Stochastic Process

\[\newcommand{\indep}{\perp \!\!\! \perp}\]

A stochastic process $X_t$ is a family/sequence of random variables indexed by time $t$. “Stochastic” is a synonym for random.

When we collect time series data set, we obtain one possible outcome, or realization, of the stochastic process. We can only see a single realization, because we cannot go back in time and start the process over again. (This is analogous to cross-sectional analysis where we can collect only one random sample.)

At a specific time point $t$, $X_t$ is a random variable with a specific density function. A stochastic process is defined in a probability space $(\Omega, \mathcal{F}, P)$.

Mean function $\mu_t$

\[\begin{align*} \mu_t = \mathbb{E}[X_t] = \int_{\mathbb{R}}x \, \mathrm dF_t(X) \end{align*}\]

$\mu_t$ depends on time $t$, as, for example, processes with a seasonal or periodical structure or processes with a deterministic trend.

Autocovariance Function $\begin{align*} \gamma(t,\tau)=\text{Cov}(X_t, X_{t-\tau}) \end{align*}$

If we define covariance stationary, then $\gamma(t,\tau)=\gamma_\tau$, means covariance only depends on the time interval apart $\tau$, not the start point $t$.

Autocorrelation Function (ACF)

\[\begin{align*} \rho_{\tau}=\frac{\gamma_{\tau}}{\gamma_0} \end{align*}\]

The autocorrelation function $\rho$ of a covariance stochastic process is normalized on the interval $[-1,1]$. $\rho$ depends only on one parameter, the lag $\tau$. The sequence, $\lbrace\rho_\tau\rbrace,$ is called correlogram.

White Noise (WN)

If $X_t$ is white noise, then

\[\begin{align*} 1.&\ \mu_t=0, and \\ 2.&\ \gamma_\tau=\left\lbrace \begin{array}{ll} \sigma^2 & \text{when $\tau=0$} \\ 0 & \text{(uncorrelated) when $\tau\neq0$} \end{array} \right. \end{align*}\]

In short, WN is with expectation 0 and finite variance. denoted as $X_t\sim \text{WN} (0, \sigma^2)$.

Note that every $\text{IID}\, (0, \sigma^2)$ sequence is $\text{WN} (0, \sigma^2)$, but not conversely.

Random Walk

\[X_t=c+X_{t-1}+\varepsilon_t\]

with a constant $c$ and white noise $\varepsilon_t$.

$Z_t=X_t-X_{t-1}=c+\varepsilon_t$ , if $c$ is not zero, then we say $X_t$ has a drift,

$c>0$ we say drift upward,

$c<0$ we say drift downward.

$\varepsilon_t$ is called disturbances or innovations in time series.

Assume that the constant $c$ and the initial value $X_0$ are set to zero, through recursive substitution we will get the representation:

\[\begin{align*} X_t=\sum^t_{i=1}\varepsilon_i \end{align*}\]

Moment functions:

$\mu_t=0$
$\text{Var}(X_t)=t\sigma^2$
$\gamma(t,\tau) = \text{Cov}(X_t,X_{t-\tau})=(t-\tau)\sigma^2$, for $\tau\lt t$.
$\rho(t,\tau)=\sqrt{1-\frac{\tau}{t}}$

Notice that autocovariance is strictly positive, and since it depends on $t$ not only on the lags $\tau$, the random walk is NOT covariance stationary.

As $\tau$ gets larger, $\rho$ gets smallers. In words, neighboring time points are more and more strongly and positively correlated as they get closer in time. On the other hand, the values of $X$ at distant time points are less and less correlated.

AR(1) process

Model set up

\[X_t = c+\alpha X_{t-1}+\varepsilon_t, \quad \varepsilon_t\sim \mathrm{WN}(0, \sigma^2).\]

In order for $\lbrace x_t\rbrace$ to converge, we need $ \vert\alpha \vert<1$.

By iterative substitutions,

\[\begin{align*} X_t = c \frac{1-\alpha^k}{1-\alpha}+\alpha^kX_{t-k}+\sum^{k-1}_{i=0}\alpha^i\varepsilon_{t-i} \end{align*}\]

For $K\to \infty$,

\[\begin{align} X_t = \frac{c}{1-\alpha}+\sum^\infty_{i=0}\alpha^i\varepsilon_{t-i} \label{eq:ma-representation} \end{align}\]

Eq. $\eqref{eq:ma-representation}$ is called the moving-average representation, MA($\infty$).

Moment functions:

Expectation: $\mu_t = \frac{c}{1-\alpha}$
Autocovariance function of lag $\tau$: $\gamma_\tau=\frac{\sigma^2}{1-\alpha^2}\alpha^\tau$
Autocorrelation function of lag $\tau$: $\rho_\tau=\alpha^\tau$

For ACF, $\rho_\tau$,

if $\alpha>0$, this function is strictly positive,
for $\alpha<0$, it alternates around zero.

When we have $\alpha>0$, for small $\alpha$, like 0.5, it converges very fast; for large $\alpha$, like 0.99, it converges quite slow.

There are cases where $\abs{\phi}\ge1$:

if $\abs{\phi}>1$ the AR(1) process is stationary but not causal.
if $\abs{\phi}=1$ we have a non-stationary process (called random walk when $\phi=1$) and we say that this process has a unit root.

Autocovariance Matrix

If $\lbrace X_t \rbrace$ is a stationary process, we define a $p\times 1$ random vector $X$:

\[X' = [X_t, X_{t+1}, \ldots, X_{t+p} ] \,.\]

We define $\Sigma_{XX}$ as the $p\times p$ autocovariance matrix of the random vector $X$ that has the following form:

\[\Sigma_{XX} = {\color{#008B45FF}\sigma^2} \begin{bmatrix} 1 & \rho(1) & \rho(2) & \cdots & \rho(p-1) \\ \rho(1) & 1 & \rho(1) & \cdots & \rho(p-2) \\ \rho(2) & \rho(1) & 1 & \cdots & \rho(p-3) \\ \vdots & \vdots & \vdots &\ddots & \vdots \\ \rho(p-1) & \rho(p-2) & \rho(p-3) & \cdots & 1 \\ \end{bmatrix}\]

Consider now the following linear combination of the elements of $X$

\[Z = a'X = \sum_{i=1}^p a_i X_i\]

where

\[a' = [a_1, a_2, \ldots, a_p]\]

Note that $Z$ is scalar here. The variance of $Z$ is

\[a'\Sigma_{XX}a \,.\]

The variance of $Z$ cannot be negative, so

\[a'\Sigma_{XX}a \ge 0\]

for all nonzero $a$.

We say that $\Sigma_{XX}$ is “positive semi-definite”. This implies

\[\sum_{r=1}^p\sum_{s=1}^p a_r\, a_s\, \rho(r-s) \ge 0 \,.\]

MA(1) process

\[\varepsilon_t = u_t + \lambda u_{t-1}\]

where $u_t\sim WN(0, \sigma^2_u).$

The memory is only one period: the MA(1) process has a non-zero first autocorrelation, with the remainder zero.

$\gamma(0) = \var(\varepsilon_t) = (1+\lambda^2)\sigma^2_u$
$\gamma(1) = \cov(\varepsilon_t, \varepsilon_{t+1}) = \lambda \sigma^2_u$
$\rho(1)=\frac{\gamma(1)}{\gamma(0)} = \frac{\lambda}{1+\lambda^2}$
$\gamma(s) = \rho(s) = 0$ for $s>1$

An MA(1) process with $\lambda\ne 1$ is serially correlated, with each pair of adjacent observations $(y_t, y_{t-1})$ correlated.

If $\lambda> 0$ the pair are positively correlated, while
if $\lambda< 0$ they are negatively correlated.

The serial correlation, however, is limited in that observations separated by multiple periods are mutually independent.

MA($q$) process

\[\varepsilon_t = \lambda_0 u_t + \lambda_1 u_{t-1} + \lambda_2 u_{t-2} + \cdots + \lambda_q u_{t-q}\]

where $\lambda_0=1.$

An MA(q) has the following moments.

$\gamma(0) = \var(\varepsilon_t) = \left(\sum_{j=0}^q\lambda^2_j\right)\sigma^2_u$
$\gamma(k) = \cov(\varepsilon_t, \varepsilon_{t+k}) = \left(\sum_{j=0}^{q-k}\lambda_{j+k}\lambda_j\right)\sigma^2_u, $ for $k\le q$
$\rho(k) = \frac{\gamma(k)}{\gamma(0)} = \frac{\sum_{j=0}^{q-k}\lambda_{j+k}\lambda_j}{\sum_{j=0}^q\lambda^2_j},$ for $k\le q$
$\gamma(k) = \rho(k) = 0,$ for $k> q$

A MA(q) has $q$ non-zero autocorrelations, with the remainder zero.

An MA(q) process $\varepsilon_t$ is strictly stationary and ergodic.

MA($\bold{\infty}$) representation

Leg $\lbrace\varepsilon_t\rbrace$ be white noise and $\lbrace\psi_j\rbrace$ be a sequence of real numbers that is absolutely summable. Then

For each $t$,
\[y_t = \mu + \sum_{j=0}^\infty\psi_j\varepsilon_{t-j}\]
converges in mean square. $\lbrace y_t\rbrace$ is covariance-stationary. (The process $\lbrace y_t\rbrace$ is called the infinite-order moving-average process (MA($\infty$)).)
The mean of $y_t$ is $\mu.$ The autocovariances $\lbrace\gamma_j\rbrace$ are given by
\[\gamma_j = \sigma^2\sum_{k=0}^\infty \psi_{j+k}\psi_k \quad (j=0,1,2,\dots).\]
The autocovariances are absolutely summable:
\[\sum_{j=0}^\infty \vert \gamma_j \vert < \infty.\]
If, in addition, $\lbrace\varepsilon_t\rbrace$ is i.i.d, then the process $\lbrace y_t\rbrace$ is (strictly) stationary and ergodic.

\[y_t = c + \phi y_{t-1} + \varepsilon_{t}\] \[y_t = \mu - \sum_{j=1}^\infty \phi^{-j}\varepsilon_{t+j}\]

AR(p) and Impulse Response Function

Consider the $p$th order autoregressive process, denoted $AR(p)$:

\[y_t = \alpha_0 + \alpha_1y_{t-1} + \alpha_2y_{t-2} + \cdots + \alpha_py_{t-p} + \varepsilon_t ,\]

where $\varepsilon_t$ is a strictly stationary and ergodic white noise process, $\varepsilon_t\sim WN(0, \sigma^2)$.

Using the lag operator,

\[y_t - \alpha_1y_{t-1} - \alpha_2y_{t-2} - \cdots - \alpha_py_{t-p} = \alpha_0 + \varepsilon_t ,\]

\[\alpha(L)y_t = \alpha_0 + \varepsilon_t,\]

where

\[\alpha(L) = 1-\alpha_1L-\alpha_2L^2-\cdots-\alpha_pL^p .\]

We call $\alpha(z)$ the autoregressive polynomial of $y_t.$

$y_t$ can be rewritten as

\[\begin{aligned} y_t &= \alpha(L)^{-1}(\alpha_0 + \varepsilon_t) \\ &= \frac{\alpha_0}{1-\alpha_1-\cdots-\alpha_p} + \alpha(L)^{-1}\varepsilon_t \\ &= \mu + \psi(L)\varepsilon_t \end{aligned}\]

where $\alpha(L)^{-1}=\psi(L)= \sum_{j=0}^\infty \psi_j L^j.$

$y_t=\mu + \psi(L)\varepsilon_t$ is called the Wold representation in lag operator.

In the Wold form, it can be shown that

\[\begin{aligned} \E[y_t] &= \mu \\ \var(\varepsilon_t) &= \gamma(0) = \left(\sum_{j=0}^\infty\psi^2_j\right)\sigma^2 \\ \gamma(k) &= \left(\sum_{j=0}^{\infty}\psi_{j+k}\psi_j\right)\sigma^2 \\ \rho(k) &= \frac{\sum_{j=0}^{\infty}\psi_{j+k}\psi_j}{\sum_{j=0}^\infty\psi^2_j} \end{aligned}\]

The coefficients of the moving average representation

\[\begin{aligned} y_t &= \mu + \psi(L) \varepsilon_t \\ &= \mu + \sum_{j=0}^\infty \psi_j \varepsilon_{t-j} \\ &= \mu + \psi_0\varepsilon_{t} + \psi_1\varepsilon_{t-1} + \psi_2\varepsilon_{t-2} + \cdots \end{aligned}\]

are known among economists as the impulse response function (IRF). (Often scaled by the standard deviation of $\varepsilon_t.$)

In linear models, the IRF is defined as the change in $y_{t+j}$ due to a shock at time $t$. This is,

\[\frac{\partial}{\partial \varepsilon_t} y_{t+j} = \psi_j,\; j=1,2,\ldots\]

This means that the coefficients $\psi_j$ can be interpreted as the magnitude of the impact of a time $t$ shock on the time $t + j$ variable. Plots of $\psi_j$ against $j$ can then be used to assess the time-propagation of shocks. This is a standard method of analysis for multivariate time series.

For a stationary and ergodic time series,

\[\lim_{j\to\infty} \psi_j = 0 ,\]

and the long-run cumulative impulse response is finite

\[\sum_{j=0}^\infty \psi_j < \infty .\]

The autoregressive-moving-average process, denoted $ARMA(p,q),$ is

\[\begin{aligned} y_t &= \alpha_0 + \alpha_1y_{t-1} + \alpha_2y_{t-2} + \cdots + \alpha_py_{t-p} \\ &\phantom{=} \quad + \theta_0\varepsilon_t + \theta_1\varepsilon_{t-1} + \theta_2\varepsilon_{t-2} + \cdots + \theta_q\varepsilon_{t-q} \end{aligned}\]

where $\varepsilon_t \sim WN(0, \sigma^2).$

It can be written using the lag operator notation as

\[\alpha(L)y_t = \alpha_0 + \theta(L)\varepsilon_t .\]

The autoregressive-integrated moving-average process, denoted $ARIMA(p,d,q),$ is

\[\alpha(L)(1-L)^d y_t = \alpha_0 + \theta(L)\varepsilon_t .\]

That is, $\Delta^d y_t$ is $ARMA(p,q).$

Partial Autocorrelation Function (PACF)

The PACF is based on estimating the sequence of AR models

\[\begin{aligned} z_t &= \phi_{11} z_{t-1} + \varepsilon_{1t} \\ z_t &= \phi_{21} z_{t-1} + \phi_{22} z_{t-2} + \varepsilon_{2t} \\ & \qquad \vdots \\ z_t &= \phi_{p1} z_{t-1} + \phi_{p2} z_{t-2} + \cdots + \phi_{pp} z_{t-p} + \varepsilon_{pt} \\ \end{aligned}\]

where $z_t = y_t-\mu$ is the demeaned data.

The coefficients $\phi_{jj}$ for $j=1,\ldots,p$ (i.e., the last coefficients in each AR(p) model) are called the partial autocorrelation coefficients.

In an AR(1) model the first partial autocorrelation coefficient $\phi_{11}$ is non-zero, and the remaining partial autocorrelation coefficients $\phi_{jj}$ for $j > 1$ are equal to zero.
Similarly, in an AR(2), the first and second partial autocorrelation coefficients $\phi_{11}$ and $\phi_{22}$ are non-zero and the rest are zero for $j > 2$.
For an AR(p) all of the first $p$ partial autocorrelation coefficients are non-zero, and the rest are zero for $j > p$.

The sample partial autocorrelation coefficients up to lag $p$ are essentially obtained by estimating the above sequence of $p$ AR models by least squares and retaining the estimated coefficients $\phi_{jj}.$

Markov Process

A stochastic process has the Markov Property if for all $t\in \mathbb{Z}$ and $k\geq 1$

\[\begin{align*} F_{t|t-1,\ldots,t-k}(x_t|x_{t-1},\ldots, x_{t-k})=F_{t|t-1}(x_t|x_{t-1}) \end{align*}\]

In other words, the conditional distribution of a Markov process at a specific point in time is entirely determined by the condition of the system at the previous date.

Two examples: random walk with independent $\varepsilon_t$ and the AR(1) process with independent white noise.

Martingale

Let $\lbrace X_t\rbrace$ denote a sequence of random variables and let $\mathcal F_{t} = \lbrace X_{t}, X_{t-1},\ldots, X_0\rbrace$ denote a set of conditioning information or information set based on the past history of $X_{t}.$ The sequence $\lbrace X_t, \mathcal F_t\rbrace$ is called a martingale if

\[E[X_t \mid X_{t-1}=x_{t-1}, \ldots, X_{t-k}=x_{t-k}] = x_{t-1}\]

or more succinctly,

\[E[X_t | \mathcal F_{t-1}] = x_{t-1}\]

For martingale, only one statement about the conditional expectation is made; while for Markov process, statements on the entire conditional distribution are made.

The most common example of a martingale is the random walk model

\[X_t = X_{t-1} + \varepsilon_t, \quad \varepsilon_t \sim WN(0, \sigma^2).\]

We have

\[E[X_t | \mathcal F_{t-1}] = X_{t-1}\]

since $E[\varepsilon_t \vert \mathcal F_{t-1}]=0.$

Martingale Difference Sequence (MDS)

Let $\lbrace \varepsilon_t\rbrace$ be a sequence of random variables with an associated information set $\lbrace\mathcal F_t\rbrace$. The sequence $\lbrace\varepsilon_t, \mathcal F_t\rbrace$ is called a martingale difference sequence (MDS) if

\[E[\varepsilon_t | \mathcal F_{t-1}] = 0 \,.\]

If $\lbrace X_t, \mathcal F_t\rbrace$ is a martingale, then its first difference sequence $\lbrace \Delta X_t, \mathcal F_t\rbrace$ is a MDS. Define

\[\varepsilon_t \equiv \Delta X_t = X_t - X_{t-1} ,\]

$\lbrace\varepsilon_t, \mathcal F_t\rbrace$ is a MDS.

To show that $\lbrace\varepsilon_t, \mathcal F_t\rbrace$ is MDS, we need

\[\begin{split} E[\varepsilon_t | \mathcal F_{t-1}] &= E[X_t|\mathcal F_{t-1}] - E[X_t|\mathcal F_{t-1}] \\ &= X_{t-1} - X_{t-1} \\ &= 0 \,. \end{split}\]

□

Useful properties of a MDS:

A MDS has mean zero.
\[E[\varepsilon_t] = E \left[E[\varepsilon_t | \mathcal F_{t-1}]\right] =0\]
By construction, a MDS is an uncorrelated process. The covariance of $\varepsilon_t$ and $\varepsilon_{t-k}$, for $k>0$, is
\[\begin{aligned} \text{Cov}(\varepsilon_t, \varepsilon_{t-k}) &= E[\varepsilon_t\varepsilon_{t-k}] \\ &= E[E[\varepsilon_t\varepsilon_{t-k} | \mathcal F_{t-1}]] \\ &= E[\varepsilon_{t-k} E[\varepsilon_t| \mathcal F_{t-1}]] \\ &= E[\varepsilon_{t-k} 0] \\ &= 0 \end{aligned}\]
The term “martingale difference sequence” refers to the fact that the summed process (assuming $S_0=0$)
\[S_t = \sum_{j=1}^t \varepsilon_j\]
is a martingale, and $\varepsilon_t$ is its first difference. We can verify that
\[\begin{split} E[S_t | \mathcal F_{t-1}] &= E[S_{t-1} + \varepsilon_t | \mathcal F_{t-1}] \\ &= E[S_{t-1} | \mathcal F_{t-1}] + E[\varepsilon_t | \mathcal F_{t-1}] \\ &= S_{t-1} + 0 \\ &= S_{t-1} \,, \end{split}\]
which satisfies the martingale property.
If $\varepsilon_t$ is i.i.d. and mean zero, then it is a MDS, but the reverse is not true.

Example:

Let $u_t$ be i.i.d. $N(0,1)$ and set
\[\varepsilon_t = u_tu_{t-1}\]

Lag Operator

The lag operator, $L$, shifts a time value $X_t$ back by one period. For example, $LX_t=X_{t-1}$ and $L^kX_t=X_{t-k}.$

Sometimes also called as “backshift operator”, and denoted as $B$.

Useful results of $L$:

$L^0=1$
The lag of a constant is the same constant:
\[La=a .\]
The lag operator is commutative:
\[L(\beta y_t) = \beta L y_t.\]
The lag operator is distributive:
\[L(x_t+y_t) = L z_t = z_{t-1} = x_{t-1} + y_{t-1} = L x_t + L y_t,\]
where $z_t=x_t+y_t.$
Factorizing polynomials
\[1-\phi_1 z - \phi_2 z^2 - \ldots - \phi_p z^p = (1-\lambda_1z)(1-\lambda_2z) \cdots (1-\lambda_pz).\]
- $\lbrace\frac{1}{\lambda_i}\rbrace_{i=1}^p$ are the $p$ roots of the polynomial.
- Some of the roots may be complex and some may be identical.
If we factor the $p$-th order lag polynomial in the same way as a real-valued polynomial:
\[\begin{split} & (1-\phi_1 L - \phi_2 L^2 - \ldots - \phi_p L^p)y_t \\ & \hspace{0.5in} = (1-\lambda_1 L)(1-\lambda_2 L) \cdots (1-\lambda_p L) y_t = w_t. \end{split}\]
If $\abs{\lambda_i}<1,$
\[(1 - \lambda_i L)^{-1} = \sum_{j=0}^{\infty} \lambda_i^j L^j, \,\,\,\,\, \forall i.\]

More properties: https://ealdrich.github.io/Teaching/Econ211C/LectureNotes/Unit1-ARMA/lagOperators.html

Autoregressive Polynomial $\begin{equation} \phi(L) = 1 - \phi_1 L - \phi_2 L^2 - \cdots - \phi_p L^p \label{lag-poly} \end{equation}$

Eq. $\eqref{lag-poly}$ is common to see in ARMA process.

Using an AR(1) model, $\begin{equation} Y_t=\phi_1 Y_{t-1} + \varepsilon_t,\label{AR1} \end{equation}$

as an example.

Let $\phi(L)=1-\phi_1 L\,,$ then we can express the innovation in $\eqref{AR1}$ as

\[\begin{equation} \varepsilon_t = Y_t-\phi_1 Y_{t-1} = (1-\phi_1L)Y_t = \phi(L) Y_t \,. \end{equation}\]

Other common lag polynomials:

General form: $\beta(L)=\beta_0 + \beta_1L + \beta_2L^2 + \cdots$
$\alpha(L)=1 + \alpha L + (\alpha L)^2 + (\alpha L)^3 + \cdots$

We also call these lag polynomials “filters.”

Product of two filters $\alpha(L)$ and $\beta(L)$ $\delta(L) = \alpha(L)\beta(L)$

If follows the same rules as common polynomials. For example, for $\alpha(L)=1+\alpha_1L$ and $\beta(L)=1+\beta_1L$, we have

\[\delta(L)=(1+\alpha_1L)(1+\beta_1L) = 1+ (\alpha_1+\beta_1)L + \alpha_1\beta_1L^2.\]

The product of two filters is called “convolution” of $\lbrace\alpha_{j}\rbrace$ and $\lbrace\beta_{j}\rbrace.$

\[\begin{gathered} \delta_0 = \alpha_0\beta_0 \\ \delta_1 = \alpha_0\beta_1 + \alpha_1\beta_0 \\ \delta_2 = \alpha_0\beta_2 + \alpha_1\beta_1 + \alpha_2\beta_0 \\ \vdots \end{gathered}\]

Inverses of $\alpha(L)$ is to find $\beta(L)$ such that $\alpha(L)\beta(L)=1$, we say $\beta(L)$ is the inverse of $\alpha(L)$, denoted as $\alpha(L)^{-1}$ or $1/\alpha(L).$

Difference Operator: $\Delta=1-L$,

The first difference of a series $X_t$ is

\[\Delta X_t = X_t - X_{t-1}\]

The second difference is

\[\Delta^2 X_t = \Delta X_t - \Delta X_{t-1} = X_t - 2X_{t-1} + X_{t-2}\]

In general, we have

\[\Delta^k=(1-L)^k .\]

The change of a series $X_t$ with frequency $s$ is

\[\Delta_s X_t = X_t - X_{t-s} = (1-L^s)X_t\]

Unit Root Tests $X_t = c+\alpha X_{t-1}+\varepsilon_t$

For $\vert \alpha \vert<1$, the process is stationary.
For $\vert \alpha \vert=1$, the process is non-stationary.

Kernel Smoothing

https://atmos.washington.edu/~breth/classes/AS552/lect/lect24.pdf

Python code: https://matthew-brett.github.io/teaching/smoothing_intro.html

GMM

gmm(g, x, t0=NULL, gradv=NULL, type=c("twoStep","cue","iterative"), wmatrix = c("optimal","ident"), vcov=c("HAC","MDS","iid","TrueFixed"), ..., data)

g A function of the form $g(\theta,x)$ and which returns a $n\times q$ matrix with element $g_i(\theta,x_t)$ for $i=1,\ldots,q$ and $t=1,\dots,n$. This matrix is used to build the $q$ sample moment conditions.
x The matrix or vector of data from which the function $g(\theta,x)$is computed. If “g” is a formula, it is an n × Nh matrix of instruments or a formula.
wmatrix Which weighting matrix should be used in the objective function.
- By default, it is the inverse of the covariance matrix of $g(\theta,x)$, optimal.
- The other choice is the identity matrix which is usually used to obtain a first step estimate of $\theta$.

Vector Autoregressive

VAR model allows feedback to occur between the variables in the model. For example, we could use a VAR model to show how real GDP is a function of policy rate and how policy rate is, in turn, a function of real GDP.

A systematic but flexible approach for capturing complex real-world behavior. Better forecasting performance. Ability to capture the intertwined dynamics of time series data.

Conventions

$\Delta$ means “change” or “difference.” If an original variable is named $y$ then its change of difference is often denoted $cy$ or $dy.$ For example, the change in $price$ might be denoted $cprice.$

If an original variable is named $y$ then its growth rate is often denoted $gy,$ so that for each $t,$ $gy_t=\log(y_t)- \log(y_{t-1})$ or $gy_t=(y_t-y_{t-1})/y_t.$