Can I use linear regression analysis if the predictors and criteria change during time? Can I use linear regression analysis to analyze the relationship between two variables if both of them are changing in time (each day)? 
Predictor = Number of unemployed people (in a country)
Criteria = Life satisfaction
We run this survey for a month and see how life satisfaction in a country changes with the growing number of unemployed people. We have data for number of unemployed people every day (single number - sometimes it grows from day to day, sometimes it stays the same), and every day a number of people participates in the survey (multiple answers per day).
If I want to model Life satisfaction ~ No_unemployed, can I do it with the classic lm() function, or do I somehow need to account for the fact that this is an example of a time-series?
What do I need to take care of when conducting these analyses?
EDIT: I am primarily interested in the technical (can I use linear regression, and in which cases), and not the conceptual aspect of this question (which other variables would be valuable to have a look at).
 A: Yes, you should be able to use linear regression analysis, however, I suspect, more work on the variables is needed.
To begin, I would, apriori, expect a Life Satisfaction (a rating index) to be a function of two drivers. First, cable news content including major announcements, and second, direct personal experience (for example, job status, sickness in the family, problems in acquiring basic necessities, lack of access to healthcare, a crime/accident event,...). 
Your successful use of unemployment data is too be expected, however, perhaps some smoothing using a weighted average series and even a lagged effect, as someone may have completed the online survey due to their current availability to be online, perhaps due to job loss. This smoothing could address your concern of observed variables changing in time each day.
Other potential explanatory variables are random except data on the crime rate, pandemic reported cases/deaths, hospital closings, and, gasoline price. So, a rule to turn-on a dummy variable (a very bad news day with, for example, reported shortages in food stocks) could be tested as an added variable to account for the variation in the Life Satisfaction Index.
Further efforts could include the construction of a quantitative misery index for other indicators available expected to induce distress (so, for example, overnight pandemic related deaths). 
The use of a Stepwise Regression routine may be of assistance in identifying contributing variables if you become so interested.
