For many time series, particularly prices in speculative...

For many time series, particularly prices in speculative markets, the random walk model has been found to give a good representation of actual data. This model is written as follows: $$ x_{t}=x_{t-1}+\varepsilon_{t} $$ Show that, if this model is appropriate, forecasts of $x_{n+l b}$ standing at time $n$, are given by $$ \hat{x}_{n+h}=x_{n} \quad(h=1,2,3, \ldots) $$

For many time series, particularly prices in speculative markets, the random walk model has been found to
give a good representation of actual data. This model is written as follows:
$$
x_{t}=x_{t-1}+\varepsilon_{t}
$$
Show that, if this model is appropriate, forecasts of $x_{n+l b}$ standing at time $n$, are given by
$$
\hat{x}_{n+h}=x_{n} \quad(h=1,2,3, \ldots)
$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Random Walk Model

This is a type of time series model where the current value is simply the previous value plus a random error term. Mathematically, it is represented as x? = x??? + ??. The key characteristic of a random walk is that its future movements are unpredictable because they depend on new random shocks, making the process inherently non-stationary.

Time Series Forecasting

Forecasting in time series involves predicting future values of a series based on its past behavior. In the context of a random walk, since the errors have an expected value of zero, the best forecast for any future time point is simply the most recently observed value. Thus, the h-step forecast for a random walk is given by the current value, irrespective of the forecast horizon.

White Noise Error Term

The error term (??) in the random walk model is typically assumed to be white noise, meaning it is a sequence of independent and identically distributed random variables with a mean of zero. This property ensures that any deviations from the current value are purely random with no predictable pattern, reinforcing the idea that the best forecast for future values is just the current observation.

Transcript

Please give Ace some feedback