Comparing MSE: Optimal Sampling Frequency and Beta Interval Angela Ryu Economics 201FS Honors Junior Workshop: Finance Duke University March 31, 2010

Data & Variables SPY TGT, XOM, WMT (3 stocks) Aug 20 2004 – Jan 7 2009 (1093 days) Sampling frequency (SF): 1 – 20 min. Beta calculation days (BI): 1 – 50 days

Recap So far, we have observed MSE varying over BI for a given sampling frequency. Similar trends were observed among 9 different stocks: relative drop in MSE around 5 – 15 Beta interval days Remaining questions: (1) (1 stock) For a stock, how do the levels of MSE vary among different sampling frequencies? (2) (2+ stocks) Are the results from different stocks comparable? If so, how? (3) Can we empirically find an “optimal” beta interval days for a given sampling frequency?

(1) Comparing MSE (1 stock) Method: plot a stock’s MSEs of different frequency. *The higher the sampling frequency, the higher the MSE!

Results from (1) As we have less data in low frequencies, it intuitively makes sense to have higher MSE. Despite the sudden drop starting at (approx.) 5 days, the increase in MSE due to lower sampling frequency dominates The lowest point of 10 min. MSE is still strictly higher than the highest point of 9 min. MSE, etc.

(2) Comparing MSE (2+ Stocks) Method: For a given sampling frequency, compare percentage changes of MSE w.r.t. MSE1 (= MSE with Beta Interval Day = 1) i.e. plot MSE / MSE1, where MSE = [MSE1 MSE2 … MSE50]

SPY vs. TGT, WMT, XOM (2 min)

SPY vs. TGT, WMT, XOM (5 min)

SPY vs. TGT, WMT, XOM (10 min)

SPY vs. TGT, WMT, XOM (15 min) *In percentage terms, change in MSE of different stocks w.r.t. BI is similar to each other.

(3) “Optimal” Beta Interval Method: For different stocks, plot Beta interval where MSE is minimum vs. Sampling Frequency. For each sampling frequency, there exist a Beta interval day with minimum MSE. X-axis: Sampling frequency (1 – 20 min) Y-axis: Beta interval days (1 – 50 days) with min. MSE Linear regression

BI with MinMSE vs. SF TGT WMT XOM ALL

Linear Regression: over SI

Linear Regression: over divided SI Replaced the outlier data of day 7 Mean: 4.625 Mean: 13.3333 *For WMT, if approximately 5 or 13 days of data is used to calculate Beta, MSE is minimized.

Further Steps Try regression on different stocks Hypothesis on optimal Beta interval: e.g. H0: “BI with MinMSE lies in [4, 14]” Relevant theses?