Econometrica Vol. 42 No. 5 1974
Testing Structural Specification Using the Unrestricted Reduced Form
Abstract
In the first section of this paper the overidentifying restrictions on a system of linear simultaneous equations are expressed in terms of restrictions on the reduced form parameters. These restrictions provide the basis of a test of the structure using only the unrestricted reduced form parameter estimates. Under Ho the test proposed is asymptotically equivalent to a likelihood ratio test. The test may be applied as a single equation or complete system procedure and it may be presented as either a x2 or an F statistic. The case is also made here for system overidentification tests rather than single equation procedures, the arguments being drawn from the statistical literature on hypothesis testing by induction. The computational advantages of the present proposals are substantial when compared to FIML based likelihood -ratio tests and Monte Carlo experiments confirm that a system version of the test performs well in large samples. The system version of the test behaves like the FIML likelihood ratio test in large sample situations both under Ho and H1. However, the Monte Carlo studies indicate that both the single equation and system versions of the test perform poorly in small samples. THE AIM OF THIS paper is to investigate the possibility of deciding on the specification of a simultaneous equation model prior to the estimation of the structure. A well established test procedure is suggested which uses OLS estimates of the reduced form parameters; it enables the null hypothesis, that the model specified is not significantly different from the model which generated the sample, to be tested. Because of the one-to-one correspondence between the overidentified structure and the restricted reduced form, it is possible to make inferences about the structure from the observed compatibility of the reduced form restrictions with the sample information. In addition, the reduced form restrictions resulting from a particular equation may be isolated and tested separately, if desired. The principle underlying the test would appear to be due to Wald [17J; namely, that if the null hypothesis is correct and the structure postulated as the maintained hypothesis was responsible for the generation of the observed sample, then the unrestricted reduced form parameter estimates will tend, if the sample size is large enough, to satisfy the reduced form restrictions advanced under the maintained hypothesis. A number of problems relating to identification of linear simultaneous equations make life a little difficult and are discussed subsequently. Now, take the linear structure
- DOI
- 10.2307/1913794
- Volume
- 42
- Issue
- 5
- Pages
- 869
- Sources
- openalex crossref bibtex:phds-export.bib