Derivations for Max Share Identification Method

Matthew DeHaven

January 16, 2023

Article explaining the mathematics behind the two key fevdid functions:

• id_fevdfd()
• id_fevdtd()

Structural Vector Autoregression (SVAR)

A SVAR(p) model with $p$ lags, for a vector of variables $y_t$, assumes the data generating process is accurately represented as, ¹

$$B_0 y_t = B_1 y_{t-1} + ... + B_p y_{t-p} + w_t$$

where $w_t$ are mean zero structural shocks, with a serially uncorrelated diagonal covariance matrix of $\Sigma_w = E[w_t w_t]$,

and $B_i$ are matrices of coefficients.

Empirically, however, only the following $A_i$ matrices of coefficients and $v_t$ vector of reduced form residuals are observed,

$$ y_t = \underbrace{B_0^{-1}B_1}_{A_1} y_{t-1} + ... + \underbrace{B_0^{-1}B_p}_{A_p} y_{t-p} + \underbrace{B_0^{-1} w_t}_{u_t} \\ y_t = A_1 y_{t-1} + ... + A_p y_{t-p} + u_t $$

Identification Problem

We observe reduced form residuals, $v_t$, but want the structural shocks, $w_t$.

$$u_t = B_0^{-1} w_t$$

At this point, WLOG, assume that $\Sigma_w = I$.

This implies that $\Sigma_u = B_0^{-1} B_0^{-1^\prime}$.

In order to identify the structural shocks, we simply need to know the structural impact matrix, $B_0$, or equivalently, the inverse $B_0^{-1}$.

Impulse Response Functions (IRF) and Forecast Errors

It will be useful for the max share method to show the impulse response and forecast error definitions

IRFs

IRFs give the responses of each variable in $y_t$ for any horizon $i$ to a one-time impulse at time $t$ to each structural shock, $w_t$, $$\Theta_i \equiv \frac{\partial y_{t+i}}{\partial w_t} \hspace{1cm} i = 0, 1, 2, ..., H$$

The IRF for the response of variable $k$ to an impulse for structural shock $j$ at horizon $i$ is denoted, $$\theta_{kj,i} \equiv \frac{\partial y_{k, t+i}}{\partial w_{jt}}$$

Let the impulse responses to the reduced form residuals be similarily denoted as, $$ \Phi_i \equiv \frac{\partial y_{t+i}}{\partial u_t} \hspace{1cm} i = 0, 1, 2, ..., H \\ \phi_{kj,i} \equiv \frac{\partial y_{k, t+i}}{\partial u_{jt}} $$

And we can map between the two using the $B_0$ impact matrix, $$ \Theta_i = \Phi_i B_0^{-1} \\ \theta_{kj,i} = [\Phi_i B_0^{-1}]_{kj} $$

Letting $\Phi_{k*,i}$ denote the $k$th row of $\Phi_i$ and $B_{0,*j}^{-1}$ denote the $j$th column of $B_0^{-1}$, then

$$\theta_{kj,i} = \Phi_{k*,i} B_{0,*j}^{-1}$$

Forecast Errors

Let $F_{t+h}$ denote $h$-step ahead forecast errors, $$F_{t+h} \equiv y_{t+h} - y_{t+h|t}$$

Which, for a VAR process, can be denoted in terms of the redcued form IRFs, $\Phi_i$, $$F_{t+h} = \sum_{i=0}^{h-1} \Phi_i u_{t+h-i} = \sum_{i=0}^{h-1} \Phi_i B_0^{-1} w_{t+h-i}$$

Forecast Error Variance (FEV)

The FEV, also known as the predicted mean squared error, is then calculated as $$ FEV_h \equiv E[F_{t+h}F_{t+h}^\prime] \\ = \sum_{i=0}^{h-1} \Phi_i \Sigma_u \Phi_i^\prime = \sum_{i=0}^{h-1} \Phi_i B_0^{-1} B_0^{-1^\prime} \Phi_i^\prime = \sum_{i=0}^{h-1} [\Phi_i B_0^{-1}] [\Phi_i B_0^{-1}]^\prime $$

The FEV for impulse $j$ contribution to variable $k$ is then denoted as, $$ FEV_{kj,h} = \sum_{i=0}^{h-1} [\Phi_i B_0^{-1}]_{jk} [\Phi_i B_0^{-1}]_{jk}^\prime \\ = \sum_{i=0}^{h-1} \Phi_{k*,i} B_{0,*j}^{-1} [\Phi_{k*,i} B_{0,*j}^{-1}]^\prime \\ = \sum_{i=0}^{h-1} \Phi_{k*,i} B_{0,*j}^{-1} B_{0,*j}^{-1\prime} \Phi_{k*,i}^\prime $$

Since $\Phi_{k*,i} B_{0,*j}^{-1}$ is a single number (not a matrix), we can rearrange to $$ = \sum_{i=0}^{h-1} B_{0,*j}^{-1\prime} \Phi_{k*,i}^\prime \Phi_{k*,i} B_{0,*j}^{-1} \\ = B_{0,*j}^{-1\prime} \sum_{i=0}^{h-1}\left[\Phi_{k*,i}^\prime \Phi_{k*,i}\right] B_{0,*j}^{-1} $$

Max Share Identification Method - Time Domain

Pick a structural shock $j$ by choosing the weights for $B_{0,*j}^{-1}$ that maximize the forecast error variance for the horizon in $[h^-, h^+]$

Solve the maximization problem: $$\max_{B_{0,*j}^{-1}} \sum_{h^-}^{h^+} \text{FEV}_{kj,h}$$

This is equivalent to solving $$\max_{B_{0,*j}^{-1}} B_{0,*j}^{-1\prime} \sum_{h^-}^{h^+} \sum_{i=0}^{h-1}\left[\Phi_{k*,i}^\prime \Phi_{k*,i}\right] B_{0,*j}^{-1}$$

Which is maximized when $B_{0,*j}^{-1}$ is the eigenvector associated with the largest eigenvalue of the matrix $\sum_{h^-}^{h^+} \sum_{i=0}^{h-1}\left[\Phi_{k*,i}^\prime \Phi_{k*,i}\right]$.

There’s a piece missing here, which is that $B_{0,*j}^{-1}B_{0,*j}^{-1}$ can be decomposed into a Choleskey matrix and rotation matrix Q, then the maximization problem becomes picking a column of Q.

Max Share Identification Method - Frequency Domain

Pick a structural shock $j$ by choosing the weights for $B_{0,*j}^{-1}$ that maximize the forecast error variance for the frequencies in $[\omega^-, \omega^+]$

Solve the maximization problem: $$\max_{B_{0,*j}^{-1}} \int_{\omega^-}^{\omega^+} \text{FEV}_{kj,\omega} d\omega$$

This is equivalent to solving $$\max_{B_{0,*j}^{-1}} B_{0,*j}^{-1\prime} \int_{\omega^-}^{\omega^+} \Phi_{k*,\omega}^\prime \Phi_{k*,\omega} \ d\omega \ B_{0,*j}^{-1}$$

where $\Phi_{k*,\omega}$ is the impulse response function for a specific frequency $\omega$. The next sections show how to calculate this value.

Solving for IRF in frequency space

Note: One approximate solution is to calculate the time domain IRF out to long horizon (say 1000 periods), then take the fourier transformation, keep only the frequencies you want, square the contributions, and maximize the real value portion.

Representing VAR(p) as a VAR(1)

First it’s easiest to work with the VAR(1) process instead of the VAR(p) ²

. . .

$$Y_t = \upsilon + A Y_{t-1} + U_t$$

. . .

MA($\infty$) Representation

Next, represent as a moving average. ³

. . .

$$y_t = C(L)^{-1} \upsilon + C(L)^{-1} u_t$$

where $C(L)^{-1} \equiv \sum_{i=0}^{\infty} \phi_i L^i$

. . .

Calculating the spectrum for an MA($\infty$)

Now, we can calculate the spectrum using ⁴

. . .

$$s_Y(\omega) = \frac{1}{2\pi} \sigma^2 C(e^{-i\omega})C(e^{-i\omega})$$

The key here is that we end up getting to, $$C(e^{-i\omega}) = (I - A_1 e^{-i\omega})^{-1}$$

which is essentially solving out for the entire path of IRFs by inverting, and making the Fourier transformation at the same time.

. . .

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Resources

Hamilton, James D. Time Series Analsysis. Chapter 6: Spectral Analysis. Pg 152 - 172. In particular, page 154 for deriving the population spectrum for a MA(infinity) process.

Kilian, Lutz and Lutekepohl Helmut. Structural Vector Autoregressive Analysis. (2017). In particular,

• Chapter 1: Introduction for SVAR notation.
• Chapter 2: Vector Autoregressive Models. Pg 19 - . Specifically, page 25 for representing a VAR(p) process as a VAR(1) process.

Angeletos, George-Marios, Fabrice Collard, and Harris Dellas. (2020). “Businsess Cycle Anatomy”. American Economic Review, 110(10): 3030 - 3070. In particular, pages 3036 - 3037 for derivations.