READING 26 : CHARACTERIZING CYCLES

Covariance Stationary Time Series

        A time series is covariance stationary if the relationships among its present and past values remain stable over time.
         The covariance between the current value of a time series and its value τ periods in the past is referred to as its autocovariance at displacement τ. Its autocovariances for all τ make up its autocovariance function.
         To convert an autocovariance function to an autocorrelation function, we divide the autocovariance at each τ by the variance of the time series.
         Autoregression is a linear regression of a time series against its own past values at a specific lag. The regression coefficient that results is referred to as the partial autocorrelation for that lag. Partial autocorrelations for all lags make up the partial autocorrelation function.

Requirements to be Covariance Stationary

To be covariance stationary, a time series must exhibit the following three properties:

1.   Its mean must be stable over time.
2.   Its variance must be finite and stable over time.
3.   Its covariance structure must be stable over time.

         If a time series is not covariance stationary, we cannot model it directly from its past values. However, we can often transform a series and find a covariance stationary underlying process —for example, by filtering out trend and seasonality.

White Noise

        White noise refers to a serially uncorrelated time series that has a mean of zero and a constant variance. A time series is serially uncorrelated if it exhibits no correlation among any of its lagged values.
         If the observations in a white noise process are independent, as well as uncorrelated, the process is referred to as independent white noise. If the process also follows a normal distribution, it is known as normal or Gaussian white noise.

Characteristics Of The Dynamic Structure Of White Noise

         Events in a white noise process exhibit no correlation between the past and present. Thus, its autocorrelations and partial autocorrelations are zero for any displacement. Although its unconditional mean is zero and its variance is constant, a process may have a conditional mean and variance that are not necessarily constant. If so, past values may be useful for forecasting.

Lag Operator

        The notation for a lag operator is L^my_t = y_t–m, where y_t is the value of a time series at time t, and m is a number of periods before time t.
         A distributed lag is a model that assigns weights to past values of a time series. A lag operator polynomial of degree m is a distributed lag that includes all lags from 1 to m.

Wold’s Theorem & General Linear Process

         Wold’s theorem, or Wold’s representation, proposes that a covariance stationary process can be modeled as an infinite distributed lag of a white noise process as follows:

         Because this expression can be applied to any covariance stationary series, it is known as a general linear process.

         Wold’s representation can be approximated with a ratio of rational distributed lags, which are distributed lags with a finite number of terms. Such an approximation has only as many terms as the two rational distributed lags in the ratio.

The Sample Mean & Sample Autocorrelation

         The sample mean of a time series is the arithmetic average of its observations. The sample autocorrelation for displacement τ is estimated by:

         The Box-Pierce Q-statistic is a test statistic for the hypothesis that the autocorrelations of a time series are jointly equal to zero. The Ljung-Box Q-statistic is similar but may be more useful with small samples. Both statistics follow a chi-squared distribution.

Sample Partial Autocorrelation

         While partial autocorrelations are theoretically based on an infinite-length time series, sample partial autocorrelations can be estimated from a finite number of time series observations.