Consider the regression model Y; = B1X11 + B2X2¡ + uj for i = 1,..., n. %3D • Specify the least squares function that is minimized by OLS.

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Consider the regression model
Y; = B1X1i + B2X2; + Ui
for i = 1, ..., n.
• Specify the least squares function that is minimized by OLS.
• Compute the partial derivatives of the objective function with respect to
B1 and B2.
• Compute the second derivatives of the objective function with respect to
B1 and B2. Is the least squares function quadratic?
• Derive an expression for B1 as a function of the data (Y;, X1i, X2i), i =
1,...,n.
• Suppose E-1 X1;X2i = 0. Show that
i=D1
n
i=1
i=1
• Suppose the equation satisfies MLR.1-MLR.4, please derive E(ß1).
• Suppose the equation satisfies MLR.5, please derive Var(Bì).
• Suppose that the model contains an intercept. Also suppose that
n
E(Xli – X1) (X2i – X2) = 0
i=1
Transcribed Image Text:Consider the regression model Y; = B1X1i + B2X2; + Ui for i = 1, ..., n. • Specify the least squares function that is minimized by OLS. • Compute the partial derivatives of the objective function with respect to B1 and B2. • Compute the second derivatives of the objective function with respect to B1 and B2. Is the least squares function quadratic? • Derive an expression for B1 as a function of the data (Y;, X1i, X2i), i = 1,...,n. • Suppose E-1 X1;X2i = 0. Show that i=D1 n i=1 i=1 • Suppose the equation satisfies MLR.1-MLR.4, please derive E(ß1). • Suppose the equation satisfies MLR.5, please derive Var(Bì). • Suppose that the model contains an intercept. Also suppose that n E(Xli – X1) (X2i – X2) = 0 i=1
Show that
βΣ(Χ - Χ1) (Y-Υ) /Σ ( Χ- X.).
i=1
i=1
How does this compare to the OLS estimator of B1 from the regression
that omits X2 ?
• Suppose you are interested in B1, the causal effect of X1 on Y. Suppose
that X1 and X2 are uncorrelated. You estimate B1 by regressing Y onto
X1 (so that X2 is not included in the regression). Does this estimator
suffer from omitted variable bias? Explain.
Transcribed Image Text:Show that βΣ(Χ - Χ1) (Y-Υ) /Σ ( Χ- X.). i=1 i=1 How does this compare to the OLS estimator of B1 from the regression that omits X2 ? • Suppose you are interested in B1, the causal effect of X1 on Y. Suppose that X1 and X2 are uncorrelated. You estimate B1 by regressing Y onto X1 (so that X2 is not included in the regression). Does this estimator suffer from omitted variable bias? Explain.
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