# How do I reproduce the cross-sectional regression in “Intraday Patterns in the Cross-section of Stock Returns”?

Recently I was trying to reproduce the results of "Intraday Patterns in the Cross-section of Stock Returns". The authors used cross-sectional regression to determine which intraday lags have predictive power.

From my understanding when doing cross-sectional regression all variables have to be from the same time period. For example, I can take the one day returns of all stocks and regress them against the number of employees in each company.

The following is a short description of how cross-sectional regression was used in the research:

For each lag, $k$, we run cross-sectional regressions of half-hour stock returns on returns lagged by $k$ half-hour periods, $$r_{i,t}=\alpha_{k,t}+\gamma_{k,t}r_{i,t-k}+u_{i,t},$$ where $r_{i,t}$ is the return on stock i in the half-hour interval $t$. The slope coefficients $\gamma_{k,t}$ represent the response of returns at half-hour $t$ to returns over a previous interval lagged by $k$ half-hour periods, the “return responses.”

If I understood well, the returns of one period were regressed against the returns of another period and a slope was obtained from each regression. Later, autocorrelation analysis has been done on the slopes.

Unless my thoughts are wrong, I don't see the point of regressing the returns of one period against another - $R^2$ values are close to zero. Here is an example:

Did I get the cross-sectional regression wrong? By the way, I was working with a relatively small number of stocks, but I thought that 38 should be enough.

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This is a great paper and potentially very relevant to all transaction-cost-sensitive equity traders. This question deserved some clean-up and improved formatting. – Tal Fishman Sep 26 '11 at 21:11

The $R^2$s are usually close to zero for single stock regressions. The big $R^2$s that a lot of asset pricing research shows is by forming portfolios. Forming portfolios cancels a lot of the idiosyncratic returns, which has a smoothing effect.

The $R^2$s should be low here, although I don't see any in the paper for you to compare. This probably means they are very low. We don't expect that the lagged 30-min return should predict much of the future 30-min return. Maybe one percent or two? Heston et al's point is that they are correlated. We can't expect that the lagged return will tell us exactly the future return (i.e., $R^2 = 100\%$), but the two are correlated.

They run the regression $$r_{i,t} = \alpha_{k,t} + \gamma_{k,t} r_{i,t-k} + u_{i,t}$$ which finds the autocorrelation between $r_{i,t}$ and its $k^{th}$ lag $r_{i,t-k}$. This seems endogenous to me because there's an omitted variable that drives both the return and it's lag (maybe some news?). This should bias their $\gamma$, but I don't know a priori which way. When Jegadeesh (1990) looked at short-term reversals he did some other tricks to get around it, but I don't see that here. Heston et al are very well respected, so there's likely something I'm missing and I'm not very familiar with the intr-day literature, although I didn't see any discussion of this in the paper. I am interested to see this in it's peer-reviewed and published form.

Regardless, their $\gamma$s aren't autocorrelated, so they should be fine using the Fama-MacBeth approach. Oh, and that you're only doing 38 stocks shouldn't affect your $R^2$, just the t-statistics on your $\gamma$s.

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I should say, that I ran regression on 38 stocks - I took returns of 38 stocks at t(k) and regressed them against t(k-n) to obtain the slope. – Dzidas Mar 13 '11 at 9:25
I seems, that I misinterpreted the way the research has been conducted - it is difficult to reveal all "tricks". Thank you for the answer. – Dzidas Mar 13 '11 at 9:35
@Dzidas -- That is how I interpreted that you did the regressions and it's correct. You can get significant regressors (i.e., high t-stats on $r_{i,t-k}$) without having a very explanatory model (i.e., high $R^2$). The $r_{i,t-k}$ can be significantly correlated with $r_{i,t}$ but still only explain a very small fraction of the $r_{i,t}$ we see. Adding more stocks won't change the explanatory power of the model, but it will allow for better $\gamma$ estimates. – Richard Herron Mar 14 '11 at 11:14
It was published in the Journal of Finance 2010 Vol 65 Issue 4. afajof.org/journal/…. – Louis Marascio Sep 13 '11 at 11:25
While it seems plausible that news or some other omitted variable may be driving consecutive returns (which is the focus of Jegadeesh (1990)), their main results concern lags of multiples of 13 (corresponding to one day). Furthermore, they are, in a sense, trying to measure the impact of this omitted variable (which they speculate is institutional flows), rather than some true lagged effect between returns at the same time of day. – Tal Fishman Sep 26 '11 at 21:20