# Solving for a difference equation for a time series variable

I am trying to solve for the values of a variable $u_t$. $u_t$ is defined as:

$(1-L-L^2)u_t = \epsilon_t$

where $L$ is the lag operator and $t=1,...,n$. $\epsilon_t$ is a random variable (normal[0,1]) that can be generated. Using the values of $\epsilon_t$, how can I solve for $u_t$.

• Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. – gung - Reinstate Monica Jul 7 '16 at 2:10
• Hey Gung- this is a general form of a problem I'm facing in a project, not from a specific textbook or course. – A Kroeger Jul 7 '16 at 2:42
• It's not stationary – Glen_b -Reinstate Monica Jul 7 '16 at 5:15

Rewriting your equation using the properties of $L$ we have $$u_t=u_{t-1}+u_{t-2}+\epsilon_t$$
This is a second order difference equation and pinning a solution down for it requires two conditions (read intializations). For example set $u_0=u_{-1}=0$ then \begin{align} u_1&=\epsilon_1\\ u_2&=\epsilon_1+\epsilon_2\\ u_3&=2\epsilon_1+\epsilon_2+\epsilon_3\\ u_4&=3\epsilon_1+2\epsilon_2+\epsilon_3+\epsilon_4\\ ... \end{align}