class arch.unitroot.cointegration.DynamicOLS(y, x, trend='c', lags=None, leads=None, common=False, max_lag=None, max_lead=None, method='bic')[source]

Dynamic OLS (DOLS) cointegrating vector estimation

  • y (array_like) – The left-hand-side variable in the cointegrating regression.

  • x (array_like) – The right-hand-side variables in the cointegrating regression.

  • trend ({"n","c","ct","ctt"}, default "c") –

    Trend to include in the cointegrating regression. Trends are:

    • ”n”: No deterministic terms

    • ”c”: Constant

    • ”ct”: Constant and linear trend

    • ”ctt”: Constant, linear and quadratic trends

  • lags (int, default None) – The number of lags to include in the model. If None, the optimal number of lags is chosen using method.

  • leads (int, default None) – The number of leads to include in the model. If None, the optimal number of leads is chosen using method.

  • common (bool, default False) – Flag indicating that lags and leads should be restricted to the same value. When common is None, lags must equal leads and max_lag must equal max_lead.

  • max_lag (int, default None) – The maximum lag to consider. See Notes for value used when None.

  • max_lead (int, default None) – The maximum lead to consider. See Notes for value used when None.

  • method ({"aic","bic","hqic"}, default "bic") –

    The method used to select lag length when lags or leads is None.

    • ”aic” - Akaike Information Criterion

    • ”hqic” - Hannan-Quinn Information Criterion

    • ”bic” - Schwartz/Bayesian Information Criterion


The cointegrating vector is estimated from the regression

\[Y_t = D_t \delta + X_t \beta + \Delta X_{t} \gamma + \sum_{i=1}^p \Delta X_{t-i} \kappa_i + \sum _{j=1}^q \Delta X_{t+j} \lambda_j + \epsilon_t\]

where p is the lag length and q is the lead length. \(D_t\) is a vector containing the deterministic terms, if any. All specifications include the contemporaneous difference \(\Delta X_{t}\).

When lag lengths are not provided, the optimal lag length is chosen to minimize an Information Criterion of the form

\[\ln\left(\hat{\sigma}^2\right) + k\frac{c}{T}\]

where c is 2 for Akaike, \(2\ln\ln T\) for Hannan-Quinn and \(\ln T\) for Schwartz/Bayesian.

See 1 and 2 for further details.



Saikkonen, P. (1992). Estimation and testing of cointegrated systems by an autoregressive approximation. Econometric theory, 8(1), 1-27.


Stock, J. H., & Watson, M. W. (1993). A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica: Journal of the Econometric Society, 783-820.


fit([cov_type, kernel, bandwidth, …])

Estimate the Dynamic OLS regression