Title: oxford / 300 qe

Front: properties of E[]
Back: E[a + bX + Y] = a + bE[X] + E[Y]

Front: joint distribution
Back: [latex]
Given $X, Y$ defined on the same probability space,
the joint distribution of $X,Y$ is defined and has range $S=\{<x,y> | x,y$ realisations of $X,Y \}$.
In the case where $X$ and $Y$ are discrete:
Letting $F_{X,Y}(x,y)$ be the joint probability mass function of X and Y.
$$cov(X,Y) = \sum_{(x,y) \in S} (x - E[X])(y-E[Y]) F_{X,Y}(x,y)$$
[/latex]

Front: covariance
Back: [latex]
$$cov(X,Y) = E[XY] - E[X]E[Y]$$
$$cov(X,Y) = E[(X-E[X])(Y-E[Y])]$$
[/latex]

Front: variance
Back: [latex]
$$var(X) = E[(X-E[X])^2]$$
$$var(X) = E[X^2]-E[X]^2$$
[/latex]

Front: properties of variance
Back: [latex]
$$var(a + bX + cY) = b^2 var(X) + 2bc*cov(X,Y) + c^2 var(Y)$$
[/latex]

Front: pearson correlation coefficient
Back: [latex]
$$p_{X,Y} = \frac{cov(X,Y)}{\sigma_X \sigma_Y}$$
[/latex]

Front: central limit theorem
Back: CLT: (Lindeberg-Levy):

Front: asymptotic theory notation
Back: [latex]
asymptotic distribution:
$$X \overset{A}{\sim} N(0,1) $$
equivalently, tends in distribution:
$$X \xrightarrow{d} N(0,1)$$
tends in probability:
$$a \xrightarrow{p} b$$
[/latex]

Front: constructing a confidence interval
Back: [latex]
Let $\beta$ be the parameter to estimate, and $b$ our estimate of $\beta$ from an iid sample of size $n$. $b$ has standard deviation $sd(b)$. By the CLT and by rearranging:
$$ \frac{b - \beta}{se(b)} \xrightarrow{d} N(0,1)$$
The problem will tell you eg $se(b)=0.04$. Assuming that 10000 is sufficiently close to $\infty$, the random interval
$$[b - 1.96\widehat{sd(b)} , b + 1.96\widehat{sd(b)}]$$
has an approximately 95\% probability of containing $\beta$. A realisation of this random interval is the confidence interval
$$[12.3 - 1.96* 0.04 , 12.3 + 1.96* 0.04]  = [12.22 , 12.37] $$
[/latex]

Front: binomial distribution
Back: [latex]
$B(n,p)$ is the discrete probability distribution of the number of successes in a sequence of $n$ independent bernouilli trials with probability of success $p$.
probability mass function:
$$F(k) = \binom{n}{k} p^k (1-p)^{n-k} $$
mean:
$$np$$
variance:
$$np(1-p)$$
[/latex]

Front: the linear regression model
Back: [latex]
the population linear least squares regression model is defied as:
$$Y=\beta_0 + \beta_1 X + u$$
where
$$\beta_0 = E[Y] - \beta_1 E[X]$$
and
$$\beta_1 = \frac{cov(X,Y)}{var(X)}$$
[/latex]
sample LRM: calulate sample analogues of the population moments. by the CLT, the sampling distribution of the estimators are approximately normal.

Front: sampling from a normal distribution
Back: [latex]
The sample mean $\bar{x}$ of a random sample of size $n$ from a $N(\mu,\sigma^2)$ has the following property:
$$\bar{x} \sim N({\mu,\frac{\sigma^2}{n}})$$
[/latex]

Front: random variable (formal)
Back: [latex]
Let a probability space be a triple $(\Omega,A,P)$ where
\bigskip
$\Omega$ is the sample space, which is the set of possible outcomes
\bigskip
$A$ is the set of events, which is a nonempty set of subsets of $\Omega$.
\bigskip
$P$ is a probability measure defined on A, which is a function with domain $A$ and range $[0,1]$.
\bigskip
\bigskip
A discrete random variable $X$ on a probability space $(\Omega,A,P)$ is a function with domain $\Omega$ and range a finite or countably infinite subset $\{x_1,x_2,...\}$ of $R$ such that
$$\forall i, \{\omega \in \Omega : X(\omega)=x_i \} \in A$$
[/latex]

Front: F-statistic
Back: [latex]
F-statistic
\bigskip
The F-statistic is a random variable measuring the "improvement in fit" when moving from a restricted model to an unrestricted model which contains the resitricted model as well as additional regressors, called restrictions.
$$F=\frac{\frac{RSS_r - RSS_u}{k_u - k_r}}{\frac{RSS_u}{n-k_u}} = \frac{RSS_r - RSS_u}{RSS_u}\frac{n-k_u}{k_u-k_r}$$
Where $RSS_r$ and $RSS_u$ are the residual sum of squares on the restricted and unrestricted models, $k$ is the number of right-hand-side terms, so that $k_u - k_r$ is the number of restrictions, and $n$ is the sample size.
\bigskip
\bigskip
[/latex]

Front: instrument validity

Front: LATE definition

Front: LATE theorem

Front: one-directional non-compliance
Back: If you can refuse treatment to those who have not been assigned to treatment, you are said to have one-directional non-compliance.This guarantees that there are no defiers and no always-takers.

Front: r^2 definition of the F-statistic
Back: [latex]
$$ F = \frac{n-k_u-k_r}{k_u-k_r}\frac{R^2_u-R^2_r}{1-R^2_u}$$
[/latex]

Front: autocovariance, autocorrelation

Front: AR(p)

Front: ADL(p,q)

Front: RMSFE

Front: stationarity

Front: random walk
Back: [latex]
The random walk model is:
$$Y_t = Y_{t-1} + u_t$$
where $E[u_t | Y_{t-1} , Y_{t-2} , ...]=0$.
The random walk model with drift is:
$$Y_t = \beta_0 + Y_{t-1} + u_t$$
Both are special cases of an AR(1):
$$ Y_t = \beta_0 + \beta_1 Y_{t-1} + u_t $$
The random walk model is a special case with $\beta_1 = 1$ and $\beta_0 = 0$.
The random walk model with drift is a special case where $\beta_1 =1$.
[/latex]

Front: stationarity of an AR(1)
Back: [latex]
$| \beta_1 | \geq 1$ and $u_t$ is stationary $\rightarrow AR(1)$ is not stationary.
$| \beta_1|  < 1$ and $u_t$ is stationary $\rightarrow$ AR(1) is stationary.
[/latex]

Front: stationarity and unit roots
Back: [latex]
$Y_t$ has a stochastic trend $\iff$ $Y_t$ has a unit root
$Y_t$ is stationary $\rightarrow$ $Y_t$ does not have a unit root
[/latex]

Front: 99% CI crictical value
Back: For a 99% CI, the critical value is 2.58

Front: f-test
Back: [latex]F-test
\bigskip
Letting $H_0$ be the hypothesis that all the coefficients on the restrictions are equal to 0.
$$H_0 \rightarrow F \sim F(k_u - k_r, n - k_u)$$
\bigskip
decision rule: Reject $H_0$ iff $F>F_{critical}.$
[/latex]

Front: r^2
Back: Where:
TSS =

Front: adjusted r^2
Back: [latex]
$$R^2_{adj} = 1 - (1 - R^2) \frac{n-1}{n-K}$$
Where K is the total number of explanatory variables, \textbf{including} the constant term.
[/latex]

Front: t test
Back: [latex]take the case where $\bar{\beta_k}=0$:[/latex]

Front: p-value
Back: under H_0, the probability of observing data at least as extreme as actual data.
in the case of a t-test:

Front: granger causality test
Back: [latex]
in an $ADL(p,q)$:
$$Y_t = B_0 + \sum_{i=1}^p{Y_{t-i} B_i} +  \sum_{i=1}^q{X_{t-i} \delta_i} + u_t$$
X granger causes Y iff $H_0$ is rejected
$$H_0: \forall i,  \delta_i = 0$$
We thus conduct an f-test with $q$ restrictions.
[/latex]

Front: first stage, 2nd stage, reduced form
Back: [latex]
Suppose Y is the outcome of interest, D is the (endogenous) treatment, and Z is the (exogenous) instrument.
\bigskip
Then the first stage is:
$$D = \beta_0 + \beta_1Z + e$$
The second stage is:
$$Y = B_0 + B_1\hat{D} + u$$
$$Y = B_0 + B_1(\hat{\beta_0} + \hat{\beta_1}Z) + u$$
The reduced form (outcome of interest as a fct of exogenous variables only) is:
$$Y = \alpha_0 + \alpha_1 Z + \phi$$
Since Z is exogenous, $\alpha_1$ in the reduced form is the causal effect of Z on Y.
[/latex]

Front: relative RMSFE

Front: categorical variable
Back: a variable that can take on one of a limited number of possible values, assigning each unit of observation to a particular group or nominal category on the basis of some qualitative property

Front: inference on an ARDL
Back: the correct model is an ECM.
Even though OLS is asymptotically normal, and consistent for the coefficient of the projection of ∆y on ∆x, the residuals are serially correlated, so OLS standard errors will not be valid.
Notwithstanding these problems, the proposed model is OK for forecasting, it will not suffer from the problem of spurious
regression.

Front: coefficient interpretation in log-log models
Back: coefficient B:
1% increase in X associated with B% increase in Y.

Front: 1.96
Back: In probability and statistics, 1.96 is the approximate value of the 97.5 percentile point of the normal distribution. 95% of the area under a normal curve lies within roughly 1.96 standard deviations of the mean, and due to the central limit theorem, this number is therefore used in the construction of approximate 95% confidence intervals

Front: t summary (QE)
Back: with QE examinable highlighted

Front: mean independence
Back: e is mean independent of X iff
E(e|X)=E(e)

Front: testing for breaks [qe]

Front: testing for breaks at an unkown time point [qe]

Front: types of nonstationarities [qe]
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 unit roots [qe]
Back: Where W(.) is a standard brownian motion distribution.

Front: ADF test [qe]

Front: attenuation bias