Inflation and Unit Labor Cost (Footnotes) Part 5

Footnotes 1 Fuhrer (1997) argues that the NKPE fails badly when a conventional output gap is employed. However, a class of modern macroeconomic models – those that add nominal frictions to a real business cycle core framework – imply a time-varying level of capacity output due to changes in productivity and other real factors that would make conventional output gaps a poor proxy for pricing pressures, while a measure of real marginal cost (or average markup) would be an appropriate indicator (see, for example, Goodfriend and King (1997)). Gali and Gertler (1999) make an explicit empirical comparison between marginal cost and output gap models of inflation dynamics. 2 In this study, which involves replication and extension of prior work, there is potential confusion between the results of the original studies and the results of replications/extensions developed along the lines of the studies. We use full names to indicate material which is exactly taken from prior studies, such as the parameter values estimated by Gali and Gertler (1999) and Smets-Wouters (2007). When we refer to data, we use the author's full names because we have carefully reconstructed and extended the data employed these authors, so that the series closely match over the estimation intervals for both prior studies with some small exceptions discussed below (see Appendix A for some additional details). Thus, when we discuss the Gali-Gertler and Smets-Wouters measures of labor cost, we are referring to their conceptual constructions and our data set. We use abbreviations GG and SW to indicate our constructions. Fundamental inflation (FI) plays an important role in our study, as it did in Gali and Gertler (1999) and Sbordone (2002). When we refer to GG fundamental inflation, this is our FI construction along Gali-Gertler lines but it differs in some details that we discuss below. The SW fundamental inflation is our construction using Smets-Wouters parameter estimates and our replication/extension of their data set. We think it is an informative construction, but it is not one which they used. 3 This specification is sometimes called the "New Keynesian Phillips Curve." As it links inflation to marginal cost rather than output or unemployment, we prefer the terminology of Kurmann (2005). This terminology accurately captures its role in NK-DSGE models as one equation in a wage-price block. 4 Textbook treatments are provided by Gali (2008), Walsh (2010), and Woodford (2003). 5 See Roberts (1995) for a comparison of specifications. 6 This near-neutrality property is shared by many models, as noted, for example, by Roberts (1995) and Goodfriend and King (1997)). It holds for state dependent pricing models (see, Dotsey, King and Wolman (1999). More recently, Cogley and Sbordonne (2008) have considered the consequences of time-varying trend inflation within a higher order approximation of the Calvo (1983) model. 7 This a second-order expectational difference equation familiar from Sargent (1978). For the specific formulas applicable in this case, see Gali and Gertler (1999, page 217). 8 Our measure of fundamental inflation differs slightly from the measure used by Gali and Gertler. Their measure is  t 1    j  0  i Et t  j while our measure is  t1    j  0  i Et t  j which   differ in the value of lagged inflation. We solve out for the effects of past inflation so as to make our results more conceptually comparable to historical shock decomposition such as those employed in Smets and Wouters (2007). Generally, the Gali-Gertler measure will track inflation more closely, as it incorporates the effects of lagged z. But given that the value of the persistence parameter is low (about one-third), the differences between their fundamental inflation and ours is practically small. Further, both measures share the same low frequency properties stressed below. 99 The relationship is less strong in the early and late periods of our sample, with much of poor performance in the later period arising from revisions in the real unit labor cost data that were not available to Gali, Gertler, and Sbordone. 10 The estimate of fundamental inflation is subject to substantial sampling variability – that is, appropriately computed confidence bands in Figure 2 are wide. Because fundamental inflation is a discounted sum of expected future values of , the sampling variability arises from uncertainty in the discount factor  and the parameters used to construct the forecasts. Because  is large, uncertainty about the long-run properties of the forecasting model are particularly important. Characterizing this uncertainty is complicated because the variables in the model,  and , are highly serially correlated so the VAR has roots near the unit circle, i.e., approximate “unit roots.” We have therefore not attempted to compute appropriate confidence bands. However, to give you some sense of the uncertainty, consider a simple version of the model in which  follows a univariate AR(1), i = t−1 + ut, in which case fundamental inflation is given by (1−)−1(1−L)−1ut. Considering uncertainty in the first term, (1−)−1, the 95% confidence interval for  computed using Stock’s (1991) method ranges from 0.861 to 1.012, and using the point estimate of  from table 1 ( = 0.876) yields a 95% confidence interval for (1−)−1 that ranges from 4.1 to 8.8, so that the scale varies by a factor of model that 2. Additional sampling uncertainty arises from the estimated values of  and , and the persistence in . Rudd and Whelan (2005) go beyond the (, )- bivariate VAR and document substantial variability in the point-estimates of fundamental inflation associated with changing the variables included the VAR; thus, the specification of the VAR is another important source of uncertainty. See also Kurmann (2005). 11 See Gali (2008), Walsh (2010), and Woodford (2003). 12 Note that their estimate is just under .04, but is for a quarterly inflation rate. As we are using annualized inflation rates, we multiply by 4 to obtain the .15 value referenced in the text. 13 The fact that actual micro prices are held fixed in nominal terms rather than adjusted mechanically according to an index has led to substantial criticism of the indexation mechanism (see, for example, Collard and Dellas (2006)). 14 Variable depreciation arising from endogenous capacity utilization and overhead production costs also augment the core neoclassical model. 15 These numbers are the posterior means of the parameters, provided in Table 2 of Smets and Wouters (2007). In their work, the parameter is called p to distinguish it from the AR parameter of other shock processes. 16 While the Smets-Wouters -DSGE model was not directly fit to match modified real unit labor cost, it was fit to match the joint behavior of its ingredients (W,n,P,y) and modified real unit labor cost is one of the many variables provided by the Smets-Wouters Dynare code that is employed to estimate and simulate the Smets-Wouters -DSGE model. 17 Gali-Gertler and Smets-Wouters also differ in some details of their measurements of the components of real marginal cost (W,P,n, and y). Gali and Gertler use measures for the nonfarm business sector for all of these components. Smets and Wouters use full-economy measures for y and P (real GDP and the GDP deflator), nominal compensation per hour in the non-farm business sector for W, and compute n (total hours of employment) using average weekly hours in the non-farm business sector multiplied by employment in the total economy (from the U.S. household survey). Finally, because n and y are in per-capita terms, Smets and Wouters divide hours and output by a measure of population; this adjustment is not necessary for Gali and Gertler (because  =1 for their measurement of  ). Edge and Gurkaynak (2010) discuss measurement problems with the population series used by Smets and Wouters. Our analysis is based on a modified version of the series that eliminates these problems. See Appendix A for details. 18 Our Figure 8 conveys similar content to Figure 4 in Smets and Wouters (2007), but is tailored to our purposes. Smets and Wouters (2007) combine price and wage markup shocks in providing the historical decomposition of inflation. 19 Recall from section 2 above that these parameters are functions of b, f, and : the implied values of , , and  are listed in Table 1. In turn, for the Smets-Wouters model, b, f, and  are functions of deeper structural parameters. 2020 To help understand the role of M for the sign of (1) consider the VAR(1) model, yt = yt−1 + t, where yt = (t t)’. In this case, the discounted sum of future expectations of y is given by Ayt, where A = (I − )−1, so that (1) is proportional to A12. When |A| is positive, the sign of (1) is determined by the sign of 12. When 12 is positive, high values of inflation today predict high values of  in the future; when 12 is negative, high values of inflation today predict low values of  in the future. In the general VAR(p) model yt = (L)yt−1 + t, the same result holds as an approximation with (1) (the sum of the coefficients) replacing . In the VAR represented by MGG-VAR, (1)12 is positive ( (1)12 = 0.03), while it is negative in the approximate VAR constructed using MSW ((1)12 = −0.15) and when using the VAR estimated directly using the Smets-Wouters data, MSW−VAR ((1)12 = −0.03). 21 The decline in labor's share is discussed, inter alia, in Fleck et al (2011). Table 1: NKPE Parameter Values Parameters in equation (2) Model Gali‐ Gertler Smets‐ Wouters Parameters in equation (3) b f     z 0.253 0.682 0.153 0.325 0.876 0.197 5.138 0.186 0.813 0.082 0.228 0.998 0.101 4.911 Notes: Gali-Gertler parameter values are from Gali and Gertler (1999, Table 2, Row 1). Smets-Wouters structural parameter values are from Smets and Wouters (2007, Table 1A) and the Dynare replication files for this paper, which are then used to calculate the parameters in equation (6). The values of , , and z reported above are four times the values reported in Gali-Gertler and Smets-Wouters because we measure inflation at annual rate. Table 2: Properties of Fundamental Inflation Using Different Parameters, Forecasting Rules, and Unit Labor Cost Measures Row  1 2 3 4 5 GG SW GG GG SW 6 7 8 9 SW GG SW GG M  (1) (1)  ( ) cor(  ,  ) A. VAR‐Based Measures of Fundamental Inflation M 1.30 0.26 2.90 GG 1.09 0.99 3.94 MGG‐VAR GG GG‐VAR SW 1.30 0.26 4.31 M  SW 2.12 ‐0.29 6.31 MSW‐VAR  ‐SW‐VAR SW 4.37 ‐2.11 12.45 M  B. DSGE‐Based Measures of Fundamental Inflation 1.89 ‐0.89 5.22 MSW SW SW SW 1.63 ‐0.34 4.63 M  GG 1.89 ‐0.89 3.46 MSW  SW GG 1.63 ‐0.34 3.01 M  GG‐VAR cor ( ,  ) 0.99 0.99 1.00 1.00 1.00 0.61 0.87 0.51 0.28 ‐0.01 0.97 0.99 NA NA 0.02 0.24 ‐0.15 0.20 Notes: GG are the Gali-Gertler parameter values shown in Table 1; SW are the corresponding Smets-Wouters values. MGG-VAR are VAR coefficients estimated from a bivariate (,GG) VAR(4) model over 1960:I-1999:IV; MSW-VAR are VAR coefficients estimated from a bivariate (,SW) VAR(4) model over 1960:I-1999:IV; and MSW is the companion matrix from the SmetsWouters DSGE model. GG is the logarithm of real unit labor cost for the non-farm business sector and SW is the Smets-Wouters modified real unit labor cost measure. See Figure 10 for a visual comparison of the two series. Figure 1: Gali and Gertler Data Inflation and Real Unit Labor Cost Notes: The solid line is inflation in the GDP price deflator measured in percentage points at an annual rate; that is, 400×ln(Pt/Pt−1), where Pt is the value of the GDP deflator. The dashed line is the deviation in real unit labor cost from its value in 1960:I in percentage points; that is, 100×  GG t GG GG  1960: is the logarithm of the index of real unit labor cost in the non-farm I  , where  business sector (see Appendix A). Figure 2: GG Fundamental Inflation Notes: The solid line is inflation shown in Figure 1. The dotted line is fundamental inflation  t from equation (5). The parameter values b, f, and  are from Gali-Gertler and given in Table 1 of this paper. Real unit labor cost, , is from Figure 1. Expected real unit labor cost, Ett+j, is computed from a bivariate VAR(4) that includes  and . Figure 3: Smets-Wouters Data Inflation and Modified Real Unit Labor Cost Notes: The solid line is inflation shown in Figure 1. The dashed line is the deviation of modified real unit labor cost (see equation (8) and Appendix A) from its value in 1960:I in   SW SW is the modified unit labor cost percentage points; that is, 100×  tSW  1960: I , where  measure developed by Smets and Wouters (see Appendix A).