ARMA methodology

ARMA models are particularly useful when information is limited to a single stationary series,8 or when economic theory is not useful. They are a "highly refined curve-fitting device that uses current and past values of the dependent variable to produce accurate short-term forecasts" (Hanke and Reitsch, 1998: 407).

The ARMA methodology does not assume any particular pattern in a time series, but uses an iterative approach to identify a possible model from a general class of models. Once a tentative model has been selected, it is subjected to tests of adequacy. If the specified model is not satisfactory, the process is repeated using other models until a satisfactory model is found. Sometimes, it is possible that two or more models may approximate the series equally well, in this case the most parsimonious model should prevail. For a full discussion on the procedure refer to Box et al. (1994), Gourieroux and Monfort (1995), or Pindyck and Rubinfeld (1998). The ARMA model takes the form:

Yt = + faYt-1 + faYt-2 +-----+ <PpYt-p + et - w^t-1 - W2et-2-----wqst-q

8 The general class of ARMA models is for stationary time series. If the series is not stationary an appropriate transformation is required.

Figure 1.9 (1,40) combination moving average Excel spreadsheet (in-sample)

where Yt is the dependent variable at time t; Yt—1, Yt—2, ■ ■ ■, Yt—p are the lagged dependent variables; ■ ■ ■ are regression coefficients; et is the residual term;

et-1, et-2, ■ ■ ■, et-p are previous values of the residual; w\, w2, ■ ■ ■ ,wq are weights.

Several ARMA specifications were tried out, for example ARMA(5,5) and ARMA(10,10) models were produced to test for any "weekly" effects, which can be reviewed in the arma.wfl EViews workfile. The ARMA(10,10) model was estimated but was unsatisfactory as several coefficients were not even significant at the 90% confidence interval (equation arma1010). The results of this are presented in Table 1.5. The model was primarily modified through testing the significance of variables via the likelihood ratio (LR) test for redundant or omitted variables and Ramsey's RESET test for model misspecification.

Once the non-significant terms are removed all of the coefficients of the restricted ARMA(10,10) model become significant at the 99% confidence interval (equation arma13610). The overall significance of the model is tested using the F-test. The null hypothesis that all coefficients except the constant are not significantly different from zero is rejected at the 99% confidence interval. The results of this are presented in Table 1.6. Examination of the autocorrelation function of the error terms reveals that the residuals are random at the 99% confidence interval and a further confirmation is given by the serial correlation LM test. The results of this are presented in Tables 1.7 and 1.8. The model is also tested for general misspecification via Ramsey's RESET test. The null hypothesis of correct specification is accepted at the 99% confidence interval. The results of this are presented in Table 1.9.

Table 1.5 ARMA(10,10) EUR/USD returns estimation

Dependent Variable: DR-USEURSP Method: Least Squares Sample(adjusted): 12 1459

Included observations: 1448 after adjusting endpoints Convergence achieved after 20 iterations

White Heteroskedasticity-Consistent Standard Errors & Covariance Backcast: 2 11

Variable

Coefficient

Std. error

t -Statistic

Prob.

0 0

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