Homework 2

Slurm and the cluster HW

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HW 2

Testing OLS Regression’s Breaking Point

The goal is to simulate how the Ordinary Least Squares linear regression estimator behaves under stress. According to the Gauss-Markov theorem, OLS is the “best” linear unbiased estimator only under assumptions. Your job is to quantify what happens when they aren’t. You’ll investigate two key properties of an estimator for a regression coefficient, \(\beta\):

  • Bias
  • Efficiency (Variance)

The Simulation Grid

You will create a simulation where you generate thousands of datasets and fit a regression model to each one. The key is to build a grid of conditions to test. Each task in your Slurm job array will handle one unique combination of these conditions.

Simulation Parameters to Vary:

  • Sample Size (n): How does the number of data points affect performance?

  • Values: 50, 100, 500, 1000, 5000

  • Degree of Heteroscedasticity (alpha): This is a violation of the “constant variance” assumption. We’ll make the error variance depend on the independent variable \(x\).

  • \(x \sim N(0,1)\)

  • \(\beta = 1\)

  • The error term \(\epsilon\) will be drawn from \(N(0, \exp(\alpha x))\).

  • Values for alpha: 0 (no violation), 0.5 (mild), 1.0 (strong), 2.0 (extreme).

  • Degree of Autocorrelation (\(\rho\)): This violates the “independent errors” assumption, common in time-series data.

  • The error term \(\epsilon_i\) will be calculated as \(\epsilon_i = \epsilon_{i −1} + w_i\), where \(w_i \sim N(0,2)\).

  • Values: 0 (no violation), 0.25 (mild), 0.5 (strong), 0.9 (extreme).

Your full experiment will be a grid of 5 (\(n\)) x 4 (\(\alpha\)) x 4 (\(\rho\)) = 80 unique scenarios. For each scenario, run \(1000\) replications. Evaluate both bias and standard error. Run your simulation on the cluster using a SLURM array job.

Include your code and a no more than two page write up of results in your git repository and push your changes.