ARIMA, Intuitively

An autoregressive model does what the name promises — regression on itself:

ŷ_t = c + φ₁·y_{t−1} + φ₂·y_{t−2} + … + φ_p·y_{t−p}

Tomorrow's value is a weighted sum of the last p values, plus noise. An AR(1) with φ = 0.8 says: today is 80% of yesterday plus a surprise. An AR(7) on daily energy load can literally learn 'Mondays behave like Mondays.'

The order p — how many lags matter — is where the PACF earns its keep: it isolates each lag's direct influence after removing the echoes through shorter lags.