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 the assumptions are not met. 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 will follow an AR(1) process, \(\epsilon_i = \rho\epsilon_{i-1} + w_i\), where \(w_i\) is Gaussian innovation noise scaled so that comparisons across \(\rho\) are meaningful.

  • 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.

Reproducibility and AI-use requirements

Use deterministic, scenario-specific seeds; save one result file per array task; and include a combine script that refuses to summarize incomplete scenarios. Report requested and observed resources from sacct, and show how you would identify and resubmit failed tasks. Your report should distinguish estimator bias, empirical variance, mean model-based standard error, and interval coverage; explain the role of each assumption violation.

An AI agent may help draft or debug code, but submit an AI_USE.md containing relevant prompts, output used, and one independent check (for example, a locally run tiny grid or a hand-checked scenario). Never submit cluster jobs without reviewing requested resources and commands.