Assignment 3
Practice with estimation and hypothesis testing
1. Download the prices from 2007-01-01 to 2021-12-31 (15 years) for any two stocks of your choice. I’ll refer to these as A and B.
2. Calculate the continuously compounded monthly returns for both.
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3. We are going to assume that the returns for each stock follows a GWN model with constant mean mean, variance, and covariance. To obtain estimates of these quantities, for the period from 2007-01-01 to 2016-12-31 (first 10 years) calculate:
• The sample mean for each stock’s returns (μ0A and μ0B)
• The sample standard deviation for each stock’s returns (σA0 and σB0 )
• The sample correlation between the two stock’s returns (ρ0AB)
4. Next, we are going to test whether the corresponding statistics during the next 5 years are the same as the ones above. For the period from 2017-01-01 to 2021-12-31, calculate:
• The sample mean for each stock’s returns (μˆA and μˆB)
• The sample standard deviation for each stock’s returns (σˆA and σˆB)
• The sample correlation between the two stock’s returns (ρˆAB)
5. Then, calculate the corresponding estimated standard errors for each of the quantities using the for- mulas shown in class (refer to page 5 of the Estimation notes). Note that all these calculations should be done using the sample standard deviation and correlation you just calculated in 4.
• The standard error for each sample mean: sˆe(μˆA) and sˆe(μˆB)
• The standard error for each sample standard deviation: sˆe(σˆA) and sˆe(σˆB)
• The standard error for the sample correlation: sˆe(ρˆAB)
6. We now have 5 null and alternate hypotheses we can test:
i. H0 :μA =μ0A vsH0 :μA ̸=μ0A
ii. H0 :μB =μ0B vsH0 :μB ̸=μ0B
iii. H0 :σA =σA0 vsH0 :σA ̸=σA0
iv. H0 :σB =σB0 vsH0 :σB ̸=σB0
v. H0 :ρAB =ρ0AB vsH1 :ρAB ̸=ρ0AB
To test these hypotheses, calculate the 5 corresponding z-statistics. For example, for the first one, it would
μˆ A − μ 0
be: z = A . And similarly for the rest. (Note: you already have alll the required values and only need
sˆe ( μˆ A )
to calculate the ratio)
7. Let’s use a 4% significance level. What would the 2 threshold values for the z-statistics be? You can use the qnorm function here.
8. Use these threshold values to conclude for each of the 5 hypothesis tests above, whether, based on the evidence, you:
• Fail to reject the null hypothesis
• Reject the null hypothesis
9. Based on your answers above, do you think:
i. The mean return for stock A has changed over this period?
ii. The mean return for stock B has changed over this period?
iii. The standard deviation of return for stock A has changed over this period?
iv. The standard deviation of return for stock B has changed over this period?
v. The correlation between returns of stocks A and B has changed over this period?
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