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If I am being asked to do an unconditional analysis of a time series variable, lets say GDP starting in 2000, what model am I supposed to estimate?

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So the simplest response would be that an unconditional model is a model that does not include any other stochastic regressors. Lets say your variable of interest is $Y_t$ then a conditional model would be

$Y_t = \alpha + X_t\beta + v_t$

where you are implicitly conditioning on $X_t$ (i.e. treating it as fixed/exogenous). In time series $X_t$ can also include lags or transformations of $Y_t$. In the example that you gave, you could think of inflation (or lags of GDP) as potential conditioning variables.

So then, given this, what is an unconditional model? Well it is a model that does not include $X_t$ and instead only has deterministic terms (such as a constant or a trend). That is to say in the context of GDP, GDP growth doesn't depend on (i.e. isn't conditional upon) another variable but rather grows at a constant rate (or has a linear trend).

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