Auto Regression

Summaries

The dependent variable is regressed against its lagged values. E.g. x_t = b0 + b1*x_t-1 + error

AR(p) is AR with order of p

Testing of AR model:

1)      Estimate the AR model with order 1 to k with AR(i) (lag i)

2)      For each order, find the correlation (using linear regression) on the residue obtained from one. E.g. for i=2, regress 1 to T-2 against 3 to T

3)      Compute t-statistics: correlation(t,t-k)/ (1/sqrt(T)) (DOF = T-2 because this is SD) (This is from the correlation plots. For i=2, plots error_i against error_i-2)

4)      1/sqrt(T) is the standard deviation of the error

5)      Any of the lag shows significant t-value means that there is auto-correlation and the model is not good

For time series to be valid, it has to satisfy covariance stationary:

1)      Constant and finite expectation value (constant over time)

2)      Constant and finite variance (constant over time)

3)      Constant and finite covariance with leading or lagging values (meaning covariances between lag 1 or lag n are the same, remember, this covariance is found by matching a pair of shifted data)

Mean Reversion:

If the mean is constant, then there has to be mean reversion. When x_t>mean reverting level, it will decrease and vice versa. At mean reverting level, x_t= x_t+1, so,

Mean reverting level = b0/(1-b1)

In-sample forecast: forecast on available data

Out-of-sample forecast: predict the future data

Use RMSE (Root mean of squared error) to decide with model is better

Longer period gives more sample points but increases the coefficient instability and non-stationarity.

*** Afterall, AR assumes the expect value is constant. And it will revert to mean if there is variation. By using the current value, we then predict the next value using the AR equation with the expectation that it will be closer to the mean.