What is the distinguishing feature of the ACF in an AR(1) process with a positive coefficient of AR component?
A It decreases exponentially as lag increases.
B It increases exponentially over time.
C It alternates between positive and negative.
D It remains constant at all lags.
The correct answer and explanation is:
The correct answer is:
A. It decreases exponentially as lag increases.
Explanation:
An AR(1) process (first-order autoregressive process) is defined by the equation: Xt=ϕXt−1+ϵtX_t = \phi X_{t-1} + \epsilon_t
where:
- ϕ\phi is the autoregressive coefficient (AR coefficient),
- ϵt\epsilon_t is white noise with zero mean and constant variance,
- ∣ϕ∣<1| \phi | < 1 to ensure stationarity.
Autocorrelation Function (ACF) of AR(1):
The autocorrelation function measures the correlation between values of the time series at different lags kk, denoted as ρ(k)\rho(k).
For an AR(1) process, the ACF has a very specific form: ρ(k)=ϕk\rho(k) = \phi^k
This means the autocorrelation at lag kk is the AR coefficient raised to the power of kk.
What happens when ϕ\phi is positive?
- Since 0<ϕ<10 < \phi < 1, raising it to increasing powers kk produces values that get smaller.
- The ACF starts at 1 at lag 0 (correlation with itself), then at lag 1 it is ϕ\phi, at lag 2 it is ϕ2\phi^2, and so on.
- Because ϕ\phi is positive, the ACF remains positive but decreases exponentially towards zero as the lag increases.
- This behavior is smooth and monotonic decay without oscillations.
What if ϕ\phi were negative?
- The ACF would alternate in sign (positive at lag 0, negative at lag 1, positive at lag 2, etc.), leading to a “zig-zag” pattern.
- This behavior matches option C (alternates between positive and negative) but only applies when ϕ\phi is negative.
Other options:
- B. It increases exponentially over time.
This is false because autocorrelation decreases as lag increases. - C. It alternates between positive and negative.
Only true if ϕ\phi is negative, not for positive ϕ\phi. - D. It remains constant at all lags.
This would describe a non-stationary process, not AR(1).
Summary:
For an AR(1) process with a positive AR coefficient, the autocorrelation function decreases exponentially as lag increases, reflecting the diminishing influence of past values on current observations over time.