Exact Test Statistics and Distributions of Maximum Likelihood Estimators that result from Orthogonal Parameters

We show that three convenient statistical properties that are known to hold for the linear model with normal distributed errors that: (i.) when the variance is known, the likelihood based test statistics, Wald, Likelihood Ratio and Score or Lagrange Multiplier, coincide, (ii.) when the variance is unknown, exact test statistics exist, (iii) the density of the maximum likelihood estimator (mle) of the parameters of a nested model equals the conditional density of the mle of the parameters of an encompassing model, also apply to a larger class of models. This class contains models that are nested in a linear model and allow for orthogonal parameters to span the difference with the encompassing linear model. Next to linear models, an important set of models that belongs to this class are the reduced rank regression models. An example of a reduced rank regression model is the instrumental variables regression model. We use the three convenient statistical properties to conduct exact inference in the instrumental variables regression model and use them to construct both the density of the limited information maximum likelihood estimator and novel exact statistics to test instrument validity, overidentification and hypothezes on all or subsets of the structural form parameters.