EMET8002 Chap.7 Unit Roots, Lag Models and Volatility
Unit Roots, Lag Models and Volatility
Testing before modelling
Time series models need stationary data.
This chapter covers the Dickey-Fuller and augmented Dickey-Fuller tests for unit roots, orders of integration, and structural break tests, then the autoregressive and distributed lag models that are fitted once the data are stationary.
Systems and volatility
Lag length is chosen with the AIC and BIC, Granger tests measure predictive content, vector autoregressions treat variables as jointly determined, and cointegrated pairs call for error correction models.
ARCH and GARCH models capture volatility that clusters over time.
Testing and forecasting in the Week 8 lab
The Week 8 lab tests GDP per capita for unit roots with the Dickey-Fuller and augmented tests, chooses the lag length of autoregressions with the AIC and BIC over one to eight lags, fits ADL(1,1) models for Australia and the United States, and forecasts 2023. Marks go to stating each null hypothesis correctly, giving the order of integration with evidence, and explaining every lag coefficient in words.
What this chapter covers
- 01
Stationarity
- 02
Dickey-Fuller and augmented Dickey-Fuller tests
- 03
Orders of integration
- 04
Structural breaks and the Chow test
- 05
AR and ADL models
- 06
Lag length with AIC and BIC
- 07
Granger causality and VARs
- 08
Cointegration and error correction
- 09
ARCH and GARCH volatility
Worked example · free
A unit root decision and a forecast
- 1Levels: −1.90 is not below −2.86, so the unit root null is not rejected.
- 1Differences: −5.40 is below −2.86, so the change is stationary and the series is I(1).
- 1Forecast change: 0.6 + 0.35 × 2.0 = 1.30.
- 1Forecast level: 150 + 1.30 = 151.3.
Key terms
- Stationary Series
- A series whose mean, variance and autocorrelation structure do not change over time.
- Granger Causality
- Predictive content of one variable's lags for another, tested by a joint F-test.
- Cointegration
- A shared stochastic trend that makes a combination of two non-stationary series stationary.
- Error Correction Model
- A model of changes that includes the lagged gap from a long-run relation.
- GARCH Model
- A model in which the error variance depends on past squared errors and its own past values.
Unit Roots, Lag Models and Volatility FAQ
Why can't a unit root be tested with an ordinary t-test?
Under the null of a unit root, the test statistic does not follow the usual distribution, so the standard critical values are wrong. The Dickey-Fuller test uses its own critical values, which statistical tables give and Stata calculates.
Does Granger causality prove that one variable causes another?
No. It shows only that the lags of one variable help predict another beyond the other regressors. The lecture states directly that Granger causality does not imply a causal relationship.
Why is GARCH usually preferred to ARCH?
GARCH lets the error variance depend on its own past values as well as past squared errors. One lagged variance summarises a long history of shocks, so GARCH needs fewer coefficients and is more parsimonious.
How do I choose the number of lags in an autoregression?
Estimate the model with increasing numbers of lags and compute the AIC and BIC for each, choosing the smallest value. The BIC tends to choose fewer lags than the AIC. Theory, partial autocorrelations and an F-test on the last lag can help, and between two good models the more parsimonious one is preferred.
What does it mean for two series to be cointegrated?
Two non-stationary series are cointegrated when they share a stochastic trend, so that a combination of them, such as the gap between long and short interest rates, is stationary. Cointegrated pairs are modelled with an error correction model that keeps the long-run relation.
Assessment move
Draw the testing sequence from memory: plot, test the trend, test for a unit root, difference, retest, then choose the model. Run it on one real series and state the order of integration with the statistics that support it. Then explain, using one example from finance, why large price changes tend to cluster and which model the course offers for that pattern.
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