Advice on method for analyzing the effects of increase in minimum wage As a hobbyist and generally curious person with a lot of free time at home, I was curious what would be a good method to analyze the effect of an increase in the minimum wage of a country if you only have access to public data?
What initially came to my mind would be to create a simple regression with three variables time, labor cost per employee and total employment numbers where

time would consists of maybe 3 years data prior and after the increase of the minimum wage
the labor cost per employee would be our predictable variable, which we will use to understand better the changes in employment numbers

However, I am having seconds thought about how accurate the output of the regression would be as there are many other factors that can influence the employment numbers. This, I decided to ask if any of you might have any idea of what other variables can be included or simply suggesting a different mathematical approach for this? Thank you in advance.
The output from the model you mention would be not accurate at all for several reasons, including amongst other reasosns:

Your post you mention you want to use labor costs as predictable variable - there is no such thing as predictable variable in econometrics. If you by that mean predicted variable (i.e. dependent variable) it would be wrong to have labor costs as dependent variable. Even if you would want to include it as a predictor looking only at labor costs is not that useful in this case since firms can shift increases in labor costs on their customers instead of reducing employment, and indeed research shows that most of the labor cost increases by minimum wages are just shifted onto consumers (e.g. see Harasztosi &amp; Lindner, 2019). Furthermore, labor costs can increase for plethora of other reasons completely unrelated to minimum wage.

Unless you live in a country with public access to firm level data on employment having 1 country and 6 years is not enough. As a rule of thumb you need 30 observations per independent regressor. Given that data for most controls you need will be avaiable at best at monthly frequency, but most likely at quarterly/yearly frequency you will simply not have enough data points for running parametric model (such as regression). You can expand your dataset by including multiple other countries, in such case having 3 years before and 3 years after might be enough.

Controlling only for 2 other variables beside for dependent variable is not enough and you will likely have model suffering from omitted variable bias.



If you want to analyze a particular increase in minimum wage quite common method in the literature is differences in differences (DiD). There are also more complex techniques, but simple regression would not cut it here (even if it is just for fun, if you care at all about quality of results) and  DiD is not that much more complex.
Your regression specification would look like:
$$y_{it}= \delta_i + \lambda_t + \rho D_{it} + X_{it}'\beta+ \epsilon_{it}$$
where $y_{it}$ will be your dependent variable, $\delta_i$ country fixed effects, $\lambda_t$ time fixed effects, $D_{it}$ is a dummy that tells you whether country $i$ did implement increase in minimum wage at time $t$ or not, and $\rho$ will tell you the effect of this policy. Finally,  $X_{it}'$ is a vector of control variables. You want to control for anything else that you think  might affect the employment in the country.
This is slightly more complex because you will need at least one extra control country that did not increased its minimum wage at the same time (although this should not be that hard to find). In addition, you might want to add more countries into the data set if for no other reason then just to have more data.
Now what you actually want as your dependent (i.e. predicted) variable is not labor costs but unemployment if that is what you are interested in. The reason for this is that research regularly shows that minimum wage increases lead to increases in consumer prices which means that most of the increase in labor costs is shifted on the shoulder of consumers  (see Harasztosi &amp; Lindner, 2019). Hence, even if your model would find large effect on labor cost it might have zero to none effect on employment in the industry. So if employment is what you are after you need employment as dependent variable.
The above would still not be likely good enough, on its own, to be publishable (there are more issues that would need to be addressed) but it would at least count as some first preliminary model from which you can draw some tentative conclusions. You might also want to have a look at some econometrics handbook before doing this to learn more about how to estimate these sort of models. For the model above I think Mostly Harmless Econometrics  by Angrist and Pischke, chapter 5 on DiD would be sufficient reading (or have a look at some online tutorials).