an essay on cointegration and error correction models

An Essay on Cointegration andError Correction Models

Robert H. Durr

For political scientists who engage in longitudinal analyses, the question ofhow best to deal with nonstationary time-series is anything but settled. Whilemany believe that little is lost when the focus of empirical models shifts fromthe nonstationary levels to the stationary changes of a series, others argue thatsuch an approach erases any evidence of a long-term relationship among thevariables of interest. But the pitfalls of working directly with integrated seriesare well known, and post-hoc corrections for serially correlated errors oftenseem inadequate. Compounding (or perhaps alleviating, if one believes in thepower of selective perception) the difficult question of whether to difference atime-series is the fact that analysts have been forced to rely on subjectivediagnoses of the stationarity of their data. Thus, even if one felt strongly aboutthe superiority of one modeling approach over another, the procedure fordetermining whether that approach is even applicable can be frustrating.

Fortunately, within the past few years we have witnessed several devel-opments that promise to assist in the evolution of stronger time-serial models.Among these developments is the creation of a simple but powerful methodfor objectively assessing whether a time-series is stationary. Another has beenthe introduction of a new class of error correction models that are designed tocapture both the short- and long-term effects that one (or more) series mayhave on another. Finally, the phenomenon of cointegration has been intro-duced along with methods for its diagnosis and proper treatment. Thesedevelopments have occurred primarily within the realm of econometric litera-ture, where applied and theoretical treatments continue to emerge at a briskpace.

This essay surveys and assesses much of what is now known aboutintegration diagnostics, cointegration, and error correction models. Analysts

Heartfelt thanks go to Jim Stimson, without whose assistance this article would not exist. Iam also grateful to Neal Beck, John Freeman, Renee Smith, and Thorn Yantek for their feedback.Any remaining errors arise from my stubbornness in the face of good advice.

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186 Political Analysis

concerned with longitudinal studies of political phenomena stand to profithandsomely from an understanding of the fundamental notions and mechanicsof cointegration and error correction. We need not wait until the econometricdust has settled to come to grips with these fundamentals.

Part 1 of this article examines the characteristics and features of errorcorrection models. The integration of a single time-series and the cointegra-tion of two or more are discussed in part 2, along with the Engle and Granger(1987) two-step technique used to model cointegrated time-series. Part 3explores the degree to which causality may be inferred in the face of coin-tegration and/or error correction. In part 4 , 1 examine the differences betweenstandard error correction models and those developed via the Engle andGranger two-step process. Finally, in part 5, 1 develop an error correctionmodel of U.S. domestic policy sentiment in an effort to demonstrate theapplicability of this methodological approach within the context of politicalanalyses.

Error Correction Models

Error correction models were first introduced by Davidson et al. (1978) in astudy of the relationship between income and consumption in the UnitedKingdom. In general terms, error correction models (or ECMs) are appropri-ate when a priori theory dictates that the dependent variable under analysisexhibits short-term changes in response to changes in the independent vari-able(s) as well as long-term levels consistent with those of the independentvariable(s). The notion of a "moving equilibrium" captures the essence of thelevels (or long-term) relationship, which is the actual error correction portionof the model. Specifically, ECMs presume that there exists an equilibriumstate in which the levels of the time-series are typically located vis-a-vis oneanother.' If a shock disturbs this moving equilibrium by forcing the seriesfarther apart (or closer together) than "normal," this equilibrium error iscorrected over the long term as the dependent process targets a new levelconsistent with the equilibrium state.

Figure 1 shows a straightforward representation of a simple error correc-tion relationship between two variables with no random disturbances. Currentchanges in series Y may be represented as a function of both the laggedchanges in series X and the degree to which the two series were outside oftheir equilibrium in the previous time period. That is,

AY, = plAX,_l - /32[r,_, - (pyX,_{) - y). (1)

1. Equilibrium is used here in a less formal sense than is typical in economic literature,implying only that those phenomena maintaining an equilibrium relationship are loosely tied toone another.

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An Essay on Cointegration and Error Correction Models 187

Time

Y,.i + 1 • AX,., - 0.5 • [/,.! - (1 • X,.,)

^^^""Error Correction' in Yt

Fig. 1. A hypothetical error correction process

- 20]

In the example shown in figure 1, j8, = 1, fi2 - 0.5, /33 = 1, and y = 20.Substituting these values into equation 1 yields

K, = AX,_, - 0.5[y,_, - X,_, - 20]. (2)

In words, equation 2 tells us that the current change in Y is equal to the laggedchange in X minus one-half of the degree to which the series are more (or less)than 20 points away from one another. Notice that the equilibrium relationshipbetween X and Y, namely, the twenty-point spread between them, is disturbedeach time X changes directions. When X "turns downward," Y continuesupward in response to the lagged positive change in X, thus creating a positivedisequilibrium. When X turns upward, on the other hand, Y continues down-ward, producing a negative disequilibrium. In either scenario, the resultingdisequilibrium is corrected at the rate of 50 percent per time period (asreflected by the error correction coefficient of eqs. 1 and 2 above, /32 = 0.5).Since the downward shifts in X at time t produce a spread of 40 pointsbetween X and Y at time / + 1, 50 percent of the difference between thisspread and the equilibrium spread is corrected at time t + 2 [i.e., .5 * (40 —20) = 10]. This error correction process continues until the time-series arereequilibrated; that is, until they are once again 20 points away from oneanother.

Perhaps the strongest feature of error correction models is their capacityto incorporate the effects of both changes in and the levels of one (or several)time-series on another. In this sense, ECMs provide time-series analysts withwhat may be a golden mean between the two widely used modeling tech-niques that focus exclusively on either levels or changes. While restricting thescope of longitudinal models to levels has been purported to capture long-term

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relationships among variables of interest, the fairly great likelihood that suchmodels will exhibit spurious associations is well known. Such type-II errors"will be the rule rather than the exception" (Granger and Newbold 1974, 117;italics in original) when analysts are examining nonstationary time-series, andnonstationarity is quite common in those series of interest to political scien-tists and economists.2 Although a number of mechanical corrections formodels in levels have been developed,3 such corrections are not as attractiveas they were in the days of less flexible computer hardware and software.

Analyses of the relationships between changes in the independent anddependent variables, on the other hand, are generally able to solve the spu-rious regression problem, but at the cost of introducing a new set of diffi-culties that may be no less troublesome. Indeed, Granger and Newbold stateexplicitly that, in the midst of their demonstration of the perils inherent inregressions of levels on levels, they nonetheless "are not advocating firstdifferencing as a universal surefire solution to any problem encountered inapplied econometric work" (1974, 118). Analyzing time-series in changesrather than levels may erase not only spurious trends (in the case of time-series only indirectly related to one another), but also evidence of a systematiccomponent shared by variables that are, in fact, directly related. And as bothKing (1989) and Beck (1991) note, modeling changes in the dependent mea-sure as a function of (lagged) changes in the independent variable(s) assumesthat the effect of changes in the explanatory series never lasts more than onetime period. If the actual data generation process involves error correction (asin fig. 1), this assumption is indeed problematic.

Figure 2 relates the changes in the Y time-series to lagged changes in X.While the relationship is clearly strong, notice that the dramatic swings in Xlead to not only dramatic shifts in Y, but also the error corrections in Y. For thisillustration, such error-correcting changes extend roughly nine time periodsbeyond the shift in X that resulted in the disequilibrium and, as the graphplainly shows, are not related to short-term changes (i.e., lagged one period)in X. Thus, modeling changes in / as a function of only changes in X misses arather substantial portion of the variation in Y over time.

Setting aside questions of methodological integrity, the decision ofwhether to focus on either levels or changes in longitudinal analyses posestheoretical problems as well. Suppose, for example, that we are interested in

2. A discussion of nonstationarity is taken up in greater detail below.3. Such corrections include the use of a lagged observation of the dependent variable on the

right-hand side of the regression equation, variable transformations resulting in generalized leastsquares (GLS) models, and instrumental variables. For a discussion of these techniques, seeOstrom 1978. See Mills 1990 for a summary of a number of studies that detail the shortcomingsof these corrections. See also King 1989 for a discussion of the weaknesses of these "standard"time-series regression techniques.

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Lagged changes in X (AX,.]I " " C h a n g e s in Y (AY)

^ ^ ^ ™ Error correction In Y

Fig. 2. Accounting for error corrections in Y: changes in Y andchanges in X lagged one period

V

the degree to which the president's willingness to grant press conferences issome function of the national economy. We may find that those months inwhich the president meets with the press often correspond to those with lowlevels of inflation and unemployment.4 Such a finding is consistent with apriori theory: namely, the president is more willing to engage the sometimeshostile news media when there is less to be defensive about and more to takecredit for. But this approach ignores the important context within whichvarying levels of inflation and unemployment may be experienced. For ex-ample, an unemployment rate of 8 percent (up from only 5 percent lastquarter) at the beginning of what has been predicted by experts to be a longand painful recession may send the president running for cover, whereas thesame level in the midst of a recovery (down from 11 percent last quarter) islikely to give the president plenty of reasons to be accessible.

Noting that the same level of unemployment produced significantly dif-ferent behavior on the part of the president, we decide to focus our analysis onchanges in inflation and unemployment instead. But does this new approachsolve the problems encountered by analyzing levels only? Imagine an eco-nomic windfall that generates a need for new members of the work force,resulting in a dramatic, one-time 4 percent decrease in unemployment (from 6percent to 2 percent) that is sustained for several years. As we would expect,in the quarter immediately following the large dip in unemployment, thepresident calls an unprecedented number of news conferences. In the face of asustained economic upswing, the president maintains a high level of acces-

4. Ignore for the sake of the pedagogic moment that inflation "levels" are most oftenexpressed as the percentage change in the Consumer Price Index.

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sibility for a long time, discovering myriad new and creative ways to holdeven more press conferences. Yet the unemployment rate has not changed in away consistent with this increase in accessibility to the media. Finally, thepresident cuts back, realizing that time with the media may be better spentattending to reelection strategy sessions. Once again, this negative change inthe number of press conferences is not associated with a corresponding posi-tive change in unemployment. Although we believe that the president's deci-sion to hold a press conference is, indeed, a function of economic conditions,we are frustrated by our marginal results.

An answer to these dilemmas may be found by suggesting that presiden-tial news conferences are related to the state of the economy as an errorcorrection process. Specifically, increases and decreases in the number ofpress conferences held from month to month are a function of both recentchanges in economic conditions and the degree to which the president's acces-sibility to the media is consistent with the current nature of the economy.Notice how this modeling approach circumvents the problems discussedabove. The number of press conferences is still expected to be related to thelevels of unemployment and inflation, but only as partners in a moving equi-librium. Thus, the coincidence of a high number of conferences and highlevels of inflation and unemployment (and vice versa) is permitted with theexpectation that the level of presidential accessibility will eventually changeso as to reattain an equilibrium state vis-a-vis the economy. Furthermore, theinclusion of the error correction component relating levels to one anotherpermits the number of press conferences to get "back on track" in the after-math of a shock.

Returning to the hypothetical error correction relationship between X andY shown in figure 1, figure 3 graphically demonstrates the capacity of the errorcorrection component (i .e. , that portion of the model designed to capture thedegree to which the levels of each series are out of equilibrium) to explainerror correction in Y. Unlike a model that focuses only on lagged changes ofthe independent variable (fig. 2), the inclusion of an error correction compo-nent permits one to capture the changes in Y driven by the degree to whichlagged levels of both variables are outside of their equilibrium. The dottedline shown in figure 3 represents this equilibrium relationship (y,_, - X,_, -20). When the lagged levels of X and Y differ by 20 points, the series may besaid to be in an equilibrium state vis-a-vis one another. When the difference isgreater or less than 20, however, the resulting disequilibrium produceschanges in Y in a direction consistent with reequilibration. Thus, the correctmodel of Y is one that incorporates both the lagged changes of X and thelagged disequilibrium in levels. As I discuss in greater detail below, thisspecification of a model that takes into account both short- and long-term

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Error Correction Component (Vj-1 - X r-1 - 20)• Changes in Y (AY) ^ — E r r o r Correction in Y

Fig. 3. Accounting for error corrections in Y: changes in Y and theerror correction component

effects addresses the methodological problems inherent in analyses of onlyshort-term or long-term relationships.

Integration, Cointegration, and the Engle and GrangerTwo-Step Method

As Beck (1991) notes, error correction models may be appropriate to a time-series analyst whether or not the series of interest are purely (unit root)integrated. Indeed, the principal strength of an error correction model—namely, its ability to capture both long- and short-term relationships amongtime-series—seems unrelated to the question of whether or not a series isstationary in its levels. But as Granger (1986) and Engle and Granger (1987)point out, error correction models are especially applicable in the face ofcointegration. In this section, I explore integration and cointegration, as wellas the Engle and Granger two-step technique for the development of errorcorrection models.

Integration

Many time-series of interest to both political scientists and economists have atendency to wander or drift in one direction or another for an extended periodof time with current values appearing to be as much a function of past valuesas anything else. Such time-series are often integrated. A time-series is inte-grated if its current value may be expressed as the sum of all previouschanges. In other words, an integrated (or additive) process is one for which

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the value of any given point is a function of all past disturbances. As such,integrated processes have infinite variance and no mean.5 Consider the follow-ing. Let

K,= y , _ , + e , , (3)

where e, represents a random shock to series Y at time /. If Yo = e0, and

y, = ^o + elt (4)

then substitution produces

yt = e0 + elt (5)

which over time yields

y, = 2 e,. (6)

Thus, an integrated time-series is one for which the effect of any changeremains within the series forever. If something pushes a time-series up 20points on some scale, the effect of that "something" is never erased. It may becountered by some other force (or forces) pushing the series downward 20points at a later time, but the influence of the initial 20-point gain is as great attime t + 100 as at time r + 1.

Setting aside (for the moment) questions concerning empirical identifica-tion, it is important to think about why a variable may or may not be inte-grated. Strictly speaking, a time-series is integrated if one (or both) of twoscenarios hold: either the influence of the series' history remains at "fullstrength" forever, or the series is a function of other integrated processes.Regarding the example of presidential approval, it seems difficult to argue thatthe first scenario is correct. Were it to be, it must be the case that the impact ofapproval-enhancing and -diminishing effects remains forever at an equal mag-nitude. Consider the impact the March, 1981, assassination attempt had onRonald Reagan's approval rating. While the initial jump of approximately 13percentage points in the Gallup polls was to be expected, does it make senseto think that the impact of the attempt on Reagan's life would be as great a

5. Of course, any finite sample—whether cross-sectional or longitudinal in nature—has anidentifiable mean as well as finite variance. But time-series are, by their nature, infinite processes.It is in this sense that integrated time-series have no constant mean and infinite variance.

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factor in his approval rating in March, 1987, as it was six years earlier? Amore defensible argument might be that the impact of events on presidentialapproval decays slowly over time without ever completely vanishing (i.e.,approval has an autoregressive error structure), or the impact of events does,indeed, vanish after only one or two time periods (a moving average errorstructure).

Approval may nonetheless be integrated if it is a linear function of one ormore processes that are themselves integrated, such as (perhaps) evaluationsof the national economy. As I discuss in greater detail below, such a relation-ship between or among integrated processes is likely to result in cointegration.In sum, the point of this digression is to remind analysts that empiricaldiagnoses of time-series data are necessarily a function of a finite sample of arealization of the process in question. While the sample of the realization mayor may not be stationary, analysts should consider whether their empiricaldiagnosis is consistent with theoretical expectations.

Cointegration

The phenomenon of cointegration may be explained by making reference totwo or more integrated time-series that never drift very far apart from oneanother, or, stated another way, that maintain some sort of equilibrium rela-tionship vis-a-vis one another. A simple example cited by Engle and Granger(1987) is the case of time-series representing short- and long-term interestrates. Whatever else moves either series through time, we would certainly besurprised if they drifted away from some typical difference in levels andmaintained such a state. Surprised and rich.

Defined somewhat more formally, a set of integrated time-series is saidto be cointegrated if some linear combination of the series in levels produces astationary series. Consider two time-series, X and Y, each of which are inte-grated of order one.6 As Engle and Granger (1987) note, it is typically thecase that a linear combination of these series will produce a third that is alsointegrated. However, there may exist a linear combination,

Z,= Y,- PX,, (7)

that is stationary. Note that Z, may be obtained by regressing Y, on X,\ that is,

Y, = PX, + Z,, (8)

6. The order of integration, typically denoted as l(d) where the series in question isintegrated of order d, implies that the series must be differenced d times to obtain stationarity.Thus, a first-order integrated series is stationary after taking first differences.

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and subtracting f3X, from both sides of the equation. If Z is stationary, we mayassume that X and Y maintain an equilibrium relationship that manifests itselfin the form of a "common stochastic trend" (Stock and Watson 1988), and thatZ therefore represents equilibrium errors.

The Engle and Granger Two-Step Method

It is helpful to note the precise nature of Z in the preceding example. Statedsimply, time-series Z may be said to represent that portion of the levels of Ythat may not be attributed to X. Suppose that the coefficient /? in equations 7and 8 is 1. Term Z, will be equal to zero, then, when Y, is equal to X,.Alternatively, we may infer that X, and Y, are in an equilibrium state vis-a-visone another when our measure of equilibrium errors—Z,—is equal to zero.Note that a "shock to the system," that is, an exogenous jolt to X or Y (or bothif sustained in varying degrees) would disturb the equilibrium relationshipbetween the two series, producing a nonzero value for Z. If we assume that Yresponds to such equilibrium errors, series Y may be a function of X in twodistinct ways: directly and indirectly via the equilibrium relationship.

Consider a scenario in which X and Y are in an equilibrium state whenthey are equal to one another, and changes in Y (Ay,) are a function of thelagged change in X (AX,_,) as well as the degree to which the levels of X andY in the previous period are out of equilibrium (Z,_ , = >',_,— X,_,). Supposethat X = Y at time t — 4, but that X receives a shock at time t - 3. The valueof Y at time / — 3 does not register this shock (either directly or indirectly),since AYt_3 is a function of AX,_4 (or X,_4 — X,_5) and the degree to which Xand Y are out of equilibrium at t — 4 (Yt_4 — Xt_4 = Z,_4 = 0). At time t — 3,however, the equilibrium relationship has been disturbed by the shock to X, sothat Y at time t — 2 is now a function of not only AY,_3 (clearly itself afunction of the shock), but also the disequilibrium at t — 3. The direct impactof the shock to X on Y is not felt beyond time t — 2, but the indirect effect stillis, depending on the rate at which disequilibria are corrected. If the disequilib-rium is corrected at the rate of 50 percent per time period, we would see that50 percent of the initial disequilibrium remains at time t — 2, 25 percent attime t — 1, 12.5 percent at time /, and so on.

If the shock to X at time / — 3 were positive, X,_3 would be greater thany,_3, producing a negative value forZ,_3 (since Z, = Y, — X,). The level of Xwould be too "high" relative to that of Y, so Y would need to increase in orderto reattain an equilibrium state. This implies that the relationship betweenchanges in Y and the levels of Z is negative—a positive Z implies that Y is toohigh (or X is too low) and Y should adjust downward, while a negative valueimplies that Y should adjust upward. Thus,

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AJ',=./(AX,_I,Zf_1), (9)

where the coefficient on Z,_, is negative.Note that any shock to the system (X,Y) will register in Z, and that Ay,

will respond to Z,_, negatively in an effort to reestablish its equilibriumrelationship with X. These expectations provide the motivation for the Engleand Granger (1987) two-step technique. If X and Y are cointegrated, Engleand Granger suggest that an appropriate model of / may be obtained byperforming the following procedure. First, Y is regressed on X in levels,obtaining the "cointegration vector" Z:

Y, = a + bX, + Z,,

Z,= Y,- bX, - a.

(10)

(11)

The second step involves the regression of changes in Y on past changes in X(to capture any short-term relationship) as well as the equilibrium errorsrepresented by the cointegration vector:

,^ -hZ,_t. (12)

A graphic representation of the cointegration vector as error correction com-ponent (using the series shown in fig. 1) is presented in figure 4. As expected,changes in Y are shown to be a negative function of the lagged levels of thecointegration regression residuals (or Z).

zo

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, 1

[ ' ,

\ . . . . . 1 vw:>_.• ^ - l I. /' - - ] 1/

/^~V \s—/ /

Lagged Y-on-X Regression Residuals

———^Changes in Y (AY) ^ ^ ^ E r r o r Correction in Y

Fig. 4. Accounting for error corrections in Y: changes in Y and Y-on-Xregression residuals (lagged)

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A discussion of how and when the standard error correction model andthe Engle and Granger two-step differ is taken up in part 4. First, however, thecausal nature of cointegration and error correction is addressed.

Cointegration, Error Correction, and Granger Causality

According to the advocates of prewhitening modeling techniques (such as thetransfer function approach discussed in Box and Jenkins 1976), one of theadvantages of their modus operandi is that it captures the causal nature ofthe relationships among the time-series under analysis. Unlike the generalizedleast squares (GLS) models employed by time-series analysts who preferposttreatment to prewhitening, transfer function models identify the degree towhich the predicted value for an output series is improved by consideringsome exogenous input in addition to the series' own past values. Such is thenature of Granger (1963) causality; that is, a series X is said to Granger causeY if our capacity to predict the current value of Y is enhanced by the consider-ation of past values of X in addition to past values of Y.7 In light of the fact thaterror correction models appear to strike a balance between existing ap-proaches to longitudinal analyses, the question of whether cointegration anderror correction imply causality naturally arises.

Granger himself provides a happy answer when he demonstrates that,"for a pair of series to have an attainable equilibrium, there must be somecausation between them to provide the necessary dynamics" (1988, 203). Tounderstand why this is the case, we need only consider the nature of errorcorrection processes vis-a-vis that of Granger causality. As I have noted, atime-series that may be represented as an error correction process is one thatmaintains an equilibrium relationship with other processes. Any shock to thisequilibrium results in changes in the error-correcting (or dependent) processso that the preshock equilibrium may be reattained. In other words, the degreeto which the time-series are in a state of disequilibrium determines a portionof the changes in the dependent process, that is, that portion necessary forreequilibration. This scenario is consistent with the concept of Granger cau-sality, which dictates that one time-series may be said to "cause" another ifour capacity to predict the latter is enhanced when we consider, in addition toits own history, that of the former as well. If time-series Y may be accuratelyrepresented as an error correction process vis-a-vis X (as in fig. 1), then X maybe said to Granger cause Y indirectly (via changes in X that produce disequi-libria to which Y responds) and possibly directly as well. Thus, error correc-tion models in general and cointegration in particular imply causality.

7. For a detailed examination of Granger causality and its applicability to analyses ofpolitical phenomena, see Freeman 1983.

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Y, =xt = x M + s u

+ 5 + 0.5 • AXM - 0.5 ' -X(.i - 3) +

Fig. 5. A simulated error correction process with random errors

Whether this argument is persuasive or not, analysts always have re-course to Granger causality tests. Figure 5 shows a simulated error correctionprocess involving two time-series that are cointegrated by construction. Theseries were generated via the following equations.

AY, = 5 + 0.5AX,_, - 0.5(r,_, - X,_, - 3) + e2r

(13)

(14)

where both e, and e2 are white-noise disturbance terms. Note from equation14 that Y and X maintain an equilibrium state when Y, is three points greaterthan X,; that is, when the expression (Y,_, — X,_, — 3) equals zero. Shocks toX will produce changes in Y in two ways, both in the short term (via therelationship AY, = 0.5AX,_,) and in the long term (via the error correctioncomponent). As such, a standard Granger causality test ought to demonstratethat both lagged changes in X and lagged values of the error correctioncomponent cause current changes in Y.

In order to demonstrate the validity of this expectation, I begin by esti-mating a single equation error correction model of Y (fig. 5) based on X.8 Theresults, as reported in table 1, lead to the following specification.

Ay, = 7.58 + 0.59AX,_, - 0.52Y^{ + 0.49X,_, + e,, (15)

which may be rewritten as

Ay, = 7.58 + 0.59AX,_, - 0.52(y,_, - 0.94A',_1) + e,. (16)

8. All statistical analyses reported in this article were generated by RATS, version 3.10.

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TABLE 1. Estimating an Error Correction Model (fig. 5 data)

Independent Variable

Changes in X laggedLevels of Y lagged ILevels of X lagged IConstant

1 period (AX,_|)period (Y,.,)period (X,_ |)

Changes in Y

OLS Coefficient

0.59-0.52

0.497.58

SE

0.110.090.091.43

(An

T-ratio

5.39-5.63

5.315.29

p-value

0.0000.0000.0000.000

Note: R2 = .72, adjusted R* = .71 . SEE = 2 44. DW = 1.97. Q = 20.14 (27 df).

Note the similarity between the actual parameters of equation 14 and thoseestimated in equation 16 (the difference in constants reflects the addition ofthe product of the equilibrium constant and the error correction coefficient tothe estimated constant; i.e., eq. 14 may be rewritten as Ay, = 6.5 + 0.5AX,_,- 0.5[y,_, — X,_t] + e,). I also employed the Engle and Granger two-steptechnique, the results of which are reported in table 2. The two-step processleads to the following error correction representation.

Ay, = O.59AX,_, - 0.52Z,_, + e,, (17)

where Z,_, represents lagged equilibrium errors and is equal to (by step one)y,_, - 14.82 - 0.93X,_,. Substitution yields

Ay, = 7.71 + 0.59AX,_, - 0.52(y,_, - 0.93A-,_,) + e,, (18)

TABLE 2. Estimating an Error Correction Model via the Engle and GrangerTwo-Step Technique (fig. 5 data)

Independent Variable

1. Cointegration Regression"Levels of XConstant

2. Error Correction Modelb

Changes in X lagged one period(AX,_,)

Levels of residuals from X-on-X re-gression lagged one period (Z,_,)

OLS Coefficient SE

0.9314.82

0.59-0 .52

Levels of

0.041.03

Changes in Y

1.430.09

7"-ratio

Y

26.3414.46

( i n

5.29-5.72

p- value

0.0000.000

0.0000.000

.89. adjusted R' = .88. SEE = 4 08. DW = 1.74.

.72. adjusted R* = .72. SEE = 2.42, DW = 1.97, Q = 19.98 (27 df).

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TABLE 3. Granger Causality Tests (fig. 5 data)

Changes in Y(AY)

Independent Variable F' p-value

Changes in X (AX) 38.98 0.000Estimated error correction component (V, - 0.94*X,) 46.47 0.000Residuals from cointegration regression (2) 46.38 0.000

•Entries are F-statistics measuring the change in R2 when lags of the independentvariable are excluded from an equation regressing changes of Y on 4 lags of itself and4 lags of the independent variable with F(4, 78)

the coefficients of which are nearly equal to those estimated via the singleequation model.

If assertions of causality are to hold, Granger tests should demonstratethat current changes in Y (AY,) are caused by lagged changes in X (AX,_,), theestimated error correction component from equation 16 (Y,_, — 0.94Y,_ (), aswell as the lagged estimate of equilibrium errors (Z) from the two-step pro-cess (y,_| — 14.82 - 0.93X,_|). All three causal scenarios are indeed con-firmed by the results presented in table 3. Each of the three separate Grangertests shows a highly significant change in the total variance in Ay explainedby models including: (1) lagged changes in X, (2) the lagged error correctioncomponent from equation 16, or (3) the lagged residuals from the cointegra-tion regression (shown in table 2). It is worth noting that, in the case of thelatter two tests, four lags each of the proposed causal variable were included(more for the sake of symmetry than anything else), yet only one lag wouldhave been sufficient to produce evidence of causality. Such a finding is to beexpected, since in both the single equation and the two-step error correctionmodels it is the equilibrium error at time / — 1 only that affects change in Y.

It is also worth noting that the causal nature of error-correcting relation-ships is particularly important when changes in independent variables do notappear to cause changes in the dependent variable directly. In the face ofcointegration, standard Granger causality tests fail to capture the causal con-nection arising from the equilibrium relationship and may, therefore, fail toindicate any causality whatsoever. As Granger suggests, "It does seem thatmany of the causality tests that have been conducted should be reconsidered"(1988, 204).

Comparing the Engle and Granger Two-Stepand Single Equation ECMs

The discussion in part 2 and the analysis in part 3 make clear that singleequation error correction models and those estimated via the Engle and

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Granger two-step technique are very similar under certain circumstances. Butthe question remains, when do they differ? When is one modeling strategy tobe preferred over another? The analysis of U.K. wage data by Hall and Henry(1988, 68-78) seems to indicate that the Engle and Granger two-step tech-nique is to be preferred, while Beck (1991), on the other hand, implies thatsingle equation ECMs are preferable in light of his assertions that they distin-guish clearly between dependent and independent variables and may be ap-propriate whether or not cointegration is present. This "disagreement," how-ever, should not be exaggerated: neither Hall and Henry nor Beck takeanything resembling a strong stand.

A major reason is undoubtedly the youthful nature of the subject matter.Analysts are still exploring the features of these approaches, and it may besome time before definite statements may be made regarding the precisedifferences between single equation error correction models and those arrivedat by the Engle and Granger two-step technique. Nonetheless, Monte Carlosimulation results suggest that, under certain conditions, the Engle andGranger two-step technique may be more vulnerable to the incidence ofaccidental associations when an integrated but unrelated time-series appearsto belong in the cointegration regression and fails to render the vector ofresiduals nonstationary.

The Problem of Accidental Association

In theory, if a linear combination of /(I) time-series produces an 1(0) orstationary series, one may conclude that the series in question are cointe-grated, that is, they share a common stochastic trend. Given the inap-plicability of standard significance tests for coefficients on the regressors of acointegration regression,9 however, as well as the fact that estimated coeffi-cients in a cointegration regression converge on their true values much fasterthan normal (Stock and Watson 1988), it is entirely possible that an integratedprocess may be incorrectly included in a cointegration regression. This sce-nario is likely when the pattern exhibited over time by an accidentally associ-ated integrated process is insufficient to render the linear combination of itselfand two or more other processes (which do share a common stochastic trend)nonstationary. To restate, cointegration may be inferred when the linear com-bination of two or more /(1) series is stationary, since the common stochastictrend is presumably "canceled out," leaving only the equilibrium errorsaround that trend. Nonetheless, the incorrect inclusion of another/(I) processmay go undetected if its own stochastic trend is not pronounced enough to

9. It is widely held thai OLS regressions involving one or more integrated processesproduce coefficient estimates with nonnormal asymptotically distributed test statistics.

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render the new linear combination of all the series nonstationary. And sincestandard significance tests in the cointegration regression are inapplicable, theanalyst can only assume that all regression coefficients are statistically distin-guishable from zero.

To test this hypothesis, a Monte Carlo test was developed for the purposeof comparing the abilities of single equation ECMs and those developed viathe Engle and Granger two-step technique to ferret out an integrated seriespresumed to belong in a model with X and Y but, in fact, unrelated.10 Thesimulation begins with the X and Y series generated via the basic equationsfrom figure 5, each with random errors. A third series, W, is included in eacherror correction model, generated as an independent random walk but pre-sumed (incorrectly, by the ubiquitous mythical analyst, Dr. Monte Carlo) tobelong in the error correction component of Y.

The simulations begin with the following equations.

Ay, = 5 + fc,AA-,_, - b2(Y,_l - b3X,_x - 3) + eu.

W, = W,_l+ ve3r

where e,, e2, and e3 are random normal series of length T, and v is a variancemultiplier for series W. The following factors are systematically varied: (1) thecoefficients measuring the short-term (£>,) impact of X on Y and the rate (b2) ofTs error corrections (/?, = 0.25, b2 = 0.75; b} = 0.5, b2 = 0.5; fc, =0.75, b2

= 0.25); (2) the strength of the association in levels between X and Y (b3 =0.5, b3 = 1.0); (3) the variance of W, expressed as an approximate ratio to thatof X and Y (1:1, 10:1); and (4) the number of time points analyzed (100, 250,500, or 1,000). The first and second factors are included to test whether therelative short- and long-term impact of X on Y affects the likelihood of a "falsepositive" for W; the third rests on the a priori assumption that, as W's varianceincreases, the probability increases that its inclusion in the cointegration re-gression (step one) of the Engle and Granger two-step technique will renderthe residuals nonstationary. Finally, the fourth factor tests whether the likeli-hood of failure increases or decreases with sample size.

The simulation involves the generation of two error correction modelsper iteration (400 iterations per permutation), each of which assumes thatchanges in Y are a function of X in the short-term and the long-term equilib-

10. Jim Stimson wrote the computer code enabling the test results reported herein. Mycontribution to the exercise—telling programmer Stimson what I wanted—was comparativelyeasy.

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rium relationship maintained among W, X, and Y in levels. The first ECM,derived via the Engle and Granger two-step technique, includes the followingregressions.

Y, = c + dxX, + d2W, + Z,.

and

ISY, = </4AX,_, -d5Zl_l +er

In order for the Engle and Granger two-step technique to fail—that is, tosuggest that W belongs in the model—the following conditions must be met:(1) a significant (p < .05) /-ratio for coefficient d2 (note that the standard/-distribution does not apply due to nonstandard asymptotic properties; I as-sume nonetheless that analysts will cheat and peek); (2) an R2 for step onegreater than or equal to .60; (3) a Durbin-Watson statistic for step one greaterthan 1.0; (4) a Dickey-Fuller /-ratio (measuring the stationarity of Z,; this testprocedure is discussed below) less than —5.0; (5) a significant /-ratio associ-ated with d4 in step two; and (6) a satisfactory Durbin-Watson (greater than1.5) for the step two regression. If all of these conditions are satisfied, theEngle and Granger two-step technique has failed to exclude W for the modelestimated of Y.

The second error correction model tested is derived via the single equa-tion approach. The following regression is estimated.

AK, = c + fc,AX,_, + b2Y,_l + byX,_l + b4W,_t + e,,

which, in error correction form, yields

In order for the single equation model to fail, the /-ratio for b4 had to bestatistically significant," and the Durbin-Watson test statistic had to be greater

11. "Not so fast," the critic may say. "The coefficient on W,_, measures the associationbetween a stationary dependent variable (AY) and a nonslationary independent variable. If thistrips us up in the cointegration regression, why not here as well?" And the thoughtful criticappears to have a good case. But as Stock and Watson (1988) show, the usual t- and F-distribu-tions will apply to OLS regression coefficients if they may be rewritten as coefficients on astationary variable. According to Stock and Watson (1988), if this transformation results in theassociation between the coefficient and a stationary process, standard tests of significance may beapplied to the coefficient. And since the error correction component is, in fact, stationary, we mayconclude that the /-ratio on the original coefficient is valid. Furthermore, similar transformations

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than 1.5. In all, 38,400 error correction models were evaluated, one each wayfor 19,200 sets of data. The results are reported in table Al of the appendix.

Perhaps the most striking finding to arise is the low instance of failuresfor both approaches. The Engle and Granger two-step technique failed only1.23 percent of the time, while the single equation models failed in fewer thanone-third of one percent of the simulations. What failures there were occurredin those models in which the rate of error correction (0.75) dominates theshort-term impact of X on Y (0.25), and in which there were many observa-tions (1,000). Among Engle and Granger models in which X and Y arestrongly associated in levels, failures were especially likely, approaching arate of one in five. The same factors were more likely to trip up the singleequation models as well, but in no instance does the failure rate for the single-step models exceed 5 percent. The relative variance of W appears to exert littleor no effect on the results.

At first blush, one may be surprised to see that the likelihood of failuresincreases with sample size. Generally, we assume that the more observationswe have, the stronger our statistical analyses will be. In the case of errorcorrection models, however, this pattern may break down (especially in Engleand Granger two-step models) due to the fact that very long, integrated time-series are more likely to assume the characteristics of stationary time-series.In such scenarios, the incorrect inclusion of a W series will be less apt toproduce a nonstationary set of first-step residuals, suggesting that cointegra-tion is present among all variables.

The Engle and Granger procedure, as executed in the Monte Carlo anal-ysis, never forces the past levels of W to make a direct, independent contribu-tion to changes in the Y series. As a result, the W series is able to slipunnoticed into a cointegration regression dominated by the X series and hidein an error correction process driven quickly and strongly by X. Under suchcircumstances, where the dependent process under analysis maintains a stronglong-term relationship with one of the independent variables, single equationerror correction models appear to have the upper hand due to the fact that suchmodels force each process to make an independent contribution to the errorcorrection presumed to occur in the dependent variable.

U.S. Domestic Policy Sentiment as anError Correction Process

I have argued elsewhere (Durr 1992) that changes in U.S. domestic policysentiment (as measured by Stimson [1991b]) can be understood as both re-sponses to shifting economic expectations and as reactions to federal policy

may be conducted so that every coefficient on a nonstationary process is associated with one thatis stationary (although some of the equations will make little theoretical sense).

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outcomes. In this part, I apply the techniques I have discussed directly to ananalysis of political behavior. First, however, I discuss the hypotheses to betested and issues of operationalization.

Policy Sentiment and Economic Expectations

I have hypothesized that perceptions of an imminent economic downturn willcause an aggregate shift toward a more conservative policy agenda. Con-versely, expectations of a healthy (or improving) economy contribute to thewillingness within the U.S public to underwrite a liberal domestic policyagenda. Central to this hypothesis is the fact that such an agenda does notcome cheap. Since the New Deal era (and perhaps to a greater extent since theGreat Society initiatives of Lyndon Johnson), the liberal policy agenda hasexplicitly cited the need for an active federal government as provider andprotector of jobs, health care, schools, housing, civil rights, and the like.

While liberal elites may push such policies at all times, such an agendacan only be supported by a healthy economy or, to be more accurate, by apopulace enjoying the fruits of a healthy economy. Thus, perceptions ofeconomic security pave the way for the implementation of rather expensive(liberal) domestic policies by fostering a willingness among the public to payfor such policies.

Does it make sense to anticipate that, given a certain set of circum-stances, individuals will part willingly with a portion of their income in orderto contribute to the construction of a "better" society? In spite of a surprisinglyhealthy body of evidence that challenges the role of self-interest in economicand political behavior (see Citrin and Green 1990), it is generally assumedrational for individuals to prefer a greater quantity of some good to a lesserquantity—regardless of circumstance.

This assumption suggests that citizens will always prefer greater wealthfor themselves, yet it stands to reason that the utility derived by an individualfrom a bundle of goods is lessened if many such bundles are already in his orher possession. In other words, it is reasonable to anticipate that a thresholdmay be met where the relative weight of the simple desire for "more" islessened with respect to an individual's decision-making processes.

The same principle applies to the nation-state. Once a threshold of eco-nomic security is met, wherein the collective judgment of citizens dictates thateconomic well-being is sufficiently guaranteed, the citizenry will be morewilling to part with the nation's wealth toward the pursuit of other goals.Should the economy of a nation-state become sufficiently threatened that thecitizens lose their sense of security and, by extension, their willingness tounderwrite the government's pursuit of liberal policy goals, on the other hand,the public will demand a temporary suspension of such goals in favor of

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An Essay on Cointegralion and Error Correction Models 205

greater attention to the economy itself. To state the case simply, a concern forthe oppressed—whether minorities and women or lakes and streams—can besuspended rather quickly when a typically secure populace suddenly feels thatit is in danger of oppression.

The expectation that a populace enjoying the fruits of economic securitywill be more willing to support a liberal policy agenda, then, need not rest onthe assumption that individuals will occasionally behave in an altruistic man-ner. Instead, the achievement of a sufficient level of economic comfort per-mits citizens to establish new objects of their self-interest. Individuals whosupport tax-and-spend policies designed to fight poverty may do so not be-cause they are saintly, but rather because they find the visual reminders ofpoverty around them particularly unpleasant. And when economic conditionspermit, the achievement of such "societal" (albeit still, perhaps, fundamen-tally self-interested) goals becomes possible.

This hypothesis, linking greater economic security with an increasedwillingness to support liberal domestic policies, echoes the fundamental ideaof Inglehart (1971). Borrowing from Maslow's (1970) notion of a needshierarchy, Inglehart presumes that "individuals pursue various goals in hier-archical order—giving maximum attention to the things they sense to be themost important unsatisfied needs at a given time" (1971, 991). Inglehart thenproceeds to develop a generational model in an effort to demonstrate that agecohorts raised in an environment of relative economic security will be lessconcerned with economic issues in favor of a more "postmaterialist" agenda.

While this work shares the starting point of Inglehart (as well as Dalton1988; Flanagan 1987; Sirgy 1986), the destination is altogether different.Rather than unidirectional, intergenerational evolution, I anticipate a cyclical(in a loose sense of the word, implying only back-and-forth movement),intragenerational dynamic. When faced with a perceived downturn in theeconomy, the same generation that some years back embraced a liberal policyagenda during good economic times is likely to swing back to the ideologicalright.

Viewed in this light, the relationship hypothesized to exist between do-mestic policy sentiment and economic expectations may be expressed as amoving equilibrium. Any shock to the system producing a shift in the aggre-gate economic outlook will produce a reequilibrating shift in policy sentimentfor the reasons outlined above. In other words, the two processes maintain anequilibrium relationship—stated simply, they will not drift "far apart" andstay away from one another for very long. Note, however, that the equilib-rium relationship hypothesized is asymmetric. Oisequilibrating shocks areexpected to result in changes to policy sentiment so that its level is consistentwith that of economic expectations.

Those comfortable within an economic paradigm will point out that the

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anticipated relationship between domestic policy sentiment and economicexpectations is nicely captured by the principle of diminishing marginal re-turns. A nation with great collective wealth will value money less and willthus be more willing to part with it toward the attainment of other goals.Should money become scarce, however, its value relative to other goods (suchas the utility gained by contributing to the betterment of strangers, for ex-ample) increases. Given the fact that most liberal policies require a redistri-bution of wealth, one would expect that in times of economic uncertaintyor insecurity citizens will be less supportive of the liberal domestic policyagenda. Thus, the same anticipated relationship between economic expecta-tions and domestic policy sentiment emerges. As economic expectations con-tribute to a greater sense of security, domestic policy sentiment will reflect agreater willingness to adopt liberal policies.

Formally, we may conceive of the desired outcomes associated withliberal policies and with conservative policies (i.e., that which permits anincrease in private consumption) in terms of utility. Assume that citizens seekto maximize some combination of liberal policy outcomes L and conservativepolicy outcomes C subject to the constraint imposed by economic expecta-tions, EE. That is, the optimal strategy seeks to maximize the function

U(L,C), (19)

subject to the constraint

PLL + PcC = EE, (20)

where PL and Pc represent the "prices" associated with L and C (the sum ofwhich cannot exceed the value derived from economic expectations). To solvethis constrained optimization problem, we may use a Lagrange multiplier h,such that

W = [/(/.,O - h(PLL + PcC - EE). (21)

Note that values that satisfy the constraint permit the maximization of theoriginal function, since h(PLL + PCC — EE) will equal zero. In order tomaximize, first derivatives are computed with respect toL, C, and h (the latteryielding the original constraint). That is,

MUL{L,O - hPL = 0, (22)

MUc(L,C) - hPc = 0, (23)

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PLL + PcC - EE = 0, (24)

where Mil refers to the marginal utility associated with an increase in theconsumption of a good (i.e., MUL[L,C] — dU[L,C]/dL). Solving equations 22and 23 for h yields

h = MUL(L,C)/PL = MUc(L,C)/Pc, (25)

which, when rearranged, produces

MUL(L,C)/MUc(L,C) = PJPC. (26)

This ratio of marginal utility derived from L and C may be treated as themarginal rate of substitution, which, according to equation 26, is equal to theratio of the "prices" of the two goods, PL and Pc. If this ratio is constant andequal to one, a consumer of liberal policy outcomes and conservative policyoutcomes gains the greatest utility from changes in "income" (i.e., the valuederived from improving or declining economic expectations) when it is dis-tributed equally between the two. If, on the other hand, fluctuations in eco-nomic expectations lead to changes in the prices associated with liberal andconservative policy outcomes, then the marginal rate of substitution willchange as well.

Recall the hypothesis that, once a threshold of economic security is met,citizens will be more willing to support liberal policy outcomes (at the ex-pense of private consumption). In an environment of economic insecurity, onthe other hand, I hypothesize that citizens will shift support away from liberalpolicies and toward those more conservative in nature. In other words, liberalpolicy outcomes are assumed to be a superior good vis-a-vis conservativepolicy outcomes beyond some threshold of economic security, whereas thesame (liberal) policy outcomes are inferior "beneath" such a threshold. Or, tostate it another way, the price ratio PJPC favors liberal policies beyond thesecurity threshold (PL < Pc), but conservative policies are favored beneaththe threshold (PL > Pc).

Consider the pattern of indifference curves shown in figure 6. Each curve(Ux, U2, and t/3) represents many combinations (or bundles) of liberal andconservative policy outcomes, among which citizens are indifferent. To maxi-mize utility, a consumer of policy outcomes will prefer those combinations thatfall on corresponding budget lines (££,, ££2> and EE3), which in this scenarioare functions of economic expectations (i.e., EE = PJL + PcC). As economicexpectations improve, citizens will be able to enjoy a combination of liberal andconservative policy outcomes found on a "higher" indifference curve.

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o

Sa3

Conservative Policy Outcomes (Private Wealth)

Fig. 6. Maximizing utility from liberal and conservative policyoutcomes subject to the constraint imposed by economicexpectations

Suppose that point Y in figure 6 demarcates a threshold of economicsecurity. Budget lines above and to the right of Yand the budget line EE2 denotea citizenry beyond such a threshold, whereas lines beneath and to the left of Y(and EE2) imply that the threshold has not been crossed. If the prices associatedwith liberal and domestic policy outcomes above and beneath this thresholdwere constant (and in this case equal to one another), then points representingoptimal combinations (X, Y, and Z) would fall on the line marked C".

But if the price ratio (and therefore the marginal rate of substitution)changes when the threshold of economic security is crossed in a way consis-tent with the relationship hypothesized above, then the optimal combinationsare more likely to be found on the line labeled C. Note that any such combina-tion below Y (such as X) favors conservative policy outcomes at the expense ofthose more liberal, whereas combinations above Y (such as Z) favor liberaloutcomes at the expense of those more conservative in nature. Thus, accord-ing to figure 6, citizens will be more likely to support liberal policies in timesof relatively positive economic evaluations, when a threshold of security hasbeen crossed.

Measuring Economic Expectations

In order to operationalize economic expectations, I begin with a survey itemfound in the University of Michigan's Index of Consumer Sentiment thatmeasures citizens' long-term business expectations. This item has a number ofvirtues for this analysis, the most important of which is its subjective, pro-spective nature. I then regress this series on four measures of the economy:

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inflation, unemployment, coincidental indicators, and leading indicators. Thisprocedure effectively partitions the variance of long-term business expecta-tions into two categories, that predicted by the economy itself and that whichis not attributable to economic conditions. The latter category, represented bythe residuals from the linear regression, appears to be a "political" componentof business expectations. Its pattern over time corresponds in particular topresidential elections, peaking each time we are told that "happy days are hereagain." In addition, the residuals correlate nicely with a direct measure ofcitizen attitudes toward the government's economic policies. Thus, my mea-sure of economic expectations is simply those values of business expectationspredicted by the objective economy.12

The resulting measure of economic expectations, along with Stimson'smeasure of domestic policy sentiment, is shown in figure 7. Both series areaggregated quarterly, beginning in the second quarter of 1968 and ending inthe first quarter of 1988.13 As hypothesized, the two series track togetherthrough time, so that optimistic (pessimistic) economic expectations corre-spond with liberal (conservative) policy preferences.

Federal Policy Outcomes

In addition to economic expectations, I have hypothesized that policy prefer-ences are also a function of the public's assessment of federal policy out-comes. Policy outcomes viewed as overly "intrusive" into the daily lives ofcitizens will produce a call for a more limited government, whereas percep-tions that the federal government is failing to do enough on behalf of itscitizens will lead to the expressed desire for a more active government. To theextent that calls for limitation or greater action are synonymous with shifts inpolicy sentiment in a conservative or liberal direction, then, the anticipatedrelationship between policy outcomes and policy preferences is negative.Once a threshold is met where citizens believe that the federal government hasgone too far in one direction or another (i.e., policy outcomes are either toointrusive or not intrusive enough), public opinion will react by moving in anopposite direction in an effort to pull in the reins of policymakers.

Note that the hypothesized relationship between policy preferences andpolicy outcomes is nonconstant. Policy outcomes are only expected to pro-duce negative reactions when they are perceived to be extreme. Otherwise, Ianticipate no causal connection. It is also important to note the distinction

12. For a more detailed treatment of this procedure, see Durr 1992.13. These beginning and ending dates are determined exclusively by (he data. The dates

represent the first and last quaners that a sufficient number of individual policy preferences exist(and are in my possession) for a reliable quarterly estimate of domestic policy sentiment.

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110

80-

75'

120

50

1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988°

Domestic Policy Sentiment Economic Expectations

Fig. 7. U.S. domestic policy sentiment and economic expectations

between the behavior of policymakers and policy outcomes. The former,argued by Stimson (199la) to be a function of policy preferences, ultimatelyresults in actual policy outcomes experienced by citizens. Given the oft-citedincremental nature of the policy process, it is entirely conceivable that afamily of policies enacted during a time characterized by a strongly supportivepublic mood might one day (perhaps many years down the road) be met withequally strong opposition.

In order to capture empirically the intrusive nature of general policyoutcomes, I have focused on matters of federal spending. Budgets and taxesrepresent "authoritative allocations," the changes and ratios of which shouldcapture the drift of public policy over time. To wit, four distinct categories offederal budgeting are examined: domestic spending (the percentage of totalfederal outlays dedicated to human resources), defense spending (one hundredminus the percentage of total federal outlays dedicated to defense), income taxrates for the wealthy (average income tax rate of all returns with adjustedgross income of $1 million or more 1986 dollars), and federal grants in aid tostate and local governments (the percentage of total state and local revenuesconsisting of federal grants in aid). The defense measure is reflected so thatincreases and decreases are consistent with those of the other measures; i.e., ahigher percentage for any measure indicates (roughly) a more intrusive federalgovernment. Should these elements of federal policy-making move togetherthrough time, one may argue that their shared pattern is indicative of the na-ture of the federal policy outcomes. Not surprisingly, they do. All four seriesincrease steadily throughout the 1960s, reach plateaus in the mid-1970s, andreverse course at the outset of the 1980s.14

Figure 8 shows a measure of federal policy outcomes obtained by averag-

14. The mean intercorrelation coefficient for the four time-series is 0.53.

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110

E 105-

100-

aCL

36%19681970 1972 1974 1976 1978 1980 1982 1984 1986 1988

Domestic Policy Sentiment Federal Policy Outcomes

Fig. 8. Domestic policy sentiment and federal policy outcomes

ing the four time-series discussed above. Corresponding in time to the peak ofpolicy intrusiveness is a dramatic conservative shift in policy preferences—precisely the type of negative reaction hypothesized. Such a shift suggeststhat U.S. public opinion reacted negatively to policies perceived as overlyintrusive. Perhaps this reaction was due to the fact that, at the same time manyof the Great Society policies enacted in the 1960s were coming to full fruition,Americans experienced worldwide inflation. Although not unprecedented,Katona (1975) argues that this was the first instance that Americans began todoubt that resources could keep pace with worldwide demand. A naturalreaction to this impression would be a desire for the U.S. government to rollback its reach and husband its resources. Such an event, then, may haveserved as a "trigger," touching off a negative reaction to the status quo ofpolicy outcomes. Since policy outcomes are only presumed to affect policysentiment in the wake of such a reaction, only that portion of the measureoccurring after the outset of fiscal year 1975 (the first quarter of which is thebeginning of the plateau shown in fig. 8) is included in the analysis. Prior toFY 1975:1, the measure takes the value of zero.15

Controlling for Other Effects

As I have noted, the measure of domestic policy sentiment under consider-ation consists of citizens' preferences across scores of distinct issue arenas.While preferences in any one issue arena are certain to be affected by a variety

IS. My sole motivation here is improving the estimation of the relationship betweeneconomic expectations and policy sentiment. Such a relationship is not strengthened by myhandling of the policy measure. Indeed, quite the contrary is true. The coefficient on economicexpectations is cut by one-third in a model that includes the policy outcomes "control variable,"but the overall fit of the model is improved substantially.

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of political events, the list of such events capable of altering preferencesacross many such issues is likely to be small. Only four events during the twodecades under analysis (1968-88) present themselves as viable candidates:the Vietnam War, Watergate, the Iran hostage crisis, and the Iran-Contraaffair. Of these, only the first two seem to be of a sufficient magnitude to affectpolicy preferences, yet there is no reason not to include dummy variables tocontrol for the possible effects of all four.

The inclusion of dummy variables in an error correction model warrantsmore care than is typically exercised. Given the nature of ECMs, one mustdecide whether any of the phenomena represented by dummies belongs in thelong-term equilibrium relationship. If so, the standard procedure of switchinga dummy variable off after the event has ended may be inappropriate. Con-sider, for example, the case of Watergate and its possible effect on policypreferences. If domestic policy sentiment is presumed to react not just in theshort term to Watergate, but also in the long run, when does one arbitrarilyswitch the dummy off? In this analysis, the answer is never. To the extent thatWatergate affected policy preferences in the long term, it is assumed that"post-Watergate" is "post-Watergate"—whether experienced in 1975 or 1985.Thus, the long-term equilibrium relationship anticipated is that between pol-icy preferences and a political environment that includes the Watergate inci-dent in its past. If such a test seems overly difficult to pass, one shouldreconsider whether the phenomenon represented by a dummy variable iscapable of anything more than a short-term reaction in the dependent variable.For the error correction model of domestic policy sentiment developed below,both the Vietnam and the Watergate dummies are presumed able to exert along-term effect and are, thus, never switched off.

Short-term relationships should also be approached with care. In light ofthe nature of error correction models, short-term associations are captured byevidence of association between changes in the independent variable andchanges in the dependent variable. In a typical time-series regression, where adummy is switched on (from zero to one) and off (from one to zero) some timelater, only two changes are present: a 100 percent increase at the outset, and a100 percent decrease at the conclusion. While some events may begin and endthis abruptly, most do not. It therefore seems unwise to force the entire short-term effect into a single time period. Instead, dummies should begin and endin ways similar to the phenomena they purport to represent.

Regarding this study, both Vietnam and Watergate were "in the back-ground" of the U.S. agenda for some time before emerging at full power;using dummies that switch from zero to one at some time point would seem tobe inappropriate. On the other hand, both the Iran hostage crisis and the Iran-Contra affair jumped quickly into the minds of Americans with no priorwarning. While both events remained at full intensity for some time, each

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eventually faded, suggesting that their respective dummies should not beabruptly turned off. Thus, the four dummy variables included in the estima-tion of an error correction model of domestic policy sentiment are as follows:(1) Vietnam, growing exponentially to a peak value of one at 1970:2 (corre-sponding to the May 4, 1970, shootings at Kent State University, an event thatbrought the war home)16 and maintaining that value throughout, (2) Water-gate, growing exponentially to a peak value of one at 1974:3 (correspondingto President Nixon's resignation in August 1974) and maintaining that valuethroughout, (3) Iran hostage crisis, jumping from zero to one at 1979:4(corresponding to the taking of hostages in November, 1979) and returning tozero exponentially, starting at 1981:1 (corresponding with the release of thehostages), and (4) Iran-Contra, jumping from zero to one at 1986:2 (corre-sponding to the November, 1986, Ronald Reagan press conference) and re-maining at one, since the analysis only extends to 1988:1. Vietnam andWatergate are tested for both short- and long-term effects on domestic policypreferences, while the Iran hostage crisis and the Iran-Contra affair are testedfor short-term effects only.

Diagnosing Integration and Cointegration

In the examples above, involving simulated data, the time-series of interestwere by construction integrated and cointegrated. Such a priori knowledgedoes not exist regarding the measures of domestic policy sentiment, economicexpectations, and federal policy outcomes. Thus, before proceeding with thedevelopment of a model, it is necessary to determine whether the series arethemselves integrated and together cointegrated.

Recall that cointegration is present when a linear combination of/(I)series produces a stationary series, and that such a series is typically repre-sented by the residuals from a cointegration regression. Testing for cointegra-tion, then, involves the analysis of a single series (the residuals), and is"failed" if it is determined that that series is integrated. As a result, theprocess for determining whether cointegration is present is fundamentallysimilar to that invoked in the analysis of a single series. Generally referred toas the Dickey-Fuller (1979) procedure, this test examines a single series(when testing for integration, the series itself; when testing for cointegration,the residuals generated by the cointegration regression) against a null hypoth-esis of integration. Stated simply, the test centers on the significance of the

16. While I am aware that some may argue an earlier date would be more appropriate,pragmatic modeling considerations must carry the day. Since my analysis begins with the secondquarter of 1968, a dummy variable for Vietnam switched to one by that point and permitted tomaintain that value throughout would be perfectly collinear with the constant term.

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coefficient associated with lagged levels in a regression of the first differences(changes) of a series. That is, one generates the regression,

AZ, = £Z,_, + €,, (27)

and tests the significance of )3. Note, however, that the asymptotic distributionof this test statistic is nonnormal, rendering useless the standard /-test signifi-cance tables. Instead, one must use the table of critical values developed byMacKinnon (1990).

Given what we know about nonstationary time-series, the reasoningbehind the Dickey-Fuller (or DF) test is straightforward. If a series is station-ary, it exhibits finite variance around an identifiable mean. Thus, such a seriesis not likely to drift away from its mean for very long. In fact, a crude (but notcrazy) test for stationarity is eyeballing the data and determining whether, atany point in the series, the best guess of its next value is the mean. Applyingthis reasoning to the DF test (eq. 27), one would expect the coefficient on thelagged levels to be negative and significant in a stationary series. That is, inthe case of a stationary time-series, positive changes (toward the mean) willfollow on the heels of "low" levels, and negative changes (toward the mean)will be produced by "high" levels. In a nonstationary or integrated process, onthe other hand, current changes will not be associated with past levels, as theseries is likely to wander in one direction or another for an extended timebefore reversing course (if ever).

In sum, the Dickey-Fuller test may be used in order to determine whethera single series is integrated or whether two or more series are cointegrated.When testing a single series, current changes are regressed on lagged levels,and the null of nonstationarity is tested by examining the (negative) signifi-cance of the coefficient on the levels. If the coefficient is both negative andsignificant, the null is rejected, and the series is presumed to be stationary.When testing for cointegration, the same procedure is used to determinewhether the residuals from the cointegration regression are stationary.

Once an appropriate DF regression has been estimated,17 it is necessaryto examine the residuals for serial correlation. If the errors are autocorrelated,the analyst must include as many lagged changes of the time-series on theright-hand side of the regression equation as are necessary to produce white-noise residuals. This procedure is generally referred to as the AugmentedDickey-Fuller (ADF) test. For example, if equation 27 results in seriallycorrelated disturbance terms, one might estimate the following equation:

AZ, = /3,Z,_, + /32AZ,_, + e,. (28)

17. For a discussion of the diagnostic decisions involved in the Dickey-Fuller test proce-dure, see MacKinnon 1990.

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If the residuals are effectively reduced to white noise, one may reliably testthe significance of the coefficient on the lagged levels against the null ofintegration. Otherwise, additional lags of AZ must be added to insure white-noise residuals.

Having argued that analysts should consider whether the time-series theyare examining can be reasonably assumed to be integrated (and/or cointe-grated) prior to empirical determination, I am obligated to do the same. Recallthat a time-series may be integrated as a result of its own nature (shockssustained by the process remain forever) or that of other processes that driveit. Federal policy outcomes, a function of the incremental nature of the policyprocess, are virtually by definition integrated. It is well known, for example,that budgets are not cooked up from scratch. Instead, policymakers typicallybegin with the last period's budget and make incremental changes as deemednecessary. In light of the particular focus in this analysis on questions of large-scale budgeting, the operationalization of federal policy outcomes discussedabove is certain to be integrated. While one may be tempted to argue that themeasure I use of policy outcomes is necessarily stationary precisely because itis expressed in terms of percentages (and is therefore bounded in variance),we should not lose sight of the fact that the data are used to represent a processknown to be integrated.

By the same token, economic expectations are not likely to fluctuatequickly around a fixed mean. Since such evaluations flow directly from eco-nomic conditions, the fact that most booms and busts come with sustainingheads of steam suggests that expectations will not shift from optimism topessimism quickly. Finally, the hypothesized relationships among policy out-comes, economic expectations, and domestic policy sentiment predict that thelatter will also be an integrated process.

The results of Dickey-Fuller tests for the time-series representing do-mestic policy sentiment, economic expectations, and federal policy outcomesare presented in table 4. 7-tests indicate that all three series have nonzeromeans in levels, suggesting the necessity of including constant terms in eachDF regression. Note also that all three series have zero means after firstdifferencing, implying that all three exhibit zero mean drift. In each of theintegration tests on levels, the constant term is nonsignificant, so the inclusionof a linear time trend is unnecessary. For domestic policy sentiment, serialcorrelation of the error terms forced two lags of changes into the equation. Inno case is it possible to reject the null of integration in levels; thus, all threeseries are integrated.

The order of integration is tested by examining the stationarity of eachseries after first differencing. As I have noted, a series that is integrated inlevels but stationary in changes is 1(1). As table 4 demonstrates, domesticpolicy sentiment, economic expectations, and federal policy outcomes allexhibit stationary changes. Results of the Dickey-Fuller tests provide strong

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TABLE 4. Characteristics of the Individual Time-Series

Mean in levels7"-test for mean = 0

Mean in first differencesT-tesl for mean = 0

ConstantLevels, lagged 1 period7*-statistich

Changes, lagged 1 periodChanges, lagged 2 periodsDurbin-WatsonBreusch-Godfrey11

Q (24 df)

Changes, lagged 1 periodr-statistic'Durbin-Watson

Q (24 df)

Domestic EconomicPolicy Sentiment Expectations

94.14

113.03*0.07

0.16

4.34

-0.05-0.82- 0 .41 *-0.28*

0.6613.77

-1.33-12.36**

2.1826.52

Series Characteristics

82.8056.86*0.04

0.07

Integration Test, Levels-1

5.68-0.07

-1.61

2.18

24.77

Integration Test, Changes'1

-1.13-9.99**

2.0126.79

PolicyOutcomes

0.3011.75*0.01

0.91

0.02-0.04

-1.38

1.99

0.98

-1.00-8.77**

2.000.10

"For the integration tests of each series in its levels, the dependent variable of the linear regression equation isthe first differences of the series.

hThis /-statistic, the Dickey-Fuller I. is used to determine whether the series is integrated. For all three series,this statistic fails to reject the null of integration.

cThe Breusch (1978) and Godfrey (1978) statistic measures (in these analyses) first-order serial correlationamong the error terms and is appropriate when a lagged observation of the dependent variable appears on theright-hand side of the regression equation. The test statistic has a chi-square distnbution with one degree offreedom.

"For the integration test of each series after taking first differences, the dependent variable is the "seconddifferences" (first differences of the first-differenced series) of each series.

eThis /-statistic, the Dickey-Fuller /, is used to determine whether the series (now represented as changes) isintegrated. For all three series, this statistic is highly significant, indicating that the series is stationary inchanges, and is thus integrated of order one in levels.

*p < .05. *'p < .01, Dickey-Fuller distribution (nonnormal).

evidence, then, that all three time-series are first-order integrated. Regardlessof whether these processes are purely integrated (i.e., having exactly a unitroot), they are clearly long memoried (see Beck in this volume), thus suggest-ing the appropriateness of an error correction model.

To test whether domestic policy sentiment, economic expectations, and

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TABLE 5. Cointegration Regression and Evidence of Cointegration

Levels of Domestic Policy Sentiment

independent Variable

Levels of economic expectationsLevels of policy outcomesLevels of Vietnam War dummy variableLinear time trend3

Constant

OLS Coefficient

0.32-23.56

6.370.04

67.30

SE

0.032.832.060.033.53

f-ratio

9.90-8.32

3.091.25

19.04

p-value

0.0000.0000.0030.210.000

Notes' R- = 0.83. adjusted R- = 0.82. standard error of the estimate = 3 19. Durbin-Watson = 1.49, Q =19.62 (24 df). Dickey-Fuller T = -6 .72 (p < .01), Durbin-Watson (Dickey-Fuller test) = 1 92. Q (Dickey-Fuller test. 24 df) = 15.66

"Since the estimated constant is significantly different from zero, a linear time trend is included in thecointegration regression so that the asymptotic distribution of the Dickey-Fuller lest statistic is not dependent onthe constant. As the coefficient on the time trend is indistinguishable from zero, the test statistic is notdependent on it. either. The Dickey-Fuller /-statistic is used to determine whether the residuals from thecointegration regression are stationary in levels. The statistic is associated with the fi coefficient from theregression

AZ, = 0Z,_, + e,.where Z represents the vector of residuals from the cointegration regression Since the statistic is negative andhighly significant (in spite of its nonnormal asymptotic distribution), the null of integration is rejected, and theseries represented in the cointegration regression are presumed to form a cointegrated vector.

federal policy outcomes are cointegrated, the levels of the first series wereregressed on those of the latter two, along with the dummies representing theVietnam War and the Watergate incident.18 If these 1(1) series are cointe-grated, the residuals from this linear combination will be stationary. As table 5indicates, they are. Since the constant term is significant, it was necessary toinclude among the regressors a linear time trend. Note also that the Watergatedummy was dropped due to its failure to register a significant association.This regression exhibits all the hallmarks of a successful cointegration regres-sion: a high R2, a low (but not too low) Durbin-Watson statistic, and, mostimportant, stationary residuals. The DF /-ratio associated with the laggedlevels of the residuals in a regression of current changes (eq. 27) is highlysignificant, permitting the conclusion that domestic policy sentiment, eco-nomic expectations, and federal policy outcomes (along with the Vietnam

18. A potential problem for error correction modelers may arise when the variables underconsideration maintain more than one equilibrating relationship and. therefore, multiple coin-tegrating vectors. In fact, a system of k time-series may include k - I distinct cointegratingvectors. For this analysis, I did not test for multiple equilibria, since I am working with only threevariables of substantive interest and since I have no a priori reason to anticipate more than oneequilibrium relationship. For an excellent treatment of the issue of multiple equilibria in general,as well as the relevant diagnostic procedures, see Ostrom and Smith in this volume.

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dummy)19 are cointegrated. This finding suggests the appropriateness of mod-eling policy preferences as an error correction process vis-a-vis economicexpectations and policy outcomes, and permits the development of an ECMvia the Engle and Granger two-step technique.

Granger Causality Tests

Before proceeding with the development of an error correction model ofdomestic policy sentiment, it is fruitful to note the consequences of relyingon conventional Granger causality tests for preliminary diagnoses of the hy-pothesized relationships among policy sentiment, economic expectations, andpolicy outcomes. As I have noted, such tests are inappropriate in light of co-integration, since they miss (or rather cancel out) the long-term causal compo-nent that ties cointegrated series to one another.

Nonetheless, having proposed that domestic policy sentiment is endoge-nous to economic expectations, an analyst accustomed to direct Granger testswould likely regress the current levels of the policy sentiment time-series on anumber of lags of itself and the same number of lags of economic expecta-tions, relying on an F-ratio to determine the degree to which the past values ofeconomic expectations improve our knowledge of current values of policysentiment beyond simply a knowledge of the past values of policy sentimentalone. If the exclusion of economic expectations from such a "model" leads toa statistically significant reduction in explanatory power, an analyst would befree to conclude that policy sentiment is endogenous to economic expecta-tions; or, stated another way, that economic expectations Granger cause policysentiment.

Results of Granger tests on policy sentiment, economic expectations,and policy outcomes are reported in table 6. Eight lags (two years of quarterlydata) are used in each test, and the findings are as expected. No evidence isgenerated in support of the hypothesized causal relationships. If one were torely on conventional causality tests, the hypothesis that domestic policy senti-ment is endogenous to either of the other phenomena under analysis would benipped in the bud.

Suppose, however, that our mythical conventional analyst remains unfet-tered by such findings, and proceeds with the development of a cross-correlation function of prewhitened time-series representing policy sentimentand economic expectations. After identifying and estimating each time-series(policy sentiment as /[ 1 ]MA[ 1 ] and economic expectations as first-order inte-

19. The three time-series of substantive interest are "equally" cointegrated when the Viet-nam dummy is excluded. Indeed, the only coefficient affected by its inclusion is the constantlerm.

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TABLE 6. Domestic Policy Sentiment Granger Causality Tests

Independent Variable (8 lags) F' p-value

Current Levels of PolicySentiment

Levels of economic expectations 1.31 0.26Levels of federal policy outcomes 1.50 0.18

Current Changes inPolicy Sentiment

Changes in economic expectations 1.05 0.41Changes in federal policy outcomes 1.77 0.10

•Entries are F-statistics measuring the change in R2 when lags of the independentvariable are excluded from an equation regressing current values of the dependentvariable on eight lags or itself and eight lags of the independent variable. Each F-testfor the levels analyses has 8 degrees of freedom in the numerator and 55 in thedenominator, for the changes analyses, 8 and 54.

grated), cross-correlations show no significant spikes at any lag (ten in eachdirection tested). Our analyst is thus unable to estimate a transfer function,given no evidence of any such process.

The point of these digressions is to underline the importance of theobservation that, under certain circumstances (namely, a finding of cointegra-tion or error correction), conventional prewhitening approaches to the study oftime-series will miss the mark. Tests of Granger causality or estimations ofcross-correlation functions will miss the long-term causal component thaterror-correcting time-series share. Furthermore, in light of the fact that anequilibrium relationship has been hypothesized to exist among the variables ofinterest, it makes little theoretical sense to rely on a modeling strategy (suchas the Box-Jenkins transfer function methodology) best suited to capturing theeffect of an input perturbation on an output series. By their nature, Grangercausality tests and transfer function analyses focus attention on short-termeffects (i.e., the relationship between current or lagged changes in the inde-pendent variable and current changes in the dependent process), whereaserror-correcting processes maintain long-term equilibrium relationships in ad-dition to—or instead of—short-term associations.

Vector Autoregressions as Model Diagnostic Tools

Having identified evidence of the anticipated long-term relationships amongdomestic policy sentiment, economic expectations, and federal policy out-comes, I now turn to the specification of short-term effects. Specifically, the

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decision of lag lengths must be addressed. How much time lapses beforechanges in economic expectations, for example, may be expected to affectchanges in policy preferences? If one so chooses, a model may be specifiedthat presumes that each variable expected to have a short-term effect registersits effect after one period, so that changes at time t in the dependent phenome-non are (partially) a function of changes in the independent phenomena attime t — 1. For those who prefer a more empirically driven process, vectorautoregressions may be used to determine which lags of which first-differenced variables belong in the "front half" of the final error correctionmodel. In addition, VARs may be used to address questions of endogeneity.As Granger (1988) argues, a finding of cointegration is tantamount to thedetermination of causality, but the direction of causality remains unsolved.

Following the procedures outlined by Engle and Granger (1987), a sys-tem of three vector autoregressions was estimated by regressing the firstdifferences of domestic policy sentiment, economic expectations, and federalpolicy outcomes on four lags (/ — 1 to t — 4) of the first differences of eachseries, as well as one lag of the levels of each (the latter being the errorcorrection component of the final model). No such VAR was estimated for theVietnam dummy, and only its lagged levels were included in the other regres-sions. Another system was estimated in which the lagged levels of all fourvariables were replaced by the vector of residuals obtained from the cointegra-tion regression. This second system, according to Engle and Granger (1987),addresses the question of endogeneity. In order to conclude that a variable isendogenous, the coefficient corresponding to the vector of residuals must benegative and significant in the VAR associated with that variable. The resultsof these analyses led to the conclusions that only domestic policy sentiment isendogenous to the other variables and only federal policy outcomes laggedone quarter exert a statistically significant effect on changes in policy senti-ment. The latter finding is consistent with the results of an alternative specifi-cation strategy, in which one lag each of all variables hypothesized to exert ashort-term impact on policy preferences was included in a first-cut errorcorrection model.

Comparing Error Correction Modelsof Domestic Policy Sentiment

Two error correction models of domestic policy sentiment were estimated,one as a single equation and the other derived from the Engle and Grangertwo-step technique. In both models, included among the independent vari-ables were the lagged changes of policy outcomes, the "political" componentof long-term business expectations, and the dummies representing Vietnam,Watergate, the Iran hostage crisis, and the Iran-Contra affair. What distin-

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TABLE 7. Estimating Error Correction Models of U.S. Domestic Policy Sentiment

Changes in Domestic Policy Sentiment

Independent Variable OLS Coefficient SE r-ratio p-value

Single Equation Error Correction Model3

Changes in policy outcomes laggedone period

Levels of domestic policy sentimentlagged one period

Levels of economic expectationslagged one period

Levels of policy outcomes laggedone period

Levels of Vietnam War dummy vari-able lagged one period

Constant

Changes in policy outcomes laggedone period

Cointegration regression residualslagged one period

-12.48

-0 .69

0.25

-13.23

5.84

42.39

5.93

0.12

0.05

2.95

2.17

8.16

-2.10

- 5 95

5.47

-4.49

2.70

5.19

0.040

0.000

0.000

0.000

0.010

0.000

Engle and Granger Two-Step ErrorCorrection Model6

-10.68

-0.71

5.84

0.11

-1.83 0.070

-6.21 0.000

•«3 = .41. adjusted R(X* distribution, 1 df).

W?2 = .41. adjusted /?? = 39. SEE = 2.85.(X2 distribution. 1 df).

.37, SEE = 2.90. DW = 2.09, Q = 21.63 (24 df). first-order ARCH = 0.04

DW = 2.02, Q = 20.97 (24 df). first-order ARCH = 0.01

guishes the models from one another is the treatment of the long-term equilib-rium relationship assumed to exist between policy preferences and economicexpectations, policy outcomes, and the Vietnam War and Watergate dummies.In the single equation model, the lagged levels of each of these variables wereincluded on the right-hand side of the equation, whereas, in the two-stepmodel, the residuals from the cointegration regression of table 5 were usedinstead.

The results of the two modeling approaches are presented in table 7.Before discussing the particulars of each model, an examination of their"common ground" is in order. Note, first, that in both models, the onlyphenomenon to register a statistically significant short-term effect on changesin domestic policy sentiment is the lagged changes of federal policy out-comes. This confirms the prior expectation that policy preferences—as mea-sured across a large number of issue domains—are relatively impervious (at

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least in the short run) to their environment. Indeed, one must remember thatthe short-term impact of policy outcomes on policy sentiment is exaggeratedin these models, since the policy measure is only "switched on" during a timewhen such outcomes are presumed to be exerting their greatest (or only)effect. Note also the similarity between the error correction coefficients. In thesingle equation model, this is the coefficient on the lagged levels of domesticpolicy sentiment; in the two-step model, it is the coefficient on the laggedlevels of the cointegration residuals. Both models indicate that equilibriumerrors are corrected at the rate of roughly 70 percent per quarter. This impliesthat domestic policy sentiment responds to equilibrium errors relativelyquickly, leaving only 30.0 percent of the disequilibrating shock present aftertwo quarters, 9.0 percent after three, 2.7 percent after four, and so on. Finally,note that the two models generate nearly identical measures of fit, as well asdiagnostics for serial correlation and heteroskedasticity.

The single equation error correction model of domestic policy sentimentsuggests the following equation.

ADPS, = 42.39 - 12.48*APOL,_, - 0.69*DPS,_,

+ 0.25*EE,_, - 13.23*POL,_,

+ 5.84*VIET,_, + e,. (29)

where DPS is domestic policy sentiment, POL is federal policy outcomes, EEis economic expectations, and VIET is the Vietnam War dummy. Rewritingthis equation in error correction form produces the following equation.

ADPS, = 42.39 - 12.48*APOL,_, - 0.69(DPS,_,

- 0.36*EE,_, + 19.17*POL,_,

- 8.46*VIET,_,) + e,, (30)

where the portion of the equation in parentheses represents the error correc-tion component of domestic policy sentiment. According to the model, do-mestic policy sentiment will be in an equilibrium relationship vis-a-vis eco-nomic expectations, federal policy outcomes, and the (post-) Vietnampolitical environment when this expression is equal to zero. Any shock to thisequilibrium relationship will be corrected by changes in domestic policysentiment at the rate of 69 percent per quarter, beginning one quarter after theshock is experienced.

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Interpreting the coefficients in the error correction component is astraightforward matter. Starting in an equilibrium state and holding all elseconstant, assume that economic expectations improve by five points (roughlyan 8 percent increase, based on the variance of the series during the timeperiod under observation), and that such a change in outlook is maintained.Since domestic policy sentiment and economic expectations maintain an equi-librium relationship when the lagged level of policy sentiment is equal toslightly more that one-third of the lagged level of expectations,20 the resultingdisequilibrium will be characterized by policy preferences that are too conser-vative (i.e., too low on a scale on which higher scores indicate more liberalvalues). In order to correct this disequilibrium, domestic policy sentiment willultimately shift in a liberal direction (increase) by 1.8 points, or slightlygreater than 6 percent. This reequilibration will take place over a number ofquarters, with the largest portion (69 percent) coming one quarter after theinitial shock. Thus, domestic policy sentiment will increase by 1.26 pointsone quarter after the jump in economic expectations, then 0.37 points (or 69percent of the remaining disequilibrium, 0.54 points), then 0.12 points, andso on.

Turning to the results of the Engle and Granger two-step estimation, wesee that changes in domestic policy sentiment may be expressed in the follow-ing manner.

ADPS, = -10.68*APOL,_, - 0.71*Z,_, + e,, (31)

where Z,_, represents the lagged residuals from the (first-step) cointegrationregression. Relying on the data in table 5, Z,_, may be expressed as

Z,_, = DPS,_, - 0.32*EE,_, + 23.56*POL,_,

- 6.37*VIET,_,. (32)

Substituting equation 32 into equation 31 (and bringing the constant outsidethe error correction component) yields

ADPS, = 47.78 - 10.68*APOL,_, - 0.71(DPS,_,

- 0.32*EE,_, + 23.56*POL,_,

- 6.37*VIET,..,) + e,. (33)

20. Plus some unknown constant, which is presumed to exist but cannot be estimated.

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The similarities in the coefficients of equations 30 and 33 indicate that the twoapproaches to error correction modeling yield approximately the same results.Thus, the only difference between the two strategies in this analysis is theestimation of the equilibrium relationship hypothesized among the variablesof interest. In the single equation model, the long-term associations in levelspresumed to exist among policy preferences, economic expectations, andpolicy outcomes were estimated directly, forcing each independent variable tomake a unique contribution to the equilibrium environment. Furthermore, thesingle equation model presumes that the equilibrium relationship among thevariables of interest is asymmetric (as hypothesized); that is, only domesticpolicy sentiment responds to equilibrium errors. In the two-step model, on theother hand, equilibrium errors were estimated first in a static linear regressionof the levels of policy preferences on the levels of the independent variables,and the final error correction model incorporated the equilibrium errors inorder to estimate the rate of equilibration when controlling for short-termeffects. This approach permits equilibrium errors to be corrected equally byany of the variables—dependent and independent—included in the first-stepregression. In light of the similar results, 1 conclude that only policy sentimentresponds in such a manner.

Conclusion

This article has explored the nature and mechanics of error correction models,with an eye toward their applicability to longitudinal studies of politicalphenomena. Along the way, I have addressed integration, cointegration, andthe Dickey-Fuller diagnostic tests, the causal nature of the equilibrium rela-tionships witnessed among cointegrated (or simply error-correcting) time-series, and the differences that distinguish the Engle and Granger two-steptechnique from single equation error correction models. Finally, I developedan error correction model of U.S. domestic policy sentiment in an effort todemonstrate the procedures involved in the evolution of such analyses.

As I stated at the outset, both applied and theoretical treatments ofcointegration and error correction continue to emerge at a brisk pace from therealm of econometrics. Future efforts are likely to focus on the issue ofaccidental associations touched on above, shedding light on these scenarios.In the interim, I have offered preliminary evidence suggesting that, undercertain circumstances, single equation error correction models are better ableto guard against accidental associations in the estimation of multiple time-series equilibrium relationships than those models arrived at via the Engle andGranger two-step technique.

Error correction models offer analysts the opportunity to explain changesin one process in terms of both the past changes and levels of other phenom-

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An Exsay on Cointegraiion and Error Correction Models 225

ena, thereby affording the opportunity to bring together theories of short- andlong-term dynamics. This fact, along with the ease with which ECMs may bedeveloped, should guarantee their wide use in political analyses.

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Englc, Robert F.. and Clivc W. J. Granger. 1987. "Co-Integration and Error Correc-tion: Representation. Estimation, and Testing." Econometrica 55:251-76.

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A P P E N D I X

This simulation involves the estimation of 38,400 error correction models, 19,200estimated as single equation models and 19,200 estimated via the Engle and Grangertwo-step technique. The single equation models take the following form:

Ay, = c + d,AX,_, - d2Y,_, - </3X,_, - dtW,_, + e,,

and the Engle and Granger two-step modeling process involves the estimation of thefollowing two equations:

Y, = c + d,X, + d2W, + Z, (Step one)

AY, = rf4AX,_, - </,Z,_, + e, (Step two).

The time-series under analysis (AK and Y, X, and W) were created via the followingequations:

AX, = 5 + b,AX,_, - b2(Y,_, - fc,X,_, - 3) + e,,

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TABLE A1. Monte Cario Results

W Varianceb, b2 by Factor (v)

Two-StepFailures'

Single EquationFailures"

.25 .75 0.5

.25 .75 0.5

.25 .75 I.O

.25 .75 1.0

.50 .50 0.5

.50 .50 0.5

.50 .50 I.O

.50 .50 1.0

.75 .25 0.5

.75 .25 0.5

.75 .25 1.0

.75 .25 1.0

Total

10

10

10

10

10

10

100250500

1,000100250500

1,000100250500

1,000100250500

1,000100250500

1,000100250500

1,000100250500

1,000100250500

1,000100250500

1,000100250500

1,000100250500

1,000100250500

1,000

001

46(11.50)001

43 (10.75)00273(18.25)00269 (17.25)00000000000000000000000000000000

237 (1.23)

00018 (4.50)00014 (3.50)00016(4.00)00011 (2.75)0000000000000000000000000000000059(0.31)

"Failure data are per 400 iterations, with failure percentages stated in parentheses.

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X, = X,_, + e2l

where e,, e,, and e3 are random normal series of length T, and the coefficients bt, b2

b3, and v assume varying values as presented in the table.

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an essay on cointegration and error correction models

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