$$ \begin{bmatrix} & P & Q & R & S \\ P & 0 & 0 & 0 & 0 \\ Q & 0 & 3.73 & 1.85 & 3.23 \\ R & 0 & 1.85 & 0.93 & 1.61 \\ S & 0 & 3.23 & 1.61 & 2.79 \end{bmatrix} $$
Correlation Calculation from Variance-Covariance Matrix
To find the pair with the highest correlation, we use the correlation coefficient formula: $ \rho_{ij} = \frac{\sigma_{ij}}{\sigma_i \sigma_j} $ where $ \sigma_{ij} $ is the covariance between variables i and j, and $ \sigma_i $ and $ \sigma_j $ are their respective standard deviations.
Extracting Variances and Standard Deviations
From the diagonal elements of the variance-covariance matrix, we get the variances ($ \sigma^2 $). The standard deviations ($ \sigma $) are the square roots of the variances.
| Variable | Variance ($ \sigma^2 $) | Standard Deviation ($ \sigma $) |
|---|---|---|
| P | 0 | $ \sqrt{0} = 0 $ |
| Q | 3.73 | $ \sqrt{3.73} \approx 1.931 $ |
| R | 0.93 | $ \sqrt{0.93} \approx 0.964 $ |
| S | 2.79 | $ \sqrt{2.79} \approx 1.670 $ |
Extracting Covariances
The off-diagonal elements represent the covariances ($ \sigma_{ij} $).
| Covariance | Value |
|---|---|
| $ \sigma_{P,Q} $ | 0 |
| $ \sigma_{P,R} $ | 0 |
| $ \sigma_{Q,R} $ | 1.85 |
| $ \sigma_{Q,S} $ | 3.23 |
| $ \sigma_{R,S} $ | 1.61 |
Calculating Pairwise Correlations
Now, we calculate the correlation for each pair listed in the options:
- P-Q: Since $ \sigma_P = 0 $, the correlation $ \rho_{P,Q} $ is 0 (or undefined).
- P-R: Since $ \sigma_P = 0 $, the correlation $ \rho_{P,R} $ is 0 (or undefined).
- Q-R: $ \rho_{Q,R} = \frac{\sigma_{Q,R}}{\sigma_Q \sigma_R} = \frac{1.85}{\sqrt{3.73} \sqrt{0.93}} \approx \frac{1.85}{1.931 \times 0.964} \approx \frac{1.85}{1.862} \approx 0.9935 $
- Q-S: $ \rho_{Q,S} = \frac{\sigma_{Q,S}}{\sigma_Q \sigma_S} = \frac{3.23}{\sqrt{3.73} \sqrt{2.79}} \approx \frac{3.23}{1.931 \times 1.670} \approx \frac{3.23}{3.225} \approx 1.0013 $
- R-S: $ \rho_{R,S} = \frac{\sigma_{R,S}}{\sigma_R \sigma_S} = \frac{1.61}{\sqrt{0.93} \sqrt{2.79}} \approx \frac{1.61}{0.964 \times 1.670} \approx \frac{1.61}{1.611} \approx 0.9994 $
Identifying the Highest Correlation
Comparing the calculated correlation coefficients:
- $ \rho_{P,Q} = 0 $
- $ \rho_{P,R} = 0 $
- $ \rho_{Q,R} \approx 0.9935 $
- $ \rho_{Q,S} \approx 1.0013 $
- $ \rho_{R,S} \approx 0.9994 $
The pair Q-S has the highest correlation coefficient ($ \approx 1.0013 $).
Important Questions from Correlation Analysis
If X ∼ N (0, 1) and Y = X2 then the correlation coefficient r (X, Y) is
Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): If the securities with less than perfect negative correlation between their price movements are combined, portfolio risk can be reduced significantly.
Reason (R): The term with negative correlation has the effect of reducing the computed value of total portfolio risk, given other terms that are positive.
In the light of the above statements, choose the most appropriate answer from the options given below:
The two-regression equation of variable \(\rm{x}\) and \(\rm{y}\) are
\(\rm{y = 0.8x + 9.8}\) and \(\rm{x = 10.2 + 0.6y}\)
The coefficient of correlation between \(\rm{x}\) and \(\rm{y}\) is
- Let $X$ be a random variable that follows $Uniform(-1, 1)$ distribution. The conditional distribution of the random variable $Y$ given $X = x$ is the $Uniform(x^2 - 0.1, x^2 + 0.1)$ distribution.
The value of $correlation(X, Y)$ is __________ . (Answer in integer) - Consider 62 pair of observations $(x_i, y_i)$, i = 1, …., 62, obtained from the distribution of $(X, Y)$. The Pearson correlation coefficient ($R$) of these observations is 0.27.
Which one of the following statements is CORRECT for a significance level $\alpha = 0.05$ and $t_{0.025,60} = 2.0$?
Hint: Test statistic $t = \frac{R\sqrt{n - 2}}{\sqrt{1 - R^2}}$
