Variance ratio

[Robotek]

Hi,

I'm experiencing some inconsistencies when running a Lo and MacKinlay (1988) VR on daily index data (2200 observations) in eViews7.

1) I first imported (excel) logarithmic returns generated from the daily index levels as first differences. This gives me VR:s of ~(0.5, 0.25, 0.12 and 0.06) at lags (2,4,8,16) and test statistics of between 13-6, assuming heteroscedastic growth, and quite much bigger assuming homoscedastic growth. Multiplying the VR:s with corresponding lag gives values around 1, so I believe actual index levels are to be imported instead of logarithmic return series.
2) Index levels give believable values on the VR (1.1-0., and low z:s, however not in my opinion suspiciously low, as there are some p-values below 0.10. The issue I'm having here is inconsistency between p-value and z. The z Lo and MacKinlay (1988) propose is supposed to be studentized maximum modulus (SMM) distributed (which is confirmed in eViews user guide II). I found the following critical values: 10%:2.23, 5%:2.49 and 1%:3.03. The lowest p-value in my test is 0.054 and it's corresponding z is 1.926 (should be 2.23>2.49 if SMM). What am I missing here? My test specifications: Original data, use unbiased variances, demean data, asymptotic normal.

Thanks for any help!

[EViews Glenn]

Can you post the workfile?

In the meantime, a couple of observations that may or may not answer the questions.

1. The description of the test statistic in the manual is pretty clear about the exact hypothesis that is being tested. By default EViews tests for whether the series itself follows a random walk so that differences of the series are viewed as the iid innovations, but you may also indicate whether the innovations are the log differences, or whether the series contains the innovations themselves. I'm not entirely sure from your brief description but it sounds a bit as though you are giving EViews the innovations themselves while testing for whether the differences are random walks (which if is indeed what you are doing, won't be true since you've introduced an MA by differencing).

2. We clearly state in the documentation that for the single Lo and MacKinley test that we use the asymptotic normality results from the Lo and MacKinley 1988 paper. For the Chow and Denning variant of the test involving multiple tests, there is a SMM distribution theoretical result, but as indicated in the documentation and the output, we employ the asymptotic approximation which sets the number of degrees of freedom to infinity. This approximation yields the standard normal approximations outlined in Lemma 1 of Chow-Denning (which are the Sidak multiple comparison adjustments).

[Robotek]

Sadly I don't have access to the workfile for a couple of days.

Thanks Glenn for your input though. Following it is quite tricky given my knowledge level in statistics. I guess my question is more routed in VR/statistical theory than eViews practice.

Bottom line what I'm looking for is critical values for the test statistics as in L&M at 1%, 5% and 10% levels. I.e. how do I interpret the z:s of the single tests without looking at the p:s.

[EViews Glenn]

It depends on what you mean by "interpret the z's of the single test without looking at the p's". Let me take a guess...

In order to discuss this, we need to be very clear about the distributions involved in the various test statistics. As I noted earlier, the SMM distribution result is not for a simple Lo and MacKinley variance ratio test at a single lag (Lo and MacKinley never discuss the SMM distribution). The asymptotic distribution of that test statistic is asymptotic normal.

So the z's of any single tests are asymptotic normally distributed. That's how we get the p-values.

The SMM distribution comes about when you have multiple lags which you use in constructing the test. As in the usual multiple comparison procedure you can't just use the lag with the largest t-statistic as the size of the test is not valid when you look at more than one statistic and find the maximum. Chow and Denning note that the distribution of the max of the absolute values of several Lo and MacKinley variance ratio tests (at different lags) follows the SMM distribution with parameters (m=number of comparisons, df=sample size).

Thus, the joint test of whether the variance ratios at any lag are significant is SMM, which we approximate using the asymptotic normal as in Chow and Denning (which assumes that the df is infinite).That's how we compute the p-value that we report. But if you want to use your SMM table to be more accurate, simply compare the Max |z| with the appropriate table entry for your alpha, m, and df.

[Robotek]

Aha, right. The C&D MVR seemed ridiculously strict to me (from RWH null perspective) as I had the idea that it is (quite) equally distributed as the L&M SVR test, when obviously the number of lags tested, as a property of SMM, remarkably ‘penalize’ the p-value of the joint max |z|.

Another thing your last reply might indicate, let me see if I understood it correctly.

The fact that I run the VR test for multiple lags, even though I’m in a way investigating individual lags separately and hence each one is a single test, would that itself make the test a C&D? And forth, all individual lag results are hence nonsense as only the joint part of the results matter? This is a bit hard for me to believe because i) articles I’ve read that take the L&M approach, as few as they are in comparison to the C&D ones, do investigate a set of lags (commonly 2,4,8,16) and ii) the ones taking the C&D approach seem to put equally much attention to interpreting VR’s of single lags as the joint test. Of course, the common argument is that the L&M is inadequate as VR should equal 1 for all q (i.e. not only at e.g. lag 2, 4, 8, 16) under the RWH. But still the test is used, often with the added conclusion “… however, too much weight shouldn’t be put to these results as VR should equal 1 at all q…”. Is it in this context you state “…you can't just use the lag with the largest t-statistic as the size of the test is not valid when…”, hence simply stating that L&M were wrong? If so, my question on critical values of z (in the asymptotic normal distribution) is motivated, and I’ll have to underline my low knowledge level in statistics by saying/asking: I like distribution tables, and I'm familiar with tables of the F-distribution, student’s t-distribution, chi-squared-distribution ... - do I use e.g. any of those when interpreting z-values from the L&M tests that are asymptotic normally distributed? I sure haven’t seen any ‘AND’ table lying around.

[EViews Glenn]

The first part of the last post is just a reflection of differing philosophies about how to discuss results. The Multiple test results using the SMM are for the joint null that all of the lags have martingale properties. The individual tests are for a single lag having martingale properties. That's the crux of the statistical context. Can you talk about individual results if you are going to do joint testing?. Sure, you can talk about any part of the results you want, but you have to be careful about making statistical significance claims on the basis of a single lag test statistic in a multiple lag testing framework. That is presumably the sense of the "too much weight" admonition. But interpretive discussion is fair game as long as you are precise about what you are claiming.

As for the substantive question, asymptotic normal basically means normally distributed for large enough samples. Use the normal tables that you ordinarily would. The statistic is probably called the Z stat for this reason.

[Robotek]

Ok, student's t-distribution seems to give critical values in line with the eViews test results on individual lags.

I'm glad you didn't fire away a straight answer to my query (well, my query was partly invalid anyways..) directly. As a result of this discussion, I look at my research with different eyes and have made quite some changes to it.

Thanks a lot!

[EViews Glenn]

Glad to hear that. Use the normal not the Student's t.

[saifsidd]

Sir
I am attaching a table here ,

Variance Ratio Test on Dow Jones _Retuns

Joint Tests Value df Probability
Max |z| (at period 2)* 2.698854 2476 0.0994

Individual Tests
Period Var. Ratio Std. Error z-Statistic Probability
2 0.377692 0.230582 -2.698854 0.0070
3 0.222188 0.313559 -2.480593 0.0131
4 0.175080 0.358817 -2.298999 0.0215
5 0.144373 0.387248 -2.209505 0.0271
6 0.117488 0.406679 -2.170046 0.0300
7 0.103088 0.420764 -2.131628 0.0330
8 0.087632 0.431436 -2.114727 0.0345
9 0.078882 0.439801 -2.094399 0.0362
10 0.071089 0.446536 -2.080260 0.0375
11 0.065093 0.452076 -2.068031 0.0386
12 0.058941 0.456713 -2.060502 0.0394
13 0.055042 0.460653 -2.051344 0.0402
14 0.046365 0.464067 -2.054953 0.0399
15 0.056019 0.467242 -2.020325 0.0433
16 0.040596 0.470658 -2.038431 0.0415

[*]can I conclude or test the hypothesis on the basis joint test only
[*]can i simply accept the hypothesis on the basis of p-value of joint test
please reply....thanks

[EViews Glenn]

The individual tests are just informative. The joint test is the relevant one...

[saifsidd]

thanks...

[Rupikhanna]

I am trying to run Variance test on RTSE stock market index (closing) data. I had initially run the test on logarithmic return figures obtained from closing data for the index I am working on, then having read this thread I realised I was making a mistake by NOT working on the index values directly (but on their first differences i.e. returns that too logarithmic). I have also understood that the Joint variance ratio test uses SMM distribution with infinite degrees of freedom for finding out the p-value and that one has to follow the SMM table to know the critical values for the joint test Z statistic to decide if the Z statistic is significant or not. I suppose, since I am running the test for lags 2, 4, 8 and 16, the number of parameters becomes 4. Checking the SMM table for critical value @ 4 number of parameters and D.o.f. = infinite, I found that the critical values are 2.806 a@ 1%, 2.234 @ 5% and 1.943 @ 10% alpha. Now the results below indicate that the joinr Z statistic is 2.686 while the Critical value @5% is 2.234 which means that the none hypothesis has to be rejected. P-value given by eviews results table however indicates that the Z-statistic is insignificant at 5% and that the none hypothesis will have to be accepted.



Null Hypothesis: CLOSE is a martingale
Date: 11/20/13 Time: 00:22
Sample: 9/01/1995 12/30/2005
Included observations: 2678 (after adjustments)
Heteroskedasticity robust standard error estimates
Compute variances assuming zero mean
Use biased variance estimates
User-specified lags: 2 4 8 16

Joint Tests Value df Probability
Max |z| (at p. 16)* 2.686043 0.0286

[EViews Glenn]

I haven't looked at this stuff in ages, but my quick check of my notes indicates that we do an adjusted Dunn-Sidak p-value following the calculations in Chow-Denning (1993, p. 389ff).