Hi, I have been trying to look into serial autocorrelation in stock prices. The tests and methods I have been able to read so far are the Durbin-Watson test and the Portmanteau test(Chi square analysis). The biggest problem I have is that these are valid for linear regression models and serial autocorrelation is defined as the correlation between the residuals of the First regression between the stock prices series. An expansion of the equations looks like thisY(t)= a + b.X(t) + residuals1(t), where:residuals1(t)=Rho*Residuals(t-1)+residuals2The second equation can be treated as a new regression system with intercept of 0 and hence can be solved on for the Autocorrelation(Rho). This is where I start pulling put my hair, because the second residuals are very high in magnitude and cloud the whole meaning of trying to do a linear fit across the first residuals.For example, performing the above test for stock prices of SPX on a 30 day window, with a lag of 1 day, I get an autocorrelation of -5%. The first residuals range between 0.1% to 0.8% of the stock prices but the second residuals are almost 1%-80% of the second residual values which is obscene to admist as a good analysis.Kindly suggest what approach could be used to analyse this autocorrelation? Also, in one of the forums I read that this factor is used to correct volatility by sqrt(1-rho^2), which vol is being talked about here? If its implied vol, how do you decide upon maturity (assuming the correction factor applies across all strikes)Thanks

QuoteOriginally posted by: surideepakHi, I have been trying to look into serial autocorrelation in stock prices. The tests and methods I have been able to read so far are the Durbin-Watson test and the Portmanteau test(Chi square analysis). The biggest problem I have is that these are valid for linear regression models and serial autocorrelation is defined as the correlation between the residuals of the First regression between the stock prices series. An expansion of the equations looks like thisY(t)= a + b.X(t) + residuals1(t), where:residuals1(t)=Rho*Residuals(t-1)+residuals2The second equation can be treated as a new regression system with intercept of 0 and hence can be solved on for the Autocorrelation(Rho). This is where I start pulling put my hair, because the second residuals are very high in magnitude and cloud the whole meaning of trying to do a linear fit across the first residuals.For example, performing the above test for stock prices of SPX on a 30 day window, with a lag of 1 day, I get an autocorrelation of -5%. The first residuals range between 0.1% to 0.8% of the stock prices but the second residuals are almost 1%-80% of the second residual values which is obscene to admist as a good analysis.Kindly suggest what approach could be used to analyse this autocorrelation? Also, in one of the forums I read that this factor is used to correct volatility by sqrt(1-rho^2), which vol is being talked about here? If its implied vol, how do you decide upon maturity (assuming the correction factor applies across all strikes)Thanks(1) I'm not really sure what you're talking about regarding 'correlation between the residuals of the first regression...', so let's start here: - Correlation between series X and Y in general is defined as the covariance of X and Y over the vol of X time the vol of Y: rho = cov(X,Y) / std(x)*std(y), assuming X and Y are stationary - Autocorrelation for a given lag, k, is merely X as the time series itself minus the last k days of the series and Y as X lagged k days. - With X and Y, the calculation for rho is straightforward. - You can then use any test for validity you'd like (Durbin-Watson, Ljung-Box, etc.) to assess significance.(2) You can't calculate correlation, or autocorrelation, of variables that aren't stationary (ie, use returns, not prices)I'm not really sure what you're doing, but I'd wager the bulk of your problem lies in you using S&P prices as opposed to returns.

Thanks for the reply CTD....I have performed the test with both Returns and prices. The magnitude of error in both cases is the same. As for the correlation formula. it is perfect as you have defined, but for serial autocorrelation we define it as the correlation between the residual terms of the regression between Y and X. As an example, if I take the correlation of the returns of the stock WAT UN (waters corp), on a weekly return basis, with a lag of 7 days, I get the correlation as you define to be +4% (on a 100 week basis) whereas the autocorrelation comes out to be -30%. Now the Durbin watson test is performed to check if the correlation is serial. you can refer to the paper on this link:http://www.fordham.edu/iped/ecga5510/serial.pdfKindly let me know if you approve of it. If yes, my doubt regarding the magnitude of error in the second regression still stands bothering me.Thanks Deepak

Dealing with autocorrelation you have to use approaches such as Cochrane-Orcutt and others. But this implies that rho is known which it isn't and second you have to make several assumptions of the autocorrelation structure. A simple rho calculation would be to use the rho obtained from the DW statistic, remembers this is only an approximation. Finally when examining equity you should use returns and not prices, and preferably a lagged return term which cures the autocorrelation seen in your series. You may add as many lagged terms you want to solve the autocorrelation problem but this creates in turn an intepretational problem.

So, assuming I accept that the autocorrelation estimate is correct (first order only, so the structure is fine); I can easily adjust the volatility predictions based on historical data.What do I do then, if I got more then one asset; can I assume that the correlations estimate is fixed, and then use the new volatilties, and change the covariance estimates accordingly?