Auto Regressive (AR) Time Series Models (Part A)

An autoregressive (AR) model is a time series regression in which the independent variable is a lagged (past) value of the dependent variable i.e.

x_t = b₀ + b₁ x _t-1 + ε_t

First order autoregressive AR (1) for the variable x is:

x_t = b₀ + b₁ x _t-1 + ε_t

Nth order autoregressive AR (n) for the variable x is:

x_t = b₀ + b₁ x _t-1 + b₂ x _t-2 ……. b_n x _t-n + ε_t

Covariance Stationary Series

Time series must be co-variance stationary for valid statistical inference. Time series is co-variance stationary when:

1. The expected value of the time series is constant and finite in all periods
2. The variance of time series is constant and finite in all periods
3. The covariance of the time series with past or future values of itself is constant and finite in all periods

Data is said to be stationary when time series variable do not exhibit any significant upward or downward trend over time whereas when time series variable does exhibit significant upward or downward trend it is said to be Non-stationary data.

Consequences of Covariance Non-Stationary

Regression results are invalid when time series is not covariance stationary. This is due to the following reasons.

1. The “t-ratios” will not follow a t-distribution
2. The estimates of b₁ will be biased and any hypothesis tests will be invalid

It is important to note that stationary in the past does not guarantee that the results will be stationary in the future too. Consequences may change with the passage of time.