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46 | 46 | "\n", |
47 | 47 | "#### Introduction: Understanding the Problem\n", |
48 | 48 | "\n", |
49 | | - "**Autocorrelation** occurs when observations in a time series are correlated with their own past values. In causal inference with time series data, this creates a fundamental challenge:\n", |
| 49 | + "Autocorrelation occurs when observations in a time series are correlated with their own past values. In causal inference with time series data, this creates a fundamental challenge:\n", |
50 | 50 | "\n", |
51 | 51 | "**What is autocorrelation?**\n", |
52 | 52 | "- Today's outcome is influenced by yesterday's (and last week's, and last month's...)\n", |
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60 | 60 | "\n", |
61 | 61 | "Standard regression assumes **independent errors** — that the unexplained variation at time $t$ is unrelated to time $t-1$. When this assumption fails (as it almost always does in time series):\n", |
62 | 62 | "\n", |
63 | | - "1. **Coefficient estimates remain unbiased** ✓ (still correct on average)\n", |
64 | | - "2. **Standard errors are WRONG** ✗ (typically too small, leading to overconfident inference)\n", |
65 | | - "3. **Hypothesis tests are invalid** ✗ (false positives, misleading p-values)\n", |
66 | | - "4. **Confidence intervals are too narrow** ✗ (underestimate true uncertainty)\n", |
| 63 | + "1. ✅ **Coefficient estimates remain unbiased:** still correct on average\n", |
| 64 | + "2. ❌ **Standard errors are WRONG:** typically too small, leading to overconfident inference\n", |
| 65 | + "3. ❌ **Hypothesis tests are invalid:** false positives, misleading p-values\n", |
| 66 | + "4. ❌ **Confidence intervals are too narrow:** underestimate true uncertainty\n", |
67 | 67 | "\n", |
68 | 68 | "This means you might conclude an intervention \"works\" when it actually doesn't, or claim high precision when you're actually quite uncertain!\n", |
69 | 69 | "\n", |
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