# What does one-sided polynomial on lag operator $L$ mean?

In some time series texts, there are some talks about one-sided polynomial on lag operator $L$. I tried looking up what this means, but I cannot find one.

So what does one-sided polynomial on lag operator, $a(L)$, mean?

## 1 Answer

The terms "one-sided" and "two-sided" lag-polynomials, are used when the text considers the option to specify an equation with both "lags and leads", i.e a relation where both past but also future values co-vary with current value. When the lag-operator is "one-sided" it contains only lags in the one direction. When it is two-sided, it contains "lags" in both directions.
In some cases, scholars have used the term "forward" operator to verbally contrast it with the "lag" operator, but my impression is that it is not so widely used.

One may wonder "so if I read "one-sided lag-polynomial", does it mean that I will consider only past values, or only future values?". The answer is that in almost all cases the invertibility condition holds, so in principle and in theory, it makes no difference. In other words, in principle the term "one-sided lag polynomial" may mean either

$$\phi(L) = 1+\phi_1L+ \phi_2L^2+...$$

or

$$\phi(L) = 1+\phi_1L^{-1}+ \phi_2L^{-2}+...$$

where $L^{-1}x_t = x_{t+1}$.

A specification with a two-sided lag polynomial may appear to violate a basic causality postulate (the arrow of time), but we should remember that estimated relationships are relationships of association, not causation.

So when the concern and goal is purely forecasting, "anything goes" -or at least this is what many Time Series Theorists and Practioners essentially say.