SECTION 1: LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

• Distinguish time series data from cross‑sectional data and understand its unique properties.

• Decompose a time series into trend, seasonality, cyclical, and irregular components.

• Understand the concept of stationarity and why it is critical for financial modelling.

• Apply the Augmented Dickey‑Fuller (ADF) test to check for stationarity.

• Interpret autocorrelation and partial autocorrelation functions (ACF/PACF).

• Apply smoothing techniques (moving averages, exponential smoothing) for trend estimation.

• Recognise common time series patterns in banking data (e.g., deposit growth, loan defaults, market returns).

• Understand the business implications of time series analysis for forecasting and risk management.

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SECTION 2: WHAT IS TIME SERIES DATA?

A time series is a sequence of data points recorded at successive equally spaced points in time. In finance and banking, almost every key metric is observed over time:

• Daily closing prices of a stock

• Monthly unemployment rates

• Quarterly bank profits

• Hourly transaction volumes

• Daily Value‑at‑Risk (VaR) estimates

Key distinction: Unlike cross‑sectional data (where observations are independent), time series data are serially dependent – the value today is related to the value yesterday. This autocorrelation violates the independence assumption of standard regression and requires specialised techniques.

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SECTION 3: COMPONENTS OF A TIME SERIES

Classical time series decomposition breaks a series into four components:

The additive model is: Y = T + S + C + I
The multiplicative model is: Y = T × S × C × I (common when seasonal amplitude grows with trend)

Business relevance: Decomposing a series helps banks separate genuine growth from seasonal blips and cyclical swings, enabling better budgeting and capacity planning.

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SECTION 4: STATIONARITY – THE FOUNDATION OF TIME SERIES MODELLING

A time series is stationary if its statistical properties (mean, variance, autocorrelation) do not change over time.

Why stationarity matters:

• Most time series models (ARIMA, GARCH) assume stationarity.

• Non‑stationary data produce spurious regression results – you may find strong relationships that are purely due to common trends (e.g., both sunspots and stock prices increase over time, but they are unrelated).

Two types of stationarity:

• Strict stationarity: The joint distribution of any set of observations is invariant to time shifts (very restrictive).

• Weak stationarity (covariance‑stationary): Mean and variance are constant, and autocovariance depends only on the lag, not on time. This is the standard requirement.

How to check for stationarity:

1. Visual inspection: Plot the series. A clear upward or downward trend or changing variance suggests non‑stationarity.

2. Augmented Dickey‑Fuller (ADF) test: A formal statistical test.

   • Null hypothesis (H₀): The series has a unit root (i.e., it is non‑stationary).

   • Alternative (H₁): The series is stationary.

   • If p‑value < 0.05, reject H₀ and conclude the series is stationary.

What to do when a series is non‑stationary:

• Differencing: Subtract the previous observation: Yt′=Yt−Yt−1Yt′​=Yt​−Yt−1​. Repeat until stationarity is achieved.

• Log transformation: Stabilises variance and makes multiplicative patterns additive.

• Detrending: Fit a trend line and analyse the residuals.

Financial example: Stock prices are generally non‑stationary (they trend upward). Returns (percentage changes) are usually stationary – this is why we model returns, not prices.

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SECTION 5: AUTOCORRELATION AND PARTIAL AUTOCORRELATION

5.1 Autocorrelation (ACF)

Autocorrelation measures the correlation between a series and its own lagged values.
For lag *k*, it is the correlation between YtYt​ and Yt−kYt−k​.

• ACF plots are used to identify the order of moving average (MA) components.

• In finance, we often see significant positive autocorrelation at lag 1 in daily returns (though usually weak) and strong autocorrelation in volatility (absolute returns).

5.2 Partial Autocorrelation (PACF)

PACF measures the correlation between YtYt​ and Yt−kYt−k​ after removing the influence of intermediate lags (1, 2, …, k‑1).

• PACF plots help identify the order of autoregressive (AR) components.

Business application: Understanding autocorrelation is crucial for risk models. If daily trading volumes are highly autocorrelated, we can use today’s volume to predict tomorrow’s, improving liquidity forecasts.

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SECTION 6: SMOOTHING TECHNIQUES FOR TREND ESTIMATION

Smoothing methods reduce noise to reveal the underlying trend – useful for identifying turning points.

6.1 Moving Average (MA)

• The simple k‑period moving average at time t is the average of the last k observations.

• Pros: Easy to understand and implement.

• Cons: All observations get equal weight; it lags behind sudden changes.

6.2 Exponential Smoothing (EWMA – Exponentially Weighted Moving Average)

• This assigns exponentially decreasing weights to older observations.
  St=αYt+(1−α)St−1St​=αYt​+(1−α)St−1​, where $\alpha$ is the smoothing factor (0 < α < 1).

• A higher α gives more weight to recent observations, making the series more responsive.

• Pros: More reactive to recent changes; widely used in volatility forecasting (e.g., RiskMetrics uses EWMA for VaR).

Financial insight: Banks use smoothing to estimate core deposit growth, excluding one‑off large withdrawals or seasonal tax payments.

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SECTION 7: BUSINESS APPLICATIONS IN BANKING AND FINANCE

Real‑world example – Interest Rate Forecasting:
Central bank policy rates are influenced by inflation and employment. Analysts decompose these series into trend and cyclical components to anticipate rate changes. If the cyclical component is rising, it may signal an upcoming rate hike – vital for fixed‑income portfolio managers.

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SECTION 8: SUMMARY FOR THE DATA PRACTITIONER

• Time series data are not independent – autocorrelation is both a feature and a challenge.

• Stationarity is a prerequisite for most models; use the ADF test and apply differencing if needed.

• Decomposition helps separate trend, seasonality, and cycles to reveal true signals.

• ACF and PACF identify temporal dependencies and guide model selection.

• Smoothing (moving averages, EWMA) provides quick trend estimates and volatility forecasts.

• In banking, time series methods are indispensable for risk measurement, liquidity planning, and economic forecasting.

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[END OF LESSON 6]