Regression Trend Test for Compensation

Business Context

Meta HR Analytics wants to determine whether employee compensation growth changed after a compensation policy update introduced in July 2023. You are given monthly median total compensation for a stable cohort of software engineers so the goal is to test whether the time trend changed, not to compare different employee mixes.

Problem Statement

Use a time-trend regression with a policy-change indicator and an interaction term to evaluate whether compensation trends changed after the policy update.

Given Data

Monthly median total compensation is shown below in thousands of dollars.

$Y_t = \beta_0 + \beta_1 t + \beta_2 D_t + \beta_3 (t \times D_t) + \varepsilon_t$

You are also given the fitted coefficient estimates and standard errors from OLS:

Use a two-sided test with $\alpha = 0.05$. For the small sample, use a critical value of $t_{0.975,8} = 2.306$.

Requirements

1. State the null and alternative hypotheses for whether the compensation trend changed after the policy update.
2. Interpret $\beta_1$, $\beta_2$, and $\beta_3$ in business terms.
3. Test whether the post-policy slope change is statistically significant.
4. Construct a 95% confidence interval for the slope change $\beta_3$.
5. Compute the pre-policy monthly trend and post-policy monthly trend.
6. Conclude whether compensation trends meaningfully changed over time.

Assumptions

• The cohort composition is stable across months.
• The linear trend is a reasonable approximation within this 12-month window.
• Residuals are independent with constant variance; ignore autocorrelation for this exercise.