Let $R_{it}$ and $R_{mt}$ be daily stock returns for some company $i$ and the daily market index returns (respectively), with $i \in \{1,...,N\}$ and $t \in \{1,...,200\}$. It is common to have $R_{it}$ as the response and $R_{mt}$ as the predictor.

Let a severe financial crisis occur at $t=150$ and this crisis lasts until $t=200$. To model whether the stock fell more than the market index, we estimate with OLS $\forall i$:

$R_{it} = \alpha_i + \beta_i R_{mt} + \gamma_i D_t + \epsilon_{it}$

where $D_t$ equals 1 over the crisis and 0 otherwise.

What I want to know, broadly, is "Which stocks were the fastest to fall?". I have ideas for sub-sample analysis that can tackle this question, but I would prefer an econometric approach.

One way to do this would be to truncate the sample to $t \in \{1,...,160\}$ and look for the most negative $\gamma_i$'s. However, I'm looking for an econometric result that will identify the stocks that (i) fell over the entire crisis period, but (ii) had a very large proportion of their fall in the first week or so of the crisis. I also want this paramaterized so that I can use it as a dependent variable in a cross sectional regression analysis to explain this phenomena.

  • 1
    $\begingroup$ have you checked this paper? $\endgroup$
    – user603
    Aug 21, 2012 at 14:10

1 Answer 1


Two ideas I can think of are 1) create a second indicator that is true for $t\in\{151,\ldots,160\}$ or 2) create a single predictor (discard your original indicator) that decays towards 0; e.g.

$ X(t) = \begin{cases} 0, & t \in \{1,\ldots,150\} \\ 1 - e^{-(T-t)/T}, & t \in \{151,\ldots,T\} \end{cases} $

You might want to scale the exponent.

If you really believe that all stocks are affected after the shock, but that some absorb this affect more rapidly go with 1). You can a) test if the new indicator is significantly negative and b) see if the original indicator now has a smaller coefficient (presumably the affect has transfered).

A lot of times in finance however the shock is like an impulse response so option 2) might more accurately describe your problem.

  • $\begingroup$ Great answer. If I go with (2), I'm worried that it doesn't actually address "Which stocks fall first?". For example the estimated coefficient will be similarly large for (i) Stocks that fell early and didn't fall that much later on, (ii) Stocks that didn't actually fall early, but fell very hard later (and they fell sufficiently at the later stage to give a similarly large coefficient estimate). What are your thoughts on this perceived complication with using (2)? $\endgroup$
    – user13253
    Aug 21, 2012 at 14:16
  • $\begingroup$ What about, instead of either options (1) OR (2), we use a combination of both, and instead of arbitrarily defining the shorter dummy, we use a variable similar to what you've specified in (2) yet decays more rapidly towards zero? $\endgroup$
    – user13253
    Aug 21, 2012 at 14:27
  • $\begingroup$ If what you describe is a genuine concern I agree with you, and you are better of using a second indicator. 2) is saying all stocks are affected by shocks but some more so than others. If you go with 1) I think you want to compare the coeffs on your original indicator between the two groups ($\beta_{shortterm} < 0$ vs $\beta_{shortterm} = 0$), to see if stocks vary only by time or also by magnitude. $\endgroup$
    – muratoa
    Aug 21, 2012 at 14:28
  • $\begingroup$ What's $\beta_{shortterm}$ ? $\endgroup$
    – user13253
    Aug 21, 2012 at 14:32
  • $\begingroup$ Combining (1) and (2) like you suggest can work. I guess your original indicator test if stocks have a fixed change after a shock (which can vary across stocks). My (2) tests if stocks have a decayed change after a shock (can also vary across stocks). My (1) tests if price changes depend on the time interval after the shock. $\beta_{shortterm}$ would be the coef you get for the indicator that is true for $t\in\{151,\ldots,160\}$. $\endgroup$
    – muratoa
    Aug 21, 2012 at 14:33

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