The presence of an irrelevant variable with a true slope coefficient can lead to bias in the estimates of the coefficients for the relevant variables.
The OLS estimator aims to minimize the sum of squared differences between the observed and predicted values of the dependent variable. If the irrelevant variable is strongly correlated with one or more of the relevant variables, it might absorb some of the variation that should be attributed to the relevant variables. This can lead to biased and inefficient estimates.
The standard errors of the coefficient estimates may increase when an irrelevant variable is included. This is because the inclusion of an irrelevant variable can increase the variance of the error term, leading to larger standard errors for all coefficient estimates.
Complete question
Suppose we estimate a model and in addition, we include an irrelevant variable which has a true slope coefficient and which has a strong positive correlation with. what is the effect of including on the ols estimate?