Notes Accompanying The One And Only..

Author

Bas Machielsen

Published

October 5, 2022

Introduction

Here are some notes accompanying the MSc thesis “The one and only: Reduced-form parameter identification for the spatial lag and canonical social interaction model”, which in turn is based on the DePaula et al. (2025) model.1

Identification Assumptions

How do we map \(\Pi\) back to \(W, \beta, \rho, \gamma\)?

Unrestricted Model (General Case)

There are no explicit closed-form equations in the paper that map \(\Pi\) back to \(W, \beta, \rho, \gamma\) for the general case.

It is explicitly stated in Section 3.4 (Estimation strategy) (p. 27):

“However, there is no clear-cut closed-form expression \(\Pi^{-1}(\cdot)\) due to the non-linear nature of \(\Pi(\cdot)\).”

Instead of using explicit equations to calculate the parameters, the DePaula et al. (2025) paper proves that the solution is unique (identification) and then suggests using a numerical minimization method (the adaptive elastic net estimator) to find them.

The logic of how one would algebraically isolate the parameters is demonstrated in Appendix C.4 (Proof of Lemma 1) (p. 51-53).

  • Equation (14) (p. 51) is the key “master equation” derived from \(\Pi(\theta) = \Pi(\theta_0)\): \[(\beta_0 - \beta)I - (\gamma + \beta_0\rho)W + \beta\rho_0 W_0 + \rho_0\gamma W_0 W = 0\]

  • On p.52, the off-diagonal elements of this equation are used (where \(I\) is zero) to isolate relationships between \(W\) and \(W_0\).

  • On p. 53, Assumption A4 (row sums equal 1) is used to identify \(\rho\) and \(W\).

Restricted Model (\(\gamma_0 = 0\))

For the restricted model discussed in Section 4, the paper does provide explicit algebraic derivations to map the parameters back, in Appendix D, p. 60.

Here are the specific steps and equations:

Step 1: Relate Reduced Form to Structure Set \(\Pi(\theta) = \Pi(\theta_0)\) and derive Equation (20): \[ \left(\frac{1}{\beta} - \frac{1}{\beta_0}\right)I + \left(\frac{\rho_0}{\beta_0}W_0 - \frac{\rho}{\beta}W\right) = 0 \]

Step 2: Use Diagonal Elements (Assumption A1’) Use known diagonal element \(W_{ll} = \kappa\) to get: \[ \left(\frac{1}{\beta} - \frac{1}{\beta_0}\right) + \left(\frac{\rho_0}{\beta_0} - \frac{\rho}{\beta}\right)\kappa = 0 \]

Step 3: Use Row Sums (Assumption A4): Use the row sums (which equal 1) to get equation (21):

\[ \left(\frac{1}{\beta} - \frac{1}{\beta_0}\right) + \left(\frac{\rho_0}{\beta_0} - \frac{\rho}{\beta}\right) = 0 \]

Step 4: Mapping: By subtracting Equation (21) from the previous equation, explicitly isolate the parameters:

  1. Identify Ratio: \(\left(\frac{\rho_0}{\beta_0} - \frac{\rho}{\beta}\right)(\kappa - 1) = 0 \implies \frac{\rho}{\beta} = \frac{\rho_0}{\beta_0}\)
  2. Identify \(\beta\): Substitute back to find \(\frac{1}{\beta} - \frac{1}{\beta_0} = 0 \implies \beta = \beta_0\)
  3. Identify \(\rho\): Since \(\beta\) is identified, \(\rho = \rho_0\) follows immediately.
  4. Identify \(W\): Substitute everything back into Equation (20) to find \(W = W_0\).

Estimation

For the restricted model (\(\gamma=0\)), we can reverse-engineer the “recipe” using the logic in Appendix D combined with the Reduced Form definition. Here is how to go from Data to Estimates (\(\hat{\rho}, \hat{\beta}, \hat{W}\)) for the restricted model (\(\gamma = 0\)).

Step 1: Estimate \(\Pi\) from the Data You start with the observable data: outcomes \(y_t\) and covariates \(x_t\). The reduced-form model is: \[ y_t = \Pi x_t + \nu_t \] Since \(y\) and \(x\) are known, you estimate the matrix \(\hat{\Pi}\) using standard Ordinary Least Squares (OLS) (equation by equation) or the penalized estimator mentioned in Section 3.4.

  • You now have a known \(N \times N\) numerical matrix \(\hat{\Pi}\).

Step 2: Invert the Matrix The theoretical link (p. 28) is \(\Pi = \beta (I - \rho W)^{-1}\). Inverting this gives: \[ \Pi^{-1} = \frac{1}{\beta}(I - \rho W) \] Rearrange this to isolate the unknowns on the right: \[ \beta \hat{\Pi}^{-1} = I - \rho W \] Let \(Q = \hat{\Pi}^{-1}\) (which you can calculate). The equation is now: \[ \beta Q = I - \rho W \quad \Longrightarrow \quad \rho W = I - \beta Q \]

Step 3: Solve for Scalars \(\beta\) and \(\rho\) You now have a system of linear equations. You need to use the assumptions (constraints) from p. 28 to solve for the two scalar unknowns.

  • (Assumption A4): One specific row \(i\) of \(W\) sums to 1. Take the sum of row \(i\) in our equation \(\rho W = I - \beta Q\): \[ \rho \sum_k W_{ik} = \sum_k I_{ik} - \beta \sum_k Q_{ik} \] Since \(\sum W_{ik} = 1\) and \(\sum I_{ik} = 1\) (the diagonal is 1, rest are 0): \[ \rho = 1 - \beta (\text{row sum of } Q \text{ at } i) \quad \text{--- (Eq. A)} \]

  • (Assumption A1’): A specific diagonal element \(W_{ll}\) is a known constant \(\kappa\) (usually 0). Take the \((l,l)\) element of our equation: \[ \rho W_{ll} = 1 - \beta Q_{ll} \] \[ \rho \kappa = 1 - \beta Q_{ll} \quad \text{--- (Eq. B)} \]

Step 4: Calculate the Values You now have two simple linear equations (A and B) with two unknowns (\(\rho, \beta\)).

  1. Substitute (Eq. A) into (Eq. B) to solve for \(\hat{\beta}\).
  2. Plug \(\hat{\beta}\) back into (Eq. A) to find \(\hat{\rho}\).

Step 5: Estimate the Network Matrix \(W\) Now that you have the scalar estimates \(\hat{\beta}\) and \(\hat{\rho}\), go back to the equation from Step 2: \[ \hat{\rho} \hat{W} = I - \hat{\beta} Q \] \[ \hat{W} = \frac{1}{\hat{\rho}} (I - \hat{\beta} \hat{\Pi}^{-1}) \]

  • Result: You have fully estimated the adjacency matrix \(W\) from the data.

Conclusion

The focus of the thesis is on identification. The central proof (Appendix D) is trying to answer: “If I find a \(\beta\) that fits the equations, is it guaranteed to be the True \(\beta_0\)?”

  • If the answer is yes (Identification), then the “Recipe” above works.
  • If the answer is no, the recipe is useless because the math would yield multiple valid answers.

The thesis spends much of its time proving that step 3 in the recipe above is solvable and has exactly one solution. Without that proof, running the calculation on data would be meaningless.

  • I still don’t get why estimation in the general case has no simple algebraic solutions.

Footnotes

  1. De Paula, , Rasul, I., and Souza, P. C. L. (2025). Identifying Network Ties from Panel Data: Theory and an Application to Tax Competition. Review of Economic Studies, 92(4):2691–2729.↩︎