causality in structural vector autoregressions: science or ......svar and other time series methods...

Causality in Structural Vector Autoregressions: Science or Sorcery?

Dalia Ghanem and Aaron Smith

May 31, 2018

Abstract

This paper offers a simple presentation of structural vector autoregressions (SVARs)

for estimating causal effects in applied economics. We emphasize connections between

SVARs and instrumental variables (IV), both of which aim to extract exogenous vari-

ation from endogenous variables. We show that the population analogue of the Wald

IV estimator is identical to the ratio of two impulse responses from an SVAR. We then

present an SVAR analysis of Roberts and Schlenker (2013), which employed IV to es-

timate supply and demand of agricultural commodities. The SVAR reveals additional

insights beyond IV. We highlight the assumptions required to gain these insights.

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... though large-scale statistical macroeconomic models exist and are by some criteria

successful, a deep vein of skepticism about the value of these models runs through that part of

the economics profession not actively engaged in constructing or using them.

— Christopher Sims, Econometrica, 1980

1 Introduction

Almost forty years ago, Sims (1980) proposed the structural vector autoregression (SVAR)

model to replace empirical macroeconomic models that had lost credibility. SVARs have

become the staple method for generating causal estimates from time series, but skepticism

lurks among many economists. The above quote from Sims’ paper now applies to the SVAR.

This paper aims to de-mystify SVARs from the point of view of an applied microeconomist.

We do so by first showing a close connection between SVARs and the linear instrumental

variables (IV) model and then applying SVAR methods to an empirical problem previously

examined using IV (Roberts and Schlenker, 2013). Using this application, we show additional

insights that can be gained from the SVAR, and we elucidate the model specification and

identification assumptions required to gain these insights.

The motivation for this paper is twofold. First, we hope to encourage the “osmosis” of

SVAR and other time series methods into applied microeconomics and microeconometrics.

With the advent of high-frequency data, many traditionally “cross-sectional” applications

now have a time series component. Methods from the time series literature are potentially

useful in these settings. Second, this paper is a step towards bridging the gap between

micro- and macroeconometrics. By doing so, we hope to encourage further developments

and applications of robust causal inference methods in settings with a time series dimension.

Recent examples of such work include Angrist et al. (2016), Montiel-Olea et al. (2016),

Bojinov and Shephard (2017), Gafarov et al. (2018) and Baumeister and Hamilton (2017).

The SVAR is an important tool to address causal questions in time series that do not

fit neatly into the potential outcomes framework with a discrete treatment variable (Ru-

bin, 1974; Imbens, 2014). Rather, there are typically multiple continuous variables that

are serially correlated and potentially mutually dependent. The structure imposed by the

SVAR on this vector of endogenous variables defines the shocks, which are interpreted as

the exogenous components of the variables in question. These shocks mark the beginning

of treatment paths and play the role of “randomly assigned” treatments. Impulse response

functions (IRFs) quantify the effects of each shock on each variable in the model over time,

and are hence referred to as “dynamic causal effects” (Stock and Watson, 2017a). As such,

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IRFs show the short- and long-run effects and therefore give a richer view of the relationship

between the shocks and the variables in the system than a single treatment effect.

We compare triangular SVARs and linear IV models to illustrate important similarities

and differences between the two models.1 Both models share a common goal, which is to

extract exogenous variation from endogenous variables. We also formalize their similarity by

showing that the population analogue of the Wald IV estimator is identical to a ratio of two

contemporaneous impulse responses from an SVAR under certain conditions. We illustrate

this result empirically.

Next, we present an SVAR model of global supply and demand of agricultural commodi-

ties, which was examined in Roberts and Schlenker (2013) using IV. To identify the impulse

response functions, we exploit the natural sequence of events enforced by the agricultural

growing season. Farmers plant crops at the beginning of the growing season, then weather

events affect yields, which subsequently influence wholesale traders’ inventory decisions and

result in an equilibrium price. In presenting the results, we address common questions re-

garding model selection choices made in the SVAR, such as the number of lags and the

functional form of the linear trends. We show how to perform and interpret robustness

checks of the baseline SVAR results and the identification assumptions.

Our empirical results illustrate the additional insights that can be gained from the SVAR

relative to IV. Most notably, the SVAR reveals how long the shocks (or treatments) persist,

which is important in interpreting the implied demand and supply elasticities. For instance,

our SVAR model contains two distinct supply shocks. The first is a weather-induced shock

that affects production for a single year and the second is a change in land allocated to crops,

which tends to persist for multiple years. The weather shock is essentially the instrument

used in Roberts and Schlenker (2013), which raises concern that those estimates may not

reflect consumer response to long-lived shocks such as those caused by climate change or

changes in government policy. Our results alleviate this concern by showing that demand

responds similarly to both shocks. On the other hand, our estimated supply elasticities do

vary depending on the persistence of the demand shocks used to identify them. Producers

may respond more to long-lived shocks than one-year shocks because persistent shocks allow

them to make capital investments to increase production. They are less likely to make such

investments if a price shock is expected to persist only for one year.

We focus this paper on point-identified triangular stationary SVARs because this is the

simplest and most common case in the SVAR literature.2 However, an interesting and grow-

1It is important to clarify that we are not referring to SVARs with external instruments here. A recentreview of this approach to estimating impulse responses can be found in Stock and Watson (2017a).

2We discuss non-stationary time series in Section 4.3.2 and partial identification in the conclusion.

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ing literature proposes methods for credible causal inference using weaker assumptions (see

the reviews in Stock and Watson (2016, 2017a,b)). For instance, Baumeister and Hamilton

(2017) present Bayesian inference on SVARs that incorporates model uncertainty. Montiel-

Olea et al. (2016) propose uniform inference procedures for SVARs identified using external

instruments. Inference procedures for partially identified SVARs, e.g. using sign restrictions,

are presented in Gafarov et al. (2018).

Before proceeding, we note the difference between Granger causality and the IRFs in an

SVAR. Granger causality pertains to prediction, whereas establishing a causal effect requires

specific assumptions about what is ceteris paribus. A variable y1 Granger-causes a variable

y2 if the mean squared error of a forecast of y2,t+s for some t, s > 0 is lower when using lags of

both y1 and y2 as predictors rather than lags of y2 only (Hamilton, 1994). Hence, including

y1 “adds” information about future y2. Our discussion of causality in SVARs does not rely

on any restrictions on which variables Granger-cause each other and which do not. Rather,

we focus on answering the question of what is held constant when computing the IRFs of an

SVAR.

The paper is organized as follows. Section 2 serves an expository purpose by explaining

the difficulty of defining treatments in the context of time series econometrics. It also reviews

some recent work that uses the potential outcomes framework in this context. Section

3 introduces the SVAR system as a model for identifying causal effects when treatment

variables are continuous. We compare and contrast the SVAR to the IV model, and address

the question of when we can interpret IRFs as causal parameters. Finally, Section 4 presents

an SVAR analysis of Roberts and Schlenker (2013) and discusses frequently asked questions

about SVAR results and their robustness. Section 5 concludes.

2 Defining the Treatment

The potential outcomes framework, also known as the Rubin Causal Model, has become

the standard lens through which microeconometricians view causality. Time series settings

rarely generate treatments that fit neatly in the potential outcomes framework. A neat fit

would require that only a subset of the observations are treated and that the treatment

timing and magnitude is exogenous to the outcome variable.

In typical time series applications, treatments are not randomly assigned across obser-

vations; every observation is “treated” and the magnitude and persistence of the treatment

varies by observation. Defining treatments and disentangling causal effects may require more

structure and assumptions in these settings. In this section, we first present two time series

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examples that fit neatly in the existing potential outcomes framework. These two examples

clarify how the typical time series setting differs from a typical potential outcomes setting.

2.1 A Discrete Treatment Event

The event study literature provides a close time-series analog to the potential outcomes

framework (MacKinlay, 1997). An event study focuses on a single treatment, or event, that

occurs at a point in time such as an earnings announcement or a scandal. The researcher

uses pre-event data to estimate counterfactual values of the outcome variable if the event

had not occurred.3 For example, Carter and Smith (2007) examine the price effect of a food

scare caused by genetically modified StarLink corn. StarLink corn is a genetically modified

(GM) variety that was only approved for animal feed and non-food industrial products. In

2000, it was found in taco shells and other foods.

To place Carter and Smith (2007) in the potential outcomes framework, denote by t∗ the

date the food scare occurred. Then, define a binary treatment variable Dt that equals one

for t = t∗ and zero otherwise. Thus, there is a single treatment applied on a single date: July

18, 2000. The outcome variable yt is the logarithm of the relative price of corn to sorghum.4

Figure 1 plots daily yt before and after the food scare. The two horizontal red lines indicate

the estimates of E[Yt|t ≥ t∗] and E[Yt|t < t∗]. The difference between them is −0.13, the

question is whether that difference estimates a causal effect.

Define Y 0t as the potential log relative price in the absence of the food scare, and Y 1

t as

the potential log relative price in the presence of the food scare. The event study approach

of Carter and Smith (2007) estimates the causal effect of StarLink contamination if

E[Yt|t ≥ t∗]− E[Yt|t < t∗]=E[Y 1t − Y 0

t |t ≥ t∗].

The term on the right hand side is the difference between the log relative price in the

presence and absence of the food scare in the same period averaged over the periods after

the food scare; it is the treatment on the treated.5 The treatment was an unexpected event,

3Event studies are sometimes referred to as regression discontinuity in time.4Sorghum is a substitute for corn with no GM varieties.5The assumption of stationarity here is of course critical to causal inference. For the above equality to

hold, we must have E[Y 0t |t ≥ t∗] = E[Y 0

t |t < t∗], which is implied by stationarity. Suppose that Yt wasnot stationary, e.g. E[Y dt |t = τ ] = µτ (d) is time-varying, then to obtain a causal effect, we have to beable to observe µτ (0) and µτ (1) simultaneously, which is not possible. We can allow for trend stationarityhowever, suppose that E[Y dt |t = τ ] = µ(d) + f(τ)′δ, where f(τ) is a vector of parametric functions of time.By including f(τ) in our model, we can identify µ(0) and µ(1) from pre- and post-treatment observations,respectively.

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Figure 1: The Effect of a Food Scare (Carter and Smith, 2007)

3.7

3.75

3.8

3.85

3.9

3.95

4

4.05

4.1

4.15

Jan-99 Apr-99 Jul-99 Oct-99 Jan-00 Apr-00 Jul-00 Oct-00 Jan-01 Apr-01 Jul-01 Oct-01

‐0.13y

Notes: Yt is the logarithm of the price ratio of corn and sorghum. Dt is a dummy variable whichequals 1 (0) for any period t after (before) the food scare occurs. The two horizontal red lines indicatethe estimates of E[Yt|t ≥ t∗] and E[Yt|t < t∗].

a ‘shock’. It is hence plausibly independent of the potential outcomes, Dt ⊥ (Y 0t , Y

1t ), which

is the definition of random assignment. This shows how exogenous events in time series

are typically unpredictable; if they were predictable then markets may respond to the event

before it occurs.

The StarLink example includes a discrete treatment, which matches nicely with the

potential outcomes framework, but the treatment did not persist forever. By 2006, testing

by the Environmental Protection Agency found that StarLink had been virtually eliminated

from the U.S. food supply. Thus, the extent of contamination dissipated over time. The

results of Carter and Smith (2007) imply that it had persistent price effects, but it would

be incorrect to assume that observations in the year 2000 experienced the same magnitude

of treatment as those in later years. This feature is common in time series settings, where

a treatment applied to one observation is still present in later observations but at a lower

intensity and its level decays to zero over time.

2.2 Multiple Discrete Treatment Events

In an example closer to that found in a typical time series setting, Angrist et al. (2016) use

the potential outcomes framework in a time-series setting with multiple discrete treatments.

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They estimate the effect of discrete changes in the Federal Funds Rate on macroeconomic

outcomes. They employ propensity score methods to account for the non-random selection

into treatment, which is necessary because the Federal Reserve changes interest rates in

response to macroeconomic conditions.

Angrist et al. (2016) define the treatment variable Dt as a vector of policy variables that

can take values d0, . . . , dJ . They observe multiple realizations of each treatment value. By

averaging over these realizations, they can estimate the treatment effect at various horizons.

Define Yt,h(d) as the potential outcome in period t+ h if policy d is implemented at time t,

and let zt denote the vector of past data on the treatments, the outcome variables, and any

covariates.

We can write the average policy effect of a change from d0 to dj conditional on zt as

E[Yt,h(dj)− Yt,h(d0)|zt] = E[Yt,h|Dt = dj, zt]− E[Yt,h|Dt = d0, zt].

This expression assumes that the potential outcomes are not confounded with the treatment

assignment, specifically Yt,h(d) ⊥ Dt|zt for all h ≥ 0 and for all d. Angrist et al. (2016) use

propensity score methods to obtain the unconditional average policy effect

E[Yt,h|Dt = dj]− E[Yt,h|Dt = d0] = E

[Yt,h

(1Dt = djP (Dt = dj|zt)

− 1Dt = d0P (Dt = d0|zt)

)](1)

The inverse probability weighting in the above equation has an important interpretation

in the context of time series. When P (Dt = dj|zt) is smaller, the occurrence of dj is less

predictable. Hence, it is more plausibly exogenous and is given a higher weight relative to

observations with larger P (Dt = dj|zt). Thus, as in the StarLink example, we observe a

connection between unpredictability and exogeneity.

2.3 The General Case

In most time series applications, the variables, including the ‘treatment variable’, are not

discrete. Non-discreteness does not create problems for causal inference as long as sufficient

assumptions can be imposed. If a non-discrete treatment variable is conditionally indepen-

dent of the potential outcomes and the linear model is correctly specified, then the average

treatment effect can be consistently estimated by ordinary least squares. If the treatment

variable is endogenous but a valid instrument exists, then an average treatment effect may

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be consistently estimated in a linear model by two stage least squares.6

Serial correlation, on the other hand, complicates causal inference because it implies

that treatments and responses persist for multiple periods. If a serially correlated treatment

variable jumps above its mean one period and remains above the mean for several periods,

then we expect economic agents to respond as though they received a single treatment that

lasted multiple periods rather than a sequence of independent treatments. Put differently,

we expect them to respond to the treatment path. In addition to the treatment potentially

lasting for multiple periods, the responses to treatment may also play out over multiple

periods. For example, in response to a gasoline price increase (treatment), consumers may

purchase a more fuel efficient vehicle if they expect prices to remain high for a long period,

but they will not buy a new car if they expect the price increase to be shortlived.7 Thus,

the response of gasoline demand to price varies depending on the persistence of the price

change. Moreover, for a price change of a given duration, the consumer responses will vary

over time. Some consumers may respond to a persistent price change by buying a smaller

car immediately; others will wait and buy a smaller car later. The SVAR provides a way to

extract average treatment paths and dynamic responses from a set of variables.

3 The Structural Vector Autoregression

In this section, we introduce the SVAR using the Roberts and Schlenker (2013) application

(hereafter RS2013) as an example. RS2013 estimate global supply and demand elasticities

for agricultural commodities using an IV approach that lends itself naturally to an SVAR.

RS2013 observe global annual prices, quantities, and yield (production per unit of land)

for corn, wheat, rice, and soybeans. They construct calorie-weighted indexes of price, quan-

tity and yield aggregated across the four commodities. These commodities constitute about

75% of the calories consumed in the world by humans, so this study directly addresses global

supply and demand for food. It is good practice in time series analysis to plot the data.

Such plots may be viewed as the counterpart of summary statistics tables in applied microe-

6In practice, linear models are used as approximations, but to make our discussion of causal effects inlinear models precise, we emphasize the role of correct specification once we have continuous treatmentvariables.

7Bojinov and Shephard (2017) propose a model-free approach to identification, estimation and inferenceon causal effects of treatment paths in time series. Inspired by a large experiment by a quantitative hedgefund, they show how to extend the potential outcomes framework to define treatment paths and potentialoutcomes in order to achieve a completely model-free approach to causal inference solely relying on randomassignment of treatment paths. Their approach is specific to the case of a large number of randomly assignedtreatment paths.

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conomics. The left panels of Figure 2 show the three variables in the RS2013 demand model,

extended to 2013.8 Yield and quantity demanded display increasing trends and price dis-

plays a decreasing trend. These patterns are consistent with long-run technological progress

that improved land productivity thereby increasing production and reducing prices. The

right panels of Figure 2 show each of these series after removing the trend using a cubic

spline with 4 knots. RS2013 argue that yield deviations from trend are driven by weather

and therefore are exogenous to prices and quantities. This argument is the basis of their

identification strategy.

RS2013 use the variables in Panels A2, B1, and C1 in their demand model, and we use

the same three variables in our example below. After introducing the SVAR, we compare

the SVAR specification to the IV approach used in RS2013. Later, in Section 4, we present

a complete SVAR model of demand and supply of agricultural commodities.

3.1 Introducing SVARs

Let qt denote log quantity demanded, pt log of price and wt yield deviations from trend,

which are a proxy for weather. A triangular SVAR with ` lags is given by the following

wt = ρ11Yt−1 + ρ12Yt−2 + · · ·+ ρ1`Yt−` + fw(t) + vwt (2)

pt = β21wt + ρ21Yt−1 + ρ22Yt−2 + · · ·+ ρ2`Yt−` + fp(t) + vpt, (3)

qt = β31wt + β32pt + ρ31Yt−1 + ρ32Yt−2 + · · ·+ ρ3`Yt−` + fq(t) + vqt. (4)

where Yt ≡ (wt, pt, qt)′ and ρij is a 3-dimensional row vector for all i and j. The terms

fw(t), fp(t), and fq(t) are fixed functions of time and capture any deterministic trends in the

above variables. The model is triangular because, conditional on the trends and the lags of

each variable, pt and qt are omitted from the yield equation and qt is omitted from the price

equation. We explain these identification assumptions in Section 3.4.

Using the standard SVAR terminology, we refer to the elements of vt as “shocks”. The

shocks represent the part of the observed variables that (i) cannot be predicted using past

8RS2013 used data from 1962-2007. We update the data through 2013. The raw data on area, productionand yield are obtained from the Food and Agricultural Organization (FAO). Production of maize, rice,soybeans and wheat are measured in tons, then converted into calories using calorie weights from RS2013.We hence convert production tons into calories. We then divide by 365*2000, the number of calories consumedby the average person in a year. Hence, the units of production in our analysis is in millions of people asin RS2013. Yield is production per area and is measured in bushels per ha. The raw price data is obtainedfrom Quandl, and it includes spot and futures prices. The spot and futures price we use are calorie-weightedaverages of the individual commodity prices. For more details on the data and variable construction, seeRS2013 and Hendricks et al. (2014).

9

Figure 2: Time Series Plots of Yield Deviations, Price and Quantity Demanded in Robertsand Schlenker (2013)

Panel A1. Yield Panel A2. Detrended Yield

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1960 1980 2000 2020Year

Panel B1. Consumption Price Panel B2. Detrended Consumption Price

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Panel C1. Quantity Demanded Panel C2. Detrended Quantity Demanded

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Notes: Panels A1, B1 and C1 present the time series plots of logarithm of global yield, the logarithmof the futures price in month of delivery, and the logarithm of global quantity demanded for a calorie-weighted index of corn, rice, soybeans, and wheat. Quantities are defined in units of 2000*365.25 calories,which corresponds to sufficient sustenance for a person for a year. Panels A2, B2, and C2 show the sameseries after detrending using a cubic spline with 4 knots.

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observations (Yt−1, ..., Yt−`) and (ii) is not affected by contemporaneous values of other vari-

ables. As such, they constitute new information that arrives in period t. Based on this view,

we see how the SVAR disentangles treatment paths that begin at different points in time. A

shock entails the beginning of a new treatment path and the lag terms capture continuation

of that path. Importantly, the errors are white noise and uncorrelated with each other, i.e.,

vt = (vwt, vpt, vqt)′|Yt−1, Yt−2, . . . , Yt−` ∼ WN(0, D), where D is a diagonal matrix.

The three equations of the SVAR can be written in matrix notation as follows

A0Yt = A1Yt−1 + A2Yt−2 + · · ·+ A`Yt−` + f(t) + vt, (5)

where A0 is a lower-triangular matrix,

A0 =

1 0 0

−β21 1 0

−β31 −β32 1

. (6)

Multiplying through by A−10 , we can write the reduced-form of the above model, which is a

VAR(`),

Yt = Π1Yt−1 + Π2Yt−2 + · · ·+ Π`Yt−` + g(t) + εt (7)

where g(t) = A−10 f(t), Πj = A−10 Aj for j = 1, . . . , ` and εt = A−10 vt.9 The parameters in (7)

can be estimated consistently by ordinary least squares (Hamilton, 1994).

We can express Yt in vector MA(∞) form as a linear function of current and past struc-

tural errors, vt,

Yt = h(t) +∞∑l=0

Ψlvt−l, (8)

where h(t) = (I −Π1L− ...−Π`L`)−1g(t).10 The MA coefficients are square summable (i.e.,

9The above VAR imposes no zero-restrictions on the Π1,. . . ,ΠL. It is worth noting here that in a bivariateVAR, when one variable (y2) does not Granger-cause the other (y1), then it implies the following zerorestrictions on the coefficient matrix on the lagged vectors (Hamilton, 1994). Specifically,[y1ty2t

]= h(t) +

[π(11)1 0

π(21)1 π

(22)1

] [y1,t−1y2,t−1

]+

[π(11)2 0

π(21)2 π

(22)2

] [y1,t−2y2,t−2

]+ · · ·+

[π(11)` 0

π(21)` π

(22)`

] [y1,t−`y2,t−`

].

In our analysis of the SVAR as a causal tool, we allow the matrices of the lags of Yt to be completelyunrestricted.

10L denotes the backshift, or lag, operator. The MA Coefficients Ψj are functions of the parameters in

11

∑∞j=0 ‖Ψj‖2 <∞) if Yt is covariance-stationary (see Hamilton (1994) for technical conditions

and Section 4.3.2 for a discussion of this assumption). Hence, the triangular SVAR allows

us to decompose a vector of endogenous time series variables into a trend plus a weighted

sum of uncorrelated white-noise shocks, or treatments.

3.2 Impulse Response Functions

IRFs characterize the response of the observed variables to a shock. In this section, we

argue that IRFs can be interpreted as average treatment effects, where the shocks are the

treatments and the current and future observed variables are the outcomes. The IRF is

defined as the partial derivative of Yt+h for some h ≥ 0 with respect to each element of vt,

i.e., ∂Yt+h/∂vjt = Ψjh for j ∈ 1, 2, 3, where j denotes the jth row.

Before explaining further how to interpret IRFs as causal parameters, we first investigate

what they are. In the remainder of this section, we present a static version of the above

model by excluding the control variables, i.e., the trends and Yt−1, . . . , Yt−`. We re-introduce

these elements in Section 4. For now, we focus on the following static SVAR, which is simply

a triangular simultaneous equations model in order to clarify the identification assumptions, 1 0 0

−β21 1 0

−β31 −β32 1

wt

pt

qt

=

vwt

vpt

vqt

(9)

where vt ∼ WN(0,Σ) and Σ is diagonal as in the above. In this simple model, we can

express the dependent variables as a linear combination of uncorrelated shocks as follows wt

pt

qt

=

1 0 0

β21 1 0

β31 + β32β21 β32 1

︸︷︷︸

∂Yt/∂v′t

vwt

vpt

vqt

. (10)

Because this model has no autocorrelation, the IRFs are zero for all h > 0.

The elements of the matrix on the right-hand side of (10) give the contemporaneous im-

pulse responses. For instance, β21 is the impulse response of a yield shock on contemporane-

ous price (∂pt/∂vwt), β32 is the impulse response of other supply shocks on contemporaneous

(7) and can be estimated consistently using a plug-in estimator. Most econometrics software packages havebuilt-in routines to compute these estimates. Alternately, they can be estimated using the local projectionsmethod of Jorda (2005).

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quantity (∂qt/∂vwt), and β31 + β32β21 is the impulse response of a yield shock on contempo-

raneous quantity (∂qt/∂vpt). IRFs give the change in the predicted value of the dependent

variables due to a unit or marginal change in the individual shocks.

3.3 IRFs as Causal Parameters

In a least squares regression, we only consider the slope coefficients as causal estimates when

the regressors are exogenous and the linear model is correctly specified. Hence, a question

about causality is a question about correct specification and exogeneity. To view the IRFs

given in the static triangular system in (10) as causal parameters, we will assume that the

triangular structure is correctly specified.

In our example, yield deviations are not determined by any other variable in the system,

so wt = vwt. As a result, the first equation in (10) is redundant from a causal perspective.11

Considering the price equation, if we assume that E[vpt|vwt] = 0, i.e. yield shocks are

exogenous in the price equation, the resulting conditional expectation for the second equation

is given by

E[pt|vwt] = E[β21vwt + vpt|vwt] = β21vwt. (11)

In this case, β21, the impulse response of pt to vwt, is the marginal effect of a yield shock

on price ∂E[pt|vwt]/∂vwt. Intuitively, since yield shocks do not affect other price shocks, the

change in price that coincides with a yield shock cannot be attributed – even partially – to

other shocks that affect price.

Similarly, for the quantity equation in (10), assuming E[vqt|vwt, vpt] = 0, i.e. all price

shocks are exogenous in the quantity equation, implies

E[qt|vwt, vpt] = (β31 + β32β21)︸︷︷︸∂E[qt|vwt,vpt]/∂vwt

vwt + β32︸︷︷︸∂E[qt|vwt,vpt]/∂vpt

vpt. (12)

It follows that lower off-diagonal elements of the coefficient matrix in (10) are marginal causal

effects.

An important byproduct of the mutual mean independence of the elements of vt is

E[Yt|vwt, vpt, vqt] = E[Yt|vwt] + E[Yt|vpt] + E[Yt|vqt], (13)

11This is not the case when the model includes lags as in (8).

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which implies that the marginal effect of conditional and unconditional expectations are

equal. For instance,

∂E[qt|vwt, vpt]∂vwt

=∂(β31 + β32β21)vwt + β32vpt

∂vwt= β31 + β32β21, (14)

∂E[qt|vwt]∂vwt

=∂(β31 + β32β21)vwt + β32

=0︷︸︸︷E[vpt|vwt]

∂vwt= β31 + β32β21. (15)

Furthermore, mutual mean independence allows the SVAR to identify the impact of

multiple contemporaneous changes, e.g.

E[pt|vwt = vw, vpt = vp]− E[pt|vwt = 0, vpt = 0]

=E[pt|vwt = vw]− E[pt|vwt = 0] + (E[pt|vpt = vp]− E[pt|vpt = 0])

=β21vw + vp. (16)

This is an important feature of SVARs in some applications, where shocks to several variables

in the system may occur at the same time, and a researcher aims to disentangle the effects

of the different shocks. In such cases, it is not sufficient to identify the effect of a change in

a single variable, but also the effect of multiple contemporaneous shocks.

Expressing causal effects as responses to shocks can seem abstract. To make them more

tangible, we place economic labels on the shocks, which is a narrative component of SVAR

analysis akin to the narrative about instrument validity that typically accompanies an IV

identification strategy. We label vwt as a weather shock, and we allow it to affect price

and quantity. We label vpt as non-weather supply shocks and vqt as demand shocks. We

assume that price does not respond to demand shocks, i.e., that supply is perfectly elastic.

This assumption is imposed by the zero element in the second row and third column of

the coefficient matrix in (10). We assume that observed weather does not respond to non-

weather supply shocks or to demand shocks. The assumption of perfectly elastic supply is

clearly false, and we will relax it when we present a full SVAR analysis in Section 4.

By observing how price and quantity respond to weather shocks, we deduce how demand

responds to a particular supply shock (weather). In particular, the elasticity of the demand

response to weather is

∂E[qt|vwt]/∂vwt∂E[pt|vwt]/∂vwt

=β31 + β32β21

β21. (17)

This ratio differs from the elasticity of the demand response to non-weather supply shocks,

14

which is

∂E[qt|vpt]/∂vpt∂E[pt|vpt]/∂vpt

= β32. (18)

Hence, because there are two supply shocks in this model, there are two demand elasticities

produced by the model. Next, we show how this analysis compares to the IV model in

RS2013.

3.4 Triangular SVAR vs. Instrumental Variables

In the previous sections, we explain how the SVAR defines exogenous components of en-

dogenous variables and hence lends the IRFs a causal interpretation. The IV model serves a

very similar purpose in terms of extracting exogenous variation from endogenous variables,

however the two models differ in the assumptions they require to achieve this target. In

this section, we compare and contrast the SVAR and IV models and show that the Wald

estimand is identical to the ratio of two impulse responses formally and empirically.

Figure 3 presents the triangular system in (9) alongside the IV model of demand in

RS2013. The second equation in the IV setup is the ‘first stage regression’ and the third

equation is the equation of interest.12 In both systems, wt is purely a shock that is uncor-

related with other shocks. Specifically, in the IV model the yield deviation is wt = uwt and

in the triangular system the yield deviation is wt = vwt. There are two differences between

the systems. First, the IV model excludes wt from the qt equation, whereas the triangu-

lar model does not. Second, the IV model allows the price and quantity shocks (upt and

uqt) to be correlated (σ23 is unrestricted), whereas the triangular structure imposes that the

variance-covariance matrix of the shocks is diagonal.13

Figure 4 illustrates the identification assumptions graphically. Panel A shows that the

parameter b32 in the IV model is the elasticity of demand; it is the change in log quantity

given a unit change in log price holding demand constant. This parameter is identified

econometrically by the instrumental variable wt, which is valid because it affects price (b21 6=0) but not the demand curve (b31 = 0), and because it is exogenous to price and quantity

12This presentation of the IV model is closely related to the SVAR approach using “external instruments”(Montiel-Olea et al., 2016). The external instrument in our example is wt, which is correlated with pt(−b21 6= 0), but not with qt directly (b31 = 0). According to Montiel-Olea et al. (2016), we can identifyb32 using wt as an “external” instrument in the two-equation SVAR of pt and qt without specifying a fulltriangular system.

13The assumption on the diagonal variance-covariance matrix is typically made while including lags of allvariables in each equation of the model, so it is not as restrictive in practice as it may seem in the staticcase.

15

Figure 3: Instrumental Variables vs. Triangular SVAR: Demand Elasticity

Panel A: IV 1 0 0−b21 1 0

0 −b32 1

wtptqt

=

uwtuptuqt

, Ω =

σ21 0 0

0 σ22 σ23

0 σ23 σ23

.Panel B: Triangular System (Static SVAR)

1 0 0−β21 1 0

−β31 −β32 1

︸︷︷︸

A0

wtptqt

︸︷︷︸

Yt

=

vwtvptvqt

︸︷︷︸

vt

, Σ =

σ2w 0 00 σ2

p 0

0 0 σ2q

.

(b12 = b13 = σ12 = σ13 = 0). In this model, a positive weather shock increases supply, which

reduces price and increases quantity demanded. The potential correlation between the first

stage error (upt) and the error in the demand equation means that price may be endogenous

to demand. Note that, in the IV formulation presented here, the supply elasticity is not

identified because there is no instrumental variable that shifts the demand curve holding the

supply curve constant.

Panel B of Figure 4 illustrates the responses to a weather shock in the SVAR. A unit

weather shock changes price by β21, and it changes quantity by β31 + β32β21 (see (10)). The

parameter β21 represents the coefficient on wt in a least squares regression of pt on wt (see

equation (3)). The parameters β31 and β32 represent the coefficients on wt and pt in a least

squares regression of qt on wt and pt (see equation (4)). Thus, the response of quantity to

a weather shock equals the sum of a direct effect (β31) and an indirect effect that works

through price (β32β21). This is also the coefficient one would obtain from a regression of qt

on wt only.

As shown in (17), the ratio of the quantity and price responses to a weather shock

corresponds to a particular elasticity of demand. Next, we show that this ratio is identical

to the IV estimate of the demand elasticity. Due to the assumptions of the IV model,

specifically the exclusion of wt from the qt equation and the uncorrelatedness of wt = vwt

and vqt, it follows that

cov(qt, wt) = b32cov(pt, wt). (19)

16

Figure 4: Triangular SVAR vs. IVPanel A: IV

PriceSupply

Yield Dev.

Price

Demand

Quantity

Quantity

wtu

ptu

qtu

21b

32b

32b21b

Panel B: SVAR

Price

Yield Dev.

Supply

Price

Demand

Quantity

Quantity

wtv

ptv

qtv

21

31

21

31 32 21

32

Solving for b32 and multiplying and dividing by var(wt), assuming it is strictly greater than

zero, yields the following

b32 =cov(qt, wt)/var(wt)

cov(pt, wt)/var(wt), (20)

which is the population analogue of the Wald estimator. The numerator is the slope coeffi-

cient from the OLS regression of qt on wt. The denominator is the slope coefficient from an

OLS regression of pt on wt.

In the triangular SVAR,

cov(pt, wt) ≡ cov(β21wt + vpt, vwt) = β21var(wt)

cov(qt, wt) = β31var(wt) + β32cov(pt, wt) = (β31 + β32β21)var(wt) (21)

17

Table 1: Demand Elasticity: Triangular System vs. IV

IV SVAR

(1) (2) (3) (4)Dependent Variable: qt pt qt qt

pt -0.063

β32︷︸︸︷0.002

(-2.22) (0.22)

wt

β21︷︸︸︷−4.856

β31︷︸︸︷0.317

β31 + β32β21∧︷︸︸︷

0.306(-5.35) (2.18) (2.28)

Sample Size 52 52 52 52Notes: (1) is estimated using 2SLS with wt as the instrument. (2)-(4) areestimated using OLS. All regressions include flexible time trends modeled usingcubic splines with four knots as in RS2013. The t statistics in parentheses arecomputed using Newey-West standard errors to correct for heteroskedasticityand first-order autocorrelation. Sample: 1962-2013.

These two equalities imply that the population analogue of the Wald estimator is given by

the following

cov(qt, wt)/var(wt)

cov(pt, wt)/var(wt)=β31 + β32β21

β21= b32. (22)

Hence, the population analogue of the Wald estimator equals the ratio of two impulse re-

sponses, specifically the impulse response of quantity and price to a weather shock.

Table 1 presents IV and SVAR estimates of the models in Figure 3 using the updated

RS2013 data. As in RS2013, we model the trend using cubic splines with four knots. To

ease comparison with RS2013, we do not include lags in the estimates in Table 1. Column

(1) reports that the IV estimate of the demand elasticity is −0.063, which is similar to the

analogous estimate of −0.055 in RS2013 (Column (1b) of their Table 1). Columns (2) and

(3) of Table 1 illustrate how to obtain estimates of the parameters in the coefficient matrix

of the triangular SVAR, specifically β21, β31 and β32, from OLS regressions.14 The estimated

response of quantity to a weather shock is presented in Column (4) and equals 0.306, which

could also be constructed from coefficients in Columns (2) and (3). The demand elasticity

computed from the SVAR as in (17) is −0.306/4.856 = −0.063.

Thus, we have shown in this section that, under the RS2013 assumption that yield

14Column (2) is also the first stage regression in the IV model.

18

deviations constitute supply shocks and are exogenous to price and quantity, the IV and

SVAR methods produce identical demand elasticity estimates. The interpretation of these

estimates differs slightly. As written in Figure 3, b32 is the demand elasticity, whereas in the

SVAR, the ratio (β31 + β32β21)/β21 is a demand elasticity. The SVAR captures the demand

elasticity with respect to a weather shock, which may differ from a demand elasticity with

respect to a different supply shock.

We require the assumption of perfectly elastic supply to obtain an estimate of the elas-

ticity of demand with respect to other supply shocks (β32). Table 1 reports this estimate as

0.002, which is not significantly different from zero. If supply is less than perfectly elastic,

which one would expect given that land is a finite resource, then this estimate is biased

upwards. Importantly, a reader could discard this estimate because she does not believe

the identification assumption of perfectly elastic supply, while keeping the demand elasticity

estimate identified by the yield shocks. In the next section, we estimate a full SVAR of

global supply and demand for agricultural commodities that relaxes this assumption. We

then compare the results to the IV estimates in RS2013.

4 SVAR Analysis of Supply and Demand of Agricul-

tural Commodities

RS2013 use IV to estimate supply and demand elasticities for agricultural commodities. In

this section, we present a triangular SVAR model of supply and demand using an updated

version of the same dataset. Quantity supplied is determined by farmer decisions about

how much cropland to plant, i.e. acreage, and by weather realizations which ultimately

determine yield. The difference between the quantity supplied and the quantity demanded

is the change in inventories. Consumption exceeds production in years when inventory is

depleted and production exceeds consumption in years when inventory accumulates. Thus,

the decision on how much inventory to hold across crop years is an important driver of prices.

Moreover, storage arbitrage links prices across crop years; the expected value of next year’s

price equals this year’s price plus the cost of storage.

We exploit the natural annual sequence of these economic decisions, illustrated in Figure

5, to propose a triangular SVAR identification strategy. In February and March, North-

ern Hemisphere farmers choose the amount of land to cultivate (acreage, at) based on last

year’s information. Due to storage arbitrage, last year’s price (pt−1) is a good proxy for the

information on which farmers base their planting decisions. Weather realizations over the

19

Figure 5: Time Line

farmers choose at given

information at t− 1, pt−1

at

yield is realized

yt

wholesale traderschoose it

it

price isdetermined

pt

summer determine the yield (yt), which in turn determines the size of the harvest in the

early fall. Wholesale traders then decide on the amount they will sell to consumers and how

to change inventory (it). These decisions jointly determine the price (pt), which we measure

in November and December.15

This time line of events motivates the following SVAR1 0 0 0

α21 1 0 0

α31 α32 1 0

α41 α42 α43 1

︸︷︷︸

A0

at

yt

it

pt

︸︷︷︸

Yt

=

ρ11 ρ12 ρ13 ρ14

ρ21 ρ22 ρ23 ρ24

ρ31 ρ32 ρ33 ρ34

ρ41 ρ42 ρ43 ρ44

︸︷︷︸

A1

at−1

yt−1

it−1

pt−1

︸︷︷︸

Yt−1

+ΓXt +

vat

vwt

vit

vdt

︸︷︷︸

vt

(23)

where Xt is a vector of cubic spline time trends. The above SVAR only includes the first

lags of all variables. We maintain the assumption that var(vt) = Σ, a diagonal matrix.

All variables are measured in logs. We define it as the log difference between production

and consumption, i.e., a log-linearized estimate of the percentage change in inventory. To

compare the variables in our model to those in RS2013, we note that production (quantity

supplied) equals acreage times yield, hence its log equals at+yt. The supply model in RS2013

is a regression of (at + yt) on an expected price (for which we use pt−1), yield (yt), and the

trend.16 Their demand equation is a regression of (at + yt − it) on pt and the trend.

15This narrative omits the fact that farmers also plant crops in the Southern Hemisphere, where theseasons are opposite to the north. This fact may invalidate the identification strategy. However, results inHendricks et al. (2014) suggest that the endogeneity bias from this assumption is small.

16RS2013 do not use actual yield yt as a control variable in their supply equation. Rather, they use ayield shock, which is log yield minus a trend. Because the supply model controls for trends, these twospecifications are identical if the model used to detrend log yield is that same as the trend specification inthe supply equation. Hendricks et al. (2014) show that the supply elasticity estimates are almost identicalacross the two specifications.

20

Figure 6: Time Series Plots of Acreage, Yield, Inventory and Consumption PricePanel A.1. Acreage Panel A.2. Detrended Acreage

6.1

6.2

6.3

6.4

6.5

Acre

age

(log

of m

illion

s of

ha)

1960 1980 2000 2020Year

-.04

-.02

0.0

2.0

4D

etre

nded

Acr

eage

(log

of m

illion

s of

ha)

1960 1980 2000 2020Year

Panel B.1. Yield Panel B.2. Detrended Yield

1.6

1.8

22.

22.

42.

6Yi

eld

(log

of p

eopl

e pe

r ha)

1960 1980 2000 2020Year

-.04

-.02

0.0

2.0

4Yi

eld

Shoc

k (lo

g of

peo

ple

per h

a)

1960 1980 2000 2020Year

Panel C.1. Inventory Panel C.2 Detrended Inventory

-.05

0.0

5In

vent

ory

(per

cent

age

diffe

renc

e)

1960 1980 2000 2020Year

-.06

-.04

-.02

0.0

2.0

4D

etre

nded

Inve

ntor

y (p

erce

ntag

e di

ffere

nce)

1960 1980 2000 2020Year

Panel D.1. Consumption Price Panel D.2. Detrended Consumption Price

66.

57

7.5

8R

eal P

rice

(log

of 2

016

cent

s pe

r bus

hel)

1960 1980 2000 2020Year

-.4-.2

0.2

.4.6

Det

rend

ed R

eal P

rice

(log

of 2

016

cent

s pe

r bus

hel)

1960 1980 2000 2020Year

21

4.1 Identification: Defining the Treatments (Shocks)

We label the shocks as follows: (i) vat is an acreage supply shock, (ii) vwt is a weather-driven

supply shock, (iii) vit is an inventory demand shock, and (iv) vdt is a consumption demand

shock. Next, we explain these labels and the assumptions underlying them.

The zeroes in the first row of A0 imply that acreage (at) is a function of lagged variables,

the trends, and the first shock (vat), but it is unaffected contemporaneously by any of the

other three shocks (vwt, vit, or vdt). This assumption relies on the sequencing of events. When

making planting decisions, farmers may be responding to demand shocks that determined last

season’s price, but they are not responding to as yet unobserved weather or demand shocks.

Once they observe this year’s weather and demand shocks, they can use that information to

determine next year’s planted acreage, but they cannot go back in time to change this year’s

acreage. Thus, we interpret the difference between actual and predicted acreage as a shock

to supply (vat) caused by, for example, a change in cost or productivity.

The second row of A0 reveals that yield (yt) is a function of lagged variables, the trends,

current acreage, and the second shock (vwt). We assume that farmers do not take actions to

increase yield in response to contemporaneous shocks in inventory or consumption demand.

This assumption follows arguments in RS2013, who argue that yield deviations from trend

are driven by weather shocks. This is why we label vwt a weather-driven supply shock.

The zero in the third row of A0 implies that vit is the part of inventory that is not

predicted by lagged variables, the trends, or quantity supplied (at and yt). Importantly,

inventory does not respond to contemporaneous demand shocks, which means that inventory

demand is perfectly inelastic with respect to price. Thus, we interpret any difference between

actual inventory and the amount predicted by quantity supplied, lags of all variables and

trends as an exogenous change in inventory demand.17 Finally, the fourth equation expresses

prices as a function of all the other variables, lags and trends. Given the quantity supplied

and the quantity put into storage, neither of which respond contemporaneously to prices,

the price adjusts to equilibrate the market. Thus, this equation is a demand function and

its error, vdt, is a consumption demand shock.

In Section 4.3, we address some questions regarding the identification and model selection

choices we make in the above, and we investigate robustness to these assumptions.

17This assumption may be violated in our empirical context. In Section 4.3, we examine the robustness ofour results to relaxing it.

22

Figure 7: SVAR Analysis of RS2013: Impulse Response Functions

-.01

-.005

0

.005

.01

0 5step

95% CI for oirf oirf

IRF: acreage -> acreage

-.01

0

.01

.02

.03

0 5step


IRF: acreage -> yield

-.02

-.01

0

.01

.02

0 5step


IRF: acreage -> inventory

-.2

-.1

0

.1

.2

0 5step


IRF: acreage -> price

-.01

-.005

0

.005

.01

0 5step


IRF: yield -> acreage

-.01

0

.01

.02

.03

0 5step


IRF: yield -> yield

-.02

-.01

0

.01

.02

0 5step


IRF: yield -> inventory

-.2

-.1

0

.1

.2

0 5step


IRF: yield -> price

-.01

-.005

0

.005

.01

0 5step


IRF: inventory -> acreage

-.01

0

.01

.02

.03

0 5step


IRF: inventory -> yield

-.02

-.01

0

.01

.02

0 5step


IRF: inventory -> inventory

-.2

-.1

0

.1

.2

0 5step


IRF: inventory -> price

-.01

-.005

0

.005

.01

0 5step


IRF: price -> acreage

-.01

0

.01

.02

.03

0 5step


IRF: price -> yield

-.02

-.01

0

.01

.02

0 5step


IRF: price -> inventory

-.2

-.1

0

.1

.2

0 5step


IRF: price -> price

Notes: The above figure presents the impulse response functions over 5 years for the SVAR given in (23).The exact values of the impulse response functions are given in Table 2. In the first row, vat is increasedby its standard deviation and the response of all variables is presented. Similarly, in the second, thirdand fourth rows, vwt, vit, and vdt are increased by their standard deviations, respectively. The plots ineach column are presented on the same scale because they show the response of the same variable todifferent shocks. The impulse response functions (oirf) are plotted in solid lines and their 95% bootstrapconfidence intervals are shaded in gray (1000 bootstrap replications).

4.2 SVAR Results

As explained in Section 3.3, the IRF is the dynamic response of the variables to each of

the shocks and is the standard way to present SVAR results. Figure 7 shows the estimated

impulse responses along with pointwise 95% confidence intervals estimated using the residual-

based bootstrap. It contains 16 plots, each showing the dynamic effect on one of the four

variables to a one standard deviation shock in one of the four treatments. Table 2 lists the

estimated impulse responses to both a one-standard-deviation shock and a one-unit shock.

The first row of Figure 7 and Table 2 show that an acreage supply shock increases acreage

23

Table 2: SVAR Analysis of RS2013: Impulse Response Functions

Response to S.D. Change Response to Unit Change

Impulse h at+h yt+h it+h pt+h at+h yt+h it+h pt+h

vat 0 0.0090 0.0061 0.0131 -0.0380 1 0.670 1.447 -4.2041 0.0024 0.0001 -0.0003 -0.0395 0.265 0.007 -0.029 -4.3722 -0.0002 0.0009 -0.0016 -0.0288 -0.018 0.097 -0.176 -3.1853 -0.0011 0.0007 -0.0019 -0.0127 -0.120 0.076 -0.206 -1.4094 -0.0010 0.0004 -0.0012 -0.0014 -0.106 0.048 -0.134 -0.1535 -0.0005 0.0002 -0.0005 0.0037 -0.055 0.022 -0.053 0.405

vwt 0 0 0.0211 0.0170 -0.0803 0 1 0.803 -3.8041 -0.0049 -0.0005 -0.0072 -0.0026 -0.234 -0.026 -0.343 -0.1212 -0.0023 0.0007 -0.0017 0.0152 -0.107 0.032 -0.081 0.7223 -0.0005 0.0000 0.0000 0.0176 -0.026 0.000 0.000 0.8324 0.0004 -0.0002 0.0008 0.0114 0.018 -0.008 0.037 0.5425 0.0006 -0.0002 0.0008 0.0045 0.028 -0.008 0.037 0.214

vit 0 0 0 0.0107 -0.0056 0 0 1 -0.5231 -0.0011 -0.0037 0.0009 -0.0085 -0.105 -0.344 0.082 -0.7972 -0.0010 -0.0016 0.0003 -0.0108 -0.090 -0.152 0.031 -1.0133 -0.0010 -0.0009 -0.0003 -0.0073 -0.090 0.081 -0.029 -0.6864 -0.0008 -0.0004 -0.0004 -0.0032 -0.073 -0.042 -0.035 -0.3045 -0.0005 -0.0003 -0.0002 -0.0004 -0.049 -0.024 -0.023 -0.041

vdt 0 0 0 0 0.1354 0 0 0 11 0.0052 0.0001 0.0068 0.0758 0.039 0.001 0.051 0.5602 0.0053 -0.0005 0.0055 0.0241 0.039 -0.003 0.041 0.1783 0.0035 -0.0002 0.0029 -0.0058 0.026 -0.001 0.021 -0.0424 0.0015 0.0001 0.0007 -0.0152 0.011 0.001 0.005 -0.1125 0.0002 0.0003 -0.0004 -0.0129 0.001 0.002 -0.003 -0.095

Notes: The above table presents the impulse responses to a standard deviation (S.D.) aswell as a unit change in each shock on all variables in the system h-steps ahead for the SVARgiven in (23). Sample: 1962-2013.

by 0.9% and decays to zero two years later.18 This shock raises expected yield by 0.6% and

inventory by 1.3%. The result that acreage supply shocks affect yield suggests that these

shocks are driven by productivity rather than cost changes. The magnitude of the inventory

response to this shock implies that much of the supply increase is saved as inventory, which

in turn implies that the shock has a long lasting effect on prices. The contemporaneous price

response is a 3.8% decrease and the effect decays to zero by year four.

Together, the responses of price and quantity to the acreage supply shock reveal a demand

18Throughout, we describe changes in the log of variables as a percentage change. Thus, we describe alog-acreage increase of 0.0090 as a 0.90% increase.

24

Table 3: Estimates of Demand and Supply Elasticities in Response to SVAR Shocks

Elasticity Quantiles of the Bootstrap Distribution

0.025 0.05 0.95 0.975

Demand Elasticityin response to vat -0.053 -0.492 -0.296 0.087 0.254in response to vwt -0.051 -0.129 -0.114 -0.013 -0.006

Supply Elasticityin response to vw,t−1 0.067 0.010 0.022 0.198 0.225in response to vdt 0.038 -0.009 0.000 0.082 0.090Notes: The quantiles of the bootstrap distribution are obtained from a residual-basedbootstrap described in Hamilton (1994) using 1,000 replications.

elasticity. Recalling that quantity consumed equals (at + yt − it), the current-year demand

elasticity is therefore

∂qdt/∂vat∂pt/∂vat

=∂at/∂vat + ∂yt/∂vat − ∂it/∂vat

∂pt/∂vat= −0.0090 + 0.0061− 0.0131

0.0380= −0.053. (24)

This is very similar to the demand elasticity estimated using the IV strategy in RS2013.

However, it is identified by shocks to land use (acreage supply) rather than the weather

supply shocks used in RS2013. Table 3 reports this and the other elasticities we generate

from our SVAR along with confidence intervals estimated using a percentile bootstrap. This

demand elasticity is not precisely estimated; it is not statistically significant.

The second row of Figure 7 shows the effects of a weather supply shock, which is the

shock that RS2013 use to identify both supply and demand elasticities. The plot in the

second row, second column shows that a typical weather shock raises yield by 2.1% and lasts

only one year. Because production equals acreage times yield and acreage is determined

before the weather shock is observed, this shock implies a 2.1% increase in production. In

response, inventory increases by 1.7% and price decreases by 8.0%. The current-year demand

elasticity is therefore

∂qdt/∂vwt∂pt/∂vwt

=∂yt/∂vwt − ∂it/∂vwt

∂pt/∂vwt= −0.0211− 0.0170

0.0803= −0.051, (25)

which is statistically significant as implied by Table 3. It is very similar to the IV estimate

of −0.063 reported in Table 1. Based on the arguments in Section 3.4, this similarity is not

surprising. The only difference here is that we have a larger SVAR system that includes

an additional variable and a lag of every variable in the system. Furthermore, the implied

demand elasticity due to yield shocks is almost identical to the one due to acreage shocks,

25

which is reported in equation (24). This is especially interesting since acreage shocks are

more persistent than yield shocks.19

In the next year after a weather supply shock, farmers respond to the resulting lower

price by planting 0.5% fewer acres and obtaining 0.1% lower yield, which implies a production

decrease of 0.6% (see Table 2 and the second row of Figure 7). This first-year supply elasticity

is therefore

∂qs,t+1/∂vwt∂pt/∂vwt

=∂at+1/∂vwt + ∂yt+1/∂vwt

∂pt/∂vwt=

0.0049 + 0.005

0.0803= 0.067. (26)

This estimate is smaller than the corresponding supply elasticity of 0.096 reported in RS2013

because we are using a slightly different price variable. We use the lagged spot price pt−1,

whereas RS2013 use a futures price. If we use the RS2013 IV approach but with pt−1 as

the price variable, we get a supply elasticity of 0.076, which is much closer to the SVAR

estimate.20 The effect on supply declines by about half in the second year and dissipates

to zero by the third year. Thus, a one-year weather shock raises supply for two years, with

most of the effect coming in the first year.

The third row of Figure 7 shows the response to an inventory demand shock. This shock

dissipates to zero by year two and has no significant effect on the other variables. Thus,

shocks to inventory demand appear not to be a major driver of global agricultural supply

and demand. Using a partially identified SVAR of the corn market, Carter et al. (2017)

find that inventory demand shocks affect United States corn prices significantly. In Section

4.3, we investigate whether a difference in identification assumptions, specifically relaxing

the exclusion restriction on price in the inventory equation (α34 = 0), explains this different

result.

The bottom row of Figure 7 shows the responses of all variables to a consumption de-

mand shock. The bottom right figure shows that an average consumption demand shock

raises the price by 13.5% and dissipates to zero by about year three after the shock. By

19The plot in the second row, second column Figure 7 shows that a typical weather shock raises yield foronly one year, whereas the plot in the top left of the same figure shows that an acreage-supply shock affectssupply for two years.

20Table A1 in the Supplementary Appendix presents the IV estimates of the supply elasticities using thefutures price (RS2013) as well as the lagged spot price, which is used in our SVAR analysis, using our updateddata set. When using the futures price as in RS2013, the IV estimate of the supply elasticity is 0.104 inour updated data set, which is similar to the estimate in RS2013. Using the lagged spot price instead ofthe futures price in the IV regression leads to an elasticity of 0.076. Hence, it is the different price variablerather than the additional six years of data that explains the difference between the supply elasticity that weobtain using the SVAR results and that obtained in RS2013. The smaller effect of a spot price shock thana futures price shock is consistent with the notion that the futures price represents a rational expectation ofnext year’s spot price and traders are expecting some mean reversion from last year to next year.

26

assumption, current-year acreage, yield, and inventory are determined before price, so they

are not affected contemporaneously by this shock. In the following year, however, producers

respond to this price by increasing acreage by 0.52%. The estimated yield response is close

to zero and statistically insignificant, so the supply response is determined almost entirely

by land use change rather than a change in intensity. The negligible yield response reinforces

our identifying assumption that yield shocks are weather driven. If farmers do not increase

yield in response to demand shocks from the previous year, it is unlikely that they would

increase yield in response to current-year demand shocks.

The first-year acreage elasticity implied by the bottom row of Figure 7 is

∂at+1/∂vdt∂pt/∂vdt

=0.0052

0.1354= 0.038. (27)

The estimated acreage response in year 2 is similar to the year 1 response. Thus, a typical

consumption-demand shock to price induces a supply response elasticity of just under 0.04

in each of the first two years after the shock. It drops by half in year 3 and is close to zero

in year 4. The cumulative acreage response to this shock over the first five years is∑5j=1 ∂at+j/∂vdt

∂pt/∂vdt=

0.0052 + 0.0053 + 0.0035 + 0.0015 + 0.0002

0.1354= 0.116. (28)

Yield does not respond to vdt at any horizon, so this estimate also equals the total production

response. Specifically, for every 1% rise in price from a consumption demand shock, farmers

produce an additional amount equal to 11.6% of current production but they spread this

increase over several years.

In sum, the yield shocks RS2013 use to identify supply and demand elasticities affect

markets for a single year and our SVAR produces elasticities similar to those obtained from

an IV estimator. We find an almost identical demand elasticity to the IV estimate when

we estimate the price response to an unpredicted acreage increase. Acreage shocks have a

longer run effect than yield shocks, so this result implies that consumers respond the same

way to one-year supply changes as to more persistent supply changes. Our estimated supply

response differs somewhat depending on the nature of the shock. Producers have a smaller

initial response but a larger cumulative response to a consumer-demand shock than to a

price increase induced by poor weather last year. In short, the SVAR produces similar

elasticities as the IV, but also helps us understand the rich dynamics between the variables

in the system. Of course, these additional insights stem from the structure we impose on

the data. In the next section, we examine the robustness of the above results to relaxing the

27

key identification and model specification assumptions.

4.3 Frequently Asked Questions About SVAR Results

The SVAR results we present above rely on several specification choices in the baseline

model in (23). These include the number of lags in the model, the type of time trend,

and the triangular structure of the A0 matrix. Similar choices must be made in any SVAR

analysis, and they are often reasonably questioned by readers. We present robustness checks

in the section as responses to common questions about model specification and identification.

4.3.1 Why Include Lags in the Model? How Many Lags?

In practice, SVARs typically include lags of all dependent variables in all equations as in

(5). As explained in Section 2, the purpose of including these lags is to decompose the

variables into a series of uncorrelated shocks. The shocks capture the beginning of new

treatment paths and the lag coefficients capture the continuing influence of these shocks in

future periods. Statistical efficiency could be improved by excluding lags with coefficients

that are known to be zero. In practice, a researcher typically does not know which lags have

zero coefficients and would therefore need to conduct multiple hypothesis tests to determine

which ones to exclude. The resulting inference would hence suffer from multiple testing and

pre-test bias, so time series researchers typically use an unrestricted model.

The estimated IRFs from a model with only one lag of each variable tend to decay

exponentially to zero following the first period after the shock, as in Figure 7. Additional

lags allow for more flexible IRFs. It is common in empirical work to choose the number of lags

that minimizes the Akaike or Schwarz information criterion (e.g., Hamilton (1994)). In our

application, these criteria both choose a single lag. Nonetheless, we investigate robustness

to the number of lags.

Panels A and B of Figure 8 present the IRF graphs for the models with two and three

lags of all variables, respectively.21 Even though the resulting IRFs are more flexible, the

results are qualitatively very similar to the SVAR(1) results. For an acreage shock, the

signs of the IRFs and their statistical significance are unchanged for both the SVAR(2) and

SVAR(3). The IRFs due to a yield shock for the SVAR(2) and SVAR(3) confirm that the

nature of the yield shock is transitory as in the SVAR(1). However, some of the IRFs of

other variables suggest a slightly longer horizon for the response of other variables to this

21For the reader’s convenience, we present the estimated IRFs in response to an S.D. and unit change inTables A2-A3 in the supplementary appendix.

28

Figure 8: SVAR Robustness Check I: Order of Structural VARPanel A. Second-Order Structural VAR

-.01

-.005

0

.005

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0 5step



-.01

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IRF: yield -> yield

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Panel B. Third-Order Structural VAR

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29

shock. For instance, the two-step-ahead acreage and inventory responses to a yield shock

are negative and statistically significant in the SVAR(2) and SVAR(3), whereas they are

negative but statistically insignificant in the baseline SVAR(1) model. For the inventory

demand shock, even though the IRFs have different shapes in the SVAR(2) and SVAR(3),

they are statistically very similar to the SVAR(1) results. Finally, for the consumer demand

shock, the only key difference in the IRFs with different orders of the SVAR is that the

one- and two-step ahead responses of inventory to a consumer demand shock is no longer

statistically significant for SVAR(2) and SVAR(3), even though the signs are the same.

Using the estimated IRFs provided in Table A2 in the supplementary appendix, we can

compute the implied demand and supply elasticities from the SVAR(2) and SVAR(3). The

implied demand elasticities identified by a yield shock are

SVAR(2): − (0.0206− 0.0173)/0.0829 = −0.040

SVAR(3): − (0.0202− 0.017)/0.0804 = −0.040

which are somewhat smaller in magnitude than the corresponding SVAR(1) estimate of

−0.051. The supply elasticity estimates implied by the yield shock are

SVAR(2): − (−0.0047− 0.0004)/0.0829 = 0.067

SVAR(3): − (−0.0051− 0.0011)/0.0804 = 0.077

which are close to the corresponding SVAR(1) estimate of 0.067. Similarly, the demand

elasticity estimates identified by the acreage shock are −0.069 and −0.075 for the SVAR(2)

and SVAR(3), compared to −0.051 in the SVAR(1). The implied supply elasticity due to a

consumer demand shock is 0.041 and 0.047 for the SVAR(2) and SVAR(3), which are slightly

larger than the SVAR(1) estimate of 0.038.

4.3.2 What About Nonstationarity, Trends, and Unit Roots?

Strict stationarity implies that the distribution of the variables is constant over time. In

linear models, it is sufficient to invoke covariance stationarity, which implies that the means,

variances, covariances, and autocovariances of the series are constant over time. There are

many forms of non-stationarity because there are many ways that covariance stationarity

can fail. A time series may display a change in mean (as exemplified in Figure 1), it may

contain a secular trend (as in Panel B.1. of Figure 6), or it may display a lack of mean

reversion.

30

A common model for time series without mean reversion is the unit root model. The

term unit root refers to the fact that some of the roots of the autoregressive polynomial

I − Π1L − ... − Π`L` in (8) equal one. Unlike the impulse responses displayed in Figure 7,

impulse responses of unit root series do not decay to zero as the horizon increases. This means

that shocks can have permanent effects. Sims et al. (1990) show that the IRF estimators are

still consistent in the presence of unit roots, so the presentation of the SVAR in Section 3

still applies. However, inference on these objects is typically non-standard.

We have assumed throughout that the variables in the SVAR are trend stationary, which

means that the series are stationary after de-trending. Specifically, de-trending by sub-

tracting a deterministic function of time from each variable produces series that are mean

reverting around zero. From (8), trend stationarity implies a square-summable MA repre-

sentation, i.e., the impulse responses decay to zero as the horizon increases. Intuitively, if

the de-trended variables tend to return to zero, then no shock can have a permanent effect

because a permanent change in the variable would move its long-run mean away from zero.

If we fail to control for deterministic trends, then these trends will dominate any statistical

analysis of the data. Consider the acreage and yield series in Panels A.1 and A.2. of Figure

6. Regressing acreage on yield produces a positive and apparently significant relationship

merely because both series increased over time. Importantly, the coefficients are essentially

the same if we re-run the regression using only two observations: the first and last years in

the sample.22 To estimate whether acreage and yield have a robust long-run correlation (not

to mention a causal relationship), we would need to observe a change in trend for one of the

variables and see whether a similar change occurs in the other variable. Otherwise we are,

in essence, doing econometrics with two observations.

This discussion highlights the difficulty in estimating long-run effects from finite samples.

Trends characterize the long-run behavior of the series, but we do not observe more than

one “long run” in any sample. Thus, in our SVAR analysis above, we restrict ourselves to

the short-run by de-trending the data so that the dominant variation is short-run variation.

In our baseline specification, we include cubic spline trends as in the IV model in RS2013.

Inspecting the trended and de-trended variables in Figure 6 illustrates how controlling for

these trends in the baseline model seems to render all variables stationary. Each de-trended

series reverts frequently to zero, which provides multiple treatment events over which to

estimate the IRFs.

22The results are acreaget = 5.61 + 0.33yieldt + et for the full sample regression and acreaget = 5.49 +0.40yieldt + et for the two-observation regression using only 1961 and 2013.

31

Fig

ure

9:SV

AR

Rob

ust

nes

sC

hec

kII

:L

inea

rT

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AR

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ear

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rendin

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-.050.05Detrended Acreage (log of millions of ha)

1960

1980

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-.1-.050.05.1Detrended Yield (log of millions of people)

1960

1980

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-.1-.050.05Detrended Inventory (percentage difference)

1960

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-.50.51Detrended Real Price (log of 2016 cents per bushel)

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elB

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ofSV

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wit

hL

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ntor

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eage

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ld

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e

32

The more standard choice in time series models is to use linear trends rather than cubic

splines. Panel A of Figure 9 presents the time series plots of all the variables in our SVAR

model after linear de-trending. This panel shows that with the exception of inventory,

the de-trended variables mean revert much less frequently than when we use the flexible

trend function. As a result, the diagonal plots in Panel B of Figure 9 show that the yield

and consumer demand shocks in the SVAR with linear trends are more persistent than

their counterparts in the baseline model, whereas acreage and inventory demand shocks are

relatively transitory as in the baseline model.23

This difference in the nature of the yield and consumer demand shocks implies different

treatment paths, and hence their IRFs estimate different causal parameters. For acreage

and inventory demand shocks, however, the IRFs should yield similar results to the baseline

model. This is exactly what we find for the IRFs of acreage shocks in the first row. For

inventory demand shocks, we similarly find that the IRFs of acreage and price due to that

shock are similar to the baseline model. However, an inventory demand shock in the SVAR

with linear trends produces a more persistent negative yield response in the future than in

the baseline model. This difference stems from the fact that yield deviations from a linear

trend are more persistent than yield deviations from a more flexible trend. It is difficult

to construct an economic story under which inventory demand shocks would have very

persistent negative yield effects without affecting acreage, which suggests that this result

stems from insufficient de-trending of the yield series. The IRFs of the yield shock and the

consumer demand shock in Panel B of Figure 9 are similar to, but more persistent than,

those in the baseline model.

We report the implied supply and demand elasticities using the formulas in (24)–(27)

and the estimated IRFs in Table A4 in the supplementary appendix. The demand elasticity

implied by the IRFs of an acreage shock in the model with linear trends is −0.084, which is

larger than the baseline estimate of −0.053, whereas the demand elasticity implied by the

IRFs of a yield shock, −0.046, is slightly smaller than the baseline model of −0.051. The

supply elasticity implied by the yield shock is 0.0027 in the linear trend model, compared to

0.067 in the baseline. The supply elasticity implied by the consumer demand shock, 0.036,

is very similar to the baseline model. Given the different treatment paths implied by the

shocks, it is not surprising that the two models yield different elasticities.

23For the exact estimated IRFs to an S.D. and unit change, the reader is referred to Table A4 in thesupplementary appendix.

33

Figure 10: SVAR Robustness Check III: Non-zero Coefficient of Price in the InventoryEquation (α34 = 0.25)

-.01

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4.3.3 What about the Triangular Structure?

The key identifying assumption of our SVAR is arguably the triangular structure of the

matrix A0 in (23). The empirical context implies the exclusion restrictions for the acreage

and yield equations due to the timing of these events, but one could argue that price should

be included in the inventory equation. We hence consider alternative specifications in which

we fix α34 to take one of three non-zero values: 0.1, 0.25 and 0.5. These conditions relax the

constraint that the demand for inventory is perfectly inelastic. They imply that an increase

in price would lead to a contemporaneous decrease in inventory as a price increase motivates

storage firms to sell the commodity out of storage.

Figure 10 presents the resulting IRF graphs for α34 = 0.25. Tables A5-A6 contain the

IRF results for α34 = 0, 0.1, 0.25, 0.5. The IRFs for acreage and yield shocks are unaffected

by the values of α34.24 Hence, the demand and supply elasticities implied by the IRFs to

acreage and yield shocks are also identical to the baseline model.

The IRFs of inventory and consumer demand shocks, however, are quite different from

24This is true by construction in this case.

34

the baseline model results in Figure 7. The baseline results indicate that prices have a

persistent response to consumption demand shocks and no response to inventory demand

shocks. In contrast, Figure 10 shows that consumption demand shocks have small price

effects and inventory demand shocks have a large persistent effect on prices. This result is

more consistent with Carter et al. (2017), who find significant price effects from inventory

demand shocks.

The acreage responses mirror the price responses. In the triangular model, future acreage

responds positively to consumer demand shocks but not to inventory demand shocks. When

we allow price to enter the inventory equation (α34 > 0), we find the opposite. Future acreage

responds positively to inventory demand shocks but not to consumer demand shocks. Thus,

the supply response identified by demand shocks is allocated differently to the two types of

demand shock depending on the assumption about the inventory demand elasticity.

Yield does not respond to either of the demand shocks at any horizon in both models.

This is consistent with the intuition in RS2013 that yield deviations from the cubic spline

trends are driven by weather shocks. In sum, depending on the assumed short-run elasticity

of inventory demand, the dynamic effects of the two types of demand shocks change, but

the dynamic effects of the two supply shocks are unaffected. A possible direction for further

analysis would be to partially identify the SVAR based on the assumption that α34 lies in a

specified range.

5 Conclusion

This paper explains the most common method to identify causal effects in time series econo-

metrics (SVAR) in the language of microeconometrics. Our presentation highlights impor-

tant differences in objectives between SVAR analysts and proponents of reduced-form causal

inference, but also reveals important similarities. SVAR models decompose variation in the

data into “exogenous” components, whereas reduced-form causal models estimate the effect

of only one component. For the sake of clarity, we have focused on the triangular identifi-

cation scheme to obtain the shocks. In an empirical application, we illustrate how such a

triangular structure may be justified from the timing of events in our empirical context and

discuss questions regarding the robustness of SVAR results. However, the assumptions of the

triangular SVAR are strong in general. Recent research has explored alternative approaches

to identification and inference that rely on weaker assumptions (Stock and Watson, 2016,

2017a,b; Montiel-Olea et al., 2016; Gafarov et al., 2018) and may be appropriate in a wide

range of applications.

35

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