Common Factor Restrictions in ARDL(p,q) Models

Derivations, Test Statistics, and General Cases

Robert Walker

2026-07-27

Outline

  1. General Framework — the ARDL(p,q) model; COMFAC definition; the Wald and Sargan F tests
  2. Specific cases — full derivations for ARDL(1,0), (1,1), (2,1), (1,2), (2,2)
  3. New cases — ARDL(3,1); general ARDL(1,q); general ARDL(p,1); ARDL(n,n) full factorisation
  4. Summary table, structural patterns, sequential testing, references

Part I — Framework

The ARDL(p,q) Model

Suppressing the intercept without loss of generality:

A(L)\,y_t = B(L)\,x_t + \varepsilon_t, \qquad \varepsilon_t \sim \mathrm{i.i.d.}(0,\,\sigma^2)

A(L) = 1 - \phi_1 L - \cdots - \phi_p L^p, \qquad B(L) = \beta_0 + \beta_1 L + \cdots + \beta_q L^q

Parameter vector \theta = (\phi_1,\ldots,\phi_p,\,\beta_0,\ldots,\beta_q)^\intercal \in \mathbb{R}^{k_u}
Free parameters k_u = p + q + 1 (the +1 from the contemporaneous term \beta_0)
OLS covariance \hat{V} = \hat\sigma^2(X'X)^{-1}, \quad \hat\sigma^2 = \mathrm{RSS}_u/(T-k_u)

Common Factor Restriction — Definition

A COMFAC restriction of order r holds when \exists polynomial C(L) = \prod_{j=1}^r(1-\rho_j L) such that

A(L) = C(L)\,A^*(L), \qquad B(L) = C(L)\,B^*(L)

with \deg A^* = p-r, \deg B^* = q-r, and maximum testable order r_{\max} = \min(p,q).

Equivalent condition: A(z) and B(z) share r common roots, i.e., \exists\, z_0 such that

A(z_0) = 0 \quad \text{and} \quad B(z_0) = 0 \quad \text{simultaneously}

Implication: Under the restriction, A^*(L)y_t = B^*(L)x_t + u_t with C(L)u_t = \varepsilon_t — a simpler model with coloured errors rather than a correctly-specified lag structure.

Important

Cochrane–Orcutt / Prais–Winsten GLS on a static regression imposes a COMFAC restriction. It should be tested, not assumed. — Hendry & Mizon (1978)

The General Wald Test Statistic

Let H_0: g(\theta) = 0 where g:\mathbb{R}^{k_u}\to\mathbb{R}^r is smooth. Define the r\times k_u Jacobian G(\theta) = \partial g / \partial\theta'.

Wald statistic (Sargan, 1980):

\mathcal{W} = \hat{g}'\;\bigl[\hat{G}\,\hat{V}\,\hat{G}'\bigr]^{-1}\;\hat{g} \;\xrightarrow{d}\; \chi^2(r), \qquad \hat{g} = g(\hat\theta),\quad \hat{G} = G(\hat\theta)

Sargan F-statistic — compares NLS restricted residuals to OLS unrestricted residuals:

F_S = \frac{(\mathrm{RSS}_r - \mathrm{RSS}_u)\,/\,r}{\mathrm{RSS}_u\,/\,(T - k_u)} \;\sim\; F(r,\,T-k_u) \quad \text{under } H_0

Note

\mathcal{W}, F_S, and the LR statistic T\log(\mathrm{RSS}_r/\mathrm{RSS}_u) are asymptotically equivalent under H_0. The F_S form is preferred when the Jacobian \hat{G} is near-singular at the null — a genuine risk in ARDL(2,2) with r=1.

Part II — Specific Cases

ARDL(1,0) — No COMFAC Possible

y_t = \phi\,y_{t-1} + \beta_0\,x_t + \varepsilon_t

r_{\max} = \min(1,0) = 0 \quad\Longrightarrow\quad \textbf{no common factor restriction exists}

  • Standard partial-adjustment specification
  • B(L) = \beta_0 has degree zero; there is no lag for B to share a root with A
  • This is the terminus of sequential testing: if testing reduces to q=0, the procedure stops
  • Serves as the restricted model when r=1 is accepted in ARDL(2,1) or ARDL(3,1)

ARDL(1,1) — The Canonical Case

y_t = \phi\,y_{t-1} + \beta_0\,x_t + \beta_1\,x_{t-1} + \varepsilon_t \theta = (\phi,\,\beta_0,\,\beta_1)^\intercal, \quad k_u = 3, \quad r_{\max} = 1

Step 1. Postulate C(L) = 1-\rho L. Since p-r=0 we require A^*(L)=1, forcing \rho = \phi.

Step 2. Require (1-\phi L)\mid B(L), i.e. B(1/\phi)=0: \beta_0 + \beta_1/\phi = 0 \implies \boxed{g(\theta) = \beta_1 + \phi\,\beta_0 = 0}

Jacobian (1\times 3): G(\theta) = \bigl(\beta_0,\;\phi,\;1\bigr)

Wald statistic: \mathcal{W} = \frac{(\hat\beta_1 + \hat\phi\,\hat\beta_0)^2}{\hat\beta_0^2\widehat{\mathrm{Var}}(\hat\phi) + \hat\phi^2\widehat{\mathrm{Var}}(\hat\beta_0) + \widehat{\mathrm{Var}}(\hat\beta_1) + 2\hat\beta_0\hat\phi\,\widehat{\mathrm{Cov}}(\hat\phi,\hat\beta_0) + 2\hat\beta_0\,\widehat{\mathrm{Cov}}(\hat\phi,\hat\beta_1) + 2\hat\phi\,\widehat{\mathrm{Cov}}(\hat\beta_0,\hat\beta_1)} \sim \chi^2(1)

Sargan F: Restricted model is y_t = \beta_0^* x_t + u_t, u_t = \phi u_{t-1} + e_t (k_r = 2): F_S = \frac{\mathrm{RSS}_r - \mathrm{RSS}_u}{\mathrm{RSS}_u/(T-3)} \sim F(1,\,T-3)

Citations: Hendry & Mizon (1978) — identification of the restriction; Sargan (1980) — formal Wald test.

ARDL(2,1) — One Restriction

y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \beta_0 x_t + \beta_1 x_{t-1} + \varepsilon_t \theta = (\phi_1,\,\phi_2,\,\beta_0,\,\beta_1)^\intercal, \quad k_u = 4, \quad r_{\max} = 1

Step 1. Postulate C(L) = 1-\rho L with A^*(L) = 1-\lambda L. The common root condition is: \rho^2 - \phi_1\rho - \phi_2 = 0 \tag{i}

Step 2. Require B(1/\rho)=0: \beta_0 + \beta_1/\rho = 0 \implies \rho = -\beta_0/\beta_1 \tag{ii}

Step 3. Substitute (ii) into (i) and multiply by \beta_1^2: \boxed{g(\theta) = \beta_1^2 + \phi_1\beta_0\beta_1 - \phi_2\beta_0^2 = 0}

This is the resultant of A(z) and B(z): it vanishes if and only if they share a common root.

Jacobian (1\times 4): G(\theta) = \bigl(\beta_0\beta_1,\;\;-\beta_0^2,\;\;\phi_1\beta_1 - 2\phi_2\beta_0,\;\;2\beta_1 + \phi_1\beta_0\bigr)

Sargan F: Restricted model is ARDL(1,0) with AR(1) errors (k_r=3): F_S = \frac{\mathrm{RSS}_r - \mathrm{RSS}_u}{\mathrm{RSS}_u/(T-4)} \sim F(1,\,T-4)

Citation: Sargan (1980, Section 2, Example 1).

ARDL(1,2) — One Restriction (q > p case)

y_t = \phi\,y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + \beta_2 x_{t-2} + \varepsilon_t \theta = (\phi,\,\beta_0,\,\beta_1,\,\beta_2)^\intercal, \quad k_u = 4, \quad r_{\max} = 1

Step 1. Since p - r = 0 the only candidate is C(L) = A(L) = 1-\phi L, so \rho = \phi.

Step 2. Require (1-\phi L)\mid B(L), i.e. B(1/\phi)=0: \beta_0 + \beta_1/\phi + \beta_2/\phi^2 = 0

Multiply by \phi^2: \boxed{g(\theta) = \phi^2\beta_0 + \phi\,\beta_1 + \beta_2 = 0}

Jacobian (1\times 4): G(\theta) = \bigl(2\phi\beta_0 + \beta_1,\;\;\phi^2,\;\;\phi,\;\;1\bigr)

Sargan F: Restricted model is ARDL(0,1) with AR(1) errors (k_r=3): F_S = \frac{\mathrm{RSS}_r - \mathrm{RSS}_u}{\mathrm{RSS}_u/(T-4)} \sim F(1,\,T-4)

Caution

The q>p case yields a restricted model whose “signal” is ARDL(0,1) rather than a purely static regression. The interpretation is less natural; most applied work focuses on p \geq q.

Citations: Sargan (1980); Banerjee, Dolado, Galbraith & Hendry (1993), Ch. 3.

ARDL(2,2) — One Restriction (r = 1)

y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \beta_0 x_t + \beta_1 x_{t-1} + \beta_2 x_{t-2} + \varepsilon_t \theta = (\phi_1,\,\phi_2,\,\beta_0,\,\beta_1,\,\beta_2)^\intercal, \quad k_u = 5, \quad r_{\max} = 2

Step 1. The shared root z_0 satisfies both: (A)\!: 1 - \phi_1 z_0 - \phi_2 z_0^2 = 0 \qquad (B)\!: \beta_0 + \beta_1 z_0 + \beta_2 z_0^2 = 0

Step 2. Multiply (A) by \beta_0 and subtract from (B); divide by z_0 \neq 0: \underbrace{(\phi_1\beta_0+\beta_1)}_{Q} + \underbrace{(\phi_2\beta_0+\beta_2)}_{P}\,z_0 = 0 \implies z_0 = -Q/P

Step 3. Substitute z_0 = -Q/P into (A) and multiply by P^2:

\boxed{g(\theta) = P^2 + \phi_1\,PQ - \phi_2\,Q^2 = 0} P = \phi_2\beta_0+\beta_2, \quad Q = \phi_1\beta_0+\beta_1

Citation: Sargan (1980, Theorem 1 — resultant characterisation).

ARDL(2,2) — One Restriction (r = 1): Jacobian

From g = P^2 + \phi_1 PQ - \phi_2 Q^2 with \partial P/\partial\phi_2 = \partial Q/\partial\phi_1 = \beta_0, all other \phi-partials zero, and \partial P/\partial\beta_2 = \partial Q/\partial\beta_1 = 1:

G(\theta) = \Bigl( \underbrace{PQ + \beta_0(\phi_1 P - 2\phi_2 Q)}_{\partial g/\partial\phi_1},\;\; \underbrace{\beta_0(2P+\phi_1 Q)-Q^2}_{\partial g/\partial\phi_2},\;\; \underbrace{(\phi_1^2+2\phi_2)P - \phi_1\phi_2 Q}_{\partial g/\partial\beta_0},\;\; \underbrace{\phi_1 P - 2\phi_2 Q}_{\partial g/\partial\beta_1},\;\; \underbrace{2P+\phi_1 Q}_{\partial g/\partial\beta_2} \Bigr)

Wald statistic: \mathcal{W} = \frac{\bigl(\hat P^2 + \hat\phi_1\hat P\hat Q - \hat\phi_2\hat Q^2\bigr)^2}{\hat G\,\hat V\,\hat G'} \sim \chi^2(1)

Sargan F: Restricted model is ARDL(1,1) with AR(1) errors (k_r = 4): F_S = \frac{\mathrm{RSS}_r - \mathrm{RSS}_u}{\mathrm{RSS}_u/(T-5)} \sim F(1,\,T-5)

Warning

When \hat Q = \hat\phi_1\hat\beta_0 + \hat\beta_1 \approx 0, the scalar \hat G\hat V\hat G' degenerates. Use the Sargan F_S form in this situation.

ARDL(2,2) — Two Restrictions (r = 2)

r_{\max} = 2. Both polynomials share all roots, forcing B(L) = \beta_0^*\,A(L): B(L) = \beta_0\bigl(1 - \phi_1 L - \phi_2 L^2\bigr) = \beta_0 - \phi_1\beta_0 L - \phi_2\beta_0 L^2

Matching coefficients of L^1 and L^2 yields two bilinear constraints: \boxed{g(\theta) = \begin{pmatrix}g_1 \\ g_2\end{pmatrix} = \begin{pmatrix}\beta_1 + \phi_1\beta_0 \\ \beta_2 + \phi_2\beta_0\end{pmatrix} = \mathbf{0}}

Jacobian (2\times 5), ordering \theta = (\phi_1,\,\phi_2,\,\beta_0,\,\beta_1,\,\beta_2)^\intercal: G(\theta) = \begin{pmatrix}\beta_0 & 0 & \phi_1 & 1 & 0 \\[4pt] 0 & \beta_0 & \phi_2 & 0 & 1\end{pmatrix}

Wald statistic (now a 2\times 2 sandwich): \mathcal{W} = \hat{g}'\,\bigl[\hat{G}\,\hat{V}\,\hat{G}'\bigr]^{-1}\,\hat{g} \sim \chi^2(2)

Sargan F: Restricted model is y_t = \beta_0^* x_t + u_t with AR(2) errors A(L)u_t = \varepsilon_t (k_r = 3): F_S = \frac{(\mathrm{RSS}_r - \mathrm{RSS}_u)/2}{\mathrm{RSS}_u/(T-5)} \sim F(2,\,T-5)

Citations: Sargan (1980, Example 2); Hendry (1995, Ch. 7).

Part III — New and General Cases

ARDL(3,1) — One Restriction (new)

y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \phi_3 y_{t-3} + \beta_0 x_t + \beta_1 x_{t-1} + \varepsilon_t \theta = (\phi_1,\,\phi_2,\,\phi_3,\,\beta_0,\,\beta_1)^\intercal, \quad k_u = 5, \quad r_{\max} = 1

Step 1. From B(z_0) = 0: \;\beta_0 + \beta_1 z_0 = 0 \implies z_0 = -\beta_0/\beta_1.

Step 2. Evaluate A(z_0) = 0: 1 + \phi_1(\beta_0/\beta_1) - \phi_2(\beta_0/\beta_1)^2 + \phi_3(\beta_0/\beta_1)^3 = 0

Multiply by \beta_1^3: \boxed{g(\theta) = \beta_1^3 + \phi_1\beta_0\beta_1^2 - \phi_2\beta_0^2\beta_1 + \phi_3\beta_0^3 = 0}

The alternating sign pattern follows the powers of z_0 = -\beta_0/\beta_1 in A(z_0).

Jacobian (1\times 5): G(\theta) = \bigl(\beta_0\beta_1^2,\;\;-\beta_0^2\beta_1,\;\;\beta_0^3,\;\;\phi_1\beta_1^2 - 2\phi_2\beta_0\beta_1 + 3\phi_3\beta_0^2,\;\;3\beta_1^2 + 2\phi_1\beta_0\beta_1 - \phi_2\beta_0^2\bigr)

Sargan F: Restricted model is ARDL(2,0) with AR(1) errors (k_r = 4): F_S = \frac{\mathrm{RSS}_r - \mathrm{RSS}_u}{\mathrm{RSS}_u/(T-5)} \sim F(1,\,T-5)

Citation: Sargan (1980) general framework; the explicit constraint follows from \mathrm{Res}(A,B) = \beta_1^p\,A(-\beta_0/\beta_1).

General ARDL(1,q) — One Restriction (new)

y_t = \phi\,y_{t-1} + \sum_{j=0}^q \beta_j\,x_{t-j} + \varepsilon_t, \qquad r_{\max} = 1

Since p=1 and p-r=0, the only candidate is C(L) = A(L), forcing \rho = \phi.

Constraint B(1/\phi)=0, multiplied through by \phi^q:

\boxed{g(\theta) = \sum_{j=0}^q \beta_j\,\phi^{q-j} \;=\; \beta_0\phi^q + \beta_1\phi^{q-1} + \cdots + \beta_{q-1}\phi + \beta_q = 0}

Linear in the \beta_j’s; polynomial of degree q in \phi.

Jacobian (1\times(q+2)): G(\theta) = \Bigl(\underbrace{\sum_{j=0}^{q-1}(q-j)\,\beta_j\,\phi^{q-j-1}}_{\partial g/\partial\phi},\;\;\underbrace{\phi^q,\;\;\phi^{q-1},\;\;\ldots,\;\;\phi,\;\;1}_{\partial g/\partial\beta_0,\;\partial g/\partial\beta_1,\;\ldots,\;\partial g/\partial\beta_q}\Bigr)

Sargan F: \;F_S \sim F(1,\,T-q-2).

q Constraint g(\theta)=0 Recovers
1 \beta_1 + \phi\beta_0 ARDL(1,1)
2 \phi^2\beta_0 + \phi\beta_1 + \beta_2 ARDL(1,2)
3 \phi^3\beta_0 + \phi^2\beta_1 + \phi\beta_2 + \beta_3 ARDL(1,3)

General ARDL(p,1) — One Restriction (new)

y_t = \sum_{j=1}^p \phi_j\,y_{t-j} + \beta_0\,x_t + \beta_1\,x_{t-1} + \varepsilon_t, \qquad r_{\max} = 1

From B(z_0)=0: z_0 = -\beta_0/\beta_1. Evaluating A(z_0)=0 and multiplying by \beta_1^p:

\boxed{g(\theta) = \beta_1^p + \sum_{j=1}^p (-1)^{j+1}\phi_j\,\beta_0^j\,\beta_1^{p-j} = 0}

This equals the resultant \mathrm{Res}(A,B) = \beta_1^p\,A(-\beta_0/\beta_1).

Sargan F: \;F_S \sim F(1,\,T-p-2).

p Constraint g(\theta)=0 Recovers
1 \beta_1 + \phi_1\beta_0 ARDL(1,1)
2 \beta_1^2 + \phi_1\beta_0\beta_1 - \phi_2\beta_0^2 ARDL(2,1)
3 \beta_1^3 + \phi_1\beta_0\beta_1^2 - \phi_2\beta_0^2\beta_1 + \phi_3\beta_0^3 ARDL(3,1)

Note

The unifying structure: the constraint for any ARDL(p,1) is obtained by evaluating A at the root z_0 = -\beta_0/\beta_1 of B and clearing denominators. The sign alternation (-1)^{j+1} arises directly from the powers of z_0 = -\beta_0/\beta_1 in A(z_0).

General ARDL(n,n) — Full Factorisation, r = n (new)

When r = n = p = q, both polynomials share all n roots: B(L) = \beta_0^*\,A(L), identifying \beta_0^* = \beta_0.

Matching coefficients of L^k for k = 1, \ldots, n: \boxed{g_k(\theta) = \beta_k + \phi_k\,\beta_0 = 0, \qquad k = 1, 2, \ldots, n}

n linear constraints. Restricted model: y_t = \beta_0^* x_t + u_t with AR(n) errors A(L)u_t = \varepsilon_t.

Jacobian (n \times (2n+1)), with \theta = (\phi_1,\ldots,\phi_n,\beta_0,\beta_1,\ldots,\beta_n)^\intercal: G(\theta) = \begin{pmatrix}\beta_0 & \cdots & 0 & \phi_1 & 1 & \cdots & 0 \\ \vdots & \ddots & \vdots & \vdots & & \ddots & \vdots \\ 0 & \cdots & \beta_0 & \phi_n & 0 & \cdots & 1\end{pmatrix}

Row k has \beta_0 in column k, \phi_k in column n+1, and 1 in column n+1+k.

Wald: \mathcal{W} = \hat{g}'\,[\hat{G}\hat{V}\hat{G}']^{-1}\,\hat{g} \sim \chi^2(n) \qquad Sargan F: F_S \sim F(n,\,T-2n-1)

n Number of constraints Restricted model
1 1 Static regression + AR(1) errors
2 2 Static regression + AR(2) errors
n n Static regression + AR(n) errors

Citations: Sargan (1980, Example 2 and general framework); Hendry (1995, Ch. 7).

Part IV — Summary

Summary of All Derived Constraints

Model r Constraint g(\theta) = 0 d.f. Sargan F
ARDL(1,1) 1 \beta_1 + \phi\beta_0 \chi^2(1) F(1,\,T-3)
ARDL(2,1) 1 \beta_1^2 + \phi_1\beta_0\beta_1 - \phi_2\beta_0^2 \chi^2(1) F(1,\,T-4)
ARDL(1,2) 1 \phi^2\beta_0 + \phi\beta_1 + \beta_2 \chi^2(1) F(1,\,T-4)
ARDL(2,2) 1 P^2 + \phi_1 PQ - \phi_2 Q^2 \chi^2(1) F(1,\,T-5)
ARDL(2,2) 2 \beta_1+\phi_1\beta_0=0\; and \;\beta_2+\phi_2\beta_0=0 \chi^2(2) F(2,\,T-5)
ARDL(3,1) 1 \beta_1^3 + \phi_1\beta_0\beta_1^2 - \phi_2\beta_0^2\beta_1 + \phi_3\beta_0^3 \chi^2(1) F(1,\,T-5)
ARDL(1,q) 1 \displaystyle\sum_{j=0}^q\beta_j\phi^{q-j} \chi^2(1) F(1,\,T-q-2)
ARDL(p,1) 1 \displaystyle\beta_1^p + \sum_{j=1}^p(-1)^{j+1}\phi_j\beta_0^j\beta_1^{p-j} \chi^2(1) F(1,\,T-p-2)
ARDL(n,n) n \beta_k+\phi_k\beta_0=0,\; k=1,\ldots,n \chi^2(n) F(n,\,T-2n-1)

For ARDL(2,2) r=1: P = \phi_2\beta_0+\beta_2, \;Q = \phi_1\beta_0+\beta_1.

Structural Patterns

Algebraic complexity scales with \min(p,q):

  • ARDL(1,1), r=1: linear constraint
  • ARDL(2,1), r=1: quadratic in (\beta_0,\beta_1)
  • ARDL(1,2), r=1: quadratic in \phi
  • ARDL(2,2), r=1: quadratic after root elimination
  • ARDL(3,1), r=1: cubic in (\beta_0,\beta_1)

Full factorisation r=r_{\max}=n:
Always n linear constraints \beta_k = -\phi_k\beta_0,
regardless of n.

Unifying characterisation (Sargan 1980):

The COMFAC condition is equivalent to the resultant condition:

\mathrm{Res}(A(z),\,B(z)) = 0

For B linear: \;\mathrm{Res}(A,B) = \beta_1^p\,A(-\beta_0/\beta_1)

For A linear: \;\mathrm{Res}(A,B) = \beta_0^q\,B(-\phi/\beta_0) \cdot (-1)^q

For r = r_{\max}: B(L) = \beta_0 A(L) directly, giving linear constraints.

Sequential Testing and Remarks

Sequential procedure (Sargan, 1980):
Test r = 1, 2, \ldots, r_{\max} in increasing order; accept the smallest r for which the null is not rejected.

Degeneracy of the Wald form. For ARDL(2,2) r=1, the denominator \hat{G}\hat{V}\hat{G}' degenerates when

\hat{Q} = \hat\phi_1\hat\beta_0 + \hat\beta_1 \approx 0 \quad (\text{i.e., } z_0 = -\hat Q/\hat P \to \infty \text{ at the estimated root})

In such cases prefer the Sargan F_S (NLS-based) or the LR statistic T\log(\mathrm{RSS}_r/\mathrm{RSS}_u).

Interpretation. Failing to reject COMFAC does not validate the AR-error model — it means only that the data are consistent with it. The unrestricted ARDL is always preferable on specification grounds:

“A common factor restriction is a testable parametric restriction, not a modelling principle.”
— Mizon (1977); Hendry & Mizon (1978)

Connection to cointegration. The ECM reparametrisation of ARDL(p,q) is always a valid linear reparametrisation; COMFAC is an additional nonlinear restriction on top of the ECM. The bounds-testing approach of Pesaran, Shin & Smith (2001) exploits the ECM form without imposing any COMFAC.

References

  • Hendry, D.F. and Mizon, G.E. (1978). Serial correlation as a convenient simplification, not a nuisance. Economic Journal, 88, 549–563. [First identification of the ARDL(1,1) restriction as the Cochrane–Orcutt imposition.]

  • Sargan, J.D. (1980). Some tests of dynamic specification for a single equation. Econometrica, 48(4), 879–897. [Foundational paper — Theorem 1 (resultant condition), Examples 1–2 (ARDL(2,1) and full factorisation).]

  • Mizon, G.E. (1977). Model selection procedures. In Artis & Nobay (eds.), Studies in Modern Economic Analysis. Blackwell.

  • Banerjee, A., Dolado, J., Galbraith, J.W., and Hendry, D.F. (1993). Co-integration, Error Correction, and the Econometric Analysis of Non-Stationary Data. Oxford University Press, Ch. 3. [Asymmetric q>p case; textbook synthesis.]

  • Hendry, D.F. (1995). Dynamic Econometrics. Oxford University Press, Ch. 7. [Full factorisation r=r_{\max} and Sargan F.]

  • Pesaran, M.H., Shin, Y., and Smith, R.J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16(3), 289–326.

  • Pesaran, M.H. and Smith, R.P. (1995). Estimating long-run relationships from dynamic heterogeneous panels. Journal of Econometrics, 68, 79–113.

  • Nickell, S. (1981). Biases in dynamic models with fixed effects. Econometrica, 49(6), 1417–1426.