Zero sample correlation does not necessarily mean there is no linear relationship - page 52
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Read and weep at....
Stationary and ergodic series have constant mat expectation, variance, autocorrelation function and are extrapolated by horizontal or nearly horizontal line.
The question is why, from a practical point of view, should we even consider QC for stationary and ergodic series?
are extrapolated with a horizontal or nearly horizontal line.
Take a series of the form x[i] = -0.5+(i%2); i=1,2...+Inf: -0.5, 0.5, -0.5, 0.5, ... Stationary, MO = 0, variance = 0.25. ACF equals 1 for zero and even lag values, and equals -1 for odd lag values. Extrapolation using any straight line will give an error variance of at least 0.25; extrapolation using the formula x_hat[i+1]=-x[i] will give zero error. :P
Take a series of the form x[i] = -0.5+(i%2); i=1,2...+Inf: -0.5, 0.5, -0.5, 0.5, ... Stationary, MO = 0, variance = 0.25. ACF equals 1 for zero and even lag values, and equals -1 for odd lag values. Extrapolation using any straight line will give an error variance of at least 0.25; extrapolation using the formula x_hat[i+1]=-x[i] will give zero error. :P
geez, well, on a Saturday night almost, this is certainly brutal, but I'll give it a try - the series extrapolated by a straight line with what slope angle?
For that process, it is fundamentally impossible to get an error variance of less than 0.25 when extrapolating a straight line, no matter what the slope and vertical offset of the straight line are. However, an autoregressive model can easily be constructed to produce zero error.
The example was given to refute your assertion about the extrapolability of any stationary and ergodic straight line process. Your statement is only true for processes with IID increments. For stationary ergodic processes that are not delta-correlated, you can construct an AR model whose error variance will be smaller than that of extrapolation using any straight line. In the case of non-linear dependencies between samples of such a process, it is also possible to construct a model that is better than a straight line.
For that process, it is fundamentally impossible to get an error variance of less than 0.25 when extrapolating a straight line, no matter what the slope and vertical offset of the straight line are. However, an autoregressive model can easily be constructed to produce zero error.
The example was given to refute your assertion about the extrapolability of any stationary and ergodic straight line process. Your statement is only true for processes with IID increments. For stationary ergodic processes that are not delta-correlated, you can build an AR model whose error variance will be smaller than that of extrapolation using any straight line. In the case of non-linear dependencies between samples of such a process, it is also possible to construct a model that is better than a straight line.
)))very funny
1. I did NOT write that a stationary and ergodic process is best extrapolated by a straight line. Don't make this up. Certainly for some stationary and erg processes, non-linear extrapolation gives better accuracy.
2. Don't care about error variance. This process, like stats and erg, is extrapolated with a horizontal or nearly horizontal straight line. Or, the line extrapolating the stat and erg process must be horizontal or nearly horizontal.
P.S. But the question remains the same - why, from a practical point of view, to calculate QC for stat and erg series at all?
My bad, I answered the "why" question.
Why - in order to identify dependencies in the data that are specific to the series.