# How should one determine the proper number of lags in a time series regression?

I am using time series data in economic model estimation. I want determine proper lag for Error Correcting Model (ECM) model for example. I can check AIC, SC and HQ criterion for determine proper lag. But I am not sure about max lag that I must start with it. What is max lag for annual, quarterly and monthly time series.

Is max lag would be changed if I used another econometric approach for example ARDL, VAR or VECM?

## 1 Answer

I don't have advice specific to error correcting model (ECM) setting, but in undergraduate applied econometric class they gave us the generic advice to continue to extend lags in the model until the residuals of the fitted model were serially uncorrelated. For example, in the US life expectancy data, residuals of male life expectancy is serially uncorrelated in the AR(5) model but not the AR(4) model. You can see this for yourself with the following Stata code:

use http://www.stata-press.com/data/r8/uslifeexp.dta
tsset year, yearly
reg le_male L(1/4).le_male
estat durbinalt, small
reg le_male L(1/5).le_male
estat durbinalt, small


The Stata documentation for the vector error-correction models also seems to roughly follow this approach but it looks like it is automated under the varsoc function and additionally the AIC, HQIC, and SBIC are all generated programmatically.

To test for cointegration or fit cointegrating VECMs, we must specify how many lags to include. Building on the work of Tsay (1984) and Paulsen (1984), Nielsen (2001) has shown that the methods implemented in varsoc can be used to determine the lag order for a VAR model with I(1) variables. As can be seen from (9), the order of the corresponding VECM is always one less than the VAR. vec makes this adjustment automatically, so we will always refer to the order of the underlying VAR. The output below uses varsoc to determine the lag order of the VAR of the average housing prices in Dallas and Houston....

We will use two lags for this bivariate model because the Hannan–Quinn information criterion (HQIC) method, Schwarz Bayesian information criterion (SBIC) method, and sequential likelihood-ratio (LR) test all chose two lags, as indicated by the “*” in the output.

. clear all

. use http://www.stata-press.com/data/r13/txhprice

. varsoc dallas houston

Selection-order criteria
Sample:  1990m5 - 2003m12                    Number of obs      =       164
+---------------------------------------------------------------------------+
|lag |    LL      LR      df    p      FPE       AIC      HQIC      SBIC    |
|----+----------------------------------------------------------------------|
|  0 |  299.525                      .000091  -3.62835  -3.61301  -3.59055  |
|  1 |  577.483  555.92    4  0.000  3.2e-06   -6.9693  -6.92326  -6.85589  |
|  2 |  590.978  26.991*   4  0.000  2.9e-06*  -7.0851* -7.00837* -6.89608* |
|  3 |  593.437   4.918    4  0.296  2.9e-06  -7.06631  -6.95888  -6.80168  |
|  4 |  596.364  5.8532    4  0.210  3.0e-06  -7.05322   -6.9151  -6.71299  |
+---------------------------------------------------------------------------+
Endogenous:  dallas houston
Exogenous:  _cons


If, as @GraemeWalsh suggests, you would like to use an auto-regressive distributed lags methodology (ARDL) you can do so without having to code it up yourself.

use http://www.stata-press.com/data/r13/txhprice
sort t
net install ardl.pkg
ardl dallas houston,  maxlag(4)

ARDL regression
Model: level

Sample:  1990m5 - 2003m12
Number of obs  = 164
Log likelihood = 313.86816
R-squared      = .96315461
Adj R-squared  = .96246376
Root MSE       = .03613756

------------------------------------------------------------------------------
dallas |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
dallas |
L1. |    .427744   .0789194     5.42   0.000      .271886     .583602
L2. |   .1747019   .0720507     2.42   0.016     .0324089    .3169948
|
houston |   .3404766   .0567884     6.00   0.000     .2283252     .452628
_cons |   .7276476   .2061803     3.53   0.001     .3204618    1.134833
------------------------------------------------------------------------------

• While the specific-to-general approach (extending the lag length in the model until the residuals are white noise) is commonly used, it is, in fact, better (more structured) to use the general-to-specific approach (begin with a larger model and reduce it's size) often recommended by David Hendry. I'd say to the OP, build an ARDL and follow Hendry's methodology, which is the soundest, in my opinion. – Graeme Walsh Apr 20 '16 at 20:09