statsmodels.tsa.ardl.ARDL¶

class statsmodels.tsa.ardl.ARDL(endog, lags, exog=None, order=0, trend='c', *, fixed=None, causal=False, seasonal=False, deterministic=None, hold_back=None, period=None, missing='none')[source]¶

Autoregressive Distributed Lag (ARDL) Model

Parameters:

endogarray_like

A 1-d endogenous response variable. The dependent variable.

lags{int, list[int]}

The number of lags to include in the model if an integer or the list of lag indices to include. For example, [1, 4] will only include lags 1 and 4 while lags=4 will include lags 1, 2, 3, and 4.

exogarray_like

Exogenous variables to include in the model. Either a DataFrame or an 2-d array-like structure that can be converted to a NumPy array.

order{int, sequence[int], dict}

If int, uses lags 0, 1, …, order for all exog variables. If sequence[int], uses the order for all variables. If a dict, applies the lags series by series. If exog is anything other than a DataFrame, the keys are the column index of exog (e.g., 0, 1, …). If a DataFrame, keys are column names.

fixedarray_like

Additional fixed regressors that are not lagged.

causalbool, optional

Whether to include lag 0 of exog variables. If True, only includes lags 1, 2, …

trend{‘n’, ‘c’, ‘t’, ‘ct’}, optional

The trend to include in the model:

‘n’ - No trend.
‘c’ - Constant only.
‘t’ - Time trend only.
‘ct’ - Constant and time trend.

The default is ‘c’.

seasonalbool, optional

Flag indicating whether to include seasonal dummies in the model. If seasonal is True and trend includes ‘c’, then the first period is excluded from the seasonal terms.

deterministicDeterministicProcess, optional

A deterministic process. If provided, trend and seasonal are ignored. A warning is raised if trend is not “n” and seasonal is not False.

hold_back{None, int}, optional

Initial observations to exclude from the estimation sample. If None, then hold_back is equal to the maximum lag in the model. Set to a non-zero value to produce comparable models with different lag length. For example, to compare the fit of a model with lags=3 and lags=1, set hold_back=3 which ensures that both models are estimated using observations 3,…,nobs. hold_back must be >= the maximum lag in the model.

period{None, int}, optional

The period of the data. Only used if seasonal is True. This parameter can be omitted if using a pandas object for endog that contains a recognized frequency.

missing{“none”, “drop”, “raise”}, optional

Available options are ‘none’, ‘drop’, and ‘raise’. If ‘none’, no nan checking is done. If ‘drop’, any observations with nans are dropped. If ‘raise’, an error is raised. Default is ‘none’.

See also

statsmodels.tsa.ar_model.AutoReg: Autoregressive model estimation with optional exogenous regressors
statsmodels.tsa.ardl.UECM: Unconstrained Error Correction Model estimation
statsmodels.tsa.statespace.sarimax.SARIMAX: Seasonal ARIMA model estimation with optional exogenous regressors
statsmodels.tsa.arima.model.ARIMA: ARIMA model estimation

Notes

The full specification of an ARDL is

\[Y_t = \delta_0 + \delta_1 t + \delta_2 t^2 + \sum_{i=1}^{s-1} \gamma_i I_{[(\mod(t,s) + 1) = i]} + \sum_{j=1}^p \phi_j Y_{t-j} + \sum_{l=1}^k \sum_{m=0}^{o_l} \beta_{l,m} X_{l, t-m} + Z_t \lambda + \epsilon_t\]

where \(\delta_\bullet\) capture trends, \(\gamma_\bullet\) capture seasonal shifts, s is the period of the seasonality, p is the lag length of the endogenous variable, k is the number of exogenous variables \(X_{l}\), \(o_l\) is included the lag length of \(X_{l}\), \(Z_t\) are r included fixed regressors and \(\epsilon_t\) is a white noise shock. If causal is True, then the 0-th lag of the exogenous variables is not included and the sum starts at m=1.

Examples

>>> from statsmodels.tsa.api import ARDL
>>> from statsmodels.datasets import danish_data
>>> data = danish_data.load_pandas().data
>>> lrm = data.lrm
>>> exog = data[["lry", "ibo", "ide"]]

A basic model where all variables have 3 lags included

>>> ARDL(data.lrm, 3, data[["lry", "ibo", "ide"]], 3)

A dictionary can be used to pass custom lag orders

>>> ARDL(data.lrm, [1, 3], exog, {"lry": 1, "ibo": 3, "ide": 2})

Setting causal removes the 0-th lag from the exogenous variables

>>> exog_lags = {"lry": 1, "ibo": 3, "ide": 2}
>>> ARDL(data.lrm, [1, 3], exog, exog_lags, causal=True)

A dictionary can also be used to pass specific lags to include. Sequences hold the specific lags to include, while integers are expanded to include [0, 1, …, lag]. If causal is False, then the 0-th lag is excluded.

>>> ARDL(lrm, [1, 3], exog, {"lry": [0, 1], "ibo": [0, 1, 3], "ide": 2})

When using NumPy arrays, the dictionary keys are the column index.

>>> import numpy as np
>>> lrma = np.asarray(lrm)
>>> exoga = np.asarray(exog)
>>> ARDL(lrma, 3, exoga, {0: [0, 1], 1: [0, 1, 3], 2: 2})

Attributes:

ar_lags: The autoregressive lags included in the model
ardl_order: The order of the ARDL(p,q)
causal: Flag indicating that the ARDL is causal
deterministic: The deterministic used to construct the model
df_model: The model degrees of freedom.
dl_lags: The lags of exogenous variables included in the model
endog_names: Names of endogenous variables.
exog_names: Names of exogenous variables included in model
fixed: The fixed data used to construct the model
hold_back: The number of initial obs.
period: The period of the seasonal component.
seasonal: Flag indicating that the model contains a seasonal component.
trend: The trend used in the model.

Methods

`fit`(*[, cov_type, cov_kwds, use_t])	Estimate the model parameters.
`from_formula`(formula, data[, lags, order, ...])	Construct an ARDL from a formula
`hessian`(params)	The Hessian matrix of the model.
`information`(params)	Fisher information matrix of model.
`initialize`()	Initialize the model (no-op).
`loglike`(params)	Log-likelihood of model.
`predict`(params[, start, end, dynamic, exog, ...])	In-sample prediction and out-of-sample forecasting.
`score`(params)	Score vector of model.

Properties

`ar_lags`	The autoregressive lags included in the model
`ardl_order`	The order of the ARDL(p,q)
`causal`	Flag indicating that the ARDL is causal
`deterministic`	The deterministic used to construct the model
`df_model`	The model degrees of freedom.
`dl_lags`	The lags of exogenous variables included in the model
`endog_names`	Names of endogenous variables.
`exog_names`	Names of exogenous variables included in model
`fixed`	The fixed data used to construct the model
`hold_back`	The number of initial obs.
`period`	The period of the seasonal component.
`seasonal`	Flag indicating that the model contains a seasonal component.
`trend`	The trend used in the model.