Testing Joint Coefficient Restrictions in a Linear Regression

Summary

The document explains how to express a joint hypothesis about selected regression coefficients as a linear restriction. For a regression coefficient vector, a general F-test restriction takes the form of a matrix multiplying that vector and equaling a target vector. To test whether the first two coefficients are both zero, the restriction matrix consists of the first two rows of the identity matrix, and the target is a zero vector.

This setup applies directly to the question’s long-horizon regression, where the dependent variable sums outcomes over multiple periods and the regressors may include several predictors. The key point is that the restriction matrix selects only the coefficients of interest; it need not test every slope. The answer gives the construction but does not discuss covariance estimation, finite-sample assumptions, or how overlapping horizons may affect inference, so those issues must be handled separately when implementing the test.

Key ideas

• A joint linear regression hypothesis can be written as a matrix restriction on the coefficient vector.
• To test selected coefficients, use rows of the identity matrix corresponding to those coefficients.
• Set the restriction target to zero when testing that the selected coefficients jointly equal zero.
• The restriction matrix can test a subset of coefficients without constraining the others.

Tags

# Deriving the long-horizon predictive regression and hypothesis testing


# Deriving the long-horizon predictive regression and hypothesis testing












I am working on the long-horizon regression,

$$y_{t,K}=\mu+\beta_1x_{1,t-1}+...+\beta_nx_{n,t-1}+e_{t} $$, where $$y_{t,K}=y_{t}+y_{t+1}+...+y_{t+K-1}$$ and there can be multiple x's.

So I am trying to joint hypothesis test, say that that a $$\beta_1 = \beta_2 = 0$$,

I need to show this in matrix form so that I can code it up.

I know it has something to do with the Kronecker product, and the vectorisation of these equations.

However, I don't know how to define the 'hypothesis matrix', so that I am only testing that $$\beta_1 = \beta_2 = 0$$

To be specific if this was a standard regression equation, to test the hypothesis that $$\beta_1 = \beta_2 ... = \beta_n = 0$$, I would just write an identity matrix.

Thanks in advance, I've been struggling with this.

## Answer by steveo'america (score 0, accepted)

https://quant.stackexchange.com/a/40844

Generally an $F$-test hypothesis in a linear regression is expressed as $$ A \beta = c $$ for some matrix $A$ and vector $c$ (typically with $c$ the zero vector). In your case you want $A$ to be the first two rows of the identity matrix, or $e_1^{\top}$ stacked on top of $e_2^{\top}$, and $c$ to be the zero vector.

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.