4
$\begingroup$

I would like to ask 5 questions about relations between these processes.

1) Could white noise be also a random walk?

2) Could random walk be also a white noise?

3) Could white noise be stationary?

4) Could random walk be stationary?

5) Could stationary process have autocorrelation?

Thank you in advance.

$\endgroup$
8
$\begingroup$

I will assume a white noise is a process $(\varepsilon_t)$ with zero mean, no autocorrelation and constant variance $\sigma^2 > 0$ while a random walk is a process $(x_t)$ defined by $$ x_{t+1} = x_t + \varepsilon_{t+1} $$ where $\varepsilon$ is a white noise.

1) No since $Var(x_{t+1}) = Var(x_t) + Var(\varepsilon_{t+1})$ is stricly increasing while the variance of a white noise is constant.

2) No same reason as above.

3) By definition, it is stationary up to order 2. A strong white noise (i.e. an i.i.d sequence) is strongly stationary.

4) No, again because $Var(x_{t+1}) > Var(x_t)$.

5) Yes, the simplest example is an AR(1) process $$ x_{t+1} = c + \varphi x_t + \varepsilon_{t+1} $$ It has autocorrelation $\rho(j) = \varphi^j$ and it is stationary if $|\varphi| < 1$.

$\endgroup$
  • $\begingroup$ As per your 3rd point, does it mean that white noise is strictly stationary ? $\endgroup$ – Neeraj Aug 26 '15 at 6:19
2
$\begingroup$

Regarding the relationship between white noise and a random walk, I would put it this way: a random walk is integrated white noise. [And vice versa we get a white noise when we differentiate/difference a random walk]. Or to put it in quant finance terms: white noise is like the daily changes in the S&P in points, a random walk is the S&P daily level itself.

So, just for fun, of these two time series, which is the white noise and which is the random walk?

[2115,2120.5,2117.1,2097.4,2113.4,2114.2,2098.5,2101.6,2099.1,2108.3,2091.3]

[5.5,-3.4,-19.7,16,0.8,-15.7,3.1,-2.5,9.2,-17]

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.