ARCH-LM Test and ARCH, GARCH, EGARCH, TGARCH

International ocean freight rates don't move by the same amount every day. During relatively calm periods, rates change only a little, but when events that shake the market occur, such as port strikes or shipping disruptions, large rises and falls can continue for days or weeks. This is a different issue from rises and falls in the average rate, because you need to look not only at the direction of change but also at how unstable the size of the change is.

The phenomenon in a time series where large changes are followed by large changes and small changes by small changes is called volatility clustering. ARCH and GARCH family models are used to describe this time-varying volatility. The ARCH-LM test is the starting point for checking whether volatility needs to be modeled, and EGARCH and TGARCH go further by allowing for the possibility that shocks of the same size affect volatility differently depending on whether they are rises or falls.

In this article we use daily log changes in an international ocean freight rate index as an example and walk through the ARCH-LM test, model estimation and interpretation, asymmetric volatility, and diagnostics and forecasting.

But even after the mean is described adequately, the size of the residual fluctuations can vary over time. For example, if the daily log change of the freight rate index usually moves within ±1% but moves by ±4% during certain periods, a single fixed error variance can hardly explain the difference.

ARCH family models estimate a structure in which the current conditional variance depends on past shocks and past variances. So the question of interest is not whether tomorrow's rate will definitely rise or fall, but rather how large the uncertainty in tomorrow's change will be.

The index is recorded as positive values, and the analysis uses daily log changes rather than the index level.

The log change can be defined as follows.

Here P_t is the freight rate index on day t. A log change of 0.02 corresponds to a rise of about 2% for small changes. In the calculations below, log changes are used as decimals; that is, a 3% drop is -0.03.

Because the first index value is used up in computing the changes, the number of change observations available for analysis can be one fewer than the number of time points in the raw data.

For example, if the changes are autocorrelated, you can specify a mean model that includes AR terms and the like.

μ_t is the conditional mean at time t, σ_t is the conditional standard deviation, and z_t is the shock standardized to mean 0 and variance 1. If the mean model is estimated appropriately, the directional autocorrelation of the residuals ε_t should not be large. But if autocorrelation remains in the squared residuals ε_t², the volatility may be changing over time.

Rather than applying a volatility model first while leaving trends or seasonality in the raw data as they are, it is good to make clear whether you are analyzing the index level or the changes and to review the mean structure first.

It squares the residuals and checks whether those values are explained by past squared residuals.

The null hypothesis is γ₁ = ⋯ = γ_q = 0, that is, there are no ARCH effects within the specified range of lags. A significant test is evidence that the size of past shocks is related to the current squared residual.

In this result, the null hypothesis of no ARCH effects can be rejected. So there are grounds for allowing time-varying volatility rather than assuming a constant residual variance. However, a significant ARCH-LM test does not guarantee that any particular GARCH model is correct.

The simplest, ARCH(1), is as follows.

Under constraints such as ω > 0 and α₁ ≥ 0, if a larger-than-expected change occurred the previous day, today's conditional variance can increase. Because squared residuals are used here, rising and falling shocks of the same size have the same effect.

ARCH is good for explaining how volatility clustering works, but representing long-lasting volatility may require many past lags. GARCH is the best-known model that addresses this.

The widely used GARCH(1,1) is as follows.

α reflects the response to new shocks, and β the degree to which volatility persists. In the basic GARCH(1,1) with nonnegativity constraints, α + β < 1 satisfies one of the usual conditions for weak stationarity.

In this example, the sum of the two coefficients is 0.94, close to 1. This shows that once freight rate volatility rises, it may not die out quickly. However, this sum of coefficients alone cannot fully describe every form of shock persistence.

Assume the previous day's unexpected shock to the change was -3% (ε_t−1 = -0.03) and the previous day's conditional variance was 0.0002.

That is, the next day's conditional standard deviation is about 1.73%. Compared with the previous day's conditional standard deviation of √0.0002 ≈ 1.41%, you can see that the expected volatility rose after the large shock.

This is not a forecast that the rate will rise or fall by 1.73%. It is an estimate of conditional uncertainty that indicates how much the change could swing.

But in real markets, rises and falls of the same size can affect subsequent volatility differently. EGARCH (exponential GARCH) models the log of the conditional variance and reflects the sign of the shock separately. A typical EGARCH(1,1) form is as follows.

Here z_t−1 = ε_t−1 / σ_t−1. Under the sign convention of this equation, when γ < 0, a negative shock of the same size raises the next period's volatility more than a positive shock.

For example, if freight rate drops turn out to raise subsequent volatility more than rises, EGARCH can describe that feature better than the basic GARCH, which treats shocks in both directions the same way. However, the sign of EGARCH coefficients can be defined differently depending on the model equation your software uses, so always check the estimated equation as well.

However, the name TGARCH refers to different models in different literature and software. Here we use the GJR-GARCH form, which allows an additional variance response to negative shocks.

I(ε_t−1 < 0) is an indicator function that equals 1 when the previous shock is negative. In this equation, the squared-shock coefficient for positive shocks is α, and for negative shocks it is α + γ. So if γ > 0, negative shocks trigger a larger volatility response.

Some software also calls the Zakoian approach, which applies the threshold effect to absolute values or the standard deviation, TGARCH. So in a report, don't write just the name 'TGARCH'; clearly present the variance equation you used.

First check whether the residuals of the mean model show ARCH effects, and look at whether persistence and asymmetry in volatility are actually observed.

For example, suppose you estimated the models on the same 1,000-day sample and the AIC values were GARCH -6,250, GJR-GARCH -6,261, and EGARCH -6,268. If the same dependent variable and sample and comparable likelihoods were used, EGARCH, with the lowest AIC, is relatively preferable. But don't settle on a model from the numerical differences alone; also check diagnostics on the standardized and squared residuals, the forecasting purpose, and estimation stability.

If the model describes the conditional variance well, no clear autocorrelation should remain in the squared standardized residuals.

Typical diagnostics include the ACF of the standardized residuals, the Ljung–Box test on the squared standardized residuals, and a test for remaining ARCH effects (ARCH-LM). If heavy tails remain in the residuals, you can consider an innovation distribution such as a standardized Student's t instead of the normal distribution.

When comparing several models, it is good to review out-of-sample forecasts over the same evaluation period. If you use a proxy for realized volatility, also consider the measurement error of the proxy and the properties of the evaluation loss function.

A conditional standard deviation of 1.7% does not mean the rate will fall by 1.7% the next day. The mean model describes the expected value of the change, and the variance model describes the uncertainty around it.

Use consistent units for the data.

The size of ω and of the variance changes depending on whether you enter log changes as decimals like 0.03 or as percentages like 3. State the units used in the results table.

Distinguish extreme shocks from structural breaks.

If the process that generates volatility itself has changed because of changes in market rules or a restructuring of logistics networks, a single GARCH model may have trouble explaining the whole period. You can also consider structural break or regime-switching models.

Account for time dependence and out-of-sample validation.

When evaluating a model, randomly shuffling observations to split them into training and evaluation data can break the time order. Estimating on the past and evaluating on a future period is recommended.

Interpreting asymmetry coefficients depends on the model equation.

EGARCH and GJR-GARCH express asymmetry in different ways. Don't compare the signs of the coefficients in isolation.

If you claim asymmetric volatility, also state which shock sign the model was defined to respond to more strongly.

In real research, you should add the coefficients' standard errors and confidence intervals, the innovation distribution used, the exact values of the diagnostic tests, and so on.

The ARCH-LM test checks whether the residuals show signs of time-varying variance; ARCH uses past shocks, and GARCH uses both past shocks and past variances. EGARCH and the TGARCH family go further to handle the possibility that rising and falling shocks affect volatility differently.

In a real analysis, a useful order is to first check the mean structure and ARCH effects, estimate an appropriate variance model, and then review the results through the standardized residuals and out-of-sample forecasts.