Financial Risk Manager Part 1
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The following sample autocorrelation estimates are obtained using 300 data points:
| Lag | 1 | 2 | 3 |
|---|---|---|---|
| Coefficient | 0.25 | -0.1 | -0.05 |
Compute the value of the Ljung-Box Q statistic.
Other
Community
NNikitesh
A
20
B
22.74
C
24
D
18.45
Explanation:
Formula:
Q(m) = n(n + 2) Σ[ρ_j² / (n - j)]
In this case, time lag m = 3
Thus,
Q(3) = 300(302)[ (0.25)²/299 + (-0.1)²/298 + (-0.05)²/297 ] = 22.74
Explanation: The Ljung-Box Q statistic is used to test whether a group of autocorrelations of a time series are different from zero. The formula is:
Q(m) = n(n + 2) Σ[ρ_j² / (n - j)] from j=1 to m
Where:
- n = number of observations (300)
- m = number of lags (3)
- ρ_j = autocorrelation coefficient at lag j
Calculation:
- n(n+2) = 300 × 302 = 90,600
- Summation term:
- For lag 1: (0.25)²/299 = 0.0625/299 ≈ 0.00020903
- For lag 2: (-0.1)²/298 = 0.01/298 ≈ 0.00003356
- For lag 3: (-0.05)²/297 = 0.0025/297 ≈ 0.00000842
- Sum = 0.00020903 + 0.00003356 + 0.00000842 ≈ 0.00025101
- Q(3) = 90,600 × 0.00025101 ≈ 22.74
The correct answer is 22.74 (Option B).
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