Question

Q4. You analyze the non-linear relationships of two financial securities by fitting both a linear and a quadratic function with EXCEL

1. linear model

ret_A = a + b1 * ret_B + error

	Coefficients	Standard Error of coefficients
A	0.0000	0.0006
b1	-1.978	0.025

and

2. Nonlinear model

ret_A = a + b1 * ret_B + b2 * ret_B² + error

variable		Coefficients	Standard Error of coefficients
a	0.0000		0.0006
b1	-1.850		0.0245
b2	4.45		0.382

1. Calculate the t-stat for the coefficient for the quadratic term for the nonlinear model. Is there a significant non-linear effect, ie what is the probability that the coefficient is zero yet you observe 4.45? (2 points)

  

2. Given ret_B = -1%, what is the predicted change in ret_A from the forecast based on linear regression model (1 point)?

3. Given ret_B = -1%, what is the predicted change in ret_A from the forecast based on non-linear regression model (1 point)?
4. 
   
5. Given ret_B = -5%, what are the predicted changes in ret_A based on linea regression? ( 1 point)?

  

5. Given ret_B = -5%, what is the prediction based on nonlinear specifications of the regression functions (2 points)?

  

6. Given ret_B = 5%, what are the predicted changes in ret_A based on linear specifications of the regression function (1 point)?

7. Given ret_B = 5%, what are the predicted changes in ret_A based on quadratic specification of the regression functions (2 points)?

Answer 1

t-stat = 4.45/0.382 = 11.649

p value for a two tailed t-test needs to be calculated, consider degrees of freedom = 100. Probability is less than 0.00001 as the critical values corresponding to such probability is lower than calculated t-statistic. Therefore the non-linear effect is significant

ret_A = -1.978*-1% = 1.978%
ret_A = -1.850*-1% + 4.45*(-1%)^2 = 1.850% + 0.0445% = 1.8945%

ret_A = -1.978*-5% = 9.89%
ret_A = -1.850*-5% + 4.45*(-5%)^2 = 9.25% + 1.1125% = 10.3625%

ret_A = -1.978*5% = -9.89%
ret_A = -1.850*5% + 4.45*(-5%)^2 = -9.25% + 1.1125% = -8.1375%