# Wald test

**WaldWald estimatorWald statisticWald statistics**

In statistics, the Wald test (named after Abraham Wald) assesses constraints on statistical parameters based on the weighted distance between the unrestricted estimate and its hypothesized value under the null hypothesis, where the weight is the precision of the estimate.wikipedia

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### Likelihood-ratio test

**likelihood ratio testlikelihood ratiolikelihood-ratio**

Together with the Lagrange multiplier and the likelihood-ratio test, the Wald test is one of three classical approaches to hypothesis testing.

The likelihood-ratio test is the oldest of the three classical approaches to hypothesis testing, together with the Lagrange multiplier test and the Wald test.

### Abraham Wald

**WaldWald, Abraham**

In statistics, the Wald test (named after Abraham Wald) assesses constraints on statistical parameters based on the weighted distance between the unrestricted estimate and its hypothesized value under the null hypothesis, where the weight is the precision of the estimate.

### Score test

**Lagrange multiplier testLagrange multiplierLagrange multiplier (LM) test**

Together with the Lagrange multiplier and the likelihood-ratio test, the Wald test is one of three classical approaches to hypothesis testing.

The main advantage of the score test over the Wald test and likelihood-ratio test is that the LM test only requires the computation of the restricted estimator.

### Structural break

**Sup-LR testStructural break testSup-LM test**

For cases 1 and 2, the sup-Wald (i.e., the supremum of a set of Wald statistics), sup-LM (i.e., the supremum of a set of Lagrange multiplier statistics), and sup-LR (i.e., the supremum of a set of likelihood ratio statistics) tests developed by Andrews (1993, 2003) may be used to test for parameter instability when the number and location of structural breaks are unknown.

### Statistics

**statisticalstatistical analysisstatistician**

In statistics, the Wald test (named after Abraham Wald) assesses constraints on statistical parameters based on the weighted distance between the unrestricted estimate and its hypothesized value under the null hypothesis, where the weight is the precision of the estimate.

### Constraint (mathematics)

**constraintconstraintsconstrained**

In statistics, the Wald test (named after Abraham Wald) assesses constraints on statistical parameters based on the weighted distance between the unrestricted estimate and its hypothesized value under the null hypothesis, where the weight is the precision of the estimate.

### Statistical parameter

**parametersparameterparametrization**

### Estimator

**estimatorsestimateestimates**

### Null hypothesis

**nullnull hypotheseshypothesis**

### Precision (statistics)

**precisionprecision matrixconcentration**

### Sampling distribution

**finite sample distributiondistributionsampling**

While the finite sample distributions of Wald tests are generally unknown, it has an asymptotic χ 2 -distribution under the null hypothesis, a fact that can be used to determine statistical significance.

### Chi-squared distribution

**chi-squaredchi-square distributionchi square distribution**

While the finite sample distributions of Wald tests are generally unknown, it has an asymptotic χ 2 -distribution under the null hypothesis, a fact that can be used to determine statistical significance.

### Statistical significance

**statistically significantsignificantsignificance level**

While the finite sample distributions of Wald tests are generally unknown, it has an asymptotic χ 2 -distribution under the null hypothesis, a fact that can be used to determine statistical significance.

### Statistical hypothesis testing

**hypothesis testingstatistical teststatistical tests**

Together with the Lagrange multiplier and the likelihood-ratio test, the Wald test is one of three classical approaches to hypothesis testing.

### Computational complexity

**complexitycomputational burdenamount of computation**

An advantage of the Wald test over the other two is that it only requires the estimation of the unrestricted model, which lowers the computational burden as compared to the likelihood-ratio test.

### Expression (mathematics)

**expressionmathematical expressionexpressions**

However, a major disadvantage is that (in finite samples) it is not invariant to changes in the representation of the null hypothesis; in other words, algebraically equivalent expressions of non-linear parameter restriction can lead to different values of the test statistic.

### Taylor series

**Taylor expansionMaclaurin seriesTaylor polynomial**

That is because the Wald statistic is derived from a Taylor expansion, and different ways of writing equivalent nonlinear expressions lead to nontrivial differences in the corresponding Taylor coefficients.

### Binomial regression

**binary response modelbinomial models**

Another aberration, known as the Hauck–Donner effect, can occur in binomial models when the estimated (unconstrained) parameter is close to the boundary of the parameter space—for instance a fitted probability being extremely close to zero or one—which results in the Wald test no longer monotonically increasing in the distance between the unconstrained and constraint parameter.

### Boundary (topology)

**boundaryboundariesboundary point**

Another aberration, known as the Hauck–Donner effect, can occur in binomial models when the estimated (unconstrained) parameter is close to the boundary of the parameter space—for instance a fitted probability being extremely close to zero or one—which results in the Wald test no longer monotonically increasing in the distance between the unconstrained and constraint parameter.

### Parameter space

**weight space**

Another aberration, known as the Hauck–Donner effect, can occur in binomial models when the estimated (unconstrained) parameter is close to the boundary of the parameter space—for instance a fitted probability being extremely close to zero or one—which results in the Wald test no longer monotonically increasing in the distance between the unconstrained and constraint parameter.

### Monotonic function

**monotonicitymonotonemonotonic**

### Maximum likelihood estimation

**maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate**

Under the Wald test, the estimated that was found as the maximizing argument of the unconstrained likelihood function is compared with a hypothesized value \theta_0.

### Likelihood function

**likelihoodlikelihood ratiolog-likelihood**

Under the Wald test, the estimated that was found as the maximizing argument of the unconstrained likelihood function is compared with a hypothesized value \theta_0.

### T-statistic

**Student's t-statistict''-statisticStudent's ''t''-statistic**

The square root of the single-restriction Wald statistic can be understood as a (pseudo) t-ratio that is, however, not actually t-distributed except for the special case of linear regression.