3. Autoregressive (AR) Model
ECO374H1
Department of Economics
Summer 2025
Cycles
ρt within A (or B or C of D) is positive (lag of 1 to 2 years)
ρt
of Yt between A and B (or C and D) is negative (lag of 4 to 5 years)
ρt
of Yt between A and D is positive (lag of 10 to 11 years)
The ACF of the Unemployed persons data changes between positive and negative, which is typical for a cycle (see code Öle 3a. AR Motivation)
Model ACF
We would like to fit on the data a time series model that can closely approximate the dynamic pattern in the data
We will show that the Autoregressive (AR) process has such ACF
Hence, an AR model component will be suitable to fit to the data and for forecasting of the series
Note the contrast with the ACF of the MA model discussed previously
AR Model
An autoregressive model of order p 1 denoted AR(p)1 is given by
where {εt} is the white noise process
We will start with AR(1) and then extend the analysis to AR(p)
For each process, we will ask three questions:
What does a time series of the given AR process look like?
What does its ACF look like?
What is the optimal forecast?
AR(1)
For simulated data from the AR(1) process
see code file 3b. AR1 Simulation (section 1. Simulated Data)
The parameter φ is called the persistence parameter since it ináuences the "persistence" of the series
The series with φ = 0.95 stays longer above or below the unconditional mean than the series with φ = 0.4
The series with φ = 1 is extremely persistent, in fact it is non-stationary
AR(1) is stationary only for jφj < 1
ACF and PACF
The ACF decreases exponentially towards zero, with faster decay for smaller φ
r1 ≠ 0 but rk = 0 for k > 1
For ACF and PACF of the AR(1) process see code file 3b. AR1 Simulation (section 2. ACF and PACF)
The same features as for positive φ also hold for negative φ but with alternating signs (section 3. Negative φ)
Forecast for h = 1
The optimal forecast of the AR(1) model is equal to the conditional expectation:
For the forecasting horizon h = 1,
Since Yt ∈ It,
Forecast Error for h = 1
The 1-perod ahead forecast error is
The forecast variance is
Density Forecast for h = 1
Assuming , the density forecast is
The 95% confidence interval is then
where 1.96 is the 95% critical value from the Normal distribution
Forecasts for h > 1
The optimal forecast for h > 1 is
Similarly, we can show that
Forecasts for h → ∞
As h → ∞, the forecast converges to
which is the unconditional mean of {Yt}, and
which is the unconditional variance of {Yt}
Hence, the AR(1) model is suitable for forecasts in the short to medium term
Convergence of its forecasts to unconditional moments still indicates "short" memory of the process, albeit relatively longer than for MA(1)
Note that these results hold only for stationary AR(1) with |φ| < 1