Homework answers / question archive / Assignments Assignment 1 (weight 40 %, page limit 3 pages) The data file CLASS ATTENDANCE xlsx contains a random sample of 680 MSc students in psychology

Assignments Assignment 1 (weight 40 %, page limit 3 pages) The data file CLASS ATTENDANCE xlsx contains a random sample of 680 MSc students in psychology

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Assignment 1 (weight 40 %, page limit 3 pages)

The data file CLASS ATTENDANCE xlsx contains a random sample of 680 MSc students in psychology. Our interest is to study the effect of class attendance (CA) on the final exam score (FSCORE) in the course Behavioral psychology. It is believed that thy effect of class attendance on the exam score depends on the student's prior level of knowledge, which is here indicated by the GPA sore (GPA) obtained during the student's Bachelor study, Thus, GPA plays the role as a so-called moderating variable The model additionally include the student's level of scholastic aptitude as measured by the ACT score (ACT).

Consider the following model formulation:

FSCORE = b₀ + b₁CA + b₂GPA + b₃(CA×GPA) + b₄ACT + b₅ACT² + u

Using a 4% significance level, do the following:

A. Estimate the model exactly as given above, Test and evaluate the effect of class attendance on the final exam score tor:

GPA = 2.19 (low level of GPA}

GPA = 2.94 (high level of GPA}

To do this, follow the procedure in the Appendix (found at the end of this document).

Specifically,

1. Derive a general expression fur the standard error associated with the lest outlined in the appendix,

2. Implement the solution and provide a thorough interpretation of the results. The solution must be based on White's heteroskedasticity robust estimator of the covariance matrix of the parameter estimates.

(Hint: since the number in the estimated covariance matrix are rather small, it is recommended to use at least 6 decimals in the computations of the standard errors)

Assignment 2 (weight 20 %, page limit 0.5 pages)

Suppose we want to perform a simulation to see if we can approach the theoretical coverage ratio, as given by 100× (1 – a)%, of a confidence interval based on the t-distribution. The way to do this was hinted in Class notes 6, where it states:

When computing a confidence interval, one is applying a procedure for which the true parameter is contained in the interval with probability 100 × (1 — a)%. That is, if we scene to compute a Large number of confidence Intervals, based on a large number of samples (drawn from the same population), we would expect the true parameter to be contained in the interval in 100× (1 — a)% of the cases.

Let X₁,.., X_n be a random simple from a normal population with mean m and variance s² (n > 1).

Then, from Assignments 4, a 100× (1 — a)% confidence interval is obtained by

X_n ± t_a/2 × S_x/Ön

A. Using Matlab, simulate the coverage ratio of a 90% confidence interval. The simulation is to be performed using the following steps:

- Draw a random sample X_i – N(2, 3²) for i = 1,...,10

- Compute the confidence interval (a lower bound and an upper bound):

- Evaluate whether the true mean m is located within the bounds of the interval.

The simplest way to perform the simulation is to wrap the three steps in a for-loop. In your code, use rng (1) and ntep – 10000 (10000 replications). Present the result and write a brief summary.

Assignment 3 (weight 40 %, page limit 5 pages)

In this exercise, we study omitted variable bias when estimating econometric models.

Suppose that the true model, satisfying MLRI through MLR4 in Wooldridge (see page 40-62), is

Given by

Y = b₀ + b₁X₁ + b₂X₂ + u

Suppose further that we by mistake estimate a simple regression model with X₁ as the only predictor

Our aim is to correctly estimate the effect of X₁ on Y. The OLS estimator of b₁ is given by the expression

b₁ = Sⁿ_i=1(X_1i – X₁)(Y₁ – Y)/Sⁿ_i=1(X_1i – X₁)²

A. (weight 50%)

Show that the expected value of b₁, conditional on X_1i and X_2i tor all i. takes the form

E(b₁) = b₁ + b₂d₁

Where d₁ is the slope parameter obtained from regressing X₂ on X₁.

(Hint you may find Exercise 3 in Assignments 6B useful, Note that, since we condition on the sample values of X₁ and X₂, we can treat d₁ and its components as constants)

B. (weight 20%}

We will now be more specific. Let the model in (1) be

Y = X₁ – X₂ + u

Suppose that X₁ has unit variance and that the covariance between X₁ and X₂ is 2. What is the bias of b₁ in the probability list?

C. (weight 30%}

We continue studying omitted variable bias. Specifically, using simulation, we will approach the bias associated with b₁ under two different conditions

Condition 1

First, we study the bias of b₁ winder due exanlition that X₁ and X₂ are independent. You are advised to use the code from Lab, Session 2, Obviously, you will need to modify the cade to handle the problem in this assignment.

To further help develop the code use the following code lines:

ars

eng(h}i

hetad @ 1

betal - 2;

theta? = 13

sigma - sqrt tdi:

roe PDS

nrer > 109503

Eim@c.cte asta fic 42, a. and

woos raerermetty Te [te 11s

xe mormragtu, 4, in, lds

wos marsendtt, sigma, fn, ite

yoo meray + betsital - petazexd + us

Present the results and provide a brief explanation.

Condition 2

Second, we study the bias associated with b₁ under the condition that X₁ and X₂ are related.

Suppose that X₂ is formed by

X₂ = 2X₁ + e

Where e is a standard normal random variable independent of all other variables. Change the code from Condition I to allow for X₂ to be modeled as given by the equation above.

As before, present the results and provide a brief explanation.

Appendix

Consider the regression model

Y = b₀ + b₁X₁ + b₂Z + b₃(X₁ × Z) + b₄X₄ + b₅X₅ + … + b_kX_k + u

In the model, the predictors of interest are X₁ and Z. The remaining predictors X₄, ... X_k are control variables, which are needed to ensure a correct model formulation. Our aim is to study the effect of X₁ on Y for some value of Z. It is easily shown that for some realization of Z (a fixed number, denoted z}, the change in the expected value of Y for a one-unit increase in X₁ is given by

b₁ + b₃z

It is of interest to perform hypothesis texting to see if the effect of X₁ on Y, given z, is statistically significant. The hypothesis to be evaluated takes the form

H₀: b₁ + b₃z = 0

H_A: b₁ + b₃z ¹ 0

The usual t-statistic becomes

t = b₁ + b₃z/SE(b₁ + b₃z)

Under the null-hypothesis, the tent statistic in (at least) approximately t-distributed with df = n – k - 1 (which is simply the number of observations minus the number of parameters in the model)

The challenge is to derive SE(b₁ + b₃z), which is done by the following two steps:

- Based on the covariance matrix of the parameter estimates, derive Var(b₁ + b₃z) using basic rules for variances and covariances.

- Obtain SE(b₁ + b₃z) from the previous step.