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.

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

**x**_{t}= b_{0}+ b_{1}x_{t-1}+ ε_{t}First order autoregressive AR (1) for the variable x is:

**x**_{t}= b_{0}+ b_{1}x_{t-1}+ ε_{t}Nth order autoregressive AR (n) for the variable x is:

**x**_{t}= b_{0}+ b_{1}x_{t-1}+ b_{2}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:

- The expected value of the time series is constant and finite in all periods
- The variance of time series is constant and finite in all periods
- 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.

- The “t-ratios” will not follow a t-distribution
- The estimates of b
_{1 }will be biased and any hypothesis tests will be invalid

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