Empirical Investigation of a Catastrophe Theory Extension of the Phillips Curve
检验了Woodcock和Davis提出的突变理论模型，该模型描述实际通胀、失业与预期通胀的关系，并使用1957-1984年美国数据通过修正最小二乘法估计参数，与传统菲利普斯曲线进行似然比检验。
A catastrophe theory model for the relationship between actual inflation, unemployment, and expected inflation that has recently been suggested by Woodcock and Davis is examined. The model's extension of traditional models and its relation to the theory of dynamic systems are pointed out. The model is empirically investigated by means of U.S. data for the period 1957-1984. Estimation equations for the parameters are derived by a modified least squares criterion. Furthermore, the traditional Phillips Curve is compared with the catastrophe model by means of a likelihood ratio test. CATASTROPHE Theory (CT) is a branch of differential topology that was originally applied in physics and biology. In the fields of economics and finance CT was applied for the first time in the crash and bull model for stock markets by Zeeman (1974). The most noteworthy articles among recent publications concerning applied CT are the business cycle model by Varian (1979) and the bankruptcy models by Ho and Saunders (1980) and Scapens, Ryan and Fletcher (1981), respectively. Since one has to consider CT models as a natural and inevitable extension of traditional models, it is remarkable that none of these applications has been the subject of empirical investigations. This apparently has been caused by the lack of an appropriate parameter estimation procedure as well as by the absence of mathematical specifications of the models. In this paper we examine the CT model by Woodcock and David (WD) (1979). Starting from the traditional inflation model based on the Phillips Curve, the model by WD is presented in section J.. Section II provides a brief outline of the applied elementary catastrophe from a dynamic system's point of view and specifies the underlying deterministic and stochastic processes. The main section of this paper, section III, is concerned with the empirical investigation of the catastrophe model by means of U.S. data for the period 1957-1984. I. The CT Extension of the Phillips Curve Since the work of Samuelson and Solow (1960) the empirically observable inverse relation between the inflation rate p and the unemployment rate u has been known as the modified Phillips Curve. Later empirical investigations have led to the conclusion that the modified Phillips Curve has shifted upwards. These shifts have been thought to be caused by the increase of an additional influential factor to the inflation rate, i.e., by the increase of the expected (anticipated) inflation. According to Friedman (1976, p. 228) the relation between p and u in period [t, t + 1] and the inflationary expectations 7r at time t for the same period is usually assumed to be a function of the form p = a + br + h(u) (1) where h (u) has been specified in various nonlinear forms. In the mid-seventies the problem of occurred in almost all western countries. In the United States, for example, the inflation rate was more than 10% per year, although the unemployment rate was above 6%. If the parameters of the traditional model (1) are assumed to be constant, this phenomenon can be explained only by an enormous increase of inflationary expectations. But a look at the data, like these shown in table 1 for the United States, seems to show that this explanation does not hold. One possible modification of the traditional model is the CT model by Woodcock and Davis (1979). They use the same explanatory variables u and S for the inflation rate, but the main difference of their CT model from the traditional one is that its surface in the (u, 7r, p)-space is not smooth throughout but rather has a significant overhanging region as depicted in figure 1. The authors describe their model as follows: The worst case (in terms of its effects on inflation) is low unemployment and a high expected inflation rate (a). It can be improved somewhat by lowering the expected rate (a b); this may be achieved by a government's adoption of an aggressive Received for publication December 22, 1983. Revision accepted for publication August 14, 1985. * University of Graz. Copyright ? 1986 9 ] This content downloaded from 40.77.167.91 on Sat, 01 Oct 2016 06:14:53 UTC All use subject to http://about.jstor.org/terms 10 THE REVIEW OF ECONOMICS AND STATISTICS TABLE 1.-U.S. DATA FROM THE EARLY SEVENTIES Average Expected Actual Unemployment Rate Inflation Rate Inflation Rate in the at t for the in the Time t Period [t, t + 12 months] Period [t, t + 12 months] Period [t, t + 12 months] June 70 5.71 3.72 4.5 Dec. 70 5.95 3.81 3.4 June 71 5.85 4.10 2.9 Dec. 71 5.58 3.24 3.4 June 72 5.19 3.81 5.9 Dec. 72 4.85 3.48 8.8 June 73 4.94 4.24 11.0 Dec. 73 5.59 5.36 12.2 June 74 7.32 6.84 9.3 Dec. 74 8.48 7.42 7.0 June 75 8.01 5.66 5.9 Dec. 75 7.68 6.03 4.8 Sources: See subsection III.A policy of jawboning to discourage price increases. To achieve a greater decline in inflation, it may be necessary to permit a politically unpopular increase in unemployment at the same time (a c d). Increasing unemployment alone, with no decrease in the expectation of future inflation, will produce only a slight decline in inflation (a e). (WD (1979), p. 117 f.) correspond to the stagflation (stagnant inflation) that has plagued Great Britain and Italy from the early 1970s until the present. To move from this region to one of lower inflation would require a drastic increase in unemployment (e f g) or, preferably, a slight increase in unemployment coupled with credible steps to reduce future inflation (e c). (WD (1979), p. 118) The main advantage of this CT model is its proper description of the phenomenon of by means of the fold in the model's surface. States around the point e in figure 1 The model's additional value lies in its indication that the sequence in which the control variables u and w are influenced (e.g., by money supply and public expenditures) can be at least as important as their quantitative levels. The importance of the sequence drastically depends on the location of the overhanging region. II. The Model and Its Relation to the Theory of Dynamic Systems In CT surfaces as shown in figure 1 are known as cusps (for CT see, for example, Poston and Stewart (1978) and Saunders (1980)). The cusp is the most popular and most frequently applied elementary catastrophe. The canonical form of the cusp's surface is