Solving the Synthetic Control Weight Problem

From Predictor Matching to Donor Weights

Once a donor pool and a set of pre-intervention predictors have been chosen, the synthetic control is defined by weights on the donor units. The goal of the weight problem is to make a weighted combination of donor predictors resemble the treated unit’s predictors. The constraints matter: they determine which combinations are allowed and therefore what “best match” means.

Let there be \(J\) predictors and \(K\) donor units. Write \(X_1\) for the \(J \times 1\) vector of predictors for the treated unit, and \(X_0\) for the \(J \times K\) matrix whose \(k\)th column contains the predictors for donor \(k\). The vector \(W=(w_1,\ldots,w_K)'\) contains the donor weights. For a fixed \(J \times J\) predictor-weighting matrix \(V\), the discrepancy between the treated predictors and the synthetic predictors is measured by a weighted quadratic distance.

The synthetic predictor vector is \(X_0W\). The constraints make it a convex combination of the donor predictor vectors: no donor receives a negative weight, and the weights add to one. Thus, the synthetic predictors lie in the convex hull of the donor predictors. This is a substantive restriction, not just a computational convenience. It prevents the solution from extrapolating by subtracting one donor’s predictors from another’s.

The matrix \(V\) defines how discrepancies across predictors contribute to the objective. For example, a diagonal entry may give greater importance to one predictor than another. In this tutorial \(V\) is treated as fixed; the central task is to solve the weight problem for that chosen matrix. The objective measures predictor mismatch under that metric. It does not guarantee exact balance, since the treated unit may lie outside the donors’ convex hull or the selected predictors may not all be balanced by one set of weights.

Quadratic-Program Form and Convexity

The original objective can be written as a quadratic function of \(W\). Define \(Q=X_0'VX_0\), \(b=X_0'VX_1\), and \(c=X_1'VX_1\). Because \(V\) is symmetric, the two cross-products in the expansion are equal:

Therefore the weight problem is a quadratic program: minimize \(W'QW-2b'W+c\) over the simplex of nonnegative weights summing to one. The constant \(c\) does not depend on \(W\), so it can be omitted when computing the minimizer. It is retained in the displayed expansion because it is part of the original mismatch.

The objective is convex. For any \(K \times 1\) vector \(u\),

This establishes that \(Q\) is symmetric positive semidefinite: symmetry follows from \(V=V'\), and the displayed inequality establishes nonnegative quadratic forms. Hence the quadratic objective is convex, and any feasible point satisfying the minimization conditions is a global minimizer. The simplex is also closed and bounded, so a minimizer exists. It need not be unique: different weight vectors can produce the same predictor combination, or the objective can be flat along a feasible direction.

How the Constraints Shape the Solution

For a general number of donors, the nonnegativity constraints mean the optimum can lie on a face of the simplex. A donor with weight zero is excluded from the synthetic combination. The first-order conditions clarify how the active donors and excluded donors differ. For the objective \(f(W)=c-2b'W+W'QW\), its gradient is \(2(QW-b)\). Introduce a multiplier \(\nu\) for the constraint \(\mathbf{1}'W=1\), where \(\mathbf{1}\) is a \(K \times 1\) vector of ones, and nonnegative multipliers \(\mu_k\) for \(w_k \geq 0\). The Karush–Kuhn–Tucker conditions are:

For any donor with positive weight, complementary slackness gives \(\mu_k=0\), so its gradient component is balanced by the common equality-constraint multiplier. For a zero-weight donor, the corresponding multiplier can be positive; the condition prevents a feasible move that adds that donor from improving the objective. Since the objective is convex and the constraints are linear, these conditions are sufficient for a global optimum as well as necessary.

What the Weights Do—and Do Not—Establish

The optimization makes the predictor-matching rule explicit. If a predictor has a large discrepancy and receives substantial weight through \(V\), the solution has an incentive to reduce that discrepancy, subject to the available donors and the simplex constraints. But the solution cannot overcome a lack of suitable donors: it may leave appreciable residual imbalance when the treated predictors are outside the donor convex hull or when matching one predictor worsens another.

A small minimized objective means close fit under the chosen predictors and \(V\). It is not, by itself, evidence that the donor combination reproduces the treated unit’s untreated post-intervention trajectory. Causal interpretation still depends on the design’s identification assumptions and on defensible choices of donor pool, predictors, and pre-intervention window. The weights solve a specified optimization problem; they do not validate those choices.

• Mistake: allowing negative weights to improve the fit. An unconstrained least-squares solution may subtract one donor from another. That is not the synthetic control problem defined here, because it permits extrapolation beyond convex combinations of donors.
• Mistake: assuming every selected predictor must match exactly. The minimizer balances the aggregate weighted discrepancy. Exact matching is possible only when the treated predictor vector can be represented by a feasible donor combination.
• Mistake: reading a positive donor weight as a causal claim. A positive weight identifies a donor’s contribution to the constructed predictor match. Causal interpretation requires more than the optimization result.