Question

Generate n = 100 observations from the autoregression

x t = − . 9 x t − 2 + w t
with σ w = 1, using the method described in Example 1.10. Next, apply the moving average filter
v t = ( x t + x t − 1 + x t − 2 + x t − 3 ) / 4
to x t , the data you generated. Now plot x t as a line and superim- pose v t as a dashed line. Comment on the behavior of x t and how applying the moving average filter changes that behavior.
Repeat (a) but with x t = cos(2 πt/ 4) .

Answer 1

To generate and analyze 100 observations from a specified autoregression and apply a moving average filter, you can use a statistical software or programming language. The resulting plots will reveal the effect of the moving average filter, which typically smooths out fluctuations to highlight trends. For a deterministic function like a cosine wave, the filter will dampen the amplitude.

Step-by-step explanation:

To generate n = 100 observations from the autoregression Xt = -0.9Xt-2 + Wt with σw = 1, you can use a statistical software or programming language like R or Python. You would start by initializing the first two observations (X1 and X2) as they will be needed for the autoregression, and then you would use a loop to calculate each consecutive value of Xt using the specified autoregression formula and drawing random samples for Wt from a normal distribution with standard deviation 1.

To apply the moving average filter Vt = (Xt + Xt-1 + Xt-2 + Xt-3)/4 to the data, you would again use a loop to calculate each Vt based on the corresponding X values. After generating both the Xt and Vt series, you could plot them using a plotting library in your software of choice, using a line for Xt and a dashed line for Vt. The moving average filter will smooth out the fluctuations in the time series data, providing a clearer view of the underlying trend.

When repeating the process with Xt = cos(2πt/4), you're dealing with a deterministic periodic function. The application of the moving average will dampen the amplitude of the cosine function, as the averaging process smooths out the high and low points of the wave over the observed time period.