# Conditional Expectation of an autoregressive (AR1) Process

Suppose X is a time series that follows the process:

$$X[t]=\rho X[t-1] + \varepsilon[t]$$

where $$\varepsilon[t] \sim \mathcal{N}(0.0, \sigma)$$, $$\varepsilon[t]$$ is IID and $$X[0] \sim \mathcal{N}(0.0, \sigma)$$,.

My interest is computing a conditional expectation using mathematical:

Allowing for notational sloppiness, I want to focus on the subset of $$X[t]$$ that satisfy:

$$\tilde{X}[t]\ < Q( X[t] < Q(0.1, X[t])$$ Where $$Q(0.1, X[t])$$ is the $$0.1$$ quantile observation of $$X[t]$$.

Then find the conditional expectation of $$\tilde{X}[t]$$:

$$\mathbb{E}[\ \tilde{X}[t] \ |\ ( \tilde{X}[t-3] < Q(0.5, \tilde{X}[t-3]))]$$

In words, I want to find the subset of the process that falls below the 0.1 quantile observation. Then find the mean of the subset that satisfies the condition before and also had a realization three periods ago that is lower than the median of all observations from three periods for this particular subset of realizations.

• Define AR1..... Jan 10 at 2:56
• For reference: Autoregressive Model wiki page.
– Syed
Jan 10 at 3:29
• Should we assume that the $\epsilon_i$'s are all independent (which is implied by the title but not made explicit in the question) and that $X_0\sim N(0,\sigma)$?
– JimB
Jan 10 at 4:29
• I want to emphasize that $X_0 \sim N(0,\sigma/\sqrt{1-\rho^2})$ (which is the value in the link given by @Syed) which means that $E(X_t | X_{t-3} < 0)$ is $-\frac{\sqrt{\frac{2}{\pi }} \rho ^3 \sigma }{\sqrt{1-\rho ^2}}$. (I haven't got a closed-form for when $1/10=0.1$ is used.)
– JimB
Jan 10 at 21:55

If in Mathematica terms by $$Q(0.05, X_t)$$ you mean

x = InverseCDF[NormalDistribution[0, σ/Sqrt[1 - ρ^2]], 1/20]


then because the marginal distributions of the $$X_i$$ are all normal with mean zero and variance $$\sigma^2/(1-\rho^2)$$, $$Q(0.05, X_t)$$ is always less than $$Q(0.10, X_{t-3})$$, then mean of $$X_t$$ conditional on $$X_t < x$$ is found with

dist = TruncatedDistribution[{-∞, InverseCDF[NormalDistribution[0, σ/Sqrt[1 - ρ^2]], 1/20]},
NormalDistribution[0, σ/Sqrt[1 - ρ^2]]]
Mean[dist]


But I suspect this is not the conditioning you want.

• You are right Jim. Thinking about it some more, I am after something that is definitely not like this. What I want to compute is a conditional expectation. I first want to focus on $X_t$ less than $Q(0.1,X_t)$ . So let's define these as: $$\tilde{X_t} < Q(0.1, X_t)$$ Then the object I am interested in can be summarised in sloppy notation as: $$\mathbb{E}_t[ \tilde{X_t} \ | \ \tilde{X}_{t-3} \ < \ Q(0.5, \tilde{X}_{t-3}) ]$$ Jan 10 at 14:45
• Sounds good. Editing your question to reflect that would be helpful to others. In short, $X_t$ and $X_{t-3}$ will have a bivariate normal distribution so that conditional expectation will be easy to obtain.
– JimB
Jan 10 at 15:49