[Math] Calculating the Variance of a Pure Random Walk

Question

I am trying to calculate the first two moments of the random walk below:

$y_t$=$y_0$+$\sum_{j=1} ^t u_j$

• Mean: E[$y_t$]=$y_0$

• Variance: E[$y_t ^2$]=t$\sigma^2$

I understand how to obtain the 1st moment because we assume the expectation of the disturbance terms is equal to 0 and the expectation of a constant is the constant.

I do not understand how to obtain the variance. I tried the following formula:

var($y_t$)=E[$y_t ^2$]-$E[y_t]^2$

I incorrectly simplified this in the following steps:

1. $var(y_t)=E[(y_0+\sum u_j)(y_0+\sum u_j)]+y_0 ^2$

2. $var(y_t)=E[y_0 ^2+2\sum u_j+\sum u_j ^2]+y_0 ^2$

3. $var(y_t)=E[y_0]+2E[\sum u_j]+E[(\sum u_j)^2]$

4. $var(y_t)=y_0$

I am hoping someone can explain that form of the variance is obtained.

This information comes from the ninth slide of the following presentation:
http://www.ncer.edu.au/events/documents/Lecture1_Intro.pdf

Answer 1

For the random walk $y_t = y_0 + \sum u_j$ where disturbance terms $u_j$s are modeled as white noise, mean can be calculated as: $E[y_t] = E[y_0 + \sum u_j] = E[y_0]+ E[\sum u_j] = E[y_0] + \sum E[u_j] = y_0 + \sum 0 = y_0$.

If var($y_t$) = E[$y_t^2$] - $E[y_t]^2$ formula is used to calculate variance of $y_t$ some simplifications can be applied as in the following steps:

1. $var(y_t) = E[(y_0+\sum u_j)(y_0+\sum u_j)] - y_0^2$
2. $var(y_t) = E[y_0^2 + 2y_0\sum u_j + \sum\sum u_j u_i] - y_0^2$
3. $var(y_t) = E[y_0^2] + 2y_0E[\sum u_j] + E[\sum\sum u_j u_i] - y_0^2$
4. $var(y_t) = E[y_0^2] + 2y_0\sum E[u_j] + \sum\sum E[u_j u_i] - y_0^2$

Since $u_j$~$WN(0,\sigma ^2)$, $u_j$ terms are uncorrelated (all covariances of them except variances will be zero). Therefore, expectations in step 4. can be replaced with following values:

• $E[y_0^2] = y_0^2$
• $E[u_j]=0,\ \forall j$
• $E[u_j u_i]=E[u_j^2]=\sigma^2,\ \forall j,i \ where \ j=i$
• $E[u_j u_i]=0,\ \forall j,i \ where \ j\not=i$

If these substitutions are applied, rest of the simplification steps can be written as follows:

5. $var(y_t) = y_0^2 + 2y_0\sum E[u_j] + \sum E[u_j^2] - y_0^2$
6. $var(y_t) = (y_0^2 - y_0^2) + (2y_0\sum 0) + (\sum\sigma^2)$
7. $var(y_t) = 0 + 0 + (\sum\sigma^2)$
8. $var(y_t) = t\sigma^2$