As indicated in the previous article, the salient features of FX data include the presence of certain statistical patterns that justify the use of specific forecasting models, depending on the distribution of the currency you are trying to analyze. For the purpose of this discussion, we will be limiting the alternative models to different variations of the GARCH family of models.
Estimation Method
To determine the appropriate degree of the autoregressive process, a series of regressions is employed to conduct the optimal lag test. Subsequently, comparisons are made using both the Akaike Information Criterion (AIC) and Schwarz Information Criterion (SIC) across lags ranging from 1 to 14. The model displaying the lowest AIC and SIC values is selected. The analysis indicated that the FX data under consideration best aligns with an AR(14) process, as evident in Table 1, which shows an illustrative result of optimal lag testing. Additionally, it is worth noting that the FX data adheres to a driftless random walk, as demonstrated by the insignificance of the constant term in the regression of the natural logarithm of FX. Consequently, the GARCH(1,1) technique is employed for modeling the error term, as seen by looking at the model’s output.
Given an assumed AR(14) process for FX returns, the estimation process proceeds with GARCH modeling at the specified parameters, as well as alternative GARCH models, which include in-mean specifications. The conditional variance is computed alongside the mean equation for the models. These models can be estimated using various statistical software available. This specific guide utilizes the Berndt, Hall, Hall, and Hausman (BHHH) optimization algorithm for GARCH model computations.
Model Specification Test
The initial step involves assessing whether the FX data exhibits the anticipated volatility characteristics as described above. If so, the next task is to determine which of the alternative GARCH models provides the most accurate description of the time-varying conditional volatility process. This selection is made through in-sample goodness-of-fit tests.
In-sample Goodness of Fit Tests
For each model, standardized residuals are estimated and subsequently examined for correlation and ARCH effects. Additionally, squared normalized residuals are tested for serial correlation. To test for ARCH effects, the LM-ARCH test is employed, while the Ljung-Box Q statistic is used to assess correlation up to order 14. Other valuable goodness-of-fit tests applied in this study include the AIC and SIC tests. These tests hold particular significance as they are not contingent on a specific distribution, making them exceptionally useful for comparing the three error term distributions used in this study.
The AIC is utilized to select the model based on the equation:
AIC = -2 * log(L) + 2 * k
Here, ‘k’ represents the number of estimated parameters, and ‘log(L)’ is the maximum likelihood of the model. The model with the lowest AIC value is preferred. Similarly, the SIC is determined by:
SIC = -2 * log(L) + k * log(T)
Where ‘T’ is the number of observations used for estimation. The model with the lowest SIC value is considered the best-fitting model.
In-sample Forecast Comparison
To evaluate the in-sample forecast performance of the conditional volatility models, forecast errors are generated in conjunction with the software mentioned earlier. Specifically, Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) are computed to identify the model that offers the most accurate FX data forecasts. These metrics are defined as follows:
RMSE = √(Σ(y – ŷ)² / n)
MAE = Σ|y – ŷ| / n
These evaluations are pivotal in determining which among the alternative GARCH models provides the most reliable forecasts given the available data and estimation results. As this initial battery of tests on the data provides you with all the information you need regarding the specific FX data of interest, you can move forward in attempting to forecast how the volatility of returns behaves and use it as a guide in FX trading decisions.