How to validate random walk model I am studying ARIMA models and find it hard to validate the model in terms of "it's a good, useful model" and "I shouldn't use that model for prediction".
So at first I started with the easiest model, a random walk, where the model uses the mean value of the series.
Let's say I have the series:
data1<-c(1,1,1,1,1,1,1,1,1,1,1,1,1)
data2<-c(1,2,1,1,1,1,1,1,1,2,2,1,1)
data2<-c(10,-5,33,1,96,45,-978,3,15,8,9,32,12)

All three are random walks, but obviously the values of the first series data1 are better to predict than for data2 or data3.
My question is, how can I quantify this kind of reliability in a metric? (Ideally in a single metric from 1.0 for data1 and ~0 for data3). Probably the variance is the metric, but how can I standardize it to tell if the predictions from the models will be "good" (1.0) or "bad" (0.0)? 
 A: First of all, you may be misunderstanding what a random walk is. It is a process $x_t$ the value of which at time $t$ is equal to the initial value plus the cumulative sum of shocks at times $1,...t$, i.e. 
$$x_t=x_0+\varepsilon_1+...+\varepsilon_t$$
where the $\varepsilon$'s are random shocks. It could also be expressed as the most recent value plus the most recent shock:
$$x_t=x_{t-1}+\varepsilon_t$$
Your variables data1 through data3 do not seem to be realizations of a random walk process. 
The forecast of value at time $t+1$ given the value at time $t$ equals to value at time $t$ in a random walk model:
$$\hat{x}_{t+1|t} = \mathbb{E}(x_{t+1}|x_t) = \mathbb{E}(x_t+\varepsilon_{t+1}|x_t) = \mathbb{E}(x_t|x_t)+\mathbb{E}(\varepsilon_{t+1}|x_t) = x_t+0 = x_t$$
This is because the shock $\varepsilon_{t+1}$ is random and the best prediction is just its mean, zero (random walk with drift would have a constant term extra).
Regarding the last question of quantifying forecasting performance, you could take a look at established measures such as root mean squared error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), mean absolute scaled error (MASE) and the like. MASE will perhaps be the closest to your requirement of fitting in between zero and one. It will not necessarily be smaller than one, but the interpretation will be quite clear: MASE>1 will signal that the out-of-sample forecast does poorer than a naive forecast performs in sample.
