You are given an annual time series with 40 consecutive values and asked to fit a fifth-order autoregressive model. How many parameters do you need to estimate?
To fit a fifth-order autoregressive (AR) model, we need to estimate the coefficients for each lag term up to lag 5.
An autoregressive model of order 5 is expressed as:
x[t] = c + a1*x[t-1] + a2*x[t-2] + a3*x[t-3] + a4*x[t-4] + a5*x[t-5] + ε[t]
Where:
- x[t] represents the value at time t in the time series.
- c is the intercept term.
- a1, a2, a3, a4, a5 represent the coefficients for each lag term.
- ε[t] is the error term (residual).
In total, there are 6 parameters to estimate for a fifth-order AR model:
- 1 parameter for the intercept (c)
- 5 parameters for the coefficients of the lag terms (a1, a2, a3, a4, a5)
Therefore, we need to estimate 6 parameters to fit the given fifth-order autoregressive model to the annual time series.