Chapter 12 — Time Series Analysis - Signals to Transformers

The Question¶

“Has birth weight changed over the years? Is there a trend?”

All previous chapters treated observations as unordered. But NSFG records the date of each pregnancy. When order matters — when today’s value depends on yesterday’s — we have a time series.

What is a Time Series?¶

A sequence of measurements indexed by time:

y_1, y_2, \ldots, y_T

(1)

where the subscript is a time index. Examples:

Monthly average birth weight (1990–2002)
Weekly pregnancy count
Annual preterm birth rate

Key difference from cross-sectional data: observations are not independent. What happened last month affects this month. Standard statistical methods that assume independence give wrong answers for time series.

Importing and Cleaning¶

NSFG’s datend variable encodes the date of each pregnancy outcome. We convert it to a year and aggregate by year.

Linear Regression on Time¶

The simplest time series model: fit a line where $x$ is time and $y$ is the outcome.

\hat{y}_t = \alpha + \beta t

(2)

This models a linear trend — a consistent upward or downward drift.

For birth weight over 1990–2002: is $\hat{\beta}$ positive (weights increasing)? negative (decreasing)? near zero (stable)?

Moving Averages¶

A moving average smooths a time series by averaging nearby points:

\text{MA}_k(t) = \frac{1}{k}\sum_{i=0}^{k-1} y_{t-i}

(3)

where $k$ is the window size. Larger windows are smoother but less responsive to changes.

Why use it? Real data is noisy. A moving average reveals the underlying trend by averaging out random fluctuations.

This is also called a low-pass filter — it passes slow (long-term) changes and blocks fast (short-term) noise.

Serial Correlation¶

Standard statistics assume observations are independent. In time series, they’re not.

Serial correlation (also: autocorrelation) measures how correlated each observation is with the observations before it:

\text{Corr}(y_t, y_{t-k})

(4)

where $k$ is the lag. If $k=1$ , we’re asking: does a high value this month predict a high value next month?

For birth weight by year: serial correlation might be low (birth weight fluctuates year-to-year without strong persistence). For economic data, serial correlation is often very high (GDP this quarter predicts next quarter).

Autocorrelation Function (ACF)¶

The ACF plots serial correlation for all lags:

\text{ACF}(k) = \text{Corr}(y_t, y_{t-k})

(5)

ACF decays quickly → stationary series (no trend, no persistence)
ACF decays slowly → non-stationary (strong trend or long memory)
ACF shows periodic pattern → seasonal effects

Missing Values¶

Real time series have gaps. In NSFG, some years have too few observations for reliable estimates. Options:

Remove sparse years
Interpolate using neighboring years
Model the missing-data mechanism explicitly

We take option 1 for simplicity.

Prediction¶

Given a fitted trend line, we can predict future values:

\hat{y}_{T+h} = \alpha + \beta(T + h)

(6)

But time series prediction has a fundamental limitation: the further ahead you predict, the more uncertainty accumulates. Always report prediction intervals, not just point estimates.

Exercises¶

Aggregate mean birth weight by year. Plot the time series.
Fit a linear trend. Is birth weight increasing or decreasing over 1990–2002?
Compute a 3-year moving average and overlay it on the original series.
Compute and plot the ACF for birth weight by year. Is there serial correlation?
Predict the mean birth weight for 2005 using the linear trend. What is your uncertainty?

Glossary¶

time series — a sequence of observations indexed by time

trend — a long-term increase or decrease in the series

moving average — smoothed series computed as the average of a sliding window

serial correlation — correlation between an observation and a lagged version of itself

ACF (Autocorrelation Function) — serial correlation plotted as a function of lag

lag — the time offset used in serial correlation (lag $k$ → correlation with $k$ steps back)

stationary — a time series with constant mean and variance (no trend or seasonality)

interpolation — estimating missing values from surrounding observed values

prediction interval — a range expected to contain a future observation with stated probability