# Hourly Returns Statistical test

I am trying to do an analysis on time zones effect on intraday returns.

As a first step, I collected hourly log returns for the past 3 years and bucketed them by hour (so that I have 24 buckets with around 700 data points)

I am now trying to see if for some hours of the day, the average log return is significantly different than 0. To do that, I performed a 1sample tstat test on each of the buckets.

Are there any additional tests that I need to do to make sure the analysis is valid? (For each bucket, data looks normally distributed and there seems to be little autocorrelation)

Thanks for the help

Edit: the asset class is FX

• Do the hourly averages themselves form a normal distribution? – barrycarter Jan 26 '17 at 18:33
• Hi Barry Both the overall distribution of all hourly returns, and the 24 individual hourly buckets look normal when plotting them. However, the pvalue of a chi-squared test for all of these is equal to 0. I read somewhere though that chi-squared tests tend to reject the null hypothesis for large sample sizes, and I have over 700 data points, so not sure what to make of that. – Karimb Jan 26 '17 at 18:44

You should consider adjusting your p-values for multiplicity. Otherwise you would expect 5% of your tests to come out significant even if the null hypothesis is true (assuming you use 5% as the significance level).

Agreed with @compwarrior. The Bonferroni correction, while conservative, is a reasonable way to test multiple hypotheses. Confidence regions are (to me) a more intuitive way of evaluating plausible values, and have a direct relationship with p-values. If what follows doesn't make sense, just consider that if zero is not in your interval than your returns are significantly different from zero

Where the (one-sided interval) t-statistic to test that the returns are greater than 0 is:

$\bar{x} \pm t_{n-1}(\alpha) \sqrt{(s^2/n)}$

The Bonferroni correction for your case (since you are testing your hypothesis on 24 data sets) would be :

$\bar{x} \pm t_{n-1}(\alpha/24) \sqrt{(s^2/n)}$

Here, $\bar{x}$ are your returns, $\alpha$ is your significance level, $t_{n-1}(\alpha)$ is the t-statistic of level alpha with $n-1$ degrees of freedom, and $s$ is your sample standard deviation.

• Thanks for your answer David. A few questions: - is that equivalent to use a 0.05/24 level of confidence for the pvalue instead of the 0.05? - also, – Karimb Jan 25 '17 at 19:42
• also, just to clarify, I am indeed testing 24 hypotheses but they are somewhat unrelated: for instance, if I look at the 8am to 9am bucket, I compute log(P9am/P8am) for the past three years, and then perform the test on this bucket only to see if the average is different than 0. I then replicate to all the other buckets. Do I still need to check for multiplicity then? – Karimb Jan 25 '17 at 20:07
• Regarding your first comment, yes it's equivalent. And for your second, If I'm understanding correctly then yes. It sounds like you're testing some hypothesis (hourly log return > 0) on 24 different datasets rather than testing 24 different hypotheses. If that's the case, then that is precisely what Bonferroni's correction was designed for. – David Kozak Jan 25 '17 at 23:47

You would have to include a lot of variables here to actually isolate timing's effect on an index's returns. Given the most-influential variable on intraday returns is market news, this would be very, very difficult.

Assuming you could correct for news, remember that equity returns as a whole are leptokurtic and therefore not perfectly normally-distributed. Also, the benchmarks you select for this will have to be carefully chosen for specific reasons and compared to each other just make sure you are working with an appropriate sample to begin with. You don't mention what they are or what the asset-class is, so I thought this was worth mentioning. Volume is another huge variable you're leaving out here. Markets opening and closing will also greatly affect returns. There are other things to look for as well within the data, but the fact that it's misspecified to begin with makes it a moot point, really.

• Thks for the answer. Some clarifications: the asset class is FX, should have specified. I just want to find evidence of the tendency for some ccies to depreciate/appreciate during local/foreign hours. What I am concerned for now is the statistical validity, i.e. is it correct to look at the historical hourly returns for 1h bucket, and from a 1sample ttest, conclude that there is significant historical appreciation during that hour (because of market news or other)? Should I look at autocorrelation/normality/smth else? Your points were really helpful tho, will look into it at a second stage – Karimb Jan 25 '17 at 1:25