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Univariate Time Series

Single-series models in two movements. First the conditional mean: Box–Jenkins SARIMA forecasting, and the change-point → Markov-switching ladder — a Bayesian AR with mean and variance shifts and a Markov-switching AR — that lets the dynamics break. Then the conditional variance, a full volatility arc: GARCH, asymmetric GJR-GARCH with Value-at-Risk backtesting, GARCH-in-mean (the risk–return tradeoff), stochastic volatility, realized volatility (HAR and Realized GARCH from intraday data), and regimes vs long memory in volatility persistence — capped by an application that prices options under GARCH and stochastic volatility. Each is built from scratch and cross-checked across Python and R.

Stationarity, and why two tests rather than one. Nearly everything below assumes the series is stationary — that its statistical behaviour does not drift, so the mean and the autocorrelations mean the same thing at the start of the sample as at the end. A trending series is not, and fitting one as though it were produces a model that mistakes the trend for memory. The usual repair is differencing: model the changes rather than the levels. Deciding whether that is needed is what the two tests do, and the reason both appear is that they are set up in opposite directions. The ADF test has non-stationarity as its null, so rejecting it is evidence for stationarity; the KPSS test has stationarity as its null, so rejecting that is evidence against. Running one gives you a result you cannot fully interpret — failing to reject a null is not evidence for it — while running both lets the two agree, which is genuine evidence, or disagree, which is a warning that the series is borderline. The other diagnostics recur too: the ACF plots the correlation of the series with itself at each lag, which is how a seasonal spike is told apart from a damped cycle, and the Ljung–Box test asks whether any autocorrelation is left in the residuals, which is the standard check that a fitted model has taken out everything it should.

What volatility actually is. Seven of the ten examples below are about volatility, so it is worth being plain about the word. A daily financial return splits into a predictable part, which is almost nothing — tomorrow’s direction is very close to unforecastable, and one of the examples below is an honest demonstration of just how close — and an unpredictable part, which is almost everything. Volatility is the scale of that unpredictable part: not which way the price will move, but how far. The empirical fact that generates this entire arc is that the scale is not constant. It moves, and it moves in a particular way — large days cluster with large days, calm with calm. That is volatility clustering, and its consequence is the asymmetry the whole section lives off: the size of tomorrow’s move is forecastable even though its direction is not.

Which is why so much machinery points at it. Direction is where the money is and cannot be predicted; magnitude can be, and it is what almost everything downstream is denominated in. A risk limit is a statement about volatility. So is a bank’s capital requirement. So, very nearly entirely, is an option price — which is why the arc ends by pricing options. The models differ mainly in where they get the current level from: GARCH computes it from past returns by a fixed recursion, stochastic volatility treats it as a hidden state with noise of its own, and realized volatility measures it directly by adding up squared intraday returns. The forecasting tournament near the end asks which wins, and the answer is worth knowing in advance: the choice of model matters much less than whether the intraday data was used at all.

Persistence, and how to read it as a half-life. The volatility arc keeps reporting one number — the persistence, α+β, and it is nearly always around 0.99. That value is easier to think about as a half-life: if a volatility shock decays by a factor α+β each day, half of it survives for log(½) / log(α+β) days. At 0.99 that is about 69 days; at 0.997, the figure most of these fits report, it is 231 days, most of a trading year. That is what near-unit-root persistence means in practice: a shock to volatility today is still half-present the better part of a year later, which is why the question of whether that persistence is real or an artefact gets a whole example of its own.

How a forecast is scored. Three loss measures appear and they are not interchangeable. RMSE is the root-mean-square error in the units of the series, and MAPE the mean absolute error as a percentage, which makes it comparable across series of different size — both are for forecasts of the level. Volatility forecasts need something else, because the thing being forecast is never observed: you compare a predicted variance against a noisy proxy for the realised one, and a symmetric loss like RMSE will systematically favour models that under-predict. QLIKE is the standard fix — a loss that penalises proportional error rather than absolute, so being wrong by a factor of two costs the same whether volatility is high or low, and which stays honest under a noisy proxy. It is the score the realized-volatility verdict rests on, and lower is better.

Value-at-Risk, and what backtesting it means. Two examples end in VaR: the loss threshold a portfolio should exceed only 1% of days (or 5%). It is a quantile of the predicted return distribution, so it tests the whole model at once — the volatility forecast and the tail shape together. Backtesting asks two separate questions of the realised exceedances. Were there the right number of them? That is Kupiec’s unconditional-coverage test. And were they spread out, or did they arrive in clusters? That is Christoffersen’s independence test — a model can take exactly 1% of hits and still be useless if they all land in one bad fortnight, which is precisely what happens to the models that ignore volatility clustering. Expected Shortfall answers the question VaR does not: given that the threshold was breached, how bad was it on average?