State-Dependent Phillips Curve

Abstract

We propose a state-dependent Phillips curve (PC) where the regime has changed endogenously. Using this framework, a free-standing PC is constructed. This study tests the robustness of the model, various types of inflation, slack measures, and various expectation measures. The PC is found to work strongly during recessionary periods but becomes weaker once an economy recovers. The latent factors that determine the regimes are highly correlated with the uncertainty measure. During recessionary periods, the uncertainty becomes negatively more certain and strengthens the relationship between inflation and labor market slack.

1. Introduction

Inflation, a long-time topic in macroeconomics, is affected not only by real variables, such as GDP and unemployment, but also by people’s expectations. The Phillips curve (PC), suggested by Phillips (1958), is among the representative tools used to express these complicated relationships. Friedman (1968) added inflation expectation to the PC, and it has since become a cornerstone in the inflation literature. However, due to the measurement of inflation expectation and insufficient theoretical background, the PC has not always been fully supported, although it has received more recent attention in the analysis of monetary policy’s effectiveness or dynamics.

Despite huge liquidity inflows and outflows in money markets since the Global Financial Crisis, the behavior of inflation remains uncertain. In his New York Times opinion, Krugman (2022) stated that we would be experiencing strong inflation by now if the theory worked. Therefore, the puzzle regarding inflation is summarized in two ways, according to Conti (2021). First, deflation was not severe during the turmoil caused by the Global Financial Crisis. Gilchrist et al. (2017) and Coibion and Gorodnichenko (2015) highlighted the role of global and financial factors as well as the relevance of inflation expectations as reasons for missing deflation. In contrast, Yellen (2015) and Conti et al. (2015) revealed that the deanchoring of expectations and inflation was not severe after liquidity measures were implemented to cope with the turmoil. This view is referred to as “missing inflation”.

This topic has been accompanied by a more specific debate, which also precedes the low-inflation period, on the usefulness of the PC for describing and forecasting inflation. In particular, a flattening of the PC since the early 1960s has been documented (Blanchard et al., 2015), while since 2008–2009, evidence of a more recent steepening has been provided by Riggi and Venditti (2015). Such nontrivial shifts in the relationship between inflation and economic activity, coupled with disappointing inflation dynamics despite economic recovery, have raised concerns about whether the PC is “dead”.

While these authors concluded that the PC was viable, higher inflation rates have still failed to materialize despite the strong improvement in economic dynamics and labor market outlooks that characterized most of the advanced economies before the outbreak of COVID-191 (see Ball and Mazumder (2019), Hindrayanto et al. (2019), Hooper et al. (2020), Coibion and Gorodnichenko (2015), Laseen and Sanjani (2016), and Bobeica and Jrocinski (2019) for debates on “dead” inflation).

Whatever the view is called, in terms of the PC, the link between changes in US inflation and the output gap (or unemployment gap) has weakened in recent decades. Over roughly the same period, there has been a positive relationship between the level of US inflation and the output gap, which is reminiscent of the original 1958 version of the PC. As Yellen (2019) remarked, “The slope of the PC—a measure of the responsiveness of inflation to [economic] slack—has diminished very significantly since the 1960s”, resulting in numerous hypotheses about the issue. Jørgenssen and Lansing (2021) summarized these hypotheses into five factors. They are (1) structural changes in the economy that have reduced the inflationary pressure associated with gap variables; (2) the successful stabilizing effects of monetary policy responding to supply shocks that push inflation and the output gap in opposite directions, thereby creating the statistical illusion of a declining gap coefficient; (3) vigilant monetary policy that has served to anchor people’s inflation expectations and thereby pushing inflation itself to a value near 2%; (4) demographic shifts or other slow-moving factors that have contributed to the mismeasurement of the gap variable; and (5) the existence of a nonlinear relationship between inflation and the gap variable, which causes the gap coefficient to reduce in magnitude when inflation is low or less volatile. This weak relationship between the output gap and inflation can be interpreted as uncertainty. Moreover, Reis (2006) demonstrated how market conditions affecting firms facing uncertainty may lead them to “sample” information more frequently and thereby increase the responsiveness of prices and inflation to aggregate demand. Similarly, Murphy (2014) demonstrated how regional economic condition uncertainty may affect firms’ pricing behavior and the aggregate responsiveness of inflation to economic slack. This uncertainty can be explained as nonlinearity and endogeneity in our proposed model.

This study examines whether the change in coefficients can be explained endogenously (rather than by external factors) and used to develop a universal PC that can be applied in any state. Thus, an endogenous PC is proposed in which the coefficient changes with the regime change. This model is robust to the choice of inflation measure, inflation expectation, and the natural rate of unemployment. Regime change is suitable for explaining the change in the relationship between inflation and unemployment slack as there has been a dramatic change in inflation during the Oil Shock, the Great Moderation, and the Great Recession. These events prevented the accurate estimation of coefficients, and removing their effects led to less volatile outcomes. Although there must be external factors to lead to a change in the PC, here, the dynamics of inflation and unemployment slack are assumed to be explainable endogenously. The main contribution of this study is proposing a novel PC model to track the inflation slack in different periods, as we introduce a time-varying feature in the coefficient. This can help us better forecast the inflation–labor market slack with given macroeconomic shocks. We leave the theoretical discussion about the model for a future study.

The regime-switching framework of the PC was studied by Amisano and Fagan (2013), who developed a time-varying transition probability Markov-switching model in which inflation is characterized by two regimes (high and low inflation). A smoothed measure of broad money growth has important leading indicator properties for switches between inflation regimes. Thus, money growth provides an important early warning indicator of risks to price stability. Nalewaik (2016) developed a regime-switching PC that focuses on wage and core PC equilibrium (PCE) inflation. Its key innovation is the addition to the models of fundamental driving variables such as labor market slack. Additionally, the evidence strongly reveals a nonlinear effect of slack on wage growth and core PCE price inflation that becomes much larger after labor markets tighten beyond a certain point. Forbes et al. (2021) proposed a nonlinear PC, especially with low inflation. This nonlinear curve is steep when output is above potential (slack is negative) but flat when output is below potential (slack is positive). Thus, further increases in economic slack have little effect on inflation. This finding is consistent with evidence of downward nominal wage and price rigidity.

A review of the key literature that examines the PC is as follows. Ball (2014) examined the recent behavior of core inflation in the US and specified a simple PC based on the assumptions that inflation expectations are fully anchored by the Federal Reserve’s target, which is that labor market slack is captured by the level of short-term unemployment, and this equation explains inflation behavior since 2000. He also proposed a more general PC in which core inflation depends on short-term unemployment and expected inflation as measured by the Survey of Professional Forecasters (SPF). This specification fits US inflation figures since 1985. Benigno and Ricci (2008) found that the curve is virtually vertical for high inflation rates but becomes flatter as inflation declines. Additionally, macroeconomic volatility shifts the PC outward, implying that stabilization policies play an important role in shaping the tradeoff. Third, nominal wages also tend to be endogenously rigid upward at low inflation rates. Fourth, when inflation decreases, the volatility of unemployment increases, while the volatility of inflation also decreases. According to Blanchard (2018), a small coefficient implies an attractive short-run tradeoff between inflation and unemployment. In the benchmark new Keynesian model, stabilizing inflation keeps the unemployment rate at the natural rate, and the natural rate, in turn, is the “constrained efficient rate”, that is, the best rate that can be achieved through policy actions. This proposition has been called the “divine coincidence”. The residual can be interpreted in two ways. First, it captures unobserved movement at the natural rate; if this is so, it implies large, high-frequency movements at the natural rate. Second, it can be interpreted as the result of misspecification, for example, the use of incorrect inflation series or dynamic specifications.

2. Model

2.1. PC

Gordon (1990) proposed the PC as follows:

π_t = α(L)π_t−1 + β(L)D_t + γ(L)z_t + ν_t

(1)

where D_t is an index of excess demand (normalized so that D_i = 0 indicates the absence of excess demand), and z_t is a vector of supply shock variables (normalized so that z_i = 0 indicates an absence of supply shocks). Following Gordon (1990)’s “triangle model”, we consider the following PC:

π_t = µ + α(L)π_t−1 + β(L)u_t−1 + γ(L)z_t−1 + ν_t

(2)

The triangle equations estimated in this study use current and lagged values of the unemployment gap as a proxy for the excess demand parameter D_t, where the unemployment gap is defined as the difference between the actual rate of unemployment and the natural rate, and the natural rate (or NAIRU) is allowed to vary over time.

The triangle approach differs from the new Keynesian PC (NKPC) approach by including long lags of the dependent variable, additional lags of the unemployment gap, and explicit variables to represent the supply shocks. These include the effects on inflation caused by changes in the relative price of food and energy, changes in the relative price of nonfood and nonoil imports, the eight-quarter change in the trend rate of productivity growth, and dummy variables for the effects of the 1971–1974 Nixon-era price controls. Thus, the NKPC of Roberts (1995) and Gali and Gertler (1999) is as follows:

πt = Etπt + 1 + λmct

(3)

where mc_t refers to the exogenous structures affecting the PC.

Additionally, following Matheson and Stavrev (2013), Blanchard et al. (2015) proposed the specification of the PC for selected countries using a time-varying approach as follows:

$${\pi }_t={\theta }_t\left(u_t-u_t^{\ast }\right)+{\lambda }_t{E}_{t-1}\left({\pi }_t\right)+\left(1-{\lambda }_t\right){\pi }_{t-1}^{\ast }+{\mu }_t{\pi }_{im,t}+{ϵ}_t$$

(4)

where E_t−1(π_t) denotes long-term expectation; ${\pi }_{t-1}^{\ast }$ is the average of the last four quarterly inflation rates; and π_im,t is import price inflation relative to headline inflation. The present study develops the PC considering commonly used terms, along with labor market slack, expected inflation, and exogenous variables to extract the optimal specification without a macroeconomic theoretical background.

2.2. State-Dependent PC

This Section utilizes the endogenous regime-switching model developed by Chang et al. (2017) for PC analysis in the US. Endogenous regime switching is used instead of conventional Markov switching because the endogenous regime-switching model allows for the implementation of the current realization of the underlying time series. With Markov switching, the future transition between low and high states is solely determined by the current state, which is unrealistic. Furthermore, the endogenous model facilitates identifying an unobservable latent factor that characterizes the regime between states. This factor can be used for interesting economic interpretations of the dynamics of relationships between series. The transition probabilities estimated from the endogenous model are determined by the current state and the underlying time series change over time, but the probabilities of the Markov-switching model between two states are always constant. Furthermore, as demonstrated by Chang et al. (2017), the Markov-switching model is a subset of the endogenous regime-switching model. Thus, the endogenous regime-switching model can be reduced to the conventional Markov-switching model.

The following specification is used to investigate the relationship between the inflation rate and the unemployment rate by adopting the endogenous regime-switching model to provide useful information on the dynamics of the PC. This leads to the following two specifications:

• Backward-looking model

π_t = µ + β(s_t)(ur_t − nur_t) + γ(s_t)E_t(π_t−1) + σ(s_t)ε_t

(5)

• Forward-looking model

π_t = µ + β(s_t)(ur_t − nur_t) + δ(s_t)E_t(π_t+1) + σ(s_t)ε_t

(6)

where π_t indicates the quarterly inflation rate at t, and E[π_t−1] and E[π_t+1] signify a four-period average of lag and lead inflation rates, respectively. ur_t and nur_t denote the unemployment rate and the natural rate of unemployment, respectively.

In these models, the state process (s_t) is determined by the autoregressive latent factor ω_t and the threshold level τ. Specifically, the regime is switched by the state process s_t = 1{ω_t ≥ τ} where 1{.}.

The latent factor ω_t follows a first-order autoregressive process as follows:

ωt = αωt − 1 + νt

(7)

for t = 1, 2, … with parameters α ∈ (−1, 1] and independent and identically distributed (i.i.d.) standard normal innovation ν_t. A state-dependent parameter θ_t ∈ {β(s_t),γ(s_t),δ(s_t),σ(s_t)} can be described as follows:

θ_t = θ(s_t) = θ^l(1 − s_t) + θ^hs_t

(8)

The regime with θ_t = θ^l or s_t = 0(ω_t < τ) is referred to as the “low regime”, whereas the “high regime” is when θ_t = θ^h or s_t = 1(ω_t ≥ τ).

The latent factor ω_t is assumed to be correlated with the previous shock. Thus, ε_t and ν_t are jointly i.i.d. as follows:

$$\left(\begin{array}{c}{ϵ}_t\\ {\nu }_{t+1}\end{array}\right)~N\left(\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right)\right)$$

(9)

When the correlation between two components ϵ_t and ν_t+1 is not zero (ρ ≠ 0), the latent factor ω_t+1 is correlated with the observed inflation rate π_t. The current inflation rate π_t affects future transitions between states, and the latent factor ω_t determines the future states s_t+1. The zero correlation between ϵ_t and ν_t+1 implies that the potential transition between states is not affected by the current inflation rate π_t. Hence, an analysis of the results in the study by Chang et al. (2017) implies that when ρ = 0 with |α| < 0, the endogenous regime switching is general enough to include the conventional Markov-switching model. Therefore, maximum likelihood estimation using a modified Markov-switching filter is considered when estimating the parameters in the model.

3. Data and Specification

This study employs four inflation measures that are widely used in the inflation literature—headline consumer price index (CPI), PCE and core CPI, and PCE. The natural rate of unemployment was obtained from the Congressional Budget Office (CBO)’s measure. The CBO no longer officially reports the short-term measure of the natural rate of unemployment, but it has released the expected rates up to 2030. The CBO’s natural rate of unemployment is the rate of unemployment arising from all sources except fluctuations in aggregate demand.

The estimates of potential GDP are based on the long-term natural rate.2

As an inflation expectation, the present study uses the survey results of the SPF and the Livingston Survey (LS). SPF provides forecasts of CPI and PCE for up to six quarters. The LS provides CPI forecasts for only 6 and 12 months ahead. Although the LS includes forecasts of longer horizons, such as 2, 3, and 10 years ahead, it appears to have a too-long horizon, so it was not used.

Here, the price index was transformed into the four-quarter log change to avoid high-frequency noise in the monthly change. Additionally, the available samples span from 1961Q1 to 2021Q3. Labor market slack was measured by the difference between the unemployment rate and the natural rate of unemployment, which was obtained from the CBO. The natural rate is occasionally revised and may thus affect the results, especially when forecasting, as described by Nalewaik (2016).

As inflation proxies, CPI all, core CPI, PCE all, core PCE, and the natural rate of unemployment are employed as the CBO’s long-term natural, short-term natural, and LS’s natural rates. For backward-looking specifications, the average of the last four quarters of inflation was utilized. However, for the forward-looking specification, the LS for one-quarter-ahead forecasting was utilized. Following Nalewaik (2016), the proxy for exogenous shock was measured as the import price shock. Forbes et al. (2021) used exchange rates, oil prices, and global value chains to describe the cost-push shock, while the import price index was utilized in this study.

Table 1. List of variables used in the estimation of the PC.

No.	Section	Variables	Description
1	Inflation	CPI	Headline
2			Core
3		PCE	Headline
4			Core
5	Employment	Unemployment rate	Level
6		Natural Rate of Unemployment	Long-term
7			Short-term
8	Inflation Expectation	SPF	Mean of 1–6 Quarters Ahead
9		LS	Mean and Median (CPI Headline only)
10	Exogenous Var.	Import Price Index	Import Price Inflation

Sources: All series are quarterly. Inflation expectations are derived from the Philadelphia Fed, and all others are from the FRED.

presents the list of variables used to estimate the PC. Combining these, we created forty specifications with the survey of inflation expectations, and eight with the previous inflation average.

Specification

To test the sensitivity of the PC depending on model specification, six different model types were tested. Based on Equations (5) and (6), we find that the predictors of inflation are unemployment gap (captured by β), inflation expectation (captured by δ), inflation lag (captured by γ), and exogenous variables (captured by λ). The full model (Model 1) considered all the predictors in the model. Subsequently, each predictor was sequentially subtracted to indicate the robustness of the model. Thus, Model 2 excluded the lagged term of inflation; Model 3 conveyed expected inflation; Model 4 utilized the exogenous variable; Model 5 utilized the lagged term and exogenous variable; and Model 6 included only the unemployment gap and lagged terms.

4. Empirical Results

Table 2. Estimation results of forward-looking CPI.

	Model 1		Model 2		Model 3
Parameters	Estimates	St. Error	Estimates	St. Error	Estimates	St. Error
µ	0.003 **	0.001	0.005 ***	0.001	0.005 ***	0.001
βl	−0.111 ***	0.036	−0.053	0.035	−0.079 **	0.035
βu	−0.588 ***	0.214	−0.677 ***	0.115	−0.253 *	0.143
δl	0.247 ***	0.049	0.678 ***	0.043	-	-
δu	0.826 ***	0.212	1.023 ***	0.046	-	-
γu	0.654 ***	0.056	-	-	0.823 ***	0.041
γl	0.208	0.196	-	-	0.909 ***	0.046
λl	0.022	0.018	0.062 **	0.027	−0.007	0.020
λu	0.290 **	0.123	0.241 ***	0.031	0.001	0.050
σl	0.007 ***	0.000	0.006 ***	0.000	0.007 ***	0.000
σl	0.023 ***	0.003	0.019 ***	0.002	0.025 ***	0.003
α	0.978 ***	0.018	0.989 ***	0.010	0.992 ***	0.007
τ	4.852 **	2.452	2.774	6.684	6.041 ***	1.569
ρ	0.561 *	0.316	0.641 ***	0.181	−0.156	0.341
Likelihood	830.995		818.317		813.224
	Model 4		Model 5		Model 6
Parameters	Estimates	St. Error	Estimates	St. Error	Estimates	St. Error
µ	0.003 ***	0.001	0.005 ***	0.001	0.006 ***	0.001
βl	−0.073 **	0.031	−0.112 ***	0.032	−0.086 **	0.034
βu	−0.503 ***	0.095	−1.021 ***	0.156	−0.186 *	0.113
δl	0.384 ***	0.044	0.676 ***	0.060	-	-
δu	0.386 ***	0.085	1.101 ***	0.048	-	-
γu	0.420 ***	0.061	-	-	0.812 ***	0.037
γl	0.638 ***	0.067	-	-	0.897 ***	0.048
λl	-	-	-	-	-	-
λu	-	-	-	-	-	-
σl	0.006 ***	0.000	0.006 ***	0.000	0.007 ***	0.000
σl	0.020 ***	0.002	0.026 ***	0.002	0.025 ***	0.003
α	0.996 ***	0.004	0.995 ***	0.004	1.000 ***	0.000
τ	3.529 ***	0.697	5.351 ***	1.317	0.143	0.963
ρ	−0.020	0.245	0.728 ***	0.135	−0.387	0.242
Likelihood	830.930		795.537		808.467

Notes: Model 1 considers all the predictors in the model, and then one of them is sequentially subtracted to indicate the robustness of the model; Model 2 excludes the lagged term of inflation; Model 3 includes the inflation expectation; Model 4 includes the exogenous variable; Model 5 includes the lagged term and the exogenous variable; and Model 6 includes only the unemployment gap and the lagged terms. Inflation is measured by core CPI. LS represents the inflation expectation. The asterisks *, **, and *** indicate statistical significance at the 10%, 5%, and 1% levels, respectively.

and

Table 3. Estimation results of backward-looking CPI.

	Model 1		Model 2		Model 3
Parameters	Estimates	St. Error	Estimates	St. Error	Estimates	St. Error
µ	0.003 ***	0.001	0.003	0.104	0.003 **	0.001
βl	−0.109 ***	0.040	−0.140	75.181	−0.050	0.046
βu	−0.406 **	0.187	−0.140	8.206	−0.221	0.222
δl	0.615 ***	0.146	0.901	2.235	-	-
Δu	0.553 **	0.275	0.897	4.336	-	-
γu	0.286 **	0.145	-	-	0.920 ***	0.040
γl	0.416	0.278	-	-	0.891 ***	0.075
λl	−0.012	0.018	0.101	5.978	−0.012	0.017
λu	0.026	0.060	−0.013	0.681	0.020	0.069
σl	0.006 ***	0.001	0.026	0.320	0.007 ***	0.000
σl	0.023 ***	0.003	0.006	0.038	0.029 ***	0.004
A	0.984 ***	0.013	0.980	0.907	0.976 ***	0.019
τ	4.235	3.672	−3.729	95.626	4.548 ***	1.380
ρ	0.074	0.457	−0.185	11.815	0.312	0.296
Likelihood	830.710		822.861		812.693
	Model 4		Model 5		Model 6
Parameters	Estimates	St. Error	Estimates	St. Error	Estimates	St. Error
µ	0.003	0.003	0.003 ***	0.001	0.005 ***	0.001
βl	−0.068	0.094	−0.114 ***	0.040	−0.091 **	0.036
βu	−0.552 ***	0.110	−0.565 ***	0.219	−0.249	0.218
δl	0.523 **	0.230	0.899 ***	0.036	-	-
δu	0.614	0.693	1.000 ***	0.061	-	-
γu	0.395	0.310	-	-	0.820 ***	0.033
γl	0.375	0.697	-	-	0.909 ***	0.053
λl	-	-	-	-	-	-
λu	-	-	-	-	-	-
σl	0.007 ***	0.000	0.006 ***	0.000	0.007 ***	0.000
σl	0.025 ***	0.004	0.025 ***	0.003	0.025 ***	0.003
α	0.980 ***	0.020	0.979 ***	0.015	0.993 ***	0.006
τ	5.130 ***	1.725	3.661 **	1.472	6.295 ***	0.611
ρ	0.647	1.825	0.201	0.189	−0.185	0.159
Likelihood	830.163		823.691		813.264

Notes: Model 1 considers all the predictors in the model and then one of them is sequentially subtracted to indicate the robustness of the model; Model 2 excludes the lagged term of inflation; Model 3 includes inflation expectation; Model 4 includes the exogenous variable; Model 5 includes the lagged term and exogenous variable; and Model 6 includes only the unemployment gap and lagged terms. Inflation is measured by core CPI. Inflation expectation follows adaptive expectation, which is measured by the moving average of four quarter’s previous inflation rates. The asterisks *, **, and *** indicate statistical significance at the 10%, 5%, and 1% levels, respectively.

present the estimation results of the PC proposed in Equation (4). Table 2 follows forward-looking behavior, so the surveys of predicted inflation from the SPF and LS were used as inflation expectations. In contrast, Table 3 used previous inflation moving averages as the inflation expectation, so it followed adaptive expectations. Among various specifications, the results in both tables used core CPI for the inflation measure and the long-term natural rate of unemployment. The inflation expectation in the forward-looking behavior equation was per the LS (median). Other specifications had consistent and robust signs of coefficients, but the significance and the magnitude of the coefficients varied depending on the model specification and variable selected figures (the results of other specifications are available upon request). Instead of giving all possible results, we summarize them as figures later in this Section.

With forward-looking behavior, the parameters describing the relationship between inflation and employment slack (β_l and β_u) were estimated negatively. As the unemployment rate surpassed the natural rate of employment, stronger downward pressure on inflation was seen. The magnitude was significantly greater in the upper regime and refers to the strong nonlinearity in the PC. We specify the two regimes later. δ is the coefficient for the expected inflation and can be interpreted as the sensitivity for the inflation anchor. Thus, if the public has more confidence in its central bank to maintain the inflation target, δ will be greater. In the upper state, the public demonstrated significantly stronger confidence in inflation expectations. However, the difference between the two states was limited when the adaptive expectation was followed. In some cases, the coefficients were not significant, indicating that the adaptive expectation may not work properly in the empirical results. γ assesses the persistence of past inflation and was significantly stronger in the lower state in all specifications. λ represents the coefficient for the exogenous variable. Import price inflation was employed, which proxies cost-push inflation. These effects were split depending on the state but were significant in the upper state where the import price inflation was positively related to inflation. In summary, the lower state can be defined as the stable one as inflation responds less sensitively against employment slack and inflation expectation, and it becomes more persistent and less sensitive against exogenous variables. However, in the lower state, the coefficient of the unanticipated shock, σ, is greater than in the upper state. This means that if there is any unexpected shock, the economy may react more strongly than in the other state. Thus, without exogenous variables or lags of inflation, δ is robustly estimated.

Using backward-looking behavior in which the average of the previous four quarterly inflation rates was utilized as the inflation expectation, most of the parameters revealed a similar result as in the former case. Thus, β_l was estimated significantly in three out of six specifications, although it still demonstrated less sensitivity to employment slack. This implies that backward-looking behavior provides quite a different perspective for policymakers. Additionally, λ’s were estimated as insignificant in both models. It appears that past inflation may reflect the changes in exogenous variables but this requires further investigation.

As we tested the various proxies of inflation, expectations, and external shocks, Table 2 and Table 3 present only 1 of 40 specifications and cannot represent all the results. Instead of reporting all specifications, we report the density of β in various specifications. Figure 1 depicts the density of β_l and β_u. In Figure 1a, with forward-looking inflation, β_l has a bimodal distribution. It has a very narrow range and is close to zero compared with the case of β_u. This means that inflation rarely reacts to labor market slack. Thus, the PC may appear dead, such as during the Great Moderation. However, β_u, as depicted in Figure 1b, is skewed to the right, but it is still negative. This relationship can be stronger in some cases, but the possibility is low. From a backward-looking perspective, the result is quite similar to the case of forward-looking inflation expectations, but the strength of the PC with it is smaller than that in the previous case.

In summary, regardless of which specification was used, the robustness results reveal hard evidence of nonlinearity in the PC, especially in the upper case. The next question is as follows: “What does this upper state represent?” To identify this, we must examine the latent factors that determine the state of the model.

As depicted in Equation (7), the latent factors follow the AR(1) process and reveal high persistence (Table 2 and Table 3). Figure 2 and Figure 3 depict the latent factor, s_t, which changes the coefficient regime in our PC specification. The gray area in the figure represents The National Bureau of Economic Research’s recession periods. The two figures correspond to the estimation results in Table 2 and Table 3.

Latent factors were found to be usually in the upper state when the economy is in recession, and this supports our interpretation of the estimation results. However, in the recessions of the 21st century, excluding the one caused by the pandemic, the state variable did not react as much as in the 20th century, especially with backward-looking behavior. From the 1960s to the 1990s, the latent factors of the PC corresponded well with the business cycle. In contrast, only the downturn during the Great Recession was well captured after the 2000s. The more important finding is that all the models revealed that the current latent factors were in the upper state, where inflation reacts more sensitively to labor market slack. However, the current state is quite different from the previous business cycle as it was caused by the pandemic. In the labor market, there was insufficient labor supply due to pandemic-related quarantines, although there was excess demand. This affected the global value chain, including the Chinese shutdown and shipment shortages, thereby creating cost-push inflation. We expect inflation to react aggressively as labor market slack increases. After the pandemic, latent factors increased to a historic level, which implies that the PC can be stronger for the next few periods.

As we constructed the PC using various measures, how well the different specifications estimate the curve can be tested for robustness. Figure 4 depicts overlapped latent factors of all specifications using models estimated in Table 2 and Table 3.

Although the latent factors are unobservable, we attempt to identify those that cause endogenous changes in the PC. One possible method involves the Chicago Board Options Exchange’s Volatility Index (VIX), which is a real-time index that represents the market’s expectations for the relative strength of near-term price changes in stocks. Once the daily VIX is transferred to the quarterly value (on average), a correlation is found between them, as depicted in Figure 5. Based on this, the state of the PC may depend on the volatility of the economy.

Our findings reveal that the PC exhibits stronger dynamics during recession periods but weakens as the economy recovers. Additionally, our latent factor determining these regimes has a high correlation with the uncertainty measure. Below, we provide a detailed explanation of this mechanism. During recessions, economic slack is pronounced, characterized by significant unemployment and underutilized capacity. In these conditions, the PC relationship is robust, with inflation responding strongly to changes in slack. This heightened sensitivity arises because prices and wages are more responsive to economic conditions when there is substantial excess capacity. During recovery periods, firms may anticipate future demand and price increases, leading to price stickiness. Wages adjust more slowly due to contracts and other rigidities, weakening the immediate impact of slack on inflation. The VIX, a measure of market volatility and uncertainty, is significantly correlated with our latent factor, which aligns with the regimes of strong and weak PC dynamics. This correlation indicates that higher economic uncertainty influences the inflation–slack relationship. This state-dependent behavior is significantly influenced by economic volatility and uncertainty, as evidenced by the high correlation of our latent factor with the VIX.

5. Concluding Remarks

We propose the PC in a regime-switching framework in which latent factors are endogenously estimated. The main contribution of the new framework is the combination of the nonlinearity of the PC with endogenously determined latent factors. Then, a free-standing PC was constructed separately from theoretical complexities. Based on the robustness of the proposed framework, this model was run with various proxies for inflation, expectation, and supply shocks. As we defined the two regimes following the business cycle, during recessionary periods, the PC strengthens, but at other times, the relationship between inflation and employment slack weakens and becomes insignificant. We demonstrated that the nonlinearity of the curve can be made endogenously, rather than by idiosyncratic shocks. We also tested and confirmed VIX as a key component of latent factors.