- Email: [email protected]
Modelling the Great Lakes freeze: forecasting and seasonality in the market for ferrous scrap
ELSEVIER
International Journal of Forecasting 12 (1996) 345-359
,
Modelling the Great Lakes freeze" forecasting and seasonality in the market for ferrous scrap 1 Kevin Albertson*, Jonathan Aylen University of Salford, Salford, M5 4WT, UK
Abstract The paper offers a methodology for modelling seasonality in a volatile commodity market. It gives a practical example of the way seasonal factors can be incorporated into industrial forecasts. Recycled ferrous scrap is a widely traded commodity used in the steel and foundry industries. This paper considers the problems of forecasting scrap prices in the US market. Scrap prices display seasonal behaviour as a result of weather and patterns of industrial production. We consider various ways of modelling this seasonality, use of seasonal vector autoregression, the concept of seasonal integration and the use of dummy variables. A seasonal vector autoregression (VAR) is developed. Here the quarterly series is decomposed into four annual series, one for each quarter. We regress each of these resultant series on its own lags and lags of other series, so developing a periodic autoregressive model. A series of tests enables us to determine the type of seasonality exhibited by the data. The simplest form of seasonal adjustment using seasonal dummy variables turns out to be the best for forecasting US scrap prices. Use of the test procedure suggests that employing seasonal dummies is the correct specification in this case. Inclusion of seasonal effects usually improves the estimation and forecasting performance of time series models. Comparison of a range of alternative forecasting models suggests a periodic autoregression only forecasts satisfactorily in the short run. A R I M A models with seasonal dummies show the best performance. A long lag length is necessary to capture long run cyclical effects. Keywords: Autoregressive modelling; ARIMA forecasting; Ferrous scrap; Seasonality; Sector modelling
* Corresponding author. Tel. (161) 745-5000; fax: (161) 745-7027. ~The financial support and advice of British Steel Engineering Steels is warmly acknowledged. We are most grateful for the research assistance of Martin Krill and the welcome comments of Professor P.H. Franses and three conscientious but anonymous referees. The David J. Joseph Company gave valuable advice on the working of the US scrap market.
1. Introduction
Iron and steel scrap is a basic raw material for the steel and foundry industries. Some 350 million t of ferrous scrap are consumed worldwide in a peak year such as 1989 with a value at current prices of US$40 billion. Scrap comes from old buildings and industrial machinery.
0169-2070/96/$15.00 © 1996 Elsevier Science B.V. All fights reserved PII S0169-2070(96)00669-3
346
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
Processors extract steel scrap from junked cars and discarded consumer durables. Scrap is also generated during manufacturing operations, both within steelworks themselves and from processes such as car industry press lines. The ferrous scrap market is arguably the world's largest market for recycled material. There is a growing international trade in scrap. The mature industrialised economies are the main exporters. Over 39 million t of scrap were traded internationally in 1992. The main trade flows are from northern Europe into Spain, Italy and Turkey and from the USA outwards to Europe and towards the newly industrialising countries of the Pacific rim. But, for the most part, scrap supplies are predominantly local, with transport costs representing 10-20% of a steel mill's scrap purchasing budget at current prices (Wulff, 1995). The market for ferrous scrap is highly competitive. World scrap prices react quickly to shifts in demand and supply (Anderson, 1987; US Bureau of Mines, 1993). The scrap industry has a tiered structure, with a multitude of small collectors supplying a much smaller number of medium sized yards, which in turn feed just a few large dealers, as described in International Iron and Steel Institute (IISI) (1987, Chapter 2). Although supply is concentrated in the hands of a few major dealers in some countries, no single buyer or seller is able to influence the world price. World scrap prices are set in the main trading area which is the north-eastern quarter of the USA. The US rust-belt is a major scrap reservoir for American mini-steelplants and for the export trade from the Eastern Seaboard and out of the Great Lakes. Prices in other major markets, such as the UK, are led by inter-dealer trading for export cargoes. Daily fluctuations of the dockside price provide a lead indicator for regular domestic contracts between dealers and scrap users of longer duration. There is no futures market for scrap, apart from a brief experiment in the 1950s on the Chicago Mercantile Exchange, as discussed by Stone (1977). The Chicago Board of Trade applied for permission to trade scrap steel in 1990, but nothing has come of the idea so far.
Interview evidence suggests some US processors speculate by accumulating or running down stocks. Otherwise scrap dealing displays the features of a typical commodity market. In Section 2 of this paper we discuss the cyclical, seasonal and trend pattern of US scrap prices. Section 3 outlines the different models we use to capture seasonal effects in the data. In Section 4 we estimate and choose between models on the basis of statistical tests. In Section 5 we report some forecast comparisons. Section 6 concludes the paper.
2. The behaviour of scrap prices
Over time, scrap price movements display three characteristics. There is a marked year-toyear cyclical fluctuation. There are seasonal patterns within each year. Finally there is an underlying trend in the behaviour of prices. We examine each of these in turn.
2.1. Cyclical behaviour Scrap is a key raw material in part of the steel industry, typically accounting for a third of the cost of a ton of wire rod made in a US ministeelworks. A strong cyclical pattern is observable in the year-to-year behaviour of scrap prices. Fig. 1 shows the longest available price
US Scrap Price: No. 1 Heavy Melting Nominaland Real (1987) US$ ,,
I
:300 240 D)
~) 180
~120 ffJ 0 1900
....
o."~)-.,~ . . . . . .
1920
,,..°.d
1940
Year
1960
2000
1980
I --real (1987) price -.nominal price
I
Fig. 1. The long run behaviour of scrap prices.
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
series (US Bureau of Mines, 1993). Typically, peak year prices are 5 years apart and closely follow the cyclical pattern of steel production in the USA. Since the USA is the world's largest free scrap market and the world's largest scrap exporter, global prices move in line with the American market. As a result, high scrap prices arising during a boom period in the USA may coincide with recession in other major steel markets. Prices are volatile over the cycle. Ferrous scrap prices regularly double and halve. US No. 1 Heavy Melting scrap sold for 56% more at the end of 1993 compared with the end of 1992, for instance.
2.2. Seasonal behaviour
There are two reasons for expecting the scrap market to be influenced by seasonal effects; the weather and industrial production. The first seasonal factor is the weather: market gossip suggests scrap prices rise the day the Great Lakes freeze in North America and drop back when navigation resumes. Portions of the Great Lakes freeze every year. Until recently, Detroit and Chicago were major supply centres for US scrap exports and the end of Great Lakes' navigation for the year curtailed supply to the world market. The frozen lakes also stop the movement of iron ore to the integrated steelplants on the lake shore. Scrap is a substitute for molten iron in basic oxygen steelmaking converters. The official shipping season ends with the closure of the Sault Ste. Marie locks between Michigan in the USA and Ontario in Canada on January 15th, and resumes with their reopening on March 25th, see Swanson (1994). An unduly severe freeze delays the re-opening of the locks which causes an unanticipated run-down of iron ore stocks prompting higher demand for scrap as a substitute raw material in the Great Lakes region. Such a severe freeze was experienced in the winter of 1993/4 and the US No. 1 Heavy Melting scrap price fell $2.33 per long ton in the first week of April 1994 when the Great Lakes reopened for navigation. In truth, navigation across the Great Lakes is
347
just one climatic factor influencing scrap supply. Severe winters affect rail transport within the United States. Flooding, bridge wash outs and blizzards all disrupt the flow of rail cars from scrap stockyards to steelmakers. Scrap inventories in the hands of dealers fall as the onset of winter inhibits collection of obsolete scrap. Conversely, the return of better weather produces a 'spring flush' of junk cars, discarded durables and demolition scrap. Summer heat is not conducive to scrap collection, but the autumn produces further material for recycling from the highways and byways. So there are spring and autumn peaks in scrap supply. The second seasonal factor is the annual rhythm of industrial production which prompts shifts in demand for scrap. Inventories of both raw materials and steel products are run down in anticipation of the end of the fiscal year in December and rebuilt as the US fiscal year gets under way in January. So there is a fall off in scrap demand in the last quarter of the year and a surge in the first quarter as steelmakers expand output and rebuild their scrap stocks. Scrap consumption drops in the USA in July and August as holidays, annual maintenance schedules and model changes reduce steel and foundary production in the summer months. The summer slow-down is a feature apparent across all OECD economies, but the rise in US scrap demand during the winter stands in contrast to the widely reported trough in virtually all aspects of economic activity in the first quarter of the year for both the USA (Barsky and Miron, 1989) and other OECD economies (Beaulieu and Miron, 1992). Seasonal influences appear in various ways. Steel scrap prices rise systematically in the winter and fall in the summer and autumn, see Marcus and Kirsis (1994, Fig. 52). We might expect the volatility of prices to vary over the year, too. That is to say, the variance of the scrap price series is not constant over the year, but depends on the season. We anticipate seasonal heteroscedasticty. Previous studies of the US scrap market such as Barnett and Crandall (1986) ignore seasonal factors, preferring to use annual data instead.
348
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
2.3. Trend
Over the long run, US scrap prices are a non-stationary series. There has been a gradual secular decline in real scrap prices. The real annual price of the US No. 1 Heavy Melting Grade has dropped by 0.65% year -1 compound since 1909. The nominal price trends upwards by 2.5% year -1 over the period 1907-1993 due to inflation. The quarterly price series used here is collected by a trade journal, American Metal Market. Scrap is traded in a wide variety of grades. The leading price is for No. 1 Heavy Melting Grade steel scrap. This is the most widely traded grade of scrap. The price for No. 1 Heavy Melting Grade steel is closely matched by equivalent grades elsewhere, such as OA scrap in Britain or No. 1 Old Scrap in Germany. American Metal Market tracks the price of this No. 1 Heavy Melting Grade in three US cities, currently Chicago, Pittsburgh and Philadelphia and publishes an average or composite price. We use the log of the No. 1 Heavy Melting Grade composite price per long ton as our series for analysis, with the aim of forecasting the future scrap price. We take an arithmetic average of monthly price data to form quarterly observations. We lack data on the volume of transactions at each price which would enable us to calculate a more appropriate weighted average. We assume the Great Lakes' freeze coincides with the first quarter and the summer production pause occurs in the third quarter of the year. The data runs from the first quarter of 1954 to the present. Before this start date, there were price and export controls in the US market. Nor can we be sure that subsequent data is unaffected by government regulation. For instance there was a brief episode of price control during the boom of 1973 and export restrictions in 1973 and 1974. Our forecasting models are estimated using quarterly data from 1954 quarter one up to 1989 quarter four. Having specified our model we use it to 'forecast' the price series for the period 1990 quarter one through to 1995 quarter one to see how well it performs. Our nominal scrap price series is characterised
by an upward jump in 1974, quarter one. This is visible in Fig. 1. The mean shift is also statistically significant. It is tempting to attribute this permanent rise in the nominal price of scrap to the rise in inflation experienced across all OECD economies after 1973. It has been suggested that the 1973 oil price shock was a once-and-for-all break in trend for the US economy at a macroeconomic level (Perron, 1989). However, 1973-74 also represented a climacteric in the fortunes of 'big steel' - both in North America and world wide (IISI, 1980). This had conflicting effects on the scrap ~market. US raw steel production peaked in 1973 and declined thereafter. By itself this would cause a fall in the derived demand for scrap with a consequent fall in price. However, the drop in steel output also prompted changes in production techniques. The switch in steelmaking processes caused demand for purchased scrap to rise. Briefly, the collapse in steel output pushed the least efficient steel plants into retirement. Typically these were open hearth works with an iron ore feedstock. These obsolete plants relied on a high level of 'circulating scrap' generated in-house as a consequence of poor production yields from older processes such as ingot casting. Electric arc furnace based mini-mills took their place. These more modern plants relied on continuous casting and generally enjoyed much higher yields of finished steel from raw materials. They relied on scrap purchased from the market. The share of electric arcs in US steel production doubled in the USA from 18% of crude steel production in 1973 to 38% by 1993. There were corresponding gains in yield among the surviving integrated plants, too. These relied on a mixed molten iron and scrap feedstock. Higher production efficiencies meant they also bought in more scrap from the market instead of generating their own supplies inhouse. Consequently, the amount of scrap purchased per ton of steel made in the US rose by 86% in the 2 decades after 1973-74 (Wulff, 1995) even though overall steel production has declined. We allow for a possible change in behaviour of the US scrap market by adding a downward mean shift dummy to our data before 1974
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
349
I Autocorrelation Functions i 1
0.8 0.6 0.4 0.2 0 -0.2
i
-0.4 -0.6
I
i
!
'
'
~ i !
i
E
-0.8
'
i
-1 2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
lag
I ==Q1 iQ2
NQ3 mQ4 i
, ,Partial Autocorrelation Functions i 1 0.8 0.6 0.4
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
lag
I NQ1 ~Q2 iQ3 ~Q4 i Fig. 2. Periodic autocorrelation and partial autocorrelation f u n c t i o n s - levels data.
quarter one. Where seasonal dummy variables are used, four seasonal mean shifts are employed instead. 3. Competing seasonal models 3.1. Deterministic seasonality Use of seasonal dummy variables is the tradi-
tional method of seasonal adjustment in forecasting models using seasonally unadjusted data, for instance the partial adjustment model Owen and Phillips (1987) use to explain British Rail intercity traffic flows. Deterministic seasonality of this sort also accounts for much of the seasonal fluctuation in macroeconomic time series such as real GDP, retail sales or industrial production (Beaulieu and Miron, 1992). Seasonal dummy
350
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
variables may be used to capture a deterministic 'shift in mean' seasonal component in the data. The use of seasonal dummy variables in isolation rests on the assumption that the seasonal component in the data is independent of the trend and cycle components and may be modelled independently. This assumption is not valid if the data is periodically integrated for instance, as Franses and Paap (1994) argue. Deterministic seasonality is just one form that seasonality may take. The arbitrary use of dummy variables to capture seasonal effects without testing for other forms of seasonality may result in mis-specification, as demonstrated for GDP data by Abeysinghe (1994). Instead, deterministic seasonality should be viewed merely as a special case of more comprehensive seasonal models which we now review briefly. 3.2. A periodic autoregressive m o d e l
A relatively new concept that has become increasingly common in the time series literature is that of a periodic autoregressive model. A simple univariate model of this form may be written: Yt = DsS, + P~,lYt-I + " " " + P s , p Y t - p + 8t ;
t = 1. . . . . N ,
(1)
where the D e are seasonal dummy variables and the autoregressive parameters, p~.pS vary with the season of the year, s = 1, 2, 3, 4, for quarterly data. Such models have been used in the context of forecasting by, for example, Osborn and Smith (1989). They find that, when there is periodicity in the data, a periodic autoregressive model performs well relative to other time series models in terms of short term forecasts. See also Tiao and Grupe (1980), Franses (1991) and Osborn (1991) among others. It is common in the literature to write the periodic autoregressive process as a vector autoregression, VAR,
where the 4.1 vector Y r = [YlrY2rY3rY4T ]' with Y,r the observation of y, in season s of year T,P=int(p/4) and the 4.4 matrices ~k are made up of the autoregressive parameters. Let us define the lag polynomial q~(L) = ~0 - qJ1L + • .. + cbeL e, where L is the lag operator such that L J Y T = YT-j. The VAR may be written • ( L ) Y r = E r. The series is stationary if the roots of the characteristic equation ]q~(z)l = 0 are all outside the unit circle. The process has as many unit roots as there are roots of the characteristic equation on the unit circle. The simple periodic model defined in (1) above encompasses the non-periodic model Y t = S s + P l Y , - I + "'" +PpYt-p + et
(3)
which, in the absence of seasonal dummy variables, we can write as ~o(L)y, = et, where ~(L) is a polynomial in the lag operator. In this case, y is stationary if the roots of the equation q~(z) = 0 lie outside the unit circle. We may respecify (1) as (3) if we may accept the null hypothesis: Ho: Ps,j = pj ; s = l , . . . , 4 ,
j = 1,..., p.
(4)
Boswijk and Franses (1993) show that the likeli2 hood ratio test of this null hypothesis has a X distribution with 3p degrees of freedom under the null hypothesis, whether or not Yt has a unit root. If y has no unit roots, this model may display deterministic seasonality. In this case the seasonal effects are constant from year to year and the seasonality can be described using seasonal dummy variables in the model. To determine if such seasonality is present, we may test the null hypothesis H0:61 ~--82 = 83 = 84
(5)
in the usual way. If we accept this null we can be confident that there is no seasonal pattern of this form in the data. 3.3. Seasonal integration
CboYv = cI)IYT-1 + " "
T=I,...
1 ,-~N,
+ ~PYT-P + ET ;
(2)
Let us again consider the characteristic equation of the non-periodic model. If the charac-
K. Albertson, J, Aylen / International Journal of Forecasting 12 (1996) 345-359
teristic equation, q~(z)= 0, has a single root of unity we say that y has a unit root at the zero frequency. Such a process must be first differenced to be reduced to stationarity. The first difference operator is defined A = ( 1 - L), with Ly, = Y,-1. Thus, if y has such a unit root, Ay~ -y, - y,_ 1 will be stationary. It may, however, be the case that the characteristic equation has more than one root on the unit circle. Let us consider the case where there are four roots on the unit circle, - 1 and +-i. In this case y requires, not first differencing, but rather seasonal differencing to reduce it to stationarity. For quarterly data, the seasonal difference operator is defined A 4 = ( 1 - L 4 ) . Thus A a y t -= y, - Yt-4" In this case y, is said to be seasonally integrated or to display stochastic seasonality. Testing for the presence of such a seasonal integration process is typically carried out using the H E G Y test of Hyllebert et al. (1990). This test, or more correctly, group of tests, is based on the decomposition (1 - L 4) = (1 - L)(1 + L)(1 - L2). We write q~*(L ) A 4 y , = rqyt. , 1 J~- "/7"2Y2,t-1 + 7r3Y3,t-2 + 7"r4Y3,t-i + e t ,
(6)
possibly including seasonal dummy variables, a linear trend, a drift term and/or lags of the dependent variable to whiten the errors. The dependent variables are defined y~, = (1 + L + L2 + L3)yt, Y2,t = - ( 1 - L + L 2 -'L3)yt and Y3,, = - ( 1 - L2)yt. The null hypothesis of a unit root at the zero frequency is given by HI: 7rI = 0 vs. H~A: 7q < 0 . Similarly, the null of unit roots at the biannual and two semiannual (seasonal) frequencies, are given by H2:~'2 = 0 vs. HaATr2 < 0, H3: "rr3 = 0 VS. H3ATr3 < 0 and H 4 : 7r4 = 0 vs. H4A: "/7"~ 0, respectively. The tests are t type tests and critical values are given in Hylleberg et al. Note that an underlying premise of the H E G Y test is that the data is generated by a non-periodic process. It has been shown, see for example Osborn et al. (1988), that the application of the test to periodic data may lead to spurious results. If the data is non-periodic and we find season-
351
al integration in the data, we may apply a seasonal A R I M A ( p , d , q ) ( P , D, Q)4 to the series. The use of the moving average, S A R I M A , error term may enable us to reduce the lag length in the autoregression. This process may be written p(L) Fp(L 4)(A d A D 4Yt) = Oq(L )OQ(L 4)Et ,
('7)
where d and D are the degree of first and seasonal differencing respectively, O, ~, 0 and O are lag polynomials of orders p, P, q and Q respectively. If seasonal differencing is not appropriate, the above seasonal A R I M A process is nested within a simple A R I M A ( p * , d , q*) model, with p* = 4P + p and q* = 4Q + q.
4. Specification, estimation and testing The most general model of those considered above is the periodic model and, indeed, the other models will be mis-specified if the data series is periodic. Hence we must first determine whether or not the parameters of the model vary seasonally. To do this we must first determine the appropriate lag length to use in the autoregression. Now, Tiao and Grupe (1980) show that when the true model is periodic, the use of sample autocorrelations and partial correlations is misleading in terms of specifying the lag length of the model. We use periodic sample correlations and autocorrelations. These are estimated by OLS regression of each of the four series (one for each quarter) on their own and each other's lags. The kth lag periodic autocorrelation coefficients may be calculated from the regression •
Ys T =
+
k
(8)
where Lq is the one quarter shift operator, fY(,-,),r; Lq(gs
,1") = ]l y4,(T_l);
s = 2, 3, 4 S: 1 .
(9)
Similarly, we estimate the lag k periodic partial autocorrelation coefficients, ~'sk, using
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
352
in Fig. 2. From an inspection of the periodic partial autocorrelation functions, we specify six lags in the periodic autoregression, PAR, defined in (1) above. An inspection of the residuals from this regression indicates that this is sufficient to whiten the model errors. This is borne out in the correlation function of the regression residuals. We may determine whether or not we need to continue to allow for periodically varying parameters using the likelihood ratio test suggested by Boswijk and Franses (1993). Before any such
k
Ys,T= o,,+Y, i=1
' ~s,iLq(Ys,T).
(10)
We may approximate the standard errors of the periodic autocorrelations and partial autocorrelations by 2/~/N (where we have N/4 years of data). The periodic autocorrelation and partial autocorrelation functions suggest that our data is integrated as there is a spike of roughly one in magnitude for each of the four quarters as shown
Autocorrelation Function First Differences 1 0.8 0.6 i
0.4 0.2
r
i
i
:
i
i
;
!
.
}
:
:
i
! ;
i
0
-0.2 -0.4
-0.6 -0.8
i
-1 1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
lag Partial Autocorrelation Function First Differences 1 0.8 0.6 0.4
0.2 0
m
m I
]
:u
NTN--
-012
m-m
m__,m--
--mm
m -
-0.4 -0.6 -0.8
-1 1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
lag Fig. 3. A u t o c o r r e l a t i o n
and partial autocorrelation
functions-
first d i f f e r e n c e d d a t a .
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
353
[ Out of Sample Forecasts I 150 130 ,-. 110 o -'-' 90 70 50 30 10 55
;
60"
I
65
I
I
70
I-,~,~
i
I
75 year
I
I
80
;
I
85
I
i
90
95
, .,,,.,,(,2.,.~,., ,~'"~'~:'"~|
150 130 c 110 o -.-, 90 70 50 30 10 55
l
60
I
65
I
70
I-.s~=
I
I
75 year • -. ~,~.~.m
I
I
80 • ~.~.o~
;
i
85
I
90
i
95
i
Fig. 4. Out of sample forecast comparisons for US scrap price.
testing, however, we must determine whether or not the error variance is the same in each quarter as, if it is not, any subsequent testing will be invalid. Using the residuals from this estimation procedure, we may estimate, 0.s the standard error in each quarter; d"1 = 0.0978, &2 = 0.0884, O'3 = 0.0847 and &4 = 0.1347, where ~', represents the estimated standard error in the sth quarter residuals. Adding the sum of squared residuals from each individual quarter and dividing by the
total degrees of freedom gives the standard error of the combined model, 6- = 0.105. A likelihood ratio test of the null hypothesis, H0:0.1 = ° 2 - % -- 0-4 against a two sided alternative gives us a test statistic, u = 7.98. This statistic has a X 2 distribution with three degrees of freedom under the null. The critical value for a test of size 5% is 7.81. Thus we may be 95% confident that there is seasonal heteroscedasticity in the data. We must correct for this heteroscedasticity as we proceed to test the hypotheses H0: "the lag
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
354
parameters vary with the seasons" vs. HA: "they do not vary" as the model is non-periodic under the null hypothesis. We do this by rewriting the PAR Yt
O's
s=l
D*
O"s + E Ps,jYt-j
+"-~s'
j=l
(11)
with D* = Ds/(r~. An F version of the likelihood ratio statistic for testing the null H0: Ps,j =Pj for j = 1 . . . . . 5, takes on the value u = 0.98, thus we accept the null and can specify the nonperiodic model which, modified for seasonal heteroscedasticity, can be written 4
P
4
Yt_ E o.O* + E Oji~=lD* + et O's s=l s j=l "= iYt-j --;-O.s
(12)
We use the H E G Y test to determine whether or not seasonal differencing, possibly in addition to first differencing, is necessary to reduce the series to stationarity as our model is non-periodic. The H E G Y regression, allowing for the heteroscedasticity, can be written 4
4
Yt A4-~ = 7r, ~] D *sYI,t-1 + 7r2 E D*iY2.t-1 s=l
s=l
4
4
+ 77"3E D*y3,t-2 + Tl'4 E D*y3,t-1 s=l s=l P* 4 Et
+ ~. ~ D*A4Y,_I +--~. i=l s=l
(13)
ors
We find that two lags on the dependent variable, A4yt/o's, are sufficient to whiten the residuals. The value of the H E G Y test statistics for the levels series are given in Table 1. It is worth noting, in passing, that if the analysis is carried out without seasonal dummy variables the H E G Y test rejects unit roots at all frequencies. The mis-specification caused by omission of seasonal effects biases the test statistics. Our preferred model at this stage includes quarterly dummies, hence we consider the critical values given in Hylleberg et al. (1990) relating to this case. Recall, however, that in addition
to seasonal dummies we have also allowed for a mean shift and seasonal heteroscedasticity. Thus we must interpret these critical values with some care as the true critical values for this situation are likely to be greater in absolute value than those given (c.f. Perron, 1990, who considers the implications of allowing for a mean shift in conducting a Dickey Fuller test for stationarity). None-the-less, it is clear that we may be confident that the series does not have unit roots at seasonal frequencies, as the test statistic values are considerably less than the critical values tabulated. The absence of unit roots at some or all seasonal frequencies is a familiar result for aggregate data, too (see for instance Beaulieu and Miron, 1993). Our results reinforce the conclusion that arbitrary use of seasonal filters is likely to result in model mis-specification when forecasting a time series. We are confident that the series may have a unit root at the zero frequency as the test statistic for this frequency is not significantly different from zero. These conclusions are borne out by an examination of the autocorrelation function of the levels and first differenced data. Based on the autocorrelation function and partial autocorrelation function of the differenced data, we specify an ARIMA(5,1,0) model. The diagnostic test statistics indicate that the residuals from this model are white noise. However, forecasting experience in the industry suggests much longer lag lengths may be necessary to capture the cyclical fluctuations in the scrap price series. For example, inspection of the autocorrelation functions show a further spike at the eleventh lag, Fig. 3, although this is not significant. We propose an additional
Table 1 H E G Y test statistics in q u a r t e r l y d a t a o n No. M e l t i n g Scrap c o m p o s i t e price
1 Heavy
Levels data R o o t , zri T e s t statistics Critical v a l u e s
zri -2.511 -3.56
7r2 -6.664 -3.49
zr3 -5.309 -4.06
rr4 -6.02 -2.72
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
355
summer appears to have no impact on prices, perhaps because seasonal supply reductions match the fall in demand. Prices tend to be lower in the autumn as supplies pick up and buyers run down stocks towards the end of the financial year (although the coefficient on quarter four is not significant so this may be a chance result). These results differ from studies of seasonality in macroeconomic variables. At an aggregate level, the run up to Christmas and the summer holiday period strongly affect macroeconomic activity, regardless of the climate. In the scrap market, climate matters.
ARIMA(12,1,7) model alongside our preferred ARIMA(5,1,0) model. Having re-specified the preferred model using first differenced data and allowing for heteroscedasticity, we may test to see if the seasonal dummy variables add significantly to the explanatory power of the model. The null hypothesis, H0:61 = 62 = ~3 : (~4 VS. a two sided alternative is rejected for any reasonable level of significance. Our final model then is the simple non-periodic ARIMA(5,1,0) with seasonal dummy variables. The estimated coefficients and heteroscedasticity adjusted standard errors are given in Table 2. These results show two of the four seasonal dummies are statistically significant. Prices tend to rise in the winter, but fall back in the spring. Recall, we are modelling using the first differences of logged data. Consequently, the coefficients in Table 2 suggest scrap prices are 2% greater than might otherwise have been expected in quarter one, but 1.3% less in quarter two. The
5. Comparison of seasonal models To determine how well our final model forecasts, compared to other seasonal models, we conduct a series of rolling regressions and calculate out-of-sample forecasts for a number of similar time series models that differ in their
Table 2 Estimation results - ARIMAd(5,1,0) with seasonal dummies on No. 1 Heavy Melting Scrap composite price, 1954ql to 1989q4 Quarterly dummy
61
6_,
63
64
Estimate Standard error
0.02008 0.00567
-0.01273 0.00536
0.00052 0.00473
-0.00407 0.00641
Lag coefficient
p~
P2
P3
P4
P~
Estimate Standard error
0.20288 0.08711
-0.03997 0.08681
0.01256 0.08788
-0.02277 0.09318
-0.22319 0.08449
Box
Pierce
Stats.
Lag
Stat.
X 2 Dof
6 7 8 9 10 11 12 1 14 15
1.17 1.91 1.92 3.11 5.95 7.89 8.00 8.64 9.46 9.59
1 2 3 4 5 6 7 8 9 10
R 2 (differenced data), 27.41; LM test for heteroscedasticity, 0.001 X~ under null; ARCH test, 0.648 X~ under null; Jarque Bera normality test, 5.421 X~ under null.
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
356
Table 3 Forecast performance of time series models of the scrap price Model
I step MAPE Q1
1 2 3 4 5 6 Model
i step RMSE
Q2
7.83 5.08 6.92 6.01 8.87 4.56
6.09 3.53 2.94 3.65 4.03 3.50
Q3 7.10 5.54 7.80 6.12 4.25 4.71
Q4 6.79 7.38 7.94 7.09 9.78 7.74
Overall 6,95 5.48 6.47 5.78 6.88 5.25
2 step MAPE
Q1
Q2
8.62 6.06 7.74 7.39 10.82 6.19
6.78 4.22 3.34 4.37 5.15 4.15
Q3
Q4
8.71 7.33 10.50 7.67 6.30 5.52
7.19 9.02 8.81 9.23 11.74 9.32
Overall 7.84 7.00 8.08 7.48 9.11 6.73
2 step RMSE
Q1
Q2
Q3
Q4
Overall
Q1
Q2
Q3
Q4
Overall
1 2 3 4 5 6
8.38 7.76 9.34 9.49 13.63 5.73
12.11 9.61 9.59 9.03 6.76 7.41
10.17 12.97 11.95 13.72 9.99 11.32
11.68 13.17 9.75 12.07 20.12 10.28
10.59 10.88 10.16 11.08 12.62 8.68
10.63 8.31 10.25 9.88 15.07 8.13
13.91 11.60 11.94 11.37 11.27 10.86
13.26 13.99 15.50 14.62 12.19 12.85
13.39 14.52 12.15 13.67 23.01 12.31
12.86 12.35 12.61 12.53 16.06 11,19
Model
4 step MAPE
4 step RMSE
Q1
Q2
Q3
Q4
Overall
Q1
Q2
Q3
Q4
Overall
1 2 3 4 5 6
9.98 12.04 14.39 12.84 16.39 9.59
16.66 14.69 8 ",a 14.~_ 13.65 12.08
12.90 17.41 10.16 17.33 23.25 15.43
14.72 17.99 19.02 17.24 26.63 17.26
13.43 15.48 13.37 15.45 20.15 13.57
12.00 13.65 17.39 15.26 20.18 12.08
19.56 16.63 9.00 15.56 17.35 14.24
14.83 20.81 11.70 20.10 26.50 19.84
15.03 22.71 22.00 22.78 33.36 23.95
15.39 18.78 16.34 18.77 25.40 18.23
Model
8 step MAPE
8 step RMSE
Q1
Q2
Q3
Q4
Overall
Q1
Q2
Q3
Q4
Overall
1 2 3 4 5 6
24.72 24.69 24.33 25.16 18.12 25.90
25.93 20.39 10.96 20.67 35.87 24.92
20.07 27.40 13.82 28.02 46.37 31.55
24.03 32.03 26.5l 32.28 42.46 33.16
23.79 26.45 19.83 26.84 34.93 28.98
25.34 26.99 30.88 28.11 27.21 30.36
26.42 25.11 17.87 23.98 37.11 26.44
22.15 29.81 17.09 29.50 47.27 32.59
27.49 32.35 29.80 32.85 44.48 35.96
25.57 28.85 25.64 29.05 39.38 31.78
Model
12 step MAPE
1 2 3 4 5 6
12 step RMSE
Q1
Q2
Q3
Q4
Overall
Q1
Q2
Q3
Q4
Overall
26.35 24.41 27.85 25.00 24.48 33.12
8.79 19.04 19.61 17.39 54.82 16.82
20.02 18.77 24.59 17.93 62.33 20.49
22.35 20.58 13.16 22.32 44.63 28.16
20.37 21.06 21.14 21.26 44.16 25.85
33.35 25.27 30.78 27.11 25.71 40.00
10.30 19.86 19.61 17.42 58.51 23.06
22.26 18.77 24.59 18.45 66.12 25.68
23.01 23.97 19.28 24.55 45.45 29.98
24.76 22.66 24.36 23.02 48.76 31.43
Model 1, ARIMAd(12,1,7); Model 2, ARIMAd(5,1,0); Model 3, ARIMA(12,1,7); Model 4, ARIMA(5,1,0); Model 5, SARIMA(5,1,0)(0,1,0),; Model 6, PARI(5,1).
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
treatment of seasonality. The models we consider are a simple ARIMA(5,1,0), an ARIMA(5,1,0) with seasonal dummies, we will denote this ARIMAd(5,1,0), a SARIMA (5,1,0)(0,1,0)4 model and a periodic autoregressive model of order 5 on the first differenced data, PARI(5,1) model. For pragmatic reasons we include a long length model, an A R I M A (12,1,7) and the related ARIMAd(12,1,7) with seasonal dummies. This range of models captures a variety of approaches to modelling seasonality, to wit, no allowance for seasonality, deterministic seasonality, stochastic seasonality and periodic seasonality. As we have seen, we may reject all these seasonal specifications except deterministic seasonality and hence we expect models incorporating this to outperform the others in forecasting. Out-of-sample forecasts were calculated for all of these models on a rolling basis. That is to say, we estimate each model for the period ending 1989q4 and generate one-step ahead forecasts for 1990ql. We extend the sample period by one observation, re-estimate the models and generate one-step ahead forecasts for 1990q2 and so forth. A similar procedure is used to estimate rolling two-step ahead forecasts. Over a variety of forecast horizons we calculate the mean absolute percentage error, MAPE, and the root mean squared error, RMSE, of the forecasts. Illustrative results are shown in Table 3. The results are tabulated according to the quarter that is forecast and the model. All models often forecast the fourth quarter less well than the other three quarters. This is shown by larger MAPE and RMSE values for the Q4 column compared to the other three columns. There are clearly effects in the market in that fourth quarter that are not captured as well as the effects in other quarters. Comparing the forecasts made by the ARIMAd(5,1,0) model with dummies, Model 2, to those made by the ARIMA(5,1,0) model without, Model 4, we see that the use of dummies generally improves the forecast accuracy in quarter one, which is the quarter in which the seasonal effect is strongest. In terms of overall accuracy the ARIMAd(5,1,0) model outper-
357
forms the ARIMA(5,1,0). The comparison between the ARIMAd(12,1,7) and the ARIMA(12,1,7) is less straightforward, but, for longer range forecasting, the model incorporating dummies tends to do better. The periodic PARI(5,1) model does better than the non-periodic models at short range forecasting but the performance of this model deteriorates as the forecast horizon lengthens. Given that we have evidence that the data is not periodic, this deterioration in accuracy is to be expected as this model is likely to be overparameterised. Conversely, the SARIMA model always performs worst of the models considered. Again, this is not surprising as we have seen that seasonal differences are not required to make the data stationary, thus this model is overdifferenced. It must also be emphasised that measuring forecasting error alone does not indicate which model best captures the 'shape' of the future out-turn. Evidence on forecasting errors needs to be read alongside visual evidence on fluctuations and turning points in the data. To this end, we plot some of the forecasts generated by the alternative models in Fig. 4. Despite poorer short range forecast performance, the ARIMAd(12,1,7) captures the longer run cyclical swings in a far more satisfactory way than other models. Even with these longer lag lengths, taking account of seasonal effects substantially improves the forecasting performance of this model, especially beyond 3 years. Omitting valid seasonal effects has implications for the estimation of the model, with consequences for long run forecasts.
6. Conclusion
This paper demonstrates that seasonal effects are at work in the US market for ferrous scrap. The significance of seasonal dummies in our preferred forecasting equation suggests seasonal weather and the annual pattern of industrial production affect this particular commodity market. Seasonal effects are strongest in the first quarter of the year. Scrap prices rise by 2% in
358
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
the winter only to fall back by 1.3% in the spring. The market folklore that the Great Lakes' freeze affects scrap prices appears to be borne out by our findings in a simple and direct way. Despite the simple specification of the seasonal component, we may have confidence that this specification is valid as it was reached on the basis of a broad general to specific methodology. There is nothing to suggest that more complicated models are required to capture the seasonal component of the data. We have considered and rejected them in favour of our simpler model. Comparison of the forecasting performance of alternative models suggests that those with shorter lag lengths perform very well within the sample period, but a model with much longer lags gives better out-of-sample forecasts. A model with long lag lengths captures the wide cyclical swings in the US scrap market. Here, as in many cases, taking account of seasonal effects improves forecasting accuracy.
References Abeysinghe, T., 1994, Deterministic seasonal models and spurious regressions, Journal of Econometrics, 61, 259272. Anderson, D.F., 1987, Scrap as a commodity, Ironmaking and Steelmaking, 14(3), 130-135. Barnett, D.F. and R. Crandall, 1986, Up from the Ashes: The Rise of the Steel Minimill in the United States (Brookings, Washington, DC). Barsky, R.B. and J.A. Miron, 1989, The seasonal cycle and the business cycle, Journal of Political Economy, 97, 503534. Beaulieu, J.J. and J.A. Miron, 1992, A cross-country comparison of seasonal cycles and business cycles, Economic Journal, 102, 772-788. Beaulieu, J.J. and J.A. Miron, 1993, Seasonal unit roots in aggregate U.S. data, Journal of Econometrics, 55, 305328. Boswijk, H.P. and P.H. Franses, 1993, Unit Roots in Periodic Autoregressions (Department of Actuarial Science and Econometrics, University of Amsterdam) mimeo. Franses, P.H., 1991, Seasonality non-stationarity and the forecasting of monthly time series, International Journal of Forecasting, 7, 199-208. Franses, P.H. and R. Paap, 1994, Model selection in periodic
autoregressions, Oxford Bulletin of Economics and Statistics, 56, 421-440. Hylleberg, S., R.F. Engle, C.W.J. Granger and B.S. Yoo, 1990, Seasonal integration and cointegration, Journal of Econometrics, 44, 215-238. International Iron and Steel Institute (IISI), 1980, Causes of the Mid-1970's Recession in Steel Demand (Committee on Economic Studies, International Iron and Steel Institute, Brussels). International Iron and Steel Institute (IISI), 1987, Scrap and the Steel Industry: Trends and Prospectsfor Solid Metallics (Committee on Raw Materials, International Iron and Steel Institute, Brussels). Marcus, P. and K.M. Kirsis, 1994, Global metallics crisis, International Iron and Steel Institute, Annual Conference, Colorado Springs, October 4. Osborn, D.R., 1991, The implications of periodically varying coefficients for seasonal time series processes, Journal of Econometrics, 48, 373-384. Osborn, D.R. and J.E Smith, 1989, The performance of periodic autoregressive models in forecasting seasonal U.K. consumption, Journal of Business and Economic Statistics, 7, 117-127. Osborn, D.R., A.P.L. Chui, J.P. Smith and C.R. Birchenhall, 1988, Seasonality and the order of integration for consumption, Oxford Bulletin of Economics and Statistics, 50, 361-377. Owen, A.D. and G.D.A. Phillips, 1987, The characteristics of railway passenger demand, Journal of Transport Economics and Policy, 17(2), 133-153. Perron, E, 1989, The great crash, the oil price shock, and the unit root hypothesis, Econometrica, 57, 1361-1401. Perron, E, 1990, Testing for a unit root in a time series with a changing mean, Journal of Business and Economic Statistics, 8, 153-162. Stone, J.K., 1977, The iron and steel industry and the commodity market, Iron and Steel Manager, February, 36-38. Swanson, S., 1994, Lake ice freezes factories' plans, Chicago Tribune, March 27, Section 7, 3. Tiao, G.C. and M.R. Grupe, 1980, Hidden autoregressivemoving average models in time series data, Biometrica, 67, 365-373. US Bureau of Mines, 1993, Metal Prices in the United States Through 1991 (United States Department of the Interior, Bureau of Mines, Washington, DC). Wulff, S.W., 1995, Ferrous Scrap: an involuntary by-product that has become an essential raw material, presented at AISE Conference on Future of Flat Rolled Steel Production, Chicago, June (The David J. Joseph Company, Cincinnati) mimeo.
Biographies: Kevin ALBERTSON is Lecturer in Econometrics in the Department of Economics at the University of Salford. Dr Albertson took his first degree in maths and operations research at the University of Canterbury in Christchurch, New Zealand before joining the New Zealand
K. Albertson, J, Aylen / International Journal of Forecasting 12 (1996) 345-359 Department of Statistics for 2 years. He subsequently took a first class Masters Degree in Economics and completed a Doctorate on econometric theory. Jonathan AYLEN was educated at the University of Sussex and Worcester College, Oxford. He is now Senior Lecturer in
359
Economics at the University of Salford, where he has taught since 1974. Mr Aylen's published papers in economics cover cost-benefit analysis, international comparisons of performance and privatisation. He has travelled widely throughout the world steel industry and advised a number of national and international organisations on steel issues.
International Journal of Forecasting 12 (1996) 345-359
,
Modelling the Great Lakes freeze" forecasting and seasonality in the market for ferrous scrap 1 Kevin Albertson*, Jonathan Aylen University of Salford, Salford, M5 4WT, UK
Abstract The paper offers a methodology for modelling seasonality in a volatile commodity market. It gives a practical example of the way seasonal factors can be incorporated into industrial forecasts. Recycled ferrous scrap is a widely traded commodity used in the steel and foundry industries. This paper considers the problems of forecasting scrap prices in the US market. Scrap prices display seasonal behaviour as a result of weather and patterns of industrial production. We consider various ways of modelling this seasonality, use of seasonal vector autoregression, the concept of seasonal integration and the use of dummy variables. A seasonal vector autoregression (VAR) is developed. Here the quarterly series is decomposed into four annual series, one for each quarter. We regress each of these resultant series on its own lags and lags of other series, so developing a periodic autoregressive model. A series of tests enables us to determine the type of seasonality exhibited by the data. The simplest form of seasonal adjustment using seasonal dummy variables turns out to be the best for forecasting US scrap prices. Use of the test procedure suggests that employing seasonal dummies is the correct specification in this case. Inclusion of seasonal effects usually improves the estimation and forecasting performance of time series models. Comparison of a range of alternative forecasting models suggests a periodic autoregression only forecasts satisfactorily in the short run. A R I M A models with seasonal dummies show the best performance. A long lag length is necessary to capture long run cyclical effects. Keywords: Autoregressive modelling; ARIMA forecasting; Ferrous scrap; Seasonality; Sector modelling
* Corresponding author. Tel. (161) 745-5000; fax: (161) 745-7027. ~The financial support and advice of British Steel Engineering Steels is warmly acknowledged. We are most grateful for the research assistance of Martin Krill and the welcome comments of Professor P.H. Franses and three conscientious but anonymous referees. The David J. Joseph Company gave valuable advice on the working of the US scrap market.
1. Introduction
Iron and steel scrap is a basic raw material for the steel and foundry industries. Some 350 million t of ferrous scrap are consumed worldwide in a peak year such as 1989 with a value at current prices of US$40 billion. Scrap comes from old buildings and industrial machinery.
0169-2070/96/$15.00 © 1996 Elsevier Science B.V. All fights reserved PII S0169-2070(96)00669-3
346
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
Processors extract steel scrap from junked cars and discarded consumer durables. Scrap is also generated during manufacturing operations, both within steelworks themselves and from processes such as car industry press lines. The ferrous scrap market is arguably the world's largest market for recycled material. There is a growing international trade in scrap. The mature industrialised economies are the main exporters. Over 39 million t of scrap were traded internationally in 1992. The main trade flows are from northern Europe into Spain, Italy and Turkey and from the USA outwards to Europe and towards the newly industrialising countries of the Pacific rim. But, for the most part, scrap supplies are predominantly local, with transport costs representing 10-20% of a steel mill's scrap purchasing budget at current prices (Wulff, 1995). The market for ferrous scrap is highly competitive. World scrap prices react quickly to shifts in demand and supply (Anderson, 1987; US Bureau of Mines, 1993). The scrap industry has a tiered structure, with a multitude of small collectors supplying a much smaller number of medium sized yards, which in turn feed just a few large dealers, as described in International Iron and Steel Institute (IISI) (1987, Chapter 2). Although supply is concentrated in the hands of a few major dealers in some countries, no single buyer or seller is able to influence the world price. World scrap prices are set in the main trading area which is the north-eastern quarter of the USA. The US rust-belt is a major scrap reservoir for American mini-steelplants and for the export trade from the Eastern Seaboard and out of the Great Lakes. Prices in other major markets, such as the UK, are led by inter-dealer trading for export cargoes. Daily fluctuations of the dockside price provide a lead indicator for regular domestic contracts between dealers and scrap users of longer duration. There is no futures market for scrap, apart from a brief experiment in the 1950s on the Chicago Mercantile Exchange, as discussed by Stone (1977). The Chicago Board of Trade applied for permission to trade scrap steel in 1990, but nothing has come of the idea so far.
Interview evidence suggests some US processors speculate by accumulating or running down stocks. Otherwise scrap dealing displays the features of a typical commodity market. In Section 2 of this paper we discuss the cyclical, seasonal and trend pattern of US scrap prices. Section 3 outlines the different models we use to capture seasonal effects in the data. In Section 4 we estimate and choose between models on the basis of statistical tests. In Section 5 we report some forecast comparisons. Section 6 concludes the paper.
2. The behaviour of scrap prices
Over time, scrap price movements display three characteristics. There is a marked year-toyear cyclical fluctuation. There are seasonal patterns within each year. Finally there is an underlying trend in the behaviour of prices. We examine each of these in turn.
2.1. Cyclical behaviour Scrap is a key raw material in part of the steel industry, typically accounting for a third of the cost of a ton of wire rod made in a US ministeelworks. A strong cyclical pattern is observable in the year-to-year behaviour of scrap prices. Fig. 1 shows the longest available price
US Scrap Price: No. 1 Heavy Melting Nominaland Real (1987) US$ ,,
I
:300 240 D)
~) 180
~120 ffJ 0 1900
....
o."~)-.,~ . . . . . .
1920
,,..°.d
1940
Year
1960
2000
1980
I --real (1987) price -.nominal price
I
Fig. 1. The long run behaviour of scrap prices.
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
series (US Bureau of Mines, 1993). Typically, peak year prices are 5 years apart and closely follow the cyclical pattern of steel production in the USA. Since the USA is the world's largest free scrap market and the world's largest scrap exporter, global prices move in line with the American market. As a result, high scrap prices arising during a boom period in the USA may coincide with recession in other major steel markets. Prices are volatile over the cycle. Ferrous scrap prices regularly double and halve. US No. 1 Heavy Melting scrap sold for 56% more at the end of 1993 compared with the end of 1992, for instance.
2.2. Seasonal behaviour
There are two reasons for expecting the scrap market to be influenced by seasonal effects; the weather and industrial production. The first seasonal factor is the weather: market gossip suggests scrap prices rise the day the Great Lakes freeze in North America and drop back when navigation resumes. Portions of the Great Lakes freeze every year. Until recently, Detroit and Chicago were major supply centres for US scrap exports and the end of Great Lakes' navigation for the year curtailed supply to the world market. The frozen lakes also stop the movement of iron ore to the integrated steelplants on the lake shore. Scrap is a substitute for molten iron in basic oxygen steelmaking converters. The official shipping season ends with the closure of the Sault Ste. Marie locks between Michigan in the USA and Ontario in Canada on January 15th, and resumes with their reopening on March 25th, see Swanson (1994). An unduly severe freeze delays the re-opening of the locks which causes an unanticipated run-down of iron ore stocks prompting higher demand for scrap as a substitute raw material in the Great Lakes region. Such a severe freeze was experienced in the winter of 1993/4 and the US No. 1 Heavy Melting scrap price fell $2.33 per long ton in the first week of April 1994 when the Great Lakes reopened for navigation. In truth, navigation across the Great Lakes is
347
just one climatic factor influencing scrap supply. Severe winters affect rail transport within the United States. Flooding, bridge wash outs and blizzards all disrupt the flow of rail cars from scrap stockyards to steelmakers. Scrap inventories in the hands of dealers fall as the onset of winter inhibits collection of obsolete scrap. Conversely, the return of better weather produces a 'spring flush' of junk cars, discarded durables and demolition scrap. Summer heat is not conducive to scrap collection, but the autumn produces further material for recycling from the highways and byways. So there are spring and autumn peaks in scrap supply. The second seasonal factor is the annual rhythm of industrial production which prompts shifts in demand for scrap. Inventories of both raw materials and steel products are run down in anticipation of the end of the fiscal year in December and rebuilt as the US fiscal year gets under way in January. So there is a fall off in scrap demand in the last quarter of the year and a surge in the first quarter as steelmakers expand output and rebuild their scrap stocks. Scrap consumption drops in the USA in July and August as holidays, annual maintenance schedules and model changes reduce steel and foundary production in the summer months. The summer slow-down is a feature apparent across all OECD economies, but the rise in US scrap demand during the winter stands in contrast to the widely reported trough in virtually all aspects of economic activity in the first quarter of the year for both the USA (Barsky and Miron, 1989) and other OECD economies (Beaulieu and Miron, 1992). Seasonal influences appear in various ways. Steel scrap prices rise systematically in the winter and fall in the summer and autumn, see Marcus and Kirsis (1994, Fig. 52). We might expect the volatility of prices to vary over the year, too. That is to say, the variance of the scrap price series is not constant over the year, but depends on the season. We anticipate seasonal heteroscedasticty. Previous studies of the US scrap market such as Barnett and Crandall (1986) ignore seasonal factors, preferring to use annual data instead.
348
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
2.3. Trend
Over the long run, US scrap prices are a non-stationary series. There has been a gradual secular decline in real scrap prices. The real annual price of the US No. 1 Heavy Melting Grade has dropped by 0.65% year -1 compound since 1909. The nominal price trends upwards by 2.5% year -1 over the period 1907-1993 due to inflation. The quarterly price series used here is collected by a trade journal, American Metal Market. Scrap is traded in a wide variety of grades. The leading price is for No. 1 Heavy Melting Grade steel scrap. This is the most widely traded grade of scrap. The price for No. 1 Heavy Melting Grade steel is closely matched by equivalent grades elsewhere, such as OA scrap in Britain or No. 1 Old Scrap in Germany. American Metal Market tracks the price of this No. 1 Heavy Melting Grade in three US cities, currently Chicago, Pittsburgh and Philadelphia and publishes an average or composite price. We use the log of the No. 1 Heavy Melting Grade composite price per long ton as our series for analysis, with the aim of forecasting the future scrap price. We take an arithmetic average of monthly price data to form quarterly observations. We lack data on the volume of transactions at each price which would enable us to calculate a more appropriate weighted average. We assume the Great Lakes' freeze coincides with the first quarter and the summer production pause occurs in the third quarter of the year. The data runs from the first quarter of 1954 to the present. Before this start date, there were price and export controls in the US market. Nor can we be sure that subsequent data is unaffected by government regulation. For instance there was a brief episode of price control during the boom of 1973 and export restrictions in 1973 and 1974. Our forecasting models are estimated using quarterly data from 1954 quarter one up to 1989 quarter four. Having specified our model we use it to 'forecast' the price series for the period 1990 quarter one through to 1995 quarter one to see how well it performs. Our nominal scrap price series is characterised
by an upward jump in 1974, quarter one. This is visible in Fig. 1. The mean shift is also statistically significant. It is tempting to attribute this permanent rise in the nominal price of scrap to the rise in inflation experienced across all OECD economies after 1973. It has been suggested that the 1973 oil price shock was a once-and-for-all break in trend for the US economy at a macroeconomic level (Perron, 1989). However, 1973-74 also represented a climacteric in the fortunes of 'big steel' - both in North America and world wide (IISI, 1980). This had conflicting effects on the scrap ~market. US raw steel production peaked in 1973 and declined thereafter. By itself this would cause a fall in the derived demand for scrap with a consequent fall in price. However, the drop in steel output also prompted changes in production techniques. The switch in steelmaking processes caused demand for purchased scrap to rise. Briefly, the collapse in steel output pushed the least efficient steel plants into retirement. Typically these were open hearth works with an iron ore feedstock. These obsolete plants relied on a high level of 'circulating scrap' generated in-house as a consequence of poor production yields from older processes such as ingot casting. Electric arc furnace based mini-mills took their place. These more modern plants relied on continuous casting and generally enjoyed much higher yields of finished steel from raw materials. They relied on scrap purchased from the market. The share of electric arcs in US steel production doubled in the USA from 18% of crude steel production in 1973 to 38% by 1993. There were corresponding gains in yield among the surviving integrated plants, too. These relied on a mixed molten iron and scrap feedstock. Higher production efficiencies meant they also bought in more scrap from the market instead of generating their own supplies inhouse. Consequently, the amount of scrap purchased per ton of steel made in the US rose by 86% in the 2 decades after 1973-74 (Wulff, 1995) even though overall steel production has declined. We allow for a possible change in behaviour of the US scrap market by adding a downward mean shift dummy to our data before 1974
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
349
I Autocorrelation Functions i 1
0.8 0.6 0.4 0.2 0 -0.2
i
-0.4 -0.6
I
i
!
'
'
~ i !
i
E
-0.8
'
i
-1 2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
lag
I ==Q1 iQ2
NQ3 mQ4 i
, ,Partial Autocorrelation Functions i 1 0.8 0.6 0.4
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
lag
I NQ1 ~Q2 iQ3 ~Q4 i Fig. 2. Periodic autocorrelation and partial autocorrelation f u n c t i o n s - levels data.
quarter one. Where seasonal dummy variables are used, four seasonal mean shifts are employed instead. 3. Competing seasonal models 3.1. Deterministic seasonality Use of seasonal dummy variables is the tradi-
tional method of seasonal adjustment in forecasting models using seasonally unadjusted data, for instance the partial adjustment model Owen and Phillips (1987) use to explain British Rail intercity traffic flows. Deterministic seasonality of this sort also accounts for much of the seasonal fluctuation in macroeconomic time series such as real GDP, retail sales or industrial production (Beaulieu and Miron, 1992). Seasonal dummy
350
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
variables may be used to capture a deterministic 'shift in mean' seasonal component in the data. The use of seasonal dummy variables in isolation rests on the assumption that the seasonal component in the data is independent of the trend and cycle components and may be modelled independently. This assumption is not valid if the data is periodically integrated for instance, as Franses and Paap (1994) argue. Deterministic seasonality is just one form that seasonality may take. The arbitrary use of dummy variables to capture seasonal effects without testing for other forms of seasonality may result in mis-specification, as demonstrated for GDP data by Abeysinghe (1994). Instead, deterministic seasonality should be viewed merely as a special case of more comprehensive seasonal models which we now review briefly. 3.2. A periodic autoregressive m o d e l
A relatively new concept that has become increasingly common in the time series literature is that of a periodic autoregressive model. A simple univariate model of this form may be written: Yt = DsS, + P~,lYt-I + " " " + P s , p Y t - p + 8t ;
t = 1. . . . . N ,
(1)
where the D e are seasonal dummy variables and the autoregressive parameters, p~.pS vary with the season of the year, s = 1, 2, 3, 4, for quarterly data. Such models have been used in the context of forecasting by, for example, Osborn and Smith (1989). They find that, when there is periodicity in the data, a periodic autoregressive model performs well relative to other time series models in terms of short term forecasts. See also Tiao and Grupe (1980), Franses (1991) and Osborn (1991) among others. It is common in the literature to write the periodic autoregressive process as a vector autoregression, VAR,
where the 4.1 vector Y r = [YlrY2rY3rY4T ]' with Y,r the observation of y, in season s of year T,P=int(p/4) and the 4.4 matrices ~k are made up of the autoregressive parameters. Let us define the lag polynomial q~(L) = ~0 - qJ1L + • .. + cbeL e, where L is the lag operator such that L J Y T = YT-j. The VAR may be written • ( L ) Y r = E r. The series is stationary if the roots of the characteristic equation ]q~(z)l = 0 are all outside the unit circle. The process has as many unit roots as there are roots of the characteristic equation on the unit circle. The simple periodic model defined in (1) above encompasses the non-periodic model Y t = S s + P l Y , - I + "'" +PpYt-p + et
(3)
which, in the absence of seasonal dummy variables, we can write as ~o(L)y, = et, where ~(L) is a polynomial in the lag operator. In this case, y is stationary if the roots of the equation q~(z) = 0 lie outside the unit circle. We may respecify (1) as (3) if we may accept the null hypothesis: Ho: Ps,j = pj ; s = l , . . . , 4 ,
j = 1,..., p.
(4)
Boswijk and Franses (1993) show that the likeli2 hood ratio test of this null hypothesis has a X distribution with 3p degrees of freedom under the null hypothesis, whether or not Yt has a unit root. If y has no unit roots, this model may display deterministic seasonality. In this case the seasonal effects are constant from year to year and the seasonality can be described using seasonal dummy variables in the model. To determine if such seasonality is present, we may test the null hypothesis H0:61 ~--82 = 83 = 84
(5)
in the usual way. If we accept this null we can be confident that there is no seasonal pattern of this form in the data. 3.3. Seasonal integration
CboYv = cI)IYT-1 + " "
T=I,...
1 ,-~N,
+ ~PYT-P + ET ;
(2)
Let us again consider the characteristic equation of the non-periodic model. If the charac-
K. Albertson, J, Aylen / International Journal of Forecasting 12 (1996) 345-359
teristic equation, q~(z)= 0, has a single root of unity we say that y has a unit root at the zero frequency. Such a process must be first differenced to be reduced to stationarity. The first difference operator is defined A = ( 1 - L), with Ly, = Y,-1. Thus, if y has such a unit root, Ay~ -y, - y,_ 1 will be stationary. It may, however, be the case that the characteristic equation has more than one root on the unit circle. Let us consider the case where there are four roots on the unit circle, - 1 and +-i. In this case y requires, not first differencing, but rather seasonal differencing to reduce it to stationarity. For quarterly data, the seasonal difference operator is defined A 4 = ( 1 - L 4 ) . Thus A a y t -= y, - Yt-4" In this case y, is said to be seasonally integrated or to display stochastic seasonality. Testing for the presence of such a seasonal integration process is typically carried out using the H E G Y test of Hyllebert et al. (1990). This test, or more correctly, group of tests, is based on the decomposition (1 - L 4) = (1 - L)(1 + L)(1 - L2). We write q~*(L ) A 4 y , = rqyt. , 1 J~- "/7"2Y2,t-1 + 7r3Y3,t-2 + 7"r4Y3,t-i + e t ,
(6)
possibly including seasonal dummy variables, a linear trend, a drift term and/or lags of the dependent variable to whiten the errors. The dependent variables are defined y~, = (1 + L + L2 + L3)yt, Y2,t = - ( 1 - L + L 2 -'L3)yt and Y3,, = - ( 1 - L2)yt. The null hypothesis of a unit root at the zero frequency is given by HI: 7rI = 0 vs. H~A: 7q < 0 . Similarly, the null of unit roots at the biannual and two semiannual (seasonal) frequencies, are given by H2:~'2 = 0 vs. HaATr2 < 0, H3: "rr3 = 0 VS. H3ATr3 < 0 and H 4 : 7r4 = 0 vs. H4A: "/7"~ 0, respectively. The tests are t type tests and critical values are given in Hylleberg et al. Note that an underlying premise of the H E G Y test is that the data is generated by a non-periodic process. It has been shown, see for example Osborn et al. (1988), that the application of the test to periodic data may lead to spurious results. If the data is non-periodic and we find season-
351
al integration in the data, we may apply a seasonal A R I M A ( p , d , q ) ( P , D, Q)4 to the series. The use of the moving average, S A R I M A , error term may enable us to reduce the lag length in the autoregression. This process may be written p(L) Fp(L 4)(A d A D 4Yt) = Oq(L )OQ(L 4)Et ,
('7)
where d and D are the degree of first and seasonal differencing respectively, O, ~, 0 and O are lag polynomials of orders p, P, q and Q respectively. If seasonal differencing is not appropriate, the above seasonal A R I M A process is nested within a simple A R I M A ( p * , d , q*) model, with p* = 4P + p and q* = 4Q + q.
4. Specification, estimation and testing The most general model of those considered above is the periodic model and, indeed, the other models will be mis-specified if the data series is periodic. Hence we must first determine whether or not the parameters of the model vary seasonally. To do this we must first determine the appropriate lag length to use in the autoregression. Now, Tiao and Grupe (1980) show that when the true model is periodic, the use of sample autocorrelations and partial correlations is misleading in terms of specifying the lag length of the model. We use periodic sample correlations and autocorrelations. These are estimated by OLS regression of each of the four series (one for each quarter) on their own and each other's lags. The kth lag periodic autocorrelation coefficients may be calculated from the regression •
Ys T =
+
k
(8)
where Lq is the one quarter shift operator, fY(,-,),r; Lq(gs
,1") = ]l y4,(T_l);
s = 2, 3, 4 S: 1 .
(9)
Similarly, we estimate the lag k periodic partial autocorrelation coefficients, ~'sk, using
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
352
in Fig. 2. From an inspection of the periodic partial autocorrelation functions, we specify six lags in the periodic autoregression, PAR, defined in (1) above. An inspection of the residuals from this regression indicates that this is sufficient to whiten the model errors. This is borne out in the correlation function of the regression residuals. We may determine whether or not we need to continue to allow for periodically varying parameters using the likelihood ratio test suggested by Boswijk and Franses (1993). Before any such
k
Ys,T= o,,+Y, i=1
' ~s,iLq(Ys,T).
(10)
We may approximate the standard errors of the periodic autocorrelations and partial autocorrelations by 2/~/N (where we have N/4 years of data). The periodic autocorrelation and partial autocorrelation functions suggest that our data is integrated as there is a spike of roughly one in magnitude for each of the four quarters as shown
Autocorrelation Function First Differences 1 0.8 0.6 i
0.4 0.2
r
i
i
:
i
i
;
!
.
}
:
:
i
! ;
i
0
-0.2 -0.4
-0.6 -0.8
i
-1 1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
lag Partial Autocorrelation Function First Differences 1 0.8 0.6 0.4
0.2 0
m
m I
]
:u
NTN--
-012
m-m
m__,m--
--mm
m -
-0.4 -0.6 -0.8
-1 1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
lag Fig. 3. A u t o c o r r e l a t i o n
and partial autocorrelation
functions-
first d i f f e r e n c e d d a t a .
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
353
[ Out of Sample Forecasts I 150 130 ,-. 110 o -'-' 90 70 50 30 10 55
;
60"
I
65
I
I
70
I-,~,~
i
I
75 year
I
I
80
;
I
85
I
i
90
95
, .,,,.,,(,2.,.~,., ,~'"~'~:'"~|
150 130 c 110 o -.-, 90 70 50 30 10 55
l
60
I
65
I
70
I-.s~=
I
I
75 year • -. ~,~.~.m
I
I
80 • ~.~.o~
;
i
85
I
90
i
95
i
Fig. 4. Out of sample forecast comparisons for US scrap price.
testing, however, we must determine whether or not the error variance is the same in each quarter as, if it is not, any subsequent testing will be invalid. Using the residuals from this estimation procedure, we may estimate, 0.s the standard error in each quarter; d"1 = 0.0978, &2 = 0.0884, O'3 = 0.0847 and &4 = 0.1347, where ~', represents the estimated standard error in the sth quarter residuals. Adding the sum of squared residuals from each individual quarter and dividing by the
total degrees of freedom gives the standard error of the combined model, 6- = 0.105. A likelihood ratio test of the null hypothesis, H0:0.1 = ° 2 - % -- 0-4 against a two sided alternative gives us a test statistic, u = 7.98. This statistic has a X 2 distribution with three degrees of freedom under the null. The critical value for a test of size 5% is 7.81. Thus we may be 95% confident that there is seasonal heteroscedasticity in the data. We must correct for this heteroscedasticity as we proceed to test the hypotheses H0: "the lag
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
354
parameters vary with the seasons" vs. HA: "they do not vary" as the model is non-periodic under the null hypothesis. We do this by rewriting the PAR Yt
O's
s=l
D*
O"s + E Ps,jYt-j
+"-~s'
j=l
(11)
with D* = Ds/(r~. An F version of the likelihood ratio statistic for testing the null H0: Ps,j =Pj for j = 1 . . . . . 5, takes on the value u = 0.98, thus we accept the null and can specify the nonperiodic model which, modified for seasonal heteroscedasticity, can be written 4
P
4
Yt_ E o.O* + E Oji~=lD* + et O's s=l s j=l "= iYt-j --;-O.s
(12)
We use the H E G Y test to determine whether or not seasonal differencing, possibly in addition to first differencing, is necessary to reduce the series to stationarity as our model is non-periodic. The H E G Y regression, allowing for the heteroscedasticity, can be written 4
4
Yt A4-~ = 7r, ~] D *sYI,t-1 + 7r2 E D*iY2.t-1 s=l
s=l
4
4
+ 77"3E D*y3,t-2 + Tl'4 E D*y3,t-1 s=l s=l P* 4 Et
+ ~. ~ D*A4Y,_I +--~. i=l s=l
(13)
ors
We find that two lags on the dependent variable, A4yt/o's, are sufficient to whiten the residuals. The value of the H E G Y test statistics for the levels series are given in Table 1. It is worth noting, in passing, that if the analysis is carried out without seasonal dummy variables the H E G Y test rejects unit roots at all frequencies. The mis-specification caused by omission of seasonal effects biases the test statistics. Our preferred model at this stage includes quarterly dummies, hence we consider the critical values given in Hylleberg et al. (1990) relating to this case. Recall, however, that in addition
to seasonal dummies we have also allowed for a mean shift and seasonal heteroscedasticity. Thus we must interpret these critical values with some care as the true critical values for this situation are likely to be greater in absolute value than those given (c.f. Perron, 1990, who considers the implications of allowing for a mean shift in conducting a Dickey Fuller test for stationarity). None-the-less, it is clear that we may be confident that the series does not have unit roots at seasonal frequencies, as the test statistic values are considerably less than the critical values tabulated. The absence of unit roots at some or all seasonal frequencies is a familiar result for aggregate data, too (see for instance Beaulieu and Miron, 1993). Our results reinforce the conclusion that arbitrary use of seasonal filters is likely to result in model mis-specification when forecasting a time series. We are confident that the series may have a unit root at the zero frequency as the test statistic for this frequency is not significantly different from zero. These conclusions are borne out by an examination of the autocorrelation function of the levels and first differenced data. Based on the autocorrelation function and partial autocorrelation function of the differenced data, we specify an ARIMA(5,1,0) model. The diagnostic test statistics indicate that the residuals from this model are white noise. However, forecasting experience in the industry suggests much longer lag lengths may be necessary to capture the cyclical fluctuations in the scrap price series. For example, inspection of the autocorrelation functions show a further spike at the eleventh lag, Fig. 3, although this is not significant. We propose an additional
Table 1 H E G Y test statistics in q u a r t e r l y d a t a o n No. M e l t i n g Scrap c o m p o s i t e price
1 Heavy
Levels data R o o t , zri T e s t statistics Critical v a l u e s
zri -2.511 -3.56
7r2 -6.664 -3.49
zr3 -5.309 -4.06
rr4 -6.02 -2.72
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
355
summer appears to have no impact on prices, perhaps because seasonal supply reductions match the fall in demand. Prices tend to be lower in the autumn as supplies pick up and buyers run down stocks towards the end of the financial year (although the coefficient on quarter four is not significant so this may be a chance result). These results differ from studies of seasonality in macroeconomic variables. At an aggregate level, the run up to Christmas and the summer holiday period strongly affect macroeconomic activity, regardless of the climate. In the scrap market, climate matters.
ARIMA(12,1,7) model alongside our preferred ARIMA(5,1,0) model. Having re-specified the preferred model using first differenced data and allowing for heteroscedasticity, we may test to see if the seasonal dummy variables add significantly to the explanatory power of the model. The null hypothesis, H0:61 = 62 = ~3 : (~4 VS. a two sided alternative is rejected for any reasonable level of significance. Our final model then is the simple non-periodic ARIMA(5,1,0) with seasonal dummy variables. The estimated coefficients and heteroscedasticity adjusted standard errors are given in Table 2. These results show two of the four seasonal dummies are statistically significant. Prices tend to rise in the winter, but fall back in the spring. Recall, we are modelling using the first differences of logged data. Consequently, the coefficients in Table 2 suggest scrap prices are 2% greater than might otherwise have been expected in quarter one, but 1.3% less in quarter two. The
5. Comparison of seasonal models To determine how well our final model forecasts, compared to other seasonal models, we conduct a series of rolling regressions and calculate out-of-sample forecasts for a number of similar time series models that differ in their
Table 2 Estimation results - ARIMAd(5,1,0) with seasonal dummies on No. 1 Heavy Melting Scrap composite price, 1954ql to 1989q4 Quarterly dummy
61
6_,
63
64
Estimate Standard error
0.02008 0.00567
-0.01273 0.00536
0.00052 0.00473
-0.00407 0.00641
Lag coefficient
p~
P2
P3
P4
P~
Estimate Standard error
0.20288 0.08711
-0.03997 0.08681
0.01256 0.08788
-0.02277 0.09318
-0.22319 0.08449
Box
Pierce
Stats.
Lag
Stat.
X 2 Dof
6 7 8 9 10 11 12 1 14 15
1.17 1.91 1.92 3.11 5.95 7.89 8.00 8.64 9.46 9.59
1 2 3 4 5 6 7 8 9 10
R 2 (differenced data), 27.41; LM test for heteroscedasticity, 0.001 X~ under null; ARCH test, 0.648 X~ under null; Jarque Bera normality test, 5.421 X~ under null.
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
356
Table 3 Forecast performance of time series models of the scrap price Model
I step MAPE Q1
1 2 3 4 5 6 Model
i step RMSE
Q2
7.83 5.08 6.92 6.01 8.87 4.56
6.09 3.53 2.94 3.65 4.03 3.50
Q3 7.10 5.54 7.80 6.12 4.25 4.71
Q4 6.79 7.38 7.94 7.09 9.78 7.74
Overall 6,95 5.48 6.47 5.78 6.88 5.25
2 step MAPE
Q1
Q2
8.62 6.06 7.74 7.39 10.82 6.19
6.78 4.22 3.34 4.37 5.15 4.15
Q3
Q4
8.71 7.33 10.50 7.67 6.30 5.52
7.19 9.02 8.81 9.23 11.74 9.32
Overall 7.84 7.00 8.08 7.48 9.11 6.73
2 step RMSE
Q1
Q2
Q3
Q4
Overall
Q1
Q2
Q3
Q4
Overall
1 2 3 4 5 6
8.38 7.76 9.34 9.49 13.63 5.73
12.11 9.61 9.59 9.03 6.76 7.41
10.17 12.97 11.95 13.72 9.99 11.32
11.68 13.17 9.75 12.07 20.12 10.28
10.59 10.88 10.16 11.08 12.62 8.68
10.63 8.31 10.25 9.88 15.07 8.13
13.91 11.60 11.94 11.37 11.27 10.86
13.26 13.99 15.50 14.62 12.19 12.85
13.39 14.52 12.15 13.67 23.01 12.31
12.86 12.35 12.61 12.53 16.06 11,19
Model
4 step MAPE
4 step RMSE
Q1
Q2
Q3
Q4
Overall
Q1
Q2
Q3
Q4
Overall
1 2 3 4 5 6
9.98 12.04 14.39 12.84 16.39 9.59
16.66 14.69 8 ",a 14.~_ 13.65 12.08
12.90 17.41 10.16 17.33 23.25 15.43
14.72 17.99 19.02 17.24 26.63 17.26
13.43 15.48 13.37 15.45 20.15 13.57
12.00 13.65 17.39 15.26 20.18 12.08
19.56 16.63 9.00 15.56 17.35 14.24
14.83 20.81 11.70 20.10 26.50 19.84
15.03 22.71 22.00 22.78 33.36 23.95
15.39 18.78 16.34 18.77 25.40 18.23
Model
8 step MAPE
8 step RMSE
Q1
Q2
Q3
Q4
Overall
Q1
Q2
Q3
Q4
Overall
1 2 3 4 5 6
24.72 24.69 24.33 25.16 18.12 25.90
25.93 20.39 10.96 20.67 35.87 24.92
20.07 27.40 13.82 28.02 46.37 31.55
24.03 32.03 26.5l 32.28 42.46 33.16
23.79 26.45 19.83 26.84 34.93 28.98
25.34 26.99 30.88 28.11 27.21 30.36
26.42 25.11 17.87 23.98 37.11 26.44
22.15 29.81 17.09 29.50 47.27 32.59
27.49 32.35 29.80 32.85 44.48 35.96
25.57 28.85 25.64 29.05 39.38 31.78
Model
12 step MAPE
1 2 3 4 5 6
12 step RMSE
Q1
Q2
Q3
Q4
Overall
Q1
Q2
Q3
Q4
Overall
26.35 24.41 27.85 25.00 24.48 33.12
8.79 19.04 19.61 17.39 54.82 16.82
20.02 18.77 24.59 17.93 62.33 20.49
22.35 20.58 13.16 22.32 44.63 28.16
20.37 21.06 21.14 21.26 44.16 25.85
33.35 25.27 30.78 27.11 25.71 40.00
10.30 19.86 19.61 17.42 58.51 23.06
22.26 18.77 24.59 18.45 66.12 25.68
23.01 23.97 19.28 24.55 45.45 29.98
24.76 22.66 24.36 23.02 48.76 31.43
Model 1, ARIMAd(12,1,7); Model 2, ARIMAd(5,1,0); Model 3, ARIMA(12,1,7); Model 4, ARIMA(5,1,0); Model 5, SARIMA(5,1,0)(0,1,0),; Model 6, PARI(5,1).
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
treatment of seasonality. The models we consider are a simple ARIMA(5,1,0), an ARIMA(5,1,0) with seasonal dummies, we will denote this ARIMAd(5,1,0), a SARIMA (5,1,0)(0,1,0)4 model and a periodic autoregressive model of order 5 on the first differenced data, PARI(5,1) model. For pragmatic reasons we include a long length model, an A R I M A (12,1,7) and the related ARIMAd(12,1,7) with seasonal dummies. This range of models captures a variety of approaches to modelling seasonality, to wit, no allowance for seasonality, deterministic seasonality, stochastic seasonality and periodic seasonality. As we have seen, we may reject all these seasonal specifications except deterministic seasonality and hence we expect models incorporating this to outperform the others in forecasting. Out-of-sample forecasts were calculated for all of these models on a rolling basis. That is to say, we estimate each model for the period ending 1989q4 and generate one-step ahead forecasts for 1990ql. We extend the sample period by one observation, re-estimate the models and generate one-step ahead forecasts for 1990q2 and so forth. A similar procedure is used to estimate rolling two-step ahead forecasts. Over a variety of forecast horizons we calculate the mean absolute percentage error, MAPE, and the root mean squared error, RMSE, of the forecasts. Illustrative results are shown in Table 3. The results are tabulated according to the quarter that is forecast and the model. All models often forecast the fourth quarter less well than the other three quarters. This is shown by larger MAPE and RMSE values for the Q4 column compared to the other three columns. There are clearly effects in the market in that fourth quarter that are not captured as well as the effects in other quarters. Comparing the forecasts made by the ARIMAd(5,1,0) model with dummies, Model 2, to those made by the ARIMA(5,1,0) model without, Model 4, we see that the use of dummies generally improves the forecast accuracy in quarter one, which is the quarter in which the seasonal effect is strongest. In terms of overall accuracy the ARIMAd(5,1,0) model outper-
357
forms the ARIMA(5,1,0). The comparison between the ARIMAd(12,1,7) and the ARIMA(12,1,7) is less straightforward, but, for longer range forecasting, the model incorporating dummies tends to do better. The periodic PARI(5,1) model does better than the non-periodic models at short range forecasting but the performance of this model deteriorates as the forecast horizon lengthens. Given that we have evidence that the data is not periodic, this deterioration in accuracy is to be expected as this model is likely to be overparameterised. Conversely, the SARIMA model always performs worst of the models considered. Again, this is not surprising as we have seen that seasonal differences are not required to make the data stationary, thus this model is overdifferenced. It must also be emphasised that measuring forecasting error alone does not indicate which model best captures the 'shape' of the future out-turn. Evidence on forecasting errors needs to be read alongside visual evidence on fluctuations and turning points in the data. To this end, we plot some of the forecasts generated by the alternative models in Fig. 4. Despite poorer short range forecast performance, the ARIMAd(12,1,7) captures the longer run cyclical swings in a far more satisfactory way than other models. Even with these longer lag lengths, taking account of seasonal effects substantially improves the forecasting performance of this model, especially beyond 3 years. Omitting valid seasonal effects has implications for the estimation of the model, with consequences for long run forecasts.
6. Conclusion
This paper demonstrates that seasonal effects are at work in the US market for ferrous scrap. The significance of seasonal dummies in our preferred forecasting equation suggests seasonal weather and the annual pattern of industrial production affect this particular commodity market. Seasonal effects are strongest in the first quarter of the year. Scrap prices rise by 2% in
358
K. Albertson, J. Aylen / International Journal of Forecasting 12 (1996) 345-359
the winter only to fall back by 1.3% in the spring. The market folklore that the Great Lakes' freeze affects scrap prices appears to be borne out by our findings in a simple and direct way. Despite the simple specification of the seasonal component, we may have confidence that this specification is valid as it was reached on the basis of a broad general to specific methodology. There is nothing to suggest that more complicated models are required to capture the seasonal component of the data. We have considered and rejected them in favour of our simpler model. Comparison of the forecasting performance of alternative models suggests that those with shorter lag lengths perform very well within the sample period, but a model with much longer lags gives better out-of-sample forecasts. A model with long lag lengths captures the wide cyclical swings in the US scrap market. Here, as in many cases, taking account of seasonal effects improves forecasting accuracy.
References Abeysinghe, T., 1994, Deterministic seasonal models and spurious regressions, Journal of Econometrics, 61, 259272. Anderson, D.F., 1987, Scrap as a commodity, Ironmaking and Steelmaking, 14(3), 130-135. Barnett, D.F. and R. Crandall, 1986, Up from the Ashes: The Rise of the Steel Minimill in the United States (Brookings, Washington, DC). Barsky, R.B. and J.A. Miron, 1989, The seasonal cycle and the business cycle, Journal of Political Economy, 97, 503534. Beaulieu, J.J. and J.A. Miron, 1992, A cross-country comparison of seasonal cycles and business cycles, Economic Journal, 102, 772-788. Beaulieu, J.J. and J.A. Miron, 1993, Seasonal unit roots in aggregate U.S. data, Journal of Econometrics, 55, 305328. Boswijk, H.P. and P.H. Franses, 1993, Unit Roots in Periodic Autoregressions (Department of Actuarial Science and Econometrics, University of Amsterdam) mimeo. Franses, P.H., 1991, Seasonality non-stationarity and the forecasting of monthly time series, International Journal of Forecasting, 7, 199-208. Franses, P.H. and R. Paap, 1994, Model selection in periodic
autoregressions, Oxford Bulletin of Economics and Statistics, 56, 421-440. Hylleberg, S., R.F. Engle, C.W.J. Granger and B.S. Yoo, 1990, Seasonal integration and cointegration, Journal of Econometrics, 44, 215-238. International Iron and Steel Institute (IISI), 1980, Causes of the Mid-1970's Recession in Steel Demand (Committee on Economic Studies, International Iron and Steel Institute, Brussels). International Iron and Steel Institute (IISI), 1987, Scrap and the Steel Industry: Trends and Prospectsfor Solid Metallics (Committee on Raw Materials, International Iron and Steel Institute, Brussels). Marcus, P. and K.M. Kirsis, 1994, Global metallics crisis, International Iron and Steel Institute, Annual Conference, Colorado Springs, October 4. Osborn, D.R., 1991, The implications of periodically varying coefficients for seasonal time series processes, Journal of Econometrics, 48, 373-384. Osborn, D.R. and J.E Smith, 1989, The performance of periodic autoregressive models in forecasting seasonal U.K. consumption, Journal of Business and Economic Statistics, 7, 117-127. Osborn, D.R., A.P.L. Chui, J.P. Smith and C.R. Birchenhall, 1988, Seasonality and the order of integration for consumption, Oxford Bulletin of Economics and Statistics, 50, 361-377. Owen, A.D. and G.D.A. Phillips, 1987, The characteristics of railway passenger demand, Journal of Transport Economics and Policy, 17(2), 133-153. Perron, E, 1989, The great crash, the oil price shock, and the unit root hypothesis, Econometrica, 57, 1361-1401. Perron, E, 1990, Testing for a unit root in a time series with a changing mean, Journal of Business and Economic Statistics, 8, 153-162. Stone, J.K., 1977, The iron and steel industry and the commodity market, Iron and Steel Manager, February, 36-38. Swanson, S., 1994, Lake ice freezes factories' plans, Chicago Tribune, March 27, Section 7, 3. Tiao, G.C. and M.R. Grupe, 1980, Hidden autoregressivemoving average models in time series data, Biometrica, 67, 365-373. US Bureau of Mines, 1993, Metal Prices in the United States Through 1991 (United States Department of the Interior, Bureau of Mines, Washington, DC). Wulff, S.W., 1995, Ferrous Scrap: an involuntary by-product that has become an essential raw material, presented at AISE Conference on Future of Flat Rolled Steel Production, Chicago, June (The David J. Joseph Company, Cincinnati) mimeo.
Biographies: Kevin ALBERTSON is Lecturer in Econometrics in the Department of Economics at the University of Salford. Dr Albertson took his first degree in maths and operations research at the University of Canterbury in Christchurch, New Zealand before joining the New Zealand
K. Albertson, J, Aylen / International Journal of Forecasting 12 (1996) 345-359 Department of Statistics for 2 years. He subsequently took a first class Masters Degree in Economics and completed a Doctorate on econometric theory. Jonathan AYLEN was educated at the University of Sussex and Worcester College, Oxford. He is now Senior Lecturer in
359
Economics at the University of Salford, where he has taught since 1974. Mr Aylen's published papers in economics cover cost-benefit analysis, international comparisons of performance and privatisation. He has travelled widely throughout the world steel industry and advised a number of national and international organisations on steel issues.