Solving the AR(1) Model – Finding its Mean, Variance and Covariance

We discussed the AR(p) model previously when discussing linear time series. In this post, we’ll learn to solve the AR(1) model, where

R_t = c₀ + c₁R_t−1 + σ*z_t

Finding the Mean of the AR(1) model

E[R_t] 
= c₀ + c₁E[R_t−1] + σ*E[z_t]
= c₀ +c₁E[R_t−1] + σ*0
= c₀ +c₁E[R_t−1]

Since AR(1) is stationary, E[R_t] = E[R_t−1]

Therefore,
E[R_t] = c₀/(1−c₁)

Let μ = c₀/(1+λ), where μ = E[R_t] and λ = −c₁

c₀ = μ*(1+λ)

Then,
R_t = μ*(1+λ) + (−λ)R_t−1 + σ*z_t
R_t − μ = −λ(R_t−1 − μ) + σ*z_t ———————— (1)

Finding the variance of the AR(1) model

Let 𝛾₀ 
= Var[R_t]
= E[(R_t − μ)²]
= E[(−λ(R_t−1 − μ) + σ*z_t)²]            (refer to (1))
= λ²E[(R_t−1 − μ)²] + 2*(−λ)*(σ)*E[(R_t−1 − μ)*z_t] + σ²E[z_t²]
= λ²E[(R_t−1 − μ)²] + σ²
= λ²*𝛾₀ + σ²

Note: E[(R_t−1 − μ)*z_t] = 0 due to the independence of z_t from all earlier time values

Therefore,
𝛾₀ = σ² / (1−λ²)

λ should be less than one in magnitude else the denominator blows up. λ is the multiplier for R_t−1, so it makes sense for λ to be smaller than one as the effect of R_t−1 dies off over time.

Finding the autocovariance of the AR(1) model

Autocovariance = Covariance in the time series context

𝛾_k 
= E[(R_t − μ)(R_t−k − μ)]
= E[(−λ(R_t−1 − μ) + σ*z_t)(R_t−k − μ)]
= E[(−λ(R_t−1 − μ)(R_t−k − μ)] + σ*E[(z_t(R_t−k − μ)]
= −λ*E[(R_t−1 − μ)(R_t−k − μ)] + 0
= −λ*𝛾_k−1  (using recursion from step 1)

Therefore, 
𝛾_k 
= (−λ)^k𝛾₀
= (−λ)^kσ²/(1−λ²)

This is known as the lag-k autocovariance coefficient. It relates the influence of an excess return value at one point in time with values k periods in the past.

Conclusion

Solution of Model

R_t = c₀ + c₁R_t−1 + σ*z_t

Mean = c₀/(1−c₁)
Variance = σ² / (1−λ²)
Covariance = (−λ)^kσ²/(1−λ²)

where λ = −c₁

Note 1

The covariance is known as the lag-k autocovariance coefficient

Note 2

Model can be re-written as
R_t − μ = −λ(R_t−1 − μ) + σ*z_t

Note 3

AR(1) model is often used to model mean reversion.

R_t − μ and R_t−1 − μ are differences from the mean of the two variables R_t and R_t−1, respectively.

If λ is positive and (R_t−1 − μ) is positive, the first term on the RHS will be negative, thus decreasing the value of R_t − μ. 

In other words, if we overshoot the mean in one period, we push R towards the mean in the next period.