The meta-Phillips Curve: Modelling U.S. inflation in the presence of regime change

A novel approach to modelling inflation dynamics is presented based on a set of Hybrid New-Keynesian Phillips Curves, distinguished by the regime duration and measures of real marginal cost, and combined into a meta-Phillips Curve using model averaging techniques. The analysis of US data over 1950q1 - 2016q1 shows that, while the importance of expectations of future inflation varies through time depending on the monetary policy regime and economic environment, future expectations make a more substantial contribution to current inflation than past inflation, and that the labour share is superior to the output gap as a measure of cyclical pressures on prices.


Introduction
The New-Keynesian Phillips Curve (NKPC) is a key relationship, widely adopted in macroeconomic models. Many studies that use the NKPC fail to accommodate an essential feature: instability. Given the deep structural changes the U.S. economy has gone through and the changing monetary policy, it is very likely that in ation dynamics have experienced major shifts. Besides, the Phillips Curve is an important ingredient in monetary policy analysis and, thus, it is only natural that shifts in monetary policy regimes will induce changes in the price-setting behaviour of rms.
This paper contributes to literature by providing a compelling characterisation of U.S. in ation dynamics using exible techniques that accommodate regime changes in an e ective way.
There is substantial empirical evidence that the in ation process has changed over time (inter alia, Kim, Manopimoke and Nelson (2014), Davig (2016), Zhang, Osborn and Kim (2008) and Kim and Nelson (1999)). We argue that in ation dynamics are endogenously determined by recent in ation experiences and that expectations of future in ation adopt according to the underlying in ation behaviour over the recent past. In addition, it is shown that changes in in ation dynamics coincide with important monetary policy regime changes. For instance, central banks' move towards a strong anti-in ation stance in 1980s is shown to have anchored in ation expectations and altered in ation dynamics. These shifts imply that the dominance of future expectations over past in ation in determining current in ation varies across time and might explain the puzzling controversy in literature regarding which of the backwardlooking and forward-looking term provides the greatest contribution in the Hybrid NKPC. Indeed, Gal and Gertler (1999), Gal and Gertler and Lopez-Salido (2005) and Sbordone (2002Sbordone ( , 2005 nd that in ation inertia is a much less signi cant contributor [2] to current in ation and conclude that forward-looking expectations are particularly re ective of the current state of in ation, while Fuhrer (1997) and Whelan (2005, 2006) demonstrate that the purely backward-looking Phillips Curve provides a good approximation to the dynamics of in ation.
The aim of this paper is to analyse U.S. in ation dynamics and accommodate structural instability arising from regime breaks and changes in the underlying drivers of price-setting decisions in a exible way. The analysis follows the approach of Lee, Morley and Shields (2015) who suggest combining models that are estimated over di erent sample periods using model averaging techniques. The paper constructs a meta-Phillips Curve, which involves estimation of a set of speci c NKPCs, estimated over di erent sample periods, combined using Model Averaging techniques.
The weights employed in combining individual Phillips Curves to obtain the \meta-Phillips Curve" are determined according to the ability of the individual Phillips Curves to explain past in ation behaviour. The fact that weights change over time provides a useful and exible structure with which we can interpret the changing in ation dynamics. The analysis shows that, despite the considerable structural instability observed, the meta-Phillips Curve provides a useful vehicle with which to explain in ation dynamics, and supports the view that forward-looking expectations play a key role in in ation determination, although the dominance of the forwardlooking term varies according to the prevailing economic environment and monetary policy in place. As we shall see, the estimated meta-Phillips Curve provides a coherent characterisation of in ation dynamics in the U.S. over the last fty years, often matching regime changes in monetary policy and central banks' reactions to economic situations.
The remainder of this paper is as follows: Section 2 discusses how the regime uncertainty embedded in in ation dynamics is accommodated through the use of [3] model averaging techniques and describes the modelling framework we use, focusing on the construction of weights. Section 3 presents the results of the estimation of the U.S. meta-Phillips Curve over the period 1959q4 2016q1, emphasising the phases of in ation dynamics in which expectations were more or less anchored, where antiin ationary policies were pursued more or less aggressively and when responses to the real economic activity became more or less acute. Section 4 concludes.
2 Modelling In ation in the Presence of Structural Change 2.1 Price-setting behaviour and its evolution over time The basic building block of our approach is based on the seminal hybrid NKPC model laid out in Gal and Getler (1999): and where t denotes in ation, E t f t+1 g represents in ation expectations conditional on the information up to time t, and x t is a proxy for the marginal cost (as a deviation from the steady-state). The derivation of the hybrid Phillips Curve asserts that the coe cients ; f and b are functions of structural model parameters: , which measures the degree of price stickiness; !, which re ects the fraction of backward-looking price setters; and , the discount factor.
The structural parameters underlying the Hybrid NKPC capture propensities of the rm that relate to their pricing behaviour but which are likely to vary over time for [4] at least three reasons: First, the commitment of central banks to maintain price stability and its strong anti-in ation stance can substantially in uence the price-setting behaviour of rms. 1 As Mishkin (2007)  Second, rms' price setting behaviour will adopt in light of recent experiences of in ation. For example, lower and more stable in ation leads to less frequent price adjustments and higher , with rms inclined to leave prices xed for longer periods of time (see for example Ball, Mankiw and Romer (1988) and Mishkin (2007)). Equally, more persistent recent in ation means past in ation contains more information that is relevant for rms' pricing decisions and in ation will be more backward-looking, with higher !, in this case (see, for example, Taylor (2000)). The increase in ! due to an increase in persistence can therefore lead to a decrease in f ; an increase in b and a fall in : Third, changes in the extent to which rms \pass through" changes in costs to prices (often known as the \pricing power" of rms) will be re ected in the deep 1 For instance, Volcker-Greenspan's adoption of a proactive stance towards managing in ation has led to a greater control over in ation expectations (see Erceg and Levin (2003) and Taylor (2000)). [5] parameters underlying the Hybrid NKPC. For example, as trade barriers decline, the increase in global competition dampens the ability of rms to increase prices so that the proportion of rms that leave prices unchanged i.e. increases resulting in an increase in f ; and a fall in b and : Indeed, an apparent attening of Phillips Curves in a number of countries in recent years is attributed to the globalisation process and the reduction of pricing power of rms (Ihrig et al. (2007) and Melick and Galati (2006)).

The meta-Phillips Curve and Model Averaging
Against this backdrop, an analysis of in ation dynamics should accommodate the possibility of structural instability arising from changes in policy regime and shifts in economic conditions, especially when data span a long period. Although many studies employ formal break-detection tests (e.g. Zhang, Osborn and Kim (2008)), we argue that unless there is a clear-cut and abrupt break in price-setting behaviour, there will be uncertainty on the time span over which a given Phillips Curve describes in ation dynamics. Here we follow the approach of Lee et al. (2015) who designed a novel and exible technique which combines di erent Taylor Rule speci cations using model averaging techniques. Speci cally, regime uncertainty can be accommodated in a \meta-Phillips Curve," constructed as a weighted average of a set of hybrid NKPC models, M jT ; each distinguished according to the sample period for which the model is relevant. The set of models characterising in ation dynamics over the period T 1 ; :::; T n is given by: where x t is the output gap as the measure of marginal cost 2 and 2 Amongst others, Gal and Gertler (1999), Gal et al. (2001) and Sbordone (2002) suggest that the [6] j = j min ; :::; j max ; t = T 1 j + 1; :::; T 1 and T = T 1 ; :::; T n : The models are distinguished by the time span over which a given Phillips Curve is assumed to hold, considered here to be in operation for j periods ending in period T .
When there is a regime break, a new regime starts afresh so that in principle j min = 1.
In practice, however, we might use a minimum sample size of 16 observations (j min = 16) so that we have enough observations for estimation purposes. The maximum period for the survival of an unchanged in ation behaviour is theoretically unlimited, although changes in monetary policy regimes might suggest that, in practice, a given in ationary regime would not last longer than ten years, i.e. using quarterly data j max = 40: 3 With these parameters, there are 40 16 + 1 = 25 models that explain data at each point in time; i.e. there are 25 candidate Phillips Curve models that di er according to their relevant sample size. The rst set of 25 models are estimated using data from period T 1 j max + 1 and ending in period T 1 . Further sets of 25 models are estimated as we roll through the sample to T 2 , allowing for considerable exibility in characterising regime change. The estimated parameters in an individual Hybrid NKPC, M jT , are denoted by^ f jt ;^ bjt and^ jt : The models M jT can be brought together in a \meta" model using methods based on Model Averaging techniques, so that analysis is not conditioned on a single model. labour share is a better measure of the marginal cost, documenting that this measure incorporates both productivity and wage pressures to in uence in ation. However, Rudd and Whelan (2007) and Neiss and Nelson (2005) condemn using the labour share which is countercyclical while basic economic theory suggests real marginal cost should be procyclical (see Mazumder (2010) for more evidence). Instead the use of output gap is encouraged. 3 In the U.S. there have been six Federal Reserve Chairs since the mid-sixties so that, even in the absence of any other information, one might anticipate that there would be breaks every six or seven years. [7] Speci cally, the considerable structural uncertainty surrounding in ation dynamics is re ected by the idea that in ation observed at time t could be explained by any of the 25 di erent models according to (2.2) if we set j min = 16 and j max = 40: The meta-Phillips Curve accommodates regime uncertainty by using a weighted average of the model parameters in (2.2). Denoting the vector of parameters in the hybrid Phillips Curve (equation (2.2)) at time t as: our aim is to compute the average of the posterior probability of the parameters of interest i.e. t under each model weighted by the corresponding posterior model probabilities. The approach is motivated by the Bayesian Model Averaging formula, taken from Draper (1995) and Hoeting et. al. (1999), and given by: Pr ( (M ), which is assumed to be true and making inferences that are based on stochastic and parameter uncertainties. Pr( jt jM ; Z t ) are computed on the basis of the individual models' Generalised Methods of Moments (GMM) estimates.

The Model Weights
Model weights are constructed according to: In practice, we can choose model weights so that they evolve over time, recursively updating them to re ect the extent to which they remain useful. A model's weight, minimum sample size of 16 observations. That is, if a break in the PC occurs. (2.5) If in ation is explained by a previously estimated NKPC, i.e. there is no break, the model just gets bigger by one additional observation while updating the weights on the di erent models recursively from one period to the next to re ect the likelihood that the models remain relevant. Thus, the transition probability is equal to 1 . If a new NKPC now explains in ation dynamics, such that a new in ation regime is \born", then the transition probability is equal to .
Taken together, (2.4) and (2.5) dictate the models' weights in each period. The models' weights for the rst set i.e. the rst period are assumed to be equal across all models.
This approach can capture the e ect of complicated structural changes that are hard to disentangle using conventional one-o structural break methods. The fact that model weights evolve over time allows for considerable exibility in the way changes can take place. In particular, the approach can accommodate periods in which the responsiveness of in ation to the di erent factors changes both gradually from one state to another and abruptly. in their online supplement on http://www.aeaweb.org/jel/app/mar14 Mav doc.zip. 5 As part of a robustness check, the standard quarterly HP smoothing parameter of 1600, has also been used delivering identical results. It should be noted that the results remain robust to alternative de nitions of the labour share gap such as the deviation of labour share from its mean [11] is the change in the (log) of business sector hourly compensation. Survey forecasts have been used to address the weak instruments problem (see next sub-section) and inlcude the mean growth rate of CPI from the Livingston Survey (1950q1 1981q4) and implied in ation forecasts based on the mean CPI from the Survey of Professional

The weak instruments problem
It is now widely acknowledged that there is a profound weak instruments problem associated with GMM estimation of the hybrid NKPC in ation model (see for example Nason and Smith (2008) and Mavroeidis (2005)). The problem is exacerbated, and estimates become even more unreliable, when the NKPC is at and in ation is driven only by cost-push shocks. If such shocks are unpredictable, no relevant predetermined instruments exist and the coe cient on expected in ation becomes unidenti ed. Mavroeidis et al. (2014) show that one type of speci cation that is better identi ed uses observable in ation forecasts as proxies for in ation expectations. 7 Following the survey approach, the mathematical expectation of in ation, E t f t+1 g in (2.1) is replaced by direct measures of expectations, e t+1jt 1 which denote the one-step-ahead as suggested by Gal and Gertler (1999 [12] survey forecast of in ation formed at time t 1. 8 9 Mavroeidis et al. (2014) suggest using e t+1jt 1 instead of e t+1jt ; which re ects the one-step ahead in ation expectation formed at time t, since e t+1jt 1 is certainly predetermined and not measured within the quarter. The fact that surveys forecasts contain information about the future beyond the information incorporated in most recent data makes them ideal proxies of the private sector's in ation expectations. 10 Following the suggestions of Zhang et al. (2009) and Mavroeidis et al. (2014), survey data are treated as endogenous. The instrument set consists of predetermined variables and includes four lags of in ation, two lags of survey forecasts, two lags of the labour share, two lags of the output gap and two lags of the wage in ation. 11 Here a parsimonious instrument set is used to avoid the potential estimation bias arising in small samples when there are too many over-identifying restrictions (Staiger and 8 Since this paper uses quarterly data, e t+1jt 1 ; captures the expectation of in ation one quarter ahead, as re ected in Livingston survey (until 1981q4) and SPF (post 1981q4).
9 This paper uses short-term one-quarter ahead in ation forecasts but recent papers also consider long-term forecast horizons which re ect a central bank's in ation goals (see for example Fuhrer (2011)). The SPF data on expected in ation over the next ten years become available in 1991. As part of a robustness check, the meta technique discussed in this paper was conducted with post-1991 data, using long run in ation expectations and the results remained unchanged. 10 Inter alia, Faust and Wright (2013) and Ang, Bekaert, and Wei (2007) demonstrate that SPF in ation forecasts exhibit superior forecasting performance than model-based forecasts, suggesting that surveys re ect information that is useful for the joint data generating process of realised and anticipated in ation (Mertens and Nason (2015)). 11 To gauge the extent to which the meta approach is sensitive to the instrument set, analysis was re-conducted using larger and smaller instrument sets. This robustness check has shown that the approach is invariant to the exact variables that enter the instrument set, with the analysis resulting in the same inferences about the existence and location of in ation regimes. [13] Stock (1987)). Heteroskedasticity and Autocorrelation Consistent (HAC) Newy-West type standard errors are computed with lag truncation parameter equal to 2.

Results
Our characterisation of U.S. in ation is based on our estimated meta-Phillips Curve, obtained as a weighted average of the various models described in (2.2), using the U.S. [15] above the coe cient on lagged in ation. As we will later see, with few exceptions, the weighted average coe cient on the forward-looking term was larger than the one on the backward-looking term, suggesting that even when structural breaks are taken into consideration, expectations are particularly important in determining current in ation. Figure 2e shows the evolution of the p-value from the test that the average coecient on the forward looking term is equal to the average coe cient on the backward looking term, against the alternative that the rst is larger than the latter. The statistic is based on the di erence between the forward and backward looking parameters in each of the 25 models at each point it time. Treating this di erence as independent observations of a variable, this statistic gives an indication of the size and statistical signi cance of the di erence between the average coe cients on the forward and backward looking terms. Having 25 models at time T , the test considers the null hypothesis that the mean of these 25 di erences is signi cantly di erent to zero using a standard t test, assuming the variable is normally distributed. Figure 2e reveals that, with few exceptions, the test rejects the null hypothesis and concludes that, on average, the coe cient on the forward-looking term is signi cantly larger than that on the backward looking terms at 1% signi cance level. The only two periods where the average coe cient on the forward looking term is not signi cantly bigger than that on the backward looking term are 1959q4 1961q4 and 1969q4 1980q4. As it is shown later on, the second period coincides with the high in ation episodes that de-anchored in ation expectations.
In contrast to Russell et al. (2010) who show that once structural breaks have been addressed, expectations in the Hybrid NKPC become insigni cant, we nd that expectations play a dominant role in in ation dynamics, albeit at di erent degrees, depending on the prevailing economic conditions and the monetary policy in place. [16] Second, the coe cient on the forcing variable is positive, indicating that the e ect of marginal cost on in ation is important. To validate the statistical signi cance of this result, a t-test was performed to test the null hypothesis that the mean of the weighted-average coe cients on the forcing term across the sample is equal to zero against a two-sided alternative. The test statistic is considerably higher than the critical value from a t-distribution with 225 degrees of freedom at the 5% signi cance level, and therefore the null hypothesis is rejected. This result is in line with the priori theory that predicts that the slope coe cient on the real economic activity measure should be positive and signi cant (Gal and Gertler (1999, p.207)).
Third, the large standard deviations demonstrate the considerable structural instability embedded in in ation dynamics, highlighting the need to accommodate structural breaks in the Phillips Curve relationship.
In the event, of course, the sample average coe cients do not convey the full detail of the meta model and the time variation in the parameters of gures 2b-2d.
On the contrary, the patterns in the time-varying coe cients of the meta-Phillips Curve can be interestingly be explored by running a simple OLS regression of the time-varying bt on factors that are thought to a ect its level. As outlined in section 2.1, in ation dynamics are likely to be a ected by the monetary policy in place and recent in ation experiences. Accordingly, we regress bt on lagged in ation, t 1 ; the variance of in ation rates over the past 12 quarters, v t ; and dummies re ecting di erent exogenous monetary policy regimes as distinguished by Lee et al. (2015) who use the meta-technique on the Taylor rule relationship. The following regression was therefore estimated over 1959q4 2016q1: where D i t is an indicator function corresponding to the i th monetary policy regime as [17] identi ed by Lee et al. (2015). [18] was signi cantly lower during Volcker's big disin ation monetary regime compared to Martin's regime. The results emphasise that the monetary policy in place a ects the size of the Phillips Curve's parameters to a great extent.

The eleven in ation regimes
Continuity in in ation regimes is characterised by a rising average sample size in gure 2a. On the contrary, a sharp decline in the average sample size is a signal that in ation dynamics changed at that time. Figure 2a suggests that in ation dynamics can be usefully grouped into eleven regimes as listed below. Associated in ation experiences and summary coe cients are presented in table 2.
Phase 1: Bretton Woods I This was a period of low in ation, attributed to the stability established under the xed exchange rate system. The monetary policy mechanism in place was automatic: Signs of overheated aggregate demand that threatened to accelerate in ation and undermine the country's competitiveness were promptly addressed by triggering a strong tightening policy. The Fed's commitment to maintain price stability reinforced its credibility and anchored in ation expectations. The fact that any shock had only transient e ects meant that rms were more capable to predict the future prospects of in ation, so that the fraction of the backward-looking rms, !, was small. As shown in gure 2b, f t exhibits a rising path, suggesting that the rst in ation regime could be described by a purely forward-looking NKPC. The weighted average coe cient on the forcing variable over this period is shown to have steadily decreased towards zero as shown in gure 2d.
Phase 2: Bretton Woods II In the second in ation regime, in ation doubled and became much more volatile as conveyed by table 2. Bordo and Eichengreen (2013) emphasise that in 1963, there has been an important perceptual shift in the assumed responsibilities of the Fed that considered itself free to pursue goals other [19] than dollar stabilisation, undermining the importance of controlling in ation. E ectively, policymakers placed high importance on stabilising the real economic activity and paid much less attention to price stability, unmooring in ation expectations (Orphanides and Williams (2012)). The loss of Federal Reserve Bank's credibility meant that rms became less forward-looking, re ected by the drop in f t from 0:994 in the early quarters of the second regime to 0:472 by the end of the regime. Table 2 shows that average t remained stable just above zero.
Phase 3: The Great In ation The third regime, (1970q1 1973q4); was marked by unusual economic turmoil. The collapse of the Bretton Woods system plagued in ation expectations while the rst oil price shock of the seventies brought in ation to unprecedentedly high levels. The Great In ation forced rms into more frequent price adjustments, causing to drop signi cantly. This change in the deep parameter can explain why the coe cient on the forward-looking term in the hybrid NKPC, f t , exhibited a downward path, while the weighted average coe cient on the forcing variable, t , increased. Table 4.2 validates the argument since average f t almost halved compared to the previous regime.

Phase 4: The Energy Crisis
The incidence of the second oil price shock marked the fourth in ation regime.
In ation rate reached double digit values forcing rms into more frequent price adjustments. The high in ation episodes can therefore explain why f t reached its minimum value just above zero over this in ation regime. A purely backward-looking Phillips Curve provides a good approximation to in ation dynamics in this regime. This is in accordance with Zhang et al. (2008) who found that forward-looking behaviour played a very small role during the volatile in ation period 1968 1981. The substantial decrease in the fraction of rms that left their prices unchanged is also re ected by the rising weighted average coe cient on the forcing varaible, t , that increased [20] from zero in 1974q4 to 0:16 in 1982q1.
Phase 5 Another important change is the well-documented attening of the Phillips Curve, as seen by the steady drop in the forcing variable coe cient, t . On average, the weighted average coe cient on the forcing variable has experienced a noticeable drop since the early eighties (with the exception of the peak in 2008q4) but the decline has become even more pronounced after 2009. Over this in ation regime, the weighted average coe cient on the forcing variable declines steadily and reaches negative values, although con dence bands include positive values. Blanchard and Gal (2007) attribute the attening of the Phillips Curve to globalisation and the reduction in the pass-through of oil prices to prices charged to consumers. The decrease in rms' pricing power and global competition results in more rms leaving their prices unchanged ( increases). Other authors, like Roberts (2006) and Borio and Filardo (2007) and [22] Musso et al.(2009) indentify that the recent attening of the Phillips Curve can also be attributed to the monetary policy in place.
Phase 11: Slow recovery The meta-approach has identi ed that the last ve quarters in the sample consitute a separate in ation regime (see gure 2a). Although the duration of this last regime is very small, a modest increase in f t and t can be observed, re ecting some early signs of recovery after the Great Recession. The meta-Phillips shows that there are periods where the size of the forward-looking term may become bigger or smaller depending on the monetary policies in place.
Nevertheless, the forward-looking term remains dominant throughout the sample as [23] its coe cient exceeds that of the backward-looking term, validating a number of studies that suggest that expectations are important drivers of current in ation.