Deriving Supply Chain Metrics From Financial Statements

Transcription

1 Deriving Supply Chain Metrics From Financial Statements Robert L. Bray and Haim Mendelson Stanford Graduate School of Business December 31, 2011 Abstract We develop an empirical supply chain model that enables empiricists to derive operational metrics from demand signal processing how demand information flows through supply chains and perform corresponding counterfactuals. We develop estimators of our firms preference parameters and the impulse response functions governing demand signal processing, translate these estimates into key supply chain measures, and study how these measures change under alternative supply chain settings. We conduct six studies with our methodology and firm-level Compustat data, analyzing (1) whether firms consider their supplier s inventory costs, (2) the convexity of production cost functions, (3) the replenishment rate of depleted inventories, (4) the drivers of the bullwhip effect, (5) the degree to which supply chain alignment mitigates the bullwhip effect, and (6) whether demand forecast accuracy constitutes a supply chain externality. Keywords: Supply chain metrics; demand signal processing; production smoothing; inventory adjustment; benchmarking. 1 Introduction This project grew from our frustrations with securing proprietary supply chain data. After extended courtships, our leads at Safeway, Falabella, and Abbott Labs all fell through before yielding any data. So it goes in supply chain management, a field hobbled by a dearth of data. For a lack of better options, Compustat an accessible, financial-statements dataset bolsters much of the field s empiricism. Nevertheless, designed to support investment decisions, these accounting data have a shallow reach within supply chain management. This article uses supply chain theory to deepen Compustat s purview used as widely as it is, Compustat merits further investment. More precisely, this work develops an approach to formulate empirical supply chain studies from operational metrics derived from Compustat. It develops a production model that translates standard supply chain theories into workhorse supply chain estimators. Our framework consists of a production model and its associated estimators. Our model generalizes those of Graves et al. (1986, 1998), Balakrishnan et al. (2004), Chen and Lee (2009), and Bray and Mendelson (2011). At the heart of our model is demand signal processing (DSP) (see Lee et al., 1997, p. 548), the translation of demand signals into order quantity signals passed upstream how demand information morphs 1

2 as it crawls up supply chains speaks volumes about operations. We capture this process by estimating the impulse response functions (IRFs) that govern DSP. From these DSP IRF estimates, researchers can derive estimates of many unobserved supply chain metrics. Additionally, to study structural supply chain changes, we estimate the fundamental preference parameters that underlie DSP. Our estimates agree with one another and our theory. We demonstrate our approach with six studies. The first examines whether downstream firms regard upstream inventory costs by measuring whether companies aim to lengthen supplier response times. The second explores the convexity of production cost functions by determining whether firms space out production. The third measures the replenishment rate of depleted inventories previous attempts failed to account for demand signal information lead times. The fourth examines the mechanism driving the bullwhip effect, studying whether firms amplify demand variability by accentuating incoming demand signals or by contributing new sources of variation to those signals. The fifth explores how supply chain coordination mitigates this bullwhip effect. And the last measures the propagation of demand forecast improvements through supply chains. Others have estimated empirical supply chain models. For example, Cohen et al. (2003, p. 1653) and Terwiesch et al. (2005) model the production policy of a computer chip manufacturer and the order policies of its customers; the former article finds the producer very conservative when commencing the order fulfillment, which undermines the effectiveness of the overall forecast-sharing mechanism, and the latter concludes that orders inflations beget order fulfillment delays, which in turn beget further orders inflations. Musalem et al. (2010) demonstrate that lost sales increases convexly in the number of stocked out goods, as suitable alternatives dwindle. Aguirregabiria (1999) estimates the parameters of a dynamic inventory policy, calculating that eliminating fixed order costs would halve the frequency of promotions. Hall and Rust (2000) show that a steel wholesaler follows a modified (S, s) order policy, where S and s decrease in the steel spot price. Finally, Holmes (2011) finds that procurement costs compel Walmart to cluster distribution networks, which partially cannibalizes sales. Also, others have studied supply chain topics with Compustat. For example, Huson and Nanda (1995) use these data to demonstrate that JIT adoption correlates with enhanced earnings. Rajagopalan and Malhotra (2001), Chen et al. (2005), and Chen et al. (2007) conclude that the Compustat s work-in-progress inventories have decreased since the 1980s, but that its finished-goods inventories have not. Gaur et al. (2005), Kesavan et al. (2008), and Roumiantsev and Netessine (2007) study how inventories relate to earnings, gross margins, and capital intensity. And with these data, Rumyantsev and Netessine (2007) test monotone statics derived from classic inventory models, and Hendricks and Singhal (2003) and Randall et al. (2006) respectively analyze procurement disruptions and inventory positioning. In what follows, 2 presents our production model; 3 describes our data, and estimates our core model parameters; 4 conducts six studies with these estimates; and 5 concludes. 2

3 2 Supply Chain Model 2.1 Setup We consider a manufacturer that produces a single good from a single input sourced from a single supplier. The manufacturer manufactures with an L-period leadtime, and the supplier sources with a L s -period leadtime. The supplier always meets orders promptly, borrowing stocked-out components from a third party: The supply chain is decoupled (Kahn, 1987; Gavirneni et al., 1999; Lee et al., 2000; Chen and Lee, 2009). The manufacturer, placing an order in each period and free to return stock, can meet any orderup-to level (Lee et al., 1997; Aviv, 2003; Chen and Lee, 2009). The manufacturer only stores finished-good inventories, starting production as soon as its inputs arrive: Hence the production quantity equals the order quantity. Following Graves et al. (1998), we impose an arbitrarily large forecast horizon, H, beyond which the manufacturer never schedules production. We impose this limit to avoid using infinite matrices, and because data are inherently finite. Our model incorporates three newsvendor-type cost structures. The first is the manufacturer s inventory cost: b per unit per period of demand backlogged, and h per unit per period of inventory stored. The second is the manufacturer s production capacity cost: q per unit per period of capacity short, and g per unit per period of capacity idle. That is, the manufacturer invests in z units of fixed capacity, for which it pays g > 0 per unit per period to maintain; with this capacity, the manufacturer produces the first z units in-house at unit cost v, and outsources the rest at unit cost v + q + g > c. The third is the supplier s inventory cost: b s per unit per period borrowed from the third party, and h s per unit per period stored. To align the supply chain, the manufacturer reimburses the supplier fraction θ of its inventory costs such transfers arise commonly in the supply chain contracting literature (see Cachon, 2003). Before delving into the details, we define six matrices C, D, R, J, K, and G the first five are (H + 1) (H + 1), and the last is (H + 1) (H L). The first matrix, a lower triangular matrix of ones, is the cumulative sum operator: the k th element of Cv, where v is a vector, is k j=1 v j. The second, which has ones in its subdiagonal and zeros elsewhere, is the delay operator: the k th element of Dv is v k 1, if k > 1, and zero otherwise (D i, the i th power of D, is the i-period delay operator.) The third, which has ones in its superdiagonal and top left corner and zeros elsewhere, repeats the first column: the k th element of v R is v max(k 1,1) (R i+1 repeats the first column i times). The fourth, which has ones in the first L s elements of the principal diagonal and zeros elsewhere, keeps the first L s rows intact, and turns the rest to zero: the k th element of Jv is v k, if L s k, and zero otherwise. The fifth, which has ones in its (H + 1 L) th row and zeros elsewhere, consolidates mass in this the (H + 1 L) th row: the k th element of Kv is j v j, if k equals (H + 1 L) th, and zero otherwise. And the last, G, stacks three matrices: an (H L) (H L) identity matrix on top, a 1 (H L) matrix of a negative ones beneath that, and a L (H L) matrix of zeros beneath that. The manufacturer faces a Martingale Model of Forecast Evolution (MMFE) demand process (see Hausman, 1969; Heath and Jackson, 1994; Chen and Lee, 2009), which generalizes most commonly-used exogenous demand processes (cf. Chen and Lee, 2009): d t µ + H e lɛ t l, (1) l=0 3

4 where d t is period-t demand, µ is the mean demand, e l is a unit vector indicating the l th position, 1 and ɛ t is a vector of length (H + 1) containing the demand signals the observed in period-t. We model ɛ t as i.i.d., mean-zero multivariate normals, with covariance matrix Σ. The manufacturer chooses an order policy from an extension of the Generalized Order-Up-To Policy (GOUTP) class of Graves et al. (1998) and Chen and Lee (2009). The GOUTPs are policies in which orders follow a linear, time-invariant function of observed demand signals 2 viz., those in which period-t orders can be expressed o t µ + H l=0 e l Aɛ t l, for some matrix A that satisfies ι A = ι, where ι is a vector of ones, and e la = 0 for all l greater than H L. The first regularity condition, making A s columns sum to one, ensures the firm eventually satisfies all demand. The second, setting A s bottom L rows to zero, simplifies the accounting (adding these dummy rows sacrifices no generality because H L is arbitrary). Aviv (2007, p. 778) describes this framework as an elegant model of... production and inventory planning. Under a GOUTP, order quantities can only depend on observed demand signals, but in practice orders will depend on additional exogenous factors, such as manufacturing input prices, labor availability, process yields, tool downtimes, and the weather. We extend the GOUTP to allow for such factors by introducing an error term, n t, to the order quantity expression. These order-shifting errors also follow an MMFE: n t = H l=0 e l η t l, where η t drawn from i.i.d. mean-zero, multivariate normals, with covariance matrix Λ is an auxiliary signal vector the firm observes in period-t. The firm makes optimal mean-square-errorminimizing demand and order forecasts, which implies E[ɛ τ ɛ t] = E[η τ η t] = E[ɛ τ η t] = 0, for τ t (see Heath and Jackson, 1994, p. 21). We further bar correlation between contemporaneous demand and order-shifting signals: E[ɛ t η t] = 0. (2) With this, we define the extended Generalized Order-Up-To Policy (XGOUTP) class as the policies with expected order quantities following linear, time-invariant functions of observed demand signals, or, equivalently, those with orders taking the form: H H o t µ + e laɛ t l + n t = µ + e l(aɛ t l + η t l ), (3) l=0 l=0 where ι A = ι, e la = 0 l > H L. Of course the GOUTPs nest within the XGOUTPs, as setting Λ = 0 yields n t = 0. The order policy in (3) induces the following end-of-period-t inventory: H H i t =m + (o t L l d t l ) = m + e lc ( (D L A I)ɛ t l + D L ) η t l, (4) l=0 where m is the mean period-end inventory level. The manufacturer shares its order forecasts, so its order signals become supplier demand signals (see Chen and Lee, 2009). Furthermore, we specify that the supplier follows a GOUTP, with mean inventory m s and impulse response matrix A s, where, like before, ι A s = ι 1 We adopt the convention of counting rows and columns from zero e.g. X 0,0 is the top-left element of matrix X. 2 Bertsimas et al. (2009) prove the optimality of linear order policies in a similar robust optimization context. l=0 4

5 and e l A s = 0 l > H L s. Thus, at the end of period-t, the supplier holds inventory: H i s t = m s + e lc(d Ls A s I)(Aɛ t l + η t l ). (5) l=0 Note, expressions (3), (4), and (5) specify that manufacturer orders, manufacturer inventories, and supplier inventories also follow MMFEs, with signal vectors ɛ o t Aɛ t + η t, ɛ i t C ( ) (D L A I)ɛ t + D L η t, and ɛ s t C(D Ls A s I)(Aɛ t + η t ), respectively. This is demand signal processing: The manufacturer translates exogenous signals ɛ t and η t into endogenous signals ɛ o t, ɛ i t, and ɛ s t, which are distributed according to mean-zero multivariate random normals with covariance matrices Σ o (A) AΣA + Λ, Σ i (A) C(D L A I)Σ(D L A I) C + CD L ΛD L C, and Σ s (A, A s ) C(D Ls A s I)[AΣA + Λ](D Ls A s I) C, respectively. The DSP is linear and time-invariant, so we can characterize it with impulse response functions (IRFs) (Phillips et al., 2008). Matrices A, C(D L A I), and C(D Ls A s I)A house these functions: their l th columns respectively give the IRFs that govern how demand signals with l-period information leadtimes, e l ɛ t, transform into signals ɛ o t, ɛ i t, and ɛ s t. Henceforth, we refer to A as the impulse response matrix. The manufacturer optimizes over this matrix viz., it optimizes over the impulse response functions that govern DSP. To recap, in period-t: (1) signals ɛ t and η resolve; (2) the manufacturer orders and receives o t input units from the supplier; (3) the manufacturer starts producing o t units, and finishes producing o t L units; (4) the manufacturer and supplier adjust their respective inventories to i t and i s t (negative quantities indicating backlogs); (5) the manufacturer and supplier pay newsvendor penalties C t (i t ) + h+( i t ) + b+(o t X) + q + (X o t ) + g+θ[(i s t ) + h s +( i s t) + b s ] and Ct s (1 θ)[(i s t) + h s +( i s t ) + b s ], respectively, where (x) + max(x, 0). 2.2 Supply Chain Equilibrium We seek the equilibrium solution of the manufacturer and supplier production problems: The manufacturer minimizes A,m,z E[C t A, A s, m, m s, z], subject to A s regularity conditions, and the supplier minimizes As,m s E[C s t A, A s, m, m s, z], subject to A s s. Inventories and order quantities, linear combinations of normally distributed signals, also follow normal distributions. Hence, the optimal production capacity and mean inventory levels, as functions of A and A s, are z(a) = Φ 1( ) q q+g Tr[Σ o (A)] 1/2, m(a) = Φ 1( ) b b+h Tr[Σ i (A)] 1/2, and m s (A, A s ) = Φ 1( ) b s b s+h s Tr[Σ s (A, A s )] 1/2 (Φ and φ are the normal distribution 5

6 CDF and PDF, respectively). Given these quantities, the firms objectives become (cf. Porteus, 2002, p. 13): Manufacturer: min A s.t. Tr[Σ i (A)] 1/2 + αtr[σ o (A)] 1/2 + βtr[σ s (A, A s )] 1/2, ( α q + g b + h φ Φ 1( q ) ) ( φ Φ 1( b ) ) 1, q + g b + h ( ) ( β θ(b s + h s ) φ Φ 1( b ) s φ Φ 1( b ) ) 1, b + h b s + h s b + h Supplier: ι A = ι, e la = 0 l > H L, min A s Tr[Σ s (A, A s )] 1/2, s.t. ι A s = ι, e la s = 0 l > H L s. Note, Tr[Σ i ], Tr[Σ s ], and Tr[Σ o ] are the variances of i t, i s t, and o t, respectfully, so the manufacturer linearly trades off between stability in its inventory, its supplier s inventory, and its orders/production. It does so with weights α and β, which respectively parameterize the newsvendor costliness, to the manufacturer, of misalignments in production capacity and upstream inventory, relative to the newsvendor costliness of misalignments in downstream inventory; α and β constitute fundamental preference parameters. The following proposition characterizes this system s unique Nash equilibrium (see the appendix for proofs). Proposition 1. The supplier sets A s to R Ls, and the manufacturer sets A to the unique fixed point of f, where: f(x) K + G ( [G g(x)g] 1 [G D L C C G g(x)k] ), g(x) D L C CD L + a(x)i + b(x)c JC, a(x) αtr[σ i (x)] 1/2 Tr[Σ o (x)] 1/2, b(x) βtr[σ i (x)] 1/2 Tr[Σ s (x, R Ls )] 1/2. Exhibit 1 depicts the optimal DSP IRFs (columns of A), and shows how our solution unifies much of the decoupled supply chain literature: When Λ = α = β = 0, our model generalizes the single-stage inventory models of Kahn (1987), Lee et al. (1997), and Graves (1999): The MMFE generalized these models demand processes and the XGOUTP their order policies. When Λ = α = 0 and β = x, our model generalizes Graves et al. (1986) s decoupled supply chain model with a more general production process and a closed-form expression for upstream inventory levels. When Λ = α = 0 and β = x, our model reduces to a generalization of Chen and Lee (2009) s ours additionally allows partially-aligned supply chains, with fractional θ. When Λ = β = 0 and α = x, our model formalizes Graves et al. (1998) s analysis, deriving its supply chain score card metrics inventory and production variability from model primitives. 6

7 When Λ = β = 0 and α = x, our model generalizes Bray and Mendelson (2011) s model, imposing no restrictions on Σ. Also, by incorporating the η t order-shifters, it offers a more nuanced perspective on the bullwhip effect see and When Λ = 0, α = x, and β = y, our model generalizes Balakrishnan et al. (2004) s, relaxing the assumption that orders follow either a weighted average or exponential smoothing of observed demand signals. 2.3 Empirical Identification Our model provides an empirical supply chain framework. By design, it incorporates internally consistent errors, and hence can support structural estimation. That is, signals ɛ o t, ɛ i t, and ɛ s t comprise both DSP components Aɛ t, C(D L A I)ɛ t, and C(D Ls A s I)Aɛ t and noise components η t, CD L η t, and C(D Ls A s I)η t. The noise components make these signals conducive to statistical modeling. Consider for a moment what it has taken to include these errors: Simply adding noise to the order quantity wouldn t have sufficed, as the order and inventory signals must be stochastic as well (we model signal processing, after all). Thus, we had to specify that the order-shifting errors, n t, also follow an MMFE, find a reasonable way to combine its signals with demand s (thereby developing the XGOUTP class), and carry them through the firms optimization problems. Our specification is parsimonious A, Σ, and Λ compactly parameterize all manufacturer behavior yet general each matrix houses (H + 1) 2 parameters. More importantly, these matrices are identified under the weak conditions that demands follow an MMFE and orders follow an XGOUTP. How? Well, first we can derive demand and order signals, ɛ t and ɛ o t, from demand and order forecasts. Then we can estimate Σ from ɛ t. After, we can consistently estimate the l th row of A by regressing the l th element of ɛ o t on ɛ t, since ɛ o t depends linearly on ɛ t. Finally, from ɛ t, ɛ o t, and A, we can get η t = ɛ o t ɛ t A, and thus Λ. The preference parameters, α and β, are also identified, but only if we additionally impose our Proposition 1 equilibrium and L s > 0. Recall, the manufacturer linearly trades off between the standard deviations of o t, i t, and i s t with weighting parameters α and β. These standard deviations identify α and β: stable o t implies high α; stable i s t implies high β; and stable i t implies low α and β. For example, we can gauge β, the importance of stability in upstream inventory relative to that in downstream inventory, by measuring whose inventories absorb production shocks: Considerate, high-β firms deplete their own inventories to shelter their suppliers, yielding variable i t and stable i s t ; self-interested, low-β firms, on the other hand, propagate production shocks, passing inventory burdens upstream, yielding stable i t and variable i s t. Also leadtimes L and L s are technically identified from the shape of A the firm moves mass from its top L s rows to its L th superdiagonal but rather than estimate them, we fix L to zero and L s to one. These are the only numbers that fit with our quarterly Compustat data: We set L to zero because cycle times should be much shorter than three months, and L s to one because suppliers may respond more sluggishly and the trivial L s = 0 case lacks flexibility. The estimates of 3.3 agree with this specification. We set forecast horizon H to seven setting it to any number between four and ten yields similar results. Finally, the basic newsvendor parameters b, h, q, g, c, b s, h s, and θ are not identified, as they only factor into the firms objective functions through α and β. 7

8 3 Estimation Procedure 3.1 Data We use Compustat data from the financial statements of public U.S. companies between 1974 and 2008 in the retailing, wholesaling, and manufacturing sectors: (SIC , , and , respectively). We proxy for demand with COGS and for orders with production (see e.g., Cachon et al., 2007; Lai, 2005; Wong et al., 2007). 3 The data are quarterly, so signals with l-period information leadtimes correspond to l quarters hence. We remove data-points corresponding to fiscal calendar or reporting schedule changes, total assets of less than a million dollars, nonpositive inventories or sales, industry changes, mergers, or acquisitions. We construct a sample from the remaining data 77,692 observations from 907 firms by selecting each firm s longest uninterrupted series, as long as the series has at least 60 observations. We then transform each firm s demands and orders by: (1) dividing by total assets; (2) deseasonlizing and detrending with linear and quadratic functions of t; (3) Winsorizing the top and bottom 1%; and (4) normalizing the demand variances to one. Exhibit 2 reports summary statistics. 3.2 Estimators Here we define estimators of a given firm s our core model parameters: impulse response matrix A, signal covariance matrices Σ and Λ, and preference parameters α and β. The estimators each have two-stages with a common first stage that estimates demand and order signals ɛ t and ɛ o t. Since we use sequential estimators, we calculate all standard errors with the block bootstrap (Berkowitz and Kilian, 2000; Hardle et al., 2003) MMFE Signal Estimators We first estimate MMFE signals from demand and order forecasts. Define forecast vectors F t H t=0 ɛ t ld l and Ft o H t=0 ɛo t l Dl note that e l F t and e l F t o respectively give the period-t mean-square-error-minimizing forecasts of the demand and order quantity of period-(t + l). Rearranging these expressions yields ɛ t = F t F t 1 D and ɛ o t = F o t F o t 1D. Thus, we can derive signal estimates from demand in order forecast estimates with: ɛ t F t F t 1 D and ɛ o t F o t F o t 1D. (6) For convenience, we house these vectors in matrices Ê [ ɛ 0,, ɛ T ] and Ê o [ ɛ o 0,, ɛ o T ]. We estimate forecasts by specifying that deseasonized demands and orders depend linearly on an underlying vector autoregressive processes, which means we can estimate mean-square-error-minimizing demand and order forecasts with fitted values of regressions of future demands and orders on contemporaneous explanatory variables (see Lütkepohl, 2005). 4 For explanatory variables, we use a firm s demands, squared demands, inventory levels, and squared inventory levels from the prior four quarters. 3 We define production with the following accounting identity: o t = d t + i t i t 1. Also, using the LIFO reserve, we translate all COGS observations to LIFO form. 4 The consistency these forecasts requires the number of time periods goes to infinity. Accordingly, we only use firms that have at least 60 observations. 8

9 3.2.2 Impulse Response Matrix Estimator Now we translate MMFE signal estimates into DSP IRF estimates with expression (2) s orthogonally moment conditions. Our system is just-identified A houses (H + 1) 2 elements and equation (2) specifies (H + 1) 2 moment conditions so we use the method of moments, defining our impulse response estimator as the unique matrix that sets (2) s sample moments to zero:  Êo Ê (ÊÊ ) 1. (7) This estimator consistently converges to the true impulse response matrix when Σ is invertible and ɛ t and η t are orthogonal: ) 1 plim  = plim Êo Ê (ÊÊ T ) = plim (T 1 (Aɛ t + η t )ɛ t Σ 1 = A + E[η t ɛ T T t] = A. This estimator imposes minimal theory just the flexible MMFE and XGOUTP specifications. In 3.2.4, we present an alternative impulse response matrix estimator that additionally imposes our Proposition 1 equilibrium result. t= Signal Covariance Matrix Estimators We estimate Σ and Λ, the other two matrices that determine firm behavior, with: Σ ÊÊ /T and Λ (Êo ÂÊ)(Êo ÂÊ) /T. (8) Proving these estimators consistency is straightforward Preference Parameter Estimators Finally we present estimators of α and β, the respective costliness, from the perspective of the downstream firm, of instability in production and supplier inventories relative to the costliness of instability in downstream inventories. Note, we could simultaneously estimate all exogenous model parameters α, β, Σ, and Λ with equation (2) s moment conditions combined with E[ɛ t ɛ t Σ] = 0 and E [ η t η t Λ ] = 0. However, dividing the problem in the fashion of Newey (1984) makes the estimation easier. That is, rather than estimate all parameters jointly, we do so sequentially, estimating α and β from the stand-alone Σ and Λ estimates of We estimate the two preference parameters from the (H + 1) 2 moment conditions of equation (2); the system is over identified, so we use the Generalized Method of Moments (GMM): ( α, β) argmin a,b 0 vec[êo Ê A (a, b, Σ, Λ)ÊÊ ] W vec[êo Ê A (a, b, Σ, Λ)ÊÊ ], (9) where vec stands for matrix vectorization, W the optimal two-stage GMM weighting matrix, and A (α 0, β 0, Σ 0, Λ 0 ) the Proposition 1 fixed point solution under parameter specification {α = α 0, β = β 0, Σ = Σ 0, Λ = Λ 0 }. Expression (9) describes a nested fixed point estimator (see Rust, 1994; Aguirregabiria and Mira, 2002; Judd and Su, 2008), as evaluating the GMM criterion function requires solving Proposition 1 s fixed point solution. 9

10 To prove that α and β are consistent, we show the sample moment conditions converge to zero only under the true preference parameter values. We first show T 1 [Êo Ê A (a, b, Σ, Λ)ÊÊ] only converges to zero if A (a, b, Σ, Λ) converges to the true impulse response matrix: plim T 1 [Êo Ê A (a, b, Σ, Λ)ÊÊ ] = plim T t=1 T 1 (Aɛ t +η t )ɛ t A (a, b, Σ, Λ)Σ = AΣ A (a, b, Σ, Λ)Σ. Since Σ and Λ are consistent, A (a, b, Σ, Λ) will converge to A (a, b, Σ, Λ). Thus, since A is one-to-one in its first two arguments, A (a, b, Σ, Λ) converges to A (α, β, Σ, Λ) if and only if (a, b) = (α, β). Finally, note that A ( α, β, Σ, Λ), the impulse response matrix implied by our exogenous parameter estimates, is a consistent estimator of A. This estimator, unlike  from 3.2.2, will always satisfy the Proposition 1 form. 3.3 Estimates Starting with the exogenous signal covariance matrices, we depict the mean and median elements of Σ and Λ in Exhibit 3. Our DSP model fits well: the error terms, η t, generally have smaller magnitudes than the explanatory variables, ɛ t. The last-minute order revisions are noisiest (Λ s top-left entry dominates) because erratic behavior is most difficult to anticipate. Finally, as expected, demand signal informativeness (variability) decreases with information leadtime. Next, in Exhibits 4 we consider α and β, the model s other two exogenous parameters. Roughly three quarters of firms exhibit positive α regard for production stability and about half exhibit positive β regard for upstream inventory stability. Most estimates fall short of one, and most β fall short of the corresponding α: Generally firms most prefer stable downstream inventories, then stable production quantities, and then stable upstream inventories. The estimates exhibit a large degree of heterogeneity: in each case the 75 th percentiles exceed the medians by more than a factor of four. We confirm these findings in and with alternative methodologies. Finally, we consider the endogenous impulse response matrices. Exhibit 5 plots the means, medians, and interquartile ranges of the elements of  and A ( α, β, Σ, Λ) across firms. While operating under different economic restrictions  unconstrained and A ( α, β, Σ, Λ) constrained to the manifold of Proposition 1 solutions the estimates agree remarkably: The mean and median  estimates match their A ( α, β, Σ, Λ) counterparts. By their own accord, the unrestricted estimates follow the Proposition 1 form. 5 Arriving at similar estimates, both with and without our equilibrium solution, suggests a sound specification. Indeed, imposing nested theoretical restrictions shows that our model s key parameters are identified under weak conditions, and that our theory explains our data well. 6 5 The congruence between b A and A (bα, b β, b Σ, b Λ) estimates also validates our L = 0 and L s = 1 specification. 6 As Keane (2010, p. 17) explains: [One should] attempt to estimate individual key equations or parameters of a model in ways that do not rely on the full set of assumptions that are required for full information maximum likelihood (FIML) estimation of the complete model. A finding that estimates are similar when these alternative approaches are used would give us more confidence in the specification of the structural model.... [Also,] if it can be shown that certain key parameters of a structural model are identified even if the functional form assumptions required for FIML estimation are relaxed, then we gain confidence that the model is a useful tool for organizing the world and making predictions.... This view seems to be widely accepted among applied researchers. 10

11 4 Applications With our empirical production model, we conduct six studies that span a wide range of topics. The first four, presented in 4.1, deduce supply chain features from demand signal processing; these studies stem from policy variable estimates Â. The last two, presented in 4.2, provide counterfactual analyses of hypothetical supply chain changes; these studies stem from preference parameter estimates α and β. 4.1 Cross Sectional Studies of Derived Supply Chain Metrics The decision variable A and signal covariance matrices Σ and Λ govern firm behavior in our model. Hence, from Â, Σ, and Λ one can derive consistent, asymptotically-normal estimates of any behavioral supply chain metric that relates to our model see Exhibit 6. To estimate these metrics, simply plug Â, Σ, and Λ into Exhibit 6 s formulas (cf. Cameron and Trivedi, 2005, p. 231); since the matrix estimates are consistent and asymptotically normal, the metric estimates will be as well. Firms can use our metric derivation procedure to benchmark their supply chain performance (or that of their competitors). This benchmarking tool is promising, as While there are many on going research efforts on various aspects and areas of SCM, so far little attention has been given to the performance evaluation, and hence, to the measures and metrics of supply chains (Gunasekaran et al., 2001, p. 71). But more interesting are the research implications. Our procedure can derive telling metrics for effectively any supply chain phenomenon that can be couched in terms of our model. Accordingly, our method opens a new avenue of research: cross-sectional studies of derived supply chain metrics. We provide four examples: measures the prevalence of supply chain coordination; addresses a long-standing empirical quandary with a more accurate measure of production cost function convexity; resolves a macroeconomic inventory anomaly by accounting for continuous firm learning; and sharpens our understanding of the drivers of the bullwhip effect Do Firms Consider Their Suppliers Inventory Costs? Supply-chain coordination has become a paramount research topic: see surveys Whang (1995), Cachon (2003), and Chen (2003). Indeed, practitioners and academics alike have placed a greater emphasis on the need to view the supply chain as a whole as the vehicle by which competitive advantage is achieved (Christopher and Peck, 2003, p. 162). Yet while it contains a considerable amount of theory, [it has] an embarrassingly paltry amount of empiricism. Thus, we have little guidance on how the theory should now proceed (Cachon, 2003, p. 111). Accordingly, As a first step towards wider implementation, this research needs to develop an empirical theoretical feedback loop... our most rewarding efforts now lie with collecting data (Cachon, 2003, p. 111). Responding to this call for empiricism are Terwiesch et al. (2005), Randall et al. (2006), and Mortimer (2008). We contribute to this stream of research by creating a supply chain metric that indicates whether firms consider supplier inventory costs (i.e., whether β is positive): Proposition 2. A firm s delayed orders metric Tr[CD(D L A AD L )] is positive if and only if it regards supplier inventory costs (i.e., β is positive) otherwise it s zero. 11

12 Proposition 2 stems from a simple notion: Firms that regard upstream inventory costs will postpone production, to a degree, to increase supplier response times. They do so by shifting A s mass downwards (A s lower rows relate to longer supplier information leadtimes); the delayed orders metric measures the reallocation of mass to longer leadtimed rows. By delaying production in this manner, the downstream firm in part sacrifices its own inventories for its supplier s. Overall, 56% of our sample exhibits a positive delayed orders metric estimate, Tr[CD(D L Â ÂDL )], which reiterates 3.3 s finding that about half our β estimates are positive. Exhibit 7 presents significantly positive mean and median delayed orders metric estimates; however, Exhibit 8 tempers these results, demonstrating that most industries median estimates are insignificant Are Production Cost Functions Convex? This section studies the convexity of production cost functions (PCFs). PCF convexity lies at the heart of a critical discrepancy between a macroeconomic finding and a microeconomic theory; Blinder and Maccini (1991, p. 73) explain: No one seemed to notice the tension that was developing between the emerging macroeconomic and microeconomic views of inventories. Macro-economists routinely thought of inventories as a destabilizing factor: in theory, the inventory accelerator created cycles that otherwise might not exist; in practice, GNP was more volatile than final sales (GNP less inventory investment). Yet the prevailing micro theory viewed inventories as a stabilizing factor, something a cost-minimizing firm could use to smooth production in the face of fluctuating sales. Could something that was stabilizing at the micro level actually destabilize the macroeconomy? It was a fascinating question that was barely explored. According to Blinder and Maccini (1991, p. 80), without PCF convexity, the main foundation of the law of supply would crumble. 7 PCF convexity also has significant operational implications. Capturing the ramp-up and shut-down costs associated with production rate changes and the limits of production capacity, it is a mainstay of the production scheduling literature (e.g. Holt et al., 1960; Beckmann, 1961; Klein, 1961; Lippman et al., 1967a,b; Kleindorfer et al., 1975; Graves, 1981; Bertrand, 1986; Graves et al., 1998). These scheduling models indicate that firms should smooth production, to a degree, to avoid costly workload spikes. However, the field of supply chain management, focusing on inventory rather than production, has deemphasized these costs: With several notable exceptions (e.g. Holt et al., 1960; Anand and Mendelson, 1997; Simchi-Levi and Zhao, 2003; Aviv, 2007; Veeraraghavan and Scheller-Wolf, 2008; Boute and Van Mieghem, 2011), supply chain models generally specify PCFs to be linear. Systematically ignoring PCF convexity may overburden supply chains, as Given capacity constraints on their logistics (receiving inbound shipments, warehousing, store deliveries, shelf restocking, etc.), it [might be] prudent for [firms] to smooth production (Cachon et al., 2007, p. 466). 7 Blinder (1986a, p. 431) elaborates: All that is necessary to create a production smoothing motive for holding inventories is that demand vary through time and either that the short-run cost function be convex (i.e., the short-run production function be concave) or that there exist costs of changing output, or both. If, in addition, there is a random element to demand, inventories will also serve as a buffer stock. These conditions appear to be very weak so weak, in fact, that it is hard to imagine how they could fail to hold. 12

13 Despite its prominence, PCF convexity has yet to be empirically established. Researchers have almost exclusively determined whether PCFs are convex by gauging whether firms smooth production. But over a decade of research proves an absence of production smoothing (see West, 1985; Eichenbaum, 1989; Blinder, 1986a; Schuh, 1996; Ramey and West, 1999). For example, production is less variable than demand in only 21% of our sample. Production smoothing s limited success, 8 suggests we need a better measure. Indeed, the following corollary of Proposition 1 shows that production smoothing can be a biased measure of PCF convexity. The corollary relates to our model s PCF, c(o t ) o t v + (q + g)(o t z) + + gz, which is strictly convex if and only if α is positive. Corollary 1. Production smoothing is neither a necessary nor a sufficient condition for a strictly convex production cost function (positive α). The bullwhip effect (see Kahn, 1987; Lee et al., 1997) and consideration for supplier inventory costs underlie Corollary 1: The absence of production smoothing does not imply a nonconvex PCF, because the bullwhip effect can mask a firm s smoothing attempts. Conversely, production smoothing does not imply strictly convex PCF, because firms also smooth production to stabilize upstream inventories. To control for these confounding factors, we develop a sharper metric: Proposition 3. A positive early production metric, Tr[CDAD L ], is both a necessary and a sufficient condition for a strictly convex production cost function (positive α). The intuition behind Proposition 3 is as follows: Firms produce early, with an intent to store final outputs, if and only if they face a strictly convex PCF. The convexity compels firms to stock up in anticipation of future demand spikes to offload work from peak production periods. But without PCF convexity, firms, no longer compelled to divert work to idle periods, schedule production so that finished goods arrive just in time. Thus, whether or not firms ramp up production early indicates PCF convexity. Producing early means turning long-information-lead-timed demand signals into short-information-lead-timed order signals, or equivalently, allocating mass to A s top-right corner. The Tr[CDAD L ] expression reports this corner s total mass. By design, the early production metric sidesteps the production smoothing metric s problems. Our early production metric provides strong evidence of PCF convexity, confirming 3.3 s positive- α results. Exhibit 9 demonstrates the economy-wide mean and median Tr[CDÂDL ] are significantly positive at p =.01. Disaggregating into manufacturer, retailer, and wholesaler subsamples yields similar results. As expected, manufacturers, for whom production is most significant, exhibit the highest early production metric estimates, and wholesalers, who handle merchandise the least, exhibit the lowest. Moreover, Exhibit 10 demonstrates these patterns hold widely across industries. Our procedure finds firms do, on average, exhibit convex PCFs; it rejects the oft-employed linear-production-cost assumption, and resolves the missing- PCF-convexity dilemma How Quickly Do Firms Adjust Inventories? Here we apply our framework to an anomaly that preoccupied economists for decades (e.g., see Carlson and Wehrs, 1974; Feldstein et al., 1976; Blinder, 1986b; Blinder and Maccini, 1991; Seitz, 1993; Ramey and West, 8 Fair (1989) found production smoothing in trade associations data, and Tomiura (2002) found it at an air conditioner manufacturer. Also Krane and Braun (1991, p. 558) found production smoothing in 28 out of 38 industries, but they explain doubts about the [production smoothing] model persist. 13

14 1999): the apparent sluggishness of inventory adjustment. Feldstein et al. (1976, p. 364) summarize the problem: The basic problem lies in the estimated speed of adjustment of actual to desired inventory stocks. Previous investigators have interpreted their parameter estimates as implying that the gap between actual and desired inventories is reduced by only about 10 percent per quarter, or 35 percent per year. Such an extremely slow adjustment seems very unlikely, especially since a major inventory correction over an entire year is equivalent to, at most, a few days of production. Our framework yields reasonable adjustment rate estimates and explains why others estimated such slow inventory realignment. 9 The slow adjustment rate estimates lead Blinder and Maccini (1991, p. 75, 93) to conclude that the standard production-smoothing/buffer-stock model of inventories [is] in deep trouble, and that The profession, it seems to [them], has devoted too many intellectual resources to using [this] problematic productionsmoothing model. Indeed, we find this production-smoothing inventory model problematic. The model in question developed by Lovell (1961), and underpinning most inventory-adjustment-rate studies (Blinder and Maccini, 1991, p. 81) hinges upon an assumption that changing inventory levels is inherently costly. But classic inventory theory cannot justify this assumption. On the contrary, Lovell (1961) s model appears to blend two legitimate considerations inventory stability and production stability (both of which our model incorporates) into stability in the change in inventory, which has no direct operational meaning. To demonstrate its problems, we apply Lovell (1961) s estimation procedure to our Compustat data. We too find unreasonably slow adjustment rates: Exhibit 11 depicts median manufacturer and retailer quarterly closure rates of about 5%, which correspond to over-three-year-long inventory gap half-lives egregious. Now we approach the problem with our empirical production model. Rather than estimate the rate of convergence to an ideal inventory level, we estimate how a given demand signal affects inventory levels over time. This scheme accommodates our various demand signal information leadtimes: How demand shocks deplete inventories, we show, depends on how far ahead firms can anticipate them information availability influences inventory adjustment. Adopting a supply chain perspective, we also measure how demand shocks deplete supplier inventories. To do all this, we estimate the IRFs that govern how demand signals with l-period information leadtimes, e l ɛ t, modify current and future inventory levels, both downstream and upstream. These IRFs equal the time series of inventory deviations attributable to demand signals that equal one. According to our framework, the l th columns of matrices C(D L A I) and C(D Ls A s I)A respectively house the IRFs that describe how l-period-information-lead-timed demand signals adjust own- and supplier inventories. Thus, we calculate C(D L Â I) and C(DLsĀ I)Â for each firm, and thereby consistently estimate each company s inventory adjustment process and that of its supplier, supposing that this supplier behaves like the average firm in our sample viz., that A s equals Ā, the average estimated impulse response matrix. Exhibit 12 plots these inventory IRFs. The zigzags make sense: the inventory IRFs first increase (firms build inventories to prepare for upcoming demand shocks); then they sharply drop (demand signals realize, depleting inventories); then they revert back to zero (firms restock depleted inventories to baseline levels). The supplier IRFs indicate that these inventory displacements propagate upstream, albeit in an attenuated 9 McCarthy and Zakrajsek (1998) and Tsoukalas (2011) also estimate reasonable adjustment rates. 14

15 fashion. More to the point, the IRFs depict reasonable adjustment rates: Generally, firms rebuild inventories depleted by a demand shock within two quarters. 10 If firms respond so quickly then why has the literature consistently estimated such slow adjustment rates? Exhibit 13 explains the discrepancy. Although firms briskly replenish stocks depleted by a given demand shock, e l ɛ t, the demand signals observed in a given period, ɛ t, can have a long-lasting effect, as they incorporate a sequence of such demand shocks the next hits as soon as the firm recovers from the last. Indeed, Exhibit 13 demonstrates that the inventory deviations attributable to a period s demand signals can last for years. Lovell (1961) s model interprets this persistence as firm sluggishness, because slow response rates are that model s only means for old signals to affect current inventories Do firms bullwhip additively or multiplicatively? The bullwhip effect, one of the most prominent findings in supply chain management, is the phenomenon in which demand variability amplifies up a supply chain like the crack of a whip (see Lee et al., 1997). For example, the bullwhip in a beer supply chain would lead sales to be more variable at a brewer than at a liquor store. Firms can bullwhip either multiplicatively, by magnifying existing demand signals, or additively, by contributing new sources of variation to those signals (cf. Fransoo and Wouters, 2000, p. 88). The multiplicative bullwhip depends on the variability and autocorrelation of demand viz., it responds to ɛ t. The additive bullwhip arises from factors other than demand, such as tool downages, input price variations, order batching, and staffing issues viz., it responds to η t. While we know firms generally bullwhip (cf. Bray and Mendelson, 2011), we don t know how. The theory permits both multiplicative and additive bullwhips: the former relates to Lee et al. (1997) s first bullwhip driver demand signal processing and the latter to their other three input rationing, order batching, and input-price fluctuations. Identifying the mechanism that underpins the bullwhip could help practitioners select the best recourse: 11 Firms address multiplicative bullwhips by stabilizing demand, sharing sales forecasts, and shortening production leadtimes (signal exposure windows), and they address additive bullwhips by making production processes more systematic, say by reducing setup costs to induce smaller production batches, or by improving preventative maintenance schedules to decrease tool downages. Our signal processing specification decomposes the total bullwhip effect into multiplicative and additive components: B Var(o t ) Var(d t ) = Tr[AΣA + Λ] Tr[Σ] = B M + B A, where B M Tr[AΣA Σ] and B A Tr[Λ]. It equivalently decomposes Bray and Mendelson (2011) s lead-l bullwhip, the amplification of uncertainty that resolves with l periods notice: B l Var(e l ɛo t ) Var(e l ɛ t) = e l (AΣA +Λ Σ)e l = B l M +Bl A, where B l M e l (AΣA Σ)e l and B l A e l Λe l. Adding the distortion mechanism a second dimension to Bray and Mendelson (2011) s information leadtime bullwhip decomposition yields three bullwhip decompositions in all: H H B = B M + B A = B l = BM l + BA. l (10) l=0 10 As an aside, our inventory IRFs settle another inventory conundrum: Blinder and Maccini (1991, p. 82) explain that A second troubling feature of empirical stock-adjustment models is that unanticipated sales shocks do not seem to lead to inventory disinvestment... This suggests that finished goods inventories play little role as a buffer stock. Exhibit 12 refutes this finding; the IRF corresponding to demand signals with zero information leadtimes illustrate that unanticipated sales do meaningfully deplete inventories. 11 cf. Table 1 of Lee et al. (1997). l=0 15

16 We estimate the terms of expression (10) at the firm level and tabulate their sample means in Exhibit 14. As expected, our total and lead-l bullwhip estimates match Bray and Mendelson (2011) s. The additive bullwhip dominates, which suggests firms should mitigate the bullwhip by stabilizing production rather than demand; they should specifically focus on last-minute production fluctuations, as the lead-0 additive bullwhip drives most of the effect. 12 The multiplicative bullwhip has no appreciable effect on the total bullwhip because a negative B 0 M component counteracts the positive B 1 M, B2 M, and following Proposition speaks to our negative B 0 M estimate: Proposition 4. If L = 0 and Σ s diagonal entries are descending, then B 0 M 0. B 3+ M components. The Proposition 4 s conditions largely hold in our data: Σ s diagonals generally decrease cf. Exhibit 3 and production leadtimes, relative to our period length, are essentially zero cf. footnote 5. When L = 0, firms have no signal exposure window, and hence no reason to amplify demand variations (i.e., to overlap autocorrelated demand signals). Rather firms use DSP in this case to mitigate the bullwhip s information distortion by increasing supplier information leadtimes viz., by transforming costly lead-0 variations into less-costly lead-1, -2, and -3+ variations, which explains the negative B 0 M and positive B 1 M, B2 M, and estimates. In this way, the multiplicative bullwhip checks the amplification of demand uncertainty. B 3+ M 4.2 Counterfactual Analyses We now use our empirical production model to conduct counterfactual experiments, simulating the effect of potential supply-chain changes. Although matrices A, Σ, and Λ characterize firm behavior, they can not underpin counterfactual analyses, because decision variable A is endogenous and the optimal inventory policy changes with the supply chain setting (cf. Lucas, 1976). Instead, we base our what-if studies on the fixed, exogenous preference parameters α and β. We study two changes firms can readily make: increasing consideration for upstream inventory costs, and improving demand forecasts. The first study demonstrates that an increase in supply chain coordination can actually increase the overall bullwhip effect (but not the lead-0 bullwhip). The second outlines a forecast-accuracy supply chain externality: Downstream forecast improvements beget upstream forecast improvements Does Supply Chain Coordination Alleviate the Bullwhip Effect? Combining the topics of and 4.1.4, we study the degree to which supply chain coordination dampens the bullwhip effect. Specifically, we simulate how the bullwhip would change if firms stopped considering supplier inventory costs i.e., if all β went to zero. We measure how the total bullwhip effect, B(A, Σ, Λ) Tr[AΣA + Λ Σ], and the lead-0 bullwhip, B 0 (A, Σ, Λ) e 0(AΣA + Λ Σ)e 0, change from a baseline scenario with parameters {α 0, β 0, Σ 0, Λ 0 } { α, β, Σ, Λ} and corresponding impulse response matrix A 0 A (α 0, β 0, Σ 0, Λ 0 ) to a zero-coordination scenario with parameters {α 1, β 1, Σ 1, Λ 1 } { α, 0, Σ, Λ} and corresponding impulse response matrix A 1 A (α 1, β 1, Σ 1, Λ 1 ). 12 This finding agrees with the standing empirical literature. Respectively in auto-component, ready-to-eat-meal, and grocery supply chains, Taylor (1999), Fransoo and Wouters (2000), and Lai (2005) each find demand signal processing a minor bullwhip driver. 16

17 Exhibit 15 plots bullwhip changes against supply chain coordination changes. While the change in the lead-0 bullwhip decreases in the change in β, the change in the overall bullwhip does not. Why? Well, recall we ve assumed L s = 1, which means the only order variabilities harmful to suppliers are those with zero period information lead times, e oɛ o t. The lead-0 bullwhip strictly amplifies these harmful variations, so of course it decreases in the degree of supplier consideration. However, the overall bullwhip responds both to order variations that harm suppliers and those that do not. Decreasing lead-0 variations sometimes requires increasing other variations by even larger amounts, so increasing supply chain coordination sometimes increases the total bullwhip. Regard for supplier inventories dampens demand uncertainty, but not necessarily demand variability Does demand forecast accuracy constitute a supply chain externality? We conjecture that suppliers will indirectly benefit when buyers improve their demand forecasts as the advanced demand information, propagating through supply chains, ultimately helps suppliers anticipate their own demands. In other words, we believe better-informed buyers will purchase more predictably, and hence make more desirable customers. In this sense, suppliers have a stake in the accuracy of their buyers forecasts. Disregarding this externality the role demand information plays in the entire supply chain could lead to forecast capability underinvestments. The following counterfactual analysis demonstrates this externality. We consider a hypothetical forecast improvement that enables firms to observe all demand signals a period earlier, allowing them to act on more current demand information (as Vanston and Vanston, 2005, p. 33 explain, the most common reasons for poor forecasts are the use of unreliable or outdated data ). 13 This forecast improvement is equivalent to incrementing signal information leadtimes by one, changing ɛ t to Dɛ t and thus Σ to DΣD. Of course this change would reduce demand uncertainty, to the benefit of the downstream firm, but might it also reduce order quantity uncertainty, to the benefit of the upstream firm? We find it does. We measure order quantity uncertainty with Aviv (2007, p. 780) s adherence-to-plans metric, U o (A, Σ, Λ) Tr[P (AΣA + Λ)], where P is a diagonal matrix with e l P e l = (1/2) l+1. This metric captures the idea that order quantity changes made at the last moment are more troublesome than those made in advance, penalizing the variability of order quantity revisions made with l-periods notice by amount (1/2) l+1. A decrease in this metric indicates a drop in order quantity uncertainty, and hence supplier inventory costs. For perspective, we also consider the change in demand uncertainty, which we capture with the adherence-to-sales-plans metric, U(Σ) Tr[P Σ]; this metric measures demand uncertainty exactly how the adherence-to-plans metric measures order uncertainty. Putting this together, we measure how the adherence-to-plans and adherence-to-sales-plans metrics change from a baseline scenario with parameters {α 0, β 0, Σ 0, Λ 0 } { α, β, Σ, Λ} and corresponding impulse response matrix A 0 A (α 0, β 0, Σ 0, Λ 0 ) to an improved-forecast scenario with parameters {α 1, β 1, Σ 1, Λ 1 } { α, β, D ΣD, Λ} and corresponding impulse response matrix A 1 A (α 1, β 1, Σ 1, Λ 1 ). Exhibit 16 demonstrates a strong forecast improvement externality: decreased demand uncertainty i.e. negative U(Σ 1 ) U(Σ 0 ) begets decreased order uncertainty i.e. negative U o (A 1, Σ 1, Λ 1 ) U o (A 0, Σ 0, Λ 0 ). 13 In the MMFE context, improving forecasts means increasing demand signal information leadtimes (cf. Heath and Jackson, 1994, p. 21). 17

18 As forecasts improve, demand uncertainty drops mechanically; what s interesting is that this uncertainty reduction propagates upstream, manifesting its self in more predictable orders. Moreover, the benefits of advanced demand information translate to the supplier efficiently: on average, the order uncertainty drop is 71% as large as the demand uncertainty drop the forecast improvements benefits suppliers greatly. 5 Conclusion Supply chain management is a field rich in theory but poor in data. Fortunately, we can use the former to compensate for the latter we can apply supply chain theory to broaden supply chain empiricism. Our model-cum-estimators provide a scaffolding to do so, a basic empirical supply chain specification. Our approach frees researchers from the shackles of proprietary data limited sample sizes, disorderly databases, capricious managers by inferring the workings of hundreds of supply chains from published data. Nevertheless, our empirical model is no panacea. It only relates to a fraction of supply chain topics: Demand signal processing its driving engine elucidates many, but not all, supply chain features. Also, relying on Compustat data aggregated both temporally and cross-sectionally limits its scope to highlevel studies (it would be more applicable with more detailed data, however). Moreover, like any structural specification, ours has its assumptions and limitations; e.g., our model disregards prices to focus on the nuances of inventory management. To recap, this work (1) creates an empirical supply chain model based on demand signal processing, (2) devises nested fixed point GMM estimators of the model s fundamental preference parameters and sequential method of moments estimators of its decision variables, the impulse response functions that govern demand signal processing, (3) estimates these quantities across firms with Compustat, demonstrating that our theory explains our data well, and (4) translates these estimates into supply chain measures that we use to conduct the following six studies: In we measure whether firms sacrifice their inventories for the sake of their suppliers. We find about half our sample does. In we develop a better indicator for production cost function convexity; our measure suggests a prevalence of convex production cost functions. In we study how quickly misaligned inventories readjust. Our estimates depart from previous findings because our model incorporates a more flexible demand learning process. In we decompose the bullwhip effect into additive and multiplicative components; we find the former drives the bullwhip and the latter tempers it. In we find that consideration for upstream inventory costs always decreases order uncertainty, but not necessarily order variability. In we identify a forecast accuracy externality: Suppliers benefit when buyers improve their demand forecasts. As these studies demonstrate, our empirical model allows researchers to study many supply chain phenomena with nothing more than Compustat. 18

19 Exhibit 1: Optimal Demand Signal Processing Impulse Response Functions These plots characterize Proposition 1 solutions for various preference parameter values. Our model reduces to those from the listed papers, under the corresponding parameter specifications and Λ = 0. A plot s l th line corresponds to the l th column of A (we start counting columns from zero), the impulse response function characterizing the transformation of demand signals with l-period information lead times, e l ɛt, into order signals, ɛo t. We consider the case in which Tr[Σi ] = Tr[Σ o ] = Tr[Σ s ]. x = 1/5 x = 1 x = 5 α = β = 0 Kahn (1987) Lee et al. (1997) Graves (1999) α = 0, β = x Graves et al. (1986) Chen and Lee (2009) α = x, β = 0 Graves et al. (1998) Bray and Mendelson (2011) α = x, β = 1 Balakrishnan et al. (2004) 19