代写NEKN96、代做Python/Java设计编程

日期：2024-09-20 11:06

Homework Assignment 1

Linear Regression

NEKN96

Guidelines

1. Upload the HWA in .zip format to Canvas before the 2nd of October, 23:59, and only

upload one HWA for each group. The .zip ffle should contain two parts:

- A report in .pdf format, which will be corrected.

- The code you used to create the output/estimates for the report. The code itself will

not be graded/corrected and is only required to conffrm your work. The easiest is to add

the whole project folder you used to the zip ffle.

1 However, if you have used online tools,

sharing a link to your work is also ffne.

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2. The assignment should be done in groups of 3-4 people, pick groups at

Canvas → People → Groups.

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3. Double-check that each group member’s name and ID number are included in the .pdf ffle.

4. To receive your ffnal grade on the course, a PASS is required on this HWA.

- If a revision is required, the comments must be addressed,  However, you are only guaranteed an additional

evaluation of the assignment in connection to an examination period.

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You will have a lot of ffexibility in how you want to solve each part of the assignment, and all things

that are required to get a PASS are denoted in bullet points:

ˆ . . .

Beware, some things require a lot of work, but you should still only include the ffnal table or ffgure

and not all intermediary steps. If uncertain, add a sentence or two about how you reached your

conclusions, but do not add supplementary material. Only include the tables/ffgures explicitly asked

for in the bullet points.

Good Luck!

1Before uploading the code, copy-paste the project folder to a new directory and try to re-run it. Does it still work?

2Make sure the repository/link is public/working before sharing it.

3Rare exceptions can be made if required. In that case,4Next is the retake on December 12th, 2024.

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Assignment

Our goal is to put into practice the separation of population vs. sample using a linear regression

model. This hands-on approach will allow us to generate a sample from a known Population Regression

Function (PRF) and observe how breakages of the Gauss-Markov assumptions can affect our sample

estimates.

We will assume that the PRF is:

Y = α + β1X1 + β2X2 + β3X3 + ε (1)

However, to break the assumptions, we need to add:

A0: Non-linearities

A2: Heteroscedasticity

A4: Endogeneity

A7: Non-normality in a small sample

A3 autocorrelation will be covered in HWA2, time-series modelling.

Q1 - All Assumptions Fulfflled

Let’s generate a ”correct” linear regression model. Generate a PRF with the parameters:

α = 0.7, β1 = −1, β2 = 2, β3 = 0.5, ε ∼ N(0, 4), Xi

iid∼ N(0, 1). (2)

The example code is also available in Canvas

Setup Parameters

n = 30

p = 3

beta = [-1, 2, 0.5]

alpha = 0.7

Simulate X and Y, using normally distributed errors

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np. random . seed ( seed =96)

X = np. random . normal (loc=0, scale =1, size =(n, p))

eps = np. random . normal (loc =0, scale =2, size =n)

y = alpha + X @ beta + eps

Run the correctly speciffed linear regression model

result_OLS = OLS( endog =y, exog = add_constant (X)). fit ()

result_OLS . summary ()

ˆ Add a well-formatted summary table

ˆ Interpret the estimate of βˆ

2 and the R2

.

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Important: The np.random.seed() will ensure that we all get the same result. In other words, ensure that we are

using the ”correct” seed and that we don’t generate anything else ”random” before this simulation.

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ˆ In a paragraph, discuss if the estimates are consistent with the population regression function.

Why, why not?

ˆ Re-run the model, increasing the sample size to n = 10000. In a paragraph, explain what happens

to the parameter estimates, and why doesn’t R2 get closer and closer to 1 as n increases?

Q2 - Endogeneity

What if we (wrongly) assume that the PRF is:

Y = α + β1X1 + β2X2 + ε (3)

Use the same seed and setup as in Q1, and now estimate both the ”correct” and the ”wrong” model:

result_OLS = OLS( endog =y, exog = add_constant (X)). fit ()

result_OLS . summary ()

result_OLS_endog = OLS ( endog =y, exog = add_constant (X[:,0:2 ])). fit ()

result_OLS_endog . summary ()

ˆ Shouldn’t this imply an omitted variable bias? Show mathematically why it won’t be a problem

in this speciffc setup (see lecture notes ”Part 2 - Linear Regression”).

Q3 - Non-Normality and Non-Linearity

Let’s simulate a sample of n = 3000, keeping the same parameters, but adding kurtosis and skewness

to the error terms:

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n = 3000

X = np. random . normal (loc=0, scale =1, size =(n, p))

eps = np. random . normal (loc =0, scale =2, size =n)

eps_KU = np. sign ( eps) * eps **2

eps_SKandKU_tmp = np. where ( eps_KU > 0, eps_KU , eps_KU *2)

eps_SKandKU = eps_SKandKU_tmp - np. mean ( eps_SKandKU_tmp )

Now make the dependent variable into a non-linear relationship

y_exp = np.exp( alpha + X @ beta + eps_SKandKU )

ˆ Create three ffgures:

1. Scatterplot of y exp against x 1

2. Scatterplot of ln(y exp) against x 1

3. plt.plot(eps SKandKU)

The ffgure(s) should have a descriptive caption, and all labels and titles should be clear to the

reader.

Estimate two linear regression models:

6The manual addition of kurtosis and skewness will make E [ε] ̸= 0, so we need to remove the average from the errors

to ensure that the exogeneity assumption is still fulfflled.

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res_OLS_nonLinear = OLS( endog =y_exp , exog = add_constant (X)). fit ()

res_OLS_transformed = OLS ( endog =np.log ( y_exp ), exog = add_constant (X)). fit ()

ˆ Add the regression tables of the non-transformed and transformed regressions

ˆ In a paragraph, does the transformed model fft the population regression function?

Finally, re-run the simulations and transformed estimation with a small sample, n = 30

ˆ Add the regression table of the transformed small-sample estimate

ˆ Now, re-do this estimate several times

7 and observe how the parameter estimates behave. Do

the non-normal errors seem to be a problem in this spot?

Hint: Do the parameters seem centered around the population values? Do we reject H0 : βi = 0?

ˆ In a paragraph, discuss why assuming a non-normal distribution makes it hard to ffnd the

distributional form under a TRUE null hypothesis, H0 ⇒ Distribution?

Hint: Why is the central limit theorem key for most inferences?

Q4 - Heteroscedasticity

Suggest a way to create heteroscedasticity in the population regression function.

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ˆ Write down the updated population regression function in mathematical notation

ˆ Estimate the regression function assuming homoscedasticity (as usual)

ˆ Adjust the standard errors using a Heteroscedastic Autocorrelated Consistent (HAC) estimator

(clearly state which HAC estimator you use)

ˆ Add the tables of both the unadjusted and adjusted estimates

ˆ In a paragraph, discuss if the HAC adjustment to the standard errors makes sense given the

way you created the heteroscedasticity. Did the HAC adjustment seem to ffx the problem?

Hint: Bias? Efffcient?

7Using a random seed for each estimate.

8Tip: Double-check by simulating the model and plotting the residuals against one of the regressors. Does it look

heteroscedastic?

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