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Assignment Task

Task

1. GMM

In the dataset hw data.csv, you have a time series which comes from the Moving Average process with q lags, MA(q) process. The MA(q) process is defined as follows:

yt = µ + ϵt + θ1ϵt −1 + ... + θqϵ− q

You will need libraries ’gmm’ and ’tseries’ to answer the following questions. Note that if you do not answer correctly point a), all consecutive questions will be wrong, hence no points can be earned.

(a) How would you identify the lag q? Identify the lag q.

(b) Derive the moment conditions function for the process you identified previously and write a corresponding function in R. Use more moment conditions than the number of coefficients that you want to estimate.

(c) Estimate the model using ’gmm’ with an identity weighting matrix. Provide the output and interpret the coefficient significance and the J-test statistic.

(d) Estimate the model using ’gmm’ with an optimal weighting matrix. Provide the output and interpret the coefficient significance and the J-test statistic.

(e) Compare the results from c) and d).

2. Delta method

Let us model the variables X1 and X2 as follows

X1 = 0.2V1 + 0.7V2 + 0.3V3 X2 = 4V2 − 3V3 V1 ∼ N(0, 1), V2 ∼ χ 2 (3), V3 ∼ U[1, 2]

The linear model is defined as:

Y = β0 + β1X1 + β2X2 + ϵ

Let us assume, that we are interested in quantity Φ:

Φ = ln(β0 + β 2 1 ) − (β 1 2 1 )

Estimate the value of Φ and also the variance of Φ, V ar(Φ). To estimate V ar(Φ) use the next methods:

(a) Delta method (by hand)

(b) Delta method (using ’deltamethod’ function)

(c) Bootstrap (with 1000, 10000 replications)

(d) Compare the results.

3. Bootstrap 

Assume a random sample from Exponential distribution. X ∼ exp(λ). (a) Generate 200 observations from the exponential distribution for λ = 2.

(b) Check the sample mean and variance and compare them to the theoretical values. (c) Plot the histogram of generated data, kernel density approximation, theoretical density of exponential distribution and exponential Q-Q plot. Discuss.

(d) Use brute force bootstrapping to obtain the Bootstrap mean, standard errors and bias-reduced estimate of the mean. Use 100,000 bootstrap replications.

(e) Now, use the ’boot’ function with 100,000 bootstrap replications.

(f) Obtain the confidence intervals, use different types and compare the results. Which type is appropriate to use in this case?

(g) Plot the bootstrapped mean. Discuss.

4. Endogeneity

(a) Discuss the nature of the endogeneity problem in the system above. You might check important correlations and you should explain the difference between x1 and x3. Do you expect to observe any bias within the OLS estimation? Explain why and discuss the source of possible bias. What solutions do you suggest?

(b) Estimate the model by OLS and interpret. 

 

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  • Uploaded By : Andrew
  • Posted on : January 08th, 2024

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