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ARCHTest
Attachment  Size  

ARCHTest.xlsx 
Calculates the pvalue of the ARCH effect test (i.e. the whitenoise test for the squared time series).
Syntax
X
is the univariate time series data (a one dimensional array of cells (e.g. rows or columns)).
Order
is the time order of the data series (i.e. whether the first data point corresponds to the earliest or latest date (earliest date=1 (default), latest date=0)).
Order  Description 

1  ascending (the first data point corresponds to the earliest date) (default) 
0  descending (the first data point corresponds to the latest date) 
M
is the maximum number of lags included in the ARCH effect test. If omitted, the default value of log(T) is assumed.
Remarks
 The time series is homogeneous or equally spaced.
 The time series may include missing values (e.g. #N/A) at either end.

The ARCH effect applies the whitenoise test on the time series squared:

The test hypothesis for the ARCH effect:
Where:
 is the null hypothesis^{i}.
 is the alternate hypothesis.
 is the population autocorrelation function for the squared time series (i.e. ).
 is the maximum number of lags included in the ARCH effect test.

The LjungBox modified statistic is computed as:
Where:
 is the maximum number of lags included in the ARCH effect test.
 is the sample autocorrelation at lag j for the squared time series.
 is the number of nonmissing values in the data sample.

has an asymptotic chisquare distribution with degrees of freedom and can be used to test the null hypothesis that the time series has an ARCH effect.
Where:
 is the Chisquare probability distribution function.
 is the degrees of freedom for the Chisquare distribution.
 This is oneside (i.e. onetail) test, so the computed pvalue should be compared with the whole significance level ().
 In practice, the selection of may affect the performance of the statistic. Several values of m are often used. Simulation studies suggest that the choice of provides better power performance.
Examples
Example 1:
Let's consider the sample time series in the spreadsheet attachment on this page.
We computed the PValue for different values of the maximum lags:
Please note the plot of the ARCH test Pvalue reaches it lowest level (floor) around , and looses its power for larger numbers due to the size of the sample.
Files Examples
References
 Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0691042896
 Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0471690740
Attachment  Size 

ARCHTest.xlsx  14.72 KB 