Research Discussion Paper – RDP 2024-05 Sign Restrictions and Supply-demand Decompositions of Infation

Economists are often interested in decomposing changes in prices or quantities into contributions from shocks to supply or demand. To give a prominent example, there has been great interest in understanding how much of the surge (and subsequent decline) in inflation in many economies post COVID-19 was due to supply or demand factors.^[1] Understanding the supply-demand composition of inflation is important in this context because the desired policy response may depend on the nature of the underlying shocks.

One common approach to decomposing changes in variables into contributions from different shocks is to estimate a structural vector autoregression (SVAR). Within this framework, disentangling the contributions of shocks requires making identifying assumptions. When interest is in decomposing changes in prices and quantities into contributions from supply and demand shocks, one such set of assumptions is to impose sign restrictions on the slopes of supply and demand curves or, equivalently, on the responses of prices and quantities to supply and demand shocks.^[2] Economic theory typically implies that supply curves are upward sloping and demand curves are downward sloping or, equivalently, that supply shocks move prices and quantities in opposite directions, while demand shocks move them in the same direction. Hence, the main appeal of these restrictions is that they are uncontroversial. There is, however, a cost associated with relying on these relatively weak assumptions – the sign restrictions only identify a set (or range) of structural parameters, such as the slopes of the demand and supply curves (i.e. they are ‘set identifying’). In other words, there are many combinations of supply and demand curves that could explain the observed data and that are consistent with the sign restrictions. In turn, this implies that the sign restrictions can – on their own – only be used to recover a set of shock contributions or decompositions.

In this paper, I characterise the informativeness of sign restrictions when attempting to quantify the contributions of supply and demand shocks to variation in prices. I answer the question: under what conditions do these sign restrictions yield economically (un)informative decompositions of price changes? Answering this question is important because – as I discuss below – researchers have been relying on these sign restrictions when estimating the drivers of inflation. I focus on two types of decomposition. The first is the historical decomposition, which is the contribution of a particular shock to the realisation of a particular variable (or its forecast error) in a given period. For example, the historical decomposition can be used to quantify the role of supply shocks in driving the post-pandemic increase in US inflation. The second is the forecast error variance decomposition (FEVD), which is the contribution of shocks to forecast error variances. To give an example, the FEVD can be used to quantify the importance of supply shocks in driving unexpected variation in inflation over a specific forecast horizon (e.g. two years) on average over time.

This paper builds on and complements existing analyses of the use of sign restrictions in identifying supply-demand systems. In the SVAR context, Uhlig (2017) discusses the use of sign restrictions to identify the slopes of supply and demand curves given data on prices and quantities. Leamer (1981) contains a similar discussion in the context of maximum likelihood estimation of simultaneous equation systems subject to inequality constraints. Baumeister and Hamilton (2015) use a model of supply and demand to illustrate the role that the commonly used ‘uniform’ prior plays in driving Bayesian posterior inference when SVARs are identified using sign restrictions. The setting of my analysis is similar – a bivariate VAR in prices and quantities identified with sign restrictions on the slopes of supply and demand curves – but I focus on FEVDs and historical decompositions as the quantities of interest. While Baumeister and Hamilton (2015) and Uhlig (2017) use the bivariate model as a ‘toy’ example to illustrate issues associated with the use of sign restrictions, my focus on this model is motivated by its recent use empirically (as discussed below). The problem of disentangling supply and demand from data on prices and quantities is also well-studied outside of the SVAR literature and in fact was the motivating problem in the development of the instrumental variables estimator in the 1920s (e.g. Stock and Trebbi 2003).^[3]

I explain that the informativeness of sign restrictions about supply-demand decompositions depends on the reduced-form correlation between price and quantity forecast errors. When this correlation is strong, the sign restrictions can allow us to draw relatively unambiguous conclusions about the contributions of shocks (in the sense that the sets of decompositions are narrow). In contrast, when this correlation is weak, the sign restrictions do not reveal much about which shock is driving variation. In the case of historical decompositions, whether the restrictions are informative also depends on the realisations of the data (or forecast errors) in the periods under consideration.

Ultimately, because the informativeness of the sign restrictions depends on features of the data, whether they allow us to draw sharp conclusions about the contributions of shocks will depend on the empirical application at hand. I estimate the contributions of supply and demand shocks to price changes in two settings.

First, I use aggregate data to estimate the contributions of aggregate supply shocks to US inflation. I follow Chang, Jansen and Pagliacci (2023) by using an SVAR that includes growth in the GDP deflator and real GDP. The SVAR is identified with sign restrictions on the slopes of aggregate supply and demand curves. Bergholt, Furlanetto and Vaccaro-Grange (2023), Bergholt et al (2024) and Giannone and Primiceri (2024) use similar models to estimate historical decompositions and/or FEVDs of inflation.^[4] A feature of these papers is that, loosely speaking, they work with a single decomposition chosen from the set of decompositions that are consistent with the data and identifying restrictions. It is therefore unclear to what extent results are driven by the selection of a single, arguably arbitrary, decomposition; there are many other decompositions that are equally consistent with the identifying restrictions and the observed data.^[5] Instead, I directly estimate sets of decompositions that are consistent with the sign restrictions. These sets transparently reflect what we can learn about the contributions of aggregate supply and demand shocks to inflation given the sign restrictions.

The estimated sets for the FEVD imply that aggregate supply shocks account for between zero and 80 per cent of the variance of one-step-ahead forecast errors, and between 20 and 60 per cent of the variance of forecast errors at longer horizons. Estimates of the historical decomposition suggest that supply shocks made a substantial contribution to the post-pandemic increase in inflation; for example, supply shocks are estimated to have contributed between 1.3 and 3.4 percentage points to year-ended growth in the GDP deflator in mid-2022. However, we cannot unambiguously conclude whether the increase was predominantly driven by supply or demand shocks. Moreover, the results are sensitive to whether the COVID-19 period is included in the sample used to estimate the reduced-form parameters; when the COVID-19 period is excluded, the reduced-form correlation between forecast errors in inflation and real GDP growth is close to zero, and the sign restrictions are largely uninformative about the drivers of inflation.

Second, I conduct an exercise based on disaggregated data that is motivated by an influential decomposition proposed by Shapiro (2022).^[6] He estimates separate VARs for different expenditure categories of goods and services making up the personal consumption expenditures (PCE) basket and computes one-step-ahead forecast errors. Given sign restrictions on slopes of supply and demand curves, if the forecast errors have the same sign, a demand shock must have occurred, and inflation in that category is classified as ‘demand driven’. If the forecast errors have opposite signs, a supply shock must have occurred, and inflation in that category is classified as ‘supply driven’. He then takes an expenditure-weighted average of inflation in supply- and demand-driven categories to arrive at a supply-demand decomposition of aggregate inflation. A feature of this approach is that it allocates the entirety of inflation in each category to either a supply or demand shock and ignores the contributions of lagged shocks and deterministic terms. Instead, I use the historical decomposition to directly quantify the contributions of supply shocks to realised inflation within each expenditure category. The exercise therefore sheds light on the extent to which the sign restrictions underlying the decomposition in Shapiro (2022) are informative about the drivers of inflation in different expenditure categories.

I find that the sign restrictions are largely uninformative about the drivers of inflation in most expenditure categories and time periods, with some exceptions. To give an example, the sign restrictions deliver sharp decompositions of inflation in ‘food produced and consumed on farms’. Intuitively, this is because there is an extremely strong negative correlation between innovations in prices and quantities in this expenditure category, so changes in prices and quantities trace out a short-run demand curve and most variation is evidently due to supply shocks. To assess whether the disaggregated data help identify the drivers of aggregate inflation, I construct a ‘bottom-up’ decomposition of aggregate PCE inflation and compare it against the decomposition obtained when using the aggregate data directly. I find that the bottom-up decomposition tends to be substantially less informative than the aggregate decomposition, in the sense that the set of values for the contribution of supply shocks tends to be much wider.

Overall, these exercises suggest that assumptions about the signs of the slopes of supply and demand curves - on their own - may not deliver unambiguous conclusions about whether price changes are driven by shocks to supply or demand. This is the case when decomposing US inflation. Any additional assumptions inherent in selecting a single model or decomposition are likely to have a strong influence on inferences about the contributions of shocks, and these inferences may not be robust to relaxing or perturbing these additional assumptions.^[7]

Outline. I describe the SVAR, historical decomposition and FEVD in Section 2 before discussing the factors that influence how informative sign restrictions are about these decompositions in Section 3. The insights from this discussion are applied in Section 4, where I estimate the contributions of supply and demand shocks to US inflation. Readers purely interested in the empirical exercises can skip to Section 4. Section 5 concludes.

Notation. I will make use of the following notation in the paper. Vectors and matrices are in bold. For a matrix X, vec(X) is the vectorisation of X, which stacks the columns of X into a vector. vech(X) is the half-vectorisation of X, which stacks the elements of X that lie on or below the diagonal into a vector. e_i is the ith column of the 2 × 2 identity matrix, I₂.

This section describes a bivariate SVAR in prices and quantities, outlines a convenient alternative parameterisation of the model and introduces the structural objects of interest, including the FEVD and historical decomposition.

2.1 SVAR and orthogonal reduced form

Assume that y_t = (p_t, q_t)′ contains data on prices p_t and quantities q_t, and is generated by the SVAR(p) process:

where: A₀ is an invertible matrix with non-negative diagonal elements; $x_t={\left({y^{\prime }}_{t-1},\mathrm{...},{y^{\prime }}_{t-p}\right)}^{\prime }$ stacks the p lags of y_t; A₊ = (A₁,...,A_p) contains the structural coefficients on x_t; and ${\epsilon }_t={\left({\epsilon }_{1t},{\epsilon }_{2t}\right)}^{\prime }$ are the structural shocks, which have zero mean and identity variance-covariance matrix. For simplicity, I abstract from the inclusion of deterministic terms (e.g. a constant).

When the SVAR is set identified, it is convenient to work in the model's ‘orthogonal reduced-form’ parameterisation (e.g. Arias, Rubio-Ramírez and Waggoner 2018):

where: B = (B₁,...,B_p) is a matrix of reduced-form coefficients; ${\Sigma }_{tr}$ is the lower-triangular Cholesky factor of the reduced-form innovation variance-covariance matrix $$ \Sigma =𝔼\left(u_t{u^{\prime }}_t\right)$$ with u_t = (u_pt, u_qt)′ = y_t – Bx_t; and Q is an orthonormal matrix in the space of 2 × 2 orthonormal matrices, $$ \mathcal{O}(2).$$

Let $\varphi ={\left(\text{vec}{\left(B\right)}^{\prime },\text{vech}{\left({\text{Σ}}_{tr}\right)}^{\prime }\right)}^{\prime }$ collect the reduced-form parameters. The two parameterisations are related by $B=A_0^{-1}{A}_{+},\Sigma =A_0^{-1}(A_0^{-1}{)}^{\prime }$ and $Q={\Sigma }_{tr}^{-1}{A}_0^{-1}$.

In the absence of identifying restrictions, any orthonormal matrix Q is consistent with the second moments of the data, which are summarised by $\varphi$. In this sense, Q is set identified and hence so are the structural parameters (A₀, A₊) (e.g. Uhlig 2005). Imposing sign restrictions restricts the values of Q that are consistent with a given value of $\varphi$; the set of such values is an ‘identified set’ (e.g. Baumeister and Hamilton 2015; Gafarov, Meier and Montiel Olea 2018; Giacomini and Kitagawa 2021). An identified set for Q will induce identified sets for other parameters that are functions of the structural parameters.

Assume that the reduced-form parameters are such that the vector moving average $\left(\text{VMA}\left(\infty \right)\right)$ representation of the model exists:^[8]

where C_h are the reduced-form impulse responses, defined by $C_h={\sum }_{l=1}^{\min \left\{h,p\right\}}{B}_l{C}_{h-l}$ for $h\ge 1$ with C₀ = I₂ Element (i, j) of $C_h{\Sigma }_{tr}Q$ is the horizon-h impulse response of variable i to structural shock j, denoted by ${\eta }_{i,j,h}\left(\varphi ,Q\right)={c^{\prime }}_{ih}\left(\varphi \right)q_j$, where ${c^{\prime }}_{ih}\left(\varphi \right)={e^{\prime }}_i{C}_h{\Sigma }_{tr}$ is row i of $C_h{\Sigma }_{tr}$ and q_j = Qe_j is column j of Q. While not the focus of this paper, the impulse responses are useful for understanding the decompositions that are of central interest.

2.2 Historical decomposition

The historical decomposition is the cumulative contribution of a particular shock to the unexpected change (i.e. forecast error) in a variable over some horizon (e.g. Antolín-Díaz and Rubio-Ramírez 2018; Baumeister and Hamilton 2018). Specifically, let H_i,j,t,t+h be the contribution of the jth shock to the unexpected change in the ith variable between periods t and t + h:

Equation (5) shows that the historical decomposition is obtained by multiplying the realisations of the structural shocks and the impulse responses to those shocks. Equation (6) represents the historical decomposition in terms of the reduced-form innovations rather than the structural shocks themselves; this representation is useful, because the reduced-form innovations are what we can recover from the data given knowledge of the reduced-form parameters.^[9] To give an example, when h = 0 and i = 1, the historical decomposition represents the contribution of shock j to the one-step-ahead forecast error in p_t.

Kilian and Lütkepohl (2017) define the historical decomposition as the cumulative contribution of all past realisations of a particular shock to the realisation of a particular variable in some period (see also Plagborg-Møller and Wolf (2022) or Bergholt et al (2024)). Following from the VMA$\left(\infty \right)$ representation in Equation (3), the contribution of shock j to the realisation of y_it is ${\Sigma }_{l=0}^{\infty }{c^{\prime }}_{il}\left(\varphi \right)q_j{\epsilon }_{j,t-l}$. Given a finite time series, this infinite sum must be truncated at l = t – 1; intuitively, we cannot completely apportion the realisation of y_it to supply and demand shocks, since the realisation of y_it will also reflect the effects of shocks that occurred before the beginning of the sample (i.e. initial conditions'), though these effects should tend to vanish over time. In terms of Equation (5), this definition of the historical decomposition corresponds to H_i,j,1,t. The difference between y_it and ${\text{Σ}}_j{H}_{i,j,1,t}$ represents the contributions of initial conditions and any deterministic terms (e.g. a constant), which I have thus far abstracted from but will feature in the empirical exercises.

It is straightforward to show that ${\text{Σ}}_j{H}_{i,j,t,t+h}={\Sigma }_{l=0}^h{e^{\prime }}_i{C}_l{u}_{t+h-l}$, which is the (h + 1)-step-ahead forecast error in variable i. In the two-variable setting, knowledge of the contribution of one shock to the (h + 1)-step-ahead forecast error means that we also know the contribution of the other shock. In what follows, I therefore focus on the contribution of the first shock to the forecast error in the first variable (p_t) as the object of interest, and denote this by $H_{t,t+h}\equiv H_{1,1,t,t+h}$.

2.3 Forecast error variance decomposition

The FEVD of variable i at horizon h with respect to shock j is the cumulative contribution of the shock to the horizon-h forecast error variance (FEV) of variable i, expressed as a fraction of the horizon-h FEV (e.g. Kilian and Lütkepohl 2017; Baumeister and Hamilton 2018; Plagborg-Møller and Wolf 2022):

FEVD_i,j,h measures by how much the FEV of variable i at horizon h is reduced by knowing the path of future realisations of structural shock j. It therefore tells us how important a particular shock is for driving unexpected variation in a particular variable over a given horizon on average over time.^[10]

In the two-variable setting, knowing the contribution of one shock to the horizon-h FEV means that we know the contribution of the other shock, since FEVD_i,1,h + FEVD_i,2,h = 1. In what follows, I therefore focus on the contribution of the first shock to the FEV of the first variable (p_t) as the object of interest, and denote this by $FEV{D}_h\equiv FEV{D}_{1,1,h}.$

This section analytically characterises identified sets for particular structural parameters under sign restrictions on the slopes of supply and demand curves, closely following Baumeister and Hamilton (2015). I use this characterisation to discuss features of the sets of decompositions that are consistent with the restrictions, including conditions under which these sets are (un)informative about the decompositions.

3.1 Sign restrictions and identified sets

Let vech$\left({\Sigma }_{tr}\right)={\left({\sigma }_{11},{\sigma }_{21},{\sigma }_{22}\right)}^{\prime }$. The correlation between the innovations in p_t and q_t is

so ${\sigma }_{21}$ controls whether the correlation is positive $\left({\sigma }_{21}>0\right)$ or negative $\left({\sigma }_{21}<0\right)$. The space of 2 × 2 orthonormal matrices can be represented as

where $\theta \in \left[-\pi ,\pi \right]$ (e.g. Baumeister and Hamilton 2015).

In the absence of identifying restrictions, the identified set for $A_0^{-1}$ (the matrix of impact impulse responses) is

Consider imposing the following pattern of sign restrictions on $A_0^{-1}$:

These restrictions require p_t and q_t to move in opposite directions in response to the first shock and in the same direction in response to the second shock. Therefore, the first shock can be interpreted as a supply shock and the second shock can be interpreted as a demand shock. Accordingly, I henceforth write ${\epsilon }_{st}\equiv {\epsilon }_{1t}$ to represent the supply shock and ${\epsilon }_{dt}\equiv {\epsilon }_{2t}$ to represent the demand shock. This means that FEVD_h represents the contribution of the supply shock to the horizon-h FEV of p_t and H_t,t+h represents the contribution of the supply shock to the (h + 1)-step-ahead forecast error in p_t.

The identified set for $\theta$ under the sign restrictions is^[11]

This identified set is identical under the following pattern of restrictions on A₀:

which imply that the first equation of the SVAR describes an upward-sloping (i.e. supply) curve and the second describes a downward-sloping (i.e. demand) curve.

The structural shocks and reduced-form innovations are related via ${\epsilon }_t=A_0{u}_t$. We can think of this system of equations as describing the ‘short-run’ supply and demand curves. The slopes of these supply and demand curves are useful for understanding the conditions under which the sign restrictions are informative about the decompositions. I therefore present the identified sets for the slopes of these curves below, but these results are not new; for similar results, see Leamer (1981), Baumeister and Hamilton (2015) or Uhlig (2017). In the absence of identifying restrictions, the identified set for A₀ is

The slopes of the supply and demand curves are the coefficients on u_qt in each equation after normalising the coefficients on u_pt to one.

Let ${\omega }^s\left(\varphi ,\theta \right)$ represent the slope of the supply curve:

It is straightforward to show that the identified set for ${\omega }^s\left(\varphi ,\theta \right)$ is

If the forecast errors are negatively correlated, the sign restrictions do not reveal anything about the slope of the supply curve. If the forecast errors are positively correlated, it is possible to bound the slope of the supply curve. The informativeness of these bounds depends on how strongly the forecast errors are correlated; as $\rho$ approaches one, the lower and upper bounds of the identified set converge to the same value. Intuitively, when the forecast errors are strongly positively correlated, the observed forecast errors approximately trace out the supply curve.

Similarly, the slope of the demand curve is

and the identified set for ${\omega }^d\left(\varphi ,\theta \right)$ is

If the forecast errors are positively correlated, the sign restrictions reveal nothing about the slope of the demand curve. In contrast, when the forecast errors are negatively correlated, the sign restrictions generate informative bounds for the slope of the demand curve, which shrink to a point as $\rho$ approaches minus one, in which case the forecast errors trace out the demand curve.

Figure 1 illustrates this reasoning using simulated data where the forecast errors are strongly negatively correlated. The strong negative correlation means that the forecast errors trace out the demand curve, but there is a wide range of supply curves that are consistent with the data. At one extreme, the observed forecast errors could have been generated by shifts in a vertical supply curve (corresponding to $\theta =\arctan \left({\sigma }_{22}/{\sigma }_{21}\right)$) along the demand curve. At the other extreme, they could have been generated by shifts in a horizontal supply curve (corresponding to $\theta =0$). Any intermediate case between these two extremes is consistent with the sign restrictions and observed forecast errors.

3.2 Historical decomposition

In this two-variable model, we can write the historical decomposition of p_t with respect to the supply shock, H_t,t+h, as a function of $\theta ,\varphi$ and ${\left\{u_l\right\}}_{l=t}^{t+h}$:

where

The identified set for $\theta$ in Equation (12) induces a set of values for the historical decompositions that are consistent with the reduced-form parameters and the observed data. I refer to this set of values as the conditional identified set for H_t,t+h:^[12]

where $L_{t,t+h}\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)={\inf }_{\theta \in I{S}_{\theta }\left(\varphi \right)}{H}_{t,t+h}\left(\theta ,\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$ is the lower bound of the conditional identified set and $U_{t,t+h}\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)={\sup }_{\theta \in I{S}_{\theta }\left(\varphi \right)}{H}_{t,t+h}\left(\theta ,\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$ is the upper bound.^[13] Any value of H_t,t+h lying within $CI{S}_{H_{t,t+h}}\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$ is equally consistent with the identifying restrictions, the joint distribution of the data (as summarised by $\varphi$) and the observed forecast errors.

The value of $\theta$ that attains the lower or upper bound of $CI{S}_{H_{t,t+h}}\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$ may differ across t. The model corresponding to the lower or upper bound may thus differ, implying that the bounds correspond to supply or demand curves with different slopes in different periods. However, because every model – and thus decomposition – in the set is equally consistent with the data, the models that attain the bounds should not be given particular prominence over any other model within the identified set.

3.2.1 Computation

$CI{S}_{H_{t,t+h}}\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$ can be computed by solving the optimisation problems that define $L_{t,t+h}\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$ and $U_{t,t+h}\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$. Focusing on the upper bound, $U_{t,t+h}\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$ corresponds to either an end point of $I{S}_{\theta }\left(\varphi \right)$ or a critical point of $H_{t,t+h}\left(\theta ,\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$ in the interior of $I{S}_{\theta }\left(\varphi \right)$. Evaluating $H_{t,t+h}\left(\theta ,\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$ at the end points of $I{S}_{\theta }\left(\varphi \right)$ is straightforward given that an analytical expression for $I{S}_{\theta }\left(\varphi \right)$ is available. Rewriting the historical decomposition as a function of q₁, the problem ${\max }_{q1}{H}_{t,t+h}\left(q_1,\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$ subject to ${q^{\prime }}_1{q}_1=1$ has first-order necessary condition $\left(1/2\right)\left({\Omega }_h\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)+{\Omega }_h{\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)}^{\prime }\right)q_1=\lambda q_1$ where $\lambda$ is the Lagrange multiplier on the constraint. Solutions to this equation are eigenvectors of $\left(1/2\right)\left({\Omega }_h\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)+{\Omega }_h{\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)}^{\prime }\right)$. I compute the (two) eigenvectors, normalise their signs so that the sign normalisation ${e^{\prime }}_1{A}_0{e}_1={\left({\Sigma }_{tr}^{-1}{e}_1\right)}^{\prime }{q}_1\ge 0$ is satisfied, and check whether the normalised eigenvector lies within $I{S}_{\theta }\left(\varphi \right)$. $U_{t,t+h}\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$ is then obtained by direct comparison of the function values at the end points and admissible critical points (if any). $L_{t,t+h}\left(\varphi ,{\left\{u_l\right\}}_{l=t}^{t+h}\right)$ is obtained similarly.

3.2.2 A special case

Consider the case where interest is in the contribution of the supply shock to the one-step-ahead forecast error in p_t, so h = 0. It can be shown directly that

This expression is equivalent to $H_{1,t}\left(\theta ,\varphi ,{\left\{u_l\right\}}_{l=1}^t\right)$ when shocks have non-zero effects only on impact, in which case ${c^{\prime }}_{1l}\left(\varphi \right)=\left(0,0\right)$ for all l > 0. We can therefore use this expression to draw some useful intuition about the conditions under which the sign restrictions are informative about historical decompositions, with the qualification that when interest is in definitions of the historical decomposition other than H_t,t (e.g. H_1,t), the intuition is approximate (or may break down) when the effects of shocks are persistent.

The set of historical decompositions always admits ‘extreme’ contributions. When $\rho <0$, $0\in I{S}_{\mathrm{\theta }}\left(\varphi \right)$ and $u_{pt}\in CI{S}_{H_{t,t}}\left(\varphi ,u_t\right)$. That is, when the forecast errors in p_t and q_t are negatively correlated, the restrictions always admit the possibility that the supply shock is entirely responsible for the unexpected change in p_t . Intuitively, this is because the restrictions always admit the two extreme models in which the supply curve is vertical or horizontal. Conversely, when $\rho \ge 0,-\pi /2\in I{S}_{\theta }\left(\varphi \right)$ and $0\in CI{S}_{H_{t,t}}\left(\varphi ,u_t\right)$. So, when the forecast errors in p_t and q_t are positively correlated, the restrictions always admit the possibility that a supply shock is responsible for none of the unexpected change in p_t . This is because the restrictions admit the two extremes where the demand curve is vertical or horizontal.

The set of historical decompositions is uninformative when innovations are weakly correlated. If $\rho =0,I{S}_{\theta }\left(\varphi \right)=\left[-\pi /2,0\right]$. In this case, both 0 and u_pt lie within $CI{S}_{H_{t,t}}\left(\varphi ,u_t\right)$. The set of contributions is therefore consistent with forecast errors being driven entirely by a demand shock or entirely by a supply shock. The intuition is that uncorrelated innovations are equally consistent with a vertical supply curve and a horizontal demand curve $\left(\theta =-\pi /2\right)$, in which case forecast errors are purely driven by demand shocks, or with a horizontal supply curve and a vertical demand curve $\left(\theta =0\right)$, in which case forecast errors are purely driven by supply shocks. Any observed forecast errors can be explained by either of these extreme cases.

The set of historical decompositions can be very informative when forecast errors are strongly correlated. When the forecast errors are strongly negatively correlated, they trace out the demand curve, and forecast errors predominantly reflect supply shocks. However, we learn very little about the shape of the supply curve; the same forecast errors could be generated by shocks to a supply curve that is very steep, very flat, or something in between (see Figure 1). If we observe forecast errors that are consistent with the historical negative relationship, then we can conclude that they were mainly driven by a supply shock (even though we do not know much about the shape of the supply curve), so the set of historical decompositions is narrow around the observed forecast error. Similar reasoning applies when the forecast errors are strongly positively correlated, except in that case the forecast errors trace out the supply curve and forecast errors are attributable to demand shocks.

To illustrate this case, the top panels of Figure 2 plot two potential shifts in the supply curve that could have generated a particular forecast error – specifically, a roughly 1.5 per cent fall in price and 1.5 per cent increase in quantity. The forecast error can, at one extreme, be completely explained by a shift in a vertical supply curve (top left panel) or, at the other extreme, a shift in a horizontal supply curve (top right panel). Shifts in supply curves with any intermediate slope are also compatible with the observed forecast error. Despite not knowing anything about the slope of the supply curve, the contribution of the supply shock to the forecast error in prices is pinned down to be (approximately) equal to the observed forecast error – that is, the supply shock contributed 1.5 percentage points to the unexpected decline in prices.

But the set of historical decompositions can also be very uninformative even when forecast errors are strongly correlated. If the forecast errors are strongly negatively correlated but we observe forecast errors that depart from the historical negative relationship, the set of historical decompositions may be wide and thus economically uninformative. The bottom panels of Figure 2 illustrate the reasoning. In the left panel, a 0.5 per cent fall in both price and quantity can be explained by the combination of a negative demand shock (a downward shift in the demand curve from the solid to dashed red line) and a negative shock to a vertical supply curve (a leftward shift in the supply curve from the solid to dashed lilac line). In this case, the contribution of the supply shock to the forecast error in p_t is 0.5 percentage points. Alternatively, as illustrated in the right panel, the forecast error can be explained by a combination of the same negative demand shock and a negative shock to a horizontal supply curve. In this case, the contribution of the supply shock to the forecast error in p_t is −0.5 percentage points. Shifts in supply curves with slopes that lie between these two extremes are also consistent with the forecast error. The conditional identified set for the historical decomposition is therefore $CI{S}_{H_{t,t}}\left(\varphi ,u_t\right)=\left[-0.5,0.5\right]$. One intermediate case (not plotted) is where the forecast error is driven entirely by a negative demand shock shifting the demand curve along a supply curve with gradient one, in which case the contribution of the supply shock is zero.

3.3 Forecast error variance decomposition

We can rewrite FEVD_h as

where

The identified set for $FEV{D}_h\left(\theta ,\varphi \right)$ is then defined as

where ${\ell }_h\left(\varphi \right)={\inf }_{\theta \in I{S}_{\theta }\left(\varphi \right)}FEV{D}_h\left(\theta ,\varphi \right)$ is the lower bound of the identified set and $u_h\left(\varphi \right)={\sup }_{\theta \in I{S}_{\theta }\left(\varphi \right)}FEV{D}_h\left(\theta ,\varphi \right)$ is the upper bound.^[14] Any value of FEVD_h lying within $I{S}_{FEV{D}_h}\left(\varphi \right)$ is equally consistent with the identifying restrictions and the second moments of the data, as summarised by $\varphi$. As in the case of the historical decomposition, the bounds of $I{S}_{FEV{D}_h}\left(\varphi \right)$ may be attained by different values of $\theta$ (i.e. models) at different horizons.

Computation of $I{S}_{FEV{D}_h}\left(\varphi \right)$ proceeds similarly to the case where the historical decomposition is the quantity of interest (described in Section 3.2.1). The problem ${\max }_{q_1}FEV{D}_h\left(q_1,\varphi \right)$ subject to ${q^{\prime }}_1{q}_1=1$ has first-order necessary condition ${\Upsilon }_h\left(\varphi \right)q_1=\lambda q_1$, where $\lambda$ is the Lagrange multiplier on the constraint. Solutions to this equation are eigenvectors of ${\Upsilon }_h\left(\varphi \right)$.^[15]

3.3.1 A special case

Consider the case where interest is in the one-step-ahead FEVD of p_t with respect to the supply shock, FEVD₀. The contribution of ${\epsilon }_{st}$ to the FEV of p_t is ${\sigma }_{11}^2{\cos }^2\theta$ (i.e. the square of the impact impulse response of p_t to a standard deviation supply shock). Since the one-step-ahead FEV of y_1t is ${\sigma }_{11}^2$, the contribution of ${\epsilon }_{st}$ to the FEV as a fraction of the total FEV is ${\cos }^2\theta$.

When $\rho <0$, the upper bound of $I{S}_{\theta }\left(\varphi \right)$ is zero, so ${\cos }^2\theta$ attains its maximum of one and $u_0\left(\varphi \right)=1$. ${\cos }^2\theta$ is strictly increasing for $\theta \in \left[\arctan \left({\sigma }_{22}/{\sigma }_{21}\right),0\right]$, so ${\ell }_0\left(\varphi \right)$ occurs at the lower bound of $I{S}_{\theta }\left(\varphi \right)$. When $\rho \ge 0,I{S}_{\theta }\left(\varphi \right)$ is bounded above by zero and has lower bound $-\pi /2.{\cos }^2\theta$ therefore achieves its minimum value of zero at $-\pi /2$ and ${\ell }_0\left(\varphi \right)=0.{\cos }^2\theta$ is strictly increasing for $\theta \in I{S}_{\theta }\left(\varphi \right)$, so $u_0\left(\varphi \right)$ occurs at the upper bound of $I{S}_{\theta }\left(\varphi \right)$. Putting this together and rewriting the bounds in terms of $\rho$, the identified set for FEVD₀ is

Echoing the analysis of the historical decomposition, the sign restrictions always admit extreme values of the FEVD. When $\rho <0$, the sign restrictions always admit the possibility that the supply shock explains all of the variance of u_pt (because the supply curve may be horizontal) and when $\rho \ge 0$ they always admit the possibility that the supply shock explains none of the variance (because the demand curve may be horizontal). As $\rho \to 0,I{S}_{FEV{D}_0}\left(\varphi \right)$ approaches [0, 1], in which case the restrictions are completely uninformative.

The sign restrictions become more informative as the magnitude of the correlation between forecast errors, $\rho$, increases. $I{S}_{FEV{D}_0}\left(\rho \right)$ collapses to one as $\rho \to -1$; intuitively, if the one-step-ahead forecast errors are close to perfectly negatively correlated, the forecast errors must be generated by supply shocks shifting the supply curve along a static demand curve (see Figure 1). Conversely, $I{S}_{FEV{D}_0}\left(\rho \right)$ converges to zero as $\rho \to 1$; if the one-step-ahead forecast errors are close to perfectly positively correlated, the forecast errors must be generated by demand shocks shifting the demand curve along a static supply curve. To conclude unambiguously which shock is driving most of the variation in prices, it is necessary that ${\rho }^2>0.5;$ when $\rho <-\sqrt{0.5}$, supply shocks account for at least half the variance of prices, whereas when $\rho >\sqrt{0.5}$ demand shocks account for at least half the variance of prices.^[16]

This section builds on the insights above – in terms of using the sign restrictions to set identify the contributions of supply and demand shocks – to decompose US inflation into its drivers. I conduct two exercises that relate to existing decompositions of US inflation. The first approach is based on aggregate data and relates closely to a decomposition in Chang et al (2023), as well as possessing similarities to decompositions in Bergholt et al (2023) and Giannone and Primiceri (2024). The second approach builds a bottom-up decomposition of aggregate PCE inflation using disaggregated data on different expenditure categories, and relates to the influential decomposition proposed in Shapiro (2022).

4.1 An aggregate decomposition

Using a two-variable SVAR identified via sign restrictions, Chang et al (2023) decompose inflation and real GDP growth in the United States into contributions from shocks to aggregate supply and demand. They present historical decompositions based on a single model chosen from the set of models that are consistent with the sign restrictions using the ‘median target’ criterion from Fry and Pagan (2011). In contrast, I characterise sets of decompositions that are consistent with the identifying restrictions.

The model's endogenous variables are $y_t={\left({\pi }_t,g_t\right)}^{\prime }$, where ${\pi }_t$ is quarterly growth in the GDP deflator and g_t is quarterly growth in real GDP.^[17] Quarterly growth rates are approximated by log differences. The model includes a constant and eight lags of the endogenous variables and is estimated using data from 1989:Q1 to 2023:Q2.^[18] I estimate the reduced-form parameters via ordinary least squares (OLS) and compute the sets of decompositions conditional on these estimates.^[19]

As above, the identifying restrictions are that aggregate supply shocks move inflation and output in opposite directions and aggregate demand shocks move them in the same direction, which is equivalent to assuming the aggregate supply curve is upward sloping and the aggregate demand curve is downward sloping. Chang et al (2023) restrict the impulse responses on impact and in the period after the shocks. To avoid making assumptions about the persistence of the effects of shocks, I impose the restrictions only on impact.

Consistent with Chang et al (2023), I estimate the contributions of both current and past supply shocks to the realisation of inflation. In the notation of Section 2.2, this means that the historical decomposition of interest is H_1,t. I present decompositions of both quarterly and year-ended inflation.^[20] As discussed above, it is not possible to allocate the entirety of inflation to either supply or demand shocks, because part of inflation is explained by the constant and initial conditions. For ease of presentation, Figure 3 – which presents the estimated sets of historical decompositions – plots only the part of inflation that can be explained by shocks occurring within the sample; that is, after removing the deterministic component. On average, deterministic terms contributed about 0.5 percentage points to quarterly inflation and 2 percentage points to year-ended inflation.

The correlation in the one-step-ahead forecast errors for ${\pi }_t$ and g_t is 0.47. The observed forecast errors can therefore be interpreted as roughly tracing out an upward-sloping (short-run) aggregate supply curve, but we learn nothing about the slope of the (short-run) aggregate demand curve. Although the set of historical decompositions tends to be wide, it still appears to deliver useful inferences about the drivers of inflation in some periods. For example, when year-ended inflation peaked in 2022:Q2 at 7.4 per cent, supply shocks are estimated to have contributed between 1.3 and 3.4 percentage points (deterministic terms contributed about 2.1 percentage points to year-ended inflation in this period). The estimates therefore suggest that supply shocks made a substantial contribution to the increase in inflation in this episode, though the estimates do not allow us to unambiguously conclude whether the increase was mostly driven by supply or demand shocks.

The moderate degree of correlation between the one-step-ahead forecast errors means that the identified set for the FEVD is fairly wide (Figure 4). Aggregate supply shocks are estimated to account for between zero and 78 per cent of the variance of one-step-ahead forecast errors, and between 20 and 60 per cent of the variance of forecast errors at the five-year horizon. The sign restrictions therefore do not shine much light on the drivers of aggregate inflation at short horizons, though they suggest that supply shocks make a nontrivial contribution at longer horizons.

4.1.1 Robustness

The VAR described above is estimated over a sample period that includes the COVID-19 pandemic, which is consistent with the exercise in Chang et al (2023). One may worry that the large swings in economic activity that occurred during the pandemic may unduly affect the estimates of $\varphi$ and thus the sets of decompositions generated by the sign restrictions. This section explores the robustness of the results presented above to excluding the data from this episode from the estimation sample. To do this, I omit the data from 2020:Q1 onwards.^[21] The data from 2020:Q1 to 2023:Q2 are subsequently used to construct the reduced-form VAR innovations and historical decompositions during the period after the onset of the pandemic.

Re-estimating the model over this period yields an estimate for $\rho$ around 0.01, which is much smaller than the estimate obtained over the full sample (0.47). The large reduction in $\rho$ results in sets of decompositions that are substantially less informative (i.e. much wider) than those obtained over the full sample. For example, the sets of historical decompositions for year-ended inflation tend to include zero and the realisation of inflation itself (Figure 5). In 2022:Q2, supply shocks are estimated to have contributed between 0.3 and 4.4 percentage points to year-ended inflation. Identified sets for the FEVD (not plotted) are also wide, spanning from zero to (almost) one at the impact horizon and from around 15 to around 80 per cent at the five-year horizon. Overall, after excluding the pandemic from the estimation sample, the sign restrictions appear to be largely uninformative about the drivers of inflation in the GDP deflator.^[22]

4.2 Disaggregated decompositions

Shapiro (2022) proposes a supply-demand decomposition of US inflation in the PCE index based on disaggregated data. The decomposition involves estimating VARs for prices and quantities consumed in different PCE expenditure groups. Given sign restrictions on the slopes of the supply and demand curves, if the price and quantity forecast errors have the same sign, a demand shock must have occurred, and the price change is classified as demand driven. If the forecast errors have opposite signs, a supply shock must have occurred, and the price change is classified as supply driven.^[23] On this basis, overall inflation is decomposed into demand- and supply-driven components.

More precisely, let the superscript k index expenditure categories. ${\pi }_t^{\left(k\right)}$ is quarterly inflation in expenditure category k and $w_t^{\left(k\right)}$ is the expenditure weight in the consumption basket. The supply-driven component of aggregate quarterly inflation is then defined as

where 1(.) is the indicator function. The demand-driven component is defined similarly.^[24]

A feature of this decomposition is that it allocates the entire price change in an expenditure category on the basis of contemporaneous demand or supply shocks even though these shocks could only contribute to the one-step-ahead forecast error (i.e. the ‘unexpected’ part of the price change). It therefore ignores the lagged effects of shocks and the contributions of deterministic terms. The decomposition may also misrepresent the direction in which shocks contribute to inflation. For example, if $u_{pt}^{\left(k\right)}<0,u_{qt}^{\left(k\right)}>0$ and ${\pi }_t^{\left(k\right)}>0$, then ${\pi }_t^{(k)}$ will contribute positively to the supply-driven component of inflation despite the model implying that a disinflationary supply shock has occurred in category k. The decomposition also allocates the entirety of the price change in a particular category to either a supply or demand shock, when it seems probable that price changes in each period are driven by a mix of supply and demand shocks.^[25]

I address these issues by directly estimating the set of historical decompositions within different expenditure categories. The concept of the historical decomposition employed is H_1,t, so the effects of all past shocks are captured. The aim of the exercise is to explore the extent to which the sign restrictions allow us to learn about the contributions of supply and demand shocks within different expenditure categories. I then use the disaggregated results to construct a bottom-up decomposition of aggregate PCE inflation and compare it against a decomposition based on the aggregate PCE data. The exercise therefore speaks to whether applying the supply-demand decomposition to disaggregated data provides useful information about the contributions of supply and demand shocks to aggregate inflation relative to applying the decomposition to the aggregate data directly.

I consider 136 expenditure categories that make up the PCE data between January 1988 and September 2023.^[26] Within expenditure category k, the endogenous variables are $y_t^{\left(k\right)}={\left({\pi }_t^{\left(k\right)},g_t^{\left(k\right)}\right)}^{\prime }$, where ${\pi }_t^{\left(k\right)}$ is monthly inflation in the PCE price index and $g_t^{\left(k\right)}$ is monthly growth in real PCE (approximated using log differences). Each VAR includes 12 lags of the variables and a constant.^[27] I estimate each VAR separately via OLS and compute sets of historical decompositions conditional on the OLS estimates of the reduced-form parameters. Though the VAR is estimated at the monthly frequency, I present historical decompositions of year-ended inflation.

As discussed in Section 3, the correlation between the forecast errors in prices and quantities is important in determining how informative the sign restrictions are about the slopes of supply and demand curves and thus about the contributions of shocks to inflation. The category with the smallest absolute correlation is ‘lubricants and fluids’ $\left(\left|\rho \right|<0.0001\right)$ and with the largest absolute correlation is ‘food produced and consumed on farms’ $\left(\rho =-0.98\right)$. Figure 6 presents the historical decompositions for these two groups.

The set of historical decompositions for lubricants and fluids inflation often includes zero and the realisation of inflation itself (after removing the contribution of deterministic terms). The sets are therefore largely uninformative about whether price changes in this category are entirely driven by supply or demand shocks, which is broadly consistent with the discussion in Section 3.

In contrast, for food produced and consumed on farms, the strong negative correlation between forecast errors in prices and quantities means that the forecast errors trace out a short-run demand curve, and the contribution of supply shocks is bounded within a narrow range in most periods. To give an example, in March 2022, year-ended inflation in this expenditure category was 30.1 per cent, 28.4 percentage points of which can be explained by shocks occurring during the sample period (i.e. the contribution of deterministic components was a bit under 2 percentage points). Supply shocks are estimated to have contributed 23.9 to 25.1 percentage points to year-ended inflation in this period.

The correlation between forecasts errors tends to be reasonably weak in most other expenditure groups considered; the absolute correlations are smaller than 0.2 in about half of the categories and smaller than 0.4 in about 80 per cent of categories (Figure 7). The sign restrictions therefore tend to be fairly uninformative about the supply-demand composition of inflation outcomes within different expenditure categories.

To summarise the overall informativeness of the restrictions, I compute an expenditure-weighted average of the lower and upper bounds of the set of historical decompositions. Figure 8 presents this bottom-up decomposition of aggregate PCE inflation alongside a decomposition constructed using the aggregate PCE data directly.^[28] Separately decomposing PCE inflation in disaggregated expenditure categories does not appear to generate more informative decompositions of aggregate PCE inflation relative to a direct decomposition based on the aggregate data. In fact, the bottom-up decomposition generates substantially wider sets than the aggregate decomposition; on average over the sample period, the width of the sets is 2.6 percentage points for the bottom-up decomposition compared with 1.1 percentage points for the aggregate decomposition.

The bottom-up decomposition clearly reveals little about the drivers of the post-pandemic surge in US inflation, admitting contributions for the supply shock ranging anywhere from around zero to the entire change (less the contribution of deterministic terms). The aggregate decomposition yields sharper conclusions; for instance, in June 2022, the aggregate decomposition suggests that supply shocks contributed between 0.7 and 3 percentage points to year-ended PCE inflation. Much like the aggregate decomposition of inflation in the GDP deflator (Section 4.1), the aggregate decomposition of PCE inflation suggests that, while supply shocks made a substantial contribution to the increase in inflation in the post-pandemic episode, the sign restrictions do not allow us to unambiguously conclude whether the increase was mostly driven by supply or demand shocks.

One reason why the bottom-up decomposition may generate less-informative decompositions of aggregate PCE inflation than using the aggregate data directly is if there is a greater degree of measurement error at finer levels of disaggregation. Intuitively, the presence of (uncorrelated) measurement error in the price and quantity data would generate one-step-ahead forecast errors that are less strongly correlated than the forecast errors that would be obtained if price and quantity were observed without error. As discussed above, weaker correlations in these forecast errors mean that the sign restrictions reveal less about the slopes of supply and demand curves and thus less about shock decompositions. Indeed, the Bureau of Economic Analysis notes that the PCE data at this level of disaggregation is of significantly lower quality than that of the higher-level categories of which they are a part, as they are more likely to be based on judgement or on less-reliable source data (Bureau of Economic Analysis 2017).^[29]

As in Section 4.1.1, re-doing this exercise with an estimation sample that excludes the COVID-19 period tends to generate wider sets, both for the bottom-up and aggregate decompositions. However, it remains the case that the bottom-up decomposition yields much wider sets than the decomposition estimated using the aggregate data directly; see Appendix B.

Sign restrictions on the slopes of supply and demand curves provide an appealing avenue for identifying the contributions of supply and demand shocks to price changes because they are based on assumptions that are relatively uncontroversial. This is probably why they have been employed by researchers seeking to understand what has driven recent inflation developments. However, it is important to recognise that – on their own – these restrictions only allow us to conclude that the contributions lie within a set. In some cases, these sets may not allow us to draw unambiguous conclusions about whether price changes are mostly driven by shocks to supply or demand. In such cases, any additional assumptions that are used to select a single model or decomposition are likely to have a strong influence on assessments about which shocks are principally contributing to inflation.

While this paper focuses on estimating the contributions of supply and demand shocks to variation in prices, the discussion applies to other settings in which a bivariate system is identified using the same pattern of sign restrictions. For instance, models of search and matching in the labour market imply that equilibrium unemployment and vacancy rates are determined by the intersection of a downward-sloping Beveridge curve and an upward-sloping job creation curve (e.g. Daly et al 2012). It may therefore be appealing to use sign restrictions on the slopes of these curves to decompose variation in unemployment and vacancy rates. The analysis in this paper is useful for understanding the conditions under which such decompositions are informative.

This paper has characterised sets of decompositions, taking as given the modelling framework, which is a bivariate SVAR in prices and quantities identified via sign restrictions on the slopes of supply and demand curves. It could be useful for further work to explore what this framework identifies in terms of the underlying structure of the economy. For example, one could consider a New Keynesian model as a data-generating process and explore what the sign-restricted supply-demand framework identifies in terms of the model's underlying structural parameters and shocks.^[30]

The sets of decompositions presented in Section 4 are constructed conditional on the reduced-form parameters $\varphi$ being fixed at their OLS estimates. These sets can be interpreted as frequentist estimators of (conditional) identified sets. There is uncertainty around the estimates of $\varphi$, which means there is also uncertainty around sets of decompositions.^[31] To quantify this uncertainty, I employ the ‘robust Bayesian’ approach to inference proposed in Giacomini and Kitagawa (2021). The appeal of this approach is its computational tractability and prior robust Bayesian interpretation in finite samples.^[32]

Standard (single prior) approaches to Bayesian inference specify a prior over $\varphi$ and a (conditional) prior over Q (or $\theta$ in the bivariate model under consideration) given $\varphi$.^[33] Because Q does not enter the likelihood, the conditional prior is not updated and posterior inferences may be (asymptotically) sensitive to the choice of conditional prior (e.g. Poirier 1998; Moon and Schorfheide 2012; Baumeister and Hamilton 2015). Giacomini and Kitagawa (2021) propose removing this source of posterior sensitivity by replacing the single conditional prior with the class of all conditional priors that are consistent with the identifying restrictions (i.e. that assign probability one to $I{S}_{\theta }\left(\varphi \right)$). This generates a class of posteriors for the parameters of interest, which can be summarised in different ways. For example, the ‘set of posterior means’ for a given parameter (e.g. an FEVD at a particular horizon) is an interval spanning the posterior means corresponding to the class of posteriors. A ‘robust credible interval’ with credibility $1-\tau$ is an interval that is assigned at least posterior probability $1-\tau$ under any posterior in the class. In practice, computing these quantities requires computing (conditional) identified sets for the parameters of interest at every draw of $\varphi$ from its posterior (given some prior). In the current setting, this is computationally simple given the strategy for computing the bounds of the (conditional) identified sets for the decompositions (e.g. Section 3.2.1).

I assume a Jeffreys' prior over $\varphi ,{\pi }_{\varphi }\propto {\left|\Sigma \right|}^{-\left(n+1\right)/2}$, which is truncated to the region where the VAR is stable. This implies that the posterior for $\varphi$ is a (truncated) normal-inverse-Wishart distribution.^[34] Figure A1 presents the set of posterior means for the historical decomposition of year-ended growth in the GDP deflator alongside a 68 per cent robust credible interval. The set of posterior means for the supply contribution is somewhat wider on average than the set conditional on the OLS estimates of $\varphi$ (presented in Section 4.1).^[35] The robust credible intervals have excluded zero since late 2021, suggesting that, after accounting for statistical uncertainty, there is strong evidence that supply shocks have contributed positively to inflation outcomes in this period. To quantify this more precisely, I compute the posterior lower and upper probabilities that the contribution of the supply shock to year-ended inflation is positive; these are, respectively, the smallest and largest posterior probabilities assigned to the hypothesis that $H_{1,t}\ge 0$ obtainable within the robust Bayesian class of posteriors. In the March quarter 2022 (when inflation peaked), the posterior lower probability that the supply shock was contributing positively to inflation was 95 per cent and the posterior upper probability was 100 per cent.

This appendix explores the robustness of the decomposition of aggregate PCE inflation in Section 4.2 to excluding the COVID-19 period from the estimation sample. The VARs are estimated using a sample that runs up to 2019:M12.^[36] The estimated sets tend to be wider than those obtained using the full estimation sample (Figure B1). For the decomposition based on aggregate data, these wider sets reflect that the estimated value of $\rho$ is substantially smaller (0.04) than in the full sample (0.18). It remains the case that the sets from the bottom-up decomposition are substantially wider than the sets obtained when using the aggregate data.

Adjemian MK, Q Li and J Jo (2023), ‘Decomposing Food Price Inflation into Supply and Demand Shocks’, Unpublished manuscript, College of Agricultural & Environmental Sciences, University of Georgia, rev 12 July 2023.

Angeletos G-M, F Collard and H Dellas (2020), ‘Business-cycle Anatomy’, The American Economic Review, 110(10), pp 3030–3070.

Antolín-Díaz J and JF Rubio-Ramírez (2018), ‘Narrative Sign Restrictions for SVARs’, The American Economic Review, 108(10), pp 2802–2829.

Arias JE, JF Rubio-Ramírez and DF Waggoner (2018), ‘Inference Based on Structural Vector Autoregressions Identified with Sign and Zero Restrictions: Theory and Applications’, Econometrica, 86(2), pp 685–720.

Aruoba SB, FX Diebold, J Nalewaik, F Schorfheide and D Song (2016), ‘Improving GDP Measurement: A Measurement-error Perspective’, Journal of Econometrics, 191(2), pp 384–397.

Bai X, J Fernández-Villaverde, Y Li and F Zanetti (2024), ‘The Causal Effects of Global Supply Chain Disruptions on Macroeconomic Outcomes: Evidence and Theory’, NBER Working Paper No 32098.

Ball LM, D Leigh and P Mishra (2022), ‘Understanding U.S. Inflation during the COVID Era’, NBER Working Paper No 30613.

Baumeister C and JD Hamilton (2015), ‘Sign Restrictions, Structural Vector Autoregressions, and Useful Prior Information’, Econometrica, 83(5), pp 1963–1999.

Baumeister C and JD Hamilton (2018), ‘Inference in Structural Vector Autoregressions When the Identifying Assumptions Are Not Fully Believed: Re-evaluating the Role of Monetary Policy in Economic Fluctuations’, Journal of Monetary Economics, 100, pp 48–65.

Beaudry P, C Hou and F Portier (forthcoming), ‘The Dominant Role of Expectations and Broad-based Supply Shocks in Driving Inflation’, in JV Leahy, MS Eichenbaum and VA Ramey (eds), NBER Macroeconomics Annual, 39, University of Chicago Press, Chicago.

Beckers B (2023), ‘Discussant Remarks by Benjamin Beckers on “Decomposing Supply and Demand Driven Inflation” by Adam Shapiro’, Discussion of ‘Decomposing Supply and Demand Driven Inflation’ presented at the annual Reserve Bank of Australia Conference on ‘Inflation’, Sydney, 25–26 September.

Beckers B, J Hambur and T Williams (2023), ‘Estimating the Relative Contributions of Supply and Demand Drivers to Inflation in Australia’, RBA Bulletin, June.

Bergholt D, F Canova, F Furlanetto, N Maffei-Faccioli and P Ulvedal (2024), ‘What Drives the Recent Surge in Inflation? The Historical Decomposition Roller Coaster’, Norges Bank Working Paper 7|2024.

Bergholt D, F Furlanetto and E Vaccaro-Grange (2023), ‘Did Monetary Policy Kill the Phillips Curve? Some Simple Arithmetics’, Norges Bank Working Paper 2|2023.

Blanchard OJ and BS Bernanke (2023), ‘What Caused the US Pandemic-era Inflation?’, NBER Working Paper No 31417.

Boissay F, F Collard, C Manea and A Shapiro (2023), ‘Monetary Tightening, Inflation Drivers and Financial Stress’, BIS Working Papers No 1155.

Braun R, A Flaaen and S Hacıoğlu Hoke (2024), ‘Supply vs Demand Factors Influencing Prices of Manufactured Goods’, Board of Governors of the Federal Reserve System, FEDS Notes, 23 February.

Bureau of Economic Analysis (2017), ‘Chapter 5: Personal Consumption Expenditures’, in ‘Concepts and Methods of the U.S. National Income and Product Accounts’, rev Dec 2023, pp 5-1–5-73.

Calvert Jump R and K Kohler (2022), ‘A History of Aggregate Demand and Supply Shocks for the United Kingdom, 1900 to 2016’, Explorations in Economic History, 85, Article 101448.

Carriero A and A Volpicella (forthcoming), ‘Max Share Identification of Multiple Shocks: An Application to Uncertainty and Financial Conditions’, Journal of Business & Economic Statistics.

Chang J-CD, DW Jansen and C Pagliacci (2023), ‘Inflation and Real GDP Growth in the U.S.— Demand or Supply Driven?’, Economics Letters, 231, Article 111274.

Chen Y and T Tombe (2023), ‘The Rise (and Fall?) of Inflation in Canada: A Detailed Analysis of Its Post-pandemic Experience’, Canadian Public Policy, 49(2), pp 197–217.

Daly MC, B Hobijn, A Şahin and RG Valletta (2012), ‘A Search and Matching Approach to Labor Markets: Did the Natural Rate of Unemployment Rise?’, Journal of Economic Perspectives, 26(3), pp 3–26.

Eickmeier S and B Hofmann (2022), ‘What Drives Inflation? Disentangling Demand and Supply Factors’, Australian National University Crawford School of Public Policy, Centre for Applied Macroeconomic Analysis, CAMA Working Paper 74/2022.

Firat M and O Hao (2023), ‘Demand vs. Supply Decomposition of Inflation: Cross-country Evidence with Applications’, IMF Working Paper No WP/23/205.

Fry R and A Pagan (2011), ‘Sign Restrictions in Structural Vector Autoregressions: A Critical Review’, Journal of Economic Literature, 49(4), pp 938–960.

Gafarov B, M Meier and JL Montiel Olea (2018), ‘Delta-method Inference for a Class of Set-identified SVARs’, Journal of Econometrics, 203(2), pp 316–327.

Giacomini R and T Kitagawa (2021), ‘Robust Bayesian Inference for Set-identified Models’, Econometrica, 89(4), pp 1519–1556.

Giacomini R, T Kitagawa and M Read (2023), ‘Identification and Inference under Narrative Restrictions’, RBA Research Discussion Paper No 2023-07.

Giannone D and GE Primiceri (2024), ‘The Drivers of Post-pandemic Inflation’, Paper presented at the ECB Forum on Central Banking 2024 ‘Monetary Policy in an Era of Transformation’, Sintra, 1–3 July.

Gonçalves E and G Koester (2022), ‘The Role of Demand and Supply in Underlying Inflation – Decomposing HICPX Inflation into Components’, ECB Economic Bulletin, 7/2022, pp 70–75.

Gordon MV and TE Clark (2023), ‘The Impacts of Supply Chain Disruptions on Inflation’, Federal Reserve Bank of Cleveland Economic Commentary No 2023-08.

Granziera E, HR Moon and F Schorfheide (2018), ‘Inference for VARs Identified with Sign Restrictions’, Quantitative Economics, 9(3), pp 1087–1121.

Hamilton JD (1994), Time Series Analysis, Princeton University Press, Princeton.

Kilian L and H Lütkepohl (2017), Structural Vector Autoregressive Analysis, Themes in Modern Econometrics, Cambridge University Press, Cambridge.

Kugler AD (2024), ‘The Outlook for the U.S. Economy and Monetary Policy’, Speech given at the Weidenbaum Center on the Economy, Government, and Public Policy, Washington University, St. Louis, 3 April.

Leamer EE (1981), ‘Is It a Demand Curve, or Is It a Supply Curve? Partial Identification through Inequality Constraints’, The Review of Economics and Statistics, 63(3), pp 319–327.

Lenza M and GE Primiceri (2022), ‘How to Estimate a Vector Autoregression after March 2020’, Journal of Applied Econometrics, 37(4), pp 688–699.

Moon HR and F Schorfheide (2012), ‘Bayesian and Frequentist Inference in Partially Identified Models’, Econometrica, 80(2), pp 755–782.

Plagborg-Møller M and CK Wolf (2022), ‘Instrumental Variable Identification of Dynamic Variance Decompositions’, Journal of Political Economy, 130(8), pp 2164–2202.

Poirier D (1998), ‘Revising Beliefs in Nonidentified Models’, Econometric Theory, 14(4), pp 483–509.

Read M (2022), ‘The Unit-effect Normalisation in Set-identified Structural Vector Autoregressions’, RBA Research Discussion Paper No 2022-04.

Rubbo E (2024), ‘What Drives Inflation? Lessons from Disaggregated Price Data’, NBER Working Paper No 32194.

Rubio-Ramírez JF, DF Waggoner and T Zha (2010), ‘Structural Vector Autoregressions: Theory of Identification and Algorithms for Inference’, The Review of Economic Studies, 77(2), pp 665–696.

Schorfheide F and D Song (forthcoming), ‘Real-time Forecasting with a (Standard) Mixed-frequency VAR during a Pandemic’, International Journal of Central Banking.

Shapiro AH (2022), ‘Decomposing Supply and Demand Driven Inflation’, Federal Reserve Bank of San Francisco Working Paper No 2022-18, rev February 2024.

Stock JH and F Trebbi (2003), ‘Retrospectives: Who Invented Instrumental Variable Regression?’, Journal of Economic Perspectives, 17(3), pp 177–194.

Uhlig H (2003), ‘What Moves Real GNP?’, Unpublished manuscript, April.

Uhlig H (2005), ‘What Are the Effects of Monetary Policy on Output? Results from an Agnostic Identification Procedure’, Journal of Monetary Economics, 52(2), pp 381–419.

Uhlig H (2017), ‘Shocks, Sign Restrictions, and Identification’, in B Honoré, A Pakes, M Piazzesi and L Samuelson (eds), Advances in Economics and Econometrics: Eleventh World Congress, Vol 2, Econometric Society Monographs, Cambridge University Press, Cambridge, pp 95–127.

Wolf CK (2020), ‘SVAR (Mis)identification and the Real Effects of Monetary Policy Shocks’, American Economic Journal: Macroeconomics, 12(4), pp 1–32.