Combinations of equity returns and yield-curve changes can be used to classify market-implied underlying macro news. The methodology is structural vector autoregression. Theoretical ‘restrictions’ on unexpected changes to this multivariate linear model allow identifying economically interpretable shocks. In particular, one can distinguish news on growth, monetary policy, common risk premia and hedge premia. Monetary and growth news capture shocks to investors’ expectations of discount rates and cash flows, respectively. The common risk premium is a price for exposure to risks that drive stock and bond returns in the same direction. The hedge premium is a price for exposure to risks that drive stock and bond returns in opposite directions. Identifying shocks helps to uncover trading opportunities, including market trends and reversion of relative market returns that were inconsistent with actual macro developments.

The main source of this post is: Cieslak, Anna and Hao Pang (2020), “Common shocks in stocks and bonds ”. Other sources are mentioned with links.

The below are condensed annotated quotes. Cursive text and text in brackets have been added for clarity.

The post ties up with this site’s summary on quantititative methods for macro information efficiency.

### What is structural vector autoregression?

“Vector autoregression (VAR) extends the idea of univariate autoregression [linear regression of one variable onto its own lagged values] to [mutiple] time-series regressions, where the lagged values of all series appear as regressors. Put differently, in a VAR model we regress a vector of time series variables on lagged vectors of these variables.” [Econometrics with R]

“Structural vector autoregressions (SVAR) are __a multivariate, linear representation of a vector of observable [variables] on its own lags and (possibly) other variables__ as a trend or a constant. Structural vector autoregressions make explicit identifying assumptions to isolate estimates of policy and/or private agents’ behavior and its effects on the economy while keeping the model free of the many additional restrictive assumptions needed to give every parameter a behavioral interpretation.” [Fernández-Villaverde, Rubio-Ramírez]

“Structural vector autoregressions offer an attractive approach to estimation. They promise to coax interesting patterns from the data that will prevail across a set of incompletely specified dynamic economic models __with a minimum of identifying assumptions__. Moreover, structural vector autoregressions are easy to estimate.” [Fernández-Villaverde, Rubio-Ramírez]

“While in a regular unrestricted VAR, we allow the data to speak and act for itself, the structural VAR, on the other hand, imposes key restrictions that set conditions as to how certain variables would behave….__We…differentiate from the VAR by adding restrictions__…The matrix of restrictions [refers to] contemporaneous shocks affecting the variables in the systems…__Restrictions are defined by economic principle__.” [Eloriaga]

A practical video on performing structural autoregression in R can be found here:

Structural vector autoregression (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. [previous SRSV post ‘Using SVAR for macro trading strategies’]

### How to use bond and equity markets to detect macro shocks?

“We propose a new approach to identify economic shocks (monetary, growth, and risk-premium news) from stock returns and Treasury yields…Our approach combines the finance perspective…that studies cash-flow and discount-rate news as drivers of asset prices with the macro view that focusses on structural disturbances…We isolate __four structural (orthogonal) shocks—growth news, monetary news, and two distinct shocks generating time-varying risk premiums__—by exploiting their differential impacts on stocks and yields across maturities.”

“Monetary and growth news capture shocks to investors’ expectations of discount rates and cash flows, respectively. __Good growth news raises both stock prices and yields, while good monetary news (easing) raises stock prices but depresses yields.__ Importantly, the two risk-premium shocks also differ in the direction of the co-movement between stocks and yields that they generate. __Shocks that produce a positive co-movement in equity and bond risk premium reflect the fact that stocks and bonds are both exposed to pure discount-rate news. Shocks that drive risk premiums on stocks and bonds in opposite directions arise from bonds providing a hedge for cash-flow risk in stocks__. We refer to these shocks as the **common premium** and the **hedging premium**, respectively. Both risk-premium shocks work to affect stock prices in the same way (positive shocks lower stock prices), but they have opposite effect on bonds (a positive common premium shock lowers bond prices and raises yields, while a positive hedging premium shock does the opposite).”

“Our **identification strategy** involves __two sets of restrictions__—on the co-movement between stocks and yields and on the effect that different economic shocks have on the yield curve across maturities. __The cross-maturity restrictions serve to separate shocks driving short-rate expectations from shocks to the risk premium__. The __co-movement restrictions further distill shocks to short-rate expectations into monetary and growth news__ and __risk-premium shocks into an exposure that is common to stocks and bonds and a hedging component__. The cross-maturity restrictions derive from the identity that long-term yields are conditional expectations of average future short rates plus the risk premium. To the extent that shocks to short-rate expectations—monetary and growth news in our setting—are mean-reverting (although can be persistent), they affect the short end of the yield curve more strongly than the long end. The strength of relative responses of yields at different maturities can thus be exploited to isolate the risk-premium shocks.”

“Accounting for two dimensions in the risk premium—the common and the hedging premium—is important for understanding the stock-bond dynamics…Countercyclical variation typically found in the equity risk premium is less clear for bonds, whose expected returns tend to vary at a frequency higher than the business cycle. Accordingly, successful predictors of bond returns have a much poorer predictive power for stock returns and vice versa.”

“We implement the above ideas via **sign restrictions**…on a…structural vector autoregressions [model]. __Sign restrictions allow us to convert reduced-form innovations in asset prices into structural shocks __that have a particular economic interpretation without imposing parametric structure of a fully-specified asset pricing model. Using information from asset prices alone, __we obtain structural shocks at the daily frequency over a period spanning three-and-a-half decades__.”

### Which macro shocks matter most?

“We analyze the overall importance of different shocks for the dynamics of stocks and yields. Using variance decompositions of daily yield changes and stocks returns, we show that __from 1983 to 2017, about 80% of the variance of the two-year yield changes is driven by monetary and growth news (roughly equal in shares)__. These proportions reverse for the ten-year yield changes of which 80% are explained by premium shocks (split into 45% and 35% contributions of the common premium and the hedging premium, respectively).”

“The __risk-premium shocks also constitute the main portion (about 55%) of the variation in stock returns__, with growth news accounting for about 25% and monetary news for less than 20% of the stock market variance. Analyzing the sources of the time-varying co-movement between stocks and yields, we attribute the change in stock-yield correlations from negative to positive in the late 1990s to a diminished role of the common premium and monetary shocks (both of which drive stocks and yields in the opposite direction), and an increased importance of growth and, in particular, hedging premium shocks (both of which drive stocks and yields in the same direction).”

“Observable macroeconomic variables explain a relatively small fraction of stock-bond correlations over time and suggest that risk premiums drive a significant part of the co-movement.”

### Explaining market returns on FOMC and payroll days

“We document a pronounced effect of risk-premium news on stocks and bonds on FOMC announcement days and over the full cycle between FOMC meetings. From 1994 to 2017, __the average close-to-close stock market return on FOMC days is nearly 30 basis points (bps) higher relative to all other days__, but the ten-year Treasury bond return is not significantly changed.”

“__Risk premiums drive the seemingly puzzling behavior of bonds vis-a-vis stocks__. We find that reductions in both sources of risk premium (the common and hedging premium) contribute to raising stock prices on FOMC days. Risk-premium shocks generate nearly 70% of the average FOMC-day increase in stock returns, while monetary easing shocks account for 25%. Importantly, individual shocks have a significant impact on bonds. Reductions in the common premium increase the FOMC-day return on the ten-year Treasury by 8 bps, and negative monetary shocks add another 3 bps. However, those gains are offset by a decline in the value of the hedging premium, which depresses bond prices, and thus makes the overall bond market response economically small and statistically insignificant.”

“FOMC-day returns are part of a regular pattern of high average stock returns earned in ‘even weeks’ in the FOMC cycle time…This pattern is causally related to the Fed’s decision making and works through the Fed being able to reduce the equity risk premium…The impact of risk-premium shocks is about 3.5 times stronger than that of monetary shocks.”

“[Also] non-farm payroll releases induce significant updates to investors’ expectations about monetary policy. Specifically, while during contractions investors read non-farm payroll numbers as revealing information about the actual state of the economy, in expansions they view them primarily as news about discount rates. This fact accounts for the stock market frequently rising on bad employment news in good times.”