AutocorrelatedRegression to Determine the Coefficient a1 in the Blundell/Ward Filtera1a0Coefficient0.09240.0062Std. Err. of Coef.0.05980.002538S&P500 TRR^20.85%0.0406Std. Err. of Est.F2.3836277Nobs95.0%SSreg0.00390.4560SSrest-values1.54392.5333Critical Value at 95.0%4.30271.9686Statistical Signifiance at 95.0%0.00000.0000Confidence Level99.6%Mean0.69%Ann Vol0.68%0.68%St Dev4.06%14.07%4.45%15.43%4.42%15.31%- Approximation as per Carol AlexanderSkewness-0.66-0.554.42%Kurtosis1.701.53VaR (Normal)-10.08%-11.13%-11.04%MaxDD (Normal)-42.31%-51.00%-50.20%T*61.7574.53MaxTuW (Normal)247.00298.12CoeffStd ErrF TestPennance3.003.000.09236737690.059832.3840.092370.059832.384Actual Max DD-50.95%S&P500 TRLagged SeriesFiltered SeriesS&P500 TR Mean 0.69% Std Dev 4.06% Normal VaR -10.08% MaxDD -42.31%Lagged SeriesFiltered Series Mean 0.68% Std Dev 4.45% Normal VaR -11.13% MaxDD -51.00%131-Jan-900.00%0.00%228-Feb-900.00%0.00%0.00%331-Mar-900.00%0.00%0.00%430-Apr-900.00%0.00%0.00%531-May-900.00%0.00%0.00%630-Jun-900.00%0.00%0.00%731-Jul-900.00%0.00%0.00%831-Aug-900.00%0.00%0.00%930-Sep-900.00%0.00%0.00%1031-Oct-900.00%0.00%0.00%1130-Nov-900.00%0.00%0.00%1231-Dec-900.00%0.00%0.00%1331-Jan-910.00%0.00%0.00%1428-Feb-910.00%0.00%0.00%1531-Mar-910.00%0.00%0.00%1630-Apr-910.00%0.00%0.00%1731-May-910.00%0.00%0.00%1830-Jun-910.00%0.00%0.00%1931-Jul-910.00%0.00%0.00%2031-Aug-910.00%0.00%0.00%2130-Sep-910.00%0.00%0.00%2231-Oct-910.00%0.00%0.00%2330-Nov-910.00%0.00%0.00%2431-Dec-910.00%0.00%0.00%2531-Jan-920.00%0.00%0.00%2629-Feb-920.00%0.00%0.00%2731-Mar-920.00%0.00%0.00%2830-Apr-920.00%0.00%0.00%2931-May-920.00%0.00%0.00%3030-Jun-920.00%0.00%0.00%3131-Jul-920.00%0.00%0.00%3231-Aug-920.00%0.00%0.00%3330-Sep-920.00%0.00%0.00%3431-Oct-920.00%0.00%0.00%3530-Nov-920.00%0.00%0.00%3631-Dec-920.00%0.00%0.00%3731-Jan-930.84%0.00%0.92%3828-Feb-931.36%0.84%1.42%3931-Mar-932.11%1.36%2.19%4030-Apr-93-2.42%2.11%-2.88%4131-May-932.67%-2.42%3.19%4230-Jun-930.29%2.67%0.05%4331-Jul-93-0.40%0.29%-0.47%4431-Aug-933.79%-0.40%4.22%4530-Sep-93-0.77%3.79%-1.23%4631-Oct-932.07%-0.77%2.36%4730-Nov-93-0.95%2.07%-1.26%4831-Dec-931.21%-0.95%1.43%4931-Jan-943.40%1.21%3.62%5028-Feb-94-2.71%3.40%-3.34%5131-Mar-94-4.36%-2.71%-4.53%5230-Apr-941.28%-4.36%1.86%5331-May-941.64%1.28%1.68%5430-Jun-94-2.45%1.64%-2.87%5531-Jul-943.28%-2.45%3.87%5631-Aug-944.10%3.28%4.18%5730-Sep-94-2.45%4.10%-3.11%5831-Oct-942.25%-2.45%2.72%5930-Nov-94-3.64%2.25%-4.24%6031-Dec-941.48%-3.64%2.00%6131-Jan-952.59%1.48%2.71%6228-Feb-953.90%2.59%4.03%6331-Mar-952.95%3.90%2.85%6430-Apr-952.94%2.95%2.94%6531-May-954.00%2.94%4.10%6630-Jun-952.32%4.00%2.15%6731-Jul-953.32%2.32%3.42%6831-Aug-950.25%3.32%-0.06%6930-Sep-954.22%0.25%4.62%7031-Oct-95-0.36%4.22%-0.82%7130-Nov-954.39%-0.36%4.87%7231-Dec-951.93%4.39%1.68%7331-Jan-963.40%1.93%3.55%7429-Feb-960.93%3.40%0.68%7531-Mar-960.96%0.93%0.97%7630-Apr-961.47%0.96%1.53%7731-May-962.58%1.47%2.69%7830-Jun-960.38%2.58%0.16%7931-Jul-96-4.42%0.38%-4.91%8031-Aug-962.11%-4.42%2.77%8130-Sep-965.63%2.11%5.99%8231-Oct-962.76%5.63%2.47%8330-Nov-967.56%2.76%8.05%8431-Dec-96-1.98%7.56%-2.95%8531-Jan-976.25%-1.98%7.08%8628-Feb-970.78%6.25%0.23%8731-Mar-97-4.11%0.78%-4.61%8830-Apr-975.97%-4.11%7.00%8931-May-976.09%5.97%6.10%9030-Jun-974.48%6.09%4.32%9131-Jul-977.96%4.48%8.31%9231-Aug-97-5.60%7.96%-6.98%9330-Sep-975.48%-5.60%6.60%9431-Oct-97-3.34%5.48%-4.24%9530-Nov-974.63%-3.34%5.44%9631-Dec-971.72%4.63%1.42%9731-Jan-981.11%1.72%1.04%9828-Feb-987.21%1.11%7.83%9931-Mar-985.12%7.21%4.91%10030-Apr-981.01%5.12%0.59%10131-May-98-1.72%1.01%-2.00%10230-Jun-984.06%-1.72%4.65%10331-Jul-98-1.06%4.06%-1.59%10431-Aug-98-14.46%-1.06%-15.82%10530-Sep-986.41%-14.46%8.53%10631-Oct-988.13%6.41%8.31%10730-Nov-986.06%8.13%5.85%10831-Dec-985.76%6.06%5.73%10931-Jan-994.18%5.76%4.02%11028-Feb-99-3.11%4.18%-3.85%11131-Mar-994.00%-3.11%4.72%11230-Apr-993.87%4.00%3.86%11331-May-99-2.36%3.87%-3.00%11430-Jun-995.55%-2.36%6.35%11531-Jul-99-3.12%5.55%-4.01%11631-Aug-99-0.49%-3.12%-0.23%11730-Sep-99-2.74%-0.49%-2.97%11831-Oct-996.33%-2.74%7.25%11930-Nov-992.03%6.33%1.60%12031-Dec-995.89%2.03%6.28%12131-Jan-00-5.02%5.89%-6.13%12229-Feb-00-1.89%-5.02%-1.57%12331-Mar-009.78%-1.89%10.97%12430-Apr-00-3.01%9.78%-4.31%12531-May-00-2.05%-3.01%-1.95%12630-Jun-002.47%-2.05%2.93%12731-Jul-00-1.56%2.47%-1.97%12831-Aug-006.21%-1.56%7.00%12930-Sep-00-5.28%6.21%-6.45%13031-Oct-00-0.42%-5.28%0.07%13130-Nov-00-7.88%-0.42%-8.64%13231-Dec-000.49%-7.88%1.34%13331-Jan-013.55%0.49%3.86%13428-Feb-01-9.12%3.55%-10.41%13531-Mar-01-6.34%-9.12%-6.05%13630-Apr-017.77%-6.34%9.21%13731-May-010.67%7.77%-0.05%13830-Jun-01-2.43%0.67%-2.75%13931-Jul-01-0.98%-2.43%-0.84%14031-Aug-01-6.26%-0.98%-6.80%14130-Sep-01-8.08%-6.26%-8.26%14231-Oct-011.91%-8.08%2.92%14330-Nov-017.67%1.91%8.26%14431-Dec-010.88%7.67%0.18%14531-Jan-02-1.46%0.88%-1.70%14628-Feb-02-1.93%-1.46%-1.98%14731-Mar-023.76%-1.93%4.34%14830-Apr-02-6.06%3.76%-7.06%14931-May-02-0.74%-6.06%-0.19%15030-Jun-02-7.12%-0.74%-7.77%15131-Jul-02-7.80%-7.12%-7.86%15231-Aug-020.66%-7.80%1.52%15330-Sep-02-10.87%0.66%-12.04%15431-Oct-028.80%-10.87%10.80%15530-Nov-025.89%8.80%5.59%15631-Dec-02-5.87%5.89%-7.07%15731-Jan-03-2.62%-5.87%-2.29%15828-Feb-03-1.50%-2.62%-1.39%15931-Mar-030.97%-1.50%1.22%16030-Apr-038.24%0.97%8.98%16131-May-035.27%8.24%4.97%16230-Jun-031.28%5.27%0.87%16331-Jul-031.76%1.28%1.81%16431-Aug-031.95%1.76%1.97%16530-Sep-03-1.06%1.95%-1.37%16631-Oct-035.66%-1.06%6.34%16730-Nov-030.88%5.66%0.39%16831-Dec-035.24%0.88%5.69%16931-Jan-041.84%5.24%1.49%17029-Feb-041.39%1.84%1.34%17131-Mar-04-1.51%1.39%-1.80%17230-Apr-04-1.57%-1.51%-1.58%17331-May-041.37%-1.57%1.67%17430-Jun-041.94%1.37%2.00%17531-Jul-04-3.31%1.94%-3.84%17631-Aug-040.40%-3.31%0.78%17730-Sep-041.08%0.40%1.15%17831-Oct-041.53%1.08%1.57%17930-Nov-044.05%1.53%4.30%18031-Dec-043.40%4.05%3.34%18131-Jan-05-2.44%3.40%-3.03%18228-Feb-052.10%-2.44%2.57%18331-Mar-05-1.77%2.10%-2.17%18430-Apr-05-1.90%-1.77%-1.91%18531-May-053.18%-1.90%3.70%18630-Jun-050.14%3.18%-0.17%18731-Jul-053.72%0.14%4.08%18831-Aug-05-0.91%3.72%-1.38%18930-Sep-050.81%-0.91%0.99%19031-Oct-05-1.67%0.81%-1.92%19130-Nov-053.78%-1.67%4.34%19231-Dec-050.03%3.78%-0.35%19331-Jan-062.65%0.03%2.91%19428-Feb-060.27%2.65%0.03%19531-Mar-061.24%0.27%1.34%19630-Apr-061.34%1.24%1.35%19731-May-06-2.88%1.34%-3.31%19830-Jun-060.14%-2.88%0.44%19931-Jul-060.62%0.14%0.67%20031-Aug-062.38%0.62%2.56%20130-Sep-062.58%2.38%2.60%20231-Oct-063.26%2.58%3.33%20330-Nov-061.90%3.26%1.76%20431-Dec-061.40%1.90%1.35%20531-Jan-071.51%1.40%1.52%20628-Feb-07-1.96%1.51%-2.31%20731-Mar-071.12%-1.96%1.43%20830-Apr-074.43%1.12%4.77%20931-May-073.49%4.43%3.39%21030-Jun-07-1.66%3.49%-2.19%21131-Jul-07-3.10%-1.66%-3.25%21231-Aug-071.50%-3.10%1.97%21330-Sep-073.74%1.50%3.97%21431-Oct-071.59%3.74%1.37%21530-Nov-07-4.18%1.59%-4.77%21631-Dec-07-0.69%-4.18%-0.34%21731-Jan-08-6.00%-0.69%-6.54%21829-Feb-08-3.25%-6.00%-2.97%21931-Mar-08-0.43%-3.25%-0.15%22030-Apr-084.87%-0.43%5.41%22131-May-081.30%4.87%0.93%22230-Jun-08-8.43%1.30%-9.42%22331-Jul-08-0.84%-8.43%-0.07%22431-Aug-081.45%-0.84%1.68%22530-Sep-08-8.91%1.45%-9.96%22631-Oct-08-16.79%-8.91%-17.60%22730-Nov-08-7.18%-16.79%-6.20%22831-Dec-081.06%-7.18%1.90%22931-Jan-09-8.43%1.06%-9.39%23028-Feb-09-10.65%-8.43%-10.87%23131-Mar-098.76%-10.65%10.73%23230-Apr-099.57%8.76%9.65%23331-May-095.59%9.57%5.19%23430-Jun-090.20%5.59%-0.35%23531-Jul-097.56%0.20%8.31%23631-Aug-093.61%7.56%3.21%23730-Sep-093.73%3.61%3.74%23831-Oct-09-1.86%3.73%-2.43%23930-Nov-096.00%-1.86%6.80%24031-Dec-091.93%6.00%1.52%24131-Jan-10-3.60%1.93%-4.16%24228-Feb-103.10%-3.60%3.78%24331-Mar-106.03%3.10%6.33%24430-Apr-101.58%6.03%1.13%24531-May-10-7.99%1.58%-8.96%24630-Jun-10-5.23%-7.99%-4.95%24731-Jul-107.01%-5.23%8.25%24831-Aug-10-4.51%7.01%-5.69%24930-Sep-108.92%-4.51%10.29%25031-Oct-103.80%8.92%3.28%25130-Nov-100.01%3.80%-0.37%25231-Dec-106.68%0.01%7.36%25331-Jan-112.37%6.68%1.93%25428-Feb-113.43%2.37%3.53%25531-Mar-110.04%3.43%-0.30%25630-Apr-112.96%0.04%3.26%25731-May-11-1.13%2.96%-1.55%25830-Jun-11-1.67%-1.13%-1.72%25931-Jul-11-2.03%-1.67%-2.07%26031-Aug-11-5.43%-2.03%-5.78%26130-Sep-11-7.03%-5.43%-7.19%26231-Oct-1110.93%-7.03%12.76%26330-Nov-11-0.22%10.93%-1.36%26431-Dec-111.02%-0.22%1.15%26531-Jan-124.48%1.02%4.83%26629-Feb-124.32%4.48%4.31%26731-Mar-123.29%4.32%3.19%26830-Apr-12-0.63%3.29%-1.03%26931-May-12-6.01%-0.63%-6.56%27030-Jun-124.12%-6.01%5.15%27131-Jul-121.39%4.12%1.11%27231-Aug-122.25%1.39%2.34%27330-Sep-122.58%2.25%2.62%27431-Oct-12-1.85%2.58%-2.30%27530-Nov-120.58%-1.85%0.83%27631-Dec-120.91%0.58%0.95%27731-Jan-135.18%0.91%5.61%27828-Feb-131.36%5.18%0.97%27931-Mar-133.75%1.36%3.99%28030-Apr-130.00%3.75%-0.64%2810.00%

is there a way to calculate autocorrelation of a return series in excel?

Why don't you just compute the sums? you would sum the product of your (X_i-Xbar) and (X_i+j - Xbar) terms and divide by the squares of differeneces from the means. That is 2 columns.

if your time descending series is in cells [A1:An] and you want lag = 1 then formula [B1] = [A2] and copy formulas down. compute correl(A:B). Note that as n-> inf. then correl. est. is normally distributed with SD = 1/N^0.5, i.e. you can calculte p-values easily.

The autocorrelation function is just the correlation of a series value with a lagged series value.

The partial autocorrelation function is the same thing, except that it is adjusted for the effect of all lower-order correlations.

For example, suppose I have a series with a first-order autocorrelation of 0.6. That is, the correlation of today's and tomorrow's value is 0.6. The first order partial autocorrelation is also 0.6 because there is no lower order to adjust for.

Suppose that the second-order autocorrelation is 0.24. That is the correlation between yesterday's and tomorrow's value is 0.24. This is actually less than the 0.36 you would expect, given the 0.6 first-order autocorrelation. So the second-order partial autocorrelation is -0.75. Yesterday is not negatively correlated with tomorrow (that is still a positive 0.24 correlation), but is is much less correlated with tomorrow than you would expect given its correlation with today and today's correlation with tomorrow.

Let p1 be the first-order autocorrelation (also the first-order partial autocorrelation). Let p2 be the second-order autocorrelation. The second-order partial autocorrelation is:

(p2 - p1^2)/(1 - p1^2)

In the example, that's:

(0.24 - 0.6^2)/(1 - 0.6^2)

= -0.12 / 0.64

= -0.1875

I did the example in my head and took (1 - 0.6)^2 as the denominator instead of (1 - 0.6^2).

Autocorrelation is as follows: positive autocorrelation means an up-tick in an asset (price) is more likely to be followed by an up-tick (same with down-ticks); negative autocorrleation means an up-tick is more likely to be followed by a down-tick (and vice-versa). Positive autocorrelation is a trend, negative autocorelation is mean-reversion.

Imagine you wondered whether, if the S&P went up one day, does that imply anything about whether it'll go up the next day.

That is, is there correlation between return on day i and return on day (i+1)?

That's autocorrelation.

You calculate it like you'd calculate any other correlation; for a series of N returns you'd generate a number like

C = Sum[ r(i) * r(i+1), { i, 1, N-1 } ] / variance( r ) / (N-1)

where r(i) = i'th return, and I'm assume that the variance of r(1->N-1) = variance of r(2->N).

As with all correlation estimates based on historical data, keep in mind that the statistical uncertainty is pretty high. You can estimate this using the Fisher transform; the transformed correlation is roughly normally distributed with a well-defined mean and variance.

Positive autocorrelation of period returns is present if a positive (negative) return in one period is followed by another positive (negative) return the next period. Negative autocorrelation is present if positive (negative) returns are followed by negative (positive) returns.

The problem with positive autocorrelation is that it will have a lower volatility than the uncorrelated series. As less volatility is usually preferred to more, the attractiveness of returns is therefore distorted (overestimated) in the presence of positive autocorrelation.

Positive autocorrelation has been observed in real estate and hedge fund time series. One reason for the existence of autocorrelation is illiquidity, which results in observed prices not being market prices reflecting all relevant information.

Several approaches to dealing with autocorrelation exist. A popular one is the Blundell/Ward filter, which uses the filter below...

r*(t) = 1/(1-a1) * r(t) - a1/(1-a1) * r(t-1)

r*(t)... the "decorrelated" return time seriesr(t)... the original return time seriesr(t-1)... the lagged (by one period9 return time seriesa1... a coefficient from the regression below...

r(t) = a0 + a1*r(t-1) + e(t)

a0... a constante(t)... the usual regression error term

This filter has the advantage that the mean return remains more or less unchanged. So calculating risk-adjusted performance (see chapter 'External Analysis') will result in much less biased results.

It turns out that the autocorrelation of the monthly returns for 1926 through 1998 is 0.081. Not terribly impressive, but positive nonetheless, meaning that a good return this month means a slightly better than average chance of a good return next month. What are the odds that this could have happened by chance? In order to determine this, we have to calculate the standard deviation of autocorrelations for a data series of 873 random data points. The formula for this is sqrt(n-1)/n, which for 873 is 0.034. Thus, the autocorrelation of 0.081 is more than twice the "random walk" standard deviation of 0.034. This in turn means that the odds of this occurring with 873 random numbers is less than one in a hundred.

So, yes, US security prices exhibit some momentum over periods of one month. OK, so stocks around the world do not do the random walk. How does this data affect the average investor? Only at the margins. Lest we get too carried away, the most impressive autocorrelations weve encountered are in the 0.2 range. That means that no more than 4% (0.2 squared, or "R-squared") of tomorrows price change can be explained by todays

IID NormalCalculations for IID Normal case as per Marco Lopez de Pradohttp://papers.ssrn.com/sol3/papers.cfm?abstract_id=2254668-1.644853627Via VBACodeMeanPhiSigmaMaxDDMaxTuWVaRMuZqCheckMaxDDt*MaxTuWPenanceMaxDDt*MaxTuWPenanceHFRIFOF Index0.005516691400.01696908073.53%25.6-0.02239496260.0055166914-0.02791165403.53%6.39961396525.598455859933.53%6.39961396525.59845585993.00HFRIFWI Index0.008881850700.02017678083.10%13.96-0.02430600050.0088818507-0.033187851203.10%3.490529715313.962118861133.10%3.490529715313.96211886113.00HFRIEHI Index0.009858566400.02644447234.80%19.47-0.03363871990.0098585664-0.043497286204.80%4.866724961519.466899845834.80%4.866724961519.46689984583.00HFRIMI Index0.009527016200.0214960733.28%13.77-0.02583087740.0095270162-0.035357893603.28%3.443491135113.773964540533.28%3.443491135113.77396454053.00HFRIFOFD Index0.005179518200.01741638393.96%30.59-0.02346788390.0051795182-0.028647402203.96%7.647716202930.590864811533.96%7.647716202930.59086481153.00HFRIDSI Index0.009621101400.01880033882.48%10.33-0.02130270420.0096211014-0.030923805502.48%2.582713871610.330855486432.48%2.582713871610.33085548643.00HFRIEMNI Index0.005182009400.00942788841.16%8.96-0.01032548710.0051820094-0.015507496501.16%2.23886033948.955441357731.16%2.23886033948.95544135773.00CLMeanSigmaZqMaxDDt*MaxTuWPenanceSR95%PM110,000,00010,000,000-16,448,536.276,763,858.640.67642.7131.0095%PM215,000,00010,000,000-16,448,536.274,509,239.090.30061.2031.5092%PM110,000,00010,000,000-14,145,516.285,002,390.770.50022.0031.0098%PM215,000,00010,000,000-21,208,971.027,497,007.530.49982.0031.50

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2254668

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