"Frontmatter". In: Analysis of Financial Time Series

SIMPLE ARMA MODELS 51operator, the model can be written as(1 − φ 1 B −···−φ p B p )r t = φ 0 + (1 − θ 1 B −···−θ q B q )a t . (2.25)The polynomial 1−φ 1 B −···−φ p B p is the AR polynomial of the model. Similarly,1−θ 1 B−···−θ q B q is the MA polynomial. We require that there are no common factorsbetween the AR and MA polynomials; otherwise the order (p, q) of the modelcan be reduced. Like a pure AR model, the AR polynomial introduces the characteristicequation of an ARMA model. If all of the solutions of the characteristic equationare less than 1 in absolute value, then the ARMA model is weakly stationary. In thiscase, the unconditional mean of the model is E(r t ) = φ 0 /(1 − φ 1 −···−φ p ).2.6.3 Identifying ARMA ModelsThe ACF and PACF are not informative in determining the order of an ARMA model.Tsay and Tiao (1984) propose a new approach that uses the extended autocorrelationfunction (EACF) to specify the order of an ARMA process. The basic idea of EACFis relatively simple. If we can obtain a consistent estimate of the AR component of anARMA model, then we can derive the MA component. From the derived MA series,we can use ACF to identify the order of the MA component.The derivation of EACF is relatively involved; see Tsay and Tiao (1984) fordetails. Yet the function is easy to use. The output of EACF is a two-way table,where the rows correspond to AR order p and the columns to MA order q. The theoreticalversion of EACF for an ARMA(1, 1) model is given in Table 2.4. The keyfeature of the table is that it contains a triangle of “O” with the upper left vertexlocated at the order (1, 1). This is the characteristic we use to identify the order ofan ARMA process. In general, for an ARMA(p, q) model, the triangle of “O” willhave its upper left vertex at the (p, q) position.For illustration, consider the monthly log stock returns of the 3M Company fromFebruary 1946 to December 1997. There are 623 observations. The return seriesand its sample ACF are shown in Figure 2.7. The ACF indicates that there are noTable 2.4. The Theoretical EACF Table for an ARMA(1, 1) Model, Where “X” DenotesNonzero, “O” Denotes Zero, and “*” Denotes Either Zero or Nonzero. This Latter CategoryDoes Not Play Any Role in Identifying the Order (1, 1).AR 0 1 2 3 4 5 6 70 X X X X X X X X1 X O O O O O O O2 * X O O O O O O3 * * X O O O O O4 * * * X O O O O5 * * * * X O O OMA