I would like to ask you, what is the correct number of Lags in ARCH LM Test? I am referring to ArchTest in FinTS package, but other ArchTest (such as the one in Eviews) provide same results. In many time series, when I choose Lags between 1:5 the p.value is usually higher than 0.05, but with increasing of Lags, p.value becomes smaller. So how to do the correct decision if for lag=1, the time series looks homoscedastic(NO ARCH Effects), but for lags=5 and lags=12 result is heteroscedastic (presence of ACH) or reverse? Thank you

Sincerely Jan

#Example code in R

ArchTest(residuals,lags=1)   # p-value = 0.008499 
ArchTest(residuals,lags=5)   # p-value = 0.08166 
ArchTest(residuals,lags=12)  #p-value = 0.2317 

1 Answer 1


Arch LM tests whether coefficients in the regression:

$$a_t^2=\alpha_0+\alpha_1 a_{t-1}^2+...+\alpha_p a_{t-p}^2+e_t$$

are zero, where $a_t$ is either observed series which we want to test for ARCH effects. So the null hypothesis is


If hypothesis is accepted then we can say that series have no ARCH effects. If it is rejected then one or more coefficients are non zero and we say that there are ARCH effects.

Here we have classical regression problem of joint hypotheses versus individual hypothesis. When more regressors are included the regression is jointly insignificant, although a few regressors seem to be significant. All the introductory books about regression usually have chapter dedicated to this. The key motive is that joint hypotheses take into account all the interactions, when individual hypotheses do not. So in this case the statistic with few lags do not take into account the effects of more lags.

When statistical tests give conflicting results, for me it is an indication that data should be reexamined. Statistical tests usually have certain assumptions, which data may violate. In your case if we look at the graph of the series, we see a lot of zeroes. enter image description here

So this is not an ordinary time series and I would hesitate to use plain ARCH model.

  • $\begingroup$ Thank you for your answer and driving my attention to regressors. It would definetely help me to explain diferrent results. Anyway I decided to go with Lag=12, as it is used in Analysis_of_Financial_Time_Series by TSAY and Modeling_financial_time_series_with_s-plus by Zivot, Wang where stationarity was made by using logarithmic differention. Thank you once more for finding time and answer my question in great detail. Good luck $\endgroup$
    – troger19
    Feb 23, 2011 at 18:19
  • $\begingroup$ Are you sure the zeroth term is in the hypothesis? We wouldn't expect $a_t^2$ to have mean $0$. (So if $\alpha_0$ was included in the test, you'd expect to reject even if there was nothing going on.) I realize a number of online sources say it's in the test but I don't think that can be right. $\endgroup$
    – Glen_b
    Oct 22, 2017 at 11:25
  • $\begingroup$ I've just gone and checked Engle's 1982 paper "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation.", Econometrica. 50 (4): 987–1008, (it's in Sect. 8 p999). ... no $\alpha_0=0$ in the LM test. Thank goodness, I was worried I'd missed something. $\endgroup$
    – Glen_b
    Oct 22, 2017 at 12:49
  • $\begingroup$ @Glen_b, thanks for spotting this. Yes, there should be no $\alpha_0$ in the hypothesis. $\endgroup$
    – mpiktas
    Oct 25, 2017 at 9:52

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