Unit 3: Time series analysis
Business Forecasting notes · PTU syllabus (MBA 964-18)
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Unit summary
Time series methods forecast by learning from the pattern of past values. This unit covers smoothing and extrapolation, seasonal adjustment, the autocorrelation function, stationarity and the random walk, linear time series models — moving average, autoregressive and ARIMA models with the Box–Jenkins methodology — and ARCH and GARCH volatility models.
After this unit you can
- Apply smoothing, extrapolation and seasonal adjustment
- Explain autocorrelation, stationarity and the random walk
- Apply the Box–Jenkins ARIMA methodology
- Explain ARCH and GARCH volatility models
PTU syllabus topics
- Smoothing and extrapolation of time series
- seasonal adjustment
- autocorrelation function
- stationarity
- random walk
- linear time series (moving average, autoregressive, ARIMA models, Box-Jenkins methodology)
- ARCH/GARCH volatility modeling
- 1Check stationarity
Difference if needed
- 2Identify p and q
ACF and PACF
- 3Estimate the ARIMA model
- 4Diagnose residuals
Should be white noise
- 5Forecast
Topic 1
Smoothing and seasonal adjustment
Simple moving average
Average of the latest n periods
Exponential smoothing
Ft+1 = α At + (1 − α) Ft
Seasonal index (ratio to moving average)
Actual ÷ centred moving average × 100, averaged by season
Example
With α = 0.3, last forecast 500 and actual 540, the next forecast = 0.3 × 540 + 0.7 × 500 = 512.
- Deseasonalised data = actual ÷ seasonal index × 100 — shows the underlying trend.
- SPSS: Analyze → Forecasting → Create Traditional Models (Expert Modeler chooses exponential smoothing or ARIMA); Seasonal Decomposition is under Analyze → Forecasting.
Topic 2
Extrapolation of trends
Straight line
Yc = a + bX
Normal equations
ΣY = na + bΣX; ΣXY = aΣX + bΣX²
With ΣX = 0
a = ΣY ÷ n; b = ΣXY ÷ ΣX²
Even number of years
Code X as −5, −3, −1, 1, 3, 5 (half-year units)
- Merits: objective, gives a trend value for every year and enables forecasting. Limitations: assumes a linear trend; adding data changes all values.
Topic 3
Autocorrelation function
- Autocorrelation: correlation of a series with its own past values at lag k.
- ACF: plot of autocorrelations at different lags; PACF: partial autocorrelation at lag k after removing intermediate lags.
- Patterns: slowly declining ACF suggests non-stationarity (trend); spikes at lag 12 in monthly data suggest seasonality.
Topic 4
Stationarity and the random walk
- Stationary series: constant mean, variance and autocorrelation over time — required for ARMA models.
- Making a series stationary: differencing (first difference Yt − Yt−1), seasonal differencing, log transformation.
- Tests: Augmented Dickey–Fuller (ADF) unit root test.
- Random walk: Yt = Yt−1 + et — the best forecast is the last value; stock prices approximately follow it.
Topic 5
Linear time series models
AR(p)
Value depends on its own past values: Yt = c + φ1 Yt−1 + … + et
MA(q)
Value depends on past errors: Yt = μ + et + θ1 et−1 + …
ARMA(p, q)
Both components for stationary series
ARIMA(p, d, q)
ARMA after d differences
SARIMA
Adds seasonal terms
AR(p)
Tails off gradually
Cuts off after lag p
MA(q)
Cuts off after lag q
Tails off gradually
ARMA
Tails off
Tails off
Topic 6
Box–Jenkins methodology
- 1Identification
Make stationary; use ACF and PACF to choose p, d, q
- 2Estimation
Estimate parameters (maximum likelihood)
- 3Diagnostic checking
Residuals should be white noise (Ljung–Box test); compare AIC and BIC
- 4Forecasting
Generate forecasts and intervals; monitor accuracy
Topic 7
ARCH and GARCH volatility models
- Volatility clustering: large changes tend to follow large changes (financial markets).
- ARCH (Engle, 1982): variance today depends on past squared errors.
- GARCH(1,1) (Bollerslev, 1986): σt² = ω + α e²t−1 + β σ²t−1 — variance depends on past shocks and past variance; α + β close to 1 means persistent volatility.
- Uses: forecasting volatility for risk management, option pricing, Value at Risk.
Key terms
- Autocorrelation
- Correlation of a series with its past values
- Stationarity
- Constant statistical properties over time
- Random walk
- Series where the next value equals the last plus noise
- ARIMA
- Autoregressive integrated moving average model
- GARCH
- Model of time-varying volatility
Quick revision
- Moving averages, exponential smoothing, seasonal indices, deseasonalising; trend extrapolation.
- ACF and PACF; stationarity, differencing, ADF; random walk.
- AR, MA, ARMA, ARIMA, SARIMA; identification patterns.
- Box–Jenkins: identify, estimate, check, forecast.
- ARCH and GARCH; volatility clustering.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.What is the ACF?
- Q2.Define a stationary series.
- Q3.What is a random walk?
- Q4.How is the order of an AR model identified?
- Q5.Name the stages of Box–Jenkins methodology.
- Q6.What does GARCH model?
Long-answer questions
- Q1.Explain smoothing, extrapolation and seasonal adjustment of time series.
- Q2.Explain autocorrelation, stationarity and the random walk.
- Q3.Explain ARIMA models and the Box–Jenkins methodology.
- Q4.Explain ARCH and GARCH volatility models.
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