ADL Models

ARMA is great for univariate situations, but we often want to account for the relationship between multiple time series

The autoregressive distributed lag model extends $\text{AR}(p)$ to include regressors

Definition: A time series $\{Y_t\}$ is said to be generated by an autoregressive distributed lag model of order $(p,q)$, denoted $\text{ADL}(p,q)$ is $Y_t=\alpha +{\sum }_{i=1}^p{\varphi }_i{Y}_{t-i}+{\sum }_{j=0}^q{\beta }_j{X}_{t-j}+{ϵ}_t$, where $\{X_t\}$ is a weakly stationary and exogeneous process

Or in lag polynomial notation, $\varphi (L)Y_t=\alpha +\beta (L)X_t+{ϵ}_t$, where $\varphi (L)=1-{\varphi }_{1}L-\cdots -{\varphi }_p{L}^p$ and $\beta (L)={\beta }_0+{\beta }_{1}L+\cdots +{\beta }_q{L}^q$

$Y_t$ is stationary if the roots of $\varphi (z)$ lie outside the unit circle

Definition: The long-run equilibrium of a time series $\{Y_t\}$ generated by an ADL process is the value $\bar{Y}$ to which $\{Y_t\}$ converges if $\{X_t\}$ is fixed at $\bar{X}$ and there are no errors

The final equation is called the long-run equilibrium relation

What is $\mathbb{E}[Y_t]$ for $\text{ADL}(p,q)$?

$\varphi (1){\mu }_Y=\alpha +\beta (1){\mu }_X$
$\mathbb{E}[Y_t]=\frac{\alpha +\beta (1){\mu }_X}{\varphi (1)}$

This is equal to the long-run equilibrium $\bar{Y}$ when we fix $\bar{X}={\mu }_X$

How do we forecast with an $\text{ADL}$ model?

Example:
Let $Y_t=\alpha +\varphi Y_{t-1}+{\beta }_0{X}_t+{\beta }_1{X}_{t-1}+{ϵ}_t$ follow an $\text{ADL}(1,1)$ model

Given $D_T=\{Y_1,\dots ,Y_T,X_1,\dots ,X_T\}$,
${\hat{Y}}_{T+1}\equiv \mathbb{E}[Y_{T+1}\mid D_T]$

We need to assume something about $X_t$ to allow us to forecast, as an example $X_t=\gamma X_{t-1}+u_t$ means $\mathbb{E}[X_{T+1}\mid D_T]=\gamma X_T$

So,
${\hat{Y}}_{T+1}=\alpha +\varphi Y_T+({\beta }_0\gamma +{\beta }_1)X_T$

We can compute higher forecasts recursively,
${\hat{Y}}_{T+h}=\alpha +\varphi {\hat{Y}}_{T+h-1}+{\gamma }^{h-1}({\beta }_0\gamma +{\beta }_1)X_T$

Example:
${\hat{X}}_{T+1}={\gamma }_2{X}_{T-1}$
${\hat{X}}_{T+2}={\gamma }_2{X}_T$

${\hat{Y}}_{T+1}=\varphi Y_T+{\beta }_1{X}_T$
${\hat{Y}}_{T+2}=\varphi {\hat{Y}}_{T+1}+{\beta }_1{\hat{X}}_{T+1}$

Causality

Establishing causal relationships through these models is difficult, so we opt for a simpler measure called Granger Causality

Definition: A time series $\{X_t\}$ Granger causes a time series $\{Y_t\}$ if past values of $X_t$ provide statistically significant information about $Y_t$

For an $\text{ADL}(1,1)$ model, if ${\hat{\beta }}_1$ is statistically significant then there is Granger causality, or in general, if any lag $X_{t-j},j\ge 1$ is significant

It is possible for two series to Granger cause each other

Given a parameter estimate with normal distribution and given standard error, we can divide to test the statistical significance

To further investigate causality, there are more specialized techniques like instrumental variables, diff-in-diff analyses, and event studies