Long-Run Effects in an ARDL(1,1) Model

The model

An autoregressive distributed lag model of order (1,1) takes the form

\[ y_t = \alpha + \beta_0 x_t + \beta_1 x_{t-1} + \gamma y_{t-1} + \varepsilon_t \]

where \(\beta_0\) and \(\beta_1\) capture the immediate and one-period lagged effects of \(x\) on \(y\), and \(\gamma\) is the autoregressive coefficient governing the persistence of \(y\).

Derivation

In the long run the system settles to a steady state in which \(y\) and \(x\) no longer change between periods. Imposing \(y_t = y_{t-1} = \bar{y}\), \(x_t = x_{t-1} = \bar{x}\), and \(\mathrm{E}[\varepsilon_t] = 0\) gives

\[ \bar{y} = \alpha + \beta_0 \bar{x} + \beta_1 \bar{x} + \gamma \bar{y} \]

Because \(x_t\) and \(x_{t-1}\) both equal \(\bar{x}\), the two \(x\) terms combine. Collecting \(\bar{y}\) on the left and factoring:

\[ \bar{y}(1 - \gamma) = \alpha + (\beta_0 + \beta_1)\bar{x} \]

Dividing through by \((1 - \gamma)\), which requires the stability condition \(|\gamma| < 1\):

\[ \bar{y} = \frac{\alpha}{1 - \gamma} + \frac{\beta_0 + \beta_1}{1 - \gamma} \cdot \bar{x} \]

Long-run multiplier

The coefficient on \(\bar{x}\) is the long-run effect of a one-unit permanent change in \(x\) on \(y\):

\[ \boxed{\frac{\partial \bar{y}}{\partial \bar{x}} = \frac{\beta_0 + \beta_1}{1 - \gamma}} \]

The numerator sums all direct effects of \(x\) — immediate and lagged. The denominator amplifies this through \(y\)’s own persistence: as \(\gamma \to 1\) the multiplier grows without bound.

Numerical example

lr_mult <- function(beta0, beta1, gamma) (beta0 + beta1) / (1 - gamma)

gammas  <- c(-0.5, 0.0, 0.25, 0.50, 0.75, 0.90)

knitr::kable(
  data.frame(gamma = gammas, lr = lr_mult(0.4, 0.3, gammas)),
  digits    = 3,
  col.names = c("γ", "Long-run multiplier"),
  align     = "cc"
)

Table 1: Long-run multiplier for selected values of γ (β₀ = 0.4, β₁ = 0.3)

γ	Long-run multiplier
-0.50	0.467
0.00	0.700
0.25	0.933
0.50	1.400
0.75	2.800
0.90	7.000

With \(\beta_0 + \beta_1 = 0.7\) fixed, the multiplier rises from 0.47 at \(\gamma = -0.5\) to 7 at \(\gamma = 0.9\), illustrating the amplification role of the autoregressive term.