Time scale regression analysis of the wage Phillips curve in the US

Transcription

1 Time scale regression analysis of the wage Phillips curve in the US Marco Gallegati Department of Economics, Faculty of Economics G. Fuá, Universitá Politecnica delle Marche, Piazzale Martelli 8, Ancona, Italy Mauro Gallegati Department of Economics, Faculty of Economics G. Fuá, Università Politecnica delle Marche, Piazzale Martelli 8, Ancona, Italy. James B. Ramsey Department of Economics, 19 West 4th Street, New York University, New York, NY 10003, USA. Willi Semmler Department of Economics Graduate Faculty, 65 Fifth Avenue, New School University, New York, NY 10003, USA and CEM University of Bielefeld, Germany. Abstract Wavelet analysis, although used extensively in many areas of applied sciences, has not yet fully entered the economics discipline, even if many economic processes are the result of the actions of several agents who simultaneously operate at different horizons and/or have different term objectives. Wavelets provide a unique decomposition of time series observations that enable one to decompose the data in ways that are potentially revealing of relationships that are at best problematical using standard methods and aggregated data. In this paper we apply wavelet analysis to one of the most studied empirical relationship in economics: the wage Phillips curve. Thus, after decomposing labor productivity, unemployment rate, wage inflation and price inflation into their time-scale components using the maximum overlap discrete wavelet transform (MODWT), we estimate the wage Phillips curve specification on a scale-by-scale basis. The results show that the nature of the relationship between variables is not the same through all timescales since the significance and the estimated size effect of the regression variables of the wage Phillips relationships differ widely by time scale. In particular, the results at the longest scales, where the variables have large coefficient values, are very highly significant and explain Preprint submitted to Elsevier 15 October 2008

2 a substantial proportion of the total variation of nominal wage changes, indicate that the medium-run may be an appropriate perspective for the wage Philips curve relationship. Key words: Wage Phillips curve; Wavelets; MODWT; Time scale decomposition; Phase shift analysis. JEL: C63, E24, E31 1. Introduction Economics is an example of a discipline in which time scale matters. Different economic decisions are likely to have different time scales. Consider, for example, traders operating in the market for securities: someone, the fundamentalists, may have a very long view and trade looking at firms or market fundamentals; some others, the chartists, may operate with a time horizon of weeks, days, hours or minutes. A corollary of this assumption is that different planning horizons are likely to affect the structure of the relationships themselves, and that such relationships might vary over different time horizons or hold at several time scales, but not at others. If this is the case, a more realistic assumption should be to separate out different time scales of variation in the data and analyze the relationships among variables at each scale level, not at the aggregate level. These different time scales of variation in the data may be expected to match more precisely than simple time scale aggregated data the different time horizons involved in the decision making process of the agents. As a consequence, standard econometric methods 1 could be subject to a subtle identification problem, as the economic relationships may be differently identified over different time horizons (Lee and Nelson, 2007). Empirical analyses of economic models give relatively little attention to the underlying time scale of these models because of the implicit assumption that agents economic decision making process is time scale independent. 2 One of Corresponding author: Tel.: ; fax: addresses: marco.gallegati@univpm.it (Marco Gallegati), mauro.gallegati@univpm.it (Mauro Gallegati), james.ramsey@nyu.edu (James B. Ramsey), semmlerw@newschool.edu (Willi Semmler). 1 Standard econometric tools generally employed in economics and finance are used to deal with intermediate and long-run time scales. Nonetheless, more and more researchers are aware of the possible dependence of economic relationships on the concept of time scale and that many periods could be the relevant time horizon in the analysis of economic relationships. 2 If this is the case economic relationships would then hold at every time scale 2

3 the subject areas of applied macroeconomics that has received most attention in the last fifty years is the Phillips curve (Phillips, 1958), that is Phillips findings that the rate of change of money wage rates can be explained by the level of unemployment (Phillips, 1958, p. 299). In the literature the dynamic adjustment process of nominal wages is generally related to unemployment rate, past or expected rates of inflation and productivity growth, that is to variables whose temporal relationship (dependence) with nominal wage changes is likely to be quite different across time scales. For example, it is clear that the strategies used by both employers and employees differ by time scale; that is, employers, for example, may adjust hours worked in the short run, to redesigning the plant in the longer run, to moving manufacturing abroad in the longest run. In each case, the relationship between the variables involved in these actions will be different. Indeed, while the unemployment rate represents a proxy for excess demand in the labor market and, thus, is likely to have short-run effects on nominal wage changes, the impact of labor productivity on nominal wages is expected to be stronger in the longer run, while the role of prices may be important at any time horizon. Thus, the true economic relationships among variables could be found at the disaggregated (scale) level rather than at the usual aggregation level. As a matter of fact, given that aggregate data may be considered the result of a time scale aggregation procedure over all time scales, the estimated relationships between the aggregates are estimates averaged over all time scales with the consequence that the effect of each regressor tends to be cancelled out by this averaging. Wavelet analysis, although used extensively in many areas of applied sciences, has not yet fully entered the economics discipline, even if many economic processes are the result of the actions of several agents who simultaneously operate at different horizons and/or have different term objectives. Wavelets provide a unique decomposition of time series observations that enable one to decompose the data in ways that are potentially revealing of relationships that are at best problematical using standard methods and aggregated data. Wavelet analysis represents a filtering method that provides an interesting alternative to time series and frequency domain methods for analyzing economic relationships, once we recognize that such relationships need not follow the same pattern at different time horizons (scales). The fundamental idea behind wavelets is to analyze data at different scales or resolutions. Wavelets are mathematical functions that transform the data into a mathematically equivalent representation and cut up data into different frequency components, with a resolution matched to its scale. The main advantage of wavelet analysis is its ability to decompose macroeconomic time series, and data in general, into a set of time scale components, each describing the time development of the signal at a particular observation scale. Even if many applied fields have been making use of wavelets for years (i.e. astronomy, acoustics, signal and image independently from the agents planning horizon. 3

4 processing, neurophysiology, fractals, turbulence, etc.), it is only recently that applications of wavelet analysis have been used in areas like economics and finance, following the papers by Ramsey and Lampart (1998a, 1998b), Ramsey (1999) and Gencay, Selcuk, and Whitcher (2006) that illustrated the potential and usefulness of the wavelet approach to the analysis of macroeconomic variables. The aim of this paper is to re-examine of one of the most controversial issues in empirical economics through the lens of wavelets. Indeed, despite the large number of papers purporting to investigate this issue, the empirical validity of the Phillips curve is still at the heart of a current macroeconomic debate (see King and Watson, 1994, Staiger et al. 1997a, 1997b, 2002, Stock and Watson, 1999, and Ball and Moffit, 2002). Both time and frequency domain techniques have been employed in empirical analyses of the Phillips curve, but separately, 3 thus neglecting any possible time scale variations in aggregate macroeconomic relationships. Hence, after decomposing labor productivity, unemployment rate, wage inflation and price inflation into their time-scale components using the maximum overlap discrete wavelet transform (MODWT), to be defined, we estimate the US wage Phillips curve on a scaleby-scale basis using time scale regression analysis as well as spectral analysis. The most significant results of this multi-scale analysis are the one-to-one relationship between wage and price inflation in the medium and long-run, and the close relationship at the medium term scale W 5 between nominal wage changes and unemployment rate and labor productivity (other than to price inflation). In particular, the results at the longest time scales, where the variables have large coefficient values, are very highly significant and explain a substantial proportion of the total variation of nominal wage changes, indicate that the medium-run may be an appropriate perspective for the wage Philips curve relationship. Moreover, when we investigate the degree of robustness of our results to changes along the two dimensions of samples and countries we find that, on one hand, the results in terms of the relative significance of the variables at the various scales are confirmed, and that, on the other hand, relationships that are similar at an aggregate level may be remarkably different at disaggregate (scale) level. The paper is structured as follows: in section 2 we provide a brief description of wavelet analysis and its main properties. Section 3 shows the results stemming from time scale regression analysis for the US, while in section 4 we provide a robustness analysis of the results obtained in the previous section. Then, in section 5 we present the results of the phase shift and spectral analyses of the 3 Both methods display some drawbacks: pure time domain analysis averages the relationships over the entire frequency range (frequency domain aggregation), while frequency domain analysis averages over the time domain; stochastic stationarity is a critical assumption. 4

5 time scale components for the US. Finally, section 6 concludes the paper. 2. Wavelet analysis Economic relationships are often analyzed in either the time or the frequency domain with the implicit assumption that such relationships are homogeneous. However, if this is not the case it may be instructive to look at economic relationships on the time-frequency plane or on multiple scales over time. There are many examples in the recent econometric literature of papers that try to capture both the short-run and long-run dynamics of wages using a vector equilibrium correction mechanism (VECM) model (see for example Marcellino and Mizon, 2000, Juselius, 2003), 4 but there is also an increasing number of researchers investigating the hypothesis that perhaps many periods, and not just two, could be the appropriate time scales in the analysis of economic relationships. 5 Moreover, the relationship between wage increases and unemployment need not be linear, as is usually assumed in the estimation of the wage Phillips curve. 6 Both aspects may be adequately taken into account using wavelet analysis. Thus, in the next section, we analyze the wage Phillips relationship on a scale-by-scale basis by decomposing the macroeconomic time series into their time scale components. 2.1 Wavelet transform versus Fourier transform Time domain analysis tells us everything about the value of a signal at a specific location, but little about its frequency content. On the other hand, frequency analysis tells us everything about the frequency structure of the signal, but nothing about the location of the events. Time-frequency analysis represents a compromise between them, as it gives an estimate of the frequency structure of a signal locally at a given time point. Wavelets provide a suite of tools for time-frequency analysis. 4 There is a marked difference between the empirical wage specifications in the US and Europe given by the presence of an error correction term in the European, but not in the US wage specifications. 5 In recent years several applications of wavelet analysis in economics and finance have been provided by ever more authors (see Kim and Haueck In, 2003, Jagric and Ovin, 2004, Crowley and Lee, 2005, Raihan et al., 2005, Crivellini et al., 2006, Gallegati et al., 2006, Gallegati, 2008, among others). 6 Indeed, both Phillips (1958) and Lipsey (1960) find a negative, nonlinear relationship between wage inflation and unemployment. 5

6 Comparing wavelet and Fourier based methods may be useful because spectral analysis has been by far the most important filter processing tool in many fields (including economics) for many years and thus it may be helpful both for understanding the methodology and interpreting the results of wavelet analysis. Fourier and wavelet analysis have some strong similarities as well as important differences. Indeed, both transforms can be viewed as a rotation in function space. The Fourier transform contains as basis functions sines and cosines, while the wavelet transform contains more complex basis functions called wavelets. But while the Fourier transform has a single set of basis functions, i.e. the sine and cosine functions, wavelets have transforms based on an infinite set of possible basis functions and thus may provide immediate access to information that can be obscured by Fourier analysis. The Fourier transform s utility lies in its ability to analyze a signal in the time domain for its frequency content. The Fourier transform decomposes a signal or a function into a sum of harmonic components (or waves) of different frequencies via a linear combination of Fourier basis functions (sines and cosines) that range over ± infinity. The transform works by first translating a function in the time domain into a function in the frequency domain. The signal can then be analyzed for its frequency content because the Fourier coefficients of the transformed function represent the contribution of each sine and cosine function at each frequency. Thus, the Fourier transform is a frequency domain representation of a signal containing the same information as the original function, but summarized as a function of frequency. Although extremely useful as an alternative representation of the original time series, the Fourier transform has some serious drawbacks. First, the transformation to the frequency domain does not preserve the time information so that it is impossible to determine when a particular event took place, a feature that may be important in the analysis of economic relationships. In other words, it has only frequency resolution but not time resolution. Second, in Fourier analysis a single disturbance affects all frequencies for the entire length of the series as all projections are globals, and thus the signal is assumed to be homogeneous over time. Thus, if the signal is a nonperiodic one, the summation of the periodic functions, sine and cosine, does not accurately represent the signal. Such a feature restricts the usefulness of the Fourier transform to the analysis of stationary processes, whereas most economic and financial time series display frequency behavior that changes over time, i.e. they are nonstationary (Ramsey and Zhang, 1996). The windowed or short-time Fourier transform (WFT or STFT) provides a partial solution to the problem of better representing a nonperiodic signal, because with the WFT the input signal is chopped up into sections, where each section is analyzed for its frequency content separately. In particular, the windowed Fourier transform uses a fixed window function with respect to 6

7 frequency and applies the Fourier transform to the windowed signal. The original signal is partitioned into small enough segments such that these portions of the non-stationary signal can be assumed to be stationary over the duration of the window function. The choice of the window length is based on the trade-off between the desired frequency resolution, which depends inversely on the duration of the window function, and the assumption of short-term stationarity. Once the window function is determined, both the time as well as frequency resolutions become fixed for all frequencies and times, respectively. As a consequence, the short-term Fourier transform does not allow any change in resolutions in terms of time or frequency. The effect of the window is to localize the signal in time. The problem with this approach is the right choice of the window and, more importantly, its constancy over time. Wavelet analysis may overcome the main problems evidenced by Fourier analysis and the short-time Fourier transform. 7 The wavelet transform uses a basis function that is dilated or compressed (through a scale or dilation factor) and shifted (through a translation or location parameter) along the signal so as to provide a time-frequency representation where all the information is associated with specific time horizons and locations in time. Hence, a wavelet is similar to a sine and cosine function in that it also oscillates around zero, but differ because, as wavelets are constructed over finite intervals of time, they are well-localized both in the time and the frequency domain. In contrast to Fourier analysis wavelets are compactly supported as are all projections of a signal onto the wavelet space which are essentially local, not global and thus need not be homogeneous over time. Indeed, much of the usefulness of wavelet analysis has to do with its flexibility in handling a variety of nonstationary signals. Wavelets, in opposition to time and frequency domain analyses, consider nonstationarity an intrinsic property of the data rather than a problem to be solved by pre-processing the data. In particular, since the base scale includes any non-stationary components, the data need neither be detrended nor differenced (Schleicher, 2002). An immediate way to see the time-frequency resolution differences between the windowed Fourier transform and the wavelet transform is to look at the basis function coverage of the time-frequency plane. Figure 1 shows the time-frequency resolution properties of the windowed Fourier transform (upper panel) and the wavelet transform (lower panel). It shows that, unlike the windowed Fourier transform which has constant resolution at all times and frequencies, the wavelet transform, through the adaptive partition of the time-frequency plane, analyzes the signal at different frequencies with different resolutions using a multiresolution analysis. Thus, in contrast to the fixed time-frequency partition of the windowed Fourier transform the wavelet trans- 7 For example, the set of frequencies used in Fourier analysis are themselves dilations of the fundamental frequency. See Gabor (1946). 7

8 Frequency Time Frequency Time Fig. 1. The windowed Fourier transform (upper panel) and the wavelet transform (lower panel) partitioning of the time-frequency plane. 8

9 form provides good frequency resolution (and poor time resolution) at low frequencies and good time resolution (and poor frequency resolution) at high frequencies. 8 In summary, the multiresolution analysis approach may overcome the resolution problem as it adaptively partitions the time-frequency plane, using short windows at high frequencies and long windows at low frequencies, and thus letting both time and frequency resolutions to vary in time frequency plane. 2.2 Some basics of wavelet analysis We begin this section by recalling the basic structure of the wavelet approach. For a more complete and insightful development of the theory and use of wavelets, see Percival and Walden (2000) and Gencay et al. (2002). At the simplest level wavelet analysis consists of a transformation of a signal obtained by projecting the signal onto a sequence of local basis functions, each expressible as g(t) = 1 s g ( ) t k s (1) where the time scale s corresponds to the width of the wavelet function g(t), and k is the wavelet shift along the time axis. There are two basic wavelet functions: the father-wavelet, φ, and the motherwavelet, ψ. The formal definition of the father wavelets is the function ( φ J,k = 2 J t 2 J ) k 2 φ 2 J (2) defined as non-zero over a finite time length support that corresponds to given mother wavelets ( ψ j,k = 2 j t 2 j ) k 2 ψ 2 j (3) with j = 1,..., J in a J-level wavelets decomposition. The former integrates to 1 and reconstructs the longest time-scale component of the series (trend), while the latter integrates to 0 (similarly to sine and cosine) and is used to 8 Because a single window is used for all frequencies in the WFT, the resolution of the analysis is the same at all locations in the time-frequency plane. 9

10 describe all deviations from trend. 9 The application of the basic wavelet functions to the signal allows one to compute the wavelet series coefficients at all scales given by w j,k = k v J,k = k ψ j,k f(t) φ J,k f(t) where the coefficients w jk and v Jk are the projections of the time series onto the mother and father wavelet functions, respectively. The two types of coefficients are sometimes referred to as wavelet and scaling coefficients, respectively. Wavelet coefficients at level j are related to changes of averages over a range of different scales (usually taken to be powers of two), and, in particular, are proportional to differences between local averages of the data taken at that scale. On the other hand, the scaling coefficients at level j are proportional to the local average of the data at that scale and therefore can be associated with averages on a specified scale. 10 A wavelet transform decomposes a given function into coefficients from which the original function can be reconstructed using an inverse wavelet transformation. The wavelet representation of the signal or function f (t) in L 2 (R) is represented by f (t) k v J,k φ J,k (t)+ k w J,k ψ J,k (t)+...+ k w j,k ψ j,k (t)+...+ k w 1,k ψ 1,k (t) (4) where J is the number of multiresolution components or scales, and k ranges from 1 to the number of coefficients in the specified components. The multiresolution decomposition of the original signal f (t) is given by the following expression f (t) V J + W J + W J W j W 1 (5) 9 The mother wavelets, as said above, play a role similar to sins and cosines in the Fourier decomposition. They are compressed or dilated, in time domain, to generate cycles fitting actual data. 10 Mother wavelet acts as a high-pass filter and delivers the detailed information of the function, while father wavelet serves as a low-pass filter and represents the coarsest information of the function. Thus, we may say that the low-pass filter averages, while the high-pass filter differences. 10

11 where V J = v J,k φ J,k (t) and W j = w j,k ψ j,k (t) with j = 1,..., J. The k k sequence of terms V J, W J,.., W j,..., W 1 in (4) represents a set of signal components that provide representations of the signal at different resolution levels 1 to J, with the detail signals W j providing the increments at each individual scale or resolution level. Thus, wavelet analysis enables us to separate the increments at each individual scale, or resolution, level, we are able to examine the relationship between the variables when the variation in each variable has been restricted to a specific indicated scale. 3. Time scale decomposition analysis A useful property of the wavelet approach, in the case of processes whose behavior is quite different across scales, is its ability to decompose economic variables into different time horizons, i.e. on a scale-by-scale basis. Thus, in this section we explore the possibility of there being differences in the regression results of the wage Phillips curve after allowing for a decomposition of the regression variables into different time scales. 3.1 Specification and estimation of the wage Phillips curve Since the publication of the original Phillips paper the relationship has been estimated using many different specification. The standard reference in the Phillips curve literature for the specification of the wage Phillips is represented by the work of Staiger et al. (1997a, 1997b). Omitting lag dynamics, 11 it may be expressed as: 12 w t = α + p t βu t + γ lp t + ɛ t (6) 11 We are aware of the fact that econometric specifications of wage equations contain lagged polynomials of the relevant regressors. Nonetheless, we choose to employ a specification containing only contemporaneous variables, and thus to keep the level of the specification of the wage Phillips curve as simple as possible, in order to focus on the usefulness of the application of the wavelet approach for the analysis of economic relationships. 12 Blanchard and Katz (1999) discuss how the formulation of the wage Phillips curve specification in equation (6) may be obtained, under certain conditions, from a standard representation of theoretical models of wage setting, like the efficiency wage (Shapiro and Stiglitz, 1984), matching (or bargaining) models (Mortensen and Pissarides, 1994). 11

12 where wage inflation w t depends on price inflation p t, unemployment rate u t and labor productivity growth lp t. 13 Thus, nominal wage growth may be determined by inflationary (or cost) pressures, demand pressure in the labor market and productivity changes. First, as wage indexation to inflation is an institutional characteristic of the wage determination process in the labor market of developed economies, when prices increase workers (or their organizations) require higher nominal wages to prevent a reduction in real wages. Secondly, since unemployment rises (falls) when supply (demand) exceeds demand (supply), the unemployment rate may be used as a proxy for excess demand and is expected to be negatively related to wage changes. 14 Third, productivity may determine wage changes through two main channels: i) because of the labor hoarding hypothesis productivity may provide an improved proxy for excess demand, and ii) in equilibrium, productivity gains arising from technical progress, if not entirely appropriated by producers in the form of higher profits, could pass to workers in the form of higher wages Time series regression analysis We start by estimating the wage Phillips curve represented in equation (6) using aggregate data for the US between 1948:1 and 2006:4. The data are quarterly seasonally adjusted data from the Bureau of Labor Statistics of the U.S.Department of Labor and the Federal Reserve Bank of St. Louis ( We use the percent civilian unemployment rate for the rate of utilization of the labor factor, the implicit price deflator of the gross national product for prices, output per hour 16 in the nonfarm business sector for labor productivity and the compensation per hour in the nonfarm business sector for wages. We indicate with w, p, and lp the current rates of wage inflation, price inflation and labor productivity growth, respectively Staiger et al. (2002) results indicate that wage Phillips curves incorporating productivity growth are characterized by greater stability across subsamples. 14 This argument is in the spirit of Phillips s and Lipsey s original work. 15 For example, Friberg and Uddén Sonnegard s (2001) estimate of an expectationsaugmented Phillips curve for Sweden shows that a higher profit share leads to higher wage increases. 16 Since the number of workers employed represents a poor measure of labor input, for the purpose of productivity measurement a better definition of labor input may be restated in terms of the number of workers adjusted by the number of hours worked. For this reason output per hour is preferred to other measures of labor productivity like, for example, output per worker. 17 All variables, except unemployment, are transformed into their growth rates as 400 ln(x t /x t 1 ). Details on variables used and transformations applied are provided in Appendix 1. 12

13 In Table 1 we present the results of the estimates of the wage Phillips curve for the US using aggregate data. The estimate confirms that a simple aggregate wage Phillips curve describes quite well the behavior of aggregate wage changes for the US. 18 All estimates are correctly signed and statistically significant at the 5 percent significant level, and roughly correspond, in terms of hierarchy, with the values obtained in previous analyses (see Chan-Lee et al. 1986, Staiger et al., 2002, Flaschel and Krolzig, 2006, and Flaschel et al. 2007). The estimated slope of the wage Phillips curve is negative and significant. The parameter estimate indicates that a higher unemployment rate exerts a down ward pressure on the wage rate, as a one percentage point increase of the unemployment rate leads to a 0.34 precent decrease in wage growth. 19 The estimated coefficient of price inflation is.79, a value statistically different from zero and significantly different from unity. Such a value, suggests that, even if changes in inflation rate do not lead to equally large changes in the rate of nominal wage increase, price inflation exerts a strong and positive influence on the wage rate. The estimated coefficient of labor productivity growth is positive and significant, and indicates that a one percentage point increase in productivity growth leads to a 0.23 percent increase in wage growth. Although productivity growth is expected to be an important determinant of wage increases, it seems to have only a small effect on nominal wage growth. In summary, the results in Table 1 are in line with those reported in previous macroeconometric studies about the U.S. and tend to support a negative relationship between the rate of change in wages and the unemployment rate of the type stated in Phillips original work (see Blanchard and Katz, 1997, Brainard and Perry, 2000, Ball and Moffitt, 2002, Staiger et al., 2002, for evidence about the U.S.) Time scale regression analysis The variables involved in the wage Phillips curve estimation are decomposed into their time-scale components applying the maximal overlap discrete 18 The goodness-of-fit of the model is assessed by the value of.49 for the R 2 statistic (which means that the basic equation explain a large part of the variance in wage inflation) and the statistical adequacy of the regression equation by a value of 1.96 of the DW test statistic. 19 Coe (1985) using semi-annual data finds an estimated slope of the Phillips curve from the wage inflation equation of.33. Staiger et al. (2002) using quarterly macro data find a coefficient estimate for the unemployment rate of Similar evidence is provided for Canada by Farés (2002) and by Fauvel et al. (2002). 13

14 Table 1 Wage Phillips curve aggregate regression for the US (1948:1-2006:4) Dependent variable: w ur p lp Adj-R 2 s.e. DW (-3.78) (14.49) (5.71) wavelet transform (MODWT) 21 and using the Daubechies least asymmetric (LA) wavelet filter of lenght L = 8, that is LA(8), based on eight non-zero coefficients (Daubechies, 1992), with periodic boundary conditions. 23 Such multi-scale decomposition is performed sequentially through an adaptive windowing, from smallest (high frequencies) to largest (low frequencies) scales, with the scale doubling in size with each step. The application of the translation invariant wavelet transform with a number of scales J = 5 produces five MODWT details vectors W 5, W 4, W 3, W 2, W 1 and one smooth vector, V 5, where each wavelet scale is associated with a particular time period 2 j As the MODWT wavelet filter at each scale j approximates an ideal high-pass filter with passband f [1/2 j+1, 1/2 j ], the level j wavelet coefficients are associated to periods [2 j, 2 j+1 ]. 25 Thus, since we use quarterly data the first detail level W 1 captures oscillations between 2 and 4 periods, while details W 2, W 3, W 4 and W 5 capture oscillations with a period 21 The MODWT transform, 22 also known as the non-decimated wavelet transform or the translation invariant DWT, is a generalization of the discrete wavelet transform whose coefficients are associated with zero phase filters, is translation invariant and can be applied to data sets of length not divisible by 2 J. 23 Daubechies (1992) has developed a family of compactly supported wavelet filters of various lengths, the least asymmetric family of wavelet filters (LA), which is particularly useful in wavelet analysis of time series because it allows the most accurate alignment in time between wavelet coefficients at various scales and the original time series. Our choice of the filter length is a compromise choice between competing requirements. 24 The maximum number of scale crystals is given by log 2 (N). With 236 observations the maximum number of scales is in theory j = 7, although in practice to obtain reasonable resolution less than 7 crystals are to be produced (see Crowley, 2005). 25 On the other hand scaling coefficients are associated with frequencies f [0, 1/2 j+1 ]. 14

15 of 4-8, 8-16, and quarters, respectively. 26 Figure 2 shows the multiresolution decomposition plot of the unemployment rate, price inflation, wage inflation and labor productivity growth performed using the wavelet basis LA(8) wavelet basis. The original signals are shown at the top of each plot. Over the signal, the smooth and detail components, with the finest detail W 1 at the top. Thus, after decomposing the regression variables over different time scale components we estimate a sequence of least squares regressions using and w[v J ] t = α J + p[v J ] t β J u[v J ] t + γ J lp[v J ] t + ɛ t (7) w[w j ] t = α j + p[w j ] t β j u[w j ] t + γ j lp[w j ] t + ɛ t (8) where w[v J ] t, p[v J ] t, u[v J ] t and lp[v J ] t represent the components of the variables at the longest scale, and w[w j ] t, p[w j ] t, u[w j ] t and lp[w j ] t represent the components of the variables at each j scale, with j=1,2,...,j. Thus, at each j scale we examine the relationship between the variables where the variation in both variables has been restricted to the indicated specific scale. In Table 2 we present the results obtained from running a sequence of time scale least squares regressions from equations (7) and (8). The degree of fit of the estimated wage Phillips curves tend to decline monotonically as the scale decreases: it is very high at the longest scales (W 5 and V 5 ), high at business cycle scales (W 3 and W 4 ), and insignificant at the shortest scales (W 1 and W 2 ). With the exception of the unemployment rate at levels 1 and 2 and of inflation rate at level 1, the regressors are always significantly different from zero. Price inflation is the most significant regression variable at each time scale, with the only exception at the detail level W 5 where the unemployment rate is the most significant regressor. In addition, as regards the size effect there is clear evidence for the variation in a progressive fashion of coefficient estimates across scales. In particular, the values of the estimated wavelet coefficients relating wage inflation to price inflation and labor productivity growth tend to increase as the time scale increases Detail levels W 1 and W 2, represent the very short-run dynamics of a signal (and contains most of the noise of the signal), levels W 3 and W 4 roughly correspond to the standard business cycle time period (Stock and Watson, 1999), while the medium-run component is associated to level W 5. Finally, the smooth component V 5 captures oscillations with a period longer than 32 quarters corresponding to the low-frequency components of a signal. 27 In the case of labor productivity growth wavelet coefficients increase monotonically at the detail levels, but flatten at the smooth component. 15

16 D1 D D2 D2 D2 D D3 D3 D3 D D D5 D5 D5 D S5 S5 S5 S Unemployment rate Wage inflation D1 D1 D4 D4 D4 Price inflation Growth rate of labor productivity Fig. 2. Multiresolution decomposition analysis for unemployment rate, price inflation, wage inflation and labor productivity growth. 16

17 The estimated coefficients of price inflation are always significant (exception is the estimated coefficient at the detail level W 1 ) and tend to increase as the wavelet scale increases, reaching values not significantly different from unity at scales corresponding to periods longer than 8 years, i.e. scales W 5 and V 5. Thus, wages seem to adjust gradually to price changes, with complete adjustment occurring only at scales corresponding to the medium and long run. This means that over long time horizons nominal wages are fully responsive to changes in the price level. The estimated coefficients of productivity growth are all positive and statistically significant at the 5 percent level, with values increasing as the wavelet scale increases. In terms of the size effect, the link between productivity growth and wage increases is weak at the shortest scales (W 1 to W 3 ), but becomes stronger at the detail levels W 4 and, particularly, W 5, where it reaches its largest (.758) and most significant value. Thus, the effects of changes in productivity growth translates to nominal wages changes more intensively in the medium than in the long-run, where wages are mainly determined by price. The pattern of unemployment rate coefficients differ from that of the other variables. At scales corresponding to periods shorter than 2 years the unemployment rate is not significant, while at intermediate scales, i.e scales corresponding to periods between 2 and 16 years, the estimated slopes of the Phillips curve are almost twice what we obtained using aggregate data. 28 The highest and most significant effect of unemployment rate on nominal wage growth is at level W 5, suggesting that the time of pass through of the unemployment rate to wage inflation is the medium run. Finally, at the longest scale, corresponding to periods longer than 16 years, the estimated coefficient indicates a huge flattening of the wage Phillips curve Summary of main findings The methodology employed by Phillips (1958) in his well-known article for testing the relationship between unemployment and wages suggests that he was convinced about the non-linearity and the long-term nature of such relationship. Indeed, as evidenced by Desai (1975) the unorthodox data transformation procedure used by Phillips 29 gives the impression of trying to extract 28 At scales W 3 to W 5, they range between.599 and.755. Such estimated coefficients resemble those reported in Staiger et al. (2002) from state data using IV estimation to correct OLS errors in variables bias. 29 The procedure consists in replacing the 53 raw observations with six averaged values and then estimating by means of a combination of least squares and graphical inspection (Gilbert, 1976, p.51). 17

18 Table 2 Wage Phillips curve time scale regression for the US (1948:1-2006:4) Dependent variable: w Scales ur p lp Adj.R 2 s.e. W (-0.72) (0.69) (4.48) W (-1.06) (4.66) (7.23) W (-6.04) (15.02) (5.94) W (-6.19) (17.58) (6.53) W (-24.40) (21.34) (15.42) V (-10.78) (88.53) (6.67) Note: t-ratios are in paranthesis a long-run relationship from an array of short-run observations. 30 Time scale decomposition analysis provides a different perspective on the debate on the short-term versus the long-term nature of the Phillips relationship. When we examine the relationship at each time scale separately instead of looking at the relationships between variables averaged over all time scales, we find results that are broadly consistent with the wage Phillips relationship only at the longest time scales. The comparison of the results at different time scales indicates that the relationship between nominal wage rate and each regressor differ widely at the various scales. Indeed, both the time of pass through of each regressor and the relative importance of the coefficients change with the time horizon considered. The most significant results of this multi-scale analysis are the one-to-one relationship between wage and price inflation in the medium and long-run, and the close relationship at the medium term scale W 5 between nominal wage changes and unemployment rate and labor productivity (other than to price inflation). The results stemming from time scale regression analysis suggest the existence of appropriate time scales from which to estimate 30 The short-run interpretation of the unemployment (inflation)-wage relationship was first proposed by Lipsey (1960) and then popularised by Samuelson and Solow (1960). 18

19 economic relationships. In particular, the results at the longest time scales, where the variables have large coefficient values, are very highly significant and explain a substantial proportion of the total variation of nominal wage changes, indicate that the medium-run may be an appropriate perspective for the wage Philips curve relationship. James comment: before section 4 try to add some comments and examples of the significance of the difference of the coefficient estimates within and across scales. 4. Robustness analysis In this section we attempt to address the question as to whether the results from time scale regression analysis reported in table 2 may be considered unique to the US and/or to the particular sample used. Indeed, as wavelets are a comparatively new, and not yet widely used, mathematical tool for the analysis of economic data and macroeconomic relationships, it may be useful to assess the robustness of the results obtained from wavelet estimates. Thus, to investigate the degree of robustness of our results in terms of significance of the estimated coefficients and of the estimated size effect we replicate the wavelet analysis performed in the previous subsection testing the sensitivity of the results to changes along several dimensions, i.e. sample and country. There are several papers comparing the results of the Phillips relationship between the US and other countries (see Beaudry and Doyle, 2000, Farés, 2002, Proaño et al., 2006), but given the similarity of the two countries economies, as well as their close and growing economic integration, Canada is frequently used in comparisons with the US. Moreover, in our case such a choice is enhanced by the fact that the empirical literature suggests that the wage Phillips curve specification of equation 6 explains aggregate wage behavior quite well in both the US and Canada and anyway better than in continental Europe where an error-correction specification is be preferred (Farés, 2002). Quarterly based aggregate data on output, employment and related variables for Canada are available from both national (Statistics Canada: Cansim) as well as international (OECD Economic Outlook) sources. Considering that in the first case quarterly data are currently available, in index form, from 1987 onwards only, 31 and that, on the other hand, the OECD data base provides private sector data from the first quarter of 1970, the choice of the 31 Quarterly data for labor productivity and related series for Canada are available at the business sector level from the Productivity Program Database of Statistics Canada. 19

20 OECD database was made based on data availability. Thus, we use for Canada quarterly, seasonally adjusted data from 1970:1 onwards from the OECD Economic Outlook database. We use the Deflator for GDP at Market Prices (index 2000=100) for prices, the total civilian unemployment rate (percent) for unemployment, compensation per hour worked (defined as the ratio of nominal labour compensation to hours worked) for wages, and output per hour worked (defined as the ratio of business sector real GDP estimate to hours worked) for labor productivity. 32 Except for the unemployment rate the measures of the variables are transformed into their growth rates as x t = 400ln(x t /x t 1 ). We conduct a robustness analysis of our wavelets estimates along two dimensions: across samples, comparing estimates of the time scale wage Phillips equations for the US over the periods 1948:1-2006:4 and 1970:1-2006:4, and across countries, comparing estimates of the time scale wage Phillips equations for the US and Canada over the period 1970:1-2006:4. In this section let us assess the sensitivity of our results to changes in these factors and check the robustness of the results presented in table Table 3 reports the estimated results of the standard wage Phillips curve specification for the US and Canada between 1970:1 and 2006:4. The results confirm that this simple empirical specification fits the aggregate wage data quite well. Moreover, a comparison between the parameter estimates of the two regressions reveals that there are both the similarities in the significance of the estimated coefficients, and differences, in the estimated size effect of the regressors. Indeed, in both countries price inflation explains wage inflation more than labor productivity growth and unemployment rate. Moreover, both unemployment rate and labor productivity growth size effect is larger in Canada than in the US, while the opposite is true for price inflation. Such a different pattern may reflect the different characteristics of the two labor markets, with the Canadian labor market displaying a stronger unionization than the US labor market. Following the methodology employed in the previous section, after decomposing all variables into their time-scale components applying the maximal overlap discrete wavelet transform (MODWT) we estimate the wage Phillips 32 The business sector is defined as the non-government part of the economy, thus business sector output estimated is total economic output of the economy (real GDP at market prices) less the activities of the general government sector, that is, general government consumption and gross capital formation. 33 Sensitivity analysis aims at discovering how sensitive the estimated results are to changes in crucial factors which is needed in order to assess the robustness of the final results. 20

21 Table 3 Wage Philips curve aggregate regressions for the US and Canada (1970:1-2006:4) Dependent variable: w ur p lp Adj-R 2 s.e. D.W. US (-1.94) (11.37) (3.48) Canada (-2.25) (10.04) (5.31) curve specification at different scales. 34 In table 4 we present the results of the time scale regressions of the wage Phillips curve for the US (upper panel) and for Canada (lower panel). The pattern of results for the US in table 4 resemble those reported in table 2 closely. 35 No significant differences emerge in terms of the significance of the regression variables, while in terms of the estimated size effect the only relevant differences regard the unemployment rate at level W 3, price inflation at levels W 4 and W 5 and labor productivity at level W 5, where the values of the estimated coefficients are close or equal to unity in all cases. In summary, as no relevant differences emerge between the significance of the regressors for the US across the two samples, we may conclude the results of our time scale regression analysis are robust to changing sample periods, at least for the post WWII period. As regards Canada, the results in table 4 indicate the presence of similarities as well as differences between the estimated regressions of the two countries. First of all, for Canada the degree of fit of the estimated wage Phillips curves declines as the scale decreases, and, second, wavelet regression coefficients are clearly significantly different across scales. 36 Significant uniformities between the two countries emerge with regard to the estimated slope of the price inflation coefficients. Indeed, the estimation at the longest scales confirm the full reaction of the wage to price changes for Canada 37 and that price inflation is the main determinant of wage inflation in the long-run just as in the US case. 34 Given the size of the OECD sample, 148 quarterly observations, we decide to set the maximum wavelet scale at 2 5 = 32 quarters. 35 The only differences relates to price inflation at level W 2 and labor productivity at level W 1 (not reported here for brevity), where the dominance of the noise component may generate spurious results. 36 Time scale regressions at levels W 1 and W 2 are presumed to contain most of the noise of the signal, and thus the results of the regression analysis at these detail levels may be disappointing (other than being not significant, sometimes they may also be incorrectly signed). 37 The estimated coefficient is equal to unity for periods longer than 32 quarters at both the W 5 and the V 5 levels. 21