Calculating and Plotting MSE

Angela Ryu

Economics 201FSHonors Junior Workshop: Finance

Duke University

March 24, 2010

Example

• XOM (Y) vs. WMT (X)• Number of days: 1093 days • Sampling interval: 5 minutes• Beta Calculation days: 30 days

Preparation (1) - Setup

• Get WMT, XOM log returns and denote Px, Py respectively.

• Sample P by 5 minutes: 76 data for each day• Take out the overnight returns • Size (Px) = Size (Py) = 76 * 1093

Preparation (2) – Calculate Beta• Days – (1:30), (2:31), (3:32), … , (1064:1093) • Case (1:30):– Take first 76 * 30 data (1 to 76*30) from Px and Py and

denote X1:30 and Y1:30 . – Calculate with (where Y = βX) – Take the mean and denote β1:30

• Case (2:31): – Exclude the day 1 data (1 to 76) and add data on the

day 31 (76*30+1 to 76*31). Get β2:31

• Repeat to get β3:32, …, β1063:1092Note: all betas are scalar

Preparation (2) – Calculate Beta

Day 1 (76 data) Day 2 (76 data)…Day 30 (76 data)Day 31 (76 data)…Day 1092 (76 data)Day 1093 (76 data)

Day 1 (76 data) Day 2 (76 data)…Day 30 (76 data)Day 31 (76 data)…Day 1092 (76 data)Day 1093 (76 data)

Case (1:30): Case (2:31):

Preparation (3) – Calculate MSE

• Calculate SE for each beta.– SE31 = (Y31 - β1:30 * X31 )2

– SE32 = (Y32 – β2:31 * X32 )2

…

– SE1093= (Y1093 – β1063:1092 * X1093 )2

Note: SEi is a vector of size 76 for all i = 31, … 1093

• Take the average to get MSE30

– MSE30 = avg [avg(SE1:30 ), …, avg(SE1064:1093)]

()2 Square each term in vector

Preparation (4) – change intervals & plot for each sampling interval

• Sampling intervals: change from 1 min to 20 mins in prep. (1)

• Beta Calculation intervals: change from 1 day to 50 days in prep (2)

• Plot for each sampling interval– X axis: Beta Cal. Interval days (from 1 to 50)– Y axis: the value of MSE

• In total, we get 20 plots

WMT(Y) vs. XOM(X) (5 min)