Question

Prove that the correlation coefficient at lag 1 for y[n]

Answer 1

Step-by-step explanation:

Actual formula is,

rk=∑nt=k+1(yt−y¯(1))(yt−k−y¯(2))∑nt=k+1(yt−y¯(1))2−−−−−−−−−−−−−−−√∑nt=k+1(yt−k−y¯(2))2−−−−−−−−−−−−−−−−√

rk=∑t=k+1n(yt−y¯(1))(yt−k−y¯(2))∑t=k+1n(yt−y¯(1))2∑t=k+1n(yt−k−y¯(2))2

Where,

y¯(1)=∑nt=k+1ytn−k and y¯(2)=∑nt=k+1yt−kn−k

y¯(1)=∑t=k+1nytn−k and y¯(2)=∑t=k+1nyt−kn−k

So,If n is large enough, we can approximate y¯(1) and y¯(2) to y¯=∑nt=1ytny¯(1) and y¯(2) to y¯=∑t=1nytn............(law of large numbers, average of large sample moves towards population mean.)

Hence,

rk≃∑nt=k+1(yt−y¯)(yt−k−y¯)∑nt=k+1(yt−y¯)2−−−−−−−−−−−−−√∑nt=k+1(yt−k−y¯)2−−−−−−−−−−−−−−−√

rk≃∑t=k+1n(yt−y¯)(yt−k−y¯)∑t=k+1n(yt−y¯)2∑t=k+1n(yt−k−y¯)2

Finally,

rk≃∑nt=k+1(yt−y¯)(yt−k−y¯)∑nt=1(yt−y¯)2.........(same variance)