Econometrics I

Question	Answer
Convergence in Probability	\[\underset{n \to \infty}{lim} P(\|X_n - X\| > \epsilon) = 0\]
Almost Sure Convergence	\[P(\{\omega: X_n(\omega) \nrightarrow X(\omega)\}) = 0\] \[Y_n = \underset{k\ge n}{sup} \|X_k - X\| \overset{p}{\to} 0\] \[P(\underset{k \ge n}{sup} \|X_k - X\| > \epsilon) = 0\]
Direction of Convergences	\[A.S. \to P \to D\] \[(R) \to P \to D\]
Additional A.S. Convergences	\[\sum_{k=1}^{\infty}P(\|X_n - X\|>\epsilon) \le \|\infty\| \implies X_n \overset{a.s.}{\to} X\] \[X_n \overset{p}{\to} X \implies \exists X_{n} \ s.t. \ X_{n_k} \overset{a.s.}{\to} X\]
Convergence in mean of order r	\[\underset{n\to\infty}{lim}E[[\|X_n - X\|]^r] = 0\]
Cauchy Sequence in Probability	\[\underset{n,n\to\infty}{lim} P(\|X_n - X_m\| > \epsilon) = 0\]
Borel-Cantelli Lemma	\[A = \cap_{n=1}^{\infty} \cup_{k\ge n} A_k\] \[\sum_{n=1}^{\infty} P(A_n) < \infty \implies P(A) = 0\]
Continuous Mapping Theorem	\[X_n \to X \implies g(X_n) \to g(X)\] in probability or A.S.
Convergence in Distribtuion	\[\int f(x)dF_n(x) \to \int f(x)dF(x)\]
Class of Generalized Distributions	\[\underset{x_n \to +\infty}{lim} G_{X_1...X_n}(x_1,...,x_n) = G_{X_1...X_{n-1}}(x_1,...,x_{n-1})\] \[\underset{x_n \to -\infty}{lim} G_{X_1...X_n}(x_1,...,x_n) =0\] \[G(-\infty)\ge 0\] \[G(\infty)\le 1\]
Helly-Bray Theorem	The class of generalized distributions is compact w.r.t. weak (distributional) convergence.
Asymptotic Tightness	\[\forall \epsilon > 0 \ \exists N \ s.t. \ inf_n(F_n(N) - F_n(-N)) > 1-\epsilon\]
Kinchin's Law of Large Numbers	Suppose \[\{X_n\}_{n=1}^{\infty}\] is i.i.d. sequence of r.v. w. \[E(X_n) = a\] and let \[S_n = \sum_{k=1}^n X_k\] Then \[\frac{S_n}{n} \overset{p}{\to} a\]
Central Limit Theorem	If \[0 < \sigma^2 < \infty\] \[\underset{n\to\infty}{lim} \underset{x}{sup} \|P(Z_n <x) - \Phi(x)\| = 0\] \[Z_n = \sqrt{n}\frac{(S_n - a)}{\sigma}\]
Convergence Properties	\[X_n \overset{p}{\to} c \equiv X_n \overset{d}{\to} c\] \[X_n \overset{d}{\to} X, \|X_n- Y_N\| \overset{p}{\to} 0,\implies Y_n \overset{d}{\to} X\] \[X_n \overset{p}{\to} X, Y_N \overset{p}{\to} c,\implies (X_n,Y_n) \overset{d}{\to} (X,c)\]
Slutsky Theorem	\[X_n \overset{d}{\to} X, Y_N \overset{d}{\to} c \implies \] \[ 1. X_n + Y_n \overset{d}{\to} X + c\] \[ 2. X_n Y_n \overset{d}{\to} X c\] \[ 3. X_n /Y_n \overset{d}{\to} X / c\]
Lindeberg-Feller CLT Condition	\[\sum_{i=1}^{k_n} E[\|\|Y_{n,i}\|\|^2]\boldsymbol{1}\{\|\|Y_{n,i}\|\|>\epsilon\} \to 0\] \[\lim_{n \to \infty} \frac{1}{s_n^2}\sum_{k = 1}^n \mathbb{E} \left[(X_k - \mu_k)^2 \cdot \mathbf{1}_{\{ \| X_k - \mu_k \| > \varepsilon s_n \}} \right] = 0\]
Delta Method	\[X_n \overset{d}{\to} X, b_n \to 0\] \[\implies \frac{g(a + b_nX_n) - g(a)}{b_n} \overset{d}{\to} X g^{\prime}(a)\]
Extremum Estimator	\[\theta_0 = argmax_{\theta \in \Theta} Q(\theta)\] \[Q(\theta) = E_{\theta_0}[g(Y, \theta)] = \int g(y, \theta)F(dy, \theta_0)\]
Uniform Convergence	\[Pr(\underset{T\to\infty}{lim} \underset{\theta\in\Theta}{sup} Q_T(\theta) = 0) = 1 \implies Q_T(\theta) \overset{a.s.}{\to} 0 \] \[\underset{T\to\infty}{lim}Pr( \underset{\theta\in\Theta}{sup} Q_T(\theta) < \epsilon) = 1 \implies Q_T(\theta) \overset{p}{\to} 0 \]
Assumptions 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)
Asymptotic Normality	\[1. \frac{\delta^2\hat{Q}}{\delta\theta\delta\theta^{\prime}} \ exists\] \[2. \frac{\delta^2\hat{Q}(\theta_T)}{\delta\theta\delta\theta^{\prime}\overset{p}{\to} A(\theta_0)}\] \[3. \sqrt{T}\frac{\delta\hat{Q}(\theta_0)}{\delta\theta} \overset{d}{\to} N(0,B(\theta_0))\] \[\implies \sqrt{T}(\hat{\theta} - \theta_0) \overset{d}{\to} N(0, A(\theta_0)^{-1\prime}B(\theta_0)A(\theta_0)^{-1}\]
Assumptions 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
MLE Objective Function	\[L(\theta) = E_{\theta_0}[log f(Y, \theta)\]
Identification	\[Pr(ln(f(Y, \theta_0) \ne ln(f(Y, \theta^*)) > 0\]
Score	\[\frac{\delta ln(f(Y, \theta)}{\delta \theta}\] Gradient of log likelihood Under typical assumptions, Expectation of 0
Information	\[Var(s(\theta, y))\] 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
Cramer-Rao Lower Bound	\[Var(\sqrt{T}(\hat{\theta} - \theta_0)) \ge I_{\theta}^{-1}\]
Asymptotic Efficiency	\[\underset{T \to\infty}{lim} Var(\hat{\theta}_T) = I_{\theta}^{-1}\]
Type I Error	Rejecting when the Null is True
Type II Error	Not rejecting the null when it is false
Significance Level	\[P_{\theta}(\delta(X) = d_1) = P_{\theta}(X \in S_1) \le \alpha \forall \theta \in \Theta_H\] \[0<\alpha<1\]
Size of the Test	\[\underset{\theta\in\Theta_h}{sup} P_{\theta}(X\in S_1)\] with fixed \[\alpha\]
Power Function	\[\beta(\theta) = P_{\theta}(\delta(X) = d_1)\]
Test Optimization	\[\underset{\phi(\cdot)}{max} \beta_{\phi}(\theta) = E_{\theta}[\phi(X)]\] \[s.t. \ E_{\theta}[\phi(X)] \le \alpha\]
Simple Distributions	Class of distributions with a single distribution
Composite Distributions	Class of distributions with multiple distribution
Likelihood Ratio Test	\[\frac{P_1(x)}{P_0(x)}\]
P-Value	Smallest Significance Level at which hypothesis would be rejected given observation \[\hat{p} = \hat{p}(x) = inf\{\alpha: x \in S_{\alpha}\}\]
Normal PDF	\[\frac{1}{\sqrt{2\pi\sigma^2} e^{-\frac{(x - \mu)^2}{2\sigma^2}\]
Bernoulli PDF	\[ q = 1-p \ if \ x = 0\] \[p \ if \ x = 1\]
Binomial PMF	\[(n k) p^kq^{n-k}\] \[(n k) = \frac{n!}{k!(n-k)!}\]
Uniform PMF	\[\frac{1}{b-a}\] in support 0 otherwise
Poisson PMF	\[\frac{\lambda^k e^{-\lambda}}{k!}\]
Cauchy PDF	\[\frac{1}{\pi \gamma[1 + (\frac{x - x_0}{\gamma})^2]}\]
Chebychev's/Markov Inequality	\[P(g(X) \ge r) \le \frac{E[g(X)]}{r}\]
Holder's Inequality	\[\|EXY\| \le E\|XY\| \le (E\|X\|^p)^{\frac{1}{p}}(E\|Y\|^q])^{\frac{1}{q}}\]
Jensen's Inequality	\[E[g(X)] \le g(E[X])\]

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	Created by Max Schnidman almost 3 years ago