Does improving your NPS correlate to improvements in other business metrics?
The short answer is Yes. The depth, degree and frequency to the correlation however is what can vary dramatically between companies.
How is it Correlated?
In it's simplest form, Net Promoter Score is a "Customer Happiness" metric. If your NPS increases it generally means that your customer base is happier and this will engage more with your product. This increased engagement may be measured as MAU (Monthly Active Users), Installs, Purchases, etc.
If you use NPS to understand your customer base, rather then as a vanity metric (see article: Is NPS a Vanity Metric?) then you will learn what your customers do not like or get frustrated with about your product or service. As you repair these breakages or release new features that improve upon these frustrations your user base as a whole will benefit. This increased happiness (increase in NPS score) will result in your users doing more, purchasing more, etc.
How is it not Correlated?
There is no guaranteed math equation behind the correlation. If your NPS score increases by 10 points this will not guarantee a 10% increase in sales or 10 basis points in your bottom line, etc.
It is also important to understand that every time your NPS improves there is no guarantee that you will even see a critical business metric improve. Sometimes the NPS score may have been dragged down by a bug or breakage that actually does not affect installs, purchases or other such metrics. This is not often the case but occasionally you will run into NPS increases that do not result in business critical stats improving.
So how should NPS be used?
How then do you use NPS data compared with Install, Revenue or other critical business stats. The answer is that you use it to measure your improvements and understand your customers. If by some fluke your company makes enough money and you don't care about customer opinion then you don't really need NPS. 99% of us out there though are not in this boat. NPS is a complimentary business statistic that should be understood and watched along with Install, Revenue and other critical stats.
When you are planning your yearly or quarterly goals you might have goals that look like, "Increase installs by 5%" or "Increase revenue per player by 10%" and so forth. A great yearly or quarterly goal to compliment the overall strategy is to have a goal that focuses around NPS. An example would be to increase the 7 Day average NPS score by "5 points". This type of goal empowers your team to understand the customers and drive new fixes and features towards making the product a better experience. When customers engage more in your product your other critical business metrics will increase as well.