Comparing the exact distribution of the t statistic to Student’s distribution when its constituent normal and chi-squared variables are dependent

The ubiquitous t-statistic is well known to follow Student's distribution in a variety of settings. Even when a statistical model is such that a derived t-statistic is not distributed exactly according to Student's distribution, it is common practice to use critical values of Student's distribution as opposed to critical values of the normal distribution which are often appropriate under asymptotic considerations. This paper examines the effect which dependence between the normal and chi-square constituent variables of the t-statistic has in respect to Student's distribution. Situations where the aforementioned dependence occurs, often arise in, for example, econometric practice, especially in respect of analysis of t- statistics obtained from simultaneous equation models and dynamic regression models.