Financial Time Series Models

A price level, its return, and the uncertainty of that return answer different questions. The price records a cumulative level. A return measures proportional change over an interval. A volatility model describes how large an unexpected return is likely to be, even when its sign remains hard to predict.

This separation is useful when quiet periods are followed by clusters of large positive and negative movements. A mean model can leave almost no linear return autocorrelation while the magnitudes remain strongly dependent. Modeling that dependence requires an equation for conditional variance, not simply more lags in the expected-return equation.

ARCH and GARCH specify how recent squared shocks and earlier variance estimates influence the next conditional variance. A complete model still needs a conditional mean and a standardized shock distribution. The variance controls scale, while the distribution controls how probability is allocated within and beyond that scale. Matching one of those features does not automatically match the other.

Worked calculation: returns and conditional variance

If a price rises from $P_{t-1}=100$ to $P_t=102$, the log return is

$$ r_t=\log(102)-\log(100)=\log(1.02)\approx0.01980. $$

For a GARCH-style variance recursion with $\alpha_0=0.01$, $\alpha_1=0.10$, $\beta_1=0.85$, previous shock $\varepsilon_{t-1}=r_{t-1}-\mu_{t-1}=0.2$, and previous variance $h_{t-1}=0.04$,

$$ h_t=0.01+0.10(0.2^2)+0.85(0.04)=0.048. $$

The conditional standard deviation is

$$ \sqrt{0.048}\approx0.219. $$

The squared previous mean-adjusted return, $\varepsilon_{t-1}^2$, contributes the immediate shock effect, while the previous conditional variance carries persistence forward. These illustrative variance parameters are separate from the price-return calculation above and use one consistent return scale throughout the recursion. This is why raw returns can have little autocorrelation even when squared returns remain strongly dependent.

The figure shows the defining visual feature of many financial return series: quiet periods and volatile periods cluster rather than appearing at a constant scale.

Prices, returns, and changing volatility

Financial time series such as prices, returns, and exchange rates often depart from simple stationary Gaussian assumptions. Common features include heavy tails, asymmetry, volatility clustering, and serial dependence in variance even when the conditional mean has little predictable structure.

Log Returns

If $P_t$ is a positive price, the log return is

$$ r_t=\log(P_t)-\log(P_{t-1}) =\log\left(\frac{P_t}{P_{t-1}}\right). $$

Dividing by the previous price makes a return comparable across different price levels. A movement from 100 to 102 and a movement from 200 to 204 both represent a 2% simple return. Log returns express that proportional change additively over time, which is convenient for aggregation. Neither operation guarantees a stable distribution: breaks, volatility changes, and dependence can remain in returns.

The figure contrasts a persistent price level with its one-period returns. The price path can drift for long periods, while returns fluctuate around a much more stable level and are therefore the more natural target for many short-horizon financial models.

For $P_{t-1}=100$ and $P_t=102$,

$$ r_t=\log(1.02)\approx0.01980. $$

The simple return is

$$ \frac{102}{100}-1=0.02. $$

For small changes,

$$ \log(1+r)\approx r, $$

so simple and log returns are numerically close. Log returns also add across time:

$$ \log\left(\frac{P_{t+h}}{P_t}\right)=\sum_{j=1}^{h}r_{t+j}. $$

A straight trend in log price represents multiplicative growth in price. Taking logs compresses the scale and turns products of gross returns into sums of log returns, but an accumulated stochastic trend can remain in the log level. The separate first-difference step that constructs a log return is what changes the target from log level to proportional increment.

ARCH and GARCH

ARCH and GARCH models describe time-varying conditional variance. A useful mean-variance decomposition is

$$ r_t=\mu_t+\sqrt{h_t}\,z_t, \qquad E(z_t)=0, \qquad \mathrm{Var}(z_t)=1. $$

The conditional mean describes the center given the past, while $h_t$ describes uncertainty around that center. A large $h_t$ does not predict a positive return: it permits larger deviations in either direction under a symmetric shock distribution. Distinguishing a variance forecast from a direction forecast is essential for interpreting both the equations and a volatility plot.

Here $\varepsilon_t=r_t-\mu_t=\sqrt{h_t}\,z_t$ is the mean-adjusted shock. With $z_t$ independent of the past and standardized to mean zero and variance 1, $h_t$ is its conditional variance. An ARCH($q$) model uses recent squared shocks:

$$ h_t=\alpha_0+\sum_{i=1}^{q}\alpha_i\varepsilon_{t-i}^2. $$

A GARCH($p,q$) model adds $p$ lagged conditional variances to the $q$ squared-shock terms:

$$ h_t=\alpha_0 +\sum_{i=1}^{q}\alpha_i\varepsilon_{t-i}^2 +\sum_{j=1}^{p}\beta_jh_{t-j}. $$

A positive intercept and nonnegative shock and lagged-variance coefficients keep the standard recursion positive. Their sizes have separate interpretations: shock coefficients control the immediate response to squared surprises; lagged-variance coefficients carry forward the previous scale estimate. The general $p,q$ naming convention varies across texts and software, so identify the squared-shock and lagged-variance sums in the actual equation before comparing reported orders.

Synthetic Volatility Example

The plot below shows a synthetic return series with volatility clustering together with its conditional volatility.

Large observations tend to occur when the conditional volatility is elevated. The sign of each return remains difficult to predict, but the scale of future fluctuations can be persistent.

These models are widely used when the forecasting target is risk or uncertainty rather than only the conditional mean.

Conditional scale, persistence, asymmetry, and tail forecasts

Financial time-series modeling is clearer when the problem is separated into layers:

1. choose a price or return representation;
2. specify the conditional mean;
3. specify the conditional variance;
4. choose an innovation distribution;
5. evaluate return, volatility, and risk forecasts with time-respecting backtests.

After fitting the mean, distinguish raw residuals from standardized residuals. A large raw residual may be ordinary during a high-variance period. Dividing by the model's conditional standard deviation asks whether it is unusually large relative to the predicted scale. Remaining dependence in squared standardized residuals suggests incomplete variance dynamics; heavy tails in their distribution suggest that the assumed shock shape is still inadequate.

Price levels and log returns

For a positive price $P_t$,

$$ r_t=\log(P_t)-\log(P_{t-1}) =\log\left(\frac{P_t}{P_{t-1}}\right). $$

Log returns are convenient for aggregation and often remove much of the non-stationary level behavior found in prices. They do not automatically remove structural breaks, changing variance, or serial dependence.

Conditional mean and variance

Write

$$ r_t=\mu_t+\sqrt{h_t}\,z_t, \qquad E(z_t)=0, \qquad \mathrm{Var}(z_t)=1. $$

Even when $\mu_t$ is constant or close to zero, $h_t$ can vary strongly. This separation explains why a return series can look unpredictable in sign while still being predictable in magnitude.

ARCH calculation

For ARCH(1),

$$ h_t=\alpha_0+\alpha_1\varepsilon_{t-1}^2. $$

With $\alpha_0=0.02$, $\alpha_1=0.35$, and previous innovation $\varepsilon_{t-1}=0.4$,

$$ h_t=0.02+0.35(0.4^2)=0.076. $$

The conditional standard deviation is

$$ \sqrt{0.076}\approx0.276. $$

A large shock raises next-period variance regardless of its sign in this symmetric ARCH model.

In ARCH(1), a large shock raises the next variance. That higher variance can produce larger subsequent shocks, which feed back through their squares. Thus a finite number of squared-shock lags in the equation does not imply a finite-duration volatility response like the shock cutoff of an MA model. The response can propagate indirectly through later random innovations.

For ARCH(1), the condition $\alpha_1<1$ is required for a finite unconditional variance under the usual assumptions.

GARCH calculation

GARCH(1,1) adds lagged conditional variance:

$$ h_t=\alpha_0+\alpha_1\varepsilon_{t-1}^2+\beta_1h_{t-1}. $$

With $\alpha_0=0.01$, $\alpha_1=0.10$, $\beta_1=0.85$, $\varepsilon_{t-1}=0.2$, and $h_{t-1}=0.04$,

$$ h_t=0.01+0.10(0.2^2)+0.85(0.04)=0.048. $$

The persistence measure is

$$ \alpha_1+\beta_1=0.95. $$

When this sum is below 1, the usual GARCH(1,1) unconditional variance is

$$ \bar h=\frac{\alpha_0}{1-\alpha_1-\beta_1} =\frac{0.01}{0.05}=0.2. $$

For forecasts beyond the next variance, the expected squared future shock equals its conditional variance under the standardized-innovation assumptions. The expected variance then moves toward $\bar h$ at a rate governed by $\alpha_1+\beta_1$. Here the sum is 0.95, so departures can decay slowly. The finite unconditional variance formula requires the denominator $1-\alpha_1-\beta_1$ to be positive; it is not a formula to use unchanged at the unit-persistence boundary.

The figure compares variance paths with different persistence levels. As $\alpha_1+\beta_1$ approaches 1, the response to a volatility shock lasts longer and the process returns more slowly toward its long-run variance.

Why squared returns matter

A volatility process can have

$$ \mathrm{Corr}(r_t,r_{t-1})\approx0 $$

while

$$ \mathrm{Corr}(r_t^2,r_{t-1}^2)>0. $$

The sign of a return can therefore be difficult to predict even when its magnitude is persistent.

The return ACF in the figure is comparatively weak, while the squared-return ACF retains visible structure. This is the diagnostic pattern that motivates conditional-variance models.

Asymmetric volatility

Squaring makes $+x$ and $-x$ contribute equally in a symmetric model. An indicator for a negative shock changes that rule: the squared size still matters, but the sign determines whether an additional coefficient contributes. This lets the model represent a larger next-period variance after a negative shock than after an equally sized positive shock, without claiming that the next return's sign is predictable.

One extension is

$$ h_t=\alpha_0+\alpha\varepsilon_{t-1}^2 +\gamma\mathbf 1(\varepsilon_{t-1}<0)\varepsilon_{t-1}^2 +\beta h_{t-1}. $$

If $\gamma>0$, a negative innovation of a given magnitude contributes more to the next variance than a positive innovation of the same magnitude.

The leverage-effect figure makes that asymmetry visible by comparing the variance response to positive and negative shocks.

Innovation distributions

The innovation distribution describes standardized shocks after their time-varying scale is removed. A Student-$t$ distribution can allocate more probability to large standardized surprises than a normal distribution. To keep $h_t$ as conditional variance, standardize the $t$ shock to variance 1; for degrees of freedom $\nu>2$, this rescales an ordinary $t_\nu$ variable by $\sqrt{(\nu-2)/\nu}$. If the required moments do not exist, variance-based interpretations must be reconsidered.

The figure emphasizes that matching conditional variance is not enough if the standardized residual distribution still produces more extremes than the assumed normal model. Tail behavior matters directly for interval and risk forecasts.

Risk forecasts

For a one-step return forecast with conditional mean $\mu_{t+1}$ and standard deviation $\sqrt{h_{t+1}}$, a Gaussian lower quantile at probability $\alpha$ is

$$ q_{\alpha,t+1}=\mu_{t+1}+z_\alpha\sqrt{h_{t+1}}. $$

The lower return quantile identifies a threshold exceeded downward with probability $\alpha$ under the forecast distribution. With the positive-loss convention, Value-at-Risk negates that return threshold; other conventions can label the probability by its complementary confidence level. Expected shortfall instead averages the relevant tail outcomes. State the convention and units, then check later exceedances and tail behavior rather than treating a fitted quantile as automatically calibrated.

The shock-response figure shows how a single large innovation affects future conditional volatility. The decay rate determines how quickly risk forecasts return toward their long-run level.

Modeling cautions

Return construction depends on the data definition. A stock split can look like an enormous loss if prices are not adjusted consistently. A missing weekend differs from an unobserved trading-day return, and decimal returns differ from percentage-point returns by a factor of 100, making variances differ by 10,000. Resolve units, corporate actions, and the trading calendar before interpreting estimated volatility parameters.

Backtests can also be biased by revised data, survivorship filtering, or parameters estimated over regimes that differ from the evaluation period. Reproduce the information set that would actually have been available at each forecast origin.

Workflow

1. Check prices, corporate actions, units, and trading dates.
2. Convert prices to returns when the target concerns relative change.
3. Plot returns, absolute returns, and squared returns.
4. Fit the conditional mean before diagnosing variance dependence.
5. Fit ARCH/GARCH or an asymmetric/heavy-tailed extension when diagnostics justify it.
6. Check standardized residuals and squared standardized residuals.
7. Backtest volatility forecasts and tail quantiles chronologically.
8. Examine stability across market regimes and innovation distributions.

Formula reference