Case Study: Model Risk and Model Valid ...
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To solve the given simultaneous equations, we can use the substitution or elimination method. Here, we’ll use the elimination method for convenience.
The given equations are:
1) \(\beta_1 + 0.6899\beta_2 = 0.5633\)
2) \(\beta_2 + 0.804\beta_1 = -0.7633\)
Let’s rearrange equation 2 to make it easier to eliminate one of the variables:
2) \(\beta_2 = -0.7633 – 0.804\beta_1\)
Now, we’ll substitute the expression for \(\beta_2\) from equation 2 into equation 1:
\(\beta_1 + 0.6899(-0.7633 – 0.804\beta_1) = 0.5633\)
Expanding and simplifying:
\(\beta_1 – 0.6899 \times 0.7633 – 0.6899 \times 0.804\beta_1 = 0.5633\)
\(\beta_1 – 0.5270 – 0.5548\beta_1 = 0.5633\)
Combining like terms:
\(0.4452\beta_1 = 1.0903\)
Dividing both sides by 0.4452:
\(\beta_1 = \frac{1.0903}{0.4452}\)
\(\beta_1 \approx 2.4493\)
Now that we have \(\beta_1\), we can find \(\beta_2\) using either equation. We’ll use equation 2:
\(\beta_2 = -0.7633 – 0.804\beta_1\)
\(\beta_2 = -0.7633 – 0.804 \times 2.4493\)
\(\beta_2 = -0.7633 – 1.9691\)
\(\beta_2 = -2.7324\)
So, the solution to the given simultaneous equations is:
\(\beta_1 \approx 2.4493\)
\(\beta_2 \approx -2.7324\)