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Econometrics I

Max Schnidman Business/Economics English 49 cards

Cards in this deck (49)

  1. 1Convergence in Probability\[\underset{n \to \infty}{lim} P(|X_n - X| > \epsilon) = 0\]
  2. 2Almost Sure Convergence\[P(\{\omega: X_n(\omega) \nrightarrow X(\omega)\}) = 0\]
    \[Y_n = \underset{k\ge n}{sup} |X_k - X| \overset{p}{\to} 0\]
    \[P(\underset{k \ge n}{sup} |X_k - X| > \epsilon) = 0\]
  3. 3Direction of Convergences\[A.S. \to P \to D\]
    \[(R) \to P \to D\]
  4. 4Additional A.S. Convergences\[\sum_{k=1}^{\infty}P(|X_n - X|>\epsilon) \le |\infty| \implies X_n \overset{a.s.}{\to} X\]
    \[X_n \overset{p}{\to} X \implies \exists X_{n} \ s.t. \ X_{n_k} \overset{a.s.}{\to} X\]
  5. 5Convergence in mean of order r\[\underset{n\to\infty}{lim}E[[|X_n - X|]^r] = 0\]
  6. 6Cauchy Sequence in Probability\[\underset{n,n\to\infty}{lim} P(|X_n - X_m| > \epsilon) = 0\]
  7. 7Borel-Cantelli Lemma\[A = \cap_{n=1}^{\infty} \cup_{k\ge n} A_k\]
    \[\sum_{n=1}^{\infty} P(A_n) < \infty \implies P(A) = 0\]
  8. 8Continuous Mapping Theorem\[X_n \to X \implies g(X_n) \to g(X)\] in probability or A.S.
  9. 9Convergence in Distribtuion\[\int f(x)dF_n(x) \to \int f(x)dF(x)\]
  10. 10Class of Generalized Distributions\[\underset{x_n \to +\infty}{lim} G_{X_1...X_n}(x_1,...,x_n) = G_{X_1...X_{n-1}}(x_1,...,x_{n-1})\]
    \[\underset{x_n \to -\infty}{lim} G_{X_1...X_n}(x_1,...,x_n) =0\]
    \[G(-\infty)\ge 0\]
    \[G(\infty)\le 1\]
  11. 11Helly-Bray TheoremThe class of generalized distributions is compact w.r.t. weak (distributional) convergence.
  12. 12Asymptotic Tightness\[\forall \epsilon > 0 \ \exists N \ s.t. \ inf_n(F_n(N) - F_n(-N)) > 1-\epsilon\]
  13. 13Kinchin's Law of Large NumbersSuppose \[\{X_n\}_{n=1}^{\infty}\] is i.i.d. sequence of r.v. w. \[E(X_n) = a\] and let \[S_n = \sum_{k=1}^n X_k\]
    Then \[\frac{S_n}{n} \overset{p}{\to} a\]
  14. 14Central Limit TheoremIf \[0 < \sigma^2 < \infty\]
    \[\underset{n\to\infty}{lim} \underset{x}{sup} |P(Z_n
  15. 15Convergence Properties\[X_n \overset{p}{\to} c \equiv X_n \overset{d}{\to} c\]
    \[X_n \overset{d}{\to} X, |X_n- Y_N| \overset{p}{\to} 0,\implies Y_n \overset{d}{\to} X\]
    \[X_n \overset{p}{\to} X, Y_N \overset{p}{\to} c,\implies (X_n,Y_n) \overset{d}{\to} (X,c)\]
  16. 16Slutsky Theorem\[X_n \overset{d}{\to} X, Y_N \overset{d}{\to} c \implies \]
    \[ 1. X_n + Y_n \overset{d}{\to} X + c\]
    \[ 2. X_n Y_n \overset{d}{\to} X c\]
    \[ 3. X_n /Y_n \overset{d}{\to} X / c\]
  17. 17Lindeberg-Feller CLT Condition\[\sum_{i=1}^{k_n} E[||Y_{n,i}||^2]\boldsymbol{1}\{||Y_{n,i}||>\epsilon\} \to 0\]
    \[\lim_{n \to \infty} \frac{1}{s_n^2}\sum_{k = 1}^n \mathbb{E} \left[(X_k - \mu_k)^2 \cdot \mathbf{1}_{\{ | X_k - \mu_k | > \varepsilon s_n \}} \right] = 0\]
  18. 18Delta Method\[X_n \overset{d}{\to} X, b_n \to 0\]
    \[\implies \frac{g(a + b_nX_n) - g(a)}{b_n} \overset{d}{\to} X g^{\prime}(a)\]
  19. 19Extremum Estimator\[\theta_0 = argmax_{\theta \in \Theta} Q(\theta)\]
    \[Q(\theta) = E_{\theta_0}[g(Y, \theta)] = \int g(y, \theta)F(dy, \theta_0)\]
  20. 20Uniform Convergence\[Pr(\underset{T\to\infty}{lim} \underset{\theta\in\Theta}{sup} Q_T(\theta) = 0) = 1 \implies Q_T(\theta) \overset{a.s.}{\to} 0 \]
    \[\underset{T\to\infty}{lim}Pr( \underset{\theta\in\Theta}{sup} Q_T(\theta) < \epsilon) = 1 \implies Q_T(\theta) \overset{p}{\to} 0 \]
  21. 21Assumptions for Extremum Estimation (Convergence in Probability)\[1. \Theta \ is \ compact\]
    \[2. \hat{Q}_T(\theta) \ continuous \ in \ \Theta\]
    \[3. \hat{Q}_T(\theta) \overset{p}{\to} Q(\theta)\] uniformly
    4. Identification (unique global maximum)
  22. 22Asymptotic Normality\[1. \frac{\delta^2\hat{Q}}{\delta\theta\delta\theta^{\prime}} \ exists\]
    \[2. \frac{\delta^2\hat{Q}(\theta_T)}{\delta\theta\delta\theta^{\prime}\overset{p}{\to} A(\theta_0)}\]
    \[3. \sqrt{T}\frac{\delta\hat{Q}(\theta_0)}{\delta\theta} \overset{d}{\to} N(0,B(\theta_0))\]
    \[\implies \sqrt{T}(\hat{\theta} - \theta_0) \overset{d}{\to} N(0, A(\theta_0)^{-1\prime}B(\theta_0)A(\theta_0)^{-1}\]
  23. 23Assumptions for MLE\[1. Y \sim F(\cdot, \theta_0)\]
    \[2. y_t \ i.i.d\]
    \[3. \theta \in \Theta \subset \boldsymbol{R}^p\]
    4. Distribution is dictated by model
  24. 24MLE Objective Function\[L(\theta) = E_{\theta_0}[log f(Y, \theta)\]
  25. 25Identification\[Pr(ln(f(Y, \theta_0) \ne ln(f(Y, \theta^*)) > 0\]
  26. 26Score\[\frac{\delta ln(f(Y, \theta)}{\delta \theta}\]
    Gradient of log likelihood
    Under typical assumptions, Expectation of 0
  27. 27Information\[Var(s(\theta, y))\]
    Unidentified models have a singular information matrix
    \[-E[\frac{\delta^2 ln(f(Y,\theta))}{\delta\theta\delta\theta^{\prime}}\] if regularity conditions are satisfied
  28. 28Cramer-Rao Lower Bound\[Var(\sqrt{T}(\hat{\theta} - \theta_0)) \ge I_{\theta}^{-1}\]
  29. 29Asymptotic Efficiency\[\underset{T \to\infty}{lim} Var(\hat{\theta}_T) = I_{\theta}^{-1}\]
  30. 30Type I ErrorRejecting when the Null is True
  31. 31Type II ErrorNot rejecting the null when it is false
  32. 32Significance Level\[P_{\theta}(\delta(X) = d_1) = P_{\theta}(X \in S_1) \le \alpha \forall \theta \in \Theta_H\]
    \[0<\alpha<1\]
  33. 33Size of the Test\[\underset{\theta\in\Theta_h}{sup} P_{\theta}(X\in S_1)\]
    with fixed \[\alpha\]
  34. 34Power Function\[\beta(\theta) = P_{\theta}(\delta(X) = d_1)\]
  35. 35Test Optimization\[\underset{\phi(\cdot)}{max} \beta_{\phi}(\theta) = E_{\theta}[\phi(X)]\]
    \[s.t. \ E_{\theta}[\phi(X)] \le \alpha\]
  36. 36Simple DistributionsClass of distributions with a single distribution
  37. 37Composite DistributionsClass of distributions with multiple distribution
  38. 38Likelihood Ratio Test\[\frac{P_1(x)}{P_0(x)}\]
  39. 39P-ValueSmallest Significance Level at which hypothesis would be rejected given observation
    \[\hat{p} = \hat{p}(x) = inf\{\alpha: x \in S_{\alpha}\}\]
  40. 40Normal PDF\[\frac{1}{\sqrt{2\pi\sigma^2} e^{-\frac{(x - \mu)^2}{2\sigma^2}\]
  41. 41Bernoulli PDF\[ q = 1-p \ if \ x = 0\]
    \[p \ if \ x = 1\]
  42. 42Binomial PMF\[(n k) p^kq^{n-k}\]
    \[(n k) = \frac{n!}{k!(n-k)!}\]
  43. 43Uniform PMF\[\frac{1}{b-a}\] in support
    0 otherwise
  44. 44Poisson PMF\[\frac{\lambda^k e^{-\lambda}}{k!}\]
  45. 45Cauchy PDF\[\frac{1}{\pi \gamma[1 + (\frac{x - x_0}{\gamma})^2]}\]
  46. 46Chebychev's/Markov Inequality\[P(g(X) \ge r) \le \frac{E[g(X)]}{r}\]
  47. 47Holder's Inequality\[|EXY| \le E|XY| \le (E|X|^p)^{\frac{1}{p}}(E|Y|^q])^{\frac{1}{q}}\]
  48. 48Jensen's Inequality\[E[g(X)] \le g(E[X])\]
  49. 49—

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Graduate-level Econometrics Flashcards
Created by
Max Schnidman
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