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CYCLES AND THE PERIODOGRAM The graph of the series in the first frame shows some cyclical pattern but it is quite well masked by the white-noise errors whose standard deviation is comparable with the amplitudes of the cycles. 025 and . 1. There are, however, smaller peaks which can only be ascribed to the white-noise term in the series. To understand these features, we use the approximate properties of the estimates, that A and Β are independent normal with means A and Β and variance ( 2 / η ) σ .
3) where, as before, a, is a sequence of independent and identically distributed (iid) variables and the functions / and g are to be determined from the data. In addition to this equation, the time series model has to specify the model for the evolution of the univariate independent series x,, that will be a univariate time series model. In the general case, we cannot say that the variable x, causes y, or vice versa because there is feedback between the two variables. The time series model will describe this situation by two dynamic regression equations.
The simplest linear prediction method is to extrapolate a deterministic curve fitted to the time series by linear regression. This curve may, for example, consist of polynomials for trend, sinusoids for cycles, and indicator variables for seasonality. Consider the monthly atmospheric carbon dioxide series shown in Figure 2. la, which may be modeled, for t = 1, 2 , . . 1) ( where _ '·' _ ί 1 I 0 in each January 1 in other months J _ ί 1 " 1 2 · ' ~ 10 in each December 1 in other months J" ( ' A Course in Time Series Analysis, Edited by Daniel Pefia, George C.