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I am wondering how would the slope and intercept change after adjusting for a confounder factor. After adjustment, would the slope be lower, or higher, and the value for the intercept?
Is there any other important numerical change when adjusting for confounding? (in standard errors, etc.)
Thank you!
The "omitted variable bias formula" as stated in Stock and Watson, Introduction to Econometrics (also available at https://www.econometrics-with-r.org/6-1-omitted-variable-bias.html) says that the slope coefficient of a regression of $y$ on $X$ only converges to the true value $\beta_1$ plus a term associated with the correlation of what is not included in the simple regression and hence left in the error term, $\rho_{Xu}$, $$ \hat\beta_1 \rightarrow_{p} \beta_1 + \rho_{Xu} \frac{\sigma_u}{\sigma_X} $$ As @whuber said, depending on your causal question and what is left in the error term and how it is associated with the variable of interest, both directions are possible.
Here are two answers discussing a specific example:
Observational vs quasi-experimental design?
The standard errors will be affected, too. For one, all versions of these (homoskedastic ones, heteroskedasticity-robust ones etc.) make use of residuals, i.e. differences between the dependent variable and fitted values. Now, if you compute fitted values from a different regression, the residuals and hence standard errors will be different, too.
In more detail, the proofs of consistency of the standard errors require that residuals are computed from a consistent estimator of $\beta$, so that omitted variable bias will also lead to inconsistent standard errors.
It depends on the substantive nature of the relationship between the confounder and the other variables. Here's one example
Let's say that you are running a model trying to predict income as a function of years of education. The model tells you education has a BIG impact on income. But what if some of that relationship doesn't have anything to do with the power of education itself - it's just because people who can afford to go to college are more likely to have richer parents and other economic resources that help them earn more. If that were true and you included a control for parental wealth in the model then what would you predict would happen to the coefficient for education? Well, it would go down, because some of the "effect" you in the previous model was really just due to the effect of the confounder.
However, there are other situations in which controlling for a confounder can make the coefficient for your key independent variable get "bigger" (i.e. further away from zero). For example, if you were estimating the effect of a tutoring program but didn't account for the fact that only people who were struggling actually received tutoring, the first model might only show a small effect, even if the program is really helpful. But when you control for prior struggles, the estimated effect gets bigger (and more accurate).
So the key to this question is to think substantively (not statistically) about how the potential confounder might be related to the other variables
