Cost-based Phillips Curve forecasts of inflation

Journal of Macroeconomics 33 (2011) 553–567

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Journal of Macroeconomics journal homepage: www.elsevier.com/locate/jmacro

Cost-based Phillips Curve forecasts of inflation Sandeep Mazumder Wake Forest University, Department of Economics, Carswell Hall, Box 7505, Winston-Salem, NC 27109, United States

a r t i c l e

i n f o

Article history: Received 28 April 2010 Accepted 8 April 2011 Available online 19 May 2011 JEL classification: C53 E31 E37 Keywords: Inflation forecasting Phillips Curve Marginal cost

a b s t r a c t It is a well-established idea that prices are a function of marginal cost, yet estimating a reliable measure of marginal cost is difficult to do. Stock and Watson (1999) use the Phillips Curve to forecast inflation for a variety of existing activity variables that researchers commonly use to proxy for marginal cost. This paper uses a similar type of approach to examine the performance of a new candidate for the activity variable, which is marginal cost measured following the theoretical methodology of Bils (1987), which we find to be simple yet powerful when implemented empirically. We then use the Phillips Curve to conduct pseudo out-of-sample inflation forecasts for the US using: output, unemployment, hours, the labor share, the capacity utilization rate, and the new measure of marginal cost. For almost all cases, forecast errors are lowest in the regressions with the new marginal cost variable, indicating that this new measure is an improvement over previous attempts to proxy for marginal cost. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction The ability to forecast inflation is of central importance for macroeconomists, who argue that there are several potential variables that determine the future path of prices. From microeconomic theory, we know that ultimately it is marginal cost that ought to drive the price level that is set by firms. This relationship between prices and marginal cost is one of the most fundamental results established in economics, which means that it is important to take the role of marginal cost seriously when it comes to understanding the behavior of prices. That being said, this paper explicitly argues that marginal cost is an excellent predictor of inflation when it comes to forecasting future price activity. Currently the workhorse of the profession when it comes to investigating inflation dynamics is the New Keynesian Phillips Curve (NKPC), a model which states that there is a structural relationship between current inflation, current marginal cost and future expected inflation. This is an important model in the inflation literature, since it is one that is derived using optimizing models with price rigidities and monopolistic competition. Empirically this model has also been shown to fit inflation data well if marginal cost is proxied by the labor income share, as shown in Gali and Gertler (1999). Since this finding, estimating labor share-NKPCs has become standard practice in the literature. However, a growing number of researchers are coming to realize that there are serious empirical flaws that exist when estimating the NKPC. For instance authors such as Rudd and Whelan (2005) have strongly argued that the labor share version of the new-Keynesian Phillips Curve is a very poor model of price inflation. Moreover, Mazumder (2010) emphasizes that the labor share really only proxies for real marginal cost under very specific and rare conditions, while Gwin and VanHoose (2008) argue that the NKPC is not very robust to alternative measures of inflation. These papers are examples of an increasing literature that casts serious doubts on the empirical validity of the NKPC.

E-mail address: [email protected] 0164-0704/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2011.04.004

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Fortunately when it comes to forecasting inflation, recent research has instead suggested that the traditional backwardlooking Phillips Curve produces promising results. For instance, Stock and Watson (1999, 2008) compute year-ahead forecasts of inflation by using the conventional unemployment Phillips Curve. This model is estimated recursively in order to compute ‘pseudo’ out-of-sample forecasts for inflation using data that is available to the forecaster. The performance of the unemployment Phillips Curve is then compared to a variety of different specifications, where many alternatives measures of the ‘activity variable’ are tested. There are several measures of economic activity that are used to proxy for marginal cost, since under certain assumptions we can expect these variables to move in the same way. For instance, the output gap and unemployment rate are widelyused activity variables that in essence are being used as proxies since we have no direct way in which to observe firms’ marginal cost. However, Gali and Gertler (1999) make a convincing argument that the labor income share ought to be used as the proxy for marginal cost when it comes to models of inflation. Indeed, this proxy of marginal cost has become the leading variable that is used in many recent papers (such as Sbordone (2002) and Gali et al. (2005)). However, Mazumder (2010) argues that the labor share of income only corresponds to the true concept of marginal cost under quite specific conditions, that are not likely to hold in reality. Specifically the labor share proxy assumes that labor can be costlessly adjusted at a fixed wage rate. However, closer examination of this idea reveals that labor – which is the product of employment and average hours – has adjustment costs to changing employment and varying wage rates that are paid to increasing hours. Therefore we cannot reasonably assume that labor input can flexibly varied at a fixed real wage rate. Fortunately we can improve upon prevailing techniques by estimating marginal cost using a methodology first proposed by Bils (1987). In particular, one can estimate real marginal cost by examining the cost of increasing output along any one input, while holding the other inputs fixed at their optimal levels. This paper takes the theoretical framework of Bils (1987) and innovates upon the empirical implementation of the theory, focusing on the manufacturing sector due to the overtime data that is required to estimate marginal cost in this setup. Therefore we can obtain what we think may be a more reasonable measure of marginal cost for the manufacturing industry, which can now be compared to the other conventional proxies for marginal cost. The metric in which this paper compares the different marginal cost measures is by computing traditional backward-looking single predictor Phillips Curve forecasts of inflation. The term ‘Phillips Curve’ is used in a variety of different ways by the profession, and for the purpose of this paper, we use a definition of the Phillips Curve similar to that used in Stock and Watson (1999, 2007, 2008). That is, the Phillips Curve is an equation which states that inflation is determined by an activity variable which proxies for marginal cost, and past lagged values of inflation, which in effect are used to substitute for inflation expectations. We then use the Phillips Curve to compute pseudo out-of-sample forecasts of US inflation, with activity variables of output, the unemployment rate, hours, the labor share of income (which is equivalent to real unit labor costs), the capacity utilization rate (which Stock and Watson (1999) identify as the best individual measure available) and the new measure of marginal cost. In addition, to check how the inflation forecasts obtained from the Phillips Curve compare to other simple benchmark models, we also compute forecasts using a univariate autoregressive model, an ARMA(1,1) model, a random walk, and forecasts obtained under Atkeson and Ohanian (2001)’s ‘‘naive’’ projections. From the results we obtain an important finding that can aid future macroeconomic research. That is, in almost all of the Phillips Curve regressions that we undertake, we obtain the lowest forecast errors in the specifications with the new measure of marginal cost as the regressor. This indicates that the simple yet powerful Bils (1987) methodology of constructing better measures of marginal cost actually helps us forecast inflation with greater accuracy. Future research can use this finding to further build upon and develop even more refined measures of marginal cost. In particular since the new variable of manufacturing’s marginal cost seems to be a better predictor of inflation than output, unemployment, hours, the labor share, or capacity utilization, this implies that an aggregate version of the new marginal cost measure could be extremely useful for forecasting economy-wide inflation, which is something that future work needs to address.

2. Phillips Curve forecasts of inflation 2.1. The new Keynesian Phillips Curve debunked Prior to Gali and Gertler (1999), researchers who estimated the NKPC with rational expectations typically proxied for real marginal cost using the output gap. However, the use of this cost proxy produces counter-intuitive negative and significant coefficients in the model, which led Gali and Gertler (1999) to search for a more appropriate proxy for real marginal cost. They argue that the labor share of income fulfills this role, and show that the labor share-NKPC performs well when looking at the US inflation dynamics from the 1960s onwards. Indeed, this seminal paper has made the labor share-NKPC standard practice among NKPC advocates, as can be seen in subsequent work such as Sbordone (2005) and Gali et al. (2005). The notion of including marginal cost in the Phillips Curve has sound microfoundations, and the intentions of searching for an adequate cost proxy should be commended. Unfortunately, many researchers strongly argue against the labor share as a measure of marginal cost. For instance, Rudd and Whelan (2005) find that the NKPC performs quite poorly when the labor share is used to proxy for marginal cost, and even go so far as to say that the labor share should not be considered in any type of monetary policy rule. Likewise Rudd and Whelan (2007) suggest that the labor share ought not to be a proxy for real marginal cost since it tends to be countercyclical, whereas from theory we would expect real marginal cost to be procyclical. In

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particular, during times of economic expansion firms are likely to be raising production, and since certain factors of production are fixed, the short-run marginal cost curve must be strictly upward-sloping. This is also exactly what Mazumder (2010) emphasizes when he estimates the NKPC with a procyclical measure of marginal cost. He argues that the NKPC simply cannot produce correct coefficient signs when a procyclical cost proxy is implemented, and that NKPC results that seem promising rely on using a countercyclical marginal cost measure. Moreover, others have also estimated the NKPC and found evidence that continues to indicate that the labor share-NKPC is a very fragile empirical model of inflation dynamics. For instance, Gwin and VanHoose (2008) find that estimates of the NKPC are highly sensitive to the choices of marginal cost and to the measure of inflation used. In terms of the econometric implementation of the NKPC, Fuhrer and Olivei (2004) also argue that the way in which the majority of the literature estimates the model (generalized methods of moments) is subject to the problem of weak instruments. Moreover correcting for the presence of weak instruments produces estimates of the discount factor that are significantly different from one. For example Fuhrer and Olivei (2004) estimate a hybrid NKPC from 1966 to 2001 with their ‘‘optimal instruments GMM’’ procedure resulting in a discount factor of 0.45 when the labor share is used as the proxy for marginal cost. Similarly, Mazumder (2010) finds a discount factor of 0.71 when estimating the purely forward looking labor share-based NKPC from 1960 to 2007 using Fuhrer and Olivei (2004)’s exact same methodology. In both cases, the discount factor on future expected inflation is significantly different from one, which in itself can be taken as evidence against the NKPC. We find the growing evidence against the use of the NKPC to be quite alarming, particularly since much of the literature continues to put a great deal of faith in this model. However, we instead argue in this paper that a simple backward-looking Phillips Curve actually produces far more promising results than the NKPC. 2.2. The backward-looking Phillips Curve According to Fuhrer (1995), the traditional backward-looking Phillips Curve is ‘‘alive and well.’’ Indeed the literature that focuses on forecasting inflation also agrees with this assessment. For example Stockton and Glassman (1987) and Stock and Watson (1999, 2008) argue that, for the most part, the Phillips Curve outperforms other multivariate models in terms of forecasts of inflation.1 This paper therefore focuses on estimating the backwards-looking Phillips Curve, which we do in a similar way to Stock and Watson (1999) (who in turn use a slightly modified version of Gordon (1982)). This Phillips Curve model states that inflation, pt, depends on an activity variable and past inflation:

pt ¼ l þ aðLÞpt1 þ bðLÞzt1 þ et

ð1Þ

where l is a constant term, a(L) and b(L) are lag polynomials written with a lag operator, zt is the activity variable (or marginal cost proxy) that drives inflation,2 and et is the error term. It is common to also incorporate a vector of supply shock variables into Phillips Curve equations such as (1), but doing so will make it harder to interpret the results that are due to the activity variable itself. Supply shocks will capture some aspects of inflation behavior that the marginal cost proxy does not capture, which then makes it harder to compare the performance of the different marginal cost measures. In other words, adding supply shocks to (1) is akin to having a multiple predictor Phillips Curve, whereas this paper is primarily interested in single predictor regressions to determine which marginal cost variable is best for forecasting future price changes. Thus we estimate (1) without supply shocks, which are also likely to be collinear with marginal cost. The degrees of a(L) and b(L) are chosen separately by AIC, with a maximum lag length of 4. In addition, the specification imposes a unit root in the autoregressive dynamics for pt. With regards to Phillips Curve forecasts of inflation, this paper uses (1) to compute h-step ahead forecasts, similar to Stock and Watson (2008). The key difference, however, is the addition of a new measure of marginal cost acting as a determinant of inflation. 2.3. Comparison of the Phillips Curve to simple benchmark models To check whether the Phillips Curve truly is a useful way to forecast inflation, we also compare the performance of this model when it comes to forecasting inflation with that of four other simple benchmark models. First, we compare the Phillips Curve forecasts against a simple univariate autoregressive model, which eliminates the need for an activity variable altogether. Second, we follow Ang et al. (2007) who argue that a basic ARMA(1,1) model is a good predictor of inflation. To do this, we follow the exact same ARMA(1,1) specification as Ang et al. (2007):

ptþ1 ¼ l þ /pt þ wt þ tþ1

ð2Þ

Intuitively, this ARMA(1,1) model states that inflation can be thought of as the sum of expected inflation and noise, and if expected inflation follows an AR(1) process, then inflation can be represented by this ARMA(1,1) model. Third, we also model inflation as a random walk where:

1 Even advocates of the NKPC concede that the backward-looking component of inflation dynamics is much too large to ignore (such as Kleibergen and Mavroeidis (2009)). 2 Which has been transformed to be I(0). This assumes that pt, and zt1 are not cointegrated, which is theoretically and empirically plausible (see Stock and Watson (1999)).

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ptþ1 ¼ pt þ tþ1

ð3Þ

Fourth, we also implement the Atkeson and Ohanian (2001) naive forecast of inflation, which simply states that inflation over the next year will be the same as it has been over the most recent four quarters. In their paper, Atkeson and Ohanian (2001) claim that this naive forecast of inflation often outperforms Phillips Curve forecasts of inflation, which essentially tells us that the current unemployment rate provides no useful information concerning future inflation. By comparing our Phillips Curve forecasts to all of these simple benchmark models, we can assess just how useful it is to have a marginal cost measure in the model when forecasting future inflation. Thereafter, out of all of the competing cost proxies, we can assess which one is the best predictor of future price changes.

3. Measures of marginal cost Arguably the variable which has most often been used to proxy for marginal cost is output, usually expressed using a ‘gap’ transformation. Under certain assumptions there is an approximate proportionate relationship between marginal cost and output.3 Theoretically we think that movements in the output gap will behave similarly to marginal cost. For instance, when production is lowered during recessions the cost of production at the margin falls, and we expect output to fall below potential. However one of the main problems of using output as a predictor of prices, is that deriving an estimate of the output gap is in itself a contentious issue with a large prevailing literature, since it is unclear how ‘potential’ or the ‘natural level’ of output should be estimated. As one example, Orphanides and van Norden (2005) compute different measures of the output gap, and argue that different estimates of the output gap have very different abilities to predict inflation. Most importantly, however, is that even if we could accurately measure potential output, the conditions under which it corresponds to marginal cost may not be satisfied. The traditional Phillips Curve states that inflation is negatively related to unemployment, which has therefore been widely-used to compute forecasts of inflation. Indeed, it is easy to think of unemployment as another approximation to marginal cost since Okun’s Law posits an empirical relationship between unemployment and output, and as just discussed we think output may approximate marginal cost. While unemployment may be viewed as a likely candidate for an activity variable, it is unclear whether unemployment adjusts quickly enough to approximate marginal cost which we believe changes over a much shorter horizon. Indeed, Stock and Watson (1999) argue that other real aggregate activity variables frequently perform better than the unemployment rate when forecasting inflation. Stock and Watson (2008) also make the argument that in periods where the Phillips Curve forecasts reasonably well, activity variables other than unemployment tend to do better.4 Stock and Watson (1999) also argue that the one individual activity variable that does seem to dominate the forecasts produced by unemployment rates, is the manufacturing capacity utilization rate. Since capacity utilization rates measure the proportion of an economy’s capacity that is actively being used in production, this in turn serves as a proxy for marginal cost. In particular, a high capacity utilization rate is indicative of a strong economy, which is also a time where we might expect marginal cost to be rising since factors of production are put into greater use. Likewise, low capacity utilization rates would be associated with declining marginal costs. Thus if we are able to accurately measure the rate of capacity utilization, it clearly also provides us with a reasonable proxy for marginal cost. An activity variable that also may adjust in a timely fashion to changes in production is average weekly hours of production. As such, one can make a case that changes in hours may be a decent approximation to changes in marginal cost, and hence could be included as an activity variable in Phillips Curve regressions. In fact, as we will discuss in just a moment, varying hours is an ideal way in which we can measure marginal cost. Therefore we shall include employees’ hours of work as an activity variable, although this measure might not fully capture marginal cost as it does not account for changes in employment and wages. One could also argue that the closest approximation to marginal cost that is used in macroeconomics is the labor income share, as proposed by Gali and Gertler (1999). Ang et al. (2007) find that among the Phillips Curve forecasts of inflation, the best results are obtained when the activity variable is the labor income share: ‘‘the recent Phillips Curve literature stresses that marginal cost measures provide a better characterization of inflation dynamics than detrended output measures. Our results suggest that the use of (the labor share as a proxy for) marginal cost leads to better out-of-sample predictive power.’’ However there are two major reasons as to why the labor income share is not well-suited to proxy for real marginal cost: first, the labor share is countercyclical whereas we expect that marginal cost is likely to be procyclical, and second the labor share proxy for marginal cost makes the unreasonable assumption that labor can be freely adjusted at a fixed real wage rate, as Mazumder (2010) explains in detail. More specifically, labor input can be thought of as employment multiplied by the number of hours that they work, and many economists have clearly pointed out that employment has adjustment costs (such as Oi (1962)), and that wages can vary in response to a change in hours. Therefore it is not reasonable to assume that labor can be flexibly varied at a fixed real wage rate. Fortunately, this is an assumption that we can improve upon in a simple and straightforward way. 3

Such as a standard sticky price framework without variable capital. The authors attribute this to the fact that in periods when the economy is quiet – that is, the unemployment rate is close to the NAIRU – unemployment does not do well in terms of forecasting inflation. 4

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4. The new measure of marginal cost 4.1. Estimation methodology This paper implements the idea set forth by Bils (1987) in order to estimate real marginal cost in a way that acknowledges the existence of adjustment costs to labor. While a few other papers recognize the need to consider adjustment costs when it comes to the labor income share, such as Muto (2009), these papers fail to recognize that the labor income share proxy of marginal cost can actually be theoretically nested into a more general expression of marginal cost. This paper is able to do this, however, by following the framework of Bils (1987). In particular this methodology states that we can measure marginal cost by examining the cost of changing output along any one input while holding all other inputs fixed at their optimal levels. Although other approaches that try to improve upon the measurement of marginal cost also exist – see Rotemberg and Woodford (1999) for a selection of these – this paper chooses to focus on one particular generalization that is simple and easy to implement. Yet this method is powerful in the sense that it allows us to derive a new estimate of marginal cost that accounts for adjustment costs of labor. Given that L = NH, where L is labor input, N is employment, and H is average weekly hours worked by employees, we have two margins along which we could measure marginal cost. Choosing N as our margin requires modeling the adjustment costs associated with the hiring and firing of workers. Therefore we choose to vary hours of work due to the absence of adjustment costs. Without time subscripts for the moment for easier notation, we can then express Bils’ (1987)’ idea as the following:

X¼

   dCosts @Costs @H ¼ jY  ; H ; N dY @H @Y

ð4Þ

where X is real marginal cost, Y is output, and ‘⁄’ terms denote optimal levels. To compute what (4) looks then like, we need     to derive @Costs and @H . For the latter derivative, we use a standard Cobb-Douglas production function5 with the exception @H @Y @Y  that labor is decomposed into employment and hours: Y = AKa(NH)1a. This then gives us the derivative of @H ¼ ð1  aÞ HY . For the derivative of Costs with respect to H, we then need a definition of the cost function. We use the same cost function as that which produces the labor share proxy of marginal cost, except that labor is decomposed into employment and hours, and we also recognize the fact that wages must be a function of hours.6 Therefore we can write the average hourly real wage rate as x(H), giving:

Costs ¼ xðHÞNH

ð5Þ

From (5) we can then compute the derivative with respect to hours as: ¼ N½xðHÞ þ x ðHÞH. Finally we can substitute our expressions for the two derivatives into (4) to get nominal marginal cost as: @Costs @H

X¼

  1 NH ½xðHÞ þ x0 ðHÞH 1a Y

0

ð6Þ

It is the presence of x0 (H) that makes the marginal cost measure expression in (6) different from previous estimates of marginal cost. Indeed if we were to set this term equal to zero, we would then get a measure of marginal cost equivalent to real unit labor costs. In other words, the labor income share can be shown to be theoretically embedded in the more general expression of marginal cost given in Eq. (6). Thus having x0 (H) – 0 means that unit labor costs are not an accurate measure of real marginal cost. From (6), everything can be simply obtained from the data except for x0 (H), which requires some sort of functional form. x(H) is estimated in a similar fashion to Mazumder (2010), which assumes that hourly wages can have a time invariant component to them as well as a component that pays overtime wages for overtime hours worked. Specifically we can then  H þ px  V, where x  is the straight-time component of the wage rate, and p is write the total real weekly pay per worker as: x the overtime premium paid on top of the straight-time wage for V overtime hours per worker. Therefore the average hourly  ½1 þ pmðHÞ, where m(H) = V/H is the ratio of overtime hours to wage rate is the total weekly pay divided by hours: xðHÞ ¼ x average hours per worker, which is clearly dependent on the number of hours worked. Using this functional form for x(H), we can now compute x0 (H) which simplifies (6) to:

X¼

  1 NH  ½1 þ pðmðHÞ þ Hm0 ðHÞÞ x 1a Y

ð7Þ

where the problem of estimating marginal cost has been further reduced in the sense that all terms are just constants or data, with the exception of m0 (H) which remains to be estimated. 5 Note that the assumption of Cobb-Douglas production is not crucial here. A more general production function of the form Y = Haf, where f represents all factors other than hours produces the exact same expression for real marginal cost as in this paper. Nonetheless, we use Cobb-Douglas production function since it is used to derive the labor share proxy for marginal cost (see Gali and Gertler (1999)). 6 There is a large labor economics literature about writing the wage as a function of hours. For an example, see Heckman (1974) for an explicit derivation of x as a function of H.

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4.2. Manufacturing data To be able to estimate (7), we require data on overtime hours and overtime premia that is paid for working extra hours than specified in a worker’s contract. For the United States, reliable overtime data are only available for the manufacturing industry, and rather than approximating for overtime hours for non-manufacturing sectors, this paper chooses to focus on the manufacturing industry. Indeed, the idea of using manufacturing variables as proxies for aggregate variable is not uncommon in the inflation literature. For example, Roberts (1997) estimates an aggregate NKPC using the manufacturing capacity utilization rate as the proxy for real marginal cost. In fact, Stock and Watson (1999) use the very same manufacturing variable when computing Phillips Curve forecasts of aggregate inflation, which again indicates that the use of disaggregated proxies for aggregate marginal cost is not unusual. Moreover, the test of relevance for marginal cost proxies in this paper lies in the relative forecasting performance which is presented in Section 6. Furthermore, attempting to approximate for non-manufacturing overtime hours is problematic once we try to determine what constitutes overtime for salaried workers as opposed to those earning an hourly wage rate. Fortunately, the manufacturing sector lends itself well when it comes to the application of hourly wages. In particular, this industry has frequent variations in hours, with workers receiving a straight-time hourly wage as well as an overtime premium for overtime hours. The data itself are taken from the Bureau of Labor Statistics and the Bureau of Economic Analysis, and are quarterly over the time period of 1960:1–2007:3. The BLS data reports overtime hours and reports wages7 with and without overtime earnings. This allows us to derive the overtime premium, p, which is defined by the Code of Federal Regulations as something that is paid when: (i) hours worked in a single day or workweek are more than required by the employee’s contract or (ii) overtime premia can be paid on ‘special’ workdays, such as weekends or holidays. p is thus obtained from the BLS earnings data, where we control for the share of overtime hours in total hours, which leaves an effective overtime premium which is close to 50 percent. 4.3. The m(H) function The theoretical expression in (7) is equivalent to that of Bils (1987). However, the empirical implementation of the model that we use is, however, different. In his paper, Bils regresses the difference of overtime hours on (H  40) and various powers of this term. There are a few problems of this method. First, there is no theoretical reason to use the change in overtime hours since we are able to estimate m0 (H) by first estimating m(H), which we demonstrate below. Second, we transform the dependent variable into percentage terms (ratio of overtime hours to average hours) which helps remove some of the trend that may appear in overtime hours across time, whereas Bils estimates the overtime hours function purely in levels. Third, Bils imposes the assumption that the usual number of hours worked per week is 40. While this may be a reasonable approximation, there is no theoretical necessity to make this assumption. Fourth, the Bils (1987) methodology assumes that the overtime function must be cubic without having any criterion to select the degree of the equation chosen. And finally, the aforementioned method uses annual data, which ignores the changes in overtime hours that is often observed within each year. These are all limitations that this paper accounts for, which leaves us with a method of computing (7) that is markedly simpler than that used in Bils (1987). To proceed, the m(H) function is then measured by taking manufacturing data from the BLS for V and H, and then plotting m against H. m is then regressed on H, and various powers of H using OLS to determine the line of best fit.8 The model which produced the best fit (as judged by the highest R2 ) is then selected as being the model which best describes the data. It turns out that this is given by the quadratic specification:

mðHÞ ¼ a þ bH þ cH2

ð8Þ

We must also note that a virtually identical marginal cost series results from the linear and cubic m(H) specifications as well. Indeed, the results of this paper are robust to the linear and cubic specifications of m(H), indicating that it is the fact that m0 (H) – 0 which is of primary importance. Results of the linear, quadratic, and cubic regressions can be seen in Table 1, and the scatter-plot of the m and H data with the line of best fit can be seen in Fig. 1. Lastly, we can use the coefficient estimates of b and c in (8) to compute a series for m0 using:

m0 ðHÞ ¼ b þ 2cH

ð9Þ

4.4. New marginal cost series Finally we are left with all of the components of (7) that are required to estimate real marginal cost, which can be seen in Fig. 2. This figure9 shows that the new measure of marginal cost moves noticeably with the business cycle: it falls sharply during each recession, and also reacts very quickly after the recession has ended by rising immediately. In other words marginal cost is markedly procyclical (see Mazumder (2010) for further evidence of the cyclicality of marginal cost), which contrasts 7 8 9

Nominal wages are then converted to real wages by deflating with aggregate prices. Where m and H are stationary variables according to augmented Dickey–Fuller unit root tests. This figure is drawn for estimates of (8) that are made for the entire sample period.

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S. Mazumder / Journal of Macroeconomics 33 (2011) 553–567 Table 1 m(H) Regressions. a

b

c

d

– (–)

(–) (–)

m(H) = a + bH Coefficient Standard error

0.9760 (.02650)

0.0264 (.0006) 0.8917

R2

m(H) = a + bH + cH2 Coefficient Standard error

8.3964 (.9497)

0.0057 (.0006)

0.4362 (.0469)

– (–)

0.9228

R2 2

m(H) = a + bH + cH + dH

3

Coefficient Standard error

30.8854 (4.4739)

0.5540 (0.0818)

2.2670 (0.3313)

0.0045 (0.0007)

0.9059

R2

Note: Estimation is OLS with HAC robust standard errors. All coefficients are statistically significant at the 1% level.

Fig. 1. m(H) = a + bH + cH2.

sharply with the labor income share which is countercyclical. And most importantly, we now have a proxy for marginal cost which behaves in a way that we might a priori expect it to.10 5. Forecasts This paper computes pseudo out-of-sample forecasts, since we are primarily interested in how inflation is forecasted in real-time, due to the practical implications that it entails. Pseudo out-of-sample forecasts simulate the experience of the forecaster in real time by determining lag length and conducting estimation in each period of the increasing window using data11 through date t, making a h-step ahead forecast12 for date t + h. Then we move forward to period t + 1, and repeat the exercise – this is then continued for the entire sample. Thus the forecasting regression based on (1) is:

phtþh ¼ l þ

p X k¼0

10

ak ptk1 þ

q X

bk ztk1 þ etþh

ð10Þ

k¼0

Assuming that demand shocks dominate the movement of marginal cost. That is, all data published at the time of writing, including data revisions since revisions to transformed marginal cost measures used in the forecasting equations are trivially small. 12 This paper looks at four-quarter ahead inflation, one of the most commonly used forecast horizons. 11

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Fig. 2. Manufacturing marginal cost.

where lag length is determined by AIC with a maximum of 4. In addition, a unit root is imposed on the lags of inflation. For example, for the four-quarter forecast of inflation from 1989:1 to 1990:1, the model is estimated, information criteria computed and lag lengths determined using data through 1989:1, and the forecast of inflation for 1990:1 is then made. We then move forward one period and re-estimate the model and information criteria using data through 1989:2, and a forecast from 1989:2 to 1990:2 is computed. Repeating this throughout the entire sample generates a sequence of pseudo out-of-sample forecasts. The critical aspect in these pseudo out-of-sample forecasts, is that all lag length selection and estimation is conducted only using data available prior to making the forecast. We therefore consider recursive forecasts, where forecasts are made in time period t. We assume that all data through t are used by the forecaster to make their projections of annual future inflation from t to t + 4, where the windows for estimation are lengthened through time. The time period 1950:1–1959:4 is used for initial parameter estimation,13 and our forecast period is 1960:1–2007:3. We also split the sample into two periods, which effectively means that two starting dates are selected for our out-of-sample forecasts: 1960:1 and 1984:1. We focus on recursive forecasts in this paper, assuming that we always start with the same initial observation of the sample in question, which enables us to capture model specification uncertainty, model instability, and estimation uncertainty. To evaluate the performance of inflation forecasts, we compute root mean squared errors (RMSE):

RMSEt1 ;t2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Xt 2  1 ¼ phtþh  phtþhjt t¼t 1 1  t2  t1

ð11Þ

where phtþhjt is the forecast of phtþh , made using data through date t, and our goal is to minimize RMSE to boost forecasting accuracy. For the purpose of this paper, we are not entirely interested in the absolute size of the RMSE, but its performance relative to forecasts made with unemployment – which is exactly the same measure used by Stock and Watson (1999).14 Hence, we report relative RMSEs, which are the size of a forecast’s RMSE relative to that achieved by using unemployment, for the same time period, data transformation, and measure of inflation. For instance, in Table 2, the very first entry in the upper left corner reports the RMSE achieved by using lags of inflation and the growth rate of output, for 1960:1–1983:4, relative to the RMSE achieved by using unemployment for the identical time period, transformation, and measure of inflation. Therefore a relative RMSE of 1 means that the forecast errors are identical to those when unemployment is used. When the relative RMSE is greater than 1, unemployment is performing better as the activity variable in terms of minimizing forecast errors, and similarly when the relative RMSE is less than 1, the competing marginal cost proxy outperforms unemployment. 13

We proxy for manufacturing weekly hours from 1950 to 1955 with the good-producing industry’s weekly hours of production workers. A test of significance of forecasts would be ideal, but is complicated by the fact the model specifications are updated in each point in time. This is something that future research may wish to address. 14

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S. Mazumder / Journal of Macroeconomics 33 (2011) 553–567 Table 2 CPI-all aggregate-level pseudo out-of-sample relative RMSEs, R2 . Forecast period

1960:1–1983:4

Single predictor Agg. Agg. Agg. Agg. Man. Man. Man. Man. Man.

gdp unemp hoursa ls gdp hours ls cub mc

Trans.

Rel. RMSE

Dln Dln Dln Dln Dln Dln Dln Dln Dln

Univariate AR ARMA(1,1) Random walk Atkeson and Ohanian (2001) Agg. Agg. Agg. Agg. Man. Man. Man. Man. Man.

gdp unemp hoursa ls gdp hours ls cub mc

Univariate AR ARMA(1,1) Random walk Atkeson and Ohanian (2001)

gap gap gap gap gap gap gap Dln gap

1984:1–2007:3

1960:1–2007:3

R

Rel. RMSE

Rel. RMSE

R

R2

0.9630 1 1.0368 0.9883 0.9855 1.0009 0.9815 1.0693 0.9228

0.7681 0.7501 0.6884 0.7534 0.7581 0.7450 0.7586 0.6404 0.7751

1.0046 1 1.0058 0.9893 1.0041 0.9874 0.9780 0.9223 0.9731

0.1402 0.1500 0.1400 0.1724 0.1431 0.1711 0.1869 0.1871 0.1771

0.9841 1 1.0152 0.9854 0.9967 0.9933 0.9765 0.9628 0.9522

0.6887 0.6778 0.6632 0.6859 0.6802 0.6818 0.6925 0.6942 0.7003

0.9988 0.9381 0.9961 1.0019

0.7521 0.7692 0.7555 0.7438

1.0061 1.0089 0.9769 0.9812

0.1489 0.1472 0.1708 0.1612

1.0000 0.9704 0.9859 0.9929

0.6795 0.6915 0.6856 0.6787

0.9836 1 1.0178 0.9949 0.9592 0.9917 0.9960 1.1465 0.9303

0.7628 0.7542 0.7088 0.7564 0.7832 0.7570 0.7455 0.6276 0.7812

0.9980 1 0.9995 0.9852 0.9804 0.9824 0.9693 0.9701 0.9678

0.1439 0.1404 0.1413 0.1657 0.1443 0.1614 0.1739 0.2169 0.1792

0.9887 1 1.0090 0.9882 0.9739 0.9857 0.9743 0.9799 0.9562

0.6881 0.6807 0.6714 0.6880 0.6982 0.6892 0.6928 0.6949 0.7044

1.0005 0.9642 1.0041 0.9910

0.7521 0.7689 0.7473 0.7503

0.9908 0.9912 0.9814 1.0060

0.1489 0.1415 0.1498 0.1382

0.9958 0.9801 0.9864 0.9947

0.6795 0.6965 0.6943 0.6823

2

2

Notes: relative RMSEs computed from forecasts, whereas R2 is from backward-looking regressions where data are known. The benchmark model from which the relative RMSEs are computed is the Phillips Curve with aggregate unemployment. For comparison purposes, the benchmark model produces root mean square forecast errors (RMSFE) of: 1.08612, 1.01625, and 1.01760 for the gap transformation for 1960:1–1983:4, 1984:1–2007:3, and 1960:1–2007:3 respectively. The absolute magnitudes of the RMSFE are comparable to these magnitudes for all tables in this paper. a Aggregate Hours data starts in 1964:1. b Manufacturing Capacity Utilization data starts in 1972:1.

6. Estimation and results 6.1. Data and estimation Prices for the whole economy are used, and the activity variables are output, unemployment, hours, labor share, capacity utilization, and the new measure of marginal cost.15 Although we argue in Section 2.1 against the use of the labor share as a marginal cost proxy, we include it in our forecasts so as to compare the performance of this proxy to competing marginal cost measures. Both aggregate and manufacturing-level activity variables are used wherever it is possible to do so. Before estimation is conducted, the activity variables are transformed in a similar manner to Stock and Watson (2003) in two ways to deal with trending or persistent variables: by ‘gaps’ and log differences.16 The gap variables are a deviation from a stochastic trend (or one can think of them as deviations from their ‘natural levels’), which are computed using a one-sided HP filter. The HP filter that is applied is one-sided, since the forecaster cannot use known future data to compute the trend of a series in the current period, which is in keeping with our out-of-sample framework.17 In addition to the gap variable, we also use the log difference (Dln) transformation, which compares the growth rates of the activity variables. See Appendix A for further details on the transformations and data employed in this paper. Quarterly data are used for the United States, from 1960:1 to 2007:3, taken from the BEA and the BLS. Monthly data are converted to quarters by taking averages for values over three months in the quarter prior to any other transformations. For instance, quarterly CPI (consumer price index) is the average of the three CPI monthly values, and quarterly CPI inflation is then the percentage growth, at an annual rate, of quarterly of CPI. That is, pt is the quarterly rate of inflation at an annual rate, 15 Data for aggregate hours and manufacturing capacity utilization rates start in 1964 and 1972 respectively. Thus RMSEs for these dates are compared to the RMSE obtained from a forecast using unemployment over the same time period. 16 We also perform unit root tests on all marginal cost variables (ADF test with maximum lag length of 4, selected by AIC), and find that all cost proxies are I(1) which prevents us from inputting these variables in level form in the model. 17 The new marginal cost measure is applied such that only data available in the period when the forecast is made, is used in that period’s forecast (where the polynomial coefficients in (8) are re-estimated each period using only data that is available at the date the forecast is made).

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S. Mazumder / Journal of Macroeconomics 33 (2011) 553–567

pt = 400ln(Pt/Pt1), where Pt is the price index. Therefore our h-period ahead forecast, computed at the one-year horizon in this paper, is then four-quarter inflation: p4t ¼ 100 lnðPt =Pt4 Þ. We consider three measures of price inflation: the CPI for all items (CPI-all), the GDP deflator, and the PCE deflator for all items (PCE-all). Forecasts were also performed for core CPI (excluding food and energy) and for core PCE, but the conclusions are almost identical to the CPI and PCE results when all items are included. Hence these core price inflation results are not presented in this paper, since we do not learn anything new with regards to the relative performance of the marginal cost proxies. In addition to computing forecast errors for the entire sample of 1960:1–2007:3, we also split the sample in order to examine changes in the forecasting performance of the Phillips Curve over time, particularly to investigate how the Phillips Curve performs in the periods of high inflation in the 1970s and early 1980s. Finally, we also compare the Phillips Curve forecasts against other simple benchmark models: a univariate autoregressive model, an ARMA(1,1) model, a random walk, and the Atkeson and Ohanian (2001) naive forecast as described in Section 2.3. The ARMA(1,1) and random walk models are estimated in an identical way to Ang et al. (2007),18 and the Atkeson and Ohanian (2001) year-ahead forecast of inflation is computed as 100[(Pt1/Pt5)  1]. 6.2. Phillips Curve forecast results First consider the forecasting results for the Phillips Curve model for CPI-all inflation, in Table 2, which also reports the R2 that is achieved when looking at the three time periods from a backward-looking perspective (i.e. when the data are known). For the entire sample range of 1960:1–2007:3 forecast errors are lowest for the model using only the new measure of manufacturing marginal cost. For the ‘gap’ transformation, the RMSEs for aggregate- and manufacturing-level output, unemployment, hours, labor share, and capacity utilization are 2–5% higher than that achieved with marginal cost. Similar numbers are obtained under the Dln transformation as well.19 Therefore we can see that in terms of forecasting inflation in the Phillips Curve, the new measure of marginal cost improves upon existing proxies of marginal cost, even on top of capacity utilization and the labor share – both of which are commonly used proxies in the literature. For the subperiod 1960:1–1983:4, the model with manufacturing marginal cost clearly performs best. RMSEs are between 3% and 23% higher with the competing marginal cost proxies than that achieved by the new measure of marginal cost. In addition, marginal cost also does well in the second subperiod, 1984:1–2007:3, although to a lesser degree than with the first subperiod. Nonetheless this new proxy for marginal cost still outperforms the frequently used labor income share (both the aggregate and manufacturing versions). We also see that the R2 drops significantly from the first subperiod to the second, providing evidence that there is time variation in the inflation process and in Phillips Curve performance. This is consistent with what other authors have found, such as Stockton and Glassman (1987) and Stock and Watson (2008), who argue that the Phillips Curve forecasts inflation best in the 1970s and early 1980s.20 From the relative RMSEs alone, it is clear that the new marginal cost measure outperforms all of the other marginal cost proxies in terms of being able to predict prices. This is in stark contrast with papers such as Staiger et al. (1997), who argue that the unemployment rate is only dominated by the rate of capacity utilization in the manufacturing industry, in terms of using single predictors to forecast inflation. The results in Table 2 provide evidence to the contrary, and in particular the new procyclical measure of marginal cost ranks highest amongst the list of marginal cost proxies. This gives us an indication that the Bils (1987) theory combined with the empirical innovations outlined in this paper may be a fruitful avenue for future macroeconomic researchers to pursue, at least when it concerns models of inflation. 6.3. Comparison of results to other benchmark models Although the new marginal cost proxy outperforms the competing variables in Phillips Curve forecasts of inflation, this result truly only becomes valuable for inflation forecasters if we compare how the model does to other simple benchmark forecasts of inflation. Table 2 also contains the results that contrast the Phillips Curve forecasts with those of the univariate autoregressive model, the ARMA(1,1) model, a random walk, and Atkeson and Ohanian (2001)’s naive forecast. We see that Phillips Curve forecasts of inflation with unemployment perform similarly to the univariate autoregressive model, suggesting that unemployment does not add much to the Phillips Curve equation. However, the new measure of manufacturing marginal cost consistently performs better than the univariate model, a result which suggests that the Phillips Curve can indeed be useful for forecasting inflation. Furthermore, this Phillips Curve model produces a lower RMSE for the entire sample than that obtained with the ARMA(1,1) or random walk model. For example, the Phillips Curve with the new marginal cost measure produces forecast errors that are over 4% lower than those obtained with the unemployment Phillips Curve. However the ARMA(1,1) and random walk forecast only reduce forecast errors by half as much (approximately 2%). 18

Estimation is maximum likelihood, conditional on a zero initial residual. Hereafter we shall only refer to gap variable results unless otherwise noted, since the results are similar whether we consider the gap or log difference transformations. 20 Indeed the size of relative RMSEs reported in this paper are consistent with the range that Stock and Watson (2008) report when comparing forecasts to a unobserved components-stochastic volatility model benchmark, in the range of 0.74–1.75. 19

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S. Mazumder / Journal of Macroeconomics 33 (2011) 553–567 Table 3 GDP-all aggregate-level pseudo out-of-sample relative RMSEs, R2 . Forecast period

1960:1–1983:4

Single predictor Agg. Agg. Agg. Agg. Man. Man. Man. Man. Man.

gdp unemp hoursa ls gdp hours ls cub mc

Trans.

Rel. RMSE

Dln Dln Dln Dln Dln Dln Dln Dln Dln

Univariate AR ARMA(1,1) Random walk Atkeson and Ohanian (2001) Agg. Agg. Agg. Agg. Man. Man. Man. Man. Man.

gdp unemp hoursa ls gdp hours ls cub mc

gap gap gap gap gap gap gap gap gap

Univariate AR ARMA(1,1) Random walk Atkeson and Ohanian (2001)

1984:1–2007:3

1960:1–2007:3

R

Rel. RMSE

Rel. RMSE

R

R2

1.0388 1 1.1155 1.0109 1.0378 1.0395 1.0108 1.1587 0.9801

0.7943 0.8158 0.7960 0.8021 0.7967 0.7994 0.8075 0.5574 0.8164

1.0302 1 1.0299 1.0226 1.0330 1.0335 1.0316 1.0311 1.0178

0.3681 0.4286 0.3685 0.3775 0.3647 0.3641 0.3867 0.3923 0.4029

1.0273 1 1.0564 1.0088 1.0280 1.0287 1.0104 0.9937 0.9813

0.7914 0.8060 0.7759 0.8016 0.7941 0.7941 0.8018 0.8083 0.8159

1.0934 0.9994 1.0558 1.0889

0.7932 0.8228 0.8003 0.7992

1.0294 0.9945 1.0501 1.0218

0.3710 0.4391 0.4138 0.4201

1.0551 0.9893 1.0378 1.0085

0.7947 0.8211 0.7888 0.8002

1.0543 1 1.1261 1.0378 1.0532 1.0315 1.0353 1.2108 1.0003

0.7951 0.8276 0.7904 0.8091 0.8031 0.8175 0.8093 0.5260 0.8150

0.9764 1 0.9895 0.9881 0.9672 0.9271 0.9847 0.9490 0.9826

0.3909 0.3445 0.3745 0.3789 0.4024 0.4509 0.3781 0.4118 0.3832

1.0439 1 1.0573 1.0181 1.0359 1.0127 1.0189 1.0060 0.9905

0.7938 0.8108 0.7826 0.8031 0.7966 0.8054 0.8033 0.8100 0.8121

1.1134 1.0515 1.0920 1.1905

0.7932 0.8016 0.7911 0.6441

0.9647 0.9866 0.9525 0.9680

0.3710 0.3821 0.4087 0.4008

1.0311 1.0084 1.0292 1.0185

0.7947 0.8088 0.8093 0.8024

2

2

Table 4 PCE-all aggregate-level pseudo out-of-sample relative RMSEs, R2 . Forecast period

1960:1–1983:4

Single predictor Agg. Agg. Agg. Agg. Man. Man. Man. Man. Man.

gdp unemp hoursa ls gdp hours ls cub mc

gdp unemp hoursa ls gdp hours ls cub mc

Univariate AR ARMA(1,1) Random walk Atkeson and Ohanian (2001)

1960:1–2007:3

Rel. RMSE

R2

Rel. RMSE

R2

Rel. RMSE

R2

Dln Dln Dln Dln Dln Dln Dln Dln Dln

1.0214 1 1.0814 1.0131 1.0032 0.9983 0.9882 1.2292 0.9478

0.8110 0.8215 0.7734 0.8093 0.8216 0.8241 0.8122 0.6877 0.8298

1.0086 1 1.0105 0.9957 0.9855 0.9735 1.0001 0.9663 0.9941

0.2970 0.3090 0.2944 0.3149 0.3140 0.3307 0.3052 0.3368 0.3092

1.0162 1 1.0480 1.0027 1.0107 1.0195 0.9888 0.9887 0.9761

0.7759 0.7844 0.7649 0.7783 0.7821 0.7781 0.7832 0.7855 0.8036

1.0338 1.0258 0.9618 0.9970

0.8063 0.8041 0.8255 0.8211

1.0105 1.0181 0.9954 0.9765

0.3020 0.3077 0.3101 0.3286

1.0245 0.9716 1.0151 0.9728

0.7720 0.8052 0.7796 0.7924

1.0247 1 1.0753 0.9806 1.0161 1.0079 0.9945 1.1916 0.9448

0.8076 0.8192 0.8202 0.8177 0.8094 0.8135 0.8090 0.6705 0.8404

1.0036 1 1.0018 1.0028 1.0057 1.0041 0.9997 0.9665 0.9913

0.3001 0.3050 0.3025 0.3011 0.2792 0.2994 0.3059 0.3289 0.3096

1.0189 1 1.0474 0.9912 1.0164 1.0081 0.9897 0.9892 0.9757

0.7741 0.7836 0.7567 0.7822 0.7743 0.7784 0.7824 0.7834 0.7960

1.0302 1.0027 1.0537 1.0263

0.8063 0.8133 0.7940 0.7998

1.0076 0.9922 1.0164 0.9859

0.3020 0.3073 0.2894 0.3156

1.0241 0.9771 1.0015 1.0196

0.7720 0.7921 0.7798 0.7706

Univariate AR ARMA(1,1) Random walk Atkeson and Ohanian (2001) Agg. Agg. Agg. Agg. Man. Man. Man. Man. Man.

1984:1–2007:3

Trans.

gap gap gap gap gap gap gap gap gap

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S. Mazumder / Journal of Macroeconomics 33 (2011) 553–567

Moreover, we see that the Atkeson and Ohanian (2001) naive forecast of inflation barely improves upon the unemployment Phillips Curve inflation forecast. In fact, several authors in the literature also similarly find Atkeson and Ohanian (2001)’s results to be highly non-robust. In particular, Sims (2002) argues that Atkeson and Ohanian (2001)’s results arise entirely due to the sample period chosen. Other papers, such as Fisher et al. (2002), Lansing (2002) and Mehra (2004), also point out that Atkeson and Ohanian’s results are dependent on the time period selected. We also find this to be true in all of the results in this paper: the naive forecast typically does better in the 1984:1–2007:3 period and much less well in the 1960:1–1983:4 time period. Indeed, Faust and Wright (2007) argue that the Atkeson and Ohanian (2001) result is not reliable in the first place as it is imprecisely estimated. Our results based on the Phillips Curve with the new marginal cost measure provide plenty of evidence that the traditional backward-looking Phillips Curve is useful for forecasting inflation, at least in certain time periods. And since we think the Phillips Curve tries to capture the relationship between changes in prices and marginal cost, it is then important to determine which activity variable is the best approximation to marginal cost.

6.4. Robustness The conclusions described above are similar if we employ GDP deflator (Table 3) or PCE-all prices (Table 4). For GDP deflator and PCE-all prices, output, unemployment, hours, and the labor share produce forecast errors in the region of 1– 5% higher than that achieved by the improved marginal cost measure for the entire sample period and 3–26% in the first subperiod, using both aggregate and manufacturing variables. In addition, observance of the R2 s points out a similar pattern to before, where the Phillips Curve forecasts inflation a great deal better in the 1970s and early 1980s than it does thereafter. In summary, the aggregate Phillips Curve results provide evidence that the new procyclical measure of marginal cost is the best predictor of prices in terms of the RMSEs of the forecasts. This can also be seen if we repeat the above exercise for manufacturing’s sectoral rate of inflation, which can be seen in Appendix B. The results here suggest that manufacturing marginal cost outperforms the competing activity variables by an even larger amount at the disaggregated level. Since this suggests that inflation can be forecasted well when the new generalized expression of marginal cost is applied to the manufacturing sector, future research could benefit by finding a way to apply this methodology to aggregate data.

7. Conclusion Many proxies for marginal cost have been used in macroeconomics, and this paper looks at what are arguably the most commonly-used ones: output, unemployment, and hours. In addition, Gali and Gertler (1999) make a strong argument that we can directly estimate marginal cost by using the labor income share. However, as Mazumder (2010) argues in explicit detail, the labor share is a poor proxy for marginal cost since it is countercyclical whereas it is likely that real marginal cost ought to be procyclical, and furthermore the labor share proxy is based on the assumption that labor can be freely adjusted at a fixed real wage rate. When we relax this assumption, account for adjustment costs to labor, and recognize that wages will be a function of the number of hours worked, we obtain a new expression for marginal cost that is based on more reasonable assumptions. This new measure of marginal cost can then be estimated with the use of manufacturing overtime data, producing a markedly procyclical series which is in stark contrast to the countercyclical labor income share. We can then compute single predictor forecasts of inflation using the Phillips Curve model with these various marginal cost proxies, in addition to the rate of capacity utilization that Stock and Watson (1999) find to work best out of all individual activity variables, to determine which variable is the best predictor of prices. This is the metric that this paper uses to determine which marginal cost measure is best, producing results that are quite striking: forecast errors are consistently lowest when the new marginal cost variable is used in the Phillips Curve. In addition, when we split the sample at the start of 1984, it is very clear that Phillips Curves based on the new marginal cost variable can account for inflation best in the 1960s, 1970s, and early 1980s. Since the new manufacturing marginal cost measure is the best predictor of inflation, this suggests that an aggregate measure of marginal cost will have important forecasting information for economy-wide inflation. Therefore future research should attempt to apply the expression for marginal cost in this paper to the aggregate level, or find a better way to estimate it from the available data. As a starting point, the basic results of this paper can be used to further investigate the use of marginal cost in inflation forecasts.

Acknowledgments I would like to thank Laurence Ball and Jon Faust for comments and suggestions on this paper. I also wish to express gratitude to an anonymous referee for valuable comments provided.

S. Mazumder / Journal of Macroeconomics 33 (2011) 553–567

565

Appendix A. Data The table below gives a summary of the data used in this paper and the transformations that are applied to the series. When the original series is monthly, data are converted into quarters by taking the average of the monthly values. This is done before any transformations are applied. Sources are the Federal Reserve Bank of St. Louis, the Bureau of Economic Analysis, and the Bureau of Labor Statistics. Short name

Transformation

Definition

Inflation series CPI-all GDP-all PCE-all

Dln Dln Dln

CPI, all items GDP deflator, aggregate PCE deflator, all items

Predictors gdp gdp unemp unemp h h ls ls cu cu mc mc

Dln gap Dln gap Dln gap Dln gap Absolute Difference gap Dln gap

Real GDP (Agg., Man.) Real GDP (Agg., Man.) Unemployment rate, total civilian 16+ (Agg.) Unemployment rate, total civilian 16+ (Agg.) Hours (average weekly) worked by production workers (Agg., Man.) Hours (average weekly) worked by production workers (Agg., Man.) Labor Share (Agg., Man.) Labor Share (Agg., Man.) Capacity Utilization (Man.) Capacity Utilization (Man.) Marginal Cost (Man.) Marginal Cost (Man.)

‘Agg.’ refers to aggregate variables, and ‘Man.’ to manufacturing sector variables. Transformations are (Xt the series used in the regression, Qt the original series):

D ln : gap :

X t ¼ ln Q t  ln Q t1 X t ¼ ln Q t  ln Q t

where ‘⁄’ is the trend. Also logs not used for unemployment gap which is already in terms of a rate. Appendix B. Manufacturing Phillips Curve results B.1. Manufacturing sector Phillips Curve In order to be able to examine the Phillips Curve for the manufacturing industry, we must adapt the model to be suitable at a disaggregated level. To do this, we rewrite the standard Phillips Curve with manufacturing variables:

pmt ¼ lm þ am ðLÞpmt1 þ bm ðLÞzmt1 þ emt

ðA:1Þ

where all the variables from the Phillips Curve are simply changed to manufacturing ones, denoted by ‘m’ superscripts. Since manufacturing prices are a function of manufacturing marginal cost, we can expect the new measure of marginal cost to have a large predictive element for the industry’s inflation rate. B.2. Data and estimation The disaggregated forecasts of inflation are conducted using two measures of inflation: GDP-man and PCE-man. GDP-man and PCE-man are both taken from the BEA data on GDP price deflators by industry, and personal consumption expenditure data categorized by major type of product. As with the aggregate Phillips Curve results, we use both aggregate and manufacturinglevel regressors where applicable. Finally, the same transformations are applied to the data as in the aggregate case. B.3. Results Examining the results for industry-level Phillips Curve real-time forecasts of manufacturing inflation for GDP-man prices (Table 5), there is convincing evidence that the new marginal cost measure is the best predictor of future price changes out of all of the competing variables. Output, unemployment, hours, the labor share, and the capacity utilization rate produce RMSEs of inflation forecasts which are substantially higher (7–24%) than that obtained by the new measure of marginal cost, when considering the entire sample period. The improvement of results by using the new measure of marginal cost are even more amplified if we consider the first subperiod of 1960:1–1983:4, where the RMSEs of competing activity variables are

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S. Mazumder / Journal of Macroeconomics 33 (2011) 553–567

Table 5 GDP-man aggregate-level pseudo out-of-sample relative RMSEs, R2 . Forecast period

1960:1–1983:4

Single predictor Agg. Agg. Agg. Agg. Man. Man. Man. Man. Man.

gdp unemp hoursa ls gdp hours ls cub mc

Rel. RMSE

R

Dln Dln Dln Dln Dln Dln Dln Dln Dln

1.0624 1 1.1427 1.0363 1.0697 1.0278 1.0266 0.9390 0.8525

Univariate AR ARMA(1,1) Random walk Atkeson and Ohanian (2001) Agg. Agg. Agg. Agg. Man. Man. Man. Man. Man.

gdp unemp hoursa ls gdp hours ls cub mc

1984:1–2007:3

Trans.

gap gap gap gap gap gap gap gap gap

Univariate AR ARMA(1,1) Random walk Atkeson and Ohanian (2001)

Rel. RMSE

0.6034 0.6330 0.6190 0.6248 0.5989 0.6111 0.6252 0.6814 0.7249

1.2489 1.0806 1.1675 0.9431

1960:1–2007:3 R

Rel. RMSE

R2

1.0568 1 1.0160 1.0475 1.0562 1.0102 1.0420 0.9794 1.0421

0.2677 0.3178 0.3109 0.2598 0.2804 0.3079 0.2522 0.3144 0.2499

1.0264 1 1.0481 1.0275 1.0309 1.0061 1.0163 0.9283 0.8490

0.4243 0.4467 0.4280 0.4237 0.4196 0.4386 0.4239 0.4812 0.5815

0.4051 0.6145 0.5991 0.6862

1.0700 1.0201 1.0444 1.0794

0.3120 0.3113 0.3088 0.2769

1.1294 1.1468 0.9798 1.1336

0.3221 0.3070 0.4631 0.3144

0.8964 1 1.0362 0.9205 0.8925 0.9035 0.8697 0.8552 0.7792

0.6490 0.6110 0.5679 0.6473 0.6544 0.6499 0.6245 0.6812 0.7044

0.9751 1 1.0300 1.0537 0.9763 1.0137 1.0403 0.9383 1.0381

0.3355 0.3208 0.2969 0.2638 0.3526 0.2966 0.2708 0.3443 0.2743

0.9524 1 1.0462 0.9558 0.9518 0.9628 0.9360 0.9085 0.8465

0.4499 0.4538 0.4373 0.4250 0.4709 0.4607 0.4284 0.4973 0.5751

1.2094 0.8864 1.0354 1.0825

0.4051 0.6200 0.5712 0.5485

1.0659 1.0489 1.0434 1.0412

0.3120 0.2510 0.2581 0.2524

1.1266 1.0106 1.0408 0.9722

0.3221 0.4465 0.4412 0.4598

2

2

10–33% higher than that of the new marginal cost variable. The results with the PCE-man can be seen in Table 6. Qualitatively, these results are extremely similar to those under the GDP-man measure of prices.

Table 6 PCE-man aggregate-level pseudo out-of-sample relative RMSEs, R2 . Forecast period

1960:1–1983:4

Single predictor Agg. Agg. Agg. Agg. Man. Man. Man. Man. Man.

gdp unemp hoursa ls gdp hours ls cub mc

gdp unemp hoursa ls gdp hours ls cub mc

Univariate AR ARMA(1,1) Random walk Atkeson and Ohanian (2001)

1960:1–2007:3

Rel. RMSE

R2

Rel. RMSE

R2

Rel. RMSE

R2

Dln Dln Dln Dln Dln Dln Dln Dln Dln

1.0050 1 1.0435 0.9978 0.9941 0.9953 0.9982 0.9347 0.8509

0.6976 0.6997 0.6482 0.7013 0.6567 0.7043 0.6979 0.7411 0.7798

0.9995 1 0.9853 0.9851 0.9046 1.0004 0.9634 0.9306 0.8991

0.2409 0.2420 0.2101 0.2572 0.3009 0.2430 0.2804 0.2938 0.3029

1.1002 1 1.0207 0.9918 1.0190 1.0004 0.9833 0.9417 0.8757

0.5688 0.5706 0.5800 0.5786 0.4939 0.5712 0.5833 0.6038 0.6229

1.0655 0.9932 0.9750 0.9980

0.6503 0.7082 0.7133 0.7031

1.0004 0.9940 0.9321 1.0320

0.2390 0.2444 0.2607 0.2388

1.0624 0.9427 0.9659 1.0295

0.4713 0.6012 0.5909 0.4825

1.0021 1 1.0715 0.9877 0.9893 0.9960 0.9905 0.9310 0.8463

0.6968 0.6968 0.6336 0.7052 0.6977 0.6982 0.7023 0.7464 0.8009

1.0003 1 0.9896 0.9978 1.0298 0.9553 0.9737 0.9476 0.8989

0.2433 0.2427 0.2468 0.2577 0.2293 0.2733 0.2675 0.2855 0.3266

1.0012 1 1.0370 0.9941 0.9978 0.9991 0.9749 0.9333 0.8744

0.5686 0.5691 0.5626 0.5757 0.5709 0.5697 0.5932 0.6119 0.6410

1.0812 0.9263 1.0248 0.9759

0.6503 0.7795 0.6871 0.7100

1.0002 1.0300 0.9701 0.9238

0.2390 0.2255 0.2623 0.2960

1.0710 0.9663 0.9994 1.0229

0.4713 0.6028 0.5710 0.5633

Univariate AR ARMA(1,1) Random walk Atkeson and Ohanian (2001) Agg. Agg. Agg. Agg. Man. Man. Man. Man. Man.

1984:1–2007:3

Trans.

gap gap gap gap gap gap gap gap gap

S. Mazumder / Journal of Macroeconomics 33 (2011) 553–567

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