Treatment Effects in Strategic Management: With an Application to Choosing Early Stage Venture Capital

This paper uses the Rubin Causal Model to formalize the treatment effects of a firm choice on its performance. Building from Porter, a firm choice can shape profitability through both strategy and operational effectiveness, but they are distinct in how they do so. The strategic treatment effect is the benefit that is predictable from a firm's characteristics (i.e., resources) and their joint configuration. The strategic determinant function is a mapping of resources to treatment effects, and the role of resource interactions in it determines the importance of coherence for a strategy. Under unconfoundedness, the strategic treatment effect, strategic determinant function, and coherence can be estimated in high-dimensional observational data using machine learning. I present an application estimating the gains from choosing venture capital as early stage financing versus other forms of capital. The results highlight the advantage of considering strategic benefits in this choice. For equity outcomes, there is no average treatment effect of early stage VC, but there is significant heterogeneity: some entrepreneurs can benefit substantially from raising early stage VC, while others be negatively affected. This heterogeneity is predictable from founder, industry and location characteristics. The estimated role of coherence in this choice is moderate. The formalizations in this paper also show that several additional assumptions are required when assessing strategic benefits compared to the usual causal inference. R code to replicate these functions will be included.

Uniform, Integral, and Feasible Proofs for the Determinant Identities

Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over GF (2) in Hrubeš-Tzameret [15]. Specifically, we show that the multiplicativity of the determinant function and the Cayley-Hamilton theorem over the integers are provable in the bounded arithmetic theory VNC 2 ; the latter is a first-order theory corresponding to the complexity class NC 2 consisting of problems solvable by uniform families of polynomial-size circuits and O (log 2 n )-depth. This also establishes the existence of uniform polynomial-size propositional proofs operating with NC 2 -circuits of the basic determinant identities over the integers (previous propositional proofs hold only over the two-element field).

Embedding and Extension Properties of Hadamard Matrices Revisited

Hadamard matrices have many applications in several mathematical areas due to their special form and the numerous properties that characterize them. Based on a recently developed relation between minors of Hadamard matrices and using tools from calculus and elementary number theory, this work highlights a direct way to investigate the conditions under which an Hadamard matrix of order n − k can or cannot be embedded in an Hadamard matrix of order n. The results obtained also provide answers to the problem of determining the values of the spectrum of the determinant function for specific orders of minors of Hadamard matrices by introducing an analytic formula.

SCHUR MULTIPLICATIVE MAPS ON MATRICES

The structure of Schur multiplicative maps on matrices over a field is studied. The result is then used to characterize Schur multiplicative maps f satisfying $f(S) \subseteq S$ for different subsets S of matrices including the set of rank k matrices, the set of singular matrices, and the set of invertible matrices. Characterizations are also obtained for maps on matrices such that Γ(f(A))=Γ(A) for various functions Γ including the rank function, the determinant function, and the elementary symmetric functions of the eigenvalues. These results include analogs of the theorems of Frobenius and Dieudonné on linear maps preserving the determinant functions and linear maps preserving the set of singular matrices, respectively.

Singular invariant hyperfunctions on the square matrix space and the alternating matrix space

Fundamental calculations on singular invariant hyperfunctions on the n ×n square matrix space and on the 2n × 2n alternating matrix space are considered in this paper. By expanding the complex powers of the determinant function or the Pfaffian function into the Laurent series with respect to the complex parameter, we can construct singular invariant hyperfunctions as their Laurent expansion coefficients. The author presents here the exact orders of the poles of the complex powers and determines the exact supports of the Laurent expansion coefficients. By applying these results, we prove that every quasi-relatively invariant hyperfunction can be expressed as a linear combination of the Laurent expansion coefficients of the complex powers and that every singular quasi-relatively invariant hyperfunction is in fact relatively invariant on the generic points of its support. In the last section, we give the formula of the Fourier transforms of singular invariant tempered distributions.