Flashcards
Econometrics II
Cards in this deck (11)
- 1Loss functionsMeasure of distance between observed values and estimates
\[L(y-\theta (x))\] - 2Squared loss functionMinimizes squared distance between observed and estimated values
Optimal is Conditional Expectation Function \[E[Y|X]\]
\[E[(y-\theta)^2|X] = E[((y-\mu_x) - (\theta - \mu_x))^2|X]\]
\[=V(y|x) + (\theta - \mu_x)\] - 3Properties of the CEF\[\theta (x) = argmin E[(y-c)^2|X]\]
\[\epsilon = Y - E[Y|X] \implies E[\epsilon |X] = 0\]
\[\implies E[X' \epsilon |X] = 0\]
\[\implies E[h(X) \epsilon |X] = 0\]
\[V(\epsilon) = E[V(Y|X)]\]
\[C(X,\epsilon) = 0\] - 4Best Linear Predictor (BLP)\[X\beta\]
\[\beta = argmin E[(Y-X\beta)^2]\]
\[E[X'(Y-X\beta)]=0\]
\[= E[X'X]^{-1} E[X'Y]\]
\[V(U) = E[V(Y|X)] + E[\omega^2]\]
Omega is difference between CEF and BLP - 5Properties of i.i.d. sampling\[E[Y_i] = \mu\]
\[V(Y_i) = \sigma^2\]
\[C(Y_i, Y_j) = 0\]
Sample average converges to population average
\[V(\bar{Y}) = \frac{\sigma^2}{n}\] - 6Mean Squared ErrorSum of Squared Bias and Variance
- 7Asymptotic properties of samples\[plim \bar{Y} = \mu\]
\[plim V(Y) = 0\]
Central Limit Theorem - 8Uniform Kernel Estimate\[frac{1/n \sum y_i \boldsymbol{1} (|x_i - x_0| \le \delta_n)}{1/n \sum \boldsymbol{1} (|x_i - x_0| \le \delta_n)}\]
Limiting Distribution:
\[N(\alpha, \beta)\] - 9Matrix Algebra of Regressions\[b_n = (X'X)^{-1}(X'Y) = Q^{-1}X'Y = AY\]
\[\hat{Y} = X(X'X)^{-1}X'Y = NY\]
\[e = Y - \hat{Y} = Y - NY = (I-N)Y = MY\] - 10Limiting distribution of beta\[N(0, E[X'X]^{-1} E[X'XU^2]E[X'X]^{-1}\]
Sandwich form, robust against HESKD.
If model HOSKD, \[\sigma^2 E[X'X]^{-1}\] - 11CRM assumtions\[1. E[Y|X] = X\beta\]
\[2. V(Y|X) = \sigma^2 I\]
\[3. Rank(X) = k\]
4. X is non-stochastic.
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- Flashcards for Graduate-level econometrics.
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- Max Schnidman
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