Prompt Title: AR(1) vs. Random Walk

Both AR(1) and random walk models are commonly used in time series analysis, but they have some important differences.

AR(1) Model: The AR(1) model is an autoregressive model of order 1, which means it describes the relationship between a variable and its own past values. Specifically, an AR(1) model can be written as:

y_t = c + ϕ y_(t-1) + ε_t

where y_t is the variable of interest at time t, y_(t-1) is its value at time t-1, ε_t is an error term at time t, c is a constant, and ϕ is a coefficient that represents the relationship between y_t and y_(t-1).

In an AR(1) model, the coefficient ϕ determines the strength of the autocorrelation between y_t and y_(t-1). If ϕ is positive, it means that y_t tends to follow the same direction as y_(t-1), while if it is negative, y_t tends to move in the opposite direction of y_(t-1). If ϕ is zero, it means that there is no autocorrelation between y_t and y_(t-1).

Random Walk Model: A random walk model, on the other hand, assumes that the variable of interest follows a random process where the change from one period to the next is based on a random shock or error term. Specifically, a random walk model can be written as:

y_t = y_(t-1) + ε_t

where y_t is the variable of interest at time t, y_(t-1) is its value at time t-1, ε_t is a random error term at time t.

In a random walk model, the error term ε_t is assumed to be normally distributed with a mean of zero and constant variance. This means that the change in y_t from one period to the next is unpredictable and can take on any value.

Key Differences: The key difference between the two models is that an AR(1) model assumes that the future value of the variable is dependent on its past values, while a random walk model assumes that the future value of the variable is based purely on randomness. In other words, an AR(1) model allows for some degree of predictability based on past values, while a random walk model does not.

Another important difference is that an AR(1) model can be stationary or non-stationary, depending on the value of the coefficient ϕ. If |ϕ| < 1, the model is stationary, meaning that the mean, variance, and autocorrelation structure of the variable do not change over time. If |ϕ| > 1, the model is non-stationary, meaning that the mean, variance, and autocorrelation structure of the variable change over time.

On the other hand, a random walk model is always non-stationary, because the variance of the variable increases over time as the number of periods increases. This means that a random walk model cannot be used for long-term forecasting, as it assumes that the future value of the variable will always be equal to its current value plus a random shock.

In summary, the main difference between AR(1) and random walk models is that AR(1) allows for some degree of predictability based on past values, while a random walk model assumes that the future value of the variable is based purely on randomness. Additionally, an AR(1) model can be stationary or non-stationary, while a random walk model is always non-stationary.