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I am wondering how to perform a paired hypothesis test in a panel regression. I have been looking in many stats books but I have not found anything.

The question goes: assume for $i=1,2,\cdots, N$ and $t=1,2,\cdots, T$, you have two panel regressions

Regression 1: $Y_i(t) - Y_0(t) = \alpha_i + \beta_{i1}(x_1(t)-Y_0(t)) + \beta_{i2}(x_2(t)-Y_0(t)) + \epsilon_i(t)$

Regression 2: $Y_i(t)-c_{i}(t)=\alpha_i^{*} + \beta_{i1}^{*}\left[\frac{x_1(t)-Y_0(t)}{a_i}\right] +\beta_{i2}^{*}\left[\frac{x_2(t)-Y_0(t)}{a_i}\right]+\epsilon_i^{*}(t)$

I have the following questions
1. Can you do a paired t-test on ($\hat{\alpha_i},\hat{\alpha_i^{*}}$), ($\hat{\beta_{i1}},\hat{\beta_{i1}^{*}}$), and ($\hat{\beta_{i2}},\hat{\beta_{i2}^{*}}$), for $i=1,2,...,N$, (separately for alphas and each of the betas ?
2. Can you do a pair test on the $\left( \mathtt{R^{2}_{i}}, \mathtt{\left(R^{*}_{i}\right)^{2}}\right) $, for $i=1,2,...,N$?

My understanding is that you cannot do these tests, since they are random variables in themselves rather than realizations of TWO fixed random variables. But doesn't this boil down if you assume if $Y_i$'s are from the same population or not?
Also $ \textbf{$R^2_i \overset{H_0}{\sim} Beta\left(\frac{p-1}{2},\frac{N-p}{2}\right)$}$, where $p=2$.

I have $\textbf{two solutions}$ in mind but I am not sure if they are fully correct:
1. Take the mean of $\hat{\alpha_i}$'s and the mean of $\hat{\alpha_i^{*}}$'s and do a paired-test on the these two means. Could use the Lindeberg-Feller Theorem, which is the CLT for non-identically distributed random variables. And you could work out a $\chi^2$ distribution asymptotically. Repeat the same procedure for each of the betas.

  1. Use $\textbf{Hotelling's T-squared test}$

Highly appreciate any help/references

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  • $\begingroup$ Can you explain the difference in notation from the $Y$ and the $y$ in each regression model? $\endgroup$
    – AdamO
    Commented Jul 15, 2015 at 19:19
  • $\begingroup$ One is the dependent and the other the independent variable. Sorry for the confusion, I changed into x the y. $\endgroup$
    – Irland
    Commented Jul 15, 2015 at 21:25
  • $\begingroup$ So you're regressing $Y_1(t)- Y_0(t)$ onto $x_1(t) -Y_0(t)$? Is there any context to the problem? The notation is confusing. $\endgroup$
    – AdamO
    Commented Jul 15, 2015 at 21:31
  • $\begingroup$ Yes. $Y_i(t)$ for $t=1,2,...,T$ represent a time series of stock returns for stock i, whereas $x_i(t)$ represent a time series of factor values that you think explain the cross-sectional returns. In the second regression, $c_i(t)$ is a time series characteristic of the stock i. $\endgroup$
    – Irland
    Commented Jul 15, 2015 at 22:07
  • $\begingroup$ How are the various components estimated in these models? Maximum likelihood? Bayesian MCMC? EM plug-in estimates from a random effects model? How many time points and how many stocks are you measuring? $\endgroup$
    – AdamO
    Commented Jul 16, 2015 at 16:34

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