# Hypothesis testing for two data series

I am currently pursuing research in management, but I have a serious problem with selecting the right statistical method.

I have quarterly data for a couple of financial ratios (for example Return on Equity) for two groups of companies: those with CEOs with MBAs and those without MBAs. I want to test whether the two groups are statistically different.

The first test which came to my mind is the student's t-test. My only concern is that this is a data series, so only the results from each quarter would be compared, and hence a t-test is not suitable.

What would be the most appropriate test for this situation?

• When you say "data series" do you mean "time series"? – Arthur Small Jan 7 '13 at 4:44
• And how long are the series? Are you interested just in their average value, or whether the direction of change over time is different for the two groups (eg those with CEOs with MBAs having return on equity that grows over time, even if in absolute terms it is lower). If so, you are in a much more complex world of time series analysis. – Peter Ellis Jan 7 '13 at 6:29

Let's suppose your data set includes $m$ firms led by MBAs, and $n$ firms led by non-MBAs, for a total of $m+n$ firms in all. Suppose you label your firms in order by some index variable $j$: let $j = 1, 2, \ldots, m$ index the firms in your first group, those led by MBAs, and let $j = m+1, m+2, \ldots, m+n$ index the remaining firms.

Suppose your data set includes data from $T$ distinct reporting periods (e.g., quarters). Let time periods be indexed by $t$, where $t = 1, \ldots, T$. Let $x_{j,t}$ denote the measured financial performance of firm $j$ during period $t$. From these numbers you want to extract some understanding of the "true" financial prowess of the firms, and to devine whether this true capacity differs in a systematic way between the subgroups.

How you address those questions depends on what you believe about the data-generating process. We might suppose that the data are generated in a fairly simple way, as a sum of three components --- intrinsic ability + macroeconomic conditions + idiosyncratic random shocks. Intrinsic ability may in turn incorporate the effect of an "MBA premium" (which could in principle be negative):

$x_{j,t} = \mu + \beta\cdot\chi(j) + \nu_j + Z_t + e_{j,t}$.

Here $\mu$ denotes the average performance of all non-MBA firms, averaged across all firms in all periods; $\beta$ is the size of the MBA premium (what you're after); $\chi(j)$ is an indicator function for the MBA-led firms, i.e, a function that takes the value $1$ if $j$ denotes and MBA-led firm, and zero otherwise; $\nu_j$ measures the firm-specific ability; $Z_t$ denotes the effect of macroeconomic conditions (varying across time, but affect all firms identically); and $e_{j,t}$ the random shock.

You want to know whether $\beta$ is non-zero.

You could estimate $\beta$ be computing the difference between the average performance of the MBA-led firms vs. non-MBA firms in each period, and then averaging across all periods. A nice thing happens: since you are taking differences in this way, the $Z_t$ terms, representing macroeconomic effects, drop out of the equation.

If you believe that the above model provides a reasonable representation of the data-generating process, and if you further believe that the error terms $e_{j,t}$ are independent and identically normally distributed, then you may test your hypothesis using a t-test.

There are many reasons to believe that this model does not, in fact, fully capture the data-generating process. Refinements of the above model would involving delving into more advanced statistical procedures, to handle issues such as autocorrelated or heteroskedastic errors, multiplicative macroeconomic effects, non-normal error processes, etc., etc.

There is not likely to be a definitive, clear answer to how far in this process you might go. Every model is an approximation, not Truth. I suggest trying the simple model, while testing for the sensitivity of your answers to its several simplifying assumptions.