# Do predicted values differ if variables included stepwise or simultaneously?

You make a linear model #1 of the following form to predict adult human height:

$$y = \beta_{0.1} + \beta_{1.1}\text{Height}_\text{dad} + \beta_{2.1}\text{Height}_\text{mom}$$ You calculate fitted values $\hat{y_1}$.

Now you add $\beta_3x_3$ to make model #2 from the same data.

$$y = \beta_{0.2} + \beta_{1.2}\text{Height}_\text{dad} + \beta_{2.2}\text{Height}_\text{mom} + \beta_{3.2}\text{Age}$$ You calculate fitted values $\hat{y_2}$.

Now you make model #3 from the same data, but model #3 uses the fitted values from model #1 instead of $Height_{dad}$ and $Height_{mom}$.

$$y = \beta_{0.3} + \beta_{4.3}\hat{y_1} + \beta_{3.3}\text{Age}$$

You calculate fitted values $\hat{y_3}$.

• In terms of prediction, what is the conceptual difference between the predicted values from models #2 and #3?
• When would model #3 be appropriate, if ever?
• Short answer is no (to the question as written in the body). – Affine Mar 9 '15 at 16:34
• @Affine: question has been improved – jtd Mar 10 '15 at 22:15

Everything depends on which parameters are relevant in explaining the dependent variable.

Case 1: Age is not relevant. $\beta_3$ = 0

In this case it shouldn't matter if you do it stepwise or simultaneously.

Case 2: Age is relevant $\beta_3\neq$ 0, but Age is orthogonal to other regressors in the model.

In this case it shouldn't also matter if you do it stepwise or simultanously.

In other cases results should be different.

I am also assuming that the estimates are consistent, i.e. this is not an errors-in-variables model (regressors are not measured with an error), or if there are other relevant regressors that are missing from the model, these are orthogonal to those that are included in the model.

• Fascinating, "But the OLS estimator $\hat{\beta_1}$ is a biased and inconsistent estimator of $\beta_1$ in the true model, except in the special case where all of the [\$\beta \ne 0 ??] omitted variables are orthogonal to all of the included variables." – jtd Mar 12 '15 at 13:11