Exactly The Same Autocovariance Function of Two Time Series A MA(2) process :
$$X_t=W_t+\frac{5}{2}W_{t-1}-\frac{3}{2}W_{t-2}$$
where $\{W_t\}\sim WN(0,1)$
And another MA(2) process :
$$X_t=W_t-\frac{1}{6}W_{t-1}-\frac{1}{6}W_{t-2}$$
where $\{W_t\}\sim WN(0,9)$
Both  time series have exactly the same autocovariance function .  What does it imply ? What is the message of the example ?
 A: 
Both time series have exactly the same autocovariance function . What
  does it imply ? What is the message of the example ?

That the autocovariance function (which is something we estimate based on the data available), in general, does not uniquely identify the process.
A related discussion can be found in Jenkins and Watts (1968), "Spectral Analysis and its Applications" (ch. 5.26, p. 170):

"A stochastic process is said to be Gaussian or Normal if the multivariate  distribution associated with an arbitrarily chosen set
  of points is multivariate Normal. Then the process is completely
  characterized by its mean, variance and acf (autocorrelation function). However, there will be a
  very wide class of non-Normal processes which have the same acf as a
  given Normal process but which differ markedly from it in other
  respects."

They offer a specific example after the above passage, and they conclude that higher-order moments should come into the picture.
A: The autocovariance function is closely related to the autocorrelation function of the deterministic sequences $\left[1 \quad \frac 52 \quad -\frac 32\right]$ and $\left[1 \quad -\frac 16 \quad -\frac 16\right]$ where the autocorrelation function of the
deterministic sequence $[x_0, x_1, \ldots, x_{N-1}]$ is defined as
$$R_x(k) = \sum_{i=0}^{N-k-1} x_i x_{i+k}, ~~ 0 \leq k \leq N-1.$$
It is not unusual for two deterministic sequences to have the same autocorrelation function.
A: Interesting question. The ACF or the autocovariance structure evidently will suggest the form of the ARIMA model BUT not the values of the coefficients of the ARIMA model.
