# Different results of Engle's Lagrange multiplier test for conditional heteroscedasticity from SAS and FinTS

To fit a simple AR(5) model, I use SAS PROC AUTOREG. I called the option ARCHTEST=(QLM) which provides Engle’s Lagrange Multiplier Test for ARCH Disturbances and the Portmanteau Q Test. The statistics SAS provided for Engle’s Lagrange Multiplier Test were:

Order   LM      p-value
1       0.018   0.893
2       0.079   0.961
3       0.881   0.830
4       1.088   0.896
5       1.105   0.954
6       4.114   0.661
7       4.179   0.759
8       4.182   0.840
9       4.528   0.873
10      4.528   0.920
11      5.565   0.901
12      6.318   0.899


I took the residuals of this model and used the ArchTest function of the FinTS package of R ... library(FinTS).

# A function to do ArchTest at multiple lags where 'data' are the residuals of
# the model.
library(FinTS)
arch.m1 = matrix(NaN,12,3)
for(i in 1:12)  {s = ArchTest(data,lags=i)
arch.m1[i,1] = s$statistic arch.m1[i,2] = s$parameter
arch.m1[i,3] = s$p.value}  The test results from the ArchTest function of the FinTS package are very different: Order LM p-value 1 0.014 0.907 2 0.092 0.955 3 0.835 0.841 4 2.194 0.700 5 3.189 0.671 6 13.013 0.043 7 19.157 0.008 8 23.161 0.003 9 18.780 0.027 10 18.467 0.048 11 22.565 0.020 12 23.828 0.021  I thought the SAS ARCHTEST=(QLM) option and the ArchTest function of the FinTS package were performing the same test but apparently not. Does anyone know what the difference is between the tests in SAS and FinTS? Note, the data used by Tsay (2005) in the example on page 102 is available here. Using the following R code: library(FinTS) intc=read.table("http://faculty.chicagobooth.edu/ruey.tsay/teaching/fts2/m-intc7303.txt", quote="\"") ArchTest(intc$V2,lags=12)


I find:

$\chi^2$= 48.0425, df = 12, p-value = 3.073e-06

However, Tsay (2005:102) reports:

$\chi^2$= 43.5041, df = 12, p-value = 0.0000

If I use:

ArchTest(log(1+intc$V2),lags=12)  Then:$\chi^2$= 43.5041, df = 12, p-value = 0.0000 However, this doesn't explain the differences observed between SAS and FinTS. Thanks for your reply @mpiktas. The statistic calculated by the FinTS library using the ArchTest function is based on the second statistic provided by @mpiktas: $$TR^2=T\left(1-\frac{SSR_1}{SSR_0}\right)$$ intc=read.table("http://faculty.chicagobooth.edu/ruey.tsay/teaching/fts2/m-intc7303.txt",quote="\"") intc.ln=log(1+intc$V2)
lag=12
mat=embed(intc.ln^2,lag+1)
m=summary(lm(mat[,1]~mat[,-1]))
SSR1=sum(residuals(lm(mat[,1]~mat[,-1]))^2)
SSR0=sum(residuals(lm(mat[,1]~1))^2)
stat=length(mat[,1])*(1-(SSR1/SSR0))


which gives $\chi^2$ = 43.5041.

If I apply the first statistic provided by @mpiktas and shown on page 102 of Tsay (2005):

$$F=\frac{(SSR_0-SSR_1)/m}{SSR_1/(T-2m-1)}$$

intc=read.table("http://faculty.chicagobooth.edu/ruey.tsay/teaching/fts2/m-intc7303.txt",quote="\"")
intc.ln=log(1+intc$V2) lag=12 mat=embed(intc.ln^2,lag+1) m=summary(lm(mat[,1]~mat[,-1])) SSR1=sum(residuals(lm(mat[,1]~mat[,-1]))^2) SSR0=sum(residuals(lm(mat[,1]~1))^2) stat=((SSR0-SSR1)/lag)/(SSR1/(length(mat[,1])-(2*lag)-1))  This version of the statistic gives$\chi^2$= 3.837. These are very different answers? I have provided the residuals for the AR(5) model from SAS below: c(0.14167,0.14019,-0.42499,0.0222,0.04388,-0.03157,0.00897, -0.04567,0.29171,-0.06735,0.05235,-0.04062,-0.18694, -0.2407,0.03766,-0.15011,0.11023,-0.30812,-0.02399, 0.13434,0.05836,0.09368,-0.01832,-0.15669,-0.00127, 0.19431,0.10366,-0.09682,-0.04597,0.07162,0.16411,-0.02181, -0.00459,-0.16325,-0.09615,-0.04581,0.10441,0.12845,-0.0398, -0.03349,-0.01575,0.06923,-0.05006,-0.05827, -0.0317,0.08914,0.07323,0.06519,0.16141,0.0537,0.08213, 0.01073,-0.06628,0.01297,0.00622,-0.10171,0.05598,-0.05615, 0.10322,-0.04991,-0.01223,-0.04975,-0.06043,0.01794, -0.00071,0.05425,-0.05462,0.01659,0.08656,0.02482,-0.04214, 0.01221)  For lag 5, SAS gives$\chi^2$= 1.105, FinTS gives$\chi^2$= 3.189, and the F-statistic version gives$\chi^2$= 0.560. References: Engle, R. F. 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50:987-1007. Tsay, R. S. 2005. Analysis of financial time series. Second edition. John Wiley & Sons, Inc., Hoboken, New Jersey, USA. ## 1 Answer I suspect that your data has small sample size. If you look at the original Engle article, the steps to calculate the statistic are the following: 1. Do initial regression and save residuals 2. Regress squared residuals on their lags. 3. Use$TR^2$as a statistic, where$T$is the sample size and$R^2$is the R-squared from the second regression. The statistic calculated in Tsay is F-statistic from the second regression: $$F=\frac{(SSR_0-SSR_1)/m}{SSR_1/(T-2m-1)},$$ where$m$is the number of lags in the second regression,$SSR_0$is the sum of total squares and the$SSR_1\$ is the sum of squared residuals from the second regression.

For comparison

$$TR^2=T\left(1-\frac{SSR_1}{SSR_0}\right)$$

So the statistics are different. I suspect that the p-values differ significantly for larger lags, because your sample size is small. This is just a guess, since you do not provide the data.

• I have added some additional commentary regarding these two statistics in question. Thanks. – RioRaider Sep 18 '12 at 19:30