If a Simple N -span moving average is applied to a time series that has a linear
ID: 3276423 • Letter: I
Question
If a Simple N-span moving average is applied to a time series that has a linear trand, say, yt = 0 + 1t + t the moving average will lag behind the observations. Assume that the observations are uncorrelated and have constant cariance. Show that at time T the expected value of the moving average is
E(MT) = 0 + 1T - ((N-1)/2) 1
241 If a simple N-span moving average is that has a linear trend, say, y, = Bo + But will lag behind the observations. Assumi uncorrelated and have constant variance. expected value moving of the average is _applied to a time series + E, the moving average that the observations are Show that at time T the ECM) = B, + B,T - N;.Explanation / Answer
weighted average of N-span is average of previous N- periods
now
y_t = b0 + b1* t + e
at T
= 1/ N ((b0 + b1* (T) ) + (b0 + b1* (T - 1) )+ ... (b0 + b1 * (T -(n-1))
= 1/N * (n *b0 + b1* ( nT - (1 + 2 + ...(n-1))
= 1/N * (n*b0 + b1* n T - b1 * n(n-1)/2) {1+2+..(n-1) = n*(n-1)/2}
= b0 + b1*T - (n -1)/2 * b1