Likelihood ratio test for model specification with boundary Null I am interested in understanding the asymptotic distribution of Likelihood ratio (LR) test statistic for model specification. I am focusing on the case in which the null hypothesis is of the form (i.e. to assess model specification):
$$
\mathcal{H}_0:\theta_j=0, \qquad \text{For any} \; \theta_j \in \theta \\
\mathcal{H}_1:\theta_j\neq 0, \qquad \text{For any} \; \theta_j \in \theta
$$
this is a test
where $\theta=\{\theta_1,\theta_2,\dots,\theta_n\} \in \Theta$ is the set of all parameters for a model $f(\theta,x)$
In particular, I am trying to understand two things:

*

*Whether the asymptotic approximation of $\chi^2_l$ remains valid
(where $l$ are the degrees of freedom)

*If so, how to correctly define the critical value at $1-\alpha$ confidence (i.e. $\chi^2_{l,1-\alpha}$)

To support my question, assume that we have a parametric model of the following form:
$$
f(\theta,x) = a_0 + a_1x^{b_1} + a_2x^{b_2}
$$
Case 1: $\mathcal{H}_0:a_0=0$
In this case, I obtain the estimates for the unrestricted case (i.e. $\hat{\theta}_{UR}$) and the estimates for the restricted case (in which $\mathcal{H}_0$ holds, i.e. $\hat{\theta}_{R}$)
In this example (and if I understand correctly), $\hat{\theta}_{UR} \in \mathbb{R}^{5}$ and $\hat{\theta}_{R} \in \mathbb{R}^{4}$ (Note: All parameters in $\theta$ can take real values, including 0). Given the Null $\mathcal{H}_0$, we lose 1 degree of freedom.
Thus when calculating the LR statistic, we need to compare it with the critical value from $\chi^2_{k-l}=\chi^2_{5-4}=\chi^2_{1}$ at $1-\alpha$ confidence. This case works as expected.
Case 2: $\mathcal{H}_0:a_1=0,a_2=0 $
Similarly, to Case 1, I obtain the estimates $\hat{\theta}_{UR},\hat{\theta}_{R}$
Here lies my question: In this case and under the null, the restricted (R) model is the following:
$$
f(\theta,x) = a_0 + 0 + 0
$$
Even though the null contains a restriction on 2 parameters (i.e. $a_1=0,a_2=0$), the role of parameters $b_1,b_2$ vanishes, as they do not have any impact over the values of f(\theta,x) because $a_1=a_2=0$. Thus:

*

*Is the dimensionality of $\theta_R$ equal to $l = 3$ (because there are 3 remain parameters with no restrictions, i.e. $a_0,b_1,b_2$) or $l=1$ in this case (because we have only 1 parameter left $a_0$ to try to "fit the data"?


*Can I still directly apply LR test in this case? It has come to my attention after going through this book, that the null in this case may be at the boundary, therefore the classic $\chi^2$ asymptotic approximation does not longer apply directly here. Also, I've seen similar questions where replies suggest the use of a chi-square mixture to compute critical values, but I am not sure how to compute the critical value
My main problem is to come up with the correct critical value for my simulation experiment. Currently, I am using $\chi^2_{l=2}$ at 95% confidence, which leads to over rejection when $\mathcal{H}_0$ is true in the data generating process (DGP)
My intuition is that, there must be some adjustment that I am not considering in this case to make either the critical value larger or use another distribution as proxy for the asymptotics, thus being able to do proper statistical inference
Thank you!

I've find the solution to this problem from Hansen (1996).
In this case, parameters $(b_1,b_2)$ are nuisance parameters, as they are relevant for hypothesis testing but become undefined under the null $\mathcal{H}_0$
The author suggests an empirical procedure to obtain Likelihood ratio test statistics from simulations given different values of nuisance parameters.
Then, the test will follow a $\chi^2$ distribution which will depend of the form of the case under study.
I hope this helps!
