5 Model vs Spec
5.1 Classification
Time series models can be broadly sorted into four categories based on whether we are dealing with stationary or non-stationary time series, or whether the model involves only one variable or multiple variables.
| Stationary | Nonstationary | |
|---|---|---|
| Univariate | ARMA | Unit root |
| Multivariate | VAR | Cointegration |
5.2 Model vs Spec
We use the word “model” rather loosely in economics and econometrics. Anything that deals with the quantified relationships between variables can be called a model. A general equilibrium model is a model. A regression is also a model.
To make things less confusing, we would use the word “model” more restrictively in this chapter. We reserve the word model to those representing the data generating processes (DGPs). That is, when we write down a model in an equation, we literally mean it. If we say \(y_t\) follows an AR(1) model:
\[ \begin{aligned} y_t &= \phi y_{t-1} + \epsilon_t,\\ \epsilon_t &\sim N(0,\sigma^2). \end{aligned} \]
We literally mean \(y_t\) is determined by its previous value and an contemporary innovation drawn from a Gaussian distribution.
A model is distinguished from a specification. Suppose \(\{y_t\}\) represent the GDP series, we can estimate a regression:
\[ y_t = \phi y_{t-1} + e_t \]
This is a specification not a model. Because the DGP of GDP data is unknown, definitely not an AR(1). We can nontheless fit this spec with the data and get an estimated \(\hat\phi\). If \(e_t\) satisfies some nice properties, for example, uncorrelated with the regressor, then we know this \(\hat\phi\) is consistent.
When we run regressions with real-life data, we are actually working with specifications. They are not the DGPs of the random variables. But they allow us to recover some useful information from the data when certain assumptions are met. Mostly we are interested in the relationships between variables. A specification describes this relationship, even though it does not describe the full DGP.
This chapter deals with models in the abstract sense. The next chapter will discuss how to fit a model or a spec with real data.