Financial Risk Manager Part 1

Financial Risk Manager Part 1

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The sample autocorrelations for a time series are estimated to be ρ^1=0.20,ρ^2=βˆ’0.03\hat{\rho}_1 = 0.20, \hat{\rho}_2 = -0.03 and ρ^3=0.05\hat{\rho}_3 = 0.05 from a sample size of 90. What is the value Ljung-Box Q statistic?_

TTanishq

Explanation:

Explanation

The Ljung-Box Q statistic (also known as the modified Box-Pierce statistic) is calculated using the formula:

Q(m)=n(n+2)βˆ‘i=1m(ρi2nβˆ’i)Q(m) = n(n + 2) \sum_{i=1}^{m} \left( \frac{\rho_i^2}{n - i} \right)

Where:

  • n=90n = 90 (sample size)
  • m=3m = 3 (time lag)
  • ρ1=0.20\rho_1 = 0.20, ρ2=βˆ’0.03\rho_2 = -0.03, ρ3=0.05\rho_3 = 0.05

Let's calculate step by step:

Step 1: Calculate the individual terms

For ρ1=0.20\rho_1 = 0.20:

0.20290βˆ’1=0.0489=0.0004494\frac{0.20^2}{90 - 1} = \frac{0.04}{89} = 0.0004494

For ρ2=βˆ’0.03\rho_2 = -0.03:

(βˆ’0.03)290βˆ’2=0.000988=0.00001023\frac{(-0.03)^2}{90 - 2} = \frac{0.0009}{88} = 0.00001023

For ρ3=0.05\rho_3 = 0.05:

0.05290βˆ’3=0.002587=0.00002874\frac{0.05^2}{90 - 3} = \frac{0.0025}{87} = 0.00002874

Step 2: Sum the terms

βˆ‘i=13(ρi2nβˆ’i)=0.0004494+0.00001023+0.00002874=0.00048837\sum_{i=1}^{3} \left( \frac{\rho_i^2}{n - i} \right) = 0.0004494 + 0.00001023 + 0.00002874 = 0.00048837

Step 3: Multiply by n(n+2)n(n + 2)

n(n+2)=90Γ—92=8280n(n + 2) = 90 \times 92 = 8280 Q(3)=8280Γ—0.00048837=4.04Q(3) = 8280 \times 0.00048837 = 4.04

Therefore, the Ljung-Box Q statistic is 4.04, which corresponds to option D.

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