What is the consequence of including irrelevant variables (those that have no direct effect on the dependent variable) in your regression model?[Fill in the blank]

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To evaluate the consequences of adding irrelevant variables to a regression model, let's examine what each option implies and connect it to core regression principles.
Option a: 'The model’s R-squared will no longer be constrained between zero and one.' In ordinary least squares, R-squared is theoretically bounded between 0 and 1, and adding more predictors cannot break that bound in a meaningful sense; it may increase, but it cannot violate the 0–1 range. Therefore this statement is incorrect because the R-squared boundary remains conceptually intact.
Option b: 'Least squares estimators will be biased.' Under the standard OLS assumptions, including irrelevant regressors that are exogenous does not bias the estimated coefficients of the included variables; it mainly affects variance. The presence of extra, unnecessary regressors does not inherently bias the coefficient estimates if regressors are uncorrelated with the error term. Thus this claim is incorrect.
Option c: 'The usual least squares standard errors produced by R are incorrect.' While multicollinearity from irrelevant variables can inflate standard errors, saying they are 'incorrect' is too strong and context-dependent. The standard errors are still the usual OLS standard errors; they may be larger due to increased variance from collinearity, but calling them outright incorrect overstates the issue. This option is not precisely correct.
Option d: 'The estimated coefficients will have higher variance.' When irrelevant variables are added, multicollinearity and reduced effective sample information tend to inflate the variances of the estimated coefficients of the included variables. This is a well-known consequence of including irrelevant regressors, leading to less precise estimates. Therefore this statement is correct.
Therefore, among the options, the one that accurately describes the consequence is the statement that the estimated coefficients will have higher variance, due to increased estimation uncertainty from unnecessary predictors.

What is the consequence of including irrelevant variables (those that have no direct effect on the dependent variable) in your regression model?[Fill in the blank]

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