Part I ยท Chapter 1
Why Time Series Matter
Time series are ordered observations indexed by time. They typically combine a slow-moving trend, recurring cycles or seasonality, and random noise โ and the order matters because today depends on yesterday.
\( y_t = \text{Trend}_t + \text{Cycle}_t + \text{Noise}_t \)
Learning objectives
Explain what makes time-series data different from cross-sectional data.
Identify trend, cycle, and noise components visually.
Understand why ordering and dependence matter.
Recognize how economic series like inflation or GDP combine these components.
Components Lab
Seed:
Live interpretation
Trend slope: โ per period
Cycle period: 24 steps (fixed)
Noise ฯ: โ
Larger trend slope โ series climbs steadily. Larger amplitude โ larger swings. Larger noise โ harder to see structure.
๐ Decomposition formula
\[ y_t = \underbrace{a + b\cdot t}_{\text{trend}} + \underbrace{A\sin(2\pi t/24)}_{\text{cycle}} + \underbrace{\varepsilon_t}_{\text{noise}} \]
a, b trend level + slope
A cycle amplitude
t time index (1, 2, โฆ, n)
ฮต_t iid noise ~ N(0, ฯยฒ)
๐ What to look for & how to read the chart
- Trend line shows the deterministic linear path; the gap between it and the observed series is cycle + noise.
- When ฯ is large, the observed line wiggles around the trend; the underlying structure is hidden by noise.
- Real economic series (GDP, inflation, unemployment) almost always combine all three components.
โ ๏ธ Pro Tip: What to Avoid
Student says
"Each observation is just a draw from the distribution, like any sample โ I'll treat them as independent."
Why this is wrong
Time-series observations are ordered and usually dependent. Treating them as iid invalidates standard errors and many test statistics.
Correct interpretation
Always check for trend, autocorrelation, and stationarity before applying methods that assume independence.
๐ Mini-quiz
๐ Key Takeaways
| Concept | Meaning | Exam tip |
|---|---|---|
| Trend | Long-run direction | Linear vs stochastic โ different treatments |
| Cycle / seasonality | Recurring pattern | Monthly = lag 12; quarterly = lag 4 |
| Noise | Unforecastable irregular shocks | Should look like white noise in residuals |
| Ordering | Time index matters | Never shuffle a time series |