Unraveling Producer Price Inflation Pass-Through: Quantification, Structural Breaks, and Causal Direction

4.1 Structural Breaks

While we are interested in quantifying by how much producer prices lead consumer prices over our full sample, the potential for the presence of structural changes in consumer price inflation necessitates consideration for subsample analysis. Rather than impose beliefs a priori as to when and how many structural breaks might be present in consumer or producer price inflation, we test for structural breaks formally by leveraging the methodology described in Bai and Perron (2003), which tests endogenously for the number of structural breaks in each time series as well as the break dates themselves. This methodology begins with a simple ordinary least-squares regression of our outcome variable of interest against a constant term (drift term) and a linear time trend ( T t ). Equation array (4.1) describes these statistical models:

Beyond the standard statistical models for estimating structural breaks, we can test for structural breaks on similar models that capture both the production view and derived-demand theories of inflation pass-through. Equation array (4.2) describes these competing views:

In theory, if producer (consumer) prices lead consumer (producer) prices, then the lag dynamics of producer (consumer) prices may contain information relevant for estimating subsequent structural breaks as well. Table 2 reports the estimated break dates with 95% confidence intervals for each equation described within arrays (4.1) and (4.2).

We observe that, in the case of equation array (4.1), there is not much difference in both the number of breaks and break dates themselves between consumer and producer price inflation. It is worth noting that the mean of each estimated break date for producer price inflation tends to precede slightly or coincide with consumer price inflation but exhibits wider confidence intervals. From equation array (4.2), we observe twice as many estimated breaks compared to equation array (4.1) with both producer and consumer price inflation rates breaking coincidentally as well. The disparities reflected in Table 2 underscore the sensitivity of estimated producer and consumer price inflation structural breaks to model specification. The natural question that arises from this is if these breaks also correspond to deviations or confirmations of producer prices leading consumer prices. To address this question, we need to implement an alternative strategy, one that entails testing for cointegration and causal direction.

4.2 Cointegration and Causal Direction

An important stylized fact about the relationship between producer and consumer prices is that they are cointegrated with one another over their full samples. However, given the multitude of breaks present across each series, and ambiguity over whether the production view theory dominates the derived demand theory of pass-through, it is possible that cointegration and causal direction of producer and consumer price inflation is heterogeneous across various subsamples. Again, refraining from assumptions on breaks in consumer price inflation, we let our results from Table 2 guide our choice of subsample analysis for cointegration and causal direction testing. Specifically, we identify the following regimes to conduct cointegration tests and causality tests: 1) 1960:Q1–1971:Q4, 2) 1972:Q1–1982:Q3, 3) 1982:Q4–1992:Q1, 4) 1992:Q2–2009:Q4, and 5) 2010:Q1–2023:Q1.

While we could test for both cointegration and causal direction separately from one another, advances in the cointegration literature make testing for both simultaneously possible via Toda and Yamamoto’s (1995) causality testing. The Toda–Yamamoto (TY) causality test considers a VAR model in levels with a maximum, r Max , cointegrating relationships, and ρ optimum lags as prescribed by the akaike information criterion (AIC) criteria. An augmented VARX ( ρ + r Max ) is then estimated where the VARX (vector autogressive model with exogenous variables) is constructed with the optimum lag length of ρ plus additional lag lengths equal to r Max treated as exogenous regressors. From a performance standpoint, because the VARX is constructed with the endogenous variables in their levels, rather than their first differences, the test is strictly more efficient than Granger’s (1969) causality tests by treating each variable as seemingly unrelated to one another. Furthermore, strictly in a bivariate VARX, the statistical evidence of causal direction (unidirectional or bidirectional) in the TY framework also confirms evidence of cointegration simultaneously.^[4] Equation (4.3) describes the general form of our bivariate VARX model estimated over our established subsamples:

where Z t = [ log ( P t C ) , log ( P t P ) ] T describes our vector of endogenous variables; Γ i describes the coefficient vector of our lagged endogenous variables from i = 1 to i = ρ lags, where ρ is the maximum number of lags prescribed by the AIC criteria; and Θ describes the coefficient vector associated with our exogenous variables. The full sample and subsample results of our TY causality tests are exhibited in Table 3.

Over the full sample, we see results not unlike other pieces in the literature that show bidirectional or mixed evidence of producer price pass-through (Akcay, 2011; Clark, 1995; Topuz et al., 2018). However, we note considerable heterogeneity over our identified subsamples. Prior to the Productivity Slowdown in our earliest regime (1960:Q1–1971:Q4), we find that producer prices lead consumer prices unidirectionally at an incredibly strong level of statistical significance (<1%). By comparison, during our second regime, which includes the Productivity Slowdown through the end of the Great Inflation Era (1972:Q1–1982:Q3), we still see producer prices lead consumer prices unidirectionally, but at a weaker statistical level (5%). In our third regime, which starts at the end of the Great Inflation and continues through the beginning of the New Economy Boom (1982:Q4–1992:Q1), we find no evidence of causality in any direction – the same is true for our final regime that starts after the Financial Crisis recovery through the present day (2010:Q1–l2023:Q1). Finally, our fourth regime (1992:Q2–2009:Q4), which encompasses both the New Economy Boom and Financial Crisis once more shows evidence of unidirectional cost-push pass-through.

Simply put, while producer prices lead consumer prices most of the time, there are significant subsamples – including the present day – where the production view theory of producer price pass-through fails to hold. The evidence put forth from Table 3 gives us a sense of when producer prices lead consumer prices but does not tell us the magnitude of pass-through itself. To quantify pass-through, we turn toward an error-correction framework, which is commonly leveraged in the oil price and exchange rate pass-through literature.^[5]

4.3 Error-Correction Model

We use a two-step error-correction framework to map short-run variation in lagged producer price inflation to the long-run CPL. The basic framework follows the work of Engle and Granger (1987) and consists of two stages. The first stage is a long-run model from which the vector of residuals post-estimation is obtained, ϵ t ˆ , and tested for a unit root. Assuming ϵ t ˆ is integrated at an order of one, or ∼ I ( 0 ) , then a second-stage model is estimated using lagged differences (or growth rates) of the regressors and a lagged error-correction term ( EC T t ), which is our lagged residual vector from our long-run model. Following Chen (2009), we estimated an augmented long-run Phillips curve and short-run Phillips within an error-correction framework. The augmented long-run Phillips curve is described by the following equation:

where β P describes the long-run pass-through (LRPT) of producer prices to consumer prices. While some may object to a “long-run” model of the price level augmented with producer prices as being relatively ad hoc, we believe this adequately captures the production view theory of inflation pass-through. For example, if producer prices lead consumer prices in logs, we have log ( P t C ) = log ( P t − 1 P ) . Under the assumption that both producer and consumer prices are cointegrated, in steady state, there is a linear contemporaneous combination of producer and consumer prices that is stationary such that log ( P t C ) − β P l og ( P t P ) = ϵ t ∼ I ( 0 ) , where β P has the same meaning as before. In a sense, this formulation when mapped to equation (4.4) describes a long-run representation of the Phillips curve considerate of the production view theory of inflation pass-through. With our long-run model defined, we also identify a corresponding augmented short-run Phillips curve described by the following equation:

Chen (2009) defined partial short-run pass-through (PSRPT) as the θ 1 coefficient associated with the first lag of producer price inflation. Additionally, Chen (2009) showed via inversion of the short-run Phillips curve that short-run pass-through (SRPT) can be constructed using the point estimates associated with LRPT ( β P ), PSRPT ( θ 1 ), and the lagged error-correction term ( ψ ). Formally, we denote SRPT as ξ and define its numerical construction by the following equation:

Another convenience afforded by the two-step error-correction framework beyond its functional form flexibility is its ability to allow for “time-varying” estimation. As shown in Chen (2009), we can define dummy variables corresponding to specific regimes or temporal breaks in consumer price inflation. This allows us to directly extend our analysis and discussion of Tables 2 and 3 by quantifying heterogeneity in SRPT, specifically in the second stage of our model. To achieve this, we define regime specific dummies, D rt , described by the following equation array:

To formally incorporate the regime-specific dummies from equation array (4.7), we rewrite equation (4.5) as the following equation:

In a sense, our short-run model now contains deterministic linear splines in accordance with our established breaks. Thus, we can estimate “time-varying” SRPT over regimes where producer prices unidirectionally cause consumer prices.^[6] In a similar vein, we can use our point estimates from our long-run model (which is unchanged, as it captures a time-invariant long-run steady state) and equation (4.8) to arrive at regime-specific SRPT coefficients as described by the following equation:

4.3.3 Robustness to Oil Prices

While our choice of controls within equations (4.5) and (4.8) are consistent with the literature, and acknowledge the importance of the monetary authority’s ability to influence macroeconomic conditions, and subsequently the price level (Caporale et al., 2002), we admit there are likely other controls that are worth exploring that could impact the price level, namely, oil price inflation, and the USD-Euro exchange rate. These potential controls have their own distinct pass-through literatures and likely contain information that could confound the true effect of producer price inflation pass-through. To explore this, consider the error-correction framework described by the following equations:

(4.11) log ( P t C ) = β 0 + β Y log ( Y t ) + β P log ( P t P ) + β M log ( M t ) + β O log ( O t ) + τ T + ϵ t ,

(4.12) π t C = α 0 + ∑ i = 1 4 ω i π t − i C + γ Y t − 1 ∼ + ∑ i = 1 4 θ i π t − i PPI + μ π t − 1 MNY + λ π t − 1 OIL + ψ ϵ t − 1 ˆ + ε t .

Note that we can also specify equation (4.11) to include discrete variation in pass-through across time analogous to equation (4.8). Equation (4.13) captures this while simultaneously controlling for oil price inflation:

(4.13) π t C = α 0 + ∑ i = 1 4 ω i π t − i C + γ Y t − 1 ∼ + ∑ i = 1 4 θ i π t − i PPI + ∑ r = 1 3 ∑ i = 1 4 δ ri ( π t − i PPI · D rt − i ) + μ π t − 1 MNY + λ π t − 1 OIL + ψ ϵ t − 1 ˆ + ε t .

Herein, we consider the possibility that long-run PPI pass-through, as well as short-run PPI pass-through can be confounded by oil prices, which has a historically strong influence on consumer price inflation in the US (Chen, 2009; Conflitti & Luciani, 2019; De Gregorio et al., 2007). The long-run relationship between consumer prices, log ( P t C ) , and oil prices, log ( O t ) is captured by the β O coefficient. In our short-run model, lagged oil price inflation pass-through, π t − 1 OIL , is controlled for and captured by the λ term.^[8]

For our study, oil prices are West Texas instruments spot crude oil prices retrieved from the Reserve Bank of St. Louis. Data go back as far as 1946:Q1, which is consistent with the window of time we conduct our study over. We stress that while it would be optimal to control for exchange rates, data on the USD-Euro spot exchange rate only go back as far as 1999; furthermore, data on the currency conversion between the USD and Euro only go as far back as 1979. As a result, and consequently, the length of exchange rate data available, and the window of time associated with our study is largely incompatible.^[9] Estimation results for equations (4.12) and (4.13) are provided in Table 6.

Table 6

Error-correction model results (additional controls)

Dependent variable: π t C
Equation	(4.12)		(4.13)
Coefficient	Estimate	Std. error	Estimate	Std. error
α 0	−0.001	(0.001)	0.001	(0.001)
ω 1	0.99***	(0.09)	0.84***	(0.13)
ω 2	0.10	(0.16)	0.05	(0.20)
ω 3	0.14	(0.14)	0.20	(0.15)
ω 4	−0.28**	(0.11)	−0.23*	(0.10)
γ	0.09**	(0.03)	0.11*	(0.05)
θ 1	0.09**	(0.03)	0.09*	(0.04)
θ 2	−0.14**	(0.05)	−0.07	(0.06)
θ 3	0.001	(0.04)	−0.01	(0.05)
θ 4	0.03	(0.03)	−0.002	(0.03)
ψ	−0.01	(0.01)	0.01	(0.02)
δ 11			0.14	(0.09)
δ 12			−0.12	(0.12)
δ 13			0.03	(0.08)
δ 14			−0.03	(0.07)
δ 21			0.13*	(0.06)
δ 22			−0.09	(0.08)
δ 23			0.01	(0.07)
δ 24			−0.01	(0.05)
δ 31			−0.02	(0.04)
δ 32			−0.04	(0.06)
δ 33			−0.04	(0.03)
δ 34			0.05	0.04
β P	1.32***		1.32***
ADF Stat.	−4.52***		−4.52***
ξ	0.07		0.10
ζ 1	—		0.24
ζ 2	—		0.24
ζ 3	—		0.07
Controls?	Yes		Yes
N	249		249
Adj. R 2	0.95		0.95

We note that controlling for oil prices and oil price inflation does not seem to alter SRPT nor PSRTP significantly. This is not too surprising when one considers that the producer price index for the United States can be thought of in the aggregate as the weighted sum of the largest commodity-specific producer price indices. Referring to the BLS page on the composition of United States PPI, one would note that the WPU05 component of PPI is defined as “fuel and related products and power,” which contains information on the prices producers pay for coal, gas fuels, electric power, gasoline, and refined petroleum among other commodities. As a result, it is likely that the aggregate producer price index is indirectly capturing some variation in oil prices on its own.^[10]

We also take note that accounting for discrete variation in pass-through via our ζ r terms is once more not informative for understanding variation in pass-through across different eras in US history. We can, however, obtain a better idea of “smoother” variation in pass-through using a rolling regression framework like that of Chen (2009). Consider a rolling configuration of equation (4.13) with windows equal to 40 quarters, or 10 years. If we roll equation (4.13) and refit our model’s coefficients accordingly, we can plot estimates for θ 1 – our partial short-run pass-through rate – over time. Figure 2 illustrates our rolling regression estimates of θ 1 .

Figure 2

Rolling PSRPT.

Note that the shaded orange bars in Figure 2 represent the regimes we discretely accounted for in Tables 5 and 6. We note that the mean PSRPT rate from our rolling regression is around 0.13, which is slightly larger compared to our estimation results. Most striking, during regimes where producer prices lead consumer prices (highlighted by the shaded orange bars in Figure 2), we see that inflation pass-through is, on average, more stable, and lower by comparison to states where there is no causal direction, or relationship present. An implication of this finding is that pass-through almost follows a regime-switching process. There are some states where producer prices lead consumer prices, and some states where they do not. As such, discrete estimation of pass-through over subsamples may not be as efficient or informative. Furthermore, while most pass-through studies (producer prices, oil, and exchange rates) leverage error-correction frameworks to quantity pass-through (oil, and exchange rate pass-through, in particular) by capitalizing on the cointegration between producer and consumer prices, the adjustment speeds from short-run to long-run, as captured by our ψ terms, are relatively weak and insignificant in most cases. Because of this, most SRPT and PSRPT rates are not numerically different from one another over our full sample. These findings motivate a more flexible framework that can account for the regime-switching nature of the production view theory of producer price inflation pass-through and rely less on the cointegrating relationships traditionally leveraged in error-correction frameworks.

4.4 Local Projection IRFs

An alternative approach for estimating pass-through is through the construction of IRFs. In essence, shocks – unit or standard deviations – also capture the pass-through of producer prices to consumer prices, or vice versa. An issue with traditional IRF projections, particularly in this setting, is that producer prices are not always leading consumer prices. In essence, there is a probability that we are in a state where the production view theory holds and a state where it does not; thus, we must account for regime-switching in producer prices for correctly identifying pass-through from our shocks.^[11] Thankfully, Jordà (2005) provided a flexible framework for circumventing these issues. We first identify two regimes as described by the following equation:

Conveniently, we have already identified these regimes via our structural break tests and TY causality tests reported in Tables 2 and 3, respectively. Formally, our local projection model takes on the following general form described by the following equation:

where Z t = [ π t C , Y t ∼ , π t P ] T describes our vector of endogenous variables carrying their same meanings as in equation (4.6), h describes our forecast horizon, ρ describes our lag length as selected by the AIC criteria, and β i , R j describes our lagged coefficient vectors corresponding to lag i and regimes j ∈ { 1,2 } .^[12] Furthermore, our state probabilities are computed via a logistic function that follows the form of equation (4.16):

where γ > 0 describes our time-invariant scaling factor.^[13] With our model definitions in mind, we can compute state-dependent nonlinear IRFs described by the following equation:

where R j captures our states such that j ∈ { 1,2 } , h captures our forecast horizon, and d i is a matrix of our shocks.^[14] Figure 3 illustrates the responses of our endogenous variables to standard deviation shocks under each regime along with shaded 90% confidence intervals.

Figure 3

Panel (a): Local projection responses to shocks when PPI leads CPI. Panel (b): Local projection responses to shocks when PPI does not lead CPI.

Panel A expresses responses under regime R 1 wherein producer prices lead consumer prices, while Panel B expresses responses under regime R 2 wherein producer prices do not lead consumer prices. We observe that in the state where producer prices lead consumer prices, CPI inflation responds slowly to standard deviation shocks PPI inflation peaking after around eight fiscal quarters ahead. We note that such shocks may seem modest with a mean response around 0.5% but persist up until 12 quarters ahead. An implication herein is that a single shock from producer price inflation is somewhat slow to permeate itself across consumer price inflation but can have a dramatic cumulative effect given the length of the shock’s persistence. Interestingly, we also note that producer price inflation responds contemporaneously to a standard deviation shock from consumer price inflation at a rate of around 1.25%, but such a shock is short-lived with producer price inflation reverting to statistical zero after only four quarters.

These two findings alone shed interesting light on the dynamics between both consumer and producer price inflation. The shocks confirm that derived-demand inflation during states where producer prices unidirectionally lead consumer prices exerts an immediate, but transitory response from producer prices. On the other hand, producer price inflation shocks take time to disseminate but are rigid and slow to revert to statistical zero.

Turning toward our regime where producer prices do not lead consumer prices, or vice versa, we see dramatically different responses from standard deviation shocks. We observe that producer price inflation shocks do not affect CPI inflation contemporaneously but after some time, can lead to small levels of deflation that persist for six quarters or so before jumping to modest levels of positive pass-through around a mean response of 2% persisting for eight quarters or so. On the other hand, producer price inflation responds significantly, albeit at a very weak rate, to a standard deviation shock from consumer price inflation; however, this effect lasts for a little less than two fiscal quarters before reverting and lingering around zero for the remainder of the projection horizon.

Panels (a) and (b) taken together reflect that knowledge of which state the economy is in is imperative – not only for forecasting and quantifying the impacts of producer price inflation shocks themselves, but also for timing policy responses to stabilize inflation over long periods. In states where producer prices lead consumer prices, our local projections suggest that policy responses to stabilize producer prices and minimize pass-through should be enacted early and executed over a long horizon (between 10 and 12 fiscal quarters). On the other end of the spectrum, in states where the production view theory of pass-through does not hold, policy responses to stabilize producer prices would likely be minimal in effect and could be quite costly from a practical standpoint.