How do AR,ARMA,ARDL and other time series models correct for omitted variable bias I have come across numerous papers that use an Auto Regressive Distributed Lag (ARDL) model of the following form:
$$
\Delta y_{t}=\alpha_{0}+\beta_1\Delta y_{t-1}+\beta_2\Delta x_{t-i}+\gamma_{1}y_{t-1}+\gamma_{2}x_{t-1}
$$
Where $y_{t}$ is one dependent variable and $x_{t}$ is one independent variable.
My question is, don't these models suffer from omitted variable bias? What allows time series studies to use only one independent variable as compared to cross sectional and panel studies that rarely ever use less than 2 independent variables? Do time series models have a property that allow researchers to use just one independent variable? I am not aware of this since I have only taken a class in cross sectional econometrics. I would greatly appreciate your help.
 A: Below are my responses to your two questions.
Don't these models suffer from omitted variable bias?


*

*ALL regression based models including time series suffer from omitted
variable bias unless regression(including time series) is based on
randomized experimental data.


What allows time series studies to use only one independent variable as compared to cross sectional and panel studies that rarely ever use less than 2 independent variables? Do time series models have a property that allow researchers to use just one independent variable?


*

*You could use one independent variable in Time series like any other
non-timeseries regression. There is nothing special in time series,
that allows you to use only one independent variable. You can use as many or less number of independent variable. 


With regards to your specific question on number of independent variables, according to this wonderful article:

"And do not try to estimate relationships for more than three
  variables in a regression (findings from Goldstein and Gigerenzer,
  2009, are consistent with this rule-of-thumb)"

The same article also provides a real world example of missing variable bias. Bottom line, use domain knowledge, available literature, experimental evidence, experts to select number of variables.
In addition, I would use Transfer Function within ARIMA framework which is a general form of ARIMA and incorporates AR/ARMA.ARDL and other time series regression.
