Title: oxford / 314 econometrics

Front: Central Limit theorem
Back: CLT: (Lindeberg-Levy):

Front: matrix product

Front: matrix product: properties

Front: df matrix

Front: symmetric matrix

Front: diagonal matrix
Back: a square matrix in which the entries outside the main diagonal are all zero

Front: scalar matrix

Front: indentity matrix

Front: matrix transposition
Back: e.g.

Front: matrix addition
Back: is defined iff A and B are the same size

Front: properties of CEF residual

Front: properties of the LRM residual

Front: inverse of a square matrix
Back: iff det(A) \neq 0

Front: determinant and inverse of a 2x2 matrix

Front: submatrix
Back: [latex]
If we are deleting one row and one column, we can write $sub(A,i,k)$ for the submatrix of A after deleting row $i$ and column $k$.
[/latex]

Front: minor
Back: [latex]
$$
minor(A,i,k) = det(sub(A,i,k))
$$
[/latex]

Front: determinant of a 3x3 matrix

Front: marginal density

Front: conditional density

Front: conditional expectation function

Front: calculus for matrices
Back: Where A is a square n*n matrix and x is a column vector of length n:
and

Front: autocorrelation

Front: Weak stationarity
Back: Intuition: first and second moments do not change over time

Front: Strict stationarity
Back: [latex]
The time series $\{ X_t , t \in \mathbb{Z} \}$ is strictly stationary iff the joint distribution of $(X_S,...,X_{S+H})$ is the same as the joint distribution of $(X_W,...,X_{W+H})$ for all $H \in \mathbb{N}$ and for all $S,W \in \mathbb{Z}$. Intuition: entire distribution does not change over time
[/latex]

Front: ergodicity
Back: [latex]A weakly stationary process $X_t$ is ergodic for the mean iff
$$ \lim_{T \rightarrow + \infty} \frac{1}{T} \sum_{t=1}^T X_t = E[X]$$
Where $E[X]$ is the (constant) ensemble expectation of $X$.
[/latex]

Front: ergodicity (general)
Back: [latex]
\textbf{ergodic theorem}. For all moments $r$, and if $E[x^r]$ has no dependence on $t$ (stationarity to order $r$), the process $X_t$ is ergodic to moment $r$ iff:
$$T^{-1}\sum_{t=1}^T x_t^r \xrightarrow{p} E[x^r]$$
[/latex]

Front: [latex]$$\gamma_h$$[/latex]
Back: [latex]this is the $h$-autocovariance$$\gamma_h = cov(x_t,x_{t-h})$$[/latex]

Front: [latex]$$\gamma_0$$[/latex]
Back: [latex]$$\gamma_0 = var(x_t)$$[/latex]

Front: sufficient condition for ergodicity of a weakly stationary process
Back: [latex]
Suppose $\{ X_t \}$ is weakly stationary. If $\sum_{h=0}^\infty | \gamma_h|$ is finite, then $\{ X_t \}$ is ergodic for the mean.
[/latex]

Front: white noise process
Back: [latex]
$x$ is a white noise process iff
\begin{align}
\forall t, E[x_t]&=0 \\
\forall t, Var[x_t]&=\sigma^2 \\
\forall s,h \text{ if } s \neq h \text{ then }cov(x_s,x_{h})&=0
\end{align}
Any white noise process is weakly stationary.
[/latex]

Front: MA(t) expression of an AR(1)

Front: [latex]MA($\infty$) representation of an AR(1)[/latex]
Back: [latex]
Assuming that $y_t$ is defined on $\mathbb{Z}$ (it has an infinite history) and that $|\phi|<1$, an AR(1) can be expressed as an MA($\infty$):
[/latex]

Front: Autocovariance of an AR(1)
Back: [latex]
Assuming that $y_t$ is defined on $\mathbb{Z}$ (it has an infinite history) and that $|\phi|<1$ OR assuming that $y_t$ is covariance stationary, we have $$cov(y_t,y_{t-s})=$$
[/latex]

Front: best linear predictor of a stationary time series
Back: [latex]Suppose that $X_t$ is a weakly stationary time series with mean $\mu$ and autocorrelation function $\rho$. The best predictor of $X_{t+s}$ of the form $a + bX_t$ is such that $b=\rho_s$ and $a=\mu(1-\rho_s)$.[/latex]

Front: rank of a matrix

Front: properties of rank
Back: We assume that A is an m × n matrix.
     * A matrix that has rank min(m, n) is said to have full rank
*
*

Front: rank of a square matrix
Back: [latex]for a square $n \times n$ matrix $A$, $det(A)\neq 0$ iff $rank(A)=n$ iff $A$ has full rank iff $A$ is invertible[/latex]

Front: definite matrices
Back: For A square n*n, and x column vector of length n,

Front: the MLR assumptions

Front: MLR: estimation of b

Front: MLR: algebra

Front: MLR: Robust Standar Errors

Front: MLR: inference

Front: MLR: inference without MLR6

Front: the instrument matrix

Front: The IV assumptions

Front: IV theorem

Front: MA stationarity
Back: any MA with white noise errors is stationary and ergodic

Front: MA(Q)

Front: Expectation of MA(Q)

Front: Variance of MA(Q)

Front: s-autocovariance of MA(Q)

Front: AR(1)

Front: AR(P)

Front: Expectation of AR(P)

Front: Variance of AR(P)

Front: ARMA(P,Q)

Front: ARMA(1,1) representations
Back: For an ARMA(1,1):
[latex]
Assuming that $y_t$ is defined on $\mathbb{Z}$ (it has an infinite history) and that $|\phi|<1$, $y_t$ can be expressed in an infinite lag representation:
[/latex]

Front: ARMA(1,1) expectation
Back: For an ARMA(1,1):
[latex]
Assuming that $y_t$ is defined on $\mathbb{Z}$ (it has an infinite history) and that $|\phi|<1$:
[/latex]

Front: ARMA(1,1) variance
Back: For an ARMA(1,1):
[latex]
Assuming that $y_t$ is defined on $\mathbb{Z}$ (it has an infinite history) and that $|\phi|<1$:
[\latex]

Front: ARMA(1,1) first order autocovariance and autocorrelation
Back: For an ARMA(1,1):
[latex]
Assuming that $y_t$ is defined on $\mathbb{Z}$ (it has an infinite history) and that $|\phi|<1$:
[\latex]

Front: OLS estimation for AR(1) without constant
Back: The OLS estimator is biased. However, it is consistent, and, furthermore, a CLT applies to it:

Front: binary response: probit model

Front: binary response: logit model

Front: probit marginal effect
Back: [latex]
$$\frac{d P(y=1 | x)}{d x_j} = \phi (x' \beta) \beta_j$$ where $\phi(.)$ is the standard normal probability density function. This depands on $x$. A typical choice is to evaluate it at $\bar{x}$:
$$\frac{d P(y=1 | x)}{d x_j} \Big|_{x=\bar{x}} = \phi (\bar{x}' \beta) \beta_j$$.
[/latex]

Front: logit marginal effect
Back: [latex]
$$\frac{d P(y=1 | x)}{d x_j} = \Lambda (x'\beta) (1-\Lambda(x'\beta)) \beta_j$$. This depends on $x$. A typical choice is to evaluate it at $\bar{x}$:
$$\frac{d P(y=1 | x)}{d x_j}  \Big|_{x=\bar{x}}= \Lambda (\bar{x}'\beta) (1-\Lambda(\bar{x}'\beta)) \beta_j$$.
[/latex]

Front: binary response model relative effects
Back: [latex]
For $P(y=1 \mid x)=G(x,\beta)$ and any $G$ such that $\lim_{-\infty} G =0$ and $\lim_{+\infty} G =1$, the relative effects do not depend on $x$: for continuous variables $x_j$ and $x_h$:
$$\frac{\frac{d P(y=1 \mid x}{d x_j}}{\frac{d P(y=1 \mid x}{d x_h}}=\frac{\beta_j}{\beta_h}$$
[/latex]

Front: algebra of binary response models
Back: The following is fully general:
[latex]
We can write this as:
$$P(y_i \mid x_i) = [G(x_i,\beta)]^{y_i} [1-G(x_i,\beta)]^{1-y_i}$$
Conditioning on $\beta$ does not change the expression:
$$P(y_i \mid x_i,\beta) = [G(x_i,\beta)]^{y_i} [1-G(x_i,\beta)]^{1-y_i}$$
[/latex]

Front: df extremum estimator

Front: nonlinear least squares

Front: maximum likelihood
Back: [latex]For $y_i = g(x_i,\theta) + e_i$, this requires specifying the entire joint distribution of $e_i$ (as opposed to just conditional 1st and 2nd moments in OLS)[/latex]

Front: binary response: ML estimation
Back: [latex]
We saw that $$P(y_i \mid x_i,\beta) = [G(x_i,\beta)]^{y_i} [1-G(x_i,\beta)]^{1-y_i}$$.
The joint density of $y_1,...,y_n$ given $x_1,...,x_n$ is then:
$$P(y_1,...,y_n \mid x_1,...,x_n, \beta) = \prod_{i=1}^n [G(x_i,\beta)]^{y_i} [1-G(x_i,\beta)]^{1-y_i}$$
Its natural logarithm is:
$$L_n(\beta) = \ln(P(y_1,...,y_n \mid x_1,...,x_n, \beta)$$
$$=\sum_{i=1}^n y_i \ln [G(x_i,\beta)] + \sum_{i=1}^n (1-y_i) \ln [1-  G(x_i,\beta)]$$
[/latex]

Front: binary response: ML estimation (logit application)
Back: ML estimation for binary response:
[latex]
For logit: $$G(x_i,\beta) = \Lambda(x_i'\beta)=\frac{e^{x_i'\beta}}{1+e^{x_i'\beta}}$$
First-order condition for $\max_\beta L_n$:
$$\frac{dL_n}{d\beta}=0$$
$$\sum_{i=1}^n [y_i - \Lambda (x_i'\beta)]x_i=0$$
[/latex]

Front: binary response: ML estimation (probit application)
Back: ML estimation for binary response:
[latex]
For probit $G(x_i,\beta)=\Phi(x_i'\beta)$
First-order condition for $\max_\beta L_n$:
$$\frac{dL_n}{d\beta}=0$$
$$\sum_{i=1}^n w_i [y_i - \Phi (x_i'\beta)]x_i=0$$
where $w_i = \frac{\phi(x_i'\beta)}{[(\Phi(x_i'\beta)(1-\Phi(x_i'\beta)]}$
[/latex]

Front: count response: poisson model
Back: [latex]When $y$ takes on values in $\mathbb{N}$, poisson regression model:
$$E[y\mid x]=exp(x^\prime\beta)$$
[/latex]

Front: poisson: coefficient interpretation
Back: [latex]
Poisson model:
$$E[y \mid x] = \exp (\beta_0+ \beta_1x_1 + ... + \beta_kx_k)$$
So:
$$\frac{d E[y\mid x]}{d x_j} = \beta_j E[y\mid x]$$ and hence
$$\beta_j  = \frac{\frac{d E[y\mid x]}{d x_j}}{E[y\mid x]}$$
Interpretation: $100\beta_j$ is the percentage change in $E[y \mid x]$ for a one-unit increase in $x_j$.
[/latex]

Front: poisson model, ML estimation
Back: [latex]
Conditional likelihood function:
$$L_n (y_1,...,y_n \mid x_1,...,x_n,\beta) = \prod_{i=1}^n \frac{\exp(-\exp({x_i'\beta}))[\exp(x_i'\beta)]^{y_i}}{y_i !}$$
Conditional log-likelihood function:
[/latex]
Maximum likelihood estimator FOC:

Front: LLN for time series

Front: AR(1) estimation and inference

Front: information criteria

Front: deterministic trend df

Front: deterministic trend properties
Back: is x_t is deterministic trend with error variance sigma^2 then

Front: deterministic trend estimation
Back: hence
Theorem:
Note that:

Front: CLT for stationary processes
Back: [latex]If $\{ X_t \}$ is weakly stationary with mean $\mu$, and $\sum_{h=0}^\infty | \gamma_h|$ is finite (implies that $X_t$ is ergodic for the mean) then as $n \rightarrow \infty$, $\sqrt n (\bar{X}_n - \mu)$ tends in distribution to $N(0,\sum_{h=-\infty}^\infty \gamma_h)$[/latex]

Front: deriving OLS estimator
Back: Note that Y'Xb and b'X'Y are scalars, hence:

Front: consistency proof
Back: Proof of
[latex]
What we are trying to prove: for all $\varepsilon > 0$, $\lim_{n \rightarrow \infty} P( |b-\beta| < \varepsilon)=1$, also written $\text{plim} \quad b = \beta$.
Write $b$ in terms of $\beta$:
$$
\begin{aligned}
b & = \beta + (X'X)^{-1}(X'U) \\
b &= \beta + (\frac{X'X}{n})^{-1 } \frac{X'U}{n} \\
\end{aligned}
$$
Take plims:
$$
\text{plim} b = \beta + \text{plim}((\frac{X'X}{n})^{-1 } (\frac{X'U}{n} )
$$
By the Slutsky theorem:
$$
\text{plim} b = \beta + \text{plim}((\frac{X'X}{n})^{-1 }) \times \text{plim}( \frac{X'U}{n} )
$$
By MLR3*:
$$
\text{plim} b = \beta + Q^{-1 } \times \text{plim}( \frac{X'U}{n} )
$$
By MLR2 and the law of large numbers:
$$
\text{plim} b = \beta + Q^{-1 } \times E[X'U]
$$
By MLR4:
$$
\text{plim} b = \beta + 0
$$
[/latex]

Front: unbiasedness proof

Front: variance of b proof

Front: gauss-markov theorem
Back: [latex]Under MLR{1 3 4 5} the OLS estimator $b$ is efficient in the class of linear conditionally unbiased estimators of $\beta$. That is, for any $c$ that is linear in $y$, if $E[c|x]=\beta$, then $Var(c| X) \geq Var(b | X)$. In particular: $$Var( c_k| X)  \geq Var(b_k | X)$$ for $(k = 1,2,..., K)$[/latex]

Front: asymptotic distribution proof
Back: Proof of
[latex]Write $b$ in terms of $\beta$:
$$
\begin{aligned}
b & = \beta + (X'X)^{-1}(X'U) \\
b &= \beta + (\frac{X'X}{n})^{-1 } \frac{X'U}{n} \\
\sqrt{n}(b-\beta) &= (\frac{X'X}{n})^{-1 } \frac{X'U}{\sqrt n}
\end{aligned}
$$
Given MLR3*, we can make use of the following theorem:
If $\frac{X'U}{\sqrt n} \xrightarrow{d} N (0, \Sigma)$ then $(\frac{X'X}{n})^{-1} \frac{X'U}{\sqrt n} \xrightarrow{d} N (0, Q^{-1}\Sigma Q^{-1})$.
Now we need to show $\frac{X'U}{\sqrt n} \xrightarrow{d} N (0, \Sigma)$ and pin down $\Sigma$.
By MLR2, we can apply the Lindberg-Levy CLT:
$$
\frac{X'U}{\sqrt n} \xrightarrow{d} N (0, V(X'U))
$$
Last step is to find $V(X'U)$.
$$
\begin{aligned} V(X'U) &= E[(X'U)(X'U)'] - E[X'U]^2 \\
&= E[(X'U)(X'U)'] \quad \text{by MLR4} \\
&= E[(X'X) (UU')] \\
&= E[(X'X) E[(UU')\mid X] ] \quad \text{law of iterated expectations} \\
&= E[(X'X) \sigma_U^2I ] \quad \text{MLR5} \\
&= \sigma_U^2 E[(X'X)] \\
&= \sigma_U^2 Q
\end{aligned}
$$
Now:
$$
\begin{aligned} (\frac{X'X}{n})^{-1} \frac{X'U}{\sqrt n} &\xrightarrow{d} N(0, Q^{-1}V(X'U)Q^{-1}) \\
(\frac{X'X}{n})^{-1} \frac{X'U}{\sqrt n} &\xrightarrow{d} N(0, Q^{-1}\sigma_U^2 QQ^{-1}) \\
(\frac{X'X}{n})^{-1} \frac{X'U}{\sqrt n} &\xrightarrow{d} N(0, \sigma_U^2 Q^{-1}) \end{aligned}
$$
And hence
$$
\sqrt{n} (b - \beta) \xrightarrow{d} N(0, \sigma_U^2 Q^{-1})
$$
[/latex]

Front: bias and inconsistency when MLR4 is violated
Back: if
[latex]$$E[U \mid X]= \eta $$[/latex]
there is bias:
if
[latex]$$E[X'U]= \gamma$$[/latex]
there is inconsistency:

Front: estimation of error variance

Front: F-statistic (matrix form)

Front: log-odds ratio for BR models
Back: [latex]
$$= \ln(\frac{P(y=1 \mid x)}{P(y=0 \mid x)})$$
[/latex]

Front: IV: the over-identified case

Front: IV theorem: over-identified case
Back: Under IV assumptions:

Front: how many minutes per question in econometrics exam
Back: 56 minutes per question
(225 minutes (180 + 45 extra) / 4 questions)

Front: taylor theorem
Back: [latex]For some point $c$ between $x$ and $x_0$. \\
So we can approximate $f(x)$ by $P_m(x)$ and the remainder will be at most $\max_c R_m(x)$.\\
First-order Taylor theorem:
$$f(x) = f(x_o) + \dot f (x_0) (x-x_0) + \frac{\dot\dot f (c)}{2}(x-x_0)^{2}$$
For some point $c$ between $x$ and $x_0$.\\
First-order taylor approximation:
$$f(x) \approx f(x_o) + \dot f (x_0) (x-x_0)$$
[/latex]

Front: Gauss-Newton method
Back: [suspended: this may have a sign error]

Front: lag operator notation of an ARMA

Front: stability and invertibility

Front: AR(infty) representation of an MA(1)

Front: unit roots properties
Back: AR(1) with unit root:

Front: order of integration
Back: [latex]
If $x_t$ has no deterministic components, and $$(1-L)^d x_t$$ has a stationary, invertible ARMA representation, then $x_t$ is said to be integrated of order $d$, denoted $x_t \sim I(d)$.
[/latex]

Front: co-integration (df)

Front: co-integration: a simple example
Back: e.g. Real Consumption and Real GDP

Front: multipliers in time series

Front: mean lag

Front: median lag

Front: examples of correlations among regressors and error term in time series

Front: testing general hypotheses in binary response models

Front: Wald test for binary response models

Front: Wald test

Front: likelihood ratio for binary response

Front: lagrange test for binary response models

Front: F-statistic vs Wald-statistic
Back: AND:

Front: MA representation of an ARMA process

Front: ARDL model

Front: ECM representation of an ARDL(1,1)

Front: DL representation of an ARDL model

Front: inference for binary response models

Front: inference for extremum estimators

Front: relationship between stationarity and stability
Back: If an ARMA process is stable, then it is weakly stationary.
The converse is not true.

Front: t summary

Front: ADF test [econometrics]

Front: testing for unit roots [econometrics]
Back: Where W(.) is a standard brownian motion distribution.

Front: types of non-stationarities [econometrics]
Back: Nonstationarities can be classified into one of four categories:
• Seasonalities
• Deterministic Trends (also known as Time Trends)
• Unit Roots (also known as Stochastic Trends)
• Structural Breaks

Front: testing for breaks [econometrics]