theme is common throughout most of the policy evaluation literature. Wherecitizen input is part of the evaluation process, its importance is usually less than thatgiven other components. For an example where this apparently was NOT the case, seePOLICEWOMEN ON PATROL (Washington: The Police Foundation, 1973).

^This response characterized many of the cities with OEO and other community orientedand based programs.

''This is often done explicitly, but also can happen implicitly. For instance, Dror inDESIGN FOR POLICY SCIENCES (pp. 89-99) discusses needed research in the policysciences and fails to elaborate the need to consider and develop more reliable andaccurate measures of citizen evaluation.

should not be surprising to political scientists, but it is a fact of political lifewhich is often ignored.

"Richard L. Cole develops a related point in his research dealing with citizen partici-pation. Richard L. Cole, CITIZEN PARTICIPATION AND URBAN PUBLIC POLICY,unpublished doctoral dissertation, Purdue University, 1973.

'In essence, effective and regular citizen input is directly related to access andpolitical power or the potential for that power. The politically disadvantaged may beable to use the evaluation process as a means to increase their overall politicalstrength within a specific geographical or political area.

"Caputo, David A., "The Evaluation of Urban Public Policy: A Developmental Modeland Some Reservations," PUBLIC ADMINISTRATION REVIEW, 33(March-April, 1973),pp. 113-119.

point needs to be fully understood. An individual's PERCEPTION of reality maybe just as important and even more relevant to policies affecting that reality thanobjective measures of that reality. Policy analysts need to develop a notion of per-ceptual reality in any evaluation attempts.

C. Design

THE USE OF TIME SERIES ANALYSIS IN THE STUDYOF PUBLIC POLICY

Virginia Gray*University of Minnesota

The purpose of this article is to demonstrate the appropriateness oftime series regression applied to the policy process. We begin by argu-ing its theoretical value, then discuss the few extant examples of suchresearch, and end with the methodological problems involved.

TfceoryMost comparative studies of policy, especially those at the sub-

national level, have been cross-sectional in design, i.e., the unit ofanalysis is the governmental system, such as a state, at a single pointin time or at repeated points in time.^ From these static analysesconclusions are sometimes drawn as to how policies can be changedover time. For example, Thomas Dye concluded that his findings fromcross-sectional correlations in the early 196O's

' 'The author is indebted to W. Phillips Shiveley, University of Minnesota, for detecting• at least one error in an earlier draft of this article.

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warn us not to be overly optimistic about the policy changeswhich can be expected from reapportionment, from the growthof two-party government in the South, or from an increase inNegro voter participation.

Economic growth rather than party competition will be the mostsignificant factor in improvements in state education, welfare,highways, and tax programs. Negroes . . . will find that eco-nomic development will define what they can provide in the waof public services as it has defined it for white policy-makers.

One might have some confidence that these empirical results aretheoretically sound if they are replicated time after time or if the samevariables which have high explanatory power for levels of expendituresalso explain change in levels of expenditures. Such is not the case,however. Both Hartwig and Sharkansky have found discrepancies inexplaining levels and change; Hofferbert has found discrepancies inrepeated cross-sections.^

The more fundamental distinction is that policy-making is a process;it occurs over time within a governmental system such as a state. Itdoes not occur across states; hence, the cross-sectional regressionequation or correlational analysis does not reflect the process fromwhich our data are gathered. On theoretical grounds, whether the theoryis the usual application of David Easton's systems theory or not, thefocus should be on explaining differences across time. A time seriesor longitudinal approach is the more appropriate technique because theunit of analysis is the system, such as a state, at time t, t-1, t-2, etc.

Examples of Longitudinal ResearchThe first place to start is with the recent article "Data Analysis,

Process Analysis, and System Change" which offers a convincingargument at the cross-national level for shifting to a longitudinaldesign.4 Their two studies (of a model from macro-economics and ofthe NPD in the Federal Republic of Germany) show that traditionalconventions of data analysis are fallacious under reasonably generalconditions. Among the more data-oriented articles one should consultat the national level is Russett's "Some Decisions in the RegressionAnalysis of Time-Series Data," in which he finds autocorrelation inhis variables, defense spending and GNP, and then he unfortunatelychooses to ignore the problem in the analysis.5

At the cross-national level one can consult Peters' "Economic andPolitical Effects on the Development of Social Expenditures in France,Sweden and the United Kingdom."^ His substantive results are thatboth economic growth variables and political variables are needed toexplain expenditures for social services in five functional areas. Hetackles the problem of autocorrelation in two ways. First, he arguesthat in the presence of autocorrelation one might add some theoretical-ly meaningful combination of variables which might eliminate theproblem. The inclusion of both economic and political variables to-gether in the regression equation eliminates autocorrelation for Franceand the United Kingdom. Secondly, from examination of the residualsfor Sweden he notices a shift point; division of the total time periodfrom I865-I965 into two smaller periods around the 1910 shift elimi-nates the autocorrelation.

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At the subnational level one might consult my "Time Series Analy-sis of State Spending" in which three models of the state policyprocess are evaluated cross-sectionally and over time.^ There areseveral interesting reversals in results from the usual cross-sectionalfindings: (1) the correlation of competition and turnout within eachstate is generally much lower than the cross-sectional results; (2) therelationship of the measure, welfare share, and per capita personalincome is negative over time but positive in cross-sectional data; (3)the relationship of the measure, education effort, and per capita per-sonal income is positive over time but negative in cross-sectionaldata. These results are from time series regression equations whichdid not exhibit autocorrelation and whose F-statistics were significantand from cross-sectional regression equations with no multicollinearityand significant F-statistics.

There are many examples one can cite from the economics literaturewhere relationships reverse according to the type of design: the re-lationship of income to birth rate is negative within household cross-sections while the relationship of income to birth rate is positivewithin business-cycle time series. Similarly, over the business cyclethe suicide rate is negatively correlated with the variables income andemployment; cross-sections show the relationship between familyincome and suicide rate to be positive.° Perhaps for this reason thestatistical theory underlying time series regression analysis has beenmost fully worked out in econometrics; here one can find discussionsof the problem of autocorrelation, tests for its presence, and a fewsolutions for eliminating it.

MethodologyOne reason that time series regression is rarely employed in politi-

cal science is that a significant problem often crops up—autocorre-lation. Consider the following regression equations:

(1) Yt = a + bXt + et

(2) Yt.i = a + bXt-l + et-1

which are rendered complex by the subscripts t indicating time, insteadof the usual i, indicating a state or other unit. Autocorrelation existswhen et is correlated, positively or negatively, with et-l, et-2, etc. Wesuspect the errors may be correlated because whatever factors producethe disturbance, e, in one year are likely to carry over into the follow-ing year. One econometrician, Kmenta, compares autoregression of thedisturbances with the effect of tapping a musical string: the sound isloudest at the time of impact but lingers on for awhile. The shorter thetime between tapping the strings, the more likely it is that the pre-ceding sound can still be heard.-^

The reason the correlation of the error terms presents a problem isbecause the assumptions of ordinary least squares (OLS) are no longermet, namely that each disturbance term represents an independentrandom drawing from a normal population with a mean of 0 and constantvariance (SD^). The parameters a and b are still unbiased and con-sistent, but OLS no longer gives efficient estimators, i.e., the sampling

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interval around each estimate is large and thus it is difficult to testhypotheses, presumably our goal in the first place.

There are several tests for the presence of serial correlation. Onecan produce die fitted residuals e from equation (1) and then applyOLS to the equation:

(3) et - pet-1 + ut

The error term uj now meets OLS assumptions of normality and inde-pendence because all the serial correlation is caught by p. If p = 0,there is little serial correlation. The most common statistic used totest for autocorrelation, Durbin and Watson's d, is just an equivalenttest for the hypothesis that p = 0.

A simple but less precise way one can test the residuals for thepresence of autocorrelation is by graphing them. If the plot of theirfrequency distribution appears toj be bell-shaped (normal), serial cor-relation is not present. If plotting the residuals on the vertical axisagainst time on the horizontal axis produces a definite trend, serialcorrelation is indicated. If fairly smooth long waves show up in thegraph, positive serial correlation is indicated; persistent alternationsbetween negative and positive values indicates negative serial cor-relation. ^^

If one can reject the hypothesis of serial correlation by any ofthese methods, then proceed by OLS. If one cannot reject the hypothe-sis, there are several solutions from the econometrics literature toconsider. Solutions like get more data" will not be discussed herebecause we are rarely in that position. The more practical solutionsare discussed and evaluated below.

One can use the maximum-likelihood method to estimate all theparameters of the equation simultaneously. ^ ^ This method will not beof much use for most political scientists because it requires an ad-vanced understanding of econometric theory and its operation requiresmuch computer time.

One can use an iterative method, i.e., from equation (3) obtain anestimate of p and construct new variables (Yt - pYt-l) and (Xt - pXt-l)and use OLS on the equation:

(4) (Yt - pYt-1) = a(l - p) + b (Xt - pXt-l) + Ut

Get the second round residuals and use them to obtain a new estimateof p; repeat the procedure until the estimates converge.'^

Finally, if one can assume p = I, then first differences can be used.Substitute for p and rearrange equation (3) so that:

(5) et - let-l = Ut

Recall that Ut possesses the OLS assumptions. Subtracting equation(2) from equation (1) yields an equation in the form of first differences:

(6) A Yt = b A Xt + Ut

Note that the intercept now drops out of the equation. The estimate ofthe intercept, if significantly different from 0, can then be interpreted

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as a linear time trend, i.e., it tells how much the variable being ex-plained changes every time i d ^ ^

Kmenta argues that die use of first differences, a fairly commonprocedure in applied economic research, is not justified unless p isbelieved to be close to 1 and that this knowledge usually does notexist. 1'̂ In the field of policy analysis, however, the belief that thetrue value of p is close to unity is quite reasonable if you examineclosely the implications of that assumption. According to Christ dieassumption that p= 1

means that when a shift occurs in a particular year, it staysforever, moving the curve or surface to a new position, fromwhich the next year's shift is received. There is no averageposition of the curve or surface, for it can be pushed back orforth quite a distance if several successive shifts should allcome in the same direction.^ 5

All studies of budgeting, in particular, and policy-making, in gener-al, at the national level and at the state level demonstrate that thedecision revolves about increases or decreases to last year's base.^^Once a new base is achieved it remains until the next year's incrementis added on. Thus, the crucial assumption in order to use first differ-ences, that p = I, is met in the type of process we are typically study-ing. The use of this fairly simple method of eliminating autocorrelationthrough first differences ought to be sufficient in most political scienceresearch on public policy.^ ^'

ConclusionThe appropriateness of a longitudinal design, rather than the typical

cross-sectional design, has been argued because the analytic model isisomorphic with the theoretic model, i.e., time series regressionfocuses on a process occurring over time. Results from the few studiesemploying this research design vary a good deal from those conductedon cross-sectional data. The major stumbling block, autocorrelation, isexplained, as well as its effects and its solutions. It is shown that themethod of first differences is suitable for most policy analysis be-cause its primary assumption is empirically justifiable.

REFERENCES

^Richard E. Dawson, "Social Development, Party Competition, and Policy, THEAMERICAN PARTY SYSTEMS, ed., William Nisbet Chambers and Walter Dean Butnham(New York: Oxford University Press, 1967); Richard I. Hofferbert, SocioeconomicDimensions of the American States: 1890-1960," MIDWEST JOURNAL OF POLITICALSCIENCE, XII (August. 1968), pp. 401-418; Ira Sharkansky. SPENDING IN THEAMERICAN STATES (Chicago: Rand McNally and Company, 1968); Richard E. Dawsonand Virginia Gray, "State Welfare Policies," POLITICS IN THE AMERICAN STATES.ed. Herbert Jacob and Kenneth N. Vines (2d ed.; Boston: Little, Brown and Company,1971).

^Thomas R. Dye, POLITICS. ECONOMICS. AND THE PUBLIC (Chicago: Rand McNailyand Co., 1966), p. 301. The fundamental question is also often posed in developmentalterms:

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Do relationships between factors such as urbanization and political participa':tion persist through different time periods, or are they related only for someparticular historical period? How are changes in levels of economic develop-ment related to changes in political composition? Or, more specifically, whatimpact is increasing industrialization likely to have upon the historical one-party domination in the Southern states?

in Dawson, pp. 205-206.

^harkansky, SPENDING; Frederick Hartwig, "Determinants of Rates of Change: TheCase of Combined State and Local Expenditures Per Pupil Between 1940 and I960,"paper prepared for delivery at the 1973 Annual Meeting of the Midwest PoliticalScience Association, Chicago, Illinois; Hofferbert, "Socioeconomic Dimensions".

'^Ronald D. Brunner and Klaus Liepelt, "Data Analysis, Process Analysis, and SystemChange," MIDWEST JOURNAL OF POLITICAL SCIENCE, XVI (November 1972),538-569.

^Bruce M. Russett, "Some Decisions in the Regression Analysis of Time-Series Data,"MATHEMATICAL APPLICATIONS IN POLITICAL SCIENCE V, eds., James F.Herndon and Joseph L. Bernd (Charlottesville: University of Virginia Press, 1971).

"B. Guy Peters, "Economic and Political Effects on the Development of Social Ex-penditures in France, Sweden and the United Kingdom," MIDWEST JOURNAL OFPOLITICAL SCIENCE, XVI (May 1972), 225-238.

^Virginia Gray, "Time Series Analysis of State Spending," paper prepared for deliveryat the 1973 Annual Meeting of the Midwest Political Science Association, Chicago,Illinois.

"For a discussion of these examples and of the problem in general, see: Dennis J.Aigner and Julian L. Simon, "A Specification Bias Interpretation of Cross-Section vs.Time Series Parameter Estimates," WESTERN ECONOMIC JOURNAL, VIll (January,1970), 149-151.

9jan Kmenta, ELEMENTS OF ECONOMETRIC^^ (New York: Macmillan Co., 1971),p. 270.

F. Christ, ECONOMETRIC MODELS AND METHODS (New York: John Wiley andSons, 1966), pp. 521-522.

, p. 483; Kmenta, pp. 282-285.

, p. 484; Kmenta, p. 288.

, p. 485.

, p. 289.

, p. 484. However, the year-to-year change (A Yj) must fluctuate randomly aboutzero, i.e., decrements as well as increments must be theoretically possible.

^"Ira Sharkansky, "Agency Requests, Gubernatorial Support and Budget Success inState Legislatures," AMERICAN POLITICAL SCIENCE REVIEW, LXII (December,1968), 1220-1231; Otto A. Davis, M.A.H. Dempster, and Aaron Wildavsky, "A Theoryof the Budgetary Process," AMERICAN POLITICAL SCIENCE REVIEW, LX (Sep-tember, 1968), 529-547; for a "revisionist" explanation of the budgetary process, seeJohn A. Wanat, "Bases of Budgetary incrementalism," AMERICAN POLITICALSCIENCE REVIEW, forthcoming.

For an elementary introduction to difference equations, see: Samuel Goldberg,INTRODUCTION TO DIFFERENCE EQUATIONS (New York: John Wiley and Sons,1958). For applications of regression analysis to difference equations, see: VirginiaGray, "Theories of Party Strategy and Public Policies in the American States,"unpublished doctoral dissertation, Washington University (St. Louis), 1972, or Gray,"Time Series".

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