RDP 2024-05: Sign Restrictions and Supply-demand Decompositions of Infation 1. Introduction

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.