# Serial Correlation

Correlation is a familiar concept used to describe the strength of the relationship between variables. Serial correlation (also known as autocorrelation) is the term used to describe the relationship between observations on the same variable over independent periods of time.

- Serial correlation is the relationship between a given variable and a lagged version of itself over various time intervals. It measures the relationship between a variable's current value given.
- Testing for Serial Correlation The above discussion suggests a very simple strategy for testing for serial correlation: check the magnitude and significance level of your estimated. Economists that deal with time-series data often prefer the sophisticated-yet-unintuitive.
- Serial Correlation and a Lagged Dependent Variable Consider the simple situation in which we have (although the result holds for more complicated models) 0 1 1, t t t y y (6) and while t satisfies all of our standard classical linear regression model assumptions, we have serial correlation, that is, 1 cov(,) 0 t t.
- The kind of serial correlation and heteroskedasticity tests used here are better regarded as specification tests. Many econometric packages provide a means of adjusting variances to adjust for.

##### Variants

### Negative Serial Correlation Emh

##### Actions

Jitsi api pricing. A statistic that serves as an estimator of the auto-correlation (auto-correlation function) of a time series. Let $ x _ {1} dots x _ {N} $be a time series. The serial correlation coefficient of order $ k $is the statistic $ r _ {k} $defined by the formula

$$ tag{* }r _ {k} = frac{frac{1}{N - k }sum _ {i = 1 } ^ { {N } - k } { xi _ {i,k} xi _ {i+} k,k } }{left [ frac{1}{N - k }left { sum _ {i = 1 } ^ { {N } - k } xi _ {i,k} ^ {2} right }frac{1}{N - k }left { sum _ {i = 1 } ^ { {N } - k } xi _ {i+} k,k ^ {2} right }right ] ^ {1/2} } ,$$

where

$$ xi _ {i,k} = x _ {i} - {frac{1}{N - k } } sum _ {i = 1 } ^ { {N } - k }x _ {i} .$$

Statistics close to (*) but of a slightly simpler form are also used as serial correlation coefficients. The set of serial correlation coefficients is called a correlogram; this term is also used for the graph of $ r _ {k} $as a function of $ k $.

Under various assumptions regarding the distribution of the $ x _ {i} $, there are exact and approximate expressions for the distribution of the serial correlation coefficients, and of their moments. Serial correlation coefficients are used in statistical problems to discover the dependence of terms in a time series.

As well as the term 'serial correlation coefficient' , the term 'sampling auto-correlationsampling auto-correlation' is used.

#### References

[1] | T.M. Anderson, 'The statistical analysis of time series' , Wiley (1971) |

[2] | M.G. Kendall, A. Stuart, 'The advanced theory of statistics' , 3. Design and analysis, and time series , Griffin (1966) |

[3] | E.J. Hannan, 'Time series analysis' , Methuen , London (1960) |

### Serial Correlation Model

**How to Cite This Entry:**

Serial correlation coefficient.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Serial_correlation_coefficient&oldid=48676