The Durbin–Wu–Hausman test (also called Hausman specification test) is a
statistical hypothesis test in
econometrics
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...
named after
James Durbin
__NOTOC__
James Durbin FBA (30 June 1923 – 23 June 2012) was a British statistician and econometrician, known particularly for his work on time series analysis and serial correlation.
Education
The son of a greengrocer, Durbin was born in W ...
,
De-Min Wu, and
Jerry A. Hausman.
[ The test evaluates the ]consistency
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
of an estimator when compared to an alternative, less efficient estimator which is already known to be consistent. It helps one evaluate if a statistical model corresponds to the data.
Details
Consider the linear model ''y'' = ''Xb'' + ''e'', where ''y'' is the dependent variable and ''X'' is vector of regressors, ''b'' is a vector of coefficients and ''e'' is the error term In mathematics and statistics, an error term is an additive type of error. Common examples include:
* errors and residuals in statistics, e.g. in linear regression
* the error term in numerical integration
In analysis, numerical integration ...
. We have two estimators for ''b'': ''b''0 and ''b''1. Under the null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
, both of these estimators are consistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
, but ''b''1 is efficient (has the smallest asymptotic variance), at least in the class of estimators containing ''b''0. Under the alternative hypothesis, ''b''0 is consistent, whereas ''b''1 isn't.
Then the Wu–Hausman statistic is:[
:
where † denotes the Moore–Penrose pseudoinverse. Under the null hypothesis, this statistic has asymptotically the ]chi-squared distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squar ...
with the number of degrees of freedom equal to the rank of matrix .
If we reject the null hypothesis, it means that b1 is inconsistent. This test can be used to check for the endogeneity of a variable (by comparing instrumental variable
In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered t ...
(IV) estimates to ordinary least squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the ...
(OLS) estimates). It can also be used to check the validity of extra instruments
Instrument may refer to:
Science and technology
* Flight instruments, the devices used to measure the speed, altitude, and pertinent flight angles of various kinds of aircraft
* Laboratory equipment, the measuring tools used in a scientific lab ...
by comparing IV estimates using a full set of instruments ''Z'' to IV estimates that use a proper subset of ''Z''. Note that in order for the test to work in the latter case, we must be certain of the validity of the subset of ''Z'' and that subset must have enough instruments to identify the parameters of the equation.
Hausman also showed that the covariance between an efficient estimator and the difference of an efficient and inefficient estimator is zero.
Derivation
Assuming joint normality of the estimators.[
:
Consider the function :
By the ]delta method
In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator.
History
The delta meth ...
:
Using the commonly used result, showed by Hausman, that the covariance of an efficient estimator with its difference from an inefficient estimator is zero yields
:
The chi-squared test is based on the Wald criterion
:
where † denotes the Moore–Penrose pseudoinverse
Panel data
The Hausman test can be used to differentiate between fixed effects model
In statistics, a fixed effects model is a statistical model in which the model parameters are fixed or non-random quantities. This is in contrast to random effects models and mixed models in which all or some of the model parameters are random ...
and random effects model
In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are dra ...
in panel analysis Panel (data) analysis is a statistical method, widely used in social science, epidemiology, and econometrics to analyze two-dimensional (typically cross sectional and longitudinal) panel data. The data are usually collected over time and over the sa ...
. In this case, Random effects (RE) is preferred under the null hypothesis due to higher efficiency, while under the alternative Fixed effects (FE) is at least as consistent and thus preferred.
See also
*Regression model validation
In statistics, regression validation is the process of deciding whether the numerical results quantifying hypothesized relationships between variables, obtained from regression analysis, are acceptable as descriptions of the data. The validation ...
* Statistical model specification
References
Further reading
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{{DEFAULTSORT:Wu-Hausman-Durbin test
Econometric modeling
Statistical tests