# Independence of observations violated

I discovered a (probably) major flaw in my research design.

I am examining investors and their investment performance. Or in other words, I want to investigate if some investors are better than others.

I collected data on particular investments of 10 different investors.

Finally I got a small dataset containing 50 investments. Now I am very concerend because of the assumption of independence of observations.

Clearly, those 50 investments aren't independent because many of them were financed by the same investor. However, isn't this exactly what I want to find out?

If I can't change my research design anymore, is there any way to deal with this issue?

Datasample: What you have there are clusters (if you use Econometrics terminology) / groups (if you use statistics terminology). So you are right that independence of observations within the same investor is a violated assumption. But there are techniques to cope with this problem by regarding your data as haing two dimensions: The investor (say $i$) and the number/identifier of $i$'s investment (say $j$, and say investor $i$ has $n_i$ investments in total). Your assumption then changes from saying 'We have variables $X_1, X_2, ... X_{50}$ which are independent' to 'we have variables $X_{1,1}, X_{1,2}, ..., X_{1,n_1}$ for the first investor, variables $X_{2,1}, X_{2,2}, ..., X_{2,n_2}$ for the second investor, ... and we have independence between investments/variables for which the investors are not the same ones. (I.e., $X_{l,p}$ and $X_{t,z}$ are independent exactly when $l \neq z$.) The techniques you need to analyze this have different names in different disciplines. In Econometrics, you will find them under the header 'panel methods'.

• thx for the great answer! I already included something like firm fixed effects by creating dummies indicating the involvement of every investor. You think that might be a proper way to deal with the situation? Jun 20, 2016 at 9:54
• If you include fixed effects, I take it you have additional measurements/regressors/explanatory variables for each tuple $(i,j)$ that you wish to hold constant? Fixed Effects are neat if (1) all relevant regressors that you want to control for are varying with the transaction (so something like 'Gender' would not work) and (2) if you believe that the slope/effect of the additional regressors/explanatory variables remains the same across different investors. Implicitly, this means that there must not be any interaction effects between the individual investor and the additional explanatory vars. Jun 20, 2016 at 11:22
• I have many other predictors (I use a lasso penalty for variable selection). But most of them are not very varying within groups (often those variables are essentially the same). Those are linguistic measures and often the measures don't change from oneobservation to the other. Jun 20, 2016 at 11:40
• Moreover, it might be important to know that my dataset is cross-sectional. Jun 20, 2016 at 11:53
• Well, if the variables that are not time varying are important for explaining what happens, you might consider including a dummy or investor as you did. Note though that this causes instability in your estimations if your sample is small. If you're familiar with asymptotic theory, it is easy to see that including an additional dummy for each investor $i$ is inconsistent unless the number of investors stays fixed while the number of investments per investor ($n_i$) goes to infinity. (Known as incidental parameter problem.) Jun 21, 2016 at 8:50

Let $s_{i}$ be an indicator as to whether investment $i$ was successful. If you're estimating the probability of a successful investment, that is, trying to estimate $E[s_i]$, that is equivalent to running a regression:

$$s_{i} = a + \epsilon_i$$

If you're concerned that the $\epsilon_i$ terms will be correlated for each investor, you can cluster standard errors at the investor level. When constructing your standard errors, loosely what will happen is that you will treat investors as independent observations rather than each investment.

## Evidence of investor skill...

This gets tricky. What can often be a challenge is constructing the null hypothesis of what would happen by chance. There would have to be more details about your exact setting to think about what makes sense...