Abstract

A.W. Phillips's discovery that inflation is negatively correlated with unemployment served as a heuristic model for conducting monetary policy; but the flattening of the Phillips curve post-1970 has divided debate on this empirical relation into two camps: "The Phillips curve is alive and well," and "The Phillips curve is dead." However, this dichotomy oversimplifies the issue. In this article, we apply spectral analysis to the U.S. inflation rate and unemployment rate to conduct a comprehensive analysis of the Phillips curve in the frequency domain. We find that in the very short run, there is no systemic relationship between inflation and unemployment; in the intermediate run, which includes the business cycle frequency, they are strongly negatively correlated; and in the very long run the Phillips curve is strongly positively sloped. Such an analysis of the frequency domain provides a natural demarcation of frequency bands that allows us to recover the Phillips curve in the time domain by applying band-pass filters. Most importantly, we show how spectral analysis can be used to identify a "supply" (permanent) and a "demand" (nonpermanent) shock in the context of a vector autoregression and that demand shocks drive the Phillips curve. Finally, the phase spectral analysis also shows that despite the existence of the Phillips curve at the business cycle frequency under a demand shock, the monetary policy implications are not obvious, due to the unclear lead-lag relationship between inflation and unemployment.

Brian Reinbold is a research associate and Yi Wen is an assistant vice president and economist at the Federal Reserve Bank of St. Louis.

INTRODUCTION

In 1958, economist A.W. Phillips discovered a strong negative correlation between the money wage rate and the unemployment rate in the United Kingdom. Shortly after his findings were published, numerous studies confirmed that this relationship held in many developed economies. For example, Samuelson and Solow (1960) demonstrated that the Phillips curve held in U.S. data, and they began to explore its policy implications. 

The profession holds that the inverse relationship between unemployment and inflation implies a tradeoff between the two: low unemployment at the cost of higher inflation or low inflation at the cost of higher unemployment. This tradeoff provides policymakers a menu of monetary policy prescriptions and also shows them how influencing nominal variables can affect the real economy. Monetary policy, for example, can adjust the money supply or nominal interest rates to affect the price level and then through the Phillips curve affect employment. Because of the explanatory power of the Phillips curve, after its introduction, economists immediately incorporated it into structural models and this literature flourished and became an indispensable part of Keynesian economics.

However, the 1970s saw the Phillips curve breakdown, and the correlation in fact became positive. The U.S. experienced higher oil prices, and these adverse supply shocks caused the Phillips curve to disappear. Economists then worked on alternative explanations to rectify this experience. One branch of research incorporated rational expectations under supply shocks and long-run neutrality of money (meaning that the Phillips curve is flat in the long run in the absence of supply shocks). Another branch had a much different approach, where prices are sticky but monetary policies are endogenous responses to output gaps and inflation (Gordon, 2011). This split in analyzing the Phillips curve led to two very different conclusions on the Phillips curve: "The Phillips curve is alive and well," and "The Phillips curve is dead." Since the 1970s, a plethora of theoretical models and regression techniques, ranging from vector autoregression (VAR) to instrumental variable models, have been developed to study the existence of the Phillips curve.

Despite the numerous econometric specifications economists have used for that purpose, very few have investigated the Phillips curve in the frequency domain using spectral analysis. The issue with pure time-domain methods is that it is difficult to distinguish short-run, intermediate-run, and long-run relationships between the inflation rate and the unemployment rate, especially when the time series are full of noise that can mask the underlying dynamics of the data. Although filters such as a band-pass filter can be used to isolate specific components of the data according to the specified frequency of fluctuation, this becomes arbitrary since there are an infinite number of specifications of the frequency bands. Furthermore, the lead-lag relations between inflation and unemployment can also be masked by noise, which makes systematic regression analyses in the time domain challenging.

Spectral analysis, on the other hand, provides a clear way to decompose a time series and its relationships with other time series into movements and comovements across a continuum of cyclical frequencies and leads and lags—all at once. That way the short-run, the intermediate-­run, and the long-run behaviors of a vector of time series and their mutual cross lead-lag correlations (or covariances) can be studied simultaneously without resorting to filters where one must specify arbitrarily the frequency intervals a priori. Thus, spectral techniques provide a more compact and complete picture of the joint dynamic behaviors of a vector of time series and ultimately provides economists an additional tool to better characterize any systematic relationships at any cyclical frequencies at any leads and lags among any specified number of economic variables.

The main purpose of this article is to demonstrate how spectral techniques can be used to analyze the U.S. Phillips curve and extract information not easily seen in the time domain. Using such techniques, we find that (i) in the very long run (such as fluctuations at frequencies lower than 0.02 cycles per quarter or 50 up to infinity quarters per cycle) the Phillips curve is positively sloped, except in the 1950s and 1960s when the Phillips curve became popular; (ii) however, in the intermediate run (i.e., around frequencies of 6 to 50 quarters per cycle), the Phillips curve is alive and well—the correlation between unemployment and inflation is always negative and significant throughout the entire postwar U.S. history, including in the 1970s and 1980s when the Phillips curve was thought to have broken down; and (iii) in the very short run (fluctuations at frequencies higher than 0.17 cycles per quarter or less than 6 quarters per cycle), there is zero correlation between unemployment and inflation throughout the entire sample. We also find the Phillips curve at the conventional business cycle frequency to be highly stable over time.

These findings explain why in the time domain it is hard to detect the existence of the Phillips curve (especially since the 1970s), because the long-run, intermediate-run, and short-run movements are mixed and thus offset each other in the time domain; in addition, the large amount of noise in the inflation rate has dominated and masked any systematic relationships the rate has with unemployment. 

Perhaps equally important and interesting is that the monetary policy implications (justified by the Keynesian models that use the Phillips curve) are not obvious, based on the phase spectrum. The conventional wisdom is that a negative correlation between inflation and unemployment automatically implies a trade-off between the two and a causal link running from inflation to unemployment. But the phase spectrum shows that in the long run, high inflation tends to "cause" high unemployment instead of low unemployment, and in the intermediate run high inflation tends to follow (instead of lead) low unemployment. Therefore, it is not clear how monetary policy that directly affects inflation could affect unemployment during the business cycle despite the strong evidence of a negatively sloped Phillips curve at the business cycle frequency. 

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