Financial Risk Manager Part 1
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An analyst has regressed the annual return on a stock (R_stock) against the annual return on the NIFTY 50 (R_index) for 30 years. The NIFTY is the index of the National Stock Exchange (NSE), India. Results are shown below. Regression equation:
R_index, t = â + b̂ × R_stock, t + ε_t
| Coefficient | Coefficient Estimate | Standard Error |
|---|---|---|
| a | 0.002 | 0.001 |
| b | 1.223 | 0.063 |
What is the 90% confidence interval for the slope coefficient? Click here to see critical values of the t-distribution._
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TTanishq
A
[1.1165; 1.3295]
B
[1.223; 1.3295]
C
[1.1158; 1.3301]
D
[0.063; 1.223]
Explanation:
Explanation
The 90% confidence interval for the slope coefficient is calculated using the formula:
CI = b̂ ± t_(α/2, n−2) × s_b̂
Where:
- b̂ = slope coefficient estimate = 1.223
- s_b̂ = standard error of slope coefficient = 0.063
- n = sample size = 30 years
- α = 1 - confidence level = 1 - 0.90 = 0.10
- α/2 = 0.10/2 = 0.05
- Degrees of freedom = n - 2 = 30 - 2 = 28
From t-distribution tables, t_(0.05, 28) = 1.701
Calculation:
- Margin of error = t × s_b̂ = 1.701 × 0.063 = 0.107163
- Lower bound = 1.223 - 0.107163 = 1.115837 ≈ 1.1158
- Upper bound = 1.223 + 0.107163 = 1.330163 ≈ 1.3301
Therefore, the 90% confidence interval is [1.1158; 1.3301].
Key Points:
- For small samples (n ≤ 30), we use the t-distribution instead of the normal distribution
- Degrees of freedom for regression slope = n - 2
- The confidence interval provides a range where we are 90% confident the true population slope coefficient lies_
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