Definition¶
- A time series is a sequence of data points collected over time.
- These data points are ordered by time, and the sequence of the data cannot be interchanged.
- Time series analysis focuses on transforming, visualizing, and modeling this type of data.
- When modeling a time series, its value at any time is random, i.e., the realization of a random variable.
- Thus, a time series is a discrete stochastic process.
Plots¶
- A time series plot (or time plot) is a graph that visualizes the evolution of a time series.
- Time, $t$, is on the horizontal axis.
- Observed values are on the vertical axis.
- Time series plots are crucial for analysis, revealing patterns such as trend and seasonality.
- The first step always is to plot the recorded data and do a visual examination, i.e., "eyeball econometrics."
- For economic time series data, usually we have observations at regular intervals, or frequencies, e.g., daily, monthly, quarterly, or annual.
Examples¶
- Daily:
- Monthly:
- Quarterly: U.S. Gross Domestic Product (GDP)
- Annual: U.S. Capital Stock
In [4]:
# Libraries
from fredapi import Fred
import pandas as pd
# Read Data
fred_api_key = pd.read_csv('fred_api_key.txt', header=None)
fred = Fred(api_key=fred_api_key.iloc[0,0])
data = fred.get_series('UNRATE').to_frame(name='UR')
print(data.head(2))
print(data.tail(2))
UR
1948-01-01 3.4
1948-02-01 3.8
UR
2025-05-01 4.2
2025-06-01 4.1
In [5]:
# Plotting
import matplotlib.pyplot as plt
# Plot
fig, ax = plt.subplots(figsize=(6.5,2.5));
ax.plot(data.UR);
ax.set_title('U.S. Unemployment Rate');
ax.yaxis.set_major_formatter('{x:,.1f}%')
ax.grid(); ax.autoscale(tight=True)
Objectives¶
- Time series plots are useful for identifying
- trends, i.e., long-term systematic patterns.
- seasons, i.e., common patterns across years.
- cycles, i.e, patterns related to economic recessions and expansions.
- structural changes to trend, seasonal, or cyclical patterns.
- outliers (anomalies).
- Descriptive statistics are used to summarize the properties of time series, e.g., Mean/Expected Value, Variance/Std. Deviation, Autocorrelation, Skewness, Kurtosis, percentiles, and min/max are all useful descriptive statistics.
In [7]:
# Import libraries
# Scientific computing, statistical functions
import scipy.stats as stats
# Mean
print(f'Most recent UR = {data.UR.iloc[-1]:.1f}%')
print(f'E(UR) = {stats.tmean(data.UR):.1f}%')
print(f'Std(UR) = {stats.tstd(data.UR):.1f}%')
print(f'Skew(UR) = {stats.skew(data.UR):.1f}')
print(f'Kurt(UR) = {stats.kurtosis(data.UR):.1f}')
print(f'min(UR) = {data.UR.min():.1f}%')
print(f'max(UR) = {data.UR.max():.1f}%')
Most recent UR = 4.1% E(UR) = 5.7% Std(UR) = 1.7% Skew(UR) = 0.9 Kurt(UR) = 1.1 min(UR) = 2.5% max(UR) = 14.8%
In [8]:
# Plot histogram
fig, ax = plt.subplots(figsize=(6.5,2.5));
ax.hist(data.UR,edgecolor='black',density='true')
ax.set_xlabel('U.S. Unemployment Rate');
ax.set_ylabel('probability')
ax.xaxis.set_major_formatter('{x:,.1f}%')
ax.grid(); ax.autoscale(tight=True)
- Most recent UR is $4.1\%$, or $1.6pp$ below historical mean of $5.7\%$
- Std. Dev. is $1.7\%$. If UR was normally distributed, then $\pm 2 SD$, or interval from $0.7\%$ to $9.1\%$, would contain $95\%$.
- But minimum UR is $2.5\%$ and maximum is $14.8\%$, which indicates UR is most likely skewed.
- $Skew(UR) = 0.9$ and we see that histogram has long right tail.
Key Concepts¶
- White Noise is a special type of time series, where the values are random and uncorrelated across time.
- Stationarity, e.g., does the data have a trend, seasonality, or is the covariance changing over time? If no to all, then it is stationary.
- If the data is non-stationary, then the underlying probabilistic structure of the series is changing over time, and there is no way to relate the future to the past.
- A Random Walk is a key model for the log price of a financial asset.
- Autocorrelation is a measure of how values in a time series are correlated with previous values.