Testing Whether Two Return Series Have Different Mean Returns
Summary
The document explains how to test whether two return series differ on average by forming their observation-by-observation differences and testing whether the mean difference is zero. This is equivalent to a one-sample test on the difference series. The intercept estimates that average difference, while its t-statistic evaluates whether the estimate is distinguishable from zero. A positive estimate favors the first series; a negative estimate favors the second.
The example uses heteroskedasticity and autocorrelation consistent standard errors to allow serial dependence in the differences. Its reported estimate is negative, but the p-value does not provide evidence of a statistically significant difference. The answer relates the approach to a Diebold–Mariano test. The interpretation depends on correctly aligning the observations and choosing suitable standard-error settings; HAC errors address dependence and heteroskedasticity, but do not by themselves establish that either series is a better trading strategy or resolve every issue raised by non-normal returns.
Key ideas
- Testing the mean of paired return differences assesses whether the series have equal average returns.
- The regression intercept is the estimated mean difference between the two series.
- A negative difference estimate favors the second series, while statistical significance indicates evidence against a zero mean difference.
- HAC standard errors can account for autocorrelation and heteroskedasticity in the differences.
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Full text
# Determining significant difference between to return series
# Determining significant difference between to return series
I want to analyse whether two return series are different. I was told to run the following regression:
```
diff = return series 1 - return series 2
constant = beta * diff
```
Where I set the constant equal to 1. I use a t-test to evaluate this regression with HAC standard errors. In R this would look like this:
```
diff <- as.numeric(series1[startRow:endRow]) - as.numeric(series2[startRow:endRow])
reg <- lm (formula = diff ~ 1, na.action = na.omit)
coeftest(reg, vcov=NeweyWest(reg, lag = 1, prewhite=FALSE), df=length(diff)-1)
```
The results are as follows:
```
t test of coefficients:
Estimate | Std. Error | t value | Pr(>|t|)
(Intercept) -8.7425e-05 | 9.3240e-05 | -0.9376 | 0.3485
```
However, I do not know how to interpret the results. What does the constant do? What does the beta mean? What does it mean when the results are significant or insignificant?
(Problems with my sample: return series are dependent & individual return series are non-normal)
## Answer by nbbo2 (score 0, accepted)
https://quant.stackexchange.com/a/19280
It is similar to a Diebold and Mariano test. It tests whether series1 minus series2 is positive or negative, while taking into account the possibility there is autocorrelation. If you had normal i.i.d data you could just look at series1-series2 and do a Student t-test as to whether the differences are on average zero or not. This is a fancy way of doing it, more general because of the HAC methodology. In your case series2 is better than series1 [i.e. on average series2[i] > series1[i] for all i] but not significantly so.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.