Structural vector autoregression may be the most practical model class for empirical macroeconomics. Yet, it can also be employed for macro trading strategies, because it helps identifying specific market and macro shocks. For example, SVAR can identify short-term policy, growth or inflation expectation shocks. Once a shock is identified it can be used for trading in two ways. First, one can compare the type of shock implied by markets with the actual news flow and detect fundamental inconsistencies. Second, different types of shocks may entail different types of subsequent asset price dynamics and may, hence, be a basis for systematic strategies.

The below is a highly simplified introduction to structural VAR and its strategies to identify specific shocks. Most quotes and the post’s basic structure are from the superb summary of Lutz Kilian of 2011. All links are at the end of the post.

*The post ties in with this site’s lecture on macro information efficiency.*

*The below are excerpts from the paper. Headings and cursive text have been added for context and convenience of reading.*

### What is structural vector autoregression (SVAR)?

“The __original meaning of a ‘structural’ model in econometrics is [ that]…it allows us to predict the effect of ‘interventions’ — deliberate policy actions, or changes in the economy or in nature of known types__. To make such a prediction, the model must tell us how the intervention corresponds to changes in some elements of the model (parameters, equations, observable or unobservable random variables), and it must be true that the changed model is an accurate characterization of the behavior being modeled after the intervention.” [Sims]

“Structural vector autoregressive (VAR) models continue to be the workhorse of empirical macroeconomics and finance…VAR models were first proposed by Sims (1980a) as an alternative to traditional large-scale dynamic simultaneous equation models…The success of such VAR models as descriptive tools and to some extent as forecasting tools is well established.” [Kilian]

*SVAR is a model class that studies the evolution of a set of connected and observable time series variables, such as economic data or asset prices…SVAR assumes that all variables depend in fixed proportion on past values of the set and new structural shocks. This means that the observable variables are endogenous while shocks are the impulses that move the system. The shocks have economic interpretation, such as unexpected policy changes or disruptions in production. A SVAR allows for as many types of shocks as there are time series variables in the set. Unlike in regression, a shock is not assigned to an observable variable: any type of structural shock can have an impact on any variable.*

*In practice one cannot directly observe these structural shocks and their impact on various observed variables. One can, however, observe the set of endogenous variables overtime. Hence, one can estimate how their present values have been related to post values and, by using the latter as predictor of the former, get a set of observable prediction errors. This looks like ordinary vector autoregression, which here is interpreted as a reduced form of the structural vector autoregresssion. The challenge is to ‘translate’ observable reduced-form forecast errors into structural shocks.*

“We call ‘structural’ a model in which we assume that the one-step-ahead prediction errors from a statistical model can be thought of as linear functions of the structural shocks.” [Lucchetti]

*The essence of SVAR is to obtain structural parameters and structural shocks based on observing the reduced form VAR. Without further well-founded economic assumptions, called restrictions, this would not be possible: the SVAR would not be identified. There is not enough information (estimable parameters) in the VAR to deduct from them all the parameters of the SVAR.*

“Such restrictions may take the form of exclusion restrictions, proportionality restrictions, or other equality restrictions. The most common approach is to impose zero restrictions on selected elements of the coefficient matrix that links structural shocks to observable variables.” [Kilian]

### How to identify meaningful shocks?

*Method 1: Short-run restrictions*

“One popular way of disentangling the structural innovations [*structural shocks*] from the reduced-form innovations [*forecast errors*] is to ‘orthogonalize’ the reduced-form errors. __Orthogonalization here means making the errors uncorrelated__. Mechanically, this can be accomplished [by forming a] the lower-triangular matrix with positive main diagonal [based on]…a Cholesky decomposition of the variance-covariance matrix [of the reduced-form shocks…

The distinguishing feature of ‘orthogonalization’ by Cholesky decomposition is that the resulting structural model is recursive…__This means that we impose a particular causal chain__…The ‘orthogonalization’ of the reduced-form residuals by applying a Cholesky decomposition is __appropriate only if the recursive structure…can be justified on economic grounds__…Unless we can come up with a convincing rationale for a particular recursive ordering, the resulting VAR impulse responses, variance decompositions, and historical decompositions are economically meaningless.” [Kilian]

“Not all structural VAR models have a recursive structure. Increasing skepticism toward atheoretical recursively identified models…stimulated a series of studies proposing explicitly structural models identified by non-recursive short-run restrictions.” [Kilian]

“There are a number of potential sources where the economic rationale of identifying restrictions…comes from. One is economic theory… we may wish to impose the structure provided by a specific economic model… [*or*] an encompassing model that includes as special cases various alternative structural models… Often there is no fully developed theoretical model available in which case identification may be achieved by…selective insights…:

**Information delays**: Information may not be available instantaneously because data are released only infrequently, allowing us to rule out instantaneous feedback [*which is why economic data surprises can be assumed to be orthogonal to concurrent market moves*]…**Physical constraints**: For example…physical investment responds with a delay [*which holds true even for the largest part of financial asset allocation*]…**Institutional knowledge**: For example, we may have information about the inability of suppliers to respond to demand shocks in the short run due to adjustment costs [*which holds true for most commodity markets*]…**Market structure**:..[*A*] common identifying assumption in empirical work is that there is no feedback from a small open economy to the rest of the world [*which for example allows distinguishing local financial shocks from global shocks*].” [Kilian]

*Method 2: Long-run restrictions*

“One alternative idea has been to impose restrictions on the long-run response of variables to shocks… For example, it has been observed that most economists agree that demand shocks such as monetary policy shocks are neutral in the long run, whereas productivity shocks are not…The structural VAR representation…[*has mathematically a*] corresponding structural vector moving average (VMA) representation [*which represents the current set of observed variables as an infinite moving average of past structural shocks*]… What does it mean to impose restrictions?…[*It*] means that __[a] variable is not affected in the long run by [ a specific] structural innovation__.” [Kilian]

“There…are serious concerns about the reliability of long-run restrictions: One weakness of VAR models identified by long-run restrictions is that they require an accurate estimate of the impulse responses at the infinite horizon.” [Kilian]

*Method 3: Identification by sign restrictions*

“Structural shocks are identified by restricting the sign of the responses of selected model variables to structural shocks… __Identification in sign-identified models requires that each identified shock is associated with a unique sign pattern__… Unlike traditional exclusion restrictions, such sign restrictions can often be motivated directly from economic theory…There is a misperception among many users that these models are more general and hence more credible than VAR models based on exclusion restrictions. This is not the case. Note that sign-identified models by construction are more restrictive than standard VAR models in some dimensions.” [Kilian]

“While it is difficult to impose sign restrictions directly on the coefficient matrix of the model, it is easy to impose them ex-post on a set of orthogonalised impulse response functions. Thus, __sign restrictions essentially explore the space of orthogonal decompositions of the shocks to see whether the responses conform with the imposed restrictions__… The steps involved in recovering the structural shocks, given a set of sign restrictions, can be summarised as follows:

Step 1: Run an unrestricted VAR in order to get [empirical coefficient estimates]

Step 2: Extract the orthogonal innovations from the model using a Cholesky decomposition. The Cholesky decomposition here is just a way to orthogonalise shocks rather than an identification strategy.

Step 3: Calculate the resulting impulse responses from Step 2.

Step 4: Randomly draw an orthogonal impulse vector…

Step 5: Multiply the responses from Step 3 times the impulse vector and check if they match the imposed signs.

Step 6: If yes, keep the response. If not, drop the draw.

Step 7: Repeat Steps 2-6.”

[Danne 2015]

*Sign restrictions are an obvious way of identifying monetary policy shocks, as they tend to drive interest rates and equity prices in different directions. They also hold potential for distinguishing growth and inflation shocks on financial markets.*

“A fundamental problem in interpreting VAR models identified based on sign restrictions is that __there is not a unique point estimate of the structural impulse response functions__. Unlike conventional structural VAR models based on short-run restrictions, sign-identified VAR models are only set identified. This problem arises because __sign restrictions represent inequality restrictions__. The cost of remaining agnostic about the precise values of the structural model parameters is that the data are potentially consistent with a wide range of structural models that are all admissible in that they satisfy the identifying restrictions. Without further assumptions there is no way of knowing which of these models is most likely… Penalty functions help in assessing worst case (or best case) scenarios, based on the set of admissible models, but the results are best thought of as providing evidence that some outcome is possible rather than that it is true or that it is the most likely outcome.” [*Kilian*]

*Method 4: Financial market information*

“[*One can*] identify monetary policy shocks in monthly VAR models based on high-frequency futures market data. Using the prices of daily federal funds futures contracts…[*one can*] measure the impact of the surprise component of Federal Reserve policy decisions on the expected future trajectory of interest rates. It is shown how this information can be used to identify the effects of a monetary policy shock in a standard VAR.” [*Kilian*]

“The procedure involves two key steps: __First… use the futures market to measure the response of expected future interest rates to an unexpected change in the Federal Reserve’s target rate__…This interpretation requires that risk premia remain unchanged…In other words, no other news move the market on that day and the policy announcement itself does not reveal information about other structural shocks. In the second step…__impose that the impulse responses of the funds rate to the monetary policy shock in the VAR model must match the response measured from the futures data__.” [*Kilian*]

Sources:

Danne, Christian (2015), “The VARsignR package”

Kilian, Lutz (2011), “Structural Vector Autoregressions”.

Lucchetti, Jack (2015), “The SVAR package”.

Pfaff, Bernhard (2016), “VAR, SVAR and SVEC Models: Implementation Within R Package vars”.