NONLINEAR DYNAMICS AND STRUCTURAL … Anderson, 1992; van Dijk and Franses, ... ﬁable cyclical behavior (Labys, Kouassi, and Terraza, 2000) and, as well, ... porting the products

NONLINEAR DYNAMICS AND STRUCTURAL CHANGE

IN THE U.S. HOG–CORN CYCLE: A TIME-VARYING

STAR APPROACH

MATTHEW T. HOLT AND LEE A. CRAIG

The linearity of the U.S. hog–corn cycle has been questioned by Chavas and Holt (1991). Even so,

attempts have not been made to model the potential nonlinear dynamics in the hog–corn cycle by using

regime-switching models. One popular alternative is Terasvirta’s smooth transition autoregressive

(STAR) model, which assumes regime switching is endogenous and potentially smooth. In this article,

we examine monthly data for the U.S. hog–corn cycle, 1910–2004. A member of the STAR family, the

time-varying STAR, is fitted to the data and its properties examined. We find evidence of nonlinearity,

regime-dependent behavior, and time-varying parameter change.

Key words: hog–corn cycle, nonlinearity, structural change, time-series models.

In recent years, there has been renewedinterest in empirical business cycle research.While the motives for this resurgence may vary,there is little doubt that two fundamentally re-lated reasons underlie much of this recent re-naissance. One is that economists have longobserved that contractionary and expansion-ary phases of the business cycle are qualita-tively different. Keynes (1936), for example,provides observations on properties of busi-ness cycles that are consistent with notionsof asymmetry, suggesting that contractions areshorter and more turbulent than expansions.An immediate implication is that the under-lying process governing business cycle behav-ior possesses features that cannot be capturedby linear models alone. But not until recentlyhave economists developed tools, most typi-cally in the form of regime switching models,capable of depicting asymmetric behavior inbusiness cycles (Neftci, 1994; Falk, 1986). This

Matt Holt is professor, Departments of Agricultural and ResourceEconomics and Economics, North Carolina State University. LeeCraig is Alumni Distinguished Professor, Department of Eco-nomics, North Carolina State University.

We thank seminar participants at North Carolina State Uni-versity, Purdue University, the Stockholm School of Economics,Ball State University, Eastern Carolina University, and the Trian-gle Universities Economic History Workshop for useful commentson earlier drafts. We also thank Atsushi Inoue, Jean-Paul Chavas,Bruno Eklund, the editor, Chris Barrett, and two anonymous ref-erees for their numerous constructive comments. We are especiallygrateful to Timo Terasvirta and Dick van Dijk for their many usefulsuggestions and patient advice. Of course the usual caveat applies.Finally, we thank the North Carolina Agricultural Research Ser-vice and the North Carolina State International Programs Officefor supporting this research.

is the second reason for renewed interest inthis line of research.

Regime switching models are generally cat-egorized as being one of two types. First,there are Markov-switching models whereinregimes are determined by an unobserved andexogenous state variable. Alternatively, thereis a class of models for which it is explic-itly assumed that the regime switch is en-dogenously determined by an observed statevariable. Models belonging to this later cate-gory include Tsay’s (1989) self-exciting thresh-old autoregression (SETAR) and the smoothtransition autoregression (STAR) proposed byTerasvirta (1994). The STAR has several ad-vantages over the SETAR including that sev-eral standard STAR models nest a SETARrepresentation. STAR models have been usedto model nonlinear features of business cyclesfor developed countries (Ocal and Osborn,2000; Skalin and Terasvirta, 1999; Terasvirtaand Anderson, 1992; van Dijk and Franses,1999).

Aside from potential nonlinearity, consid-erable research has also focused on struc-tural change and time-varying parameters intime-series models (Stock and Watson, 1996).Structural breaks and parameter time varia-tion may occur because of institutional change,an evolving policy environment, or techno-logical innovation. Recently, nonlinear mod-els have been combined with specificationsthat facilitate structural change and parame-ter time variation. Lundbergh, Terasvirta, and

Amer. J. Agr. Econ. 88(1) (February 2006): 215–233Copyright 2006 American Agricultural Economics Association

216 February 2006 Amer. J. Agr. Econ.

van Dijk (2003); Skalin and Terasvirta (2002);and van Dijk, Strikholm, and Terasvirta (2003)for example, combine the time-varying autore-gression (TVAR) of Lin and Terasvirta withSTAR models to obtain a time-varying STAR(TV-STAR) model.

In spite of the emerging popularity of non-linear models in general and STAR models inparticular for depicting aggregate business cy-cles, comparatively little research has focusedon modeling similar attributes for primarycommodity prices, a surprising result giventhat many commodity prices exhibit identi-fiable cyclical behavior (Labys, Kouassi, andTerraza, 2000) and, as well, may be associ-ated with nonlinearity (Davidson, Labys, andLesourd, 1998). There is a need then to inves-tigate the potential of STAR models for cap-turing essential features of primary commodityprice dynamics. Perhaps the longest and mostwidely recognized example of cyclic behaviorin primary commodity prices is the hog–corncycle. Beginning with Haas and Ezekiel (1926),Coase and Fowler (1937), and Ezekiel (1938),numerous studies have attempted to charac-terize the hog–corn relationship, typically byusing linear models (Harlow, 1960; Jelavich,1973; Larson, 1964; Shonkwiler and Spreen,1986; Hayes and Schmitz, 1987). Alternatively,Chavas and Holt (1991) used quarterly data,1910–84, to show that the U.S. hog–corn cyclemight be associated with deterministic chaos.

In this article, we employ for the first timea STAR framework, and more specifically, aTV-STAR, to investigate fundamental aspectsof the U.S. hog–corn relationship. Our work-ing hypothesis is that the hog–corn cycle ex-hibits nonlinear features and time-varying pa-rameters (technical change), and that thesefeatures may be adequately characterized bya smooth transition model. There are severalreasons to believe a priori that a TV-STARframework might prove fruitful. First, as al-ready noted, prior research has found evi-dence of nonlinearities in the hog–corn cy-cle (e.g., Chavas and Holt, 1991). Second, dueto the inherent biological nature of hog pro-duction, it is far easier to sell breeding stockwhen expected profits are low than it is torebuild breeding herds when expected prof-its are large. Third, even if all agents in thepork market are fully rational, it is still possi-ble to observe periodic behavior (Rosen, 1987;Rosen, Murphy, and Scheinkman, 1994), andperhaps even highly complex behavior (Brockand Hommes, 1997). Fourth, and aside fromany price expectation issues, natural animal

population dynamics are capable of giving riseto complex (i.e., nonlinear) behavior in live-stock markets (Chavas and Holt, 1993). Fi-nally, in the postwar period, there has beenconsiderable technological innovation in hogproduction, including movement to total con-finement operations, the advent of nearly con-tinuous breeding-farrowing rotations, the nowwidespread use of antibiotics and growth hor-mones, and enhanced feed conversion andcarcass quality through genetic improvementsand dietary refinement. In the empirical analy-sis, we employ a dataset consisting of monthlyobservations for hog and corn prices, 1910–2004. Among other things, the large sampleaffords sufficient observations to isolate anypotential nonlinear effects and, as well, to ex-plore possibilities for structural change and pa-rameter time variation.

The remainder of the article is organized asfollows. In the next section, we provide a briefoverview of the history of the U.S. hog–corncycle. We then discuss the data and describethe STAR testing-modeling-evaluation cycle.Following this, we summarize model estimateswhen the STAR framework is applied to thehog–corn data and evaluate the results. In thepenultimate section, the dynamics of the esti-mated nonlinear model are explored by usingvarious techniques including sliced spectra andgeneralized impulse response function analy-sis. The final section concludes.1

Corn, Hogs, and the Emergenceof the Hog–Corn Cycle2

In the economics literature, the expression“hog cycle” refers to the correlated—possiblylagged—component of the swings over time inthe hog–corn price ratio. The combination ofswine physiology, with its affinity for corn, andthe inherent logic of supply and demand in-sured that, as long as markets existed for bothcorn and hogs beyond the farm gate, therewould inevitably be hog cycles, and indeed,such cycles were recognized in American agri-culture, albeit initially at the local level, as earlyas 1818 (Buley, 1980). The subsequent histori-ography of American agriculture owes much tothe hog–corn nexus, and can be summarized bythe observations of nineteenth-century British

1 In the article, a number of intermediate results have been omit-ted for space reasons; they are, however, in a technical appendixby Holt and Craig available at http://agecon.lib.umn.edu.

2 This section is an abbreviated version of the history of thehog–corn cycle found in Craig and Holt (2005).

Holt and Craig Nonlinearity and Structural Change in the Hog–Corn Cycle 217

journalist traveling in the United States: “Thehog is regarded as the most compact formin which the Indian corn crop of the Statescan be transported to market,” as quoted inCronon (1991) (pp. 208–09). By the end ofthe nineteenth century, the combination of thetransportation revolution and the economicrelationship between corn and hogs had gen-erated something like a national hog cycle.Economists began to analyze the cycle in thetwentieth century, and two classic articles werededicated to the topic (Coase and Fowler, 1937;Ezekiel, 1938). Economists continue to ana-lyze the cycle’s causes and consequence (see,most recently, the review in Chavas and Holt,1991). The existence and importance of thehog cycle in American economic history stemsfrom at least three related factors: the capacityof swine to convert corn into meat, the impor-tance of swine in American agriculture, andthe sheer size of the U.S. market.

Despite shortcomings as a staple in hu-man diets—corn is low in glutenin andniacin (Collins, 1993; Brinkley, 1994)—cornhas proved to be an ideal device for deliveringcarbohydrates to livestock, and hogs provedto be particularly efficient in converting car-bohydrates into meat. With the rise of a na-tional market in the United States, countlesstravelers and foreign observers noted the hogbecame corn incarnate (Craig and Holt, 2005).In addition to their ability to convert corn intomeat, hogs possessed several biological charac-teristics that contributed to their importance inthe early farm economy relative to, say, cattle.These included early onset of breeding (withinone year of birth), short gestation periods (fourmonths), and large litter size.

Once a sufficient combination of urban-ization and transportation development oc-curred, farmers began producing pork for themarket as opposed to on-farm consumption.Originally valued for its ability to forage, thehog’s subsequent lofty economic status re-quired an off-farm market and transportation“revolution.” Urban consumers provided theultimate demand for farm-produced fat andprotein, but they could be supplied from thehinterland only so long as the cost of trans-porting the products did not itself consumetheir value. As the frontier moved west and thecountry urbanized back east, improvements ingraded roads, followed by the emergence ofcanals and later railroads, farmers further outin the hinterlands not only had relatively low-cost access to urban consumers and world mar-kets, but they also increasingly specialized in

a relatively few products and increased theirscale of production in those lines (Craig andWeiss, 1993).

Without low-cost transportation, early hogcycles were typically quite local in nature, usu-ally centered on a nearby market town, whichdepending on its location, might occasionallybe tied to a broader market, which itself re-flected a trans-village cycle (Buley, 1980). Al-though Cincinnati was the original “Porkopo-lis,” the railroad helped shift the center ofthe trade to Chicago, where, by the early1870s, packers slaughtered more than a mil-lion hogs annually (Cronon, 1991, pp. 230–31).Chicago’s rise marked the rise of a nationaland international market in meat. It was onlywith this transregional integration of commod-ity markets that the multitudinous local cyclesbecame singular in the late nineteenth cen-tury. Integration itself resulted from an arrayof long-run economic changes that included,among other things, the railroad, urbanization,and, importantly, refrigeration. The nationalrail grid was in place at the end of the nine-teenth century, and it was at that time that me-chanical refrigeration came to play an impor-tant role in the process (Goodwin, Grennes,and Craig, 2002). This combination of eco-nomic changes, many components of which in-volved enormous fixed costs, was itself sup-ported by improvements in U.S. financial mar-kets, and these improvements also directlyimpacted farm production, facilitating expan-sion among other things (Craig, Goodwin, andGrennes, 2004; Craig and Holt, 2005).

Taken together, by the first decade of thetwentieth century, these changes manifestedthemselves in a national hog–corn cycle thatfundamentally differed from the older localand regional cycles. In particular, the adoptionof mechanical refrigeration allowed farmers toexpand or smooth production across the year.For meat and dairy products, in particular, re-frigeration broke the tyranny of the seasons.In turn, processors of those agricultural prod-ucts, themselves located in the urban areas towhich the raw materials were initially shipped,could exploit economies of scale and scope andbecome relatively big businesses in their ownright. The hog–corn nexus proved to be a cru-cial link in this chain.

Following the integration of the prairie withthe east coast and from there the rest of theworld, the hog farmer faced what he perceivedto be an iron law of hog–corn economics, andit was this law that ultimately manifested itselfin the hog cycle. The law was that a hog was


nothing more than “fifteen or twenty bushelsof corn,” or that a bushel of corn could beconverted into ten pounds (net) of hog. Thus,twenty bushels of corn spread over a year orso, depending on the breed, might reasonablyyield 200 pounds of pork of various cuts—roughly the average for hogs slaughtered dur-ing the postbellum era (Cuff, 1992). The ruleon the farm thus became that as long as theprice of corn in bushels was less than ten timesthe price of hogs in pounds, it was profitable tofeed corn to hogs.

This relationship created the hog–corn cycle.If the supply and demand for hogs were suchthat the price of corn was less than (roughly)ten times that of pork, farmers would feed all oftheir corn to, and if possible purchase more tobe fed to, their maturing hogs, and breed more.In the absence of any other factors—such asweather shocks, that might improve or worsenthe next corn crop—this behavior tended toput upward pressure on the price of corn, andproductive resources that might have gone toother farm products went in search of morecorn. In addition, as the number of hogs on themarket increased, the price of hogs would de-crease. As the price ratio fell below the magicnumber, farmers would cut back on hog pro-duction; corn inventories would begin to accu-mulate; and the cycle would begin again.

At this level of analysis, the hog–corn re-lationship appears to be a simple exercise incomparative statics: a decrease in the priceof an input (corn) leads to a decrease in themarginal cost of production, and hence the av-erage variable cost, of an output (hogs), and ina competitive (i.e., price-taking) market, firmsincrease production. And as each does so, mar-ket supply increases and the result is a decreasein market price. But with respect to the hog–corn relationship in particular and agriculturalcommodities more generally, especially in thepast before technology divorced productionfrom the antediluvian rhythm of the seasons,the decision to supply a product months intothe future was made today based on yester-day’s price (and expectations about the fu-ture, of course). The result was not necessarilya new set of (assumed to be) stable equilib-ria, but rather a series of potentially unstabledisequilibria.

To see how this might occur, consider thatfarmers necessarily had to make a decisionabout corn acreage in the spring. If this year’scrop proved to be in relatively short supply as aresult of planting decisions which were made inresponse to last year’s price before hog produc-

ers began to bid it up, then that would tend toput more upward pressure on this year’s cornprice. As the increase in hog production, whichitself began before the run up in corn prices, si-multaneously began to put downward pressureon hog prices, the hog–corn price ratio wouldfall below the crucial ten-to-one ratio. At thatpoint, farmers would begin to slaughter in-creasingly younger hogs—even those well be-low the age of maturity—because the marginalcost of continuing to feed them would exceedthe expected price. This step, however, onlyexacerbated the downward swing in the cycle,and so on it went. Graphing this behavior inprice and quantity space yielded the famousdiagram of a series of disequilibria, and be-cause of the diagram’s shape the underlyingtheory came to be called the “cobweb theo-rem.” Depending on the behavior of the otherfactors influencing the hog and corn markets,the path of this divergence might be arrestedas quickly as one or two years or it might con-tinue for four or five years. Eyeballing the hogand corn price data, Shannon (1945) (p. 167)observed that as the national cycle emerged atthe end of the nineteenth-century, the peak-to-peak duration typically lasted four to six years.It soon captured the attention of economists.

Early studies of the hog–corn cycle in thetwentieth century were implicitly based on alinear model of the relationship between thetwo markets (Coase and Fowler, 1937; Ezekiel,1938); however, statistical techniques at thetime prohibited an explicit test of the markets’dynamics. With the evolution of econometrics,it followed that subsequent efforts to do so em-ployed linear models (Harlow, 1960; Jelavich,1973; Larson, 1964). The fundamental problemassociated with the use of such models in thehog–corn cycle pervades models of populationdynamics more generally. Specifically, animals(porcine or otherwise) may be slaughtered, inresponse to market signals, literally over night;but producing them, again in response to mar-ket signals, takes considerably longer. There-fore, one might logically expect this inherentasymmetry to be better represented by somenonlinear form.

Furthermore, the very nature of theserelationships, linear or otherwise, which them-selves are manifestations of the underly-ing structure of the corn and hog markets,would be expected to change through time.In addition, transportation improvements andrefrigeration, public policies, and a host oforganizational and technological innovationsspecific to corn and hog markets, have all been


observed over time. There is also a reason tosuspect that structural change has been, at sev-eral junctures, a key feature of the hog–cornrelationship, and as change has not been uni-form in the two markets over time, one mightexpect the “magic ratio” to have changed aswell. To obtain a better idea of the relation-ship between hog and corn prices over time, itis useful to examine in more detail the basicdata, the topic of the next section.

Data Description

The data used in the empirical analysis con-sist of monthly prices of hogs relative tocorn for the 1910—2004 period. Averageprices received by farmers, U.S., for hogs(all grades) in dollars per hundredweightare available on a monthly basis, season-ally unadjusted, from the U.S. Departmentof Agriculture’s (USDA) National Agricul-tural Statistical Service (NASS) for the 1910–2004 period. Likewise, the average price ofcorn, in dollars per bushel, received by farm-ers, U.S. (all grades), is also available on amonthly basis, seasonally unadjusted, fromNASS for the same period. Data through1992 were obtained from the USDA-NASSdata archive at Cornell University’s Mannlibrary (http://usda.mannlib.cornell.edu/data-sets/crops/92152/). Prices for 1993–2004 were

Figure 1. Observed data and stochastic extrapolations of the TV-STAR and linear AR modelsof the U.S. hog–corn ratio (horizon thirty-six years)

obtained from NASS monthly prices receivedbulletins.

The hog–corn price ratio data, in log-levelsform, are plotted in the left-hand panel figure 1.The plot is suggestive—there appears to bea substantial cyclical feature to these data.Indeed, it was exactly this observation thatcaptured the attention of Coase and Fowler(1937), and Ezekiel (1938) in the 1930s. As ob-served from figure 1, there has been an up-ward trend in the ratio since the mid to late1940s, and the ratio also appears to have be-come more variable since the early 1970s. Fi-nally, although difficult to discern from thegraph, there is a substantial seasonal compo-nent to the series. In the empirical application,the hog–corn ratio is converted to natural log-arithms in an attempt to mitigate some of theobserved heteroskedasticity in each model’sresiduals.

Based on the plot in figure 1, there is somequestion as to whether the hog–corn series pos-sesses a unit root. To further investigate thisissue, several tests were performed. First, non-parametric bootstrap versions of augmentedDickey–Fuller tests were employed by usingtwelve lags of the hog–corn ratio. Results showthat, with or without trend, the null hypothe-sis of a unit root is rejected at the 0.001 level.


Of course standard unit root tests are of ques-tionable value when nonlinear STAR-typemodels are considered (Skalin and Terasvirta,2002). Therefore, bootstrap-based tests similarto those developed by Eklund (2003) were im-plemented, wherein the null is a linear modelcontaining a unit root and the alternative isa first-order approximation of a STAR modelspecified in the levels. Again, using twelve lagsboth with and without a trend, the null hypoth-esis of a unit root is rejected at the 0.001 levelin all instances. Additional results are avail-able at the website. It is therefore reasonableto specify any statistical model of the hog–corndata, including a STAR model, in levels form,an issue to which we now turn.

STAR Models and the STAR Modeling Cycle

The Basic STAR Model

In this section, we describe the basic model-ing framework used to examine the hog–corncycle as might be applied to a time series ofmonthly observations. The STAR model ofTerasvirta (1994) is used throughout. Accord-ingly, a STAR model of order p and augmentedwith (monthly) seasonal dummies is specifiedas

�yt = � ′1xt (1 − G(�12 yt−d ; �, c))

+ � ′2xt G(�12 yt−d ; �, c) + εt

(1a)

or, alternatively,

�yt = �′1xt + �′

2xt G(�12 yt−d ; �, c) + εt(1b)

where yt is the log-level of the hog–corn price ratio; � is a first differenceoperator; xt = (1, x′

t , D′t )

′, where xt =(�yt−1, . . . , �yt−p, yt−1)′; Dt = (D∗

1t, D∗2t, . . . ,

D∗11t)

′ = (D1t − D12t, D2t − D12t, . . . , D11t −D12t)

′, where D�t, � = 1, . . . , 12 are seasonaldummy variables with D�t = 1 when time tcorresponds to month � and zero otherwise;� i = (�i0, �i1, . . . , �ip)′, i = 1, 2 are parametervectors, and �1 = �1, �2 = (�2 − �1); and εt isa white noise process, εt ∼IID(0, �2). Based onthe unit root tests reported above, we followSkalin and Terasvirta (2002) by includingthe lagged level term yt−1 as an additionalexplanatory variable, and thereby allow forthe possibility of a moving equilibrium (i.e.,a reparameterized model in levels form). In(1) G(�12yt−d; � , c) is the so-called transitionfunction; by construction it is bounded be-tween zero and one, and therefore allows thestructure of the model to change, in a possiblysmooth manner, with the value of �12yt−d, that

is, lagged annual differences of the hog–cornratio. In other words, the model’s structurewill vary depending on whether the hog–corncycle is in approaching a peak (i.e., �12yt−d >c) or a trough (i.e., �12yt−d < c) regime.

In what follows, we specify transition func-tion G(�12yt−d; � , c) to be a logistic functionof �12yt−d = yt−d − yt−12−d of the form3

G(�12 yt−d ; �, c)

= [1 + exp{−�(�12 yt−d − c)/

�(�12 yt−d)}]−1, � > 0

(2)

where �(�12yt−d) is the sample standard de-viation of �12yt−d. Here d is referred to asthe delay parameter. The combination of (1)and (2) leads to a logistic STAR (LSTAR)model. In (2), �12yt−d is the transition vari-able and � and c are, respectively, slope andlocation parameters. For the LSTAR, c is inter-preted as the threshold between two regimesin that G(c; � , c) = 0.5, with G(·) chang-ing smoothly from zero to one (i.e., fromone regime to another) as �12yt−dt increases.Here � is called the smoothness parameter. As� → ∞, G(�12yt−d; � , c) approaches a Heavi-side indicator function It = | (�12yt−d > c), de-fined as It = | (A) = 1 if A is true and It = | (A) =0 otherwise. In other words, as � → ∞, theregime switch becomes instantaneous. There-fore, when � is very large, the LSTAR given by(1) and (2) becomes a SETAR. Also, as � →0, the LSTAR converges to an autoregression(AR) model of order p, or AR(p). Finally, inspecifying (2) another possibility is to assume,as in Lin and Terasvirta (1994), that, in lieu of�12yt−d, t, t = 1, . . . , T is the transition variable.Replacing �12yt−d with t in (2) results in anAR model with parameters that time vary in apotentially smooth manner, that is, the TVARmodel.

Testing Linearity and Parameter Constancy

As the foregoing discussion makes clear, lin-ear AR models are nested within the LSTARframework. It is therefore desirable to test theLSTAR against an AR specification. A fun-damental problem with using the LSTAR to

3 Seasonal first differences are used here as the transition vari-able as we are primarily interested in nonlinearities associatedwith the hog–corn cycle—the transition variable should reflectsustained periods of expansion or contraction. We therefore omitmonthly first differences as potential transition variables in thatthey are normally too noisy to provide a consistent signal aboutthe cycle’s regime. See, for example, Skalin and Terasvirta (1999)and van Dijk, Strikholm, and Terasvirta (2003). For this reason, wespecify the transition variable as �12yt−d.


test linearity is that an AR model may beachieved in one of two ways: an AR model ob-tains if � = 0 or, alternatively, if the restrictions� ′

1 = � ′2 are imposed on (1) (Terasvirta, 1994).

The problem, therefore, is that testing H0 : � =0 against H1 : � �= 0 within the LSTAR resultsin a nonstandard test, that is, a test with uniden-tified nuisance parameters under the null (i.e.,the autoregressive coefficients and the locationparameter). One approach to dealing with thisproblem, proposed by Luukkonen, Saikkonen,and Terasvirta (1988), is to replace transitionfunction G(�12yt−d; � , c) in (1) with a suitableTaylor series approximation. The reparameter-ized model is no longer associated with an iden-tification problem, and linearity testing pro-ceeds by using standard Lagrange multiplier(LM) tests.

Let st denote either �12yt−d or t in (2), thatis, let G(st; � , c) denote the transition function.Then, assuming delay parameter d is known,one linearity test is obtained by replacing G(st;� , c) in (1b) by a first-order Taylor series ap-proximation, which yields the following artifi-cial regression

�yt = �′0xt + �′

1xt st + vt(3)

where parameters �i, i = 0, 1 are functionsof original parameters in (1b) such that when� = 0, �0 �= 0 and �1 = 0. In this case, a linear-ity test involves testing H01 : �1 = 0 against thealternative that H01 is not true. This nonlinear-ity test, called the LM1 test, may be conductedby using either an asymptotic � 2 test with (p +2 + 11) degrees of freedom or an appropriateF version of the test.4

As Luukkonen, Saikkonen, and Terasvirta(1988) note, the LM1 statistic has low powerin cases where only the intercept varies acrossregimes. A test that apparently does havepower in this situation involves a third-orderTaylor series approximation for G(st; � , c) in(1b). The following artificial regression obtains

�yt = �′0xt + �′

1xt st + �′2xt s

2t + �′

3xt s3t + vt .

(4)

Now a test of linearity involves testing H03 :�1 = �2 = �3 = 0 against the alternative that

4 The F version of the LM1 test statistic is obtained as follows.Estimate (3) by imposing the restrictions associated with H01.Denote the resulting sum of squared residuals by SSR0. Then,estimate (3) unrestricted and compute SSR1. The test statisticis then LM1 = [(SSR0 − SSR1)/(p + 2 + 11)]/[SSR1/(T − 2(p +2 + 11))]. Under H01, the test statistic is distributed asymptoticallyas an F distribution with (p + 2 + 11) and T − 2(p + 2 + 11) degreesof freedom. In the empirical analysis, we rely on the F version ofthe LM test in question, as it typically has better size propertiesthan its � 2 counterpart.

H03 is not true. This test, denoted the LM3 test,may be conducted by using either an asymp-totic � 2 test with 3(p + 2 + 11) degrees of free-dom or its F test counterpart. An “economy”version of the LM3 statistic is derived by in-cluding only s2

t and s3t as additional regressors

in (3). The artificial regression in this case is

�yt = �′0xt + �′

1xt st + �2s2t + �3s3

t + vt .(5)

A test of the null hypothesis He03 : �1 = 0, and

�2 = �3 = 0 yields the LMe3 test.

In practice, when st is taken to be �12yt−d (asopposed to t), delay parameter d is unknown,and therefore must also be determined aspart of the testing procedure. As in Terasvirta(1994), d is determined by repeating the LM1

and LM3 tests for all values of d such that1 ≤ d ≤ Dmax, Dmax being the maximal laglength considered. If H01 (H03) is rejected for

more than one value of d, then d may bedetermined by choosing the value associatedwith the smallest overall p-value. On the otherhand, if none of the p-values for LM1 (LM3)indicate rejection of H01 (H03), then the linearAR model is not rejected.

Model Diagnostics—Autocorrelation

Once a candidate LSTAR model is cho-sen, parameter estimates are obtained by us-ing standard nonlinear estimation techniques.5

And once the model has been estimated, itsability to adequately characterize the datashould be evaluated by employing a batteryof diagnostic tests. Of particular interest aretests of the hypothesis of no remaining auto-correlation in the model’s residuals and testsof hypotheses of no remaining nonlinearity orof no parameter nonconstancy.

To illustrate, consider a test of the hypothesisof no remaining autocorrelation. As such, let

F(xt ; �) = � ′1xt (1 − G(st ; �, c))

+ � ′2xt G(st ; �, c)

denote the skeleton of the model, where � =(� ′

1, � ′2, � , c)′. Eitrheim and Terasvirta (1996)

5 While estimation of an LSTAR model involves, in principle,a straight-forward application of nonlinear least squares, certainissues do require additional consideration. For example, reason-able estimates of starting values may be obtained by doing a two-dimensional grid search over the � and c parameters. As well,estimates may be obtained by concentrating the sum of squaresfunction. Finally, the � parameter is generally not estimated withprecision, especially when the true value of � is large. Of coursesuch a result does not necessarily militate against nonlinearity, asthe asymptotic distribution of the speed of adjustment parameter� is, in any event, nonstandard under the hypothesis that � = 0.Regarding these issues and more, see van Dijk, Terasvirta, andFranses (2002) (pp. 19–21).


propose testing the hypothesis of no remainingautocorrelation up to and including order q byestimating the auxiliary regression

εt = �′1∇F(xt ; �)′ +

q∑i=1

�i εt−i + t(6)

where ∇F(xt ; �) = (∂ F(xt ; �)/∂�).6 The LMtest statistic is computed in the usual fashionas TR2, where R2 is the r-squared coefficientfrom the auxiliary regression in (6). Under thenull hypothesis of no remaining autocorrela-tion, that is, under H0 : �1 = · · · = �q = 0,the resulting test statistic has an asymptotic� 2 distribution with q degrees of freedom. AnF-version of the test may also be constructed.

TV-STAR, MRSTAR, and AdditiveSTAR Models

As already suggested, there may be occasionswhere the LSTAR’s parameters are not con-stant through time, due perhaps to institutionalor technological change, a possibility that isplausible for the hog–corn ratio. In this case,it may be better to specify a model for �yt thatincludes both regime switching and noncon-stant parameters, a TV-STAR model.

The TV-STAR is expressed as

�yt = [� ′1xt (1 − G1(�12 yt−d ; �1, c1))

+ � ′2xt G1(�12 yt−d ; �1, c1)]

× (1 − G2(t ; �2, c2))

+ [� ′3xt (1 − G1(�12 yt−d ; �1, c1))

+ � ′4xt G1(�12 yt−d ; �1, c1)]

× G2(t ; �2, c2) + εt

(7a)

where G1(·) and G2(·) are logistic functionsas in (2). Clearly if � 2 = 0, or if �1 = �3

and �2 = �4, the two-regime LSTAR ob-tains. If t in transition function G2(·) in (7)is replaced by a second endogenous transitionvariable, �12yt−d, the Multiple Regime STAR,or MRSTAR model of van Dijk and Franses(1999) obtains.

6 In practice, the estimated residuals, the εt , may not be exactlyorthogonal to the gradient vector ∇F(xt ; ). This could happen dueto numerical inaccuracy if the STAR model in question is difficultto estimate. As a result the empirical size of the test could increase.To address this potential problem, Eitrheim and Terasvirta (1996)propose first regressing εt on ∇F(xt ; ), and then using the resid-uals from this regression in (6) for all subsequent tests of remain-ing nonlinearity, parameter nonconstancy, autocorrelation, etc. In-deed, this is the procedure used in implementing all diagnostic testsreported subsequently in the analysis of the hog–corn data.

Several strategies may be used to test anLSTAR versus a TV-STAR. First, once a can-didate STAR model has been estimated, G2(t;� 2, c2) in (7) may be replaced by a suitable Tay-lor series expansion. For example, if a third-order Taylor series is used the approximationto (7) is

�yt = �′1xt + �′

2xt G1(�12 yt−d ; �1, c1)

+ �′1xt t + �′

2xt t2 + �′

3xt t3

+ (�′

4xt t + �′5xt t

2 + �′6xt t

3)

× G1(�12 yt−d ; �1, c1) + �t .

(8)

The null hypothesis of no time variation isH0 : �1 = · · · = �6 = 0, with the LM test con-structed by running a regression similar to thatin (6) wherein the residuals from the estimatedLSTAR (TVAR) are regressed on the gradient

vector ∇F(xt ; ) and additional regressors

xt t, xt t2, xt t3, xt t G1, xt t2G1, xt t3G1, G1 =G1(�12 yt−d ; �1, c1) (van Dijk and Franses,1999). The LM test statistic is then constructedeither as an asymptotic � 2 test with 6(p +2 + 11) degrees of freedom or as a comparablydefined F test. This testing strategy is the“Specific-to-General” procedure outlined byLundbergh, Terasvirta, and van Dijk (2003).

An alternative approach, also suggested byLundbergh, Terasvirta, and van Dijk (2003),is to test a TV-STAR directly against a linearmodel, the “Specific-to-General-to-Specific”approach. In this case, the transition func-tions in (7) are approximated directly by, say,a first-order Taylor series expansion. Doing sogives

�yt = �′0xt + �′

1xt�12 yt−d

+ �′2xt t + �′

3xt�12 yt−d t + vt

(9)

and testing the null hypothesis HTV−STAR0 :

�1 = �2 = �3 = 0 yields the LMTV−STAR test,which may be conducted by using either anasymptotic � 2 test with 3(p + 2 + 11) degreesof freedom or an F test.

Lundbergh, Terasvirta, and van Dijk (2003)describe several additional tests of interestnested within (9) when HTV−STAR

0 is rejected.

Specifically, in this case test HSTAR0 : �1 = �3 =

0, the LMSTAR test, in which case (9) reducesto a TVAR under HSTAR

0 . Also, test HTVAR0 :

�2 = �3 = 0, the LMTVAR test, wherein (9)reduces to a STAR model under HTVAR

0 . If

both HSTAR0 and HTVAR

0 are rejected, thenthe TV-STAR model is retained. Alterna-tively, if HSTAR

0 is rejected but HTVAR0 is not,


then a STAR model is indicated. The op-posite conclusion obtains (i.e., a TVAR isselected) if HTVAR

0 is rejected but HSTAR0

is not. Lundbergh, Terasvirta, and van Dijk(2003) report simulation evidence assessingthe relative merits of the two testing strate-gies (i.e., Specific-to-General versus Specific-to-General-to-Specific), and suggest that itmay be desirable to employ both in the modelselection stage.

Finally, Eitrheim and Terasvirta propose analternative to the TV-STAR (or MRSTAR):the additive STAR. In this case, (1b) is mod-ified by appending a second additive STARcomponent. That is,

�yt = �′1xt + �′

2xt G1(�12 yt−d ; �1, c1)

+ �′3xt G2(s2t ; �2, c2) + εt

(10)

is an additive STAR model where eithers2t = �12yt−d or s2t = t. The foregoingtests for remaining nonlinearity of the TV-STAR/MRSTAR type may be modified to testadditive STAR effects by simply excluding re-gressors in artificial regression (5) involving

G1.7

Heteroskedasticity Robust Tests

When performing LM tests of remaining resid-ual autocorrelation, unspecified heteroskedas-ticity may result in spurious rejection of thenull hypothesis. Ignored heteroskedasticity inLM tests of linearity, parameter constancy,and model misspecification may have simi-lar effects. It is therefore desirable to havetest statistics that are robust in the presenceof heteroskedasticity. Wooldridge (1990) hasdeveloped a simple set of procedures for ob-taining heteroskedasticity robust LM tests ina general setting. Details on implementingheteroskedasticity robust tests in a STAR-type framework are provided in van Dijk,Terasvirta, and Franses (2002).

While it seems advantageous to computerobust LM tests if there is evidence of het-eroskedasticity, a note of caution is in order.Lundbergh and Terasvirta (1998) provide sim-ulation evidence showing that in certain in-stances robustification reduces the power oflinearity tests. In other words, robustification

7 In fact, the additive STAR model need not be viewed as asimple alternative to the TV-STAR (MRSTAR). As van Dijk,Strikholm, and Terasvirta (2003) illustrate, it is possible to com-bine additional additive components with a TV-STAR. This lateroption may be especially useful if, say, observed seasonality hasundergone several changes during the sample period.

may make it difficult to detect nonlinearitywhen in fact it truly exists. Here we simplypresent both standard and robustified versionsof LM linearity tests. Final model specificationsare then determined through careful evalu-ation of each candidate model’s propertiesat the estimation and misspecification testingstages.

Modeling the Hog–Corn Ratio

In this section, we present results on the es-timation of a provisional linear model fittedto the hog–corn ratio data. We then presentresults of linearity tests, estimates of a candi-date TVAR model, results of additional modelmisspecification tests, and finally estimates ofa TV-STAR model. To conserve space, param-eter estimates for the various models are notpresented; they are, however, available in Holtand Craig (2005).

Linear Model Results

A linear AR model is first fitted to the data.To account for seasonality, we include elevenmonthly dummy variables, as previously de-fined. The Akaike information criterion (AIC)is used to choose the lag length. Allowing up to48 lags, the AIC is minimized at lag 11, imply-ing a total of 1,103 usable observations. Sev-eral diagnostics for the best-fitting linear ARmodel are reported in the left-most column oftable 1. LM test results show that even with11 lags, the linear model apparently does notcapture all of the residual autocorrelation. LMtests also reveal substantial evidence of ARCHeffects. Based on the Lomnicki–Jarque–Bera(LJB) test (Lomnicki, 1961; Jarque and Bera,1980), the residuals associated with the ARmodel fail to satisfy normality. As indicated bythe excess kurtosis measure reported in table 1,the error distribution for the linear model hasthicker tails than that implied by normality.

Linearity, Parameter Constancy,and TV-STAR Test Results

In testing nonlinearity, we use various lags ofseasonal first differences of the hog–corn ratio,�12yt−d, d = 1, . . . , Dmax, where Dmax = 6.8 Ofcourse to test parameter constancy, we use alinear trend.

8 Tests were performed initially by using Dmax = 12, but all resultsfollowing d = 6 were found to be statistically insignificant and aretherefore not reported.


Table 1. Diagnostic Tests for Estimated Models for the U.S. Hog–Corn Ratio

Measure AR TVAR TV-STAR

T 1,103 1,103 1,103No. of Parameters 24 50 100R2 0.144 0.263 0.345�ε 0.082 0.076 0.072�ε,NL/�ε,L — 0.927 0.878AIC −4.961 −5.063 −5.091SIC −4.700 −4.519 −4.001SK 0.113 (0.125) 0.074 (0.318) 0.018 (0.803)EK 3.566 (0.000) 3.566 (0.000) 2.068 (0.000)LJB 605.081 (0.000) 597.924 (0.000) 199.950 (0.000)ARCH(4) 17.348 (0.000) 23.765 (0.000) 19.582 (0.000)ARCH(6) 11.674 (0.000) 16.074 (0.000) 13.107 (0.000)LMSC(6) S 3.502 (0.002) 1.044 (0.395) 0.940 (0.466)LMSC(6) R 2.800 (0.010) 0.759 (0.602) 0.803 (0.567)LMSC(12) S 2.020 (0.020) 1.348 (0.185) 1.519 (0.111)LMSC(12) R 1.694 (0.063) 1.168 (0.301) 1.267 (0.232)LMSC(18) S 2.215 (0.002) 1.382 (0.131) 1.282 (0.191)LMSC(18) R 2.176 (0.003) 1.205 (0.249) 1.058 (0.391)LMSC(24) S 2.510 (0.000) 1.237 (0.199) 1.079 (0.361)LMSC(24) R 2.253 (0.001) 1.047 (0.401) 0.900 (0.603)

Note: T denotes sample size, R2 the unadjusted R2, and �ε the residual standard error. �ε,NL/�ε,L is the ratio of the residual

standard error from the respective nonlinear (STAR) model relative to the linear (AR) model. SK is skewness, EK is excess kurtosis,

and LJB is the Lomnicki–Jarque–Bera test of normality of the residuals. ARCH is the LM test of no autoregressive conditional

heteroskedasticity (ARCH), and LMSC(�) denotes the F variant of standard (S) and heteroskedasticity robust (R) versions of the LM test

of no remaining autocorrelation in the residuals up to and including lag � . Numbers in parentheses after values of the test statistics are p-values.

Results for the LM3 and LM1 “Specific-to-General” linearity tests applied to the ARmodel, both standard and robustified, are pre-sented in table 2, along with comparable resultsfor parameter constancy. Tests were performedby using 11 lags of the hog–corn ratio alongwith eleven monthly dummy variables. As well,linearity (parameter constancy) tests are per-formed using only monthly dummy variablesand only lagged dependent variables. Whilethere is some evidence in favor of STAR-typenonlinearity for several values of delay param-eter d, the most striking test results in table 2are those for parameter constancy. Regardlessof the test used, the null hypothesis of param-eter constancy is soundly rejected when all re-gressors are included. An essentially identicalresult is obtained when only seasonal dummyvariables or only lagged dependent variablesare included.

The overall picture that emerges fromtable 2 then is that of some support for STAR-type nonlinearity, but overwhelming supportfor the notion that the model’s parametershave not remained constant through time. Formany of the reasons mentioned in earlier sec-tions, including institutional and technologicalchange, this result is not surprising. For ex-ample, technological change has occurred inhog production (i.e., multiple farrowings per

year, total confinement operations, improvedgenetics, etc.) that has caused seasonality inprices (production) to be less pronounced overtime. Similarly, corn yields have risen dramat-ically over the century along with the abilityto dry and store large quantities of grain. Aswell, since the 1930s various government pro-grams have, at times, substantially impactedcorn prices and production. All of these fac-tors, and more, have likely contributed to theobserved parameter instability in the linearmodel of the hog–corn ratio.

Results for the “Specific-to-General-to-Specific” testing sequence are presented intable 3. In this case results for the LMTV−STAR

test indicate that linearity is overwhelminglyrejected for all values of d considered, with theminimum p value occurring at d = 1. Further-more, there is clear evidence for d = 1 and6 that both the LMSTAR and LMTVAR statis-tics may be rejected at conventional levels forboth the standard and robustified tests; in otherwords, for these values of d a TV-STAR modelis retained.

A Provisional TVAR Model

Based on the combined results in tables 2 and3, we first fit a TVAR model to the data by us-ing nonlinear least squares. Details of various


Tabl

e2.

Res

ults

ofSt

anda

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dH

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for

Non

linea

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Pro

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orn

Rat

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All

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gre

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mm

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bu

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ran

siti

on

Va

ria

ble

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LM

3L

M1

LM

3L

M1

LM

3L

M1

LM

3L

M1

LM

3L

M1

LM

3L

M1

�1

2y t

−15

.48

E−0

60

.00

30

.38

50

.54

50

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70

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20

.33

40

.45

43

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E−0

60

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50

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90

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8�

12y t

−22

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E−0

50

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10

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50

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00

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30

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80

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90

.31

50

.00

20

.00

70

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30

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1�

12y t

−30

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30

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90

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60

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50

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60

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40

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50

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90

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0�

12y t

−40

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00

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40

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70

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−50

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−63

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43

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40

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41

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50

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20

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3.9

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2.4

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3.4

8E

−19

1.9

7E

−23

5.4

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−12

2.8

2E

−16

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−06

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are

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91

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20

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ith

11

lag

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ffi

rst

dif

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sa

nd

sea

son

al

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LM

3d

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ote

sth

eli

ne

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stb

ase

do

nth

eth

ird

-ord

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ylo

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(4),

wh

ile

LM

1is

the

lin

ea

rity

test

ba

sed

on

the

firs

t-o

rde

rT

ay

lor

seri

es

in(3

).

residual diagnostic tests applied to this modelare recorded in the middle column of table1. Results show there is an improvement infit for the TVAR model relative to the linearAR specification: the standard deviation of theresiduals from the TVAR is over 7% smallerthan that of the AR model. Moreover, unlikefor the AR model, the residuals of the TVARmodel show no evidence of remaining serialcorrelation. Overall, the TVAR model fits thedata better than does the constant parameterAR.

Diagnostic tests for remaining nonlinearity(d = 1, . . . , 6) and for parameter constancy forthe TVAR, notably the LMe

3 and LM1 tests,were obtained for the TVAR. While these re-sults are not reported to save space (they areavailable in Holt and Craig), they indicate thatthe TVAR is rejected against the TV-STAR ford = 1 and 6, a result consistent with Specific-to-General-to-Specific testing results presentedin table 3. The results also show that there isno evidence of remaining parameter noncon-stancy. Based on these additional tests and theevidence in table 3, we next fit a TV-STAR tothe hog–corn data.

A TV-STAR Model

Results in table 3 and those just discussed forthe TVAR suggest several possibilities for thedelay parameter in a TV-STAR, most notablyd = 1 or 6. To this end TV-STAR modelswith both d = 1 and 6 were fitted. Preliminaryevidence, including model fit and diagnosticstatistics and a post-sample forecasting exer-cise, indicated that the TV-STAR with d = 1was preferred. We therefore focus our remain-ing attention on results for the TV-STAR withtransition variable �12yt−1. Consequently, theTV-STAR model fitted to the hog–corn data isspecified as

�yt = [� ′1xt

(1 − G1(�12 yt−1; �1, c1))

+ � ′2xt G1(�12 yt−1; �1, c1)]

× (1 − G2(t∗; �2, c2

))+ [� ′

3x(1 − G1(�12 yt−1; �1, c1))

+ � ′4xt G1(�12 yt−1; �1, c1)]

× G2

(t∗; �2, c2

) + εt

(11)

where xt = (1, x′t , D′

t )′, xt = (�yt−1, . . . ,

�yt−12, yt−1)′, Dt is a vector of seasonal dum-mies, and t∗ = t/T , t = 1, . . . , T . Results ofseveral misspecification tests for the estimated


Table 3. Results of Standard and Heteroskedasticity Robust LM Tests for Nonlinearity, Specific-to-General-to-Specific Procedure, for Monthly Hog–Corn Ratio

Standard Tests Robust TestsTransitionVariable, st LMTV−STAR LMSTAR LMTV−AR LMTV−STAR LMSTAR LMTV−AR

�12yt−1 8.19E−22 1.26E−06 4.90E−21 2.76E−10 0.046 1.33E−12�12yt−2 1.02E−21 1.51E−06 1.07E−21 3.88E−10 0.096 4.68E−13�12yt−3 5.87E−21 6.05E−06 9.35E−21 3.54E−09 0.056 1.45E−12�12yt−4 1.90E−18 4.48E−04 1.01E−18 1.11E−07 0.217 3.57E−10�12yt−5 2.61E−18 5.58E−04 3.46E−17 1.25E−07 0.113 1.23E−09�12yt−6 7.23E−21 7.11E−06 1.13E−17 1.03E−08 0.030 3.70E−10

Note: Numbers are p-values of F variants the LM-type tests for specification of TV-STAR-type models described by Lundbergh, Terasvirta, and van Dijk

(2003) applied to the U.S. hog–corn ratio, 1913:02–2004:12. The tests are applied to an AR model with 11 lags of first differences and seasonal dummies, and

are based on the auxiliary regression in (10).

TV-STAR are recorded in table 1. Based on therelative standard error and the AIC, the TV-STAR represents an improvement over boththe AR and TVAR specifications.9 The LJBstatistic implies that this model also fails thenormality assumption; however, excess kurto-sis has now been reduced substantially relativeto the other models. Evidence of significantARCH effects remains. As well, the TV-STARis associated with no significant autocorrela-tion at any lag considered. Diagnostic tests forremaining additive nonlinearity and parame-ter constancy, although not presented to con-serve space (they are available in Holt andCraig), indicate there is no evidence of remain-ing nonlinearity of the additive type. The esti-mated TV-STAR model therefore appears todo an adequate job of capturing the nonlinear-ity and time variation in the hog–corn series.

The estimated transition functions for theTV-STAR are

G1(�12 yt−1; �1, c1)

=[

1 + exp

{− 500.0

(�12 yt−1 + 0.081

(0.003)

) /��12 yt−1

}]−1

(12)

and

9 This relative ranking was also maintained in a post-sample fore-cast evaluation. Specifically, the models were estimated initially byusing data through 1989 and then reestimated recursively on arolling window of data for each month through June, 2004. Foreach window, 1-step-ahead to 18-step-ahead forecasts for the levelof the series were obtained, resulting in a total of 163 forecasts ateach forecast horizon. At one and two month horizons, all mod-els perform equally well in terms of root mean square forecasterror. Beyond this horizon, however, the AR model exhibits infe-rior forecasting performance. And beginning with the nine-monthhorizon the TV-STAR model consistently outperforms the TVAR.Additional details are provided in Holt and Craig.

G2

(t∗; �2, c2

)=

[1 + exp

{− 2.364

(1.342)

(t∗ − 0.449

(0.069)

) /�t

}]−1

(13)

where heteroskedasticity consistent standarderrors are reported in parentheses. The esti-mated location parameter c1 in (12) is reason-ably close to zero, implying that regimes whereG1(�12yt−1) = 1 and G1(�12yt−1) = 0 are as-sociated with positive and negative changesin the hog–corn ratio over the past twelvemonths. As illustrated in the upper panel offigure 2, where each circle denotes at least oneobservation, the transition between the tworegimes is rather abrupt, as would be suggestedby the large estimate of � 1 in (12). BecauseG1(�12yt−1) is simply a monotonic transfor-mation of �12yt−1, it follows that periods forwhich G1(�12yt−1) = (1) 0 are roughly asso-ciated with peaks (troughs) in the hog cycle.In what follows, we refer to the G1(�12yt−1 >c) regime as the “peak” regime and theG1(�12yt−1 < c) regime as the “trough” regime.

As depicted in the lower panel of figure 2,the structural change implied by the TV-STARmodel is rather smooth. The estimate of lo-cation parameter c2 suggests that structuralchange is centered on t∗ = 0.45, which corre-sponds with May, 1954. This result correspondswith the long-run transformation of U.S. agri-culture, as described above, which greatly ac-celerated in the postwar period.

To further explore the implications ofthe TV-STAR model, the time-varying andregime-dependent intercept terms are plottedin the top panel of figure 3, along with theobserved data. As expected, intercept termsare higher for the peak regime (approximately0.51 at t∗ = 0) than for the trough regime


Figure 2. G1(Δ12 yt−1), as a function of the transition variable, Δ12 yt−1 (top panel) and G2(t∗)over time (bottom panel)

(approximately 0.28 at t∗ = 0). The variationfrom high-to-low intercept terms in the toppanel of figure 3 therefore provides a graph-ical depiction of the hog–corn cycle over time.Because the transition variable is �12yt−1, high(low) values of the ratio are immediately fol-lowed by peak (trough) periods in the hog–corn cycle. This result suggests, for example,that periods of relative scarcity in the cornmarket are followed by sell-offs in the hogmarket—a response to higher feed prices—which in turn induces a shift to a contractionaryregime. The implied long-run (deterministic)equilibrium for the TV-STAR model is plot-ted in the lower panel of figure 3. The grad-ual increase in the long-run equilibrium valuesfor regimes mirrors the perceptible increasein the hog–corn ratio over time, and thereforethe slow departure from the historical “ten-to-

one rule” for profitability in raising hogs, whichcharacterized the market before the 1930s.10

The results indicate that, with the exceptionof early-to-mid 1930s (when the cycles wherebrief) and the late 1980s and early 1990s (whenthey were longer), there has been roughly athree-to-five year hog–corn cycle, a result con-sistent with previous research (e.g., Jelavich,1973). There is also some evidence that the du-ration of the cycle, and especially troughs, hasdecreased since the late 1960s.

10 In part this result likely reflects the somewhat diminished im-portance of corn, a carbohydrate, as a dominant variable factorof production in raising hogs during the postwar period. Specifi-cally, protein sources such as soybean meal have gained in relativeimport over this period.


Figure 3. Observed data and moving intercept (top panel) and observed data and movinglong-run equilibrium (bottom panel)

Model Dynamics

As the foregoing makes clear, there are fea-tures of the hog market consistent with bothnonlinear dynamics and structural change. Itis therefore desirable to characterize the dy-namic behavior of the estimated TV-STARmodel in some consistent and reasonablytransparent ways, the focus of this section.

Deterministic Extrapolation

We consider first a deterministic extrapolationof the model to obtain insights into its im-plied behavior. This is done by iterating theskeleton of the model, that is, the deterministicpart of the model, ahead without introducingstochastic shocks. We start the extrapolation byusing the final values of the sample data as ini-tial values. Iterating the model ahead for thirty-six years, we find that the realizations convergeto a unique seasonal pattern associated withthe seasonal dummy variables. The results areplotted in the right-hand panel of figure 1. The

long-run seasonal peak occurs in August, witha ratio of 23.2-to-1, and the long-run seasonallow occurs for April, with a ratio of 19.3-to-1.

Table 4. Roots of Characteristic Polynomi-als for Select Values of Transition FunctionsG1(Δ12 yt−1) = 0 and G2(t∗) = 0

Root Modulus (Half-Life) Period

Regime: G1(�12yt−1) = 0, G2 (t∗) = 00.95 ± 0.15 0.96 (16.57) 38.82

−0.92 ± 0.15 0.93 (9.70) 2.11

Regime: G1(�12yt−1) = 1, G2 (t∗) = 00.99 ± 0.13 1.00 (2330.78) 49.580.76 ± 0.52 0.92 (8.55) 10.57

−0.16 ± 0.88 0.90 (6.27) 3.59

Regime: G1(�12yt−1) = 0, G2 (t∗) = 1−0.67 ± 0.59 0.90 (6.05) 2.59

Regime: G1(�12yt−1) = 1, G2 (t∗) = 10.91 ± 0.12 0.92 (7.88) 48.26

Note: Only roots with modulus ≥ 0.90 are reported. Period is denoted in

months.


Figure 4. Sliced spectra, G1(Δ12 yt−1) = 0 (solid line) and G1(Δ12 yt−1) = 1 (dashed line), forselect periods: (a) G2(t∗) = 0, (b) G2(t∗) = 0.5, and (c) G2(t∗) = 1

Characteristic Roots and Sliced Spectra

It is possible at each point in the sample periodand for various values of the transition func-tions G1(�12yt−1) and G2(t∗), to compute theroots of the characteristic polynomial of themodel (Terasvirta, 1994). Roots of the char-acteristic polynomial (with modulus ≥ 0.90),along with period lengths and half-lives, arereported in table 4 for G1(�12yt−1) = 0 andG1(�12yt−1) = 1 for G2(t∗) = 0, and likewisefor G2(t∗) = 1. Of interest is that, early in theperiod, the dominant root is associated with acomplex pair and a modulus near one whenG1(�12yt−1) = 1. Late in the sample data (i.e.,

G2(t∗) = 1), both the dominant roots are repre-sented by complex pairs, with moduli less thanone and with relatively short half-lives. There-fore, late in the sample, there is a tendency forthe model to return rather quickly to its long-run equilibrium level, as illustrated by the de-terministic extrapolation in figure 1. This find-ing is consistent with accelerated informationflows (i.e., market coordination), which in parthave been induced by the rapid rise of verti-cal integration in the hog market (i.e., growercontracts).

A similar related picture emerges by consid-ering the sliced spectra for the model at severaldifferent time periods (Skalin and Terasvirta,


Figure 5. Mean paths for generalized impulse response functions of the TV-STAR model. (a)G2(t∗) = 0, (b) G2(t∗) = 0.5, and (c) G2(t∗) = 1

1999). Specifically, the spectra are determinedat G1(�12yt−1) = 0 and G1(�12yt−1) = 1 forG2(t∗) = 0, G2(t∗) = 0.5, and G2(t∗) = 1. Theresults are presented in figure 4. For G2(t∗) = 0the spectrum has an initial peak at a frequencycorresponding to 48 months for peaks and ata frequency corresponding to 36.8 months fortroughs, results similar to those reported intable 4. These values correspond closely towhat might be called a “hog cycle frequency.”For G2(t∗) = 0.5, which corresponds to Jan-uary, 1959 in the sample, the initial peak in thespectrum occurs at 42.4 months for peaks andat 22.51 months for troughs. Finally, forG2(t∗)= 1, the initial peak occurs at 34.5 monthsfor peaks and at 19.35 months for troughs(figure 4).

Of interest is that the periodicity of thecycle in both peak and trough regimes hasshortened throughout the sample period, andespecially for troughs. This in part must re-flect the fact that building livestock inven-tories is more readily accomplished laterin the sample period than earlier, a resultof the adoption of nearly continuous farrowingschedules and total confinement operations. Inaddition, increased spatial integration of com-modity markets has facilitated greater mar-ket coordination, which has also reinforced ashortening of the periodicity of the cycle. Inshort, the flow of slaughtered hogs and infor-mation pertaining to them is more nearly in-stantaneous later in the sample period. Alsoof interest is that the contribution of the hog


cycle frequency becomes less prominent andthe seasonal effect more prominent throughtime for peak regimes. Taken together, theseresults confirm the ability of the estimated TV-STAR model to capture fundamental aspectsof the hog–corn series through time, and espe-cially the asymmetric adjustments during peakand trough periods that are apparently a fea-ture of the data.

Generalized Impulse Response Functions

To obtain additional information about thedynamic properties of the model, shockpropagation is examined by computing gen-eralized impulse response functions (GIs)proposed by Koop, Pesaran, and Potter (1996).Details on GI computation are available inHolt and Craig. Here we compute GIs for threeSTAR models: the one observed prior to thestructural change [G2(t∗) = 0]; the one ob-served when structural change is at the mid-way point [G2(t∗) = 0.5], and the one obtainedwhen structural change is complete [G2(t∗) =1]. Mean paths for the GIs, conditional on his-tories for G1(�12yt−1) > 0.5 (peaks) and forG1(�12yt−1) ≤ 0.5 (troughs), and conditionalon positive and negative shocks, are presentedin figure 5.

Several features of interest are revealed inthese plots. First, comparing panels (a) and (c),it is clear that shocks have a somewhat big-ger effect at the end of the structural changethan at the beginning. For example, follow-ing the structural change the largest effect ofa positive shock in a peak period occurs atmonth 2, with an oscillatory decline towardzero thereafter. But before structural changetakes place, the largest response occurs instan-taneously. As well, the persistence seems morehighly amplified toward the middle and the endof structural change (figures 5(b) and (c)) thenat the beginning (figure 5(a)). While there arepotentially many reasons why hog–corn mar-kets are now more responsive to shocks thanin previous periods, in part this must be re-lated to (1) the quantity and speed with whichinformation is now disseminated, processed,and acted upon; and (2) the fact that hog pro-duction in particular has become a more highlyintegrated process, implying that a smallernumber of agents is coordinating productionand marketing decisions. By the end of thestructural change the responses are relativelysymmetric to positive and negative shocks inboth regimes, although this is not the case priorto the structural change being completed, and

especially so for troughs. Finally, and most im-portantly, there is a distinct difference in shocktransmission between the two regimes. Shocksduring peaks are larger in magnitude and havemore persistence, at least initially, than doshocks during troughs, a feature that remainseven after the structural change. This effectis especially noticeable for negative shocks(figure 5), where there are distinct differencesin the mean paths even at horizons of up tothirty months. Of course this result is expectedbecause, as noted previously, it is easier to liq-uidate herds than it is to build them in responseto changes in expected profits.

Conclusions

In this article, we have explicitly modeled po-tential nonlinear features of the U.S. hog–corncycle in combination with structural change.While previous research has found evidence ofnonlinearity in the hog–corn cycle, no prior at-tempts have been made to explicitly model theimplied nonlinearity. We do so here by using aclass of endogenous regime switching modelsbelonging to the family of STARs. The timeseries of monthly observations on the hog–corn ratio used in the empirical analysis spansthe 1910–2004 period. Not only does this pe-riod include a number of complete cycles, butalso it encompasses many historical and in-stitutional changes that might lead to struc-tural instability. At the beginning of the sampleperiod, a national cycle was just emerging aslocal and regional markets became more inte-grated. Moreover, breeding cycles were suchthat a new crop of pigs would typically be pro-duced only once or at most twice a year. Thissituation began to change rather rapidly in thepostwar period as producers switched to totalconfinement operations and to nearly continu-ous breeding-production cycles. Consolidationof this sort was especially prevalent from the1950s onward. We might therefore suspect thatthese effects would have a substantial impacton seasonal production patterns, and thereforeon seasonal price patterns.

In modeling the data, we followed the basictesting, estimation, and evaluation cycle pro-posed initially by Terasvirta (1994). The re-sults of various linearity tests suggested that aTV-STAR model is appropriate for modelingthe hog–corn cycle. A TV-STAR model thatuses �12yt−1 as a transition variable was subse-quently fitted to the data, and found—basedon comparisons with linear AR and TVAR


models and, as well, model diagnostics—to bea suitable specification in nearly every respect.We proceeded to analyze various features ofthis model.

A careful examination of the TV-STAR’sproperties yielded several interesting featuresof the hog–corn cycle. First, the cycle ap-pears to have occurred with a somewhat reg-ular three-to-five year frequency during thesample period. The early 1930s emerged asa time of high activity, with cycles occurringmuch more frequently. Moreover, structuralchange has apparently occurred rapidly sincethe 1950s. As well, the role of the cycle it-self seems to have diminished somewhat bythe end of the sample period. Finally, calcula-tion of generalized impulse response functionsshowed that the response of the model to ashock is quantitatively and qualitatively differ-ent in the two regimes. In the end, our researchsuggests that the hog–corn cycle itself is not astationary process, but rather a feature of thesemarkets that has, itself, evolved through timeas dictated by institutional and technologicalchange.

[Received December 2004;accepted May 2005.]

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