Suppose that a regression relationship is given by the following: γ=β0+β1 X1+β2 X2+εIf the simpl

step 1 You can say only when e, beta, sub 2, x sub 2 equals 0, will this be correct. And the slope esti

Suppose that a regression relationship is given by the...

Suppose that a regression relationship is given by the following: $$ \gamma=\beta_0+\beta_1 X_1+\beta_2 X_2+\varepsilon $$ If the simple linear regression of $Y$ on $X_1$ is estimated from a sample of $n$ observations, the resulting slope estimate is generally biased for $\beta_1$. However, in the special case where the sample correlation between $X_1$ and $X_2$ is 0 , this will not be so. In fact, in that case the same estimate results whether or not $X_2$ is included in the regression equation. a. Explain verbally why this statement is true. b. Show algebraically that this statement is true.

Suppose that a regression relationship is given by the following:
$$
\gamma=\beta_0+\beta_1 X_1+\beta_2 X_2+\varepsilon
$$
If the simple linear regression of $Y$ on $X_1$ is estimated from a sample of $n$ observations, the resulting slope estimate is generally biased for $\beta_1$. However, in the special case where the sample correlation between $X_1$ and $X_2$ is 0 , this will not be so. In fact, in that case the same estimate results whether or not $X_2$ is included in the regression equation.
a. Explain verbally why this statement is true.
b. Show algebraically that this statement is true.

Key Concepts

Simple vs. Multiple Regression Estimation

In multiple regression, each coefficient measures the effect of its corresponding variable while holding other variables constant, which typically requires controlling for other relevant variables. In simple regression, where only one explanatory variable is considered, the coefficient can be biased if another relevant variable is omitted. However, if the omitted variable is uncorrelated with the included variable, its omission does not affect the slope estimate. Thus, when X? and X? are uncorrelated, the slope obtained from a simple regression of the dependent variable on X? equals the slope from the multiple regression that also includes X?.

Omitted Variable Bias

Omitted variable bias occurs when a regression model excludes one or more relevant variables that are correlated with the included regressor(s). This omission leads the estimated coefficients to capture not only the direct effect of the included variables on the dependent variable but also some of the effect of the missing variables. In the context of this problem, when X? is omitted from the regression, its influence on the dependent variable may be inadvertently attributed to X? unless X? and X? are uncorrelated.

Correlation Among Regressors

The correlation between explanatory variables plays a crucial role in determining the presence of omitted variable bias. When two regressors, such as X? and X?, are correlated, excluding one from a regression can cause the coefficient of the other to absorb effects via the correlation structure. However, if the sample correlation between X? and X? is zero, there is no shared variation between the two, meaning that omitting X? does not distort the estimated relationship between X? and the dependent variable.

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