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3 votes

Simultaneous Causality and Variance within a supply and demand model

$$(1) P_i = \beta_0 +\beta_1Q_i + \beta_2X_i +u_i$$ $$(2) Q_i = \delta_0 +\delta_1P_i + \delta_2W_i +v_i$$ Plugging (2) into (1) and simplifying yields, $$P_i = \beta_0 +\beta_1(\delta_0 +\...
Michael Gmeiner's user avatar
3 votes
Accepted

Measurement Error - Multivariate Case

To derive this you'll want to use the Frisch-Waugh-Lovell theorem. Using the true variable, $x_2$, let $\widetilde{x_2}$ be the residual from a regression of $x_2$ on $x_1$, $$x_2 = \delta_0 +\delta_1 ...
Michael Gmeiner's user avatar
2 votes

Comparing degree of dispersion without calculating variance

The following might help, although whether it's simpler than calculating the variances will depend on the particular functions. Suppose the two distributions are of random variables $x_1$ and $x_2$. ...
Adam Bailey's user avatar
  • 8,529
2 votes

Why can we replace dependent variable y with the residuals e?

You are not replacing $y$ with $e$. We are replacing $y$ with $\hat{y}+\hat{e}$, which is the fitted value of $y$, given by $X\hat{\beta}$, plus the estimated value of the residual, given by $y-X\hat{\...
user21356's user avatar
1 vote

Derivation of autocovariances Lewis (2021) RES

A useful implication of conditional mean independence is: $E[\varepsilon_{it}|\varepsilon_{ks}]=0 \implies E[\varepsilon_{it}\varepsilon_{ks}]=0,$ and more generally, $E[\varepsilon_{it}|\varepsilon_{...
Bertrand's user avatar
  • 3,371
1 vote

Why can we replace dependent variable y with the residuals e?

$$\big[ I-W(W'W)^{-1}W'+WC\big]y = \big[ I-W(W'W)^{-1}W'+WC\big](W\beta + e)$$ $$=\big[ I-W(W'W)^{-1}W'+WC\big]W\beta + \big[ I-W(W'W)^{-1}W'+WC\big] e$$ Analyzing the first term, $$\big[ I-W(W'W)^{...
Alecos Papadopoulos's user avatar
1 vote

Sample weights in Stata: fweight vs. pweight

mean command with pweight gives you mean and sd estimates, which in turn gives you estimate of the coefficient of variation. <...
Paul's user avatar
  • 503

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