Estimating a VAR with Constraints on Fitted Values

Summary

The document poses a constrained estimation problem for a vector autoregression of order one. The objective is to choose the intercept and coefficient matrix by minimizing the sum of squared residuals across equations, while requiring linear combinations of the model’s predicted values to satisfy inequalities. The constraints apply to fitted observations rather than directly to the estimated coefficients.

The question mentions MATLAB’s fseminf function as an attempted solution and asks whether a better implementation exists. However, it provides no answer, estimation results, or comparison of optimization methods. It is therefore useful mainly as a formulation of constrained least squares for a VAR; practical implementation details, feasibility conditions, and the statistical consequences of imposing prediction constraints remain unresolved.

Key ideas

• A VAR(1) can be estimated by minimizing summed squared errors across its equations.
• Constraints may be imposed on linear combinations of fitted values rather than on coefficients.
• The posed constraints are inequalities that must hold for predicted observations.
• The document does not provide a solution or evaluate MATLAB optimization methods.

Tags

# How to estimate constrained a constrained VAR(1) with MATLAB?


# How to estimate constrained a constrained VAR(1) with MATLAB?












Suppose I want to estimate the following VAR(1) model: $$ Y_t = \mu + \Phi Y_{t-1} + \varepsilon_t $$ where $Y_t=(y_{1t}, y_{2t},…,y_{kt})'$, $\mu=(\mu_1,…,\mu_{k})’$ and $\Phi$ a matrix of coefficients. I’m interested in obtaining the coefficients in $\Phi$ such that the resulting vector of predicted values $\hat{Y}_t = (\hat{y}_{1t}, \hat{y}_{2t},…,\hat{y}_{kt})’$ obeys some constraints. Just to give an unrealistic example I want to estimate the $\Phi$ matrix via least squares such that $3\hat{y}_{1t} + 2\hat{y}_{2t}\geq 0$ and $\hat{y}_{3t}+\hat{y}_{4t}+\hat{y}_{5t}\geq0$.

How can I do it? In particular how can I implement it in MATLAB?

EDIT : so far my approach has been to minimise the sum of squared errors obtained by every equation of the VAR(1). Suppose I have a bivariate VAR(1), my problem has been : $$ \min_{\mu,\Phi} e_1’e_1 + e_2’e_2 $$ $$ \text{s.t. constraint 1,2,3...} $$ which I tried to solve with the fseminf function in MATLAB. Is there some better way?

EDIT 2: Notice that the constrain is on the fitted values, not on the estimated coefficients

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.