factors play less important role in production level adjustments than in price
adjustments. Yet, this would contradict the well-known hypothesis that in short
run quantities are more flexible than prices. In order to assure that the results are
robust and not biased we further proceed with the estimation of fixed effect logit
model utilizing the panel data methodology, which allows correction for
unobserved heterogeneity and lack of independence across observations.
70
Appendix A shows that the panel data utilized in the estimation comprise two
main containerboard grades – linerboard and corrugating medium.
As we can see from the estimation results (see Tables 18-21) inventory
coefficients in all models have an expected negative sign and are statistically
significant at 5% level. The sign of the coefficient also proves that an increase in
previous month’s inventory level diminishes the probability of an increase in price
and output levels and vice versa. The demand variable, previous month’s sales
level, is also significant in all models.
The magnitude of the coefficients is
consistently smaller than of the inventory variable. It may indicate that changes
in inventory levels play a more important role in price changes than increases or
decreases in sales.
Table 18. Fixed Effect Logit Model Estimation Results for Panel Data (NSA)
Variable (seas. dd.)
Price increase
Price increase
Lagged inventories
-.0023913
-2.65
-0039899
-3.80
.0026902
3.25
Lagged sales
Price
increase
-.003982
-3.79
.0027462
3.29
Table 19. Fixed Effect Logit Model for Panel Data (NSA)
Variable
Price decrease
Lagged inventories
.0080552
7.11
Lagged sales
71
Price
decrease
.0089036
7.07
-.001413
-1.83
Price
decrease
.0088793
7.06
-.0014425
-1.86
Table 20. Fixed Effect Logit Model Estimation Results for Panel Data (SA)
Variable
Lagged inventories
Price
increase
-.0050559
-6.37
Lagged sales
Price
increase
-.0058152
-6.53
.0011952
2.06
Price
increase
-.0055102
-6.04
.0016912
2.80
Price
increase
-.0044197
-4.76
.0014415
2.36
Table 21. Fixed Effect Logit Model Estimation Results for Panel Data (SA)
Variable
Lagged inventories
Prod.
increase
-.0012764
-2.24
Prod.
increase
-.0017456
-2.63
.0007315
1.37
Lagged sales
Prod.
increase
-.0016462
-2.47
.0008398
1.54
Prod.
increase
-.0019714
-2.78
.0009534
1.73
Estimation of non-adjusted data with monthly dummy variables added in
the model allows for testing the asymmetric price adjustment hypothesis. As we
can see from the estimation results, the inventories’ coefficients are significant in
the models estimating probability of both price increases and decreases. Yet
the magnitude of the inventory parameter is consistently smaller in the models
estimating the probability of upward price movements. These findings show that
in containerboard industry we might have reverse case of the price asymmetry
when price is more flexible downward than upward.
Finally, we test the hypothesis that short-term quantitative adjustments are
more apparent than short-term price adjustments.
Interestingly, we find that
linerboard and corrugating medium price level changes play a greater role in
absorbing temporary increases in demand. The coefficients of inventory and
72
sales variables are much larger in the models estimating the probability of price
changes than those in the models of probability of output changes.
73
CONCLUSION
This study is an attempt to examine simple and advanced forecasting methods
performance while modeling linerboard price behavior. The comparison of out
of sample forecast performance for 2000 shows that Holt-Winters exponential
smoothing renders adequate performance in short term forecast. In long term
forecasting, the VAR model not only does better than all other techniques, but
also outperforms published forecasts by demonstrating a significantly improved
accuracy.
The ARIMA model forecasts are also quite close to those of Holt-
Winters method and VAR.
Inconclusive results of the study could be explained by the haphazard
pattern of linerboard price behavior. Due to the complicated nature of price
movement, or when abrupt price changes are followed by prolonged periods of
no change, it is not feasible to agree on one best model. Under these
circumstances forecast performance strongly depend on particular time periods,
for which forecasts are produced as well as on forecasting horizon. Hence,
mixed forecasts, combining different techniques, are likely to render better results
in price forecasting in containerboard industry.
Further, the influence of inventory on probability of price/output changes,
price vs. output flexibility, and price asymmetry is investigated. According to our
results, inventories render significant influence on the probability of price
changes. The inventory coefficients are highly significant and have consistently
expected signs. Also, quite unexpectedly we discover that in containerboard
74
industry prices are more responsive to changes in demand than output.
It
means that the industry tends to keep prices at planned levels and adjust output
in short run. Finally, prices demonstrate upward, and not downward, stickiness.
In future research we would like to further examine containerboard
industry price movements by analyzing corrugating medium and recycled
linerboard price behavior. Standard univariate and multivariate techniques as
well as mixed forecast will be utilized for this purpose. Aggregation bias is one of
the issues that we are going to address with regard to the second part of the
paper. Finally, further data collection is necessary to add new grades for panel
data estimation that allows for better checks of data robustness.
75
APPENDIX A. Variables Description
Panel 1.
Delivered price
Production
Inventory
Export
Sales
Description
Linerboard
Unbleached kraft linerboard #42
Total linerboard domestic and export
production
Total linerboard inventory at mills and box
plants
Total linerboard export
Pulp PPI
Total linerboard sales calculated as:
production – export – Δ in total inventories.
Softwood pulp PPI (base – December 1982)
Delivered price
Corrugating medium
Corrugating medium #26
Production
Inventory
Export
Sales
Pulp PPI
Total corrugating medium domestic and
export production
Total corrugating medium inventory at mills
and box plants
Total corrugating medium export
Total corrugating medium sales calculated
as: production – export – Δ in total
inventories.
Hardwood pulp PPI (base – December 1982)
Unit
USD/short
ton
1000 short
ton
1000 short
ton
1000 short
ton
1000 short
ton
USD/short
ton
1000 short
ton
1000 short
ton
1000 short
ton
1000 short
ton
Panel 2.
Linerboard.
PPI
Production
Mill inventory
Linerboard
Total linerboard domestic and export
production
Linerboard inventory at mills
Export
Total linerboard export
Sales
Total linerboard sales calculated as:
production – export – Δ in mill inventories
index
1000 short
ton
1000 short
ton
1000 short
ton
1000 short
ton
Corrugating medium.
Corrugating medium
Total corrugating medium domestic and
index
1000 short
PPI
Production
76
Mill inventory
export production
Corrugating medium inventory at mills
Export
Total corrugating medium export
Sales
Total corrugating medium sales calculated
as: production – export – Δ in mill inventories.
PPI
Production
Mill inventory
Sales
Recycled containerboard
Recycled containerboard
Recycled containerboard domestic
production
Recycled containerboard inventory at mills
Recycled containerboard sales calculated
as: domestic production – Δ in mill
inventories.
77
ton
1000 short
ton
1000 short
ton
1000 short
ton
index
1000 short
ton
1000 short
ton
1000 short
ton
REFERENCES
Alavalapati, J., Admowicz W., and M. Luckert (1997). “A Cointegration Analysis of
Canadian Wood Pulp Prices”, American Journal of Agricultural Economics, 79,
pp. 975-986.
Akaike, H. (1969). “Fitting autoregressive models for prediction”, Annals of the
Institute of Statistical Mathematics, 21, pp. 243-247.
Arstrong, J.S., and F. Collopy (1992). “Error measures for generalizing about
forecasting methods: Empirical comparisons”, International Journal of
Forecasting 8, pp. 69-80.
Barro, R. and H. Crossman (1971). “A General Disequilibrium Model of Income
and Employment”, American Economic Review, 62, pp. 82-93.
Baudin, A. (2001). “Inventory and Price Variations of Coniferous Sawnwood in the
UK Market,” Supply Chain Management for Paper and Timber Industries, Vaxjo,
pp. 77-98.
Blinder, A.S. and L.J. Maccini (1991). “Taking Stock: A Critical Assessment of
Recent Research on Inventories”, Journal of Economic Perspectives, 5(1), pp. 7396.
Booth D.L., Kanetkar V., Vertinsky I., and D. Whistler (1991). “An Empirical Model
of Capacity Expansion and Pricing in an Oligopoly with Barometric Price
Leadership: A Case Study of the Newsprint Industry in North America,’ Journal of
Industrial Economics v. XXXIX(3),pp. 255-276.
Box, GE.P. and G. Jenkins, Time series Analysis, Forecasting and Control, San
Francisco, CA: Holden-Day.
Brannlund, R., Lofgren K.G., and S. Sjostedt (1999). “Forecasting Prices of Paper
Products: Focusing on the Relations between Autocorrelation Structure and
Economic Theory”, Journal of Forest Economics 5(1), pp. 23-44.
78
Buongiorno J, Farimani M., and W.J. Chuang (1982). “Econometric Model of Price
Formation in the United States Paper and Paperboard Industry”, Wood and Fiber
Science 15(1), pp. 28-39.
Buongiorno J. and H.C. Lu (1989). “Effect of Costs, Demand, and Labor
Productivity on the Prices of Forest Products in the United States”, Forest Science
35(2), pp. 349-363.
Buongiorno, J. and J.K. Gilless (1980). “Effect of Input Costs, Economies of Scale,
and Technological Change on International Pulp and Paper Prices”, Forest
Science 26(2), pp. 261-275.
Carlson J. and W. Dunkelberg (1989). “Market Perceptions and Inventory-PriceEmployment Plans”, The Review of Economic and Statistics, 71(2), pp. 318-324.
Carlton D.W. (1979). “Contract Price Rigidity and Market Equilibrium”, The Journal
of Political Economy, 87(5), pp. 1034-1062.
Cheng, B. (1999). “Causality Between Taxes and Expenditures: Evidence from
Latin American Countries”, Journal of Economics and Finance 23(2), pp. 184-192.
Dickey, D., Bell, W., and R. Miller, (1986). “Unit Roots in Time series Models: Tests
and Implications”, American Statistician, 40, pp. 12-26.
Ekstein, O. and G. Fromm (1968). “The Price Equation,” American Economic
Review, 58(5), pp. 1159-1183.
Fildes, R., Hibon M., Makridakis S., and N. Meade (1998). “The Accuracy of
Extrapolative Forecasting Methods: Additional Empirical Evidence”, International
Journal of Forecasting 14, pp. 339-358.
Geweke, J., Meese R., and W. Dent (1984). “Comparing Alternative Tests of
Causality in Temporal Systems: Analytic Research Results and Experimental
Evidence”, Journal of Money, Credit, and Banking, 16, pp. 403-434.
Granger C.W.J. (1969). “Investigating Causal Relationship by Econometrics and
Cross Sectional Method”, Econometrica 37, pp. 424-438.
79
Hay, G. (1970), “Production, Price, and Inventory Theory, “American Economic
Review, 60, pp. 531-545.
Heckelman, J. (1997). “Determining Who Voted in Historical Elections: An
Aggregate Logit Approach”, Social Science Research, 26, pp. 121-134.
Kawasaki, S. McMillian J. and K. Zimmermann (1982). “Disequilibrium Dynamics:
An Empirical Study”, The American Economic Review, 72(5), pp. 992-1004.
Kawasaki, S. McMillian J. and K. Zimmermann (1983). “Inventories and Price
Inflexibility”, Econometrica, 51(3), pp. 599-610.
Koenig, H. and M. Nerlove (1986). “Price Flexibility, Inventory Behavior and
Production Responses”, in. Heller W, Star R. and D. Starrett (eds), Equilibrium
Analysis: Essays in Honor of Kenneth J. Arrow, vol. II (Cambridge: Cambridge
University Press).
Maccini, L.J. (1976). “An Aggregate Dynamic Model of Short-Run Price and
Output Behavior”, The Quarterly Journal of Economics, 90(2), pp. 177-196.
Maccini, L.J. (1977), “An Empirical Model of Price and Output Behavior”,
Economic Enquiry, 15, pp. 493-512.
Maccini, L.J. (1978). “The Impact of Demand and price Expectations on the
Behavior of Prices”, The American Economic Review, 68(1), pp. 134-145.
Makridakis, S. and M. Hibon (1979). “Accuracy of Forecasting: an Empirical
Investigation”, Journal of the Royal Statistical Society, A142, pp. 97-145.
Makridakis, S. and M. Hibon (1997). “ARMA Models and the Box-Jenkins
Methodology”, Journal of Forecasting 16, pp. 147-163.
McFetridge, D.G. (1973). “The Determinants of Pricing Behavior: A Study of the
Canadian Cotton Textile Industry”, Journal of Industrial Economics, 22(2), pp. 141152.
80
McIntosh J., Schiantarelli F., Breslaw J., and W. Low (1993). “Price and Output
Adjustment in a Model with Inventories: Econometric Evidence from Categorical
Survey Data”, The Review of Economic and Statistics, 75(4), pp. 657-663.
Melinvaud, E. The Theory of Unemployment Reconsidered, London: Blackwell,
1977.
Newbold, P., and C.W. Granger (1974). “Experience with Forecasting Univariate
Time series and the Combination of Forecasts”, Journal of the Royal Statistical
Society A137, pp. 131-165.
Ronning, G. and M. Kukuk (1996). “Efficient Estimation of Ordered Probit Models”,
Journal of American Statistical Association, September 1996.
Stier J. C. (1985). “Implications of Factor Substitutions, Economies of Scale, and
Technological Change for the Cost of Production in the United States Pulp and
Paper Industry”, Forest Science 31(4), pp. 803-812.
Stigler , G.J. and J.K. Kindahl (1970). “The Behavior of Industrial Prices”, Columbia
University Press, New York.
Toppinen, A., Laaksonen, S., and R. Hanninen (1996). “Dynamic Forecasting
Model for the Finnish Pulp Export Price”, Silva Fennica, 30(4).
Torma, A. (2001). “What goes Around Comes Around or Does it?”, Pulp and
Paper International, 1, pp.28-29.
Venieris Y.P. (1973). “Sales Expectations, Inventory Fluctuations, and Price
Behavior”, Southern Economic Journal, 40(2), pp. 246-261.
81