Curva de Phillips

Two Econometric Replications: The Historic Phillips and Lipsey-Phillips Curves Nancy J. Wulwick

Introduction: On Replication

“To replicate” has two meanings in science. In one sense, to replicate means to repeat the results of experiments under varying experimental circumstances. Scientists in that context are testing their explanations of established phenomena. To replicate also means to repeat entire experiments. By duplicating, or successfully checking experiments, scientists establish that there is a phenomenon to be explained (Collins 1991; Cartwright 1991; Backhouse 1992). This essay is concerned with replication in economics in the sense of repeating entire experiments or, their economic counterpart, econometric studies.

The traditional view in the experimental sciences is that the possibility of repeating experiments forms the basis of objective knowledge. How much repetition has actually occurred in the experimental sciences is a controversial matter. Certainly scientists have checked experiments that produced dubious results and treated a failed repetition as a “strong negative criterion” bearing against the original research (Mulkay and Gilbert 1991, 164). Notable failures to repeat experiments have marked the history of science (Harr6 198 1,175-76; Frank 1986,140-62,175-76; Chamberlain 1990, 136-45; Hoover 1993,259).

Correspondence may be addressed to Professor Nancy J. Wulwick, Department of Economics, State University of New York, PO Box 6000, Binghamton NY 13902-6000. I thank T. Mayer, M. Lovell, J. J. Thomas, H. Collins, and the Binghamton University Economics Department Seminar for their comments.

History of Political Economy 2813 @ 1996 by Duke University Press.

392 History of Political Economy 28:3 ( 1 996)

The credo of experimental repetition never has taken hold in economics. The formal education of economists generally ignores the matter of keeping records that might be obtained quickly and made available to other people. Econometrics courses teach students necessary techniques and mathematical proofs of important theorems, but the standards of workmanship do not include the preparation of data-a basic compo- nent of the curricula in the hard sciences. In light of their education, it is not surprising that economists rarely check the econometric work of their colleagues and that attempts to do so have been disappointing (Dewald, Thursby, and Anderson 1986; Fuess 1993, 12; Feigenbaum and Levy 1993,225).

Some economists have opposed the idea of having to repeat econometric studies (Collins 1993; Mirowski and Sklivas 1991; Mirowski 1993). They have argued that checking published research wastes time. Nobody reads, much less cites, the great mass of economic articles. Any mistakes in the few important papers, the critics have argued, will be discovered as other researchers use the findings in their own work. From that perspective, that no one until now has published a check on Lipsey’s famous Phillips curve paper remains an unexplained anomaly.’

Critics of checking published research also have argued that conditions required to repeat studies do not exist. They suggest three main problems with exact replication: The time-series data cited in the original study are continuously updated. The algorithms used in computing hardware and software are not completely standardized and change over time. The interests of the original researchers and the checkers conflict since refutations are easier to publish, so original researchers often with- hold their data (Anderson and Dewald 1994, 80).2 In the hard sciences, despite those problems, the importance of other researchers being able to reproduce one’s own experiments is unquestioned (Mulkay and Gilbert 1991). The three problems loom large in economics mainly because its researchers customarily neglect to maintain careful, detailed, complete reports of their econometric studies. This essay attempts to repeat two

1 . There is no evidence in the English-language journal articles from 1960 to 1980 that cite Lipsey 1960 of any checks on it (Social Science Citation Index-five-year cumulative indexes). Not check of Lipsey 1960 is commonly known to those who have done research on the Phillips curve.

2. Admittedly, a universal command to disseminate original data conflicts with the implicit property rights of original researchers and thus the incentive to do research. The scholarly community recognizes instances when withholding data is appropriate.

Wulwick / Two Econometric Replications 393

crucial econometric studies: A. W. H. Phillips’s (1958a) estimation of a negative relation between the rates of inflation and unemployment by means of a sophisticated application of a time-honored, simple statistical method, and the replication of the Phillips curve by R. G. Lipsey, who applied to Phillips’s data ordinary least squares, which economists by 1960 had accepted as the standard statistical method (Lipsey 1960).3 I find that Phillips’s original study can be repeated closely, while Lipsey’s results are in certain important respects not reproducible. Yet economists have tended to favor Lipsey’s study because Lipsey used a more familiar method of estimation. I trace the difference in the repeatability of the two studies to the research practices of the two economists. Phillips, with his engineering background, kept careful records of his work. Lipsey, whose major contributions were in economic theory, apparently tried many for- mulations and published the “best” without reporting how he arrived at it.

This article falls into two main parts. The first part attends to Phillips’s estimation and the next deals with Lipsey’s replication. Checking econometric studies requires knowledge of the context in which the researchers worked. Thus, this article explores the researchers’ technical backgrounds and modes of thinking, the relevant contemporary scientific literature, the available computing equipment, and the statistical conventions of the 1950s. In expounding the rationale behind the two studies, this article explains the research problem that Phillips and Lipsey each sought to solve and why they thought that their solutions were appropriate (Popper 1968,3547). I report what the researchers said they did in their studies, what they actually did, what they could have done but did not do, and what difference their choices made to the final results. This methodolog- ically motivated exercise of checking Phillips’s and Lipsey’s empirical work deepens our understanding of the nature of the Phillips curve as a hypothetical construct.

Repeating Phillips’s Study

A. W. H. Phillips, as his mentor James Meade remarked, “should not go down to history just as ‘the Phillips curve’ chap” (letter to the author, 3 February 1986). The curve was only a major incident in the

3. I develop the distinction between the estimating methods of Phillips 1958a and Lipsey 1960 originally made by C. L. Gilbert (1976).

394 History of Political Economy 28:3 (1996)

general progress of Phillips’s studies on time-series modeling and classical control systems, which are also identified with his name (Salmon 1982, 622). His contributions to economics lay in the field of dynamic macroeconomic control, which he approached from the perspective of an electrical engineer, his original profession.

Phillips (b. 19 14-d.1975), who grew up in New Zealand, possessed an ability to innovate that was “in line with the New Zealand worship of improvisation” (Bergstrom 1978, xi). He became an apprenticed elec- trician before finishing high school and, while working as an electrical fitter in private homes, found himself having to concoct makeshift arrangements to get desired r e s ~ l t s . ~ Having learned differential equations through a correspondence course with the British Institute of Technol- ogy, he passed the examinations of the Institute of Electrical Engineers shortly after coming to London in the late 1930s. During the Depression, engineering education promoted social engineering, which may have influenced Phillips to pursue a sociology degree at the London School of Economics (LSE) on a veteran’s scholarship after serving in World War I1 (Montgomery and Becklund 1938, 133). While an undergraduate, Phillips built a hydraulic machine of the macroeconomy with the help of W. Newlyn (Phillips 1950; Newlyn 1992). Phillips’s demonstration of the innovative machine to the economics department won Phillips an assistant lectureship in economics at the LSE in 1950.

Using the hydraulic simulator to clarify his ideas, Phillips applied principles of engineering control to the control of the macroeconomy (Phillips 1953; Wulwick 1989, 84-85, 88). In engineering terminology, the economy was a negative-feedback system that eliminated deviations between the system’s actual and equilibrium state^.^ Phillips identified various types of feedback mechanisms to correct deviations in output, including a proportional control dependent on the size of the deviation. In light of the Samuelson-Hansen microeconomic stability equation, Phillips treated price flexibility as a proportional stabilizer. A nonlinear curve, later known as the Phillips curve, showed the rate of change of the price level (or the money-wage level) as a varying proportion of the difference between the actual and the equilibrium level of output

4. Oral communication to the author from M. Perlman at the annual meetings of the History of Economics Society, Philadelphia, 26-29 June 1993.

5. M. Friedman learned of adaptive expectations, which obey a feedback-based rule, from Phillips (Friedman’s address in honor of his eightieth birthday to the annual meetings of the Western Economic Association, San Francisco, California, 9-13 July 1992).


(Phillips 1953, 3 1; 1954, 307). The curve’s hyperbolic shape indicated money-wage inflexibility in the presence of demand-deficient unemployment and accentuated the effects on wage inflation of increased excess demand for labor. Phillips’s election in 1958 to the LSE’s Tooke Chair of Economics and Statistics (held previously by Friedrich von Hayek) had little to do with the Phillips curve, to which he gave an empirical grounding during England’s national debate over the control of “creeping inflation” (Phillips 1958a; letter to the author from J. Meade, 3 February 1986).

Stressing that control of the economy required quantification, Phillips was one of the pioneers of time-series estimation (Phillips 1958b). He defined conditions that, if satisfied, permitted the consistent estimation of a distributed lag equation in which the regressors figured as “causal” variables, in the sense developed by Clive Granger and Christopher Sims (Phillips 1956). He proposed a method of obtaining consistent and ef- ficient estimates of the parameters of simultaneous equation systems in which the disturbances are moving averages of random elements (Phillips [ 19661 1978). Controlling the economy meant modifying the structure of the system with the aim of reducing the variance of target variables (Phillips and Quenouille 1960; Phillips 1968). Phillips-reviving an old idea of econometrics now known as the Lucas critique-remarked that this definition of control implied that one could not predict the effects of a control from an econometric model without first identifying the changes in the underlying structural model that would be caused by the control.

Phillips used the computer at the National Physical Laboratory at Ted- dington (a section outside central London) to simulate the time-forms of lags in the responses of production, given changes in aggregate demand (Phillips 1957; Kendall 1960, 17-20). Phillips’s junior colleague R. G. Lipsey recalls that, of the members of the LSE economics department, “No one, except Bill [Phillips] had seen a[n electronic] computer in 1958. . . . Bill was exceptional in knowing about, let alone using, computers” (letter to the author from R. G. Lipsey, 28 July 1993). Nev- ertheless, Phillips did his Phillips curve estimates on a Marchant mechanical desk calculator that ran on electrical power. Why did Phillips, who in 1958 was in a rush to go on sabbatical in Australia, avoid using an electronic computer to speed his research?

Birkbeck College and University College London (UCL), which like LSE belonged to the University of London, had large-scale, automatic, high-speed, general purpose computers endowed by their users with gen-


ders and names6 Data inputs were punched onto cards using machines like typewriters and fed into card readers connected to the computer. There were no terminal hookups, so Phillips or a research assistant would have had to take the work some two miles up the road to Birkbeck or UCL.7 The computers, which allowed only one user at a time, were not in continuous use due to a shortage of operating staff and space and were down a substantial part of their scheduled computing time (Williams and Campbell-Kelly 1989, xi, 247,474-75). Those inconveniences were not surprising given that the stored-program electronic computer emerged in detailed form in the United States during World War 11, and England initially lagged behind the United States in computer technology and usage (Norberg 1992).

Today’s automatic computers would find the least-squares solution given fifty-three observations and two variables in about a minute; the same problem would take as long as an hour on the mechanical electrical- powered desk calculator if the researcher checked the results. Yet few statistical programs for use on automatic computers were readily available at that time.* Writing the least-squares program was time consum- ing, since researchers had to write the program in the machine language appropriate for their computers after having checked the chain of reason- ing on a mechanical electrical-powered desk calculator. And researchers expected automatic computers to be no more, if not less, reliable than

6. In early 1958, the physics department of University College had an All Purpose Electronic X-ray APE(X)C Computer. Birkbeck College had the APE(X)C and the British Tabulating Ma- chine HEC computers. In 1960, Phillips’s research assistant carried out least-squares estimates on Joyce May (LSE Library Archives, Collection Miscellaneous 857, section 2/3; letter from A. Raspin, Archivist of the LSE Library, 25 July 1995).

A drawback tooral history is that historical agents haveconflicting memories. Nancy Wulwick (1989, 180), citing an anonymous referee, stated that Phillips could have used the University of London Ferranti mainframe computer. However, R. A. Buckingham, director of the University of London Computer Unit in the late 1950s recalls that the Ferranti MR I1 Mercury was not installed in Gordon Square (near Birkbeck College) until the end of 1958 (letter to the author from R. W. Baker, Director of the University of London Computer Centre, 14 March 1994; enclosing copy of letter to R. W. Baker from R. A. Buckingham, 1 March 1994).

7. Wulwick (1989, 180), citing an anonymous referee, refers to Phillips’s research assistant. Memories conflict as to whether he had an assistant at the time (letters from D. Laidler, 28 July 1993; and M. Steuer, 21 July 1993).

8. Wulwick (1989, 180), citing an anonymous referee, implies that statistical software programs were available to Phillips. R. A. Buckingham recalls that a statistical program for least squares was available for use on the Ferranti MR I1 Mercury computer only after Phillips published his curve estimates (copy of letter from R. A. Buckingham to R. W. Baker, 1 March 1994, which R. W. Baker sent to the author).


mechanical desk calculators (Williams and Campbell-Kelly 1989, 199- 200).9 A major source of computing error was due to round-off error, which arose because the machines stored numbers composed of a relatively small, fixed number of digits (say only up to ten digits as compared to more than thirty-two today). Round-off error tended to be greater when researchers wrote a program that solved the normal equations of least squares by the Doolittle method or some other popular elimination method of inverting matrices. lo Rounding errors at the initial stages of the elimination method pyramided and amplified through subsequent stages of the method (Longley 1967; Goldstine and von Neumann 1963,14). At least the mechanical desk calculator permitted the meticulous Phillips to be flexible about the number of significant digits.

As an engineer, a control theorist, and a time-series expert, Phillips was imbued with the culture of applied science (Montgomery and Beck- lund 1938, 124-25, 156). That professional culture was governed by an ideology that fostered universalism, objectivity, accountability, and procedures for communication (Mulkay 199 1, 62-78). Good applied science was supposed to involve careful, detailed technical statements of the problem at hand and the means to solve it. No wonder then that Phillips (1958a) reported exactly how he constructed his curve.

Phillips constructed time series to represent annual rates of money- wage inflation and unemployment for the United Kingdom for 1861- 1957 (appendix 1; figures 1-3b). He fit a hyperbolic curve to the scatter diagram of money-wage inflation and unemployment for 1861-1913 by means of the time-honored method of regression and then superimposed the curve fitted to the 1861-1913 data to the scatter diagrams for 1913-48 and 1948-57. The 1861-1913 period, which contained 6.5 trade cycles (figures 4-10), confirmed Phillips’s preconception of a negative relation between inflation and unemployment (Wulwick 1989, 174-80).

Phillips noticed that the residual variation of the trade-cycle data from the fitted hyperbolic curve varied inversely with the change of unemployment, forming counterclockwise loops about the curve. To reflect

9. Rumor has it that it is impossible to replicate the econometric results done in the 1950s by economists in Manchester working on one of the first British stored-program automatic electronic computers (letter from D. Laidler to the author, 28 July 1993).

10, Given the model Y = b? + e, one found the solution 6 of the s,et of normal equations X ’ X b = X ’ Y . One solved for b by finding the inverse of X ’ X , that is, b = ( X ’ X ) - ’ X ’ Y . The Doolittle method is an abbreviated way of arriving at the same end product (Graybill 1961, 149-52; National Physical Laboratory 1961.7).


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Figure 1 Phillips’s figure 1, 1861-1931. Mean coordinates: (1.52, 5.06), (2.35, 1.55), (3.48, 0.85), (4.49, 0.35), (5.95, -0.18), (8.37, -0.35).

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Figure 2 Phillips’s figure 9, 191348 .

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Figure 2 Phillips’s figure 9, 191348 .

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Figure 3a Phillips’s figure 1 1, seven-month lead in unemployment, 1 948-57.

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Figure 3b Phillips’s figure 10, 1948-57.


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Figure 4 Phillips’s figure 2, 1861-68.

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Figure 6 Part of Phillips’s figure 4, 1879-86.

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Figure 10 Phillips's figure 8, 1909- 13.

the effects of the level and the change of unemployment on money-wage inflation, Phillips suggested a model with which engineers are familiar, the generalized hyperbola, I

y + a = bxC + k (l/x'") ( d x l d t ) + el (1) a , b , m > O k , c c O

(Phillips 1958a, 291 ; Gilbert 1976; Lipka 191 8, viii). The simple form of equation (1) was consistent with Phillips's interpretation of wage inflation as mainly the outcome of the competitive bidding of labor (Phillips 1958a, 284, 298; Desai 1984, 255-57). Unemployment rather than inflation figured as the right-hand-side variable since Phillips viewed unemployment as a proxy for aggregate demand, which could be stabilized by monetary policy (Phillips 1954, 315; 1958a, 299; Wulwick 1989,

The computing programs available in the 1950s could not solve most nonlinear problems (Goldstine and von Neumann 1963,2). No computing programs existed that could use the least-squares criterion to search for the unknown parameters of an equation like (l), which cannot be

185-86).

1 1 . Equation ( 1 ) can be written as the generalized hyperbola y + a = ( I / x a ) [ b + ( k / x m - ) ( d x l d t ) ] , a = -c


linearized in terms of logarithms. That technical problem led Phillips to improvise by omitting the variable ( k / x " ) (dx ld t ) , which left the equation in loglinear form

+ a ) = log(b) + c log(x) + e2 a , b > O c t O

e2 = el + ( k / x " ) (dx ld t ) .

Phillips estimated equation (2).

the term ( k l x " ) ( d x l d t ) from the estimating equation. Phillips, the time-series expert, offered three justifications for omitting

The Negligible Bias in the Estimate of a

The expected estimate of b in equation (2), assuming equation (1) to be the true equation, is

E ( 6 ) = b + k cov [ x c , (l/x") (dx ld t ) ] / var(xC).

The estimate of b would be biased if the covariance and, therefore, the correlation r [ x c , (1 / x m ) ( d x l d t ) ] were nonzero.'* Phillips claimed that "it could easily be shown that (l/x") ( d x l d t ) is uncorrelated with x or any power of x provided that x is, as in this case, a trend-free variable" (Phillips 1958, 290). Indeed, by plotting x , Phillips could have shown easily with his data that unemployment lacked a time trend; Phillips even may have estimated the correlation r ( x , dx ld t ) , which comes to only 0.002.13 It would have been difficult to prove mathematically that ( l /xm) ( d x l d t ) is uncorrelated with any power c of x , but Phillips could have known from his experience of dealing with data or could have ascertained with the data at hand that the expression (l/x") ( d x l d t ) and x c are virtually uncorrelated, given arbitrary values of c and m (see table 9). Thus with good reason, Phillips could have tolerated the omitted variable bias in the estimate of 6.

Reducing the Bias in the Estimate of a

There remained the possibility of bias in the estimate of a due to the presence of the additive error term, e2, in equation (2) (Gilbert 1976,

12. E ( 6 ) = b + C C O V [ X C , l / x m ( d x / d t ) ] . 13. r ( x , d x / d r ) = cov [ x , ( d x l d r ) ] /mJvar(dxldt) . If r = 0, then cov 1. . .] = 0.

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Figure 11 Averages superimposed upon the 6.5 business cycles, 1861-1913. Mean coordinates: (1.52,5.06), (2 .35 , l . 55) , (3.48,O. 85), (4.49,O. 35), (5.95, -0. 18), (8.37, -0.35).

52-53). Phillips averaged his data to reduce the omitted variable bias, explaining that “since each interval [of my figure 113 includes years in which unemployment was increasing and years in which it was decreas- ing the effect of changing unemployment on the rate of change of wage rates tends to be cancelled out by . . . averaging, so that each cross gives an approximation to the rate of change of wages which would be associated with the indicated level of unemployment if unemployment were held constant at that level” (Phillips 1958a, 290-91). Accordingly, equation (2) estimated the relation holding between y and x when the omitted variable dx /d t x O.I4

‘The Imprecision of k

Phillips observed from the graphs of five of the complete business cycles (figures 4-5,7-10) that “there is a close relation between the deviations of

14. H. Brauchli (1972) and M. Desai (1975,9-10), as well as A. C. Chiang (1984,493-94), present thc formal basis for the argument.


the points from the fitted curve [or yi - j ] and the first central differences of the employment figures [dx/dt], though the magnitude of the relation does not seem to have remained constant over the whole period” (1958a, 291 n. 1). Given his experienced eye for data patterns, Phillips suspected the numerical estimate of the negative relation between the observed rate of inflation and unemployment would be imprecise.’’ Hence Phillips had a third reason that justified estimating equation (2).

An economist in the 1950s who relied upon a mechanical desk calculator and, like Phillips, was faced with fifty-three x , y observations would have been prone to group and average the data in order to reduce the number of computations used to solve the normal equations of least squares. G. Routh, an economist at the National Institute of Social and Economic Research, used the method of averages in his replication of the Phillips curve for 186 1-1 9 13 given alternative data transformations (Routh 1959). The method of averaging also had a theoretical rationale, since the least-squares line shows the conditional mean of y given x , E ( y I x ) (Pearson [ 19051 1956,483-85; Stigler 1986). Moreover, Phillips, as a trained engineer, was familiar with data averaging as a tool to filter out stochastic disturbances (Phillips and Quenouille 1960,335-37). Ex- temporizing, Phillips used averaging to help find the estimate of a that minimized the omitted variable bias (Gilbert 1976,53).

Phillips divided the fifty-three observations into six groups, which was consistent with traditional practice.I6 He divided the x-axis on the scatter diagram (figure 11) into six fixed bandwidths b, (z = 1, . . .6 ) defined by x = 0-2,2-3,3-4,4-5,5-7, and 7-1 l.I7 The average values of unemployment X in the bandwidths were defined as X, = C x , / n , and the average values of money-wage inflation 7 in the vertical arrays by

15. As it turned out, the standard crror of the estimate of k by means of modern nonlinear least squares is relatively small (table 4, equation Ib).

16. Sturges’s rule to determine the number of groups states that when n is the number of groups and N the number of observations, then 2“-’ = N (Sturges 1926.65). The fifty-three observations call for seven groups (n = 6.73).

17. Wulwick 1989 states, “since Phillips left no papers, we do not know if he tried out alternative intervals’’ (180). For a long time, researchers understood that Phillips left no papers (telephone conversations with the author and A. Sleeman, 16 June and 23 October 1995). According to C. A. Blyth, Phillips’s biographer, “as far as I can find out his widow has no papers” (letter to the author, 6 October 1986). In 1994, Phillips’s widow donated papers to the LSE archival collection (Collection Miscellaneous, 857). None of the papers in the collection concern the Phillips curve, except for a large table of data, which does not contain any information about the choice of intervals.


I

y z = C y , / n , . The least-squares line was computed on the basis of the averaged data X, , y,,

Phillips’s choice of bandwidths preserved the negative sign of the first derivative across the graph of averages d y l d x < 0. The resulting graph of averages supported Phillips’s hypothesis by conveying the image of a strong, negative, highly nonlinear relation between the average rate of money-wage inflation and unemployment.

Statisticians solved the problem of estimating the constant 6 of the generalized hyperbola in various ways (Lipka 19 18, 140; Wolfenden 1942, 329). In Phillips’s study, “the constants b and c were estimated by least squares using the values of y and x corresponding to the crosses in the four intervals between 0 and 5 percent unemployment [z = 1 . . .4] the constant a being chosen by trial and error to make the curve pass as close as possible to the remaining two crosses in the intervals between 5 and 11 percent unemployment” (Phillips 1958a, 290). In light of Phillips’s report, I repeated Phillips’s estimates of b, c, and a by means of this iterative procedure (see table 10):

Round I , Step I I estimated the constants bl and C I in the equation’*

logy, =b1 +cllogT,+e, a = O z = 1 . . . 4 (2b)

for the four crosses of the graph of averages in the northeast quadrant. The results were bl = 13.08 and t1 = -2.35.

Step 2 Given hI and &, I predicted wage inflation for the two crosses in the southeast quadrant of the graph of averages and found the average absolute deviation D of the predicted ypz from the actual average inflation rate yz, where D = lrpz - y,1/2 for z = 5,6. For Round 1 , D1 = 0.4.

Round 2, Step I

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I arbitrarily set 6 = DI = 0.4.

Step 2 With 6 = 0.4, I reestimated b and c of equation (2a) for the four crosses in the northeast quadrant (z = 1 . . .4). Given the new estimates 6 2 = 10.46 and t2 = - 1.76, I predicted wage inflation for the two crosses in the southeast quadrant (z = 5,6) and found the average

18. Desai, who attempted to find 6, stopped at Round 1 (1975; 1 I ) .


absolute deviation 0 2 = I. 21, which is less than D,. Hence I went on to Round 3.

Rounds 3, . . . , n decreased in value, I increased by 0.1. Thus in Round 3, 2i = 0.5; in Round 4, 2i = 0.6; and so on.

The penultimate round yielded 6 = 0.9 as the estimate associated with the smallest D-value. Setting 6 = 0.9 and estimating Phillips's equation (table 4, equation 2a) yields estimates that are identical to Phillips's estimates (Phillips 1958a, 290).

Historians of economics have thought that Phillips could not have estimated his preferred nonlinear equation ( 1) for technological reasons (Gilbert 1976; Wulwick 1989). In fact, Phillips had the know-how after estimating equation (2) to estimate equation (1). Using the estimates of (l), he could have specified the equation

As long as 0 2 . . .

Next, he could have searched for value of & that permitted equation ( la) to fit the fifty-three observations most closely (see table 11). Phillips, having searched in the range 0 5 h 5 2, would have arrived at least-squares estimates of his preferred equation (1) (table 4, equation la; figure 12). The differences between the estimates of Phillips's equation (1) given the best technique available in 1958 and modern nonlinear least squares are minor (table 4, equations 1 a, 1 b). The striking difference is the weeks that Phillips would have needed to estimate equation (1) compared to the minutes that it takes contemporary economists.

The technological limits to nonlinear least-squares estimation com- pelled Phillips to rely heavily on graphical discourse (Phillips 1958a, 287-90). Phillips saw that the Phillips curve for 1861-1913 (table 4, equation 2a) fit the data of the 1929-37 trade cycle (Phillips 1958a, 295; figure 2). Modern nonlinear least-squares estimation indicates no relation between inflation and unemployment for 1929-37 or other periods during the interwar years. l 9 Next, Phillips superimposed the Phillips curve for 1861-1913 onto the post-World War I1 data that formed a clock-

19. For the nonlinear least-squares trials for 1929-37. the program did not reach a solution after 100 iterations with 4 groups of starting parameters: -a: - 1 , 1.0, -0.5; b: 7.2. 13.7; c: -2, -2, - 1 , -2. A negative linear relation between money wage inflation and unemployment appears in the data for the mid-1920s to the late 1930s. Phillips's discussion of the interwar period (1958a, 295) admilscost push factors, which may explain why he did not use the interwar data points in estimation (letter to the author from C. L. Gilbert, 27 January 1986).


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Figure 12 superimposed on his actual equation 2.

Phillips’s hypothetical estimates of his preferred equation 1

wise loop about the curve (figure 3b; table 4, equation 2a). Proposing that the clockwise loop arose because of a time lag in the response of money wages to unemployment, Phillips introduced a seven-month lead in unemployment that eliminated the loop and visibly tightened the fit of the curve for 1861-1913 to the modern data (figure 3a; appendix 1, especially table 2). Modern nonlinear least squares indicates no significant difference between the estimated coefficients of the Phillips curve with and without the seven-month unemployment lead (table 4, equations 2c-d). Nevertheless, the modem nonlinear least-squares estimates of the Phillips curve for 1861-1913 and 1948-57 show that Phillips’s experience in time-series simulation served him well, with the curve appearing as stable as he had surmised (table 4, equations 2b, 2d, 2e).20

Lipsey’s Replication of the Phillips Curve

Phillips’s contemporaries objected to the statistical methods of double- averaging by which Phillips constructed his curve (Knowles and Winsten

20. The Phillips’ curve (equation 2) using Phillips’s data sources and definitions is stable for the postwar period up to the early 1970s but is sensitive to slight changes in the sample period.


1959). They suspected that the Phillips curve relation arose merely as an artifact of the method of averages. Lipsey (1960) attempted to replicate the Phillips curve by means of ordinary least squares, which became the standard method of analysis in economics by 1960. Lipsey’s claim to have replicated the Phillips curve by means of a standard technique convinced many economists to accept the Phillips curve as an empirical entity.21 Economists have referred to Lipsey’s article in Economicu, a heavily cited journal, as a “seminal [empirical ] contribution,” a “brilliant piece,” and a “careful reconsideration,” which offers a “more useful” statistical analysis than the original Phillips paper (Perry 1966, 8; Laidler and Parkin 1975, 753; Santomero and Seater 1978, 500; Phelps 1968, 681; 1987, 858; Wulwick 1987, 841-42). Lipsey’s article, in the context of increasingly available electronic computers, inspired a booming industry in Phillips curve estimation. Between 1965 and 1980, Lipsey’s article received over 230 citations, almost five times as many journal citations as G . L. Perry’s (1964) comparable article on the Phillips curve for the United States (Social Science Citation [Cumulative] Index). Many contemporary economists learned of the Phillips curve from Lipsey ’s 1960 article, which remained on graduate reading lists for a long time. Yet, word circulated informally among economists that the information supplied by Lipsey (1960) about his data and methods was insufficient for later researchers to be able to repeat or approximate his estimates of the Phillips curve.22

Why has no economist published a paper about a failed attempt to repeat Lipsey’s exercise until now? Detailed checking of an econometric exercise is expensive and the benefits are highly uncertain (Mayer 1993, 272). Confronting the professionally active Lipsey with the news that his famous Phillips curve estimates are nonrepeatable is awkward so- cially. An article in a technical journal saying that Lipsey’s 1960 study is nonrepeatable, appearing after citations to that study have ebbed, might

21. Phillips used multiple regression to estimate the relation between money wage inflation and the unemployment rate for Australia from 1947-58 (Phillips 1959, 6-7; Sleeman 1983, 25-26). Phillips smoothed the data using 4-quarter moving averages.

22. I received the reports of failed attempts to reproduce Lipsey 1960 from C. L. Gilbert at a conference in 1989 on “Appraising Economic Theories” (sponsored by the Latsis Foundation, Capri, 15-18 October), and from M. Desai in his letter to me dated 28 February 1994. Allen Sleeman at the 1992 annual meetings of the Western Economic Association (San Francisco, 11-14 July) passed me a message (still in my possession) saying, “No one has been able to reproduce Lipsey 1914-17.” Sleeman in telephone conversations with me (I6 June, 23 October 1995) confirmed the message.

Wulwick / Two Econometric Replications 41 1

receive few citations (Feigenbaum and Levy 1994). However, it is important to show that seminal findings cannot be repeated if economics is to maintain its claim to objectivity.

Lipsey (b. 1928) entered LSE as a graduate student in 1953, became an assistant lecturer of economics in 1955, and obtained his Ph.D. degree in 1957. The graduate students in economics at LSE specialized in one of several fields, including analytical and descriptive economics (A & D), the field that Lipsey chose. Lionel Robbins, who was convener of the economics department, headed the A & D section and its series of seminars, which were important theoretical events for the Publications by the faculty associated with the A & D section conveyed the interests of the Robbins seminars in methodological questions, the descriptive realism of assumptions, and deductive analyses (Lipsey and Lancaster 1956; Lipsey 1957; Archibald and Lipsey 1958, 1960). Later publications criticized Robbins’s approach to economics. G. C. Archibald, who became an LSE economics lecturer, intended to show that “the Robbins-Samuelson pro- gramme” of comparative statics, which obtained qualitative predictions without quantitative information, produced models that were “almost empty” (Archibald 1962, 9). The introduction to Lipsey’s first edition of his textbook (1963) on positive economics, which stressed the neces- sity of quantitatively testing theories, was partly in reaction to Robbins’s Austrian deductivism and disparagement of quantitative economics. In- deed, an anti-Robbins movement motivated by what the junior members of the A & D section perceived in part as technical and methodological differences between Robbins and themselves was underway by 1957 (De Marchi 1988). As a forum for the movement, Lipsey with the support of Archibald initiated an informal staff seminar on “Methodology and Measurement,” later adding the term “Testing” to its title. Lipsey chaired the M2T seminars until he left LSE in 1963 when the seminar series came to an end.

Lipsey and Archibald insisted that testable economic hypotheses must be expressed in statistical terms. However, the LSE economics department, which awarded most of the M2T group their Ph.D. degrees, did not offer classes in econometrics until 1961 (the statistics department covered least-squares analysis; Gilbert 1989, 110-1 1). Indeed Lipsey, like most economists in the United States and the United Kingdom in the

23. The section about the Lipsey-Robbins connection draws from Dc Marchi 1988 and an excerpt from a letter to Neil De Marchi from David Laidler, 26 November 1985, which De Marchi showed the author.


1950s, never received much formal training in econometrics. As Lipsey recalls, “Most of us learned our econometrics from Jack Johnston’s text in the early 1960s which was the first such book accessible to ordinary economists. . . . I was unusual among my colleagues in the economics department at LSE in having had two full years of conventional statistics courses in which we covered the two volumes of Smith and Duncan’s [ 19441 text which was widely used at the time. This gave me a reasonable ground in classical statistics but the kind of issues faced in an econometrics text were untouched” (letter to the author from R. G. Lipsey, 28 July 1993). “The M2T seminar was a good less attracted to econometrics” than the title of the series suggests (letter to the author from G . C . Archibald, 21 July 1993). The M2T seminars at which Lipsey and M. Steuer presented the results of their econometric research on the Phillips curve were atypical of the series (Lipsey 1960; Lipsey and Steuer 1961).

The goals of Lipsey’s 1960 paper were consistent with standard scientific method. Scientists require the guidance of a theoretical framework to permit the interpretation of quantitative observations. So Lipsey, with the help of Archibald, attempted to construct a framework for the Phillips curve based on Walrasian principles (Lipsey 1960, 12- 19; Wul- wick 1987, 843). Scientists also require that quantitative results be repeatable using acceptable methods of measurement. Lipsey commented that he “was appalled at the [Phillips’s] scientific approach, and tried to refute it, Popperwise” (quoted in Blyth 1975,306). In particular, Lipsey attempted to repeat Phillips’s measures of the relation between money- wage inflation and unemployment using “standard statistical methods” (Lipsey 1960, 2-3). In addition, Lipsey’s multiple regression equations treated money-wage inflation in the familiar manner as the sum of cost push and demand pull inflation.24

Like Phillips, Lipsey performed the least-squares calculations on the electrical-powered Marchant mechanical calculator, which limited users to about two linear multiple regressions a day (letters to the author from M. Steuer, 21 July 1993; from R. G. Lipsey, 28 July 1993). Economists who did multiple regressions on such calculators recall the experience as being tedious. As Archibald commented, “One had to find out what a matrix was, and learn about the inverse, and methods for calculating it” (letter to the author from G. C . Archibald, 21 July 1993). The method of

24. The contrast between the views of inflation held by Phillips and Lipsey may have been forced upon the two economists by their preferences for contrasting methods of estimation (letter to the author from C. L. Gilbert, 27 January 1986).


inverting the y matrix using elimination procedures that were especially adaptable for use on mechanical desk calculators was “ironically called Doolittle” (letter to the author from G. C. Archibald, 21 July 1993).

Lipsey’s estimates of the equations,

and

y = a + b x - ’ + - d i + e j , (4)

for 1862- 19 13 are virtually identical to the author’s estimates (table 5 , equations 3 4 ) . However, Lipsey’s estimates for the twentieth century are not repeatable given the data (shown in appendix 2) that Lipsey said he used.

For the years 1923-39 and 1948-57, Lipsey estimated the equation

(table 5 , equation 5L). On the basis of his estimates, Lipsey inferred “that, on the average experience of the post-1922 period, other things being equal, times of falling unemployment were associated with lower ( ‘ y ’ s ] than were times of rising unemployment. It would appear then that Phillips’s loops [figures 4-10] have changed directions” (Lipsey 1960’27).

The change in the direction of the loops about the fixed Phillips curve from counterclockwise to clockwise had various policy implications. Phillips interpreted x as a proxy for inflationary expectations. Clockwise loops would make expansionary policies look less risky to policy mak- ers who feared inflation (letter to the author from M. C . Lovell, 18 January 1994). Lipsey interpreted as a proxy for the regional dispersion of ~nemployment .~~ With the loops swinging counterclockwise, “the upswing was associated with increasing degrees of sectoral inequalities in unemployment” (Lipsey 1960, 27). In that case, regional policy that decreased inequality in unemployment reduced the national rate of inflation (Wulwick 1983; 1987,848 n. 25). The goals of regional and national economic policies would conflict in the presence of clockwise loops. The

0

25. Phillips briefly surmised, “The extremely uneven geographical distribution of unemployment may also have been a factor tending to increase the rapidity of wage changes during the upswing of business activity” (1958a. 295). Lipsey then developed the idea (1960, 17-23; Wulwick 1987, 847-48).


policy implications help explain the prevalence of the loops about the fixed Phillips curve as a research topic.

Yet, my estimate of the effect of unemployment changes on wage inflation was imprecise when based on the data Lipsey said he used (table 5 , equation 5) . W. G. Bowen and R. A. Berry, two of the many economists with research agendas shaped by Lipsey’s reported findings, note that Lipsey reported a squared partial correlation coefficient for y and i of “only 0.3” (Bowen and Berry 1963, 170). I arrived at a correlation coefficient of 0.02 using Lipsey’s reported data (table 7)- more evidence of the unimportance of unemployment changes as an influence on wage inflation.

Phillips’s archival papers contain an alternative money-wage inflation series for 1921-57. The series is based on the definition of wage inflation later used by Lipsey and Steuer (appendix 1, equation [A1.3]; Lipsey and Steuer 1961, 141). Substituting the alternative wage inflation series in place of the series that Lipsey (1960) said he used results in a coefficient on unemployment that is positive and statistically significant (table 6, equation 5). The estimated squared partial correlation coefficient (rang- ing between 0.28 and 0.35) is close to Lipsey’s estimate (table 7).26

Lipsey estimated equation

for the periods 1923-29, 1929-39, and 1948-57 (table 5, equation 6L). D. J. Smyth, another economist influenced by Lipsey (1960), noted that Lipsey “found the relationship between y and to be positive for 1923- 29, negative for 1929-39, and positive for 1948-57; that is, he found an anti-clockwise loop for 1929-39 as before World War I, but clockwise loops for the other two sub-periods, 1923-29 and 1948-57” (Smyth 1979, 230; Lipsey 1960,27). My estimates of d for those years do not support Lipsey’s conclusion about the sign for the post-World War I1 period. Using either Lipsey’s reported wage-inflation series or the alternative

26. Archibald, who reestimated equation (5) for the 1948-27 time period, remarked that “it seems, however, that his [Lipsey’s] coefficient for . . . [XI was not in fact significant” (Archibald 1969, 128). Lipsey’s estimates of the equation pertained to 1923-39 and 1948-57. Archibald apparently estimated the equation just for 1948-57. Archibald remarked that “I have absolutely no records of any econometric work aside from what was published . . . I think everyone, editors included, was very careless about such matters in those days” (letters to the author from G. C. Archibald, 21 July 1993 and 25 April 1994). Archibald thought he made satisfactory arrangements to preserve the data. Upon returning to England from abroad, he found that the data were gone.


wage inflation series results in estimates of d of about zero for each subperiod (table 5 , equation 6; table 6, equation 6). Lipsey did not report statistical levels of significance, but I assume that he would not have put forward his argument about the clockwise loops had he found the estimates of d to be very imprecise for the three subperiods, which is the case with all my estimates.

Of the estimated coefficients of equation (6), Lipsey reported only the estimate of d (table 5, equation 6L). Given the information based only on Lipsey’s estimates of equation ( 5 ) , B. Hines, K. Cowling, and D. Metcalf note that the Phillips curve relation “appeared to break down in the post-war period” (table 5 , equation 5L; Hines 1964,241; Cowling and Metcalf 1967, 31). In contrast, my estimates of equation (6) using Lipsey’s reported data suggest considerable stability of the Phillips curve during 1862-57 (table 5, equation 6).

There are three likely reasons why economists have failed to repeat Lipsey’s least-squares estimates.

Keypunching and Computational Errors

Researchers sometimes made mistakes punching cards with data that they did not catch during proofreading. Moreover, Lipsey or a research assistant probably solved the normal equations of least squares on the Marchant mechanical electrical-powered desk calculator by an elimination method such as D~olittle.~’ Elimination methods of inverting the matrix to solve the normal equations of least squares presented opportu- nities for computational mistakes (Graybill 1961, 150-52). Overlooking mistakes when researchers self-checked their work was a familiar hazard in the days before full automation.

Truncation and Rounding Error

To save time, Lipsey and his research assistant may have kept a small number of digits throughout the elimination method. Accumulation of truncation and rounding errors might affect every digit and even the signs of the least-squares estimates. Lipsey, who neglected to report the degree of rounding in the computations, perhaps was unaware of the literature on

27. June Wickens, the wife of A. G. Hines. may have been Lipsey’s research assistant in 1960 (letter to the author from M. Desai, 28 February 1994). Wickens does not have details on the Lipsey data (telephone conversation between A. Sleeman and the author, 16 June 1995).


error-in-estimates due to round-off error (Hotelling 1943; von Neumann and Goldstine 1947; Turing 1948).

The Wrong Data

We cannot expect Lipsey, who moved to Canada in the late 1960s and changed institutional affiliation several times, to have kept personal records of his old data, which could not be conveniently stored on durable disks as today. Nevertheless, we can show that Lipsey did not use the data sources and/or the definitions of variables that he reported to use in his study. Lipsey’s careless treatment of data seems paradoxical in light of the attraction of the M2T group to Popper and testing. “The scien- tifically significant physical efect may be defined as that which can be regularly reproduced by anyone who carries out the appropriate experiment in the way prescribed,” Popper explained. “No serious physicist would offer for publication . . . one for whose reproductions he could give no instructions. The ‘discovery’ would be only too soon rejected as chimerical, simply because attempts to test it would lead to negative results” (1 959,4546) .

Price Inflation Data

Interwar Period Lipsey reported only one source of his p-data for 1862-1957, the 1950 Phelps, Brown, and Hopkins retail price index (Lipsey 1960, 9). The index ends at 1938; given Lipsey’s definition of price inflation as the relative first central difference, that index yields a time series for price inflation ending with 1938. Yet Lipsey’s estimates for the interwar period extended to 1939. Lipsey may have extrapolated from the Phelps, Brown, and Hopkins data but neglected to report it, or used an entirely different time series.

0

Postwar Period Lipsey’s article did not cite the source of the price inflation data for 1948-57. Lipsey suggested that I refer to the price index published in the econometric study of the Phillips curve 1870-1958 by Lipsey and his LSE colleague, M. Steuer (1961, table 7, column 4).28

28. Phone conversation between R. G. Lipsey and the author, week of 5 June 1989. According to my records, Lipsey offered tocheckon the Phillips data when he went to London in September. However, I did not hear from Lipsey again on the data problem. Lipsey-Steuer (1961) used the same data for unemployment 1925-38 as Phillips (Lipsey-Steuer 1961, table 7, column 2). The wage index 1925-38 in Lipsey-Steuer (given differences in rounding when adjusting for


Lipsey and Steuer cite the Prest price index as the source of their price index for the postwar period, but the Prest index ends in 1946 (Lipsey and Steuer 1961, 155 n.C; Prest 1948,58-59).

Money-Wage Inflation Data

In the 1950s and 1960s, journals often published the data used in econometric studies. Phillips, Routh, and Lipsey each published the money- wage inflation data for 1948-57 that they used to estimate the Phillips curve (Phillips 1958a, 298; Routh 1959,306; Lipsey 1960,30). Routh’s wage inflation data reportedly obtained from Phillips was identical to Phillips’s published data. Table 8 compares Phillips’s published data to Lipsey’s published data. Phillips’s data (column 3) are based, as he reported, on the money-wage index of the Ministry of Labour Gazette (column 6) given his definition of money-wage inflation for the modem period as the year-to-year percentage change (as in column 5; appendix 1 , equation [A1.2]). Lipsey’s data (table 8, column 2) are not based, as Lipsey reported, on Phillips’s money-wage index (column 6) given Lipsey’s definition of money-wage inflation as the relative first central difference (as in column 4; appendix 1 , equation [A1 .1]). The Phillips archives contain a table of data on which Phillips scribbled a formula (appendix 1 , equation [A 1.31) he intended to apply to his money-wage index (table 8, column 6) in order to obtain an alternative wage-inflation series. Phillips’s handwritten tables of data pertained to the early, exploratory stage of his research. The tables contain alternative series for inflation and unemployment as well as variables and a few errors that do not appear in his 1958 publication (appendix 1 , note 5). Applying Phillips’s handwritten formula to the wage index (column 6) results in a series (column 2) that differs from Phillips’s handwritten alternative wage-inflation series (column 1) . Presumably Phillips or a research assistant made a mistake in computing the alternative inflation series. The alternative, erroneous wage inflation series that appears on Phillips’s exploratory data sheets is identical to the wage inflation series that appears in Lipsey’s published replication of the Phillips curve (Lipsey 1960, 30).*’

changes in base years) seems to be the wage index that Phillips used (table 7, column 1). The Lipsey-Steuer unemployment data 1948-57 do not match Phillips’s data.

29. This fact is consistent with Lipsey’s memory of having used data for money-wage inflation that Phillips had and that Lipsey and Steuer (1961) used (telephone conversation with R. G . Lipsey, week of 5 June 1989).


Steuer, recalling the statistical practices of his early years as an economist, explained why finding out what data Lipsey used and how it was constructed is difficult. The problem is that “the description of the data is so vague, the price indices uncertain, lags, etc., . . . I do not for a moment think anybody cooked the books. What I think happened was that lots of variations were tried and the most plausible was published without keeping a good record of how the most plausible was put together” (letter to the author from M. Steuer, 21 July 1993).

Conclusion

Phillips (1958a) fit a nonlinear curve to money-wage inflation and unemployment data for 1861-1913 by means of a time-honored, simple regression method and superimposed the fitted curve to data for 1914- 1957. I repeated Phillips’s estimates of the Phillips curve for 1861-1913 and confirmed the presence of counterclockwise loops around the curve due to the influence of unemployment changes on wage inflation. Lipsey (1960) said he applied multiple regression to Phillips’s data in order to replicate Phillips’s estimates. No problem arises in repeating Lipsey’s replication of Phillips’s curve and loops for 1861-1913. But I could not approximate Lipsey’s estimates of the Phillips curve for 19 14-57 and arrived at different signs and levels of statistical significance in respect to the loops.

Three possibilities might explain why I could not repeat the estimates of Lipsey (then a novice at econometrics): computational and keypunching mistakes, accumulating round-off error during subroutines, and data problems. I found definite evidence of data problems. That I have applied Phillips’s data for 1861-1939 and 1948-57 is clear since my data sources, scattergraphs, and estimates match those of Phillips. I have shown that for Lipsey to have obtained the results that he got requires that he used different data from that which he claimed to have used.

The crux of the repeatability of the Phillips and Lipsey exercises is in the accuracy of the records that they kept of their work. Phillips’s training in electrical engineering probably accustomed him to keep detailed project reports. As a matter of course, he kept records of each step taken on the Phillips curve project, as well as steps the project did not but could have taken. Phillips also was a time-series expert who carefully defined and constructed his variables from the raw statistics. Lipsey received training mainly in economic theory at LSE in the 1950s. Self-taught in


elementary econometrics, the youthful Lipsey and his peers were unac- customed to doing empirical research. By their own admission, neither Lipsey nor his colleagues in the early years of their professional lives carefully made records of the steps taken and the data used in their econometric research .30

Let us imagine a counterfactual historical scenario. Say that Lipsey or someone else around the same time meticulously replicated the Phillips curve by ordinary least squares. What difference would that have made to economics? Lipsey’s study was only one of a multitude of Phillips curve replications. But it was the first replication by means of ordinary least squares. The Lipsey replication was timely since its publication occurred soon after the appearance of Phillips’s paper with its eye-catching hyperbola. The replication aroused special interest because the author claimed to have used Phillips’s own data. It was Lipsey’s replication that clinched the institutionalization of the Phillips curve. By 1970, that replication received more than three times as many journal citations as Perry’s (1964) replication and ten times as many as W. G. Bowen and R. A. Berry’s ( 1963) replication (Social Science Citation [Cumulative] Index).

The research of the community of economists is a cumulative en- deavor. Citations suggest that Lipsey’s 1960 paper had an impact on the subsequent research of many economists. Economists who knew the estimates of Lipsey’s equations based on the data he said he used might well have changed the emphasis, focus, or interpretation of their own research. There are two major issues that economists with an interest in the Phillips curve might have handled differently given the revised estimates of Lipsey’s equations-the existence of the Phillips curve and the loops about the fixed curve. My estimates on Lipsey’s reported data for the modem period show little evidence of the existence of clockwise loops or the temporal instability of the Phillips curve coefficients outside the interwar period.

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Appendix 1. Phillips’s Raw Statistics and Data

The money-wage index and the unemployment data come from the sources cited by Phillips (1958a, 285, 290-93, 295-96). The series on money-wage inflation and unemployment for 1861-191 3 are identical to the data given to me as Phillips’s data by C. L. Gilbert, who himself obtained the data by referring to Phillips’s sources and definitions. All references to the Phillips archives pertain to Phillips’s table of data in section 2/1 of the archives.

Data Sources

Money-Wage Index w

1861-1920: Phelps, Brown, and Hopkins 1950,276,281, table d, column 2 (Phillips 1958,293); identical to the Phillips archives, column (a). 192147 Ministry of Labour Gazette, April 1958, p. 133, index of hourly wage rates for end December, except for June 1947 (Phillips 1958, 293), differ from the figures in Phillips’s ,archives, column (a), labeled “Weekly Wage Index, M.O.L., 2nd December.” 1948-57 Ministry of Labour Gazette, 1950-58, index of weekly rate of money wages, December (Phillips 1958, 296) is identical to column (a) of Phillips’s archives.

Unemployment x

1860-1920 Beveridge 1944, 312-14, table 22, column labeled “Em- ployment Rate (T. u.)” (Phillips 1958,290,293) is identical to Phillips’s archives, column (d).


1921-39 Ministry of Labour Gazette, January 1940, p. 2, column labeled “Percentage unemployed: Great Britain and Northern Ireland” (Phillips 1958, 293) is identical to Phillips’s archives, column (d).

194045 Phillips cited the Ministry of Labour Gazette, January 1940 and following (Phillips 1958,293). The Gazette January 1941-February 1949 shows some monthly, quarterly, or annual numbers in categories of the labor force. The data are sparse, so Phillips would have had to inter- polate to arrive at annual data. My source is column (d) in the Phillips archives labeled “MoLG (Quarterly)” for 1940-44 and “I.L.O. St. Year- book” for 1945.

I94648 Phillips cited the International Labour Office (ILO) Yearbook of Labour Statistics (Phillips 1958,293). The ILO Yearbooks revise their data. In the absence of a specific citation, I used the data in Phillips’s archives column (d) for 194647 labeled “I.L.O. St. Yearbook.” I used the unemployment figure for 1948 in the Phillips’s archives (column [a], “G.B. unlagged”), which is close to the figure in the ILO Yearbook 1951.

1949-57 Phillips cited The Ministry of Labour Gazette. He averaged the monthly percentages for each year and added 0.1 to approximate the unemployment rate in the United Kingdom (Phillips 1958, 295-96). In the absence of data for January 1949 in the 1950 Gazette, I referred to the unemployment rate in Phillips’s archives for 1949 (column [a], “G.B. unlagged”), which is close to the average of the February-December 1949 rates in the 1950 Gazette. The 1950-57 mean data based on the Gazettes are identical to the data in the Phillips archive labeled “G.B. Unlagged” (except for a difference of 0.01 percentage points at 2 dates). The data in the archive labeled “G.B. Lagged 7 months” 1950-57 is identical to that based on the Gazettes (except for a difference of 0.01 percentage points at one date).

Definitions

Money-Wage Inflation

1861-1920 First, the central difference of the wage index w (as a proxy for the absolute rate of change of wage rates during a year) is expressed

426 History of Political Economy 28:3 (1 996)

as a percentage of the index number:

y = est [(dw/dt)(l/w,)J = ( [wf+l - wf-1]/2w,) 100 (Al.1)

(Phillips 1958, 290, n. 1).

1921-57 Year-to-year percentage change of the wage index w :

y = est [(dw/dt)(l/w,)J = ( [ w , - w,-l J/w,-l) 100 (A1.2)

(Phillips 1958, 293). The Phillips archives show an alternative series defined by

(see note B).

Changes in Unemployment

First, the central difference of the employment figures, the “best simple approximation to the average rate of change of unemployment during a year,” are

d x / d t = est(du/dt) = - uf-l]/2) (A1.4)

(Phillips 1958,291, n. 1).

Seven-Month Lead of Unemployment

Phillips transformed the unemployment data 1948-57, so that “the rate of change of wage rates during each calendar year is related to unemployment [negatively] lagged seven months, i.e., to the average of the monthly percentages of unemployment from June of the preceding year to May of that year” or

rn +6

x E (1/12) c u (A 1.5) rn -6

(Phillips 1958,297).


Table 1 Phillips's Data 1860-1947 ~~

Money-Wage Money-Wage Unemployment Unemployment Index Inflation Change

1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881' 1882' 1883' 1884' 1885' 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897

68 68 68 70 73 75 78 77 75 75 78 80 89 96

I00 100 99 98 95 93 93 93 93 94 94 93 93 93 93 96

100 100 100 99 99 98 99

100

0.0 1.4706 3.57 14 3.4247 3.3333 1.282 1

- 1.948 1 - I .3333

2.0 3.205 1 6.875 8.9888 5.7292 2.0

-0.5 -1.0101 -2.0408 -2.6316 - 1.0753

0.0 0.0 0.5376 0.53 19

-0.53 19 -0.5376

0.0 0.0 1.6129 3.6458 2.0 0.0

-0.5 -0.505 1 -0.505 1

0.0 1.0101 1.5

3.7 6.05 4.7 1.95 1.8 2.65 6.3 6.75 5.95 3.75 1.65 0.95 1.15 1.6 2.2 3.4 4.4 6.25

10.7 5.25 3.55 2.35 2.6 7.15 8.55 9.55 7.15 4.15 2.05 2.1 3.4 6.2 7.7 7.2 6.0 3.35 3.45

2.1 0.5

-2.05 - 1.45

0.35 2.25 2.05

-0.175 - 1.5 -2.15 - 1.4 -0.25

0.325 0.525 0.9 1 . 1 1.425 3.15

-0.5 -3.575 - 1.45 -0.475

2.4 2.975 1.2

-0.7 -2.7 -2.55 - 1.025

0.675 2.05 2.15 0.5

-0.85 - 1.925 - 1.275 -0.2


Table 1 (continued)


1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 191 1 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921* 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1 9343 1935

102.0 104.0 108.0 107.0 107.0 106.0 105.0 105.0 107.0 107.0 107.0 107.0 107.0 108.0 11 1.0 115.0 115.0 124.02 135.29 160.1 205.2 268.33

35 8.5 31 1 55.5 286.371 12 1

99.5 98.0

101.0 101.5 100.5 99.0 98.0 98.0 97.5 95 .O 93.5 93.5 90.0 91.5

1.9608 2.8846 1.3889

-0.4673 -0.4673 -0.9434 -0.4762

0.9524 0.9346 0.0 0.0 0.0 0.4673 1.85 19 3.1532 1.739 1 3.9 130 8.1818

13.3333 2 1.8332 26.3736 28.57 14 2.5 157

-22.1865 - 17.7686 - 1.5075

3.06 12 0.495 1

-0.9852 - 1.4925 - 1.0101

0.0 -0.5 102 -2.5641 - 1.5789

0.0 - 3.7433

1.6667

2.95 2.05

' 2.45 3.35 4.2 5 .O 6.4 5.25 3.7 3.95 8.65 8.7 5.1 3.05 3.15 2.1 3.25 1 .o 0.45 0.6 0.7 2.5 2.55

17.0 14.3 11.7 10.3 11.3 12.5 9.7

10.8 10.4 16.1 21.3 22.1 19.9 16.7 15.5

-0.7 -0.25

0.65 0.875 0.825 1.1 0.125

-1.35 -0.65

2.475 2.375

- 1.775 -2.825 -0.975 -0.475

0.05


Table 1 (continued)


1936 94.0 1937 98.5 1938 99.5 1939 104.0 I940 116.0 1941 126.5 1942 134.0 1943 140.0 1944 146.0 1945 153.5 1946 169.5 1947 176.5/103.0

2.7322 4.7872 1.0152 4.5226

1 1.5385 9.05 17 5.9289 4.4776 4.2857 5.137

10.4235 9.3215

13.1 10.8 12.9 10.5 6.83 2.925 1.025 0.675 0.6 1.2 2.5 3.2

1. Phillips’s figure 4a was based on Bowley’s wage data. Bowley’s indices w for 1880-87 were 72, 72, 75, 75, 75, 73, 72, 73 (1914 = 100)-implying inflation rates y of 2.0833, 2.0, 0.0, - 1.333, -2.0548 (Phillips 1958,291, n. 3; Bowley 1937,30; Phillips Archives). Phillips did not use the Bowley data when estimating the Phillips curve 1861-1913 (Wulwick 1989, 176-79; 1994,8687). 2. There is an alternative money-wage inflation series for 1921-57 in the archives (column labeled “y%”) that is identical to the series arrived at by applying equation (A1.3) to the money-wage index, except for 1934 and 1947-57 (see table 8). The y-series for 1921-39 is:

0.1 1, 1.65, 2.7,4.67, 1.01,4.42. 3. Probably a transcription error accounts for Phillips using the money-wage index for June instead of December 1934 of 93.5, giving him zero wage inflation for 1934 (Phillips 1958, figure 9; Phillips Archives, columns [a], [c]).

-24.95, -19. 13, -1.52, 3.015.0.494, -0.99, -1.504, -1.015,0, -0.51, -2.6, -1.59.0,

Table 2 Phillips’s Unemployment Data 1948-1 957

No Lead Seven-month lead2 X X dxldr

1948 1.79 1.76 1949 1.52 1.59 1950 1.62 1.62 195 1 1.31 I .47

-0.16 -0.20 -0.0 1 -0.33

430 History of Political Economy 28:3 (1 996)

Table 2 (continued)

No Lead X X

Seven-month lead2 dddt

1952 2.08 1.61 1953 1.73 2.03 1954 1.43 1.61 1955 1.18 1.275 1956 1.28 1.16 1957 1.56 1.48

0.73 -0.28 -0.32 -0.26

0.06 0.28

~~

1. The money-wage data for 1948-57 appears in table 8. 2. In the absence of monthly data for 1947-48 in the M.O.L. Gazettes, the data for the seven-month leads in unemployment and unemployment changes come from the Phillips Archive (columns [b]-“G.B. lagged 7 months” and “x . . . 7 month lead”). The archival papers show that Phillips examined time series labeled “UK lagged 6 months I.L.O.” and “G.B. lagged 6 months,” which scarcely differ from “G.B. lagged 7 months.” Comparing Phillips’s figure 1 1 and my figure 3a, both with unemployment leading inflation, shows that Phillips’s figure indicates the inflation-unemployment data fitting a little more tightly around the Phillips curve (fitted to the 1861-1913) data) than my figure. The grapher working on Phillips’s article either had imprecise instruments or moved x leftward in order to give the impression of a tighter fit than the data warranted.

0

Appendix 2. The Data According to Lipsey 1960

Sources

Unemployment and Money-Wage Inflation (see appendix 1). Price Index. 1861-1938: Phelps, Brown, and Hopkins (1950,276,28 1)

Table D, column 3. Lipsey (1960, 9, n. 1) cites only the Phelps, Brown, and Hopkins series that ends in 1938.1948-57: Lipsey and Steuer (1961, 153) table 7, column 4.

Definitions

Price Inflation, Rate of Change of Unemployment, Money-Wage Infla- tion. 1862-1957. Equation (Al.l), appendix 1 (Lipsey 1960,7, 10).


Table 3 Lipsey's Data 1861-1914

Price Price Unemployment Money-Wage Index Inflation Change Inflation '

1861 129 1862 130 1863 133 1864 133 1865 130 1866 131 1867 131 1868 130 1869 128 1870 127 1871 130 1872 138 1873 141 1874 133 1875 128 1876 127 1877 127 1878 120 1879 116 1880 121 1881 119 1882 118 1883 118 1884 112 1885 105 1886 103 1887 101 1888 101 1889 103 1890 103 1891 103 1892 104 1893 103 1894 98 1895 96 1896 96 1897 98 1898 101

1.538462 1.12782

- 1.1 2782 -0.76923 1

0.38 1679 -0.38 1679 - 1.153846 - 1.17 1875

0.787402 4.230769 3.985507

- 1.773050 -4.8872 I8 -2.34375 -0.39370 1 -2.755905 -4.583333

0.43 1034 1.239669

- 1.260504 -0.423729 -2.542373 -5.80357 1 -4.2857 15 - 1.94 1 748 -0.990099

0.990099 0.970874 0.0 0.4 8 5 4 3 7 0.0

-2.9 1262 1 -3.571429 - 1.04 1667

1.04 1 667 2.55 1020

8.264461 -43.61703 - 74.35 897

19.44445 84.90566 32.53968 - 2.592596

-25.21008 -57.33333 - 84.84849 -26.3 1579

2 8.26087 32.8 125 40.90909 32.35294 32.38636 50.39999 -4.672897 - 68.095 24 -40.84507 -20.2 1277

92.3077 4 1.60839 14.03509

-7.329843 - 37.76224 -6 1.44579 -50.0000 1

32.14286 60.29412 34.67742 6.493506

- 1 1.80555 -32.08333 -38.0597 -5.797099

0.495050 -23.72882


Table 3 (continued) ~~

Price Price Unemployment Money-Wage Index Inflation Change Inflation '

1899 99 1900 105 1901 104 1902 104 1903 105 1904 106 1905 106 1906 107 1907 110 1908 107 1909 108 1910 110 1911 112 1912 115 1913 118 1914 115 1922 107 1923 105 1924 107 1925 104 1926 103 1927 98 1928 98 1929 97 1930 93 1931 86 1932 84 1933 83 1934 83 1935 86 1936 88 1937 93 1938 92 1939 94 1948 97.14 1949 100 1950 103.49

2.020202 2.3 80952

-0.480769 0.480769 0.95238 1 0.47 1698 0.47 1698 1.869 159 0.0

1.388889 1.81 81 82 2.232 143 2.608696 0.0

-0.934579

- -

0.0 -0.467290 - 1.923077 -2.9 1262 1 -2.55 102 -0.5 10204 -2.57732 -5.913979 -5.232558 - 1.785714 -0.6024 1

1.807229 2.906977 3.977273 2.150538 0.543478

NA NA

3.174999 6.135857

-12.19512 26.53061 26.1 194 19.64286 22.0

1.953 125 -25.7 1429 - 17.56757

62.65823 27.45665

-20.40230 -55.39215 -3 1.9672 1 - 15.07937

2.38095 - -

- 17.09402 - I .94 1746

9.7345 12 - 6.400002 -8.762886

3.24074 25.48077 33.85093 14.0845 1

-3.167420 - 13.56784 - 13.17365 -1 1.6129 - 17.93893 -0.925929 - 1.16279 1

-28.90476 -46.92738 -5.592 I04 -6.48 1483

0.765306 1.732673

-0.246305 - 1.24378 I - 1.262626 -0.5 10204 -0.255 102 - 1.538462 -2.105263 -0.8021 39

0.27 807 3 1.1 11464 2.186872 3.722984 2.79 19 2.7638 19 7.9400 12 2.803738 3.2 1 1009 7.456 14


Table 3 (continued)

Price Price Unemployment Money-Wage Index Inflation Change Inflation’

1951 112.70 7.147294 17.55725 7.936508 1952 119.60 4.113714 10.096 1 6 4.4776 12 1953 122.54 2.623633 -18.78613 3.623 188 1954 126.03 2.265333 - 19.40560 5.555556 1955 128.25 3.715399 -6.382978 7.142858 1956 135.56 4.45 1906 15.03906 6.325302 1957 140.32 2.832809 32.6923 1 4.2857 1958 143.51

1 . Lipsey used the same money wage inflation data as Phillips for 1861-1913.

Appendix 3. Phillips’s and Lipsey ’s Econometric Estimates

Table 4 Phillips’s Estimates Compared to the Author’s Estimates’

eauation U b C k m R2 DW la

1 b2 186 1-1 9 13

186 1- 19 13

1 c3

1861-19 13 1948-57

2a4 1861-1913 z = 1..4 0 < x, < 5

2b5 1861-1913

-0.9

- 1.002 (0.5 15) (0.058)

-1.194 (0.621) (0.059)

-0.9

-0.883 (0.5 89) (0.14)

9.638 -1.394 -1.317 1.1 - -

8.732 -1.292 -2.526 1.187 0.79 1.22 (0.747) (0.253) (1.15) (0.382) (0.0) (0.0) (0.0) (0.034) (0.003)

10.367 -1.313 -1.214 0.682 0.79 (0.759) (0.26) (0.91) (0.543) (0.0) (0.0) (0.0) (0.185) (0.21)

9.638 -1.394 - - - -

0.64 0.78 8.939 -1.384 - -

(0.944) (0.3 18) (0.0) (0.0) (0.0)


Table 4 (continued)

equation a b C k m R2 DW 2c3 1948-57

2d6 1948-57

2e3 1861-1913 1948-57

- 10.95 (3.28) (0.1 )

- 10.192 (3.317) (0.015)

-1.16 10.363 (0.677) (0.842) (0.092) (0.0)

-1.655 - - 0.37 1.55 (0.77) (0.06) (0.064)

-1.545 - - 0.27 1.47 (0.86) (0.13) (0.1 1)

-1.328 - - 0.72 1.01 (0.289) (0.0) (0.0)

1. All least-squares estimates result from using micro-TSP 7.0h. The first row of bracketted numbers under the estimated coefficients are standard errors and the second row are levels of statistical significance. The signs of 2 appear as if a were on the right-hand side of the equations. Solutions may be sensitive to starting parameter values. 2. In the absence of a special test to see if least-squares estimates in equations ( 1 a) and (1 b) are statistically different, treat the estimates of equation (la) as the null hypothesis. The F-ratio given the sum of squared residuals from the two equations, five variables and 48 degrees of freedom is 1 S76. The observed significance level of 0.185 supports the hypothesis of a minor difference between the two estimates (Pindyck and Rubinfeld 1991, 110-1 1).

The estimates of k in Gilbert (1976, tables 1 and A2) and Wulwick (1989, 176, equation 7a) differ from the estimate in equation ( 1 b) because the two earlier studied defined x in terms of equation A I . 1 instead of A2 (appendix 1 ). 3. With Phillips’s seven-month lead in unemployment. 4. The estimates are based on the Woods data without the Bowley substitution (Wulwick 1989, 176-78). Phillips did not use Bowley data in obtaining numerical estimates of the inflation- unemployment relation (Wulwick 1994, 87). 5. These estimates of equation (2) are identical to those of Gilbert (1976), who used nonlinear least squares, and are close to those of Oliver, who used maximum likelihood (Oliver 1986, 223). 6. Without the lead in unemployment. For 1948-57, adding ( x m ) ( d x / d r ) or dx/dr as independent variables made all the estimated coefficients in equation (2)-with or without the lead in x-highly insignificant.

In the absence of a special test to see if the estimates of equation (2) 1948-57 with and without the lead in unemployment are statistically different, treat the “no lead” estimates as the null hypothesis. The F-ratio given the sum of squares residuals from the two equations, two variables, and eight degrees of freedom is 0.69. The observed significance level of 0.53 supports the hypothesis of an insignificant statistical difference between the two estimates (Pindyck and Rubinfeld 1991, 110-1 I ) .

0


Table 5 Estimates, Given the Definitions of Variables in Lipsey 1960’

Lipsey’s Least-Squares Estimates Compared to the Author’s

eauation a b C d e R2 DW 3L2 -1.23 6 3.05 -.021 - 0.79 n.a. 1862-1 9 13

33 -1.232 5.996 3.061 -.021 - 0.79 1.15 1862-1913 (.463) (2.389) (2.423) (.004) (0.0)

(.011) (.016) (.213) (0.0)

4L -0.94 4.92 3.66 -0.016 0.2 0.82 n.a. 1862-1 9 13

44 -0.936 4.916 3.673 -0.016 0.198 0.82 1.12 1862-1913 (.45) (2.285) (2.292) (.004) (.074) (0.0)

(.043) (0.037 (.116) (0.0) (0.01)

5LS 1923-39 1948-57

5 1923-38 1949-57

6L 1923-29

6L 1929-39

6L 1948-57

6 1923-29

6 1929-38

66 1949-57

0.74 0.43 11.18 0.038 0.69 0.91 1.7 (2.1) (6) (.012) (0.08) (0.0)

[.84] [.076] [.005] [O.O]

.847 1.617 4.622 0.008 .618 0.94 1.7 (.289) (1.572) (2.82) (.012) (.078) (0.0) (.008) (.316) (.l 16) (.485) (0.0)

n.a. n.a. - 1.91 n.a. n.a. n.a.

n.a. n.a. - -6.25 n.a. n.a. n.a.

n.a. n.a. - 3.28 n.a. n.a. n.a.

0.306 7.61 1 - 0.017 0.742 0.61 1.8 (4.803) (5 1.487) (.031) (.349) (0.36) (.953) (.892) (.631) (.123)

-0.96 29.291 - -0.003 0.481 0.92 1.64 (2.322) 33.273 (.059) (.324) (.001) (.692) (.402) (.954) (.188)

-2.379 7.246 - -0.014 0.76 0.81 2.1 (2.142) (3.095) (.021) (.22) (0.031 (.317) (.066) (0.52) (.02)


Table 5 (continued)

equation U b C d e R2 DW 66 1862-1 9 13

6 1862-19 13 1949-57

6 1923-38 1949-57

-1.519 (.268)

(0.0)

- 1.496 (.267)

(0.0)

0.627 (.266) (.023)

8.385 -

(0.0) (.742)

- 8.5 (.684)

(0.0)

3.793 -

(.878) (.0003)

-0.0 17 (0.004 (0.0)

-0.014 (.004)

(0.0)

0.007 (.012) ( 3 5 )

~

0.186 0.81 1.06 (.075) (0.0) (.016)

0.282 0.86 1.19 (.06) (0.0)

(0.0)

0.6 0.93 1.47 (0.08) (0.0)

( 3 5 )

1. The following are the corresponding equation numbers in Lipsey 1960: 3:7,4:9-10.5: 12-13. 6: 14. Estimates in the table marked L are from Lipsey 1960. 2. All estimates are based on the Woods data without the Bowley substitution (Wulwick 1994, 86-87). 3. My equation (3) estimates are close to Gilbert’s estimates (Gilbert 1976, table 1: a4). 4. My equation (4) estimates are identical to Sleeman’s estimates (Sleeman 1983,28). Wulwick (1994. 87) discusses the effects of including the two unemployment variables. 5. The author found the statistical levels of significance. The range of the Durbin-Watson statistic corresponds to Lipsey’s report of “no evidence of significant auto-correlation of the residuals for lags of one . . . period at the 5 per cent probability level” (1960, 26). 6. Had Lipsey reported these estimates, an economist in 1960 well trained in econometrics would have accepted that the coefficient estimates for 1862-1913 fit the 1949-57 data. The null hypothesis is that the coefficient estimates remain the same. The F-ratio for the variance of 1949- 57 residuals (actual wage inflation less inflation predicted from the 1862-19 13 coefficients) given four variables and five degrees of freedom is 1.66. The observed level of significance is 0.32.

Table 6 Money-Wage Inflation 192 1-57 in Phillips’s Archival Notes

The Author’s Estimates Given the Alternative Definition of

eauation a b C d e R2 DW 5 0.877 -0.397 7.983 0.063 0.704 0.87 1.78 1923-38 (.467) (2.54) (4.551) (.019) (.126) (0.0) 1949-57 (.075) (.877) (.095) (.004) (0.0)

6 -2.3 30.84 - 0.048 0.5 0.25 1.8 1923-29 (9.97) (106.89) (.064) (.724) (0.81)

(.833) (.792) (0.5 1) (.539)


Table 6 (continued)

equation a b C d e R2 D W 6 - 1.82 38.51 - 0.038 0.57 0.81 2.5 1929-38 (3.69) (52.88) (0.5 1) (0.5 1) (0.01)

(0.64) (.494) (0.3 1) (0.3 1)

6 -4.07 9.8 - 0.054 0.695 0.73 0.88 1949-57 (3.73) (5.394) (.036) (.392) (0.07)

(.326) (. 1 29) (.191) (.136)

Table 7 the Author's Estimates

Lipsey's Estimates of Correlations of Variables Compared to

Estimated Squared Partial Correlations' Lipsey 's Author's

y-Data of 1923-39/1948-57 1 862- 19 13 1923-3811 949-57

Table 2A Table 2B

0.2 1 -

y , x-4 - - - 0.12 - 0.13

y , x 0.30 0.28 0.022 0.022 0.28 0.35

y , P 0.76 0.13 0.73 0.76 0.55 0.61

Y , x 0.38 0.74 0.47 - y , x- ' - - - 0.05 - 0.00 1

0

0

Estimated Squared Correlations between Independent Variables 0

-T, P 0.47 0.10 ;r, P 0.09 0.01 0 0

0.48

0.05

I. Lipsey 1960, 26. The formula for the squared partial correlation coefficient r 2 given five variables is r:(3),s42 = Rt,2345 - R:,24s/1 - R;,24s (Croxton and Cowdon 1955,551).

2. The estimate of the squared partial correlation coefficient between y and x using the formula in the textbook from which Lipsey learned statistics is 0.01 (Smith and Duncan 1944,467).

0


Table 8 Money-Wage Data

1947 - 1948 3.73 1949 1.82 1950 4.4 1951 10.61 1952 5.28 1953 2.9 1954 4.88 1955 6.58 1956 7.31 1957 5.5 1958 -

-

3.8 1 I .85 4.48

6.15 2.94 4.26 6.7 1 7.5 5.28

10.0

-

-

3.9 1.9 4.6

10.5 6.4 3 .O 4.4 6.9 7.9 5.4 -

-

2.8037 3.21 10 7.456 1 7.9365 4.4776 3.6232 5.5556 7.1429 6.3253 4.2857

-

3.8835 1.8692 4.5872

10.5263 6.3492 2.985 1 4.3478 6.9444 7.7922 5.4217

103 107 109 114 126 134 138 144 154 166 175 181

Definitions and Sources: Column 1. Lipsey 1960, 30, table 2. Identical to Phillips's archive, section 2/1, column-y. Column 2. Processed by applying equation A1.3 (appendix 1) to Column 6; Column 3. Phillips 1958,298, table 1 ; Column 4 . Processed by applying equation A 1.1 (appendix 1) to column 6.; Column 5. Processed by applying equation A 1.2 (appendix 1) to column 6; Column 6. Minisrry of Labour Gazerre (see Sources, appendix 1). Identical to the Phillips archives (section 2/1, column [a]) and to Lipsey and Steuer ( 1 961, 153. table 7, column 1).

Analysis of Table 8 Columns

Range 1.82-10.61 1.85-10 1.9-10.5 10.051-10.871 10.081-11. 121 Mean 5.3 5.3 5.49 10.004( 10. 191 Median 5.08 4.88 5 .O I . 161 I. 181

1 2 3 2- 1 3- 1

Appendix 4. Reconstructing the Steps of Phillips's Estimation Procedure

Table 9 Equation 1 Correlations between uc and ( l l x " ) ( d x l d t )

Parameters Correlation

C m 1 0 0.002' 1 0 0.0032 1.39 0.19 0.0032


Table 9 (continued)

Parameters Correlation

2 2 0.0 1 62 3 3 0.0302

10 10 0.0272

1. Without the variable k, equation ( 1 ) . 2. With the variable k = 0.03, equation ( 1 ) .

Table 10 Reconstructing How Phillips Estimated Equation (2)

13.08 10.46 10.19 9.99 9.83 9.72 9.64 9.58

-2.35 - 1.76 - 1.67 - 1.59 - 1.52 - 1.45 - 1.39 - 1.34

0.0 0.4 0.5 0.6 0.7 0.8 0.9 1 .o

0.4 0.2 0.17 0.13 0.09 0.05 0.016 0.01 8

Table 11 Equation ( 1 )

Reconstructing How Phillips Could Have Estimated

Estimating m and k m R2 Sum of Squared Residuals Standard Error of

Regression

0 .2 .7 .9

1 .O 1.1 1.2 1.3 1.5 2 .o

.306

.329

.376

.385

.387

.388

.387

.385

.378

.330

61.37 59.23

-1 55.19 54.36

-1 54.15 55.08

t 54.15 54.36

-r 55.18 59.12

1.09 1.07 1.03 -1 1.022 1.020 4 1.019 1.020 1.022 1.03 t 1.07

Ifhi = I . l , t h e n i = -1 .315 .

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