Preparing the common identification framework
Drawing the interest-rate and S&P 500 histories before the rotations enter.
Estimating a VAR gives forecast surprises and their covariance, but it does not by itself tell us which surprise is a monetary-policy shock or a stock-market shock. This Atlas holds one deliberately simple two-variable VAR fixed, samples 100 sign-normalized, covariance-equivalent shock decompositions, and asks what extra information each identification method contributes. The goal is to make the identifying assumption and the kind of answer it delivers visible.
Read the setup as a controlled experiment. We estimate one reduced-form VAR, keep everything it learns from the data fixed, and vary only the structural interpretation. That separation is the central idea behind every method page.
The VAR produces forecast errors, lag dynamics, and a residual covariance matrix.
Each angle gives different candidate shocks and IRFs while fitting the same residual covariance.
A timing, sign, historical, proxy, variance, or distributional idea filters or selects rotations.
The interest-rate series gives the policy side of the example. The S&P 500 series gives a stock-market side. For the VAR we work with the rate and monthly S&P 500 log growth, but the first picture keeps the original market level visible too.
Read this chart as the visible observed history before the VAR converts the series into forecast errors.
A VAR(4) is a compact forecasting system for the two observed variables. "4" means each equation uses four monthly lags of the interest rate and four monthly lags of S&P 500 log growth.
The residual ut is whatever the lagged system did not predict.
These ut shocks are useful, but they are not yet structural. They are still mixtures of the two deeper shocks we want to think about.
The chart shows what the VAR did not predict. These residuals are the objects the SVAR will decompose.
We assume there is a true impact matrix B0. It maps two structural shocks, εpolicy,t and εstock,t, into the reduced-form interest-rate and S&P 500 residuals ut. The first shock is the monetary policy shock; the second is the stock-market shock that we want to disentangle from it.
Rows are reduced-form residuals. Columns are structural shocks. Thus b12 is the impact of the stock-market shock on the interest-rate residual, while b21 is the impact of the policy shock on the S&P residual.
The goal is to estimate B0 and recover the true structural shocks εt = B0-1ut.
We do not observe the structural shocks directly, so we normalize their covariance to identity: E[εtεt'] = I. This says they are mutually uncorrelated and each has unit variance. The unit-variance convention fixes shock scale; it does not fix the columns' economic labels or signs.
That normalization is not enough to pin down the shocks uniquely. There are infinitely many ways to split up ut into candidate uncorrelated shocks et(θ) = B(θ)-1ut. The derivation below shows why the same reduced-form covariance is compatible with many rotated decompositions.
1. Factor the residual covariance. Let Σu = E[utut']. Choose any convenient factor P, such as a Cholesky factor, with Σu = PP'.
2. Rotate that factor. For an orthogonal rotation R(θ), define B(θ) = PR(θ). Because R(θ)R(θ)' = I, every rotated impact matrix reproduces exactly the same residual covariance.
3. Recover a different candidate shock pair. Each rotation gives et(θ) = R(θ)'P-1ut, and every pair has identity covariance.
Identification is the missing step Covariance tells us which rotations fit, but economic or statistical information must tell us which columns deserve structural labels.
Use the rotation slider to choose an angle θ. Each angle defines a candidate impact matrix B(θ) and therefore a candidate decomposition et(θ) = B(θ)-1ut. No matter which angle you pick, the recovered shocks remain uncorrelated. This is the key identification problem: the uncorrelated-shocks assumption alone leaves infinitely many admissible candidates.
Move the dial to see the same reduced-form residuals become different candidate structural shocks.
Impulse response functions are a visual summary of the structural moving-average representation. They show how each variable moves over subsequent horizons after a one-time structural shock today. The bridge from the fitted VAR to these responses is the moving-average representation.
The reduced-form coefficients Φh stay fixed. Rotating B(θ) changes the structural responses Ψh(θ).
Move the rotation dial and use the blue line to follow one admissible IRF story. Every displayed set remains consistent with uncorrelated candidate shocks.
This is the picture you will keep seeing below. Its 100 pale lines sample the sign-normalized IRFs implied by the shared shock-covariance normalization before any extra identifying restriction is imposed.
Choose a rotation θ to highlight one member of the cloud. The pale lines show the sampled candidates, and the highlighted line shows which IRF belongs to the active B(θ). The identification sections below reuse this same cloud and ask which additional economic idea should select from it.
A forecast-error variance decomposition asks how much of variable i's horizon-H forecast uncertainty is attributed to structural shock j. It uses the same structural responses Ψh(θ) as the IRF plots, but squares and adds them over horizons.
Here the table focuses on the interest-rate variable. The two columns show how its forecast-error variance is split between the policy and stock-market shocks at the current rotation.
At this point every sampled rotation is admissible because every one recovers unit-variance, mutually uncorrelated candidate shocks. Each sampled rotation also implies different IRFs and FEVD shares. Identification begins when we add one economic idea and turn it into a loss, score, or zero-violation rule over the same grid of 100 rotations.
FEVD itself is a downstream diagnostic: it says how an already selected structural story allocates forecast uncertainty. The max-share method later turns one FEVD share into an objective, which is an extra identifying assumption rather than a fact supplied by the reduced-form VAR.
The table below previews what changes from method to method: which empirical variation is used, what restriction or objective is imposed, how the rule selects from the rotation grid, and whether the output is a single candidate or a set of admissible candidates.
The common rotation grid is a teaching device. It lets you compare methods by holding the reduced form fixed and changing only the additional identifying information.
The fitted VAR supplies residuals, dynamics, and Σu. It does not supply economic shock labels.
Look for the timing claim, sign, dated event, proxy moment, target objective, distributional shape, or volatility shift that orients the rotation.
A method may select one rotation, retain a set, identify only one column, or recover anonymous columns that still require economic labels.
Core distinction Recovery asks which rotation fits the maintained information; labeling asks why a recovered column has an economic name; inference asks how uncertain the resulting object is.
Use the Atlas as an entry point into the identification literature, not as a substitute for the estimators, rank conditions, and uncertainty analysis used in research.
Sims (1980) made unrestricted multivariate dynamics, innovations, IRFs, and variance decompositions central to empirical macroeconomics. The enduring boundary is that an orthogonalization is not an economic interpretation by itself: causal shock labels require additional restrictions or information.
Rubio-Ramirez, Waggoner, and Zha (2010) express observationally equivalent structural parameters as orthogonal rotations and give global rank conditions and algorithms for exact restrictions. The Atlas reduces that general geometry to one angle in a two-shock system so every candidate can be displayed.
Short-run and long-run zeros, signs and narratives, external instruments, declared objectives, non-Gaussianity, and volatility shifts add different kinds of information. Applied papers often combine them, for example signs with proxies or conventional restrictions with statistical identification.
The display fixes a bivariate VAR, samples only 100 rotations, and reports no sampling, prior, or weak-identification uncertainty. It also assumes the chosen information set can recover the relevant shock. Real applications must address specification, invertibility, normalization, labeling, and inference.
The code examples use simulated bivariate data so readers can run the same identification ideas without relying on private project data. The panels below show the runnable Matlab bridge: the all-method runner, the shared setup, and small plotting utilities. Each method page contains its own identification loss and selection logic.
apps/atlas/source/matlab.atlas_run_all_method_demos to recreate every method figure set.atlas_setup.m, and the method scripts you want.Use the ZIP button below for one complete runnable folder with every method. If an embedded preview blocks downloads, open the page in a regular browser.