Testing Linear Restrictions in Econometrics: LR, Wald, and LM Tests Explained

In econometrics, we are often interested in testing whether specific relationships hold between estimated coefficients. When such a relationship can be expressed as a linear equation in the coefficients, it is called a linear restriction. A classic example arises from the Cobb-Douglas production function, where economists want to know whether the production process exhibits constant returns to scale.

This article explains the three main procedures used to test linear restrictions: the Likelihood Ratio test, the Wald test, and the Lagrange Multiplier test.

Table of Contents

Key Definitions

Linear Restriction

A linear restriction is a condition imposed on the coefficients of a regression model, based on economic theory or prior belief, that states the coefficients (or a linear combination of them) must satisfy a specific value or relationship. For example, in the Cobb-Douglas production function, the restriction $\alpha + \beta = 1$ states that the labour elasticity and capital elasticity of output must sum to one. A restriction is called “linear” because it is expressed as a linear equation in the coefficients, rather than involving them in a nonlinear form.

Unrestricted Model

An unrestricted model is the original model estimated without imposing any restriction on its coefficients. It is called “unrestricted” because no theoretical condition, such as constant returns to scale, has been forced onto the estimation.

Restricted Model

A restricted model is a model obtained by imposing the linear restriction directly onto the unrestricted model, typically by substituting one coefficient in terms of the others and rewriting the equation. This reduces the number of free parameters to be estimated. Comparing how well the restricted model fits the data relative to the unrestricted model is the basis for testing whether the imposed restriction is valid.

The Cobb-Douglas Production Function

Consider the following production function:

$Y = A L^{\alpha} K^{\beta}$

where $Y$ is output, $L$ is labor, $K$ is capital, $\alpha$ is the labor elasticity of output, and $\beta$ is the capital elasticity of output.

We want to test whether:

$\alpha + \beta = 1$

This restriction implies that the production function exhibits constant returns to scale. Equivalently, this restriction can be written as:

$\alpha = 1 - \beta$

Setting Up the Unrestricted and Restricted Models

To make the model linear, we take the natural log of both sides and add an error term, giving the unrestricted model:

$\ln Y = \ln A + \alpha \ln L + \beta \ln K + u$

This is Model (1), the unrestricted model.

Now, substituting the restriction $\alpha = 1 - \beta$ into the unrestricted model:

$\ln Y = C + (1-\beta)\ln L + \beta \ln K + u$

$\ln Y = C + \ln L - \beta \ln L + \beta \ln K + u$

$\ln Y - \ln L = C - \beta \ln L + \beta \ln K + u$

Defining $Y^{*} = \ln Y - \ln L$ , $K^{*} = \ln K - \ln L$ , and $C = \ln A$ , we obtain:

$Y^{*} = C + \beta K^{*} + u$

This is Model (2), the restricted model.

The core question is not simply whether we can impose a restriction, but whether that restriction is valid. If imposing the restriction does not significantly worsen the fit of the model, we accept it as valid. If it makes the fit considerably worse, we reject it. The measure used to judge “how much worse” the fit is allowed to get is based on the likelihood function, which indicates how likely a model is to be correct given the data.

Taking twice the difference between the log-likelihood of the unrestricted and restricted models produces a statistic that follows a chi-square distribution, with degrees of freedom equal to the number of restrictions imposed. This is the basis of the Likelihood Ratio (LR) test.

There are two further approaches that approximate the likelihood ratio test while requiring the estimation of only one model. Estimating only the unrestricted model and using a formula to approximate the full LR test is called the Wald test. The familiar t-test on OLS coefficients is a specific form of the Wald test. Estimating only the restricted model, and testing whether relaxing the restriction would improve the fit, is called the Lagrange Multiplier (LM) test.

1. The Likelihood Ratio (LR) Procedure

Step 1: State the hypotheses

$H_0$ : There is no significant difference between the unrestricted and restricted models (the restriction is valid, i.e., $\alpha + \beta = 1$ , or equivalently $\alpha = 1 - \beta$ ).

$H_1$ : There is a significant difference between the unrestricted and restricted models (the restriction is not valid).

Step 2: Estimate both models and obtain the residual sums of squares

Estimate the restricted and unrestricted models, and obtain their residual sum of squares, denoted $RSS_{UR}$ and $RSS_{R}$ , where:

$RSS_{UR} < RSS_{R}$

$RSS_{UR}$ is the residual sum of squares of the unrestricted model, and it is always the smaller value. $RSS_{R}$ is the residual sum of squares of the restricted model, and it is always the larger value.

Step 3: Calculate the F-statistic

$F = \dfrac{(RSS_R - RSS_{UR}) / (K_{UR} - K_R)}{RSS_{UR} / (n - K_{UR})}$

where $K$ refers to the number of parameters in each model.

Step 4: Find the F-critical value

The F-critical value depends on the numerator degrees of freedom, $K_{UR} - K_R$ , and the denominator degrees of freedom, $n - K_{UR}$ .

Step 5: Compare the F-statistic with the F-critical value

If $F\text{-statistic} > F\text{-critical}$ , we reject $H_0$ and conclude that the restriction is not valid.

If $F\text{-statistic} < F\text{-critical}$ , we do not reject $H_0$ and conclude that the restriction is valid.

2. The Wald Test

The Wald test is used to test linear restrictions in a regression model by estimating only the unrestricted model, and then using a t-test to determine whether the restriction holds.

Step 1: Estimate the unrestricted model

$\ln Y = C + \alpha \ln L + \beta \ln K + u$

To test the restriction of constant returns to scale (CRS):

Step 2: State the hypotheses

$H_0$ : $\alpha + \beta = 1$

$H_1$ : $\alpha + \beta \neq 1$

Step 3: Calculate the t-statistic

$t = \dfrac{\hat{\alpha} + \hat{\beta} - 1}{\sqrt{ \text{var}(\hat{\alpha}) + \text{var}(\hat{\beta}) + 2\,\text{cov}(\hat{\alpha}, \hat{\beta}) }}$

Step 4: Find the t-critical value

The t-critical value is found with $df = n - k$ , where $n$ refers to the number of observations and $k$ refers to the number of parameters.

Step 5: Compare the t-statistic with the t-critical value

If $t\text{-statistic} > t\text{-critical}$ , we reject $H_0$ and conclude that the restriction is not valid, and that the production function does not exhibit constant returns to scale.

3. The Lagrange Multiplier (LM) Procedure

Unlike the LR and Wald tests, the LM test estimates only the restricted model.

Given the unrestricted model:

$\ln Y = C + \alpha \ln L + \beta \ln K + u$

Our restriction is $\alpha = 1 - \beta$

Step 1: State the null and alternative hypotheses

$H_0$ : $\alpha + \beta = 1$ / Restriction is valid

$H_1$ : $\alpha + \beta \neq 1$ / Restriction is not valid

Step 2: Estimate the restricted model and obtain residuals

Estimate the restricted model and obtain the residuals $\hat{u}_R$

$Y^{*} = C + \beta K^{*} + u$

Step 3: Regress residuals on the explanatory variables of the unrestricted model and obtain $R^2_R$

After obtaining the residuals from the restricted model, regress them on the explanatory variables of the unrestricted model and obtain $R^2_R$

$\hat{u}_R = \gamma_0 + \gamma_1 \ln L + \gamma_2 \ln K + \varepsilon$

Step 4: Calculate the chi-square statistic

$\chi^2 = nR^2$

This statistic follows a chi-square distribution with $h$ degrees of freedom, where $h$ refers to the number of restrictions imposed. In our example, $h = 1$ .

Step 5: Find the chi-square critical value

The chi-square critical value is found with $h$ degrees of freedom.

Step 6: Compare the chi-square statistic with the chi-square critical value

If $\chi^2\text{-statistic} > \chi^2\text{-critical}$ , we reject $H_0$ and conclude that the restriction is not valid.

Conclusion

All three procedures address the same underlying question — whether a restriction imposed on a model’s coefficients is statistically valid — but they differ in what needs to be estimated. The Likelihood Ratio test requires estimating both the restricted and unrestricted models. The Wald test requires only the unrestricted model. The Lagrange Multiplier test requires only the restricted model. In large samples, all three tests are asymptotically equivalent and tend to lead to the same conclusion.

Reference:

D. Asteriou, Applied Econometrics.

Share this article

About the author

Muhammad Minhaj Akhtar

Muhammad Minhaj Akhtar is a Lecturer in Economics at Government Graduate College Jauharabad, Pakistan. He holds an M.Phil. in Economics from Quaid-i-Azam University, Islamabad, and an MSc in Economics from the University of Sargodha, where he earned a Silver Medal. His academic passion lies in Econometrics, with a strong focus on applying empirical methods to real-world economic issues. Through MinhajMetrixHub, he shares learning resources, research guidance, and practical econometric insights for students and researchers.

Chow Test for Structural Change: Concept, Steps and a Real-World Example

Structural change is one of the most important issues that

September 9, 2026 No Comments

Econometrics

Multiple Linear Regression Using Matrix Algebra: Demand Function Example

Introduction: Multiple Linear Regression Using Matrix The matrix approach to

August 25, 2026 No Comments

Econometrics

Data Transformations in Applied Economic Research: A Complete Guide

Data transformations are one of the most important steps in

August 15, 2026 No Comments

Econometrics

How AI and Machine Learning Are Transforming Economic Forecasting

Economic forecasting has always been one of the toughest jobs

August 14, 2026 No Comments

Econometrics

Simple Linear Regression Using Matrix: Full Numerical Example

Simple Linear Regression Using Matrix: Full Numerical Example Introduction: Why

August 6, 2026 No Comments