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Interpreting multiple regression coefficients

You fit y=β0+β1x1+β2x2+εy = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \varepsilon by OLS.

Interpret β1\beta_1 precisely. Why can β^1\hat{\beta}_1 differ in sign from the simple regression of yy on x1x_1 alone? What changes in log specifications?

Show a hint

The phrase to make precise is "holding x2x_2 fixed." The Frisch–Waugh–Lovell theorem says β^1\hat{\beta}_1 can be obtained from a regression involving residuals.

Your answer

This one is open-ended. Work it through, then check your reasoning against the full solution.

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