RDP 2024-05: Sign Restrictions and Supply-demand Decompositions of Infation 2. SVAR and Decompositions

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 $FEV{D}_h\equiv FEV{D}_{1,1,h}.$

Footnotes

This representation exists if the eigenvalues of the VAR ‘companion matrix’ lie inside the unit circle (e.g. Hamilton 1994; Kilian and Lütkepohl 2017). [8]

Since ${\epsilon }_t=A_0{u}_t=Q^{\prime }{\text{Σ}}_{tr}^{-1}{u}_t$, structural shock j is ${e^{\prime }}_j{\epsilon }_t={q^{\prime }}_j{\text{Σ}}_{tr}^{-1}{u}_t.$ [9]

As discussed in Plagborg-Møller and Wolf (2022), when all shocks are invertible (which is the maintained assumption here), the information set ${\left\{y_{\tau }\right\}}_{-\infty <\tau <t}$ coincides with the information set ${\left\{{\epsilon }_{\tau }\right\}}_{-\infty <\tau <t}$. [10]