Examining Nonlinear Business Cycle Effects on U.S. Unemployment and Capacity
Utilization
by
Peter Joseph Santoro
A Thesis Submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
December 5, 2012
Approved by
John D. Jackson, Chair, Professor, Economics
Henry Thomson, Professor, Economics
Hyeongwoo Kim, Associate Professor, Economics
ii
Abstract
This thesis develops nonlinear smooth transition autoregressive (STAR) models to capture the
business cycle dynamics of the U.S. rates of unemployment and capacity utilization. This choice of model
is motivated by a general inquiry as to whether the two series behave in a similar fashion during periods
of recession, expansion, or somewhere in between. While STAR models for unemployment have been
developed by several authors, capacity utilization has not (to our knowledge) been analyzed in the same
context. Our results for unemployment are in accord with those of previous studies: namely, that
unemployment follows essentially two business cycle regimes of expansion and contraction. By contrast,
our results for capacity utilization indicate that it follows a threeregime process: those of recession, high
growth, and normal growth. The model suggests that dynamic adjustments are roughly equal for the first
and third regimes, but different for the second.
iii
Acknowledgements
My sincere thanks to Auburn University?s Department of Economics, under whose auspices I
have been able to thrive continually as a student. Thanks are especially due to Dr. John Jackson for
providing me plenty of opportunity and wisdom from start to finish. My thanks also to Drs. Hyeongwoo
Kim and Henry Thompson for their invaluable and muchappreciated guidance. Dr. Thomas Beard has
been an endless source of optimism and inspiration, which was appreciated during difficult times. Finally,
my heartfelt gratitude goes to my parents, for their unwavering support over the years.
iv
Table of Contents
Abstract??????????????????????????????????????..ii
Acknowledgements?????????????????????????????????...iii
List of Tables????????????????????????????????????.v
List of Figures???????????????????????????????????...vi
I. Introduction???? ????????????????????????????????.1
II. Theory?????????????????????????????????????...6
III. Description of the Data??????????????????????????????..10
IV. Analysis and Methodology?????????????????????????????11
Testing for Common Stochastic Trends under the Assumption of Linear Adjustment????.11
A Review of Models of Discrete RegimeSwitches?? ???????????????...22
Smooth, Continuous Regime Switches and the STAR Modeling Procedure???..????...23
V. STAR Model Estimation? ?????????????????????????????.32
VI. Conclusion?????? ?????????????????????????????.40
References????????????????????????????????? ??? ?42
v
List of Tables
Table 1. DickeyFuller Regression Results??????????????????????...Pg. 49
Table 2. Results of EngleGranger Tests for Cointegration??????????? ?????..Pg. 50
Table 3. Template for Lagrange Multiplier Tests of Linearity?????????????? ?..Pg. 51
Table 4. Results of LM Tests for Linearity??????????????????????...Pg. 52
vi
List of Illustrations
Illustration 1. Unemployment and Capacity Utilization (1948Q12009Q3)??????????. Pg. 53
Illustration 2. Unemployment and Capacity Utilization over the Business Cycle????????Pg. 54
Illustration 3. Residuals of EngleGranger Tests of Cointegration?????????????... Pg. 55
Illustration 4. Comparison of Residuals between AR and LSTAR Models??????????..Pg. 56
Illustration 5. Unemployment Transition Function???????????????????... Pg. 57
Illustration 6. Capacity Utilization Transition Function??????????????????Pg. 58
1
CHAPTER I
INTRODUCTION
The rates of capacity utilization and unemployment are economic variables that measure the
degree of resource utilization for overall production in the economy. The fluctuations of the two series
have long been regarded by economists as reflections of the ebb and flow of shortrun macroeconomic
activity. As such, they provide relatively timely signals of the trajectory on which the economy is moving.
More important than the trajectory, however, is the speed with which these variables might tend in one
direction or another, as more rapid movements generally prefigure periods of boom and slowdown in the
shortrun. Such movements are referred to as business cycles. The prospect of business cycles engenders
uncertainty on the part of the economy?s producers, and hence may undermine the longrun growth trend
toward which the economy is tending. Therefore, insofar as both unemployment and capacity utilization
reflect producers? expectations, they abide in their importance as indicators of macroeconomic stability.
If, however, unemployment and capacity utilization offer a snapshot of an economy?s shortrun
path, do they necessarily give the same picture? Do they follow similar dynamics during periods of
expansion or contraction? How do they adjust to demand and supply shocks? Do their relative magnitudes
portend the same situation, given changes in their exogenous components like population growth and the
state of technology? Finally, given the peaks and troughs of business cycles, how smoothly does? and for
how long before ? each of the two measures revert to their longrun averages? It is these questions,
particularly the last one, which this thesis will attempt to address.
This thesis will seek to test empirically whether the local dynamics of the two series are
fundamentally similar over different phases of the business cycle. To that end, a univariate, nonlinear
2
smooth transition autoregressive (STAR) model, formalized by Timo Terasvirta (1994), will be
constructed for the two series. STAR models are designed to model smooth, continuous transitions from
one regime to another. In contrast to linear, autoregressive (AR) models whose parameters are
deterministic, STAR models append a nonlinear function to the autoregressive lag polynomial, rendering
the AR parameters stochastic and regimedependent. Understood in its usual economic context, the
parameters are dependent on the underlying structure of the economy over time. By convention, the
nonlinear function will take either a logistic or exponential form, and must be designed by the model
builder to contain information about the regime at discrete moments in time. They have proved useful in
modeling situations where there is reason to believe, whether from economic theory or by a priori beliefs
of the datagenerating process, that local, dynamic adjustment depends on the regime ? an expanding or
contracting economy, for instance ? within which adjustment takes place. This phenomenon has come to
be called ?asymmetry?, and has been noted by economists for at least as far back as Keynes (1936).
Asymmetry is but one form of nonlinearity. Pesaran and Potter (1992) offer a good, succinct
definition of nonlinearity when they describe linear models: ?One can think of a dynamic system as being
linear if its global properties can be completely characterized by its local behavior.? If global and local
behavior cannot be characterized by the same dynamic process, then linear models are bound to be
inadequate. The reasoning behind this choice of model will become clearer as this thesis progresses.
STAR models have been applied to several economic variables. Terasvirta and Anderson (1992)
used STAR models to verify the existence of business cycle asymmetries in industrial production series
for several countries. Granger and Terasvirta (1993) reinforced Terasvirta?s and Anderson?s work by
applying the models to the same variables for countries the latter did not consider. Ocal and Osborn (2000)
applied STAR models to the UK series for consumption and production, again to compare possible
nonlinearities over the business cycle. Awokuse and Christophoulos (2009) analyzed the nonlinear
relation between exports and GDP growth using STAR models. STAR models have been applied to U.S.
unemployment by Bianchi and Zoega (1998), Skalin and Terasvirta (2002), van Dijk, et.al (2002), and
3
Terasvirta, Tjostheim, and Granger (2010). To our knowledge, STAR models have yet to be applied to
capacity utilization. This is surprising given the significance ascribed by much of the economics
profession to capacity utilization as a business cycle indicator; see Corrado and Mattey (1997) for a
survey. Following many of these authors, this thesis will test using STAR models whether capacity
utilization and unemployment follow similar paths across the business cycle
. Our results for the unemployment series are on par with those of the studies cited above. The
results suggest that the shortrun dynamics of unemployment are governed according to a tworegime
framework. Depending on whether the economy is growing or contracting, the shortrun adjustment
follows a stochastic process that changes monotonically according to the overall state of the economy.
This scenario has ample theoretical backing in the literature. There is, for example, the famous search
model of Diamond (1982) where opportunities for work surface stochastically according to a Poisson
process. Cabellero and Hammour (1994) followed up on Diamond?s work by testing it empirically. In a
different area, Bentoila and Bertola (1990) exposited the theory of job creation and destruction in light of
technological advancement. Hamermesh and Pfann (1996) studied the disproportion of hiring and firing
costs as a source of friction in the labor market.
Data for capacity utilization, heretofore unanalyzed in this context of business cycle asymmetry,
revealed quite a different scenario. Our results suggest that the stochastic component of the series is better
characterized by a threeregime process, with dynamic adjustment being roughly equal for the first and
third regimes, but different for the second. This result lends itself to some significant economic
interpretations. It suggests that dynamic adjustment follows the same process when the economy has been
rooted in its recessionary trough for some extended period and when it has been expanding persistently
toward its business cycle peak, again for an extended period. The dynamics are different when the
economy at the moment is characterized by some intermediate scenario, namely, when the economy is
beginning to experience an uptick in activity, compared with a relative lull that preceded it. The work of
Bansak, Morin, & Starr (2007) point to possible explanations for the kind of behavior exhibited by
4
capacity utilization indicated by our STAR model. They discuss different decisions on the part of plant
managers in light of new information about future demand. One explanation is that a firm will change
output and capacity using existing technology in light of a demand shock. The shock triggers a dynamic
adjustment, whereby a new level of utilization is reached when adjustment is complete, at which it then
remains steady in much the same way as was the case before the shock. Perhaps the difference in
dynamics of the middle regime can also be attributed to differences over time of the relative costs of
?excess capacity? (the degree to which firms operate their capital below their full potential). In addition,
there are differences in the scale and magnitude of the expansionary periods, which are also likely to
influence the dynamics of the series during such periods. We note that Smyth and Jackson (1984) found
evidence to that effect in a ratchet model framework.
While there is, of course, no shortage of interpretations which can be adduced, this thesis will be
largely empirical and datadriven in its intent. It will follow in the same vein as that begun by Terasvirta
and Anderson (1992), and followed by, among others, Granger and Terasvirta (1993), Ocal and Osborn
(2000) and Akram (2005) in characterizing the local dynamics of macroeconomic series over the business
cycle. It is worth noting that, for a long time, research in macroeconomics operated under the assumption
that local and global properties of time series could be captured by the same specification. This
assumption motivated the classification of series as being either stationary or nonstationary. Models were
then applied to a series depending on this classification. It is no longer taken for granted that local and
global properties can be characterized in the same way, though it is very often the case that they can be.
Unemployment and capacity utilization may prove to be two exceptions; that the former is in fact an
exception is a hypothesis about which a lot of evidence has already been accumulated. One of the
operative assumptions behind the building of nonlinear models is that the series being considered are
globally stationary, but locally nonstationary; see Pesaran and Potter (1992) for an intuitive discussion.
This study will seek to reinforce this hypothesis concerning the two series it will study.
5
This thesis will proceed as follows. Chapter II will outline the theoretical background that relates
unemployment and capacity utilization. This background is grounded in a traditional Keynesian
framework. Chapter II will also review some of the conflicting economic interpretations of the business
cycle, which have become more salient in light of the most recent 20072009 recession. Chapter III will
give a description of the data. Chapter IV provides the formal background of the STAR methodology.
Preceding this, however, is an attempt to explain how and why traditional tests for common stochastic
trends have proved inadequate for data that exhibit some form of nonlinearity. In Chapter V,
unemployment and capacity utilization will be subjected to the STAR modeling framework crystallized
by Terasvirta (1994). Results will then be reported. Chapter VI gives the conclusion, which includes some
proposals for future research. All tables and figures referenced can be found at the end of the paper.
6
CHAPTER II
THEORY
The economic theory that relates capacity utilization and unemployment is grounded in traditional
Keynesian models of aggregate demand and supply. Keynes (1936) defined an employment function that
corresponds to aggregate levels of output, reflected in the aggregate demand function. Finn (1995)
reviews the comparative static model that locates some full employment level of output, , at the kink of
the discontinuous aggregate supply function. The supply function is perfectly horizontal up until ,
beyond which it slopes upward at an increasing rate to indicate that demand for higher real wages is
higher, now that labor is working overtime. The second kink occurs at some output level, , which
indicates full capacity, at which point the supply function becomes vertical, and therefore increases in
aggregate demand through increases in consumption, investment, government spending, the money
supply, or some combination thereof, can only result in a higher price level, but not higher output. Finn
notes that the Federal Reserve?s measure of capacity utilization can be understood to equal the ratio of
actual output ??effective demand? in Keynes?s terms ? to the full capacity level of output, namely, .
However, is greater than , and the space between them refers to that volume of output resulting from
overtime on the part of labor. Moreover, can be viewed as ?excess capacity?, that is, the amount by
which production is lower than that capable at full capacity. A more appropriate measure of slack in the
labor market according to this model might be where Y is the current level of output as
dictated by the level of current effective demand.
This model presents the shortrun comparative statics whereby cyclical fluctuations in capacity
utilization and unemployment correspond to deviations of actual, effective demanddriven output, from
7
the fullemployment level of output. The longrun phenomenon of growth stems from increases in
exogenous factors like the size of the workforce, the capital stock, and technology, all of which are
assumed fixed in the shortrun. Both capacity utilization and employment, therefore, are determined by
where the aggregate demand curve intersects the aggregate supply curve, and by the region within which
the point of intersection exists.
The economic factors that influence aggregate demand are consumption (and the marginal
propensity to consume), government spending (net of taxes), the money supply, and investment. However,
it is the fourth factor, namely the volume of investment, that is the more volatile among the others and
that serves as the driver of the cyclical fluctuations of demand and thus of employment and capacity. For
Keynes, the equilibrium solution occurs when the interest rate equals a measure he called the ?marginal
efficiency of capital?, a term, vague in definition, that runs throughout The General Theory. Because the
marginal efficiency of capital is a function of many things, Keynes believed that cyclical fluctuations in
output stayed within a normal range. In Keynes?s words, ?there is some recognizable degree of regularity
in the timesequence and duration of the upward and downward movements? (1936, pg. 314).
There is however, an exception to this regularity, namely, the phenomenon of ?the crisis?. The
marginal efficiency of capital is determined, as Keynes (1936, pg. 316) put it, ?by the uncontrollable and
disobedient psychology of the business world,? and that, ?when disillusion falls upon an overoptimistic
and overbought market, it should fall with sudden and even catastrophic force?. The resulting uncertainty
as to when the market will begin to recover induces a ?liquiditypreference? which, in the Keynesian
story, lays the ?liquidity trap? that renders monetary policy of money expansion impotent when interest
rates are sufficiently low. This becomes a seemingly intractable problem as long as the postcrisis level of
aggregate demand remains rooted in its trough. Thus, for Keynes, the business cycle moves according to
the volume of investment which behaves in a cyclical fashion, except in times of crisis, when the
uncertainty as to the future yield of investment becomes so grave as to precipitate a downturn in
investment that cannot be offset by an increase in the marginal propensity to consume. The fall in the
8
cycle is sudden, whereas, in the time when the economy starts to recover, there is ?no such sharp turning
point.?
Modern studies of asymmetry (e.g., Neftci, 1984, Terasvirta and Anderson, 1992) cite Keynes as
one of the first to propose the idea. An earlier study by Mitchell (1927) relied less on theory and more on
statistics to validate it, ultimately to mixed results. Keynes was silent as to whether the crisis would have
permanent or temporary effects of the overall economy. The asymmetry phenomenon was also not
presented in The General Theory as though it were a characteristic of all business cycles, a fact
overlooked by those modern studies that cite Keynes as one of the progenitors of this idea. Nonetheless, it
should be noted that the discontinuous property of the aggregate supply function suggests differences in
regime according to the region of the aggregate supply function within which the economy is operating.
This chapter will close with a brief discussion of some of the conflicting economic interpretations
of business cycles, particularly in light of the most recent 20072009 recession. One of the canonical
explanations offered by economists for the onset of recessions has been declines in worker productivity.
Traditional labor market models consist, first, of a demand function which is derived from the marginal
product of labor (MPL), and second, a supply function which is based on the marginal rate of substitution
(MRS) between labor and leisure that characterize the behavior of a representative household.
Equilibrium in the labor market is defined by equality of the MPL with the MRS. Declines in the
marginal product of labor not only indicate lower productivity, but also induce the demand shortages that
explain unemployment. Declines in both productivity and employment are two of the most salient
features of economic recessions, the former effecting the latter.
This explanation of unemployment as one of productivity decline is often associated with the
seminal work of Kydland and Prescott (1982) on real business cycle theory. Their work analyzes
deviations from the equilibrium conditions of different input markets. In the labor market, for instance,
positive (negative) labor deviation is defined by the level of employment that is above (below) the level
consistent with the marginal product of labor. If one considers total productivity as determined by a
9
production function of labor and capital, positive (negative) productivity deviations are defined as the
level of output that is above (below) the level generated by the capital and labor inputs. This deviation is
referred to as the ?Solow residual? and is often used to measure changes in productivity. In short, real
business cycle theory says that changes in productivity go hand in hand with disequilibrium in the
markets for labor and capital.
Recently, questions have been brought to bear on this theory, particularly in light of the recent
20072009 recession. In the United States, postwar recessions ? excluding the 2007 recession ? have, on
average, seen a 2.4 percent labor deviation and a 2.2 percent productivity deviation. The 2007 recession,
by sharp contrast, has seen a 12.9 percent labor deviation, along with a mere .1 percent productivity
deviation. For a good overview of how the 2007 recession differs from other postwar recessions, see
Ohanian (2010). In other words, productivity has barely dipped in this most recent and very deep
recession, which, as noted by Ohanian, can be completely characterized by a labor market slump from
which we are still recovering as of the fall of 2012.
Thus, the salience of the 20072009 recession lies in there being a relative lack of productivity
decline that is conspicuously coupled with a labor market which has been persistently marred by high
unemployment. As noted, this is an aberration from previous postwar recessions (Ohanian, 2010). It is
this persistence that this thesis will seek to both account for and quantify over the postwar period. An
assumption is that the peaks and troughs of the business cycle make for the appropriate backdrop against
which one can account for how slowly or swiftly employment and capacity recover in the face of a
recession. As noted, while the traditional Keynesian framework does not explain the nature of recovery, it
does suggest through its discontinuous aggregate supply function that the macroeconomy is governed by
several regimes based on the level of overall output. However, it is an open question as to whether all
recessions are followed by swift, or slow, rises in employment and capacity. This thesis shall seek to
contribute to filling in this gap.
10
CHAPTER III
DESCRIPTION OF THE DATA
The empirical analysis is based on quarterly data for the U.S. civilian rate of unemployment, and
for the capacity utilization series, both for the period 1948:Q1 ? 2009Q3. The unemployment series was
obtained from the St. Louis Federal Reserve Databank (FRED). Data for capacity utilization is available
from the Federal Reserve Board of Governor?s official website.
Figure 1 plots the two quarterly series over this period (Panel A for capacity utilization, Panel B
for unemployment). Figure 2, Panel B plots the two overlapping, showing that they are roughly the mirror
image of each other. Figure 2, Panel A displays capacity utilization and employment overlapping; the
vertical axes are drawn differently to make comparison more visible. A cursory evaluation of Figure 2
suggests that, from their trough, recessionary levels, adjustments of capacity utilization tend to be
smoother and more secular compared to employment. Examples of this are the recessions of the early
1970s and 2000s. An exception is the relative smoothness with which both variables recovered from their
respective troughs following the early 1980s recession. The period of the mid to late1990s is shaded
orange on the graph in order to emphasize the anomalous situation seen during this period: despite strong
economic growth, capacity utilization saw a persistent decline going right into the early 2001 recession.
Discussion of this apparent anomaly will take place in the next chapter.
11
CHAPTER IV
ANALYSIS AND METHODOLOGY
This chapter is divided into three parts. Each dovetails into the other in that an ancillary purpose
of this chapter is to chronicle how the STAR method was chosen in the effort to test the hypothesis stated
in Chapter I. The first will report results of traditional linear tests for common stochastic trends between
unemployment and capacity utilization. This first part will review some of the theories concerning the two
series. The review is placed in this chapter because these theories have direct bearing on the assumptions
the researcher must make in determining the kind of tests that will be implemented. Corrado and Mattey
(1997) contend that, ?much of the variance in capacity utilization is common to other ?business cycle
clocks? such as unemployment.? Tests for verifying this claim fall generally under the rubric of
?cointegration?, developed by Engle and Granger (1987). Cointegration as originally defined states that
the two or more variables are cointegrated if they follow the same stochastic trend; accordingly, their
divergences from one another are merely temporary. The second part will review some of the models
which have been developed to incorporate changes in the regime such that the parameters of the data
generating process change. It is important to remember that regimechange is different from what is
referred to in the literature as ?structural change?. The latter is conventionally understood to result from an
exogenous shock that leaves a permanent effect on the series? evolution. The third and final section
presents the STAR methodology, specifically the wellspecified procedure that leads up to estimation.
Part 1: Testing for Common Stochastic Trends under the Assumption of Linear Adjustment
In accordance with the framework laid out by Engle and Granger (1987), our analysis begins by
determining the order of integration d, labeled , where d is the number of times the difference
12
operator must be applied to a time series before it can be characterized as a (covariance) stationary
process. That most annual macroeconomic variables are has been well known since the seminal
work of Nelson and Plosser (1982), who applied DickeyFuller tests of stationarity to fourteen
macroeconomic variables and found that all but one contained a unit root. (It is worth mentioning that the
one exception was the U.S. unemployment rate.) This result ran counter to the thenprevailing view that
most macroeconomic series were stationary around a deterministic trend, which is termed ?trend
stationary?. Trend stationarity of a variable, say y, implies that the residuals of the regression
are stationary. The Nelson and Plosser results indicate that most variables are stationary when
differenced once, but random walks when analyzed in levels. This finding opened up a new issue that
researchers face when applying regressions of individual time series, namely, the issue of ?spurious
regression?.
The concept of spurious regression goes as far back as Yule?s (1926) discussion on spurious
correlation, when he showed that significant levels of correlation between two unrelated random walks
could result even in large samples. The of a regression of two such variables should tend to zero as the
sample size increases. However, because both variables contain a deterministic component, spurious (or
false) correlation could be reflected in the form of significant tstatistics, along with an significantly
different from zero. When a regression between time series (expressed in levels) yields these two
results, along with a low DurbinWatson (DW) statistic, the researcher should be cautious in drawing
conclusions. The crux of the problem is that two variables may well follow a shared deterministic trend ?
and hence the signs of correlation ? but not the same stochastic trend.
The spurious regression problem was revisited by Granger and Newbold (1974), when they
showed how autocorrelated residuals, in addition to causing inefficient OLS estimates, can also render
invalid the traditional ttests for significance. They took two random walks,
,
13
and generated artificial data for and according to different distribution parameters, thus allowing x
and y to follow two different stochastic paths. They then regressed y on x plus a constant and found a high
but a low DW statistic. When regressing in first differences, the was close to zero and the DW
was close to 2, thus showing that y and x were in fact unrelated, despite the significant results of the
regression that was run in levels. In short, one of the rules of thumb in applying OLS to time series is that,
if DW, this is a symptom of a spurious regression.
Understanding the concept of spurious regression is helpful in thinking about the way in which
the theory of cointegration was originally developed. In light of Granger?s and Newbold?s findings, an
obvious question arises: what if a regression of one random walk on another yields a residual series that is
stationary? The residuals would be, by definition, a meanzero process bounded by a finite variance. This
would indicate that the variables do not drift too far apart from one another, and converge in the longrun.
This condition gives rise to the concept of cointegration. The economic interpretation of cointegration is
that some longrun, equilibrium relationship exists between two or more variables. Moreover, these
cointegrated variables would share a common trend, such as some leading economic indicator like money
growth or interestrate spreads. The formal notation of cointegration is summarized as follows: if
and there exists some such that , in which , then this is
evidence that share a common trend and are cointegrated in the longrun. Engle and Granger
(1987) define cointegration more generally, stating that if, for all the components of some vector ,
there exists a vector so that , then the components of that vector are
cointegrated. In other words, for cointegration to exist, the components must be integrated of the same
order d, and a linear combination of those components must be integrated of an order less than d. An
exception would be a case where one of the variables is and the rest are . It is possible that a
linear combination of these two is in which case we would have what is called ?multicointegration?.
It is also possible that d is a fraction, in which case the series is ?fractionally integrated?, characterized as
being not quite in possession of a unit root, but, at the same time, highly persistent; this persistence is
14
reflected in an autocorrelation function that declines at a slow, hyperbolic rate (Hjalmarsson & Osterholm,
2010). For now, we will confine our attention to processes whose orders of integration are discrete
integers . Engle?s and Granger?s definition will become useful as we try to ascertain the
existence of a cointegrating relationship between the two variables of this study, capacity utilization and
unemployment, a task to which we will soon turn.
First, we will review some of the modern theories of the two variables, beginning with
unemployment. The two most prominent theories of unemployment are the ?natural rate? hypothesis
advanced by Friedman (1968), and the hysteresis hypothesis propounded by Blanchard and Summers
(1987). These two theories have important implications on time series models of unemployment and the
assumptions on which they are based.
Friedman extended the concept of the ?natural rate of interest? of Wicksell, which says that there
is some natural rate of interest that the Fed cannot influence without inducing inflation, by applying the
same idea to unemployment, which the Fed might also try to influence. Friedman was clear on the point,
often misunderstood, that the natural rate of unemployment does not stay the same, but can rise largely
because, says Friedman, of government policies that create frictions in the labor market. Nevertheless,
the natural rate theory suggests that, ceteris paribus, unemployment does revert back to some natural rate.
This precludes unemployment from being guided by a unit root process that is defined as having a mean
rate that varies with time.
In contrast to Friedman, Blanchard and Summers (1987) were writing in the wake of high
unemployment in Europe that had been persisting for nearly twenty years. They offered a framework that
is often called the ?insideroutsider? theory of employment, whereby, in the event of a shock that causes
some (the outsiders) to lose jobs, firms retain those more valuable workers (the insiders) who would then
wield bargaining power in order to ensure their own job security. Such bargaining activity pushes the
equilibrium wage higher, thus obviating competition for jobs that outsiders present, in light of their recent
joblessness. Blanchard and Summers point out that, from 1980 to 1986, unemployment rates in three
major European countries ? the U.K., France, and West Germany ? had nearly doubled through steady
15
increases over those six years, without much sign of any meanreverting tendency. Such a persistent rise
could not be explained by either monetarist or Keynesian theories. The contribution of Blanchard?s and
Summers? research is that it has shed light on the phenomenon of ?persistence? (longmemory) , which
describes how unemployment rates are in part functions of historical shocks, the effects of which never
fully die out. This is now regarded as one of the more important stylized facts of unemployment of which
theorists and modelbuilders alike must take account.
Turning to capacity utilization, this measure has long been of interest to macroeconomists,
particularly in its being viewed as a leading indicator of inflation (for a brief survey, see Finn (1995)).
Garner (1994), for example, argued that both the unemployment rate and capacity utilization, as measures
of ?resource tightness? for labor and capital respectively, give ?consistent signals about U.S. inflationary
pressure?. Garner also argued for the existence of a ?stableinflation? rate of capacity utilization? ? a
?natural rate of capacity utilization?, as it were ? of about 82%. Several other studies (e.g., Davies, 1994),
many of which just barely predate the investment boom of the late 1990s, reinforce the view of capacity
utilization as a harbinger of inflation.
Recently, however, this view has been questioned. Bansak, Morin, and Starr (2007) speculate that,
during the investment boom of the late 1990s, new technology might have enhanced capacity in a way
that made excess capacity cheaper, and hence afford firms the opportunity to operate at lower average
capacity so as to be able to handle any potential upswings in demand. By the same token, expansion of
capacity is also made cheaper, so that firms need not raise their levels of excess capacity. The effects of
technology on capacity utilization are far from obvious. However, Bansak, et.al. found using panel data
of 111 manufacturing industries that technological change tends to dampen the levels of utilization among
manufacturers.
Should the socalled ?technologyboom? of the late 1990s be seen as a ?structural break? in terms
of how firms utilize their capacity levels against the backdrop of the overall economy? Even in the
absence of the results of Bansak, et al., few would deny the marked differences in how capital is utilized
today, versus how they had been utilized in the 1940s and 1950s, with assembly lines of workers,
16
complemented by fixed largescale machinery. Further, theory must account for how capacity utilization
had either remained flat or had been declining during the late ?1990s, when investment boomed and
unemployment fell below 4% (see Figure 2, Panel A). On the whole, one would expect unemployment
and capacity utilization to be negatively correlated; perhaps anomalously, they had been trending in the
same direction during much of this boom period.
Because theory is either silent or conflicted as to the true statistical nature of these two variables,
the applied researcher?s efforts are hampered by a kind of a priori uncertainty as to whether to consider
unemployment and/or capacity utilization as stationary or not. Generally, one should not proceed to test
for cointegration if one or more variables is , given how different the datagenerating process would
be from that of an variable. It is clear that, in testing for stationarity, the results one arrives at depend
crucially on the decisions made by the empirical researcher, such as the number of autoregressive lags to
include, the sample size, the sample period, and the model specification. There are several ways to deal
with this uncertainty. Osterholm (2010) and Emerson (2011), to give two examples, both test for
cointegration ? the latter for the U.S., the former for Sweden ?between rates of unemployment and labor
force participation by applying the DFGLS test of Elliot, Rothenberg, and Stock (1996). This test is
useful for nearintegrated variables for which the ADF tests have low power; in other words, the latter set
of tests tends to accept the null hypothesis too often. The DFGLS is designed to take local dynamics into
account. Mustafa and Rahman (1995) test for cointegration between U.S. capacity utilization and U.S.
inflation. Their ADF results suggest that capacity utilization is nonstationary; however, they use monthly
data, which in general are ?noisier? (more local variation), but at the same time augment the local trend
effects due to the data?s low frequency. Their sample size of eleven years is also relatively small.
Transformation of the data is another way to overcome uncertainty. Koop and Potter (1999) remind us
that working with series like unemployment and capacity utilization is especially difficult because, on the
one hand, they are ?bounded variables? (i.e., they are percentages) that preclude global unit root behavior,
but on the other hand, they are nearintegrated variables that exhibit strong, local persistence that
confounds traditional unit root tests ? and thus traditional tests for cointegration. They suggest taking a
17
(nonlinear) logistic transformation, , in order to dampen the local trendfollowing
behavior.
Presented in Table 1 are our results of the ADF tests on both capacity utilization and
unemployment. We underscore how pronounced the uncertainty is by reporting the results of different
sample periods. All equations include a constant, and different lag lengths, given by , as recommended
by the Akaike Info Criterion (AIC) and the Schwarz Info Criterion (SIC) are also reported. The ADF
regression equation is the following:
?
It is well known that having too many lags will decreases the power of the test, and having too few will
create enough autocorrelation to bias the test in favor of rejecting the null hypothesis of a unit root.
However, for our data, it appears that adding more lags seems to produce higher pvales and hence lower
rejections rates of the null. What the results also reflect is that, as you increase the sample size but keep
the number of lags low, there is a higher probability of rejecting the null.
The reader will be able to note the ambiguity of these results. For unemployment, a sample size
of forty years and a specification that includes nine lags yields a conclusion according to which we cannot
reject the null, that is, unemployment has a unit root and is therefore nonstationary. By contrast, a sample
size encompassing ten years, with a lag length of two suggests rejection of the null, i.e., unemployment is
stationary. The results on capacity utilization provide roughly the same picture.
The augmented DickeyFuller results reinforce the a priori uncertainty discussed earlier.
Unreported here, stationarity tests using the DFGLS procedure of Elliot, et.al (1996) give evidence that
both series possess a unit root (nonstationary). It is the results of this procedure that are often reported in
empirical work to support the hypothesis that the series in question possesses a unit root. We now, in the
face of this uncertainty, take the step taken by many of the authors cited above and proceed to test for
cointegration using the twostep EngleGranger (1987) procedure.
18
The first step is to estimate the longrun equation, the second, to test the residuals for stationarity.
One of the pitfalls of this twostep procedure is that, in the absence of theory, there is little guidance in
deciding which variable to set on the lefthand side. We run the longrun equation twice, once with
unemployment (denoted ) as the dependent variable and once with capacity utilization ( ) as the
dependent variable,
?
[ ] [ ]
?
[ ] [ ]
where OLS standard errors are given in parentheses below the coefficient estimates, tvalues are given in
brackets, is the coefficient of determination, which tells us the portion of the variation in the
dependent variable that is explained by the regression, is the standard deviation of the residuals, and
is the DurbinWatson statistic of firstorder autocorrelation in the residuals. As neither of the
dependent variables follows a global trend, the addition of a linear time trend did little to contribute to the
fit of the regressions. Nor did it assist in raising the DurbinWatson statistic. Also, given that for both
regressions, and that our tstatistics are all highly significant, these two equations bear the
traits of the aforementioned spurious regression.
Figure 3 plots the residuals of the two longrun equations (Panel A for the equation with capacity
utilization as the dependent variable, Panel B for that with unemployment as the dependent variable). It is
evident that the two series exhibit strong persistence, but globally behave in a cyclical fashion. It is clear a
priori that feedback from unit root tests depends crucially on how pronounced the cyclicality is.
19
We proceed to the second step and test for the presence of a unit root in the residuals by applying
the augmented EngleGranger tests ? the counterpart to the ADF tests ? and using the critical values
calculated by Mackinnon (1991). A different set of critical values is needed because the residuals we are
testing are estimates and thus follow a distribution different from that followed by actual observations.
We specify the EG tests in a way to ensure that the stochastic disturbances (the epsilon term) are as close
to being white noise as possible. Because of the highly persistent nature of the two series, it is likely that
any autocorrelation in the residuals of the EG test may go beyond the first order. Thus, we implement the
BreuschGodfrey LM (k) test, which tests the joint hypothesis that all the autoregressive coefficients of
the residuals up to lag k are all zero. Once we achieve white noise in our specification, we can compare
the test statistic of our estimate against the EngleGranger critical values. Table 2 reports the results of the
EG tests. We also report some of the summary statistics of the epsilon term of the tests, because they are
informative in helping us ascertain the existence of white noise.
Neither of the teststatistics are significant, from which we conclude that there is no evidence of
cointegration between the two series. Both DurbinWatson statistics indicate that there is no firstorder
autocorrelation. (again) denotes the standard error of the regression; it is placed alongside the summary
statistics because it can be thought of as the standard deviation of the epsilon term. As to the white noise
of the epsilon term, the BreuschGodfrey test recommends a lag of k = 3 for both equations. Both the
mean and median are very close to 0. However, both the skewness and kurtosis cast considerable doubt
on the normality assumption of epsilon. A skewness that is significantly different than zero, and a kurtosis
significantly different from three, resulted in a high JarqueBera statistic in both cases (not reported, the J
B stat tests for the null of normality), and hence a very low pvalue. The two high levels of kurtosis are
conspicuous; they suggest there are many outliers on both tails. The equation in which unemployment is
on the lefthand side generates errorterms that are closer to being normally distributed.
We will now, in light of our results, review the limitations of the twostep EngleGranger
procedure. First, it has been amply noted by many authors that the longrun equation is static, that is, there
are no lagged dependent variables, and hence in most cases there is little information about shortrun
20
dynamics. In the Engle and Granger (1987) framework, the first step is to establish that there is a longrun
relationship, which is then followed by estimation of an ?errorcorrection model?. It is in this latter model
where the information on shortrun dynamics can be seen more clearly. Engle and Granger (1987)
proposed a famous theorem known as the ?Granger representation theorem?, which says that if two
variables are cointegrated in the longrun, then their short run dynamics can be well captured by an error
correction model of the following general form:
? ?
where ? is the lagged term of the ?equilibrium error?, that is, the error term of the longrun equation,
? . In the bivariate case, ? ? ? , where ? is the cointegrating parameter which was
estimated in the first step. This errorcorrection specification is quite useful. First, note that is
necessarily negative because the sign of the equilibrium error at time ?t1? should then be of the opposite
sign at time ?t?. can also tell us in percentage terms how much of the discrepancy between the long and
shortterm relationship is corrected within one time period. If, for example, ? and our data are
quarterly, then about 12.54% of the discrepancy is corrected within one quarter, or three months. In short,
the estimated coefficient on the lagged equilibrium error is a measure of the persistence of the
disequilibrium, or, how long it takes for the two variables to converge to their longrun values, ceteris
paribus.
Despite its usefulness, one cannot build errorcorrection models if one cannot establish the
existence of a longrun relationship between the variables. To this end, an unbiased and efficient estimate
of is essential. One of the first criticisms of the EngleGranger approach is the small sample biases in
estimates of . Banerjee, et al. (1986) showed this using Monte Carlo evidence and suggested that a
dynamic equation serve as the cointegrating regression. Cointegration has been defined as a special kind
of longrun relationship, and a sample consisting of monthly data spanning eight years (96 observations)
might not produce as efficient an estimate as a sample of quarterly data spanning fifteen years (60
21
observations). Numerous other methods of estimating longrun relations between time series using
dynamic specifications have been proposed (see Maddala and Kim, 1998, for a survey).
Another, more practical issue is the decision of which variable to classify as dependent. Of course,
this is not an issue when economic theory already has an answer. Further, in the case of a bivariate
cointegrating regression, should be unique asymptotically, and the fit of the regression depending on
which variable to place on the lefthand side should not be very different. In the equations we have
estimated, the two values are identical but both standard errors and our estimates are quite different.
Perhaps the issue can be explained in part by the findings of Ng and Perron (1997), when they studied the
normalization problem in cointegrating regressions containing two variables. They showed that, in a
bivariate cointegrating equation, having the less integrated variable as the explanatory variable creates a
bias that makes the superconsistency property of the OLS estimators discovered by Stock (1987) no
longer valid. They advised selecting as the regressand the variable that is less integrated. To consider
variables whose order of integration is a noninteger would take us into the realm of the ARFIMA
(autoregressive fractionally integrated moving average) models introduced by Granger and Joyeux (1980).
We will sidestep those models and consider instead what Ng and Perron recommended: they suggest
taking each series in first differences and ranking them according to their spectral density at zero
frequency. Eviews 6 includes a menu option on spectral estimation when using its builtin function for the
PhilipsPerron unit root test. In accordance with Ng and Perron, we apply the quadratic kernel to the least
squares residual, and used the Andrews procedure to select the bandwidth. The bandwidth on is
1.96, showing a slight increase over that for of 1.72. This suggests that capacity utilization is
integrated of a lower order (i.e., closer to zero) than unemployment, and that therefore, according to Ng
and Perron, the equation with capacity utilization on the lefthand side contains less of a bias than does
the equation with unemployment on the lefthand side.
We close this section by mentioning that some of these issues are circumvented by applying the
VARbased test of cointegration of Johansen (1995). Unreported here, the results from the Johanssen
procedure found evidence of cointegration between the two variables. We then followed the methodology
22
proposed in Osterholm (2010) of carrying out a Waldtype test whereby the following restrictions on the
cointegrating vector are implemented: and . If these restrictions are not rejected,
then the result of finding cointegration is attributed to the fact that one of the series is stationary. This is
particularly relevant when working with nearintegrated series: namely series that are not exactly in
possession of a unit root, but at the same time highly persistent. Both Wald tests were implemented and
neither of them could be rejected.
Part 2:A Review of Models of Discrete RegimeSwitches:
Time series models that incorporate regimedependent autoregressive processes were popularized
mainly by the threshold autoregressive (TAR) models of Tong (1990). A good description of the TAR
model is given in Enders and Siklos (2001): ?The basic TAR model?allows the degree of autoregressive
decay to depend on the state [that is, the regime] of the variable of interest.? If we take a simple, AR(1)
model,
and modify it in such a way that, for every ?t?, is generated by one of two linear models, with the first
order lag determining that model, , the AR(1) model becomes,
where Is an indicator function, whereby if the condition of holds, then and equals zero
otherwise. This is the simplest form of the TAR specification; more complicated ones include more
regimes. This general concept of regimedependence has been extended to unit root theory by Enders and
Granger (1998), when they analyzed interest rate spreads. It has also been applied to cointegration theory
by Balke and Fomby (1997), Enders and Siklos (2001) and Granger and Yoon (2002). Enders and
Granger (1998) allowed for asymmetric equilibrium adjustment to short and longterm interest rates by
attaching a Heaviside function to the coefficient of the firstorder lag to the variable denoting interestrate
23
spreads. Balke and Fomby (1997) theorized a scenario where equilibrium adjustment between two or
more cointegrated variables might not be active within a given threshold and, moreover, the equilibrium
adjustment process depends on the sign of the equilibrium error. They suggest using Wald tests to
determine whether such a specification is correct when doing applied work (their paper lacked an
empirical component). Enders and Siklos (2001) applied the TARlike specification to DickeyFuller
equations, when they, too, evaluate the relationship between short and longterm interest rates. Granger
and Yoon (2002) develop the idea of ?hidden cointegration?, whereby different errorcorrection equations
are specified for positive and negative equilibrium errors. They suggest that failure to detect cointegration
using the EngleGranger procedure might mask the relationship two or more variables might have in how
they respond to positive or negative shocks in their common factor restriction. In other words, there might
be a common attractor between the two variables in the event of a positive or negative shock.1
In each of these studies, however, it is important to note (at least) two common threads. First, it is
assumed that switches in regime are sudden, discrete, and discontinuous events. What these models say is
that, given the value of the indicator function, which tells us whether the adjustment process is within
some specified range, the variable will be set on some given regime. The second is the persistent problem
of how to estimate the cointegrating vector when the dynamic adjustment process is nonlinear. Of course,
when studying interest rates or exchange rates, economic theory already has something to say about long
run relationships and one therefore need not estimate the longrun equation. Since Engle and Granger
(1987), a lot of work has focused on how to circumvent the problem of testing for cointegration without
having to estimate the cointegrating vector. The literature on this topic is too vast to summarize but
perhaps its best reference work is Maddala and Kim (1998).
1 Hidden cointegration is, conceptually, perhaps the most straightforward way to test for nonlinear cointegration.
However, computationally it is somewhat cumbersome, in that you must difference each series you are working
with, separate out the positive and negative values, and calculate the cumulative functions for each. The
difference in these cumulative functions is what substitutes for the equilibrium error term in the traditional ECM
model. We estimated the two ECM equations and found some evidence that capacity utilization and
unemployment respond in a similar way to negative shocks.
24
Part 3: Smooth Continuous RegimeSwitches and the STAR Modeling Procedure
What if the transition from one regime to another is not discrete and sudden, but continuous and
smooth? There have been several models that include smooth transition functions , of which one of the
more famous is the smooth transition autoregression (STAR) model that has been popularized by
Terasvirta and Anderson (1992). A STAR model takes the following general form:
where , , , and . is a
continuous transition function of some lagged value of , where d is referred to as the delay parameter.
In the STAR framework, the delay parameter is determined empirically through a sequence of Lagrange
multiplier tests (about which more will be said in due course). One of the better known transition
functions is the logistic transition function. STAR models that incorporate the logistic function are
denoted ?LSTAR?. We will consider first the logistic function, G, which takes the following form:
[ ? ]
where is often referred to as the slope parameter, which measures the speed with which the series
moves from one regime to another. is a stochastic transition variable that determines the regime of the
series at time t. Finally, is a constant term and is dependent on , where + 1 is the number of
regimes. The most common model is one in which it is assumed that ; such models have been
applied by Terasvirta and Anderson (1992) to industrial production series for different countries, and by
Skalin and Terasvirta (2002) to unemployment series, also for different countries. This model is used to
characterize asymmetric behavior of a tworegime scenario, whereby the dynamic adjustment process is
different depending on the regime. In the context of business cycles, the tworegime scenario often
considered is that between expansion versus contraction of the overall economy. It should be noted that
there is a lot of flexibility available to the researcher in choosing the transition variable . It can be a
25
variable other than the dependent variable or a linear combination of other variables. A deterministic time
function could even serve; this latter choice makes for a model, abbreviated TSTAR, which has been
applied by Dueker, Owyang, and Sola (2010) to the unemployment rate in order to allow for variation in
the natural rate of unemployment. One will also note in that, when , and the LSTAR
equation thereby degenerates into a linear model. On the other hand, as , the model becomes the
TAR model which, as discussed previously, includes the Heaviside function as an indicator of regime.
If we restrict ourselves to the case where , the transition function now takes
the following form:
[ ]
It is observed that, as , and as ; in both cases, the transition
function will degenerate into a constant. is thus a continuous function bounded between 0 and 1. Those
autoregressive coefficients of the terms that are products of the transition variable will change smoothly
as the transition variable moves within these two extremes.
An alternative, exponential specification takes the following form:
[ ]
STAR models that have this function serve as the transition variable are denoted ESTAR. In contrast to
the LSTAR model, the exponential function is symmetric around , because
, and . The ESTAR can be used in situations where large deviations from a long
run relationship initiate the meanreversion process, but small deviations do not. A common application is
exchange rates, where small deviation from purchasing power parity will not be corrected through
arbitrage due to the presence of transaction costs. Large deviations, however, are far more likely to induce
arbitrage behavior. It is the transaction costs, such as the premium on a futures contract, that are the
frictions that preclude PPP from holding when discrepancies from equilibrium are small. In short, the
26
speed of mean reversion is dependent on the size of the discrepancy. Small to midsize discrepancies are
eliminated much more slowly than large discrepancies.
Our task is to test the hypothesis of linearity of our two series against a STAR alternative. This
amounts to testing for restrictions on coefficients. A conventional way of carrying this out involves
estimating a restricted and unrestricted model, finding for each the residual sum of squares, and
calculating an Fstatistic to which we compare the critical values of an Fdistribution. Another way to test
a hypothesis would be to estimate only an unrestricted model, and perform a ttest on the restriction. This
is known as the Wald procedure. The Ftest requires the estimation of both models, and the Wald
procedure only requires estimation of the unrestricted model. A third method, the Lagrange multiplier test,
requires only that we estimate the restricted model. Our hypothesis tests place linearity as the null
hypothesis, on which we impose the restriction that the coefficients on nonlinear terms are all zero. The
alternative hypothesis is that the data generating process follows a nonlinear STAR specification, which is
our unrestricted model. We thus elect to apply the LM test in that it affords us the convenience of only
needing to estimate the restricted, linear model.
The LMtest essentially calls for regressing the OLS residuals from the linear portion of the
model, on the partial derivatives of the dependent variable with respect to each of the model?s parameters.
Suppose we had the following2:
We find / , and then estimate an auxiliary regression,
where is the coefficient for each of the partial derivatives of with respect to
is a vector of ones since / . After estimating the auxiliary regression, our test
2 This example is taken from Enders (2004, pg. 410)
27
statistic is , which asymptotically follows a chisquare distribution with degrees of freedom, p
being the number of parameters of the equation. If our test statistic exceeds the chisquare critical value,
we reject the null of linearity.
Now, let us implement the same procedure to an LSTAR model:
[ ]
where , , and . Our null hypothesis is that the
true model is linear, enforcing the restriction that . However, there is a problem. As noted, at
. But the partial derivatives of with respect to each of the parameters, evaluated at ,
are 0 for . In other words, those latter three parameters are unidentified under the null, and as
such are classified as ?nuisance parameters?.
Therefore, the LM test cannot be applied for hypothesis testing purposes. Normally, in this case,
the applied researcher might be compelled to follow either the Ftest or Wald procedures mentioned
earlier. This would be cumbersome because, on the one hand, both procedures would involve estimation
of a nonlinear equation, which is generally more difficult compared to linear ones. On the other hand, she
would also have to run these procedures for a number of models allowing for different nonlinear
alternatives, then running the diagnostic tests for each, and then, finally, using judgment as to which
model provides the best fit. All of this, of course, would have to take place in the absence of any guidance
economic theory might provide as to the nature of the nonlinear relationship.
Despite the bleakness of the situation, there is a framework which has been designed to deal
effectively with the nuisance parameter problem. Testing for linearity using Lagrange multiplier tests was
considered by Luukkonen, et al. (1988) and then more fully in Granger and Terasvirta (1993). The former
suggested taking a Taylor expansion of the transition function around , and reparameterizing the
28
auxiliary equation by substituting the resulting Taylor expansion for the transition function. Let us
reformulate our LSTAR model in this way:
where ( ) ( ) is the logistic transition
function. If we apply a firstorder Taylor expansion to and reparameterize, our auxiliary equation
becomes,
where ( ) , is our transition function (a lagged value of ) and
, where R is the remainder term from the Taylor expansion. Under the null, however,
=> . Our null can now be specified:
[ ]
The test statistic, denoted , with the subscript reflecting the fact that the test statistic is based on a
Taylorexpansion of the first order, is . We reject the null of linearity if the test statistic is
greater than the chisquare critical value.
As noted, we select as the transition variable a lagged value of , where d is the delay
parameter. In this case the regressors are lags, either in levels or differences. Further, as noted by
Luukkonen, et al. (1988), the specification does not have power in the case in which only the
intercept differs across regimes. A test that does is the test, which requires a thirdorder Taylor
expansion of G. Let [ ] , where
In order to take a thirdorder Taylorexpansion, we take the first three partial derivatives
of with respect to , evaluated at
29



Now, we insert these into the form of the Taylorexpansion:
( ) ( )
We reparameterize our nonlinear transition function such that both the linear and cubed terms are
included, leaving out the square term since the second derivative is zero. We are now ready to test for
linearity.
However, an additional complication arises in deciding what conclusion to draw if we were to
reject linearity. As noted, the alternative has nested within it several nonlinear specifications. Thus,
rejecting the null does not necessarily mean that we can accept the alternative, which we have arbitrarily
set up as being of the LSTAR form. This point is discussed in detail in Granger and Terasvirta (1993, pg.
101), and becomes clearer as we remind ourselves that, under the LM procedure, we are not estimating
the nonlinear equation.
If linearity were to be rejected, how might the mystery of the exact nonlinear nature of the data
generating process be solved? Fortunately, there is a procedure, discussed in Granger and Terasvirta
(1993, pg. 150) and Terasvirta (1994). We have already taken the first three partial derivatives of the
logistic transition function, and found that the second one, evaluated at is zero. If we were to
do the same thing for the exponential transition function, defined above, we would find that the first and
third derivatives, again evaluated at , are both zero but the second is not zero. These facts will
have some bearing when we test the restricted models. Thus, in testing for linearity, we nest the squared
term between the linear and cubic terms. As noted by Enders (2004, pg. 412), ?the key insight in
Terasvirta (1994) is that the auxiliary equation for the ESTAR model is nested within that for an LSTAR
30
model. If the ESTAR is appropriate, it should be possible to exclude all of the terms in the cubic
expression.? Our new, reparameterized test equation for linearity is the following:
where ( ) It is to this equation, newly freed of any nuisance parameters and based
on the Taylor expansion, that we shall apply our tests for linearity. Our test for linearity tests the joint
hypothesis that the coefficients of the terms that include as factors are all zero:
. Due to the size distortions of the chisquare distribution for small samples, we will instead calculate
test statistics following an Ftest. Table 3 gives the template, which includes the regression specification
along with the degrees of freedom for each of the hypotheses. The delay parameter, d, is determined from
the data, and should be large enough to capture adequately the transition from expansion to contraction, or
vice versa. The hypothesis is run for a number of different delay parameters. If linearity is rejected for
more than one value of d, the decision rule is to select that value with the lowest corresponding pvalue.
We conduct the test for values of d ranging from 2 to the largest lag, p.
The question of which kind of nonlinearity ? ESTAR or LSTAR ? to accept as our alternative
model in the event of rejecting the null remains unanswered. Different sets of decision criteria have been
adopted in different studies. We will use those criteria adopted by Terasvirta and Anderson (1992),
Granger and Terasvirta (1993), and Akram (2005). The sequence of LM tests is based on the following
hypotheses:
31
If (which is run for a number of different delay parameters) is rejected most strongly (yields the
smallest pvalue), an ESTAR or LSTAR(2) model is appropriate. Otherwise, the LSTAR method should
be selected. These decision criteria were recommended by Terasvirta (1994).
To summarize the testing procedure, a threestep approach is followed:
1. Estimate the linear AR(p) model, both to determine the lagorder p and obtain the residuals
2. Test linearity using a sequence of LMtests on the auxiliary equation for various delay
parameters and for the various hypotheses, thus establishing both the proper delay parameter
and the nonlinear functional form in the event of rejecting linearity.
3. If linearity is rejected, estimate the appropriate STAR model using nonlinear least squares
(NLS).
It will be clear once we complete the second step whether a nonlinear STAR model is appropriate.
If it is appropriate for both series, we will be able to compare the nonlinear behavior of the two series by
graphing each of their transition variables, both across time and across different values of their respective
transition variables. The purpose of the former is to verify that the estimated transition function
corresponds to the timing of the business cycle (for which we use NBER data). The purpose of the latter
is to attain an understanding of how smooth the transitions between one regime and another are, which is
largely based on how large the estimated slope parameter, gamma, is, as larger values indicate very swift
transitions. We subject unemployment and capacity utilization to the STAR analysis in the next chapter.
32
CHAPTER V
MODEL ESTIMATION
The first step of the STAR modeling procedure is to estimate the linear portion of the model, both
to determine the lag order p and obtain the residuals. Following Terasvirta, Tjostheim, and Granger
(2010), our AR models will be regressed in first differences, with a constant and firstorder lag included
on the righthandside. Also following these authors, we will also, in the second stage of the modeling
procedure, nominate as possible transition variables lags of ?long differences?, with the lags ranging from
2 to 9: [ ].
We begin by estimating the linear portion of the unemployment series, denoted , by fitting an
AR model with a lag of 9 as recommended by the AIC:
?
where OLS standard errors are given in parentheses below the parameters, ? is the residual standard
deviation, and are, respectively, skewness and excess kurtosis in the residuals, LBQ(p) is the p
value for the LjungBox Qstatistic for nonstationarity of the residuals based on an autocorrelation
function of an order up until and including p, (a very high pv u i i ic tiv of ?whit oi ?
33
ARCH(q) is the Engle LM test for ARCH effects of order q, and is the BreuschGodfrey LM test
for autocorrelation in the residuals up till and including lag j. The numbers inside the parentheses
following the LM tests are the corresponding pvalues. We note that the residuals are positively skewed,
with a high kurtosis. We also note strong ARCH effects, as all ARCH tests for lags lower than 10 yield a
pvalue less than . Nonetheless, the above model seems adequate in that it appears entirely free of
autocorrelation.
Let us now construct the AR model for capacity utilization, denoted :
)
?
The AIC recommends twelve lags for the capacity utilization series. We have arrived at results similar to
those for the unemployment series, namely, that there is evidence of ARCH effects in the residuals, and
the kurtosis for the residual series is very high. The latter effect may be attributed to a few high residuals
in the beginning of the series. Like the unemployment series, the BreuschGodfrey tests indicate that there
is no autocorrelation, thus making the above, linear equation a suitable restricted model against which we
will test for a STAR alternative. We note in passing that we attempted to include quarterly dummies in
each of the AR equations in the hope of capturing the wellknown seasonality properties of the two series.
None of the quarterly dummies proved significant.
The second step is to conduct the sequence of LM tests for the coefficient restrictions of the
reparameterized equation on page 30. As noted, the purpose of the LM tests is twofold: to determine the
correct delay parameter d, and to infer the nonlinear form of the data. Table 3 reports the series of null
hypotheses (the restriction to be tested), their corresponding specifications and degrees of freedom. For
34
the regression equations, ? are the estimated residuals from the linear, AR regressions. The idea is to test
whether the residuals exhibit the hypothesized nonlinearity that was not captured in the AR reregressions.
Table 4 presents the results from the LM tests; the values reported are pvalues of the Fdistribution. For
the unemployment series, a delay parameter of two was rejected with the lowest pvalue for all
hypotheses, though values of eight and nine are close contenders. We suspect that a delay parameter of
two is perhaps too small, in that it may not contain timely information about the regime of which the
transition function is designed to take account. The results for capacity utilization are more interesting and
equivocal: a delay parameter of two was rejected most strongly for , but a delay parameter of nine was
rejected about as strongly for , in that both yielded roughly the same pvalue. Strongest rejection of
the latter hypothesis suggests that either an ESTAR or LSTAR (2) specification is needed. Therefore, we
will attempt estimation of three models for capacity utilization: an LSTAR(1) model with a delay
parameter of two, an LSTAR(2) model, and an ESTAR model, both including a delay parameter of nine.
Proceeding to the third step of model estimation, we have successfully estimated an LSTAR(1)
model for unemployment. However, the model incorporating the delay parameter of eight provided a
slightly better fit than the model in which the delay parameter of two was used:
{ [(
) ]}
( )
35
? ? ?
where pJB is the pvalue for the JarqueBera test of normality of the residuals, with the null hypothesis
being that the residuals are normally distributed. ? is the standard error of the linear model. (q) is a
modified version of the BreuschGodfrey Lagrange multiplier test of no autocorrelation of up till lag q.
The pvalues reported above are based on the Fversion of the test. The null hypothesis is that there is no
autocorrelation. The test is modified for nonlinear specifications, such that, instead of regressing the
residuals on lagged dependent variables, they are regressed on the numerical derivatives of the lefthand
side expression with respect to each of the model?s parameters (Terasvirta, et.al. (2010, pg. 382)). The
lefthand expression in matrix form is:
c
And the derivatives on which the residuals are regressed are given by:
where c . The residuals are also regressed on their own lags up to order q.
The LSTAR(1) equation for unemployment shows significant improvement in its over its
linear version, from .382 to .496 (about a 30% improvement). Further, the standard error of the LSTAR
equation is about 8% less than that for the AR(9) linear specification. These results are on par with
Terasvirta, et.al. (2010, pg. 406), who estimated an LSTAR(1) model for the monthly unemployment
series. While the JB pvalue remains low, it also shows significant improvement over the linear model,
which yielded a value less than . The JB is sensitive to violations of the wellknown characteristics
of the normal distribution, namely, that there are no outliers (and thus a low kurtosis) and no skewness in
the distribution. It is evident that the LSTAR model has taken more account of some of the outliers in the
residuals of the linear model. Finally, the modified LM tests of no autocorrelation yield satisfactory
results in accepting the null.
36
Moving on to capacity utilization, we have found that the LSTAR(1) and ESTAR models have
consistently failed, in that they invariably resulted in exploding standard errors. Estimation of STAR
models becomes quite difficult when the true value of the smoothness parameter, gamma, is large. The
STAR model reduces to a switching TAR model when gamma is sufficiently large. This is a common
problem in STAR models, especially when the sample size is small, and is clearly indicated by an
unreasonably high standard error in the estimate of gamma. There is also the issue of setting initial
starting values for the coefficients, which is often required when estimating equations by nonlinear least
squares. If starting values are far off the mark, the builtin algorithms will have difficulty minimizing the
sum of squared residuals function. In such cases, standard errors and tstatistics will often either fail to
converge, or find merely a local as opposed to global minimum in the SSR function.
The LSTAR(2) specification, with the delay parameter of nine as indicated by our LM tests,
converged with much greater ease, and yielded the following:
+
+
{ [( ?
) ]}
( ) ( )
? ? ?
Compared to unemployment, the LSTAR(2) equation for capacity utilization showed less improvement
over its AR(12) specification. The did increase from .299 to .374, about a 25% improvement.
However, the standard deviation of the residuals for the nonlinear model only showed about a 2%
37
improvement over the linear model. The JB pvalue remains low, and there is evidence that some
autocorrelation remains in the residuals.
Figure 4 compares the residuals of the linear AR model with those of the STAR models for both
the series. Panel A compares the residuals between the linear and nonlinear specifications for
unemployment, Panel B for capacity utilization. It is seen that the residuals? divergence from zero peak at
a lower spot than do the residuals for the linear models. This may indicate either that the strong ARCH
effects or seasonality found in the linear models may have been attenuated by the transition function.
Figure 5 plots the transition function for unemployment , both over time and over different values
of the transition variable. Panel A displays the latter, Panel B, the former. Panel B includes shaded areas
representing NBERdated recessions. The transition function shows some consistency in its adjustments
during period of recessions. The dot plot, shown in Panel A, of the transition function is plotted against
the transition variable on the horizontal axis, where each dot represents at least one observation, and
reflects the monotonic increase characteristic of the stochastic process underlying the LSTAR(1) model.
Figure 6 does the same for the capacity utilization series. The graph to the left shows a situation
quite different from that seen in Figure 5. It shows that there are three regimes: the white region toward
the right, and the two orange regions on both sides of the white region. What this scenario indicates is that
capacity utilization follows the same dynamic adjustment process when the economy is near its trough,
and when it is quickly approaching its business cycle peak (the two orange regions). It follows a different
process when the economy is in some intermediate state (the white region). That the white region is
toward the right indicates that this different dynamic adjustment process occurs as the economy is closer
to its peak than to its trough. This makes empirical sense, in that, at this region, uncertainty as to the
economy?s recovery process will have been alleviated as it nears the white region.
To recapitulate what the different models between the two series suggest, the LSTAR(1) model
contains one constant term in the transition function, indicating that there are two regimes. This constant
38
term is understood to be the threshold at which the series changes in its dynamic adjustment. The gamma
parameter, often referred to as the ?smoothness parameter? or ?slope parameter?, reflects how smoothly
this change in regime takes place, with larger values of gamma indicating that the transition takes place
more or less instantaneously, and smaller values suggest that transition is more gradual. The LSTAR(2)
model contains two constant terms, indicating three regimes. As gamma approaches infinity, the model
becomes a TAR model. The LSTAR(2) model also nests a restricted threeregime selfexciting TAR
model (SETAR), where the restriction is that the linear models in the outer regimes are identical.
This chapter will close with a few observations on the STAR model process. While STAR models
are, of course, not without their faults, their power lies in their being able to model how macroeconomic
time series vary according to different states in the economy according to their transition variables. This
power is augmented by the fact that regimechanges need not be discrete (that is, marked by a zero or one
in an indicator function, as is the case with switching TAR models), and no a priori information as to the
timing of the regime need be available. Also, there is flexibility in the number of regimes by which the
datagenerating process is governed. There is also considerable flexibility available to the researcher in
choosing the transition variable, which can be any variable or a combination of variables. However, some
common sense must be exercised in that the transition variable must contain timely and accurate
information about the regimes. The LM tests for linearity are by no means flawless, in that linearity may
well be rejected equally decisively with several transition variable candidates. The protocol in this case is
to experiment with the different transition variables, and compare their relative fits. Nonetheless, the
series of LM tests will often prove to yield valuable information ? as they had in our case ? as to the
nonlinear nature of the variable.
Perhaps the chief disadvantage of STAR models is that they are extremely sensitive to
specification error. This makes them generally quite difficult to estimate. Nonlinear least squares
estimation will often fail to converge if the transition variable is misspecified, or, if it does converge, it
will often do so at a local as opposed to global minimum of the sum of squared residuals function, in
39
which case, neither the coefficient estimates, standard errors, nor tstatistics will be asymptotically valid.
This is the most difficult feature of nonlinear least squares estimation. Often, it will be necessary to carry
out a grid search for the two parameters of the transition function, and iterate estimation of the AR
parameters using starting values for the c until the sum of squares are minimized. For an intuitive
discussion, see Griffiths, Hill, and Judge (1993). Clearly, there is the problem of how to choose the initial
values, which are usually far from obvious. In addition, in the case where there are many parameters for
lag variables, having redundant variables may confound results, and there is often no a priori knowledge
as to which of the lag terms are redundant. Maximum likelihood is an alternative method. However, ML
is sensitive to the assumption of white noise error terms, for which STAR models may not be adequate in
the first place.
When estimating STAR models, one should first look at the standard errors to make sure they are
reasonable. If they are too high, they are likely to explode as redundant variables are successively deleted,
which would then render the entire model undefined. It is common practice (for instance, van Dijk, et.al.
(2002)), to identify and then successively delete redundant parameters when their tstatistics are less than
one in absolute value
40
CHAPTER VI
CONCLUSION
This thesis has compared the local dynamics of the U.S. rates of unemployment and capacity
utilization over different phases of the business cycle. To this end, we have successfully modeled the two
series via the STAR specification of Terasvirta (1994). These specifications allow for regime changes ?
that is, changes in the dynamic properties according to broader economic influences ? as well as for a
continuous transition process. It has been demonstrated that the asymmetry phenomenon is everpresent
in unemployment in the context of business cycles, though its severity varies according to the depth of the
trough and the height of the peak. It has also been demonstrated that there is more interregime symmetry
in the adjustment of capacity utilization, with two identical processes within the two outer regimes of
strong growth and recession, but different within the intermediate regime of slow to moderate growth.
Our results for the unemployment series are on par with previous work, including that of Skalin
and Terasvirta (2002), van Dijk. et. al. (2002), and Terasvirta, Tjostheim, and Granger (2010). This thesis
makes a contribution in modeling U.S. capacity utilization for the first time in a STAR context. It was
found that capacity utilization is best characterized by a threeregime, LSTAR(2) process, whereby the
local dynamics of the two outer regimes are more or less identical, yet different from the middle regime.
This finding may prove to be the basis of future research in comparing how input markets react to broader
economic influences.
Our STAR models for the two series indicate that capacity utilization reacts more quickly to
shocks than unemployment does and that the former exhibits the same local behavior at both the peak and
trough of the business cycle, but different behavior in the intermediate phase. This is empirical evidence
that capacity utilization is governed by a threeregime process. By contrast, unemployment follows a two
41
regime process, with different local adjustment depending on whether the economy is expanding or
contracting. In terms of the stationary versus nonstationary dichotomy, which has long been operative in
macroeconomic research, these results cast the dichotomy in a different light. It was shown in Chapter 4
that, over the sample period of 19482009, unemployment and capacity utilization are stationary
processes. The meanreversion feature by which the stationary process is characterized actually reflects
the two series? reversion to their local, regimedependent equilibrium, from which it then diverges to a
new equilibrium when the economy switches to a different regime, ceteris paribus. The stationarity of the
two series as verified by DickeyFuller regressions is a reflection of the cyclical nature of this multiple
equilibrium process, as opposed to a single equilibrium to which the series reverts with varying speed in
the event of a shock. This, we speculate, may be the reason why the results of the stationarity tests yielded
different conclusions for different time periods. This may also explain why linear tests for cointegration
between two persistent (fractionally integrated) variables like unemployment and capacity utilization,
tend to suggest either spurious OLS estimates (in the case of Engle and Granger (1987)), or spurious
cointegration (in the case Johanssen (1995)).
In closing, it is worth praising STAR models for their flexibility. There is, however, a tradeoff
between their flexibility and their difficulty, the latter occurring primarily at the nonlinear estimation
stage. They have been largely implemented in a univariate setting, though multivariate STAR models are
as of now a burgeoning field. This thesis also demonstrates that univariate STAR models may be a useful
alternative to methods of finding nonlinear cointegration, which remains a nascent field in
macroeconometrics.
42
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49
TABLES
Table 1:
Augmented DickeyFuller Regression Results
Table 1: Results of ADF Regressions of Unemployment and Capacity Utilization
Unemployment
Capacity Utilization
Information Criterion
Information Criterion
Sample Period
# of Years
in Sample
AIC
SIC
AIC
SIC
Lag Length ?Stat Lag Length ?Stat
Lag Length ?Stat Lag Length ?Stat
1948:Q11952:Q4 5 1 1.56
1 1.56
4 0.68
1 1.5
1948:Q11957:Q4 10 2 3.38**
2 3.38**
9 1.99
1 2.86*
1948:Q11967:Q4 20 9 1.95
1 3.38**
6 2.20
1 3.36**
1948:Q11987:Q12 40 9 2.02
1 3.17**
6 3.25**
1 4.18***
1948:Q12007:Q4 60 12 2.69* 1 3.73*** 12 2.73* 1 4.63***
Notes: AIC is the Akaike Info Criterion
SIC is the Schwarz Info Criterion
Critical Values are adapted from Fuller (1976, pg. 373)
* refers to rejection of null of nonstationarity at the 10% level
** at 5%
*** at 1%
50
Table 2:
EngleGranger Tests of Cointegration Results*
?
Dependent Variable ? tStat DW
? Mean Median Skewness Kurtosis
Capacity Utilization 0.067 2.223 2
3.98 8.63E16 1.04 0.64 3.03
Unemployment 0.054 2.38 2.02 1.18 1.99E15 0.012 0.11 2.76
*Critical Values taken from MacKinnon
(1991)
10% Critical Value: 3.4959
5% 3.7809
1% 4.3266
? ? ? ?
51
Table 3:
Template for Lagrange Multiplier Tests of Linearity
Hypothesis Regression Equation ChiSquare Degrees of Freedom FDistribution Degrees of Freedom
? ?
? ?
? ?
?
52
Table 4:
Results of LM Tests for Linearity
Table 4: LM Tests for STAR Nonlinearity for Quarterly Unemployment and Capacity Utilization: 1948Q12009Q3
Transition Variable Quarterly Unemployment Series
Quarterly Capacity Utilization Series
...
Hypothesis
Hypothesis
2 2.00* 5.5 1.36 8.55
1.19 4.61 2.99 2.3
3 3.99 8.62 1.86 .436
1.48 1.49 7.53 2.8
4 2.06 6.67 2.64 .0110
2.55 5.42 3.26 1.4
5 8.96 9.27 6.65 .994
2.61 1.44 3.44 1.9
6 5.07 3.19 4.67 .153
4.06 3.07 4.54 9.3
7 1.67 6.01 7.95 .128
1.41 6.97 5.11 .379
8 4.31 3.01 3.29 .0231
4.19 2.7 2.61 .023
9 8.93 4.37 2.09 .201
6.1 1.31 3.34 4.3
10    
2 8.04 2.27 .432
11    
5.75 2.26 2.37 .691
12     8.29 6.76 3.58 .099
d value with smallest 2 2 2 2 2 2 9 4
pvalue
53
ILLUSTRATIONS
Figure 1:
Quarterly U.S. Unemployment and Capacity Utilization
60 %
65 %
70 %
75 %
80 %
85 %
90 %
95 %
19 50 19 60 19 70 19 80 19 90 20 00
P a n e l A :
Q u a r t e r l y U . S . C a p a c i t y U t i l i z a t i o n
2%
3%
4%
5%
6%
7%
8%
9%
10 %
11 %
19 50 19 60 19 70 19 80 19 90 20 00
P a n e l B
Q u a r t e r l y U . S . U n e m p l o y m e n t
54
Figure 2:
(Un)employment and Capacity Utilization:
Over Time and Over the Business Cycle
8 0%
8 4%
8 8%
9 2%
9 6%
1 00 %
6 0%
6 5%
7 0%
7 5%
8 0%
8 5%
9 0%
9 5%
1 00 %
1 95 0 1 96 0 1 97 0 1 98 0 1 99 0 2 00 0
C a p a c i t y U t i l i z a t i o n E m p l o y m e n t
P a n e l A :
Q u a r t e r l y U . S . E m p l o y m e n t a n d C a p a c i t y U t i l i z a t i o n O v e r t h e B u s i n e s s C y c l e
e
m
p
l
o
y
m
e
n
t
c
a
p
a
c
i
t
y
u
t
i
l
i
z
a
t
i
o
n
S h a d e d b l u e r e p r e s e n t s N B E R  d a t e d r e c e s s i o n s
2%
4%
6%
8%
1 0%
1 2%
6 0%
6 5%
7 0%
7 5%
8 0%
8 5%
9 0%
9 5%
1 95 0 1 96 0 1 97 0 1 98 0 1 99 0 2 00 0
U n e m p l o y m e n t
C a p a c i t y U t i l i z a t i o n
P a n e l B :
Q u a r t e r l y U . S . U n e m p l o y m e n t a n d C a p a c i t y U t i l i z a t i o n
u
n
e
m
p
l
o
y
m
e
n
t
c
a
p
a
c
i
t
y
u
t
i
l
i
z
a
t
i
o
n
55
Figure 3:
Residuals of EngleGranger Test
 16
 12
8
4
0
4
8
19 50 19 60 19 70 19 80 19 90 20 00
P a n e l A :
R e s i d u a l s o f E n g l e  G r a n g e r T e s t :
D e p e n d e n t V a r i a b l e : C a p a c i t y U t i l i z a t i o n
3
2
1
0
1
2
3
4
19 50 19 60 19 70 19 80 19 90 20 00
P a n e l B :
R e s i d u a l s o f E n g l e  G r a n g e r T e s t
D e p e n d e n t V a r i a b l e : U n e m p l o y m e n t
56
Figure 4:
Comparison of Residuals between AR and LSTAR(q) models
 1. 2
 0. 8
 0. 4
0. 0
0. 4
0. 8
1. 2
19 50 19 60 19 70 19 80 19 90 20 00
A R ( 9 ) R e s i d u a l s
L S T A R ( 1 ) R e s i d u a l s
P a n e l A :
U n e m p l o y m e n t R e s i d u a l s f o r A R ( 9 ) a n d L S T A R ( 1 ) M o d e l s
6
4
2
0
2
4
6
8
1 95 0 1 96 0 1 97 0 1 98 0 1 99 0 2 00 0
L i n e a r A R ( 1 2 ) R e s i d u a l s
N o n l i n e a r L S T A R ( 2 ) R e s i d u a l s
P a n e l B :
C a p a c i t y U t i l i z a t i o n R e s i d u a l s f o r A R ( 1 2 ) a n d L S T A R ( 2 ) M o d e l s
57
Figure 5:
Transition Function for the LSTAR model of Unemployment:
0. 0
0. 2
0. 4
0. 6
0. 8
1. 0
 1. 3  0. 8  0. 3 0. 3 0. 8 1. 4
P a n e l A :
T r a n s i t i o n F u n c t i o n o f U . S . Q u a r t e r l y U n e m p l o y m e n t
G r a p h e d A g a i n s t T r a n s i t i o n V a r i a b l e
V
a
l
u
e
o
f
T
r
a
n
s
i
t
i
o
n
F
u
n
c
t
i
o
n
V a l u e o f T r a n s i t i o n V a r i a b l e
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
1 9 5 0 1 9 6 0 1 9 7 0 1 9 8 0 1 9 9 0 2 0 0 0
L o g i s t i c T r a n s i t i o n F u n c t i o n : U n e m p l o y m e n t
P a n e l B :
T r a n s i t i o n F u n c t i o n o f U . S . Q u a r t e r l y U n e m p l o y m e n t
G r a p h e d A g a i n s t T i m e
V
a
l
u
e
o
f
T
r
a
n
s
i
t
i
o
n
F
u
n
c
t
i
o
n
Y e a r
S h a d e d B l u e R e p r e s e n t s N B E R  d a t e d R e c e s s i o n s
58
Figure 6:
Transition Function for the LSTAR(2) model of Capacity Utilization:
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
1 9 5 0 1 9 6 0 1 9 7 0 1 9 8 0 1 9 9 0 2 0 0 0
L o g i s t i c T r a n s i t i o n F u n c t i o n : C a p a c i t y U t i l i z a t i o n
P a n e l B :
T r a n s i t i o n F u n c t i o n o f U . S . Q u a r t e r l y C a p a c i t y U t i l i z a t i o n
G r a p h e d A g a i n s t T i m e
V
a
l
u
e
o
f
T
r
a
n
s
i
t
i
o
n
F
u
n
c
t
i
o
n
Y e a r
S h a d e d B l u e R e p r e s e n t s N B E R  d a t e d R e c e s s i o n s
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
1
0
. 0
1
5
7
0
3
. 0
9
9
9
0
0
1
. 6
0
1
3
0
0
1
. 0
2
4
5
0
0
0
. 3
4
6
3
0
0
0
. 2
9
0
3
0
0
0
. 8
0
4
8
0
0
1
. 4
1
9
1
0
0
2
. 3
0
9
4
0
0
4
. 7
0
0
4
0
0
P a n e l A :
T r a n s i t i o n F u n c t i o n o f U . S . Q u a r t e r l y C a p a c i t y U t i l i z a t i o n
G r a p h e d A g a i n s t T r a n s i t i o n V a r i a b l e
V
a
l
u
e
o
f
T
r
a
n
s
i
t
i
o
n
F
u
n
c
t
i
o
n
V a l u e o f T r a n s i t i o n V a r i a b l e