One of the reasons why there are so many different models is that "having impact" is not that easy to define. It is not to test whether a variable has "any" effect on another variable, so researchers first have to come up with a model (based on theoretical considerations) that they can estimate and test. And of course the tests are different depending on the model they assume.
Basic model
One of the most basic (but powerful and widely used) concepts for these type of hypotheses is Granger-causality. Simply put, variable $A_t$ is said to Granger-cause variable $B_t$, if $A_t$ is useful in forecasting $B_t$. So you might try to find out which macro variables Granger-cause your indices. However to do formal tests you need to restrict the previous definition a bit.
Let us say, that variable $A_t$ Granger-causes variable $B_t$ if
$$E[B_t|B_{t-1}, B_{t-2}, ...] \neq [B_t|A_{t-1}, B_{t-1}, A_{t-2}, B_{t-2}, ...]$$
or in words, knowing the past values of $A_t$ helps us get a better prediction of the current value of $B_t$. The nice thing is, that the above hypothesis is easy to test: you just have to fit a linear model.
So if your index variable is $Y_i$, and your explanatory variables are $\{X_{1t}, \dots X_{mt}\}$, then you just have to estimate the simple linear model
$$ Y_t = \alpha + \sum_{s=1}^S \gamma_s Y_{t-s} \sum_{k=1}^m \beta_{ks} X_{k, t-s} + \epsilon_t $$
by OLS. Then $X_k$ Granger-causes $Y$ (in the expectation sense), if $\beta_{k1}, \dots \beta_{kS}$ are jointly significant.
Practically to test which explanatory variables Granger-cause $Y_t$ you have to do the following:
- Run a linear regression with $Y_t$ as the dependent variable and some lags of $Y_t$ and the explanatory variables at the right hand side
- Test whether all the lags of a given explanatory variable are jointly significant
- Repeat it for each of your indices
By estimating these models you also get an answer to your second question (you just have to look at the coefficients).
Another way to go would be to estimate a VAR model where you include all of your indices and explanatory variables, and then see whether the lags of a given explanatory variable for a given index are jointly significant. This will have a slightly different interpretation: you would test whether your explanatory variables Granger-cause your indices even if you condition on all the indices. In your case I would be careful with it due to the relatively low number of observations.
Practical concerns
Deciding how many lags to include in the regression might be tricky. It depends on the amount of data you have and the model you have in mind. For quarterly data, you should probably include at least 4 lags.
As have panel data you could be inclined to include a country dummy to get fixed effects estimates. However because you have lagged dependent variables on the right hand side, your estimates would be biased if you did this. There are ways around it, but they are quite advanced.
If you have questions how to implement it in practice feel free to ask for details.
