Dear stateliest fellows,

I am very sorry to bother you about this question. Since I am got trapped by the logic chaos for several days. However, I want to have a try since my supervisor and my econometric teachers can’t solve this problem for me.


I will try to express my confusion clearly using below examples



Take an OLS model for example Y= Xβ+ε. We can use a monte carlo exercise to study finite OLS features as in Hayashi (2000) book. Here, we know both the distribution of X and error term.
Thus, we can simulate a series of random samples, not only random in error term but also in X. Then for each sample we can calculate both the t ratio and the sample beta hat value. Then we can actually get two unconditional distributions from the simulation , one is for t ratio and the other is for the beta hat estimator.



I think my confusion clearly is why it’s better to make inference from the distribution of t ratio, instead of the unconditional distribution of beta. For example, if we draw a random sample , we can calculate a corresponding beta hat value, and then we calculate its corresponding t statistics to decide whether we should reject null hypothesis or not. However, for me, it’s seems nature to make inference from the unconditional distribution of beta (if we have the chance to know this distribution) we got from simulation as well. After all, we are making inference for the true value of beta, so using unconditional distribution of beta seems nature. For example, if when the absolute value of random drawn sample beta is great than a 5% threshold from the unconditional distribution of beta, then we reject it, otherwise we accept it. Take it for example, if when the sampling beta is greater than 3, we reject it, here 3 is the 5% threshold from the simulated unconditional distribution of beta.



But this two methods will not give us the same result. For example, even we have the same value of beta hat for one random experiment, given different data in X, the t ratio will be different. But given the same beta hat value, if we make inference according to the unconditional distribution of beta, the result is always fixed not vary with data in X. My current explanation is that making inference purely according to the distribution of simulated beta distribution actually ignore the useful information contained in X and assume we have no knowledge about data in X. Because for the same calculated beta, the data in X can vary. We can also say, on average of data in X we reject when beta hat value is greater than 3 when we make inference using unconditional distribution of beta (this is the case when we don’t know the components of beta, i.e. which X is used to obtain the calculated beta).



For my understanding, the t ratio is supreme than the unconditional beta distribution for inference is because it make use of the conditional distribution and we do know under which X we calculate the beta hat. And conditional distribution is the updated version of unconditional distribution. When we know which data in X, use the distribution under this data in X will be more relevant. Thus , constructing your statistics using the conditional distribution is more accurate. And you shouldn’t bother to get the unconditional distribution of beta to make inference. Because the inference under unconditional distribution is actually inferencing average on X, not conditional on a specific X data set.



I really hope anyone can follow my logic here to help me disentangle my confusions.



I really appreciate your help!