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Auto-Correlation In Time Series

Auto-Correlation In Time Series

Auto-Correlation (Box-Jenkins, 1976) In Time Series is basically the correlation between successive values of the same series at different time lags. It seeks to answer the question, does the error terms in a time series transfer from one period to another?.

Basically, when errors are corrected over time, they are said to be Auto-Correlated. The errors in the first observation, if correlated could also lead to bigger errors in the second Value of the Time Series leading to greater errors, as the series slides down in time. This could lead to the following problems in your calculations especially when using Ordinary Least Square Methods (OLS)

1. Inefficient Ordinary Least Squares Estimates and any forecast based on those estimates. Estimated regression coefficients are still unbiased, but they no longer have the minimum variance property.

2. Exaggerated goodness of fit
3. Standard errors that are too small and The MSE may seriously underestimate the true variance of the errors.

4. T-statistics that are too large. a regression coefficient appears to be statistically significant when it is not. Statistical intervals and inference procedures are no longer strictly applicable.

Randomness is key to knowing if a univariate statistical process is in control and auto-correlations could be used to know if the data was gotten from a random process also hinting us on the type of model to be used ( non-linear or time series model)

Having assumptions of constant location and scale, randomness, and fixed distribution being reasonable we could model the time series using the random No-Trend model given by

Y(t) = B° + e(t)

Where e(t)  is the error term and B° the mean of the series.

For a Time Series Y(t), the Auto-correlation (A) at lags (l)  is computed with the mathematical representation below.

A=E[(Y(t) - μ(t)) (Y(t-l) - μ(t))]/[E(Y(t) - μ(t))]

Where μ(t) is the mean of the series

To check and test for Auto-Correlation the following test could be carried out

Residual plots - Plot e against t and look for clusters of successive residuals on one side of the zero line.

⏩A Durbin-Watson test.

⏩A Lagrange Multiplier Test.

⏩Mordified Box Pierce Test And Related Rule Of Thumb

⏩Ljung-Box Q test

⏩Hildreth-Lu Procedure

⏩Cochrane-Orcutt Procedure

A correlogram. A pattern in the results is an indication for autocorrelation. Any values above zero should be looked at with carefully.

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