Causality in Structural Vector Autoregressions: Science or Sorcery?

Transcription

1 Causality in Structural Vector Autoregressions: Science or Sorcery? Dalia Ghanem and Aaron Smith May 31, 218 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 variation 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 (213), which employed IV to estimate supply and demand of agricultural commodities. The SVAR reveals additional insights beyond IV. We highlight the assumptions required to gain these insights. 1

2 ... 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, Introduction Almost forty years ago, Sims (198) 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, 213). 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 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. (216), Montiel-Olea et al. (216), Bojinov and Shephard (217), Gafarov et al. (218) and Baumeister and Hamilton (217). 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 (Rubin, 1974; Imbens, 214). 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, 217a). As such, 2

3 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 commodities, which was examined in Roberts and Schlenker (213) 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 regarding 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 (213), 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- 1 It is important to clarify that we are not referring to SVARs with external instruments here. A recent review of this approach to estimating impulse responses can be found in Stock and Watson (217a). 2 We discuss non-stationary time series in Section 4.3 and partial identification in the conclusion. 3

4 ing literature proposes methods for credible causal inference using weaker assumptions (see the reviews in Stock and Watson (216, 217a,b)). For instance, Baumeister and Hamilton (217) present Bayesian inference on SVARs that incorporates model uncertainty. Montiel- Olea et al. (216) 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. (218). 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 y 1 Granger-causes a variable y 2 if the mean squared error of a forecast of y 2,t+s for some t, s > is lower when using lags of both y 1 and y 2 as predictors rather than lags of y 2 only (Hamilton, 1994). Hence, including y 1 adds information about future y 2. 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 (213) 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 observations; 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 4

5 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 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 (27) 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 2, it was found in taco shells and other foods. To place Carter and Smith (27) in the potential outcomes framework, denote by t the date the food scare occurred. Then, define a binary treatment variable D t that equals one for t = t and zero otherwise. Thus, there is a single treatment applied on a single date: July 18, 2. The outcome variable y t is the logarithm of the relative price of corn to sorghum. 4 Figure 1 plots daily y t before and after the food scare. The two horizontal red lines indicate the estimates of E[Y t t t ] and E[Y t t < t ]. The difference between them is 3, the question is whether that difference estimates a causal effect. Define Yt as the potential log relative price in the absence of the food scare, and Yt 1 as the potential log relative price in the presence of the food scare. The event study approach of Carter and Smith (27) estimates the causal effect of StarLink contamination if E[Y t t t ] E[Y t t < t ]=E[Y 1 t Y 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, 3 Event studies are sometimes referred to as regression discontinuity in time. 4 Sorghum is a substitute for corn with no GM varieties. 5 The assumption of stationarity here is of course critical to causal inference. For the above equality to hold, we must have E[Yt t t ] = E[Yt t < t ], which is implied by stationarity. Suppose that Y t was not stationary, e.g. E[Yt d t = τ] = µ τ (d) is time-varying, then to obtain a causal effect, we have to be able to observe µ τ () and µ τ (1) simultaneously, which is not possible. We can allow for trend stationarity however, suppose that E[Yt d 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() and µ(1) from pre- and post-treatment observations, respectively. 5

6 Figure 1: The Effect of a Food Scare (Carter and Smith, 27) y Jan-99 Apr-99 Jul-99 Oct-99 Jan- Apr- Jul- Oct- Jan-1 Apr-1 Jul-1 Oct-1 Notes: Y t is the logarithm of the price ratio of corn and sorghum. D t is a dummy variable which equals 1 () for any period t after (before) the food scare occurs. The two horizontal red lines indicate the estimates of E[Y t t t ] and E[Y t t < t ]. a shock. It is hence plausibly independent of the potential outcomes, D t (Yt, Yt 1 ), 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 26, 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 (27) imply that it had persistent price effects, but it would be incorrect to assume that observations in the year 2 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 Multiple Discrete Treatment Events In an example closer to that found in a typical time series setting, Angrist et al. (216) use the potential outcomes framework in a time-series setting with multiple discrete treatments. 6

7 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. (216) define the treatment variable D t as a vector of policy variables that can take values d,..., d J. They observe multiple realizations of each treatment value. By averaging over these realizations, they can estimate the treatment effect at various horizons. Define Y t,h (d) as the potential outcome in period t + h if policy d is implemented at time t, and let z t 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 d to d j conditional on z t as E[Y t,h (d j ) Y t,h (d ) z t ] = E[Y t,h D t = d j, z t ] E[Y t,h D t = d, z t ]. This expression assumes that the potential outcomes are not confounded with the treatment assignment, specifically Y t,h (d) D t z t for all h and for all d. Angrist et al. (216) use propensity score methods to obtain the unconditional average policy effect E[Y t,h D t = d j ] E[Y t,h D t = d ] = E [ ( 1{Dt = d j } Y t,h P (D t = d j z t ) 1{D )] t = d } P (D t = d z t ) The inverse probability weighting in the above equation has an important interpretation in the context of time series. When P (D t = d j z t ) is smaller, the occurrence of d j is less predictable. Hence, it is more plausibly exogenous and is given a higher weight relative to observations with larger P (D t = d j z t ). Thus, as in the StarLink example, we observe a connection between unpredictability and exogeneity. (1) 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 independent 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 7

8 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 (213) application (hereafter RS213) as an example. RS213 estimate global supply and demand elasticities for agricultural commodities using an IV approach that lends itself naturally to an SVAR. RS213 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, quantity 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- 6 In practice, linear models are used as approximations, but to make our discussion of causal effects in linear models precise, we emphasize the role of correct specification once we have continuous treatment variables. 7 Bojinov and Shephard (217) propose a model-free approach to identification, estimation and inference on causal effects of treatment paths in time series. Inspired by a large experiment by a quantitative hedge fund, they show how to extend the potential outcomes framework to define treatment paths and potential outcomes in order to achieve a completely model-free approach to causal inference solely relying on random assignment of treatment paths. Their approach is specific to the case of a large number of randomly assigned treatment paths. 8

9 conomics. The left panels of Figure 2 show the three variables in the RS213 demand model, extended to Yield and quantity demanded display increasing trends and price displays 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. RS213 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. RS213 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 RS213. Later, in Section 4, we present a complete SVAR model of demand and supply of agricultural commodities. 3 Introducing SVARs Let q t denote log quantity demanded, p t log of price and w t yield deviations from trend, which are a proxy for weather. A triangular SVAR with l lags is given by the following w t = ρ 11 Y t 1 + ρ 12 Y t ρ 1l Y t l + f w (t) + v wt (2) p t = β 21 w t + ρ 21 Y t 1 + ρ 22 Y t ρ 2l Y t l + f p (t) + v pt, (3) q t = β 31 w t + β 32 p t + ρ 31 Y t 1 + ρ 32 Y t ρ 3l Y t l + f q (t) + v qt. (4) where Y t (w t, p t, q t ) and ρ ij is a 3-dimensional row vector for all i and j. The terms f w (t), f p (t), and f q (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, p t and q t are omitted from the yield equation and q t 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 v t as shocks. The shocks represent the part of the observed variables that (i) cannot be predicted using past 8 RS213 used data from We update the data through 213. The raw data on area, production and 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 RS213. We hence convert production tons into calories. We then divide by 365*2, the number of calories consumed by the average person in a year. Hence, the units of production in our analysis is in millions of people as in RS213. Yield is production per area and is measured in bushels per ha. The raw price data is obtained from Quandl, and it includes spot and futures prices. The spot and futures price we use are calorie-weighted averages of the individual commodity prices. For more details on the data and variable construction, see RS213 and Hendricks et al. (214). 9

10 Figure 2: Time Series Plots of Yield Deviations, Price and Quantity Demanded in Roberts and Schlenker (213) Panel A1. Yield Panel A2. Detrended Yield Yield (log of people per ha) Yield Shock (log of people per ha) Year Panel B1. Consumption Price Year Panel B2. Detrended Consumption Price Real Price (log of 216 cents per bushel) Year Detrended Real Price (log of 216 cents per bushel) Year Panel C1. Quantity Demanded Panel C2. Detrended Quantity Demanded Quantity (log of millions of people) Year Detrended Quantity (log of millions of people) Year Notes: Panels A1, B1 and C1 present the time series plots of logarithm of global yield, the logarithm of the futures price in month of delivery, and the logarithm of global quantity demanded for a calorieweighted index of corn, rice, soybeans, and wheat. Quantities are defined in units of 2*3655 calories, which corresponds to sufficient sustenance for a person for a year. Panels A2, B2, and C2 show the same series after detrending using a cubic spline with 4 knots. 1

11 observations (Y t 1,..., Y t l ) and (ii) is not affected by contemporaneous values of other variables. 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., v t = (v wt, v pt, v qt ) Y t 1, Y t 2,..., Y t l W N(, D), where D is a diagonal matrix. The three equations of the SVAR can be written in matrix notation as follows A Y t = A 1 Y t 1 + A 2 Y t A l Y t l + f(t) + v t, (5) where A is a lower-triangular matrix, A = 1 β 21 1 β 31 β (6) Multiplying through by A 1, we can write the reduced-form of the above model, which is a VAR(l), Y t = Π 1 Y t 1 + Π 2 Y t Π l Y t l + g(t) + ε t (7) where g(t) = A 1 f(t), Π j = A 1 A j for j = 1,..., l and ε t = A 1 v t. 9 The parameters in (7) can be estimated consistently by ordinary least squares (Hamilton, 1994). We can express Y t in vector MA( ) form as a linear function of current and past structural errors, v t, Y t = h(t) + Ψ l v t l, (8) l= where h(t) = (I Π 1 L... Π l L l ) 1 g(t). 1 The MA coefficients are square summable (i.e., 9 The above VAR imposes no zero-restrictions on the Π 1,...,Π L. It is worth noting here that in a bivariate VAR, when one variable (y 2 ) does not Granger-cause the other (y 1 ), then it implies the following zero restrictions on the coefficient matrix on the lagged vectors (Hamilton, 1994). Specifically, ] [ = h(t) + y 2t [ y1t π (11) 1 π (21) 1 π (22) 1 ] [ ] y1,t 1 + y 2,t 1 [ π (11) 2 π (21) 2 π (22) 2 ] [ ] y1,t y 2,t 2 [ π (11) l π (21) l π (22) l ] [ ] y1,t l. y 2,t l In our analysis of the SVAR as a causal tool, we allow the matrices of the lags of Y t to be completely unrestricted. 1 L denotes the backshift, or lag, operator. The MA Coefficients Ψ j are functions of the parameters in 11

12 j= Ψ j 2 < ) if Y t is covariance-stationary (see Hamilton (1994) for technical conditions and Section 4.3 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 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 Y t+h for some h with respect to each element of v t, i.e., Y t+h / v jt = Ψ jh for j {1, 2, 3}, where j denotes the j th 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 Y t 1,..., Y t l. 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 β 21 1 β 31 β 32 1 w t p t q t = v wt v pt v qt (9) where v t W N(, 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 w t p t q t = 1 β 21 1 } β 31 + β 32 β 21 {{ β 32 1 } Y t/ v t Because this model has no autocorrelation, the IRFs are zero for all h >. v wt v pt v qt. (1) The elements of the matrix on the right-hand side of (1) give the contemporaneous impulse responses. For instance, β 21 is the impulse response of a yield shock on contemporaneous price ( p t / v wt ), β 32 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 have built-in routines to compute these estimates. Alternately, they can be estimated using the local projections method of Jorda (25). 12

13 quantity ( q t / v wt ), and β 31 + β 32 β 21 is the impulse response of a yield shock on contemporaneous quantity ( q t / v pt ). 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 (1) 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 w t = v wt. As a result, the first equation in (1) is redundant from a causal perspective. 11 Considering the price equation, if we assume that E[v pt v wt ] =, i.e. yield shocks are exogenous in the price equation, the resulting conditional expectation for the second equation is given by E[p t v wt ] = E[β 21 v wt + v pt v wt ] = β 21 v wt. (11) In this case, β 21, the impulse response of p t to v wt, is the marginal effect of a yield shock on price E[p t v wt ]/ v wt. 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 (1), assuming E[v qt v wt, v pt ] =, i.e. all price shocks are exogenous in the quantity equation, implies E[q t v wt, v pt ] = (β 31 + β 32 β 21 ) v wt + β 32 v }{{}}{{} pt. (12) E[q t v wt,v pt]/ v wt E[q t v wt,v pt]/ v pt It follows that lower off-diagonal elements of the coefficient matrix in (1) are marginal causal effects. An important byproduct of the mutual mean independence of the elements of v t is E[Y t v wt, v pt, v qt ] = E[Y t v wt ] + E[Y t v pt ] + E[Y t v qt ], (13) 11 This is not the case when the model includes lags as in (8). 13

14 which implies that the marginal effect of conditional and unconditional expectations are equal. For instance, E[q t v wt, v pt ] v wt = {(β 31 + β 32 β 21 )v wt + β 32 v pt } v wt = β 31 + β 32 β 21, (14) = {}}{ E[q t v wt ] = {(β 31 + β 32 β 21 )v wt + β 32 E[v pt v wt ]} = β 31 + β 32 β 21. (15) v wt v wt Furthermore, mutual mean independence allows the SVAR to identify the impact of multiple contemporaneous changes, e.g. E[p t v wt = v w, v pt = v p ] E[p t v wt =, v pt = ] =E[p t v wt = v w ] E[p t v wt = ] + (E[p t v pt = v p ] E[p t v pt = ]) =β 21 v w + v p. (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 v wt as a weather shock, and we allow it to affect price and quantity. We label v pt as non-weather supply shocks and v qt 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 (1). We assume that observed weather does not respond to nonweather 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[q t v wt ]/ v wt E[p t v wt ]/ v wt = β 31 + β 32 β 21 β 21. (17) This ratio differs from the elasticity of the demand response to non-weather supply shocks, 14

15 which is E[q t v pt ]/ v pt E[p t v pt ]/ v pt = β 32. (18) Hence, because there are two supply shocks in this model, there are two demand elasticities produced by the model. RS213. Next, we show how this analysis compares to the IV model in 3.4 Triangular SVAR vs. Instrumental Variables In the previous sections, we explain how the SVAR defines exogenous components of endogenous 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 RS213. 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, w t is purely a shock that is uncorrelated with other shocks. Specifically, in the IV model the yield deviation is w t = u wt and in the triangular system the yield deviation is w t = v wt. There are two differences between the systems. First, the IV model excludes w t from the q t equation, whereas the triangular model does not. Second, the IV model allows the price and quantity shocks (u pt and u qt ) to be correlated (σ 23 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 b 32 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 w t, which is valid because it affects price (b 21 ) but not the demand curve (b 31 = ), and because it is exogenous to price and quantity 12 This presentation of the IV model is closely related to the SVAR approach using external instruments (Montiel-Olea et al., 216). The external instrument in our example is w t, which is correlated with p t ( b 21 ), but not with q t directly (b 31 = ). According to Montiel-Olea et al. (216), we can identify b 32 using w t as an external instrument in the two-equation SVAR of p t and q t without specifying a full triangular system. 13 The assumption on the diagonal variance-covariance matrix is typically made while including lags of all variables in each equation of the model, so it is not as restrictive in practice as it may seem in the static case. 15

16 Figure 3: Instrumental Variables vs. Triangular SVAR: Demand Elasticity 1 b 21 1 b 32 1 w t p t q t Panel A: IV u wt σ1 2 = u pt, Ω = σ2 2 σ 23. u qt σ 23 σ3 2 Panel B: Triangular System (Static SVAR) 1 β 21 1 w t v wt σw 2 p t = v pt, Σ = σp 2 β 31 β 32 1 q t v qt σq 2 }{{}}{{}}{{} Y t v t A. (b 12 = b 13 = σ 12 = σ 13 = ). In this model, a positive weather shock increases supply, which reduces price and increases quantity demanded. The potential correlation between the first stage error (u pt ) 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 β 21, and it changes quantity by β 31 + β 32 β 21 (see (1)). The parameter β 21 represents the coefficient on w t in a least squares regression of p t on w t (see equation (3)). The parameters β 31 and β 32 represent the coefficients on w t and p t in a least squares regression of q t on w t and p t (see equation (4)). Thus, the response of quantity to a weather shock equals the sum of a direct effect (β 31 ) and an indirect effect that works through price (β 32 β 21 ). This is also the coefficient one would obtain from a regression of q t on w t 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 w t from the q t equation and the uncorrelatedness of w t = v wt and v qt, it follows that cov(q t, w t ) = b 32 cov(p t, w t ). (19) 16

17 Figure 4: Triangular SVAR vs. IV Panel A: IV Price Supply u wt Yield Dev. b 21 u pt Price b 21 b 32 b 32 u qt Quantity Demand Quantity Panel B: SVAR v wt Yield Dev. Price 21 Supply v pt Price Demand v qt Quantity Quantity Solving for b 32 and multiplying and dividing by var(w t ), assuming it is strictly greater than zero, yields the following b 32 = cov(q t, w t )/var(w t ) cov(p t, w t )/var(w t ), (2) which is the population analogue of the Wald estimator. The numerator is the slope coefficient from the OLS regression of q t on w t. The denominator is the slope coefficient from an OLS regression of p t on w t. In the triangular SVAR, cov(p t, w t ) cov(β 21 w t + v pt, v wt ) = β 21 var(w t ) cov(q t, w t ) = β 31 var(w t ) + β 32 cov(p t, w t ) = (β 31 + β 32 β 21 )var(w t ) (21) 17

18 Table 1: Demand Elasticity: Triangular System vs. IV IV SVAR (1) (2) (3) (4) Dependent Variable: q t p t q t q t ˆβ {}} 32 { p t -.63 (-22) (2) w t ˆβ {}} 21 { ˆβ {}} 31 {.317 β 31 + β 32 β {}}{ (-5.35) (28) (28) Sample Size Notes: (1) is estimated using 2SLS with w t as the instrument. (2)-(4) are estimated using OLS. All regressions include flexible time trends modeled using cubic splines with four knots as in RS213. The t statistics in parentheses are computed using Newey-West standard errors to correct for heteroskedasticity and first-order autocorrelation. Sample: These two equalities imply that the population analogue of the Wald estimator is given by the following cov(q t, w t )/var(w t ) cov(p t, w t )/var(w t ) = β 31 + β 32 β 21 β 21 = b 32. (22) Hence, the population analogue of the Wald estimator equals the ratio of two impulse responses, 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 RS213 data. As in RS213, we model the trend using cubic splines with four knots. To ease comparison with RS213, we do not include lags in the estimates in Table 1. Column (1) reports that the IV estimate of the demand elasticity is.63, which is similar to the analogous estimate of.55 in RS213 (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 β 21, β 31 and β 32, from OLS regressions. 14 The estimated response of quantity to a weather shock is presented in Column (4) and equals.36, which could also be constructed from coefficients in Columns (2) and (3). The demand elasticity computed from the SVAR as in (17) is.36/4.856 =.63. Thus, we have shown in this section that, under the RS213 assumption that yield 14 Column (2) is also the first stage regression in the IV model. 18

19 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, b 32 is the demand elasticity, whereas in the SVAR, the ratio (β 31 + β 32 β 21 )/β 21 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 elasticity of demand with respect to other supply shocks (β 32 ). Table 1 reports this estimate as, 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 RS SVAR Analysis of Supply and Demand of Agricultural Commodities RS213 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, Northern Hemisphere farmers choose the amount of land to cultivate (acreage, a t ) based on last year s information. Due to storage arbitrage, last year s price (p t 1 ) is a good proxy for the information on which farmers base their planting decisions. Weather realizations over the 19

20 Figure 5: Time Line a t y t i t p t farmers choose a t given information at t 1, p t 1 yield is realized wholesale traders choose i t price is determined summer determine the yield (y t ), 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 (i t ). These decisions jointly determine the price (p t ), which we measure in November and December. 15 This time line of events motivates the following SVAR 1 a t α 21 1 y t α 31 α 32 1 i t } α 41 α 42 α 43 {{ 1 p t }}{{} A Y t ρ 11 ρ 12 ρ 13 ρ 14 = ρ 21 ρ 22 ρ 23 ρ 24 ρ 31 ρ 32 ρ 33 ρ 34 } ρ 41 ρ 42 ρ 43 {{ ρ 44 } A 1 a t 1 y t 1 i t 1 p t 1 } {{ } Y t 1 +ΓX t + v at v wt v it v dt } {{ } v t (23) where X t 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(v t ) = Σ, a diagonal matrix. All variables are measured in logs. We define i t 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 RS213, we note that production (quantity supplied) equals acreage times yield, hence its log equals a t +y t. The supply model in RS213 is a regression of (a t + y t ) on an expected price (for which we use p t 1 ), yield (y t ), and the trend. 16 Their demand equation is a regression of (a t + y t i t ) on p t and the trend. 15 This narrative omits the fact that farmers also plant crops in the Southern Hemisphere, where the seasons are opposite to the north. This fact may invalidate the identification strategy. However, results in Hendricks et al. (214) suggest that the endogeneity bias from this assumption is small. 16 RS213 do not use actual yield y t as a control variable in their supply equation. Rather, they use a yield shock, which is log yield minus a trend. Because the supply model controls for trends, these two specifications are identical if the model used to detrend log yield is that same as the trend specification in the supply equation. Hendricks et al. (214) show that the supply elasticity estimates are almost identical across the two specifications. 2

21 Figure 6: Time Series Plots of Acreage, Yield, Inventory and Consumption Price Panel A. Acreage Panel A. Detrended Acreage Acreage (log of millions of ha) Year Detrended Acreage (log of millions of ha) Year Panel B. Yield Panel B. Detrended Yield Yield (log of people per ha) Yield Shock (log of people per ha) Year Year Panel C. Inventory Panel C Detrended Inventory Inventory (percentage difference) Detrended Inventory (percentage difference) Year Panel D. Consumption Price Year Panel D. Detrended Consumption Price Real Price (log of 216 cents per bushel) Detrended Real Price (log of 216 cents per bushel) Year Year 21

22 4 Identification: Defining the Treatments (Shocks) We label the shocks as follows: (i) v at is an acreage supply shock, (ii) v wt is a weather-driven supply shock, (iii) v it is an inventory demand shock, and (iv) v dt is a consumption demand shock. Next, we explain these labels and the assumptions underlying them. The zeroes in the first row of A imply that acreage (a t ) is a function of lagged variables, the trends, and the first shock (v at ), but it is unaffected contemporaneously by any of the other three shocks (v wt, v it, or v dt ). 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 (v at ) caused by, for example, a change in cost or productivity. The second row of A reveals that yield (y t ) is a function of lagged variables, the trends, current acreage, and the second shock (v wt ). 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 RS213, who argue that yield deviations from trend are driven by weather shocks. This is why we label v wt a weather-driven supply shock. The zero in the third row of A implies that v it is the part of inventory that is not predicted by lagged variables, the trends, or quantity supplied (a t and y t ). 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, v dt, 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. 17 This assumption may be violated in our empirical context. In Section 4.3, we examine the robustness of our results to relaxing it. 22

23 Figure 7: SVAR Analysis of RS213: Impulse Response Functions IRF: acreage -> acreage.3 IRF: acreage -> yield IRF: acreage -> inventory IRF: acreage -> price IRF: yield -> acreage.3 IRF: yield -> yield IRF: yield -> inventory IRF: yield -> price IRF: inventory -> acreage.3 IRF: inventory -> yield IRF: inventory -> inventory IRF: inventory -> price IRF: price -> acreage.3 IRF: price -> yield IRF: price -> inventory 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, v at is increased by its standard deviation and the response of all variables is presented. Similarly, in the second, third and fourth rows, v wt, v it, and v dt are increased by their standard deviations, respectively. The plots in each column are presented on the same scale because they show the response of the same variable to different shocks. The impulse response functions (oirf) are plotted in solid lines and their 95% bootstrap confidence intervals are shaded in gray (1 bootstrap replications). 4 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 residualbased 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

24 Table 2: SVAR Analysis of RS213: Impulse Response Functions Response to S.D. Change Response to Unit Change Impulse h a t+h y t+h i t+h p t+h a t+h y t+h i t+h p t+h v at v wt v it v dt Notes: The above table presents the impulse responses to a standard deviation (S.D.) as well as a unit change in each shock on all variables in the system h-s ahead for the SVAR given in (23). Sample: by.9% and decays to zero two years later. 18 This shock raises expected yield by.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 18 Throughout, we describe changes in the log of variables as a percentage change. Thus, we describe a log-acreage increase of.9 as a.9% increase. 24