The statistical modeling (proper) period: 1927-present

The formulation of explicit statistical models for time series began with the classic papers of Yule (1927) (Autoregressive (AR( p)) scheme):

p

yt = ao + X akVt-k + Ef, Et ~ NI(0, о2), t = 1, 2,…,

k=1

where "NI" stands for "Normal, Independent" and Slutsky (1927) (Moving Average (MA(q)) scheme):

yt = Yo + X YkE-k + Et, Et ~ NI(0, о2), t = 1, 2,….

k=1

Viewing these formulations from today’s vantage point, it is clear that, at the time, they were proposed as nothing more than convenient descriptive models for time series data. Their justification was based exclusively on the fact that when simulated these schemes gave rise to data series which appear to exhibit cycles similar to those observed in actual time series data. It should be noted that, at the time, the difference between regular cycles due to seasonality and irregular cycles due to positive dependence was insufficiently understood.

The first attempt to provide a proper probabilistic foundation for these schemes is undoubtedly that of Wold (1938) who successfully fused these schemes with the appropriate probabilistic concepts necessary to model the chance regularity patterns exhibited by certain time series data. The appropriate probabilistic con­cepts were developed by Kolmogorov (1933) and Khinchine (1932). H. Cramer (1937) was instrumental in providing the missing link between the empirical literature on time series and the mathematical literature on stochastic processes; the probabilistic framework of modeling time series. Wold (1938), in his Ph. D. under Cramer, proposed the first proper statistical framework for modeling stationary time series. The lasting effect of Wold’s work comes in the form of (i) his cele­brated decomposition theorem (see Section 4), where under certain regularity restrictions a stationary process can be represented in the form of a MA(^):

yt = Y0 + X t-k + et, X | Yk | < <*>, et ~ NI(0, о2), t = 1, 2,…,

k=1 k=1

and (ii) the ARMA( p, q) model:

p q

yt = a0 + X aкУ-k + X Yket-k + et, ^ ~ NI(0, о2), t = 1, 2,…,

k=1 k=1

appropriate for time series exhibiting stationarity and weak dependence.

Wold’s results provided the proper probabilistic foundations for the empirical analysis based on both the periodogram and the correlogram; the autocorrelations are directly related to the coefficients (a 0, a 1,…, ap, y1, Y2,…, Yq) and the

periodogram can be directly related to the spectral representation of stationary stochastic processes; see Anderson (1971) for an excellent discussion.

The mathematical foundations of stationary stochastic processes were streng­thened as well as delineated further by Kolmogorov (1941) but the ARMA( p, q) formulation did not become an empirical success for the next three decades because the overwhelming majority of times series appear to exhibit temporal heterogeneity (nonstationarity), rendering this model inappropriate. The state of the art, as it relates to the AR(p) family of models, was described at the time in the classic paper by Mann and Wald (1943).

The only indirect influence of the ARMA( p, q) family of models to econometric modeling came in the form of an extension of the linear regression model by adding an AR(1) model for the error term:

yt = pTxt + Ut, Ut = put-1 + et, I P I < 1, et ~ NI(0, о2), t = 1, 2,…, T.

(28.1)

This model provided the basis of the well known Durbin-Watson bounds test (see Durbin and Watson, 1950, 1951), which enabled econometricians to test for the presence of temporal dependence in the context of the linear regression model. This result, in conjunction with Cochrane and Orcutt (1949) who suggested a

way to estimate the hybrid model (28.1), offered applied econometricians a way to use time series data in the context of the linear regression model without having to worry about spurious regressions (against which Yule (1926) cautioned time series analysts).

The important breakthrough in time series analysis came with Box and Jenkins (1970) who re-invented and popularized differencing as a way to deal with the apparent non-stationarity of time series:

Adyt, where A := (1 – L), d is a positive integer, t = 1, 2,…, in order to achieve (trend) stationarity and seasonal differencing of period s:

AsVt := (1 – Ls)Vt = (Vt – Vt-s^

to achieve (seasonal) stationarity. They proposed the ARMA( p, q) model for the differenced series Adsyt := y*t, giving rise to the ARIMA(p, d, q) model:

p q

y* = ao + X ay« + X Yk£t-k + Et, Et ~ NI(0, о2), t = 1, 2,…

k=1 k=1

Box and Jenkins (1970) did not just propose a statistical model but a modeling strategy revolving around the ARIMA( p, d, q) model. This modeling procedure involved three stages. The first was identification: the choice of (p, d, q) using graphical techniques, the autocorrelations (correlogram) and the partial auto­correlations. The second stage was the diagnostic checking in order to assess the validity of the assumptions underlying the error term. The third stage was a for­mal forecasting procedure based on the estimated model. It must be noted that up until the 1970s the empirical forecasting schemes were based on ad hoc moving averages and exponential smoothing. The Box-Jenkins modeling strategy had a lasting effect on econometric modeling because it brought out an important weak­ness in econometric modeling: the insufficient attention paid to the temporal dependence/heterogeneity exhibited by economic time series; see Spanos (1986, 1987) for further discussion. This weakness was instrumental in giving rise to the LSE modeling methodology (see Hendry, 1993, for a collection of papers) and the popularization of the vector autoregressive (VAR( p)) model by Sims (1980).

The next important development in time series modeling came in the form of unit root testing in the context of the AR(p) model proposed by Dickey and Fuller (1979, 1981). The proposed tests provided the modeler with a way to decide whether the time series in question was stationary, trend nonstationary or unit root nonstationary. Since the 1930s it has been generally accepted that most eco­nomic time series can be viewed as stationary around a deterministic trend. Using the Dickey-Fuller testing procedures, Nelson and Plosser (1982) showed that, in contrast to conventional wisdom, most economic time series can be better described by the AR( p) model with a unit root. These results set off an explosion of time series research which is constantly revisiting and reconsidering the initial results by developing new testing procedures and techniques.

Phillips (1986, 1987) dealt effectively with the spurious correlation (regression) problem which was revisited in Granger and Newbold (1974), by utilizing and extending the Dickey and Fuller (1979, 1981) asymptotic distribution results. The same results were instrumental in formulating the notion of cointegration among time series with unit roots (see Granger, 1983; Engle and Granger, 1987; Johansen, 1991; Phillips, 1991) and explaining the empirical success of the error-correction models proposed by the LSE tradition (see Hendry, 1993).

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