Stationary residuals. Spurious regression?

[John Riveros]

Greetings Stata Users.

I'm trying to estimate the public spending role over economic growth. So basically i'm using an inverse Wagner Law to see how those two variables are correlated.

I have checked either GDP (PIB) and Public Spending, and both of them have unit roots. Integrated in order 2.
This is consistent with higher R^2 value and probably a spurious regression that shows ahead.

reg PIB_nominal_bm total_gasto_nom

      Source |       SS           df       MS      Number of obs   =        28
-------------+----------------------------------   F(1, 26)        =  11070.25
       Model |  2.0580e+30         1  2.0580e+30   Prob > F        =    0.0000
    Residual |  4.8334e+27        26  1.8590e+26   R-squared       =    0.9977
-------------+----------------------------------   Adj R-squared   =    0.9976
       Total |  2.0628e+30        27  7.6400e+28   Root MSE        =    1.4e+13

---------------------------------------------------------------------------------
 PIB_nominal_bm |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
----------------+----------------------------------------------------------------
total_gasto_nom |   5.04e+09   4.79e+07   105.22   0.000     4.94e+09    5.14e+09
          _cons |   3.52e+13   4.05e+12     8.68   0.000     2.69e+13    4.35e+13
---------------------------------------------------------------------------------

Then next codes were

predict u, res
dfuller u

Dickey-Fuller test for unit root                   Number of obs   =        27

                               ---------- Interpolated Dickey-Fuller ---------
                  Test         1% Critical       5% Critical      10% Critical
               Statistic           Value             Value             Value
------------------------------------------------------------------------------
 Z(t)             -3.732            -3.736            -2.994            -2.628
------------------------------------------------------------------------------
MacKinnon approximate p-value for Z(t) = 0.0037

Maybe i'm reading wrong the p-value, but i suppose the null hypothesis it's unit roots behaviour.


So when i ran the stationary test of the residuals of that regression. dfuller test says it's stationary. So i'm a bit confused, does this mean that relation is not spurious and it's telling me the long-term relationships?.

[Sebastian Kripfganz]

You might want to try an augmented Dickey-Fuller test instead by adding first-differenced lags of the error term with the lags() option of the dfuller command. You could obtain the optimal lag order by calling

varsoc u

Notice that the varsoc command delivers the optimal lag order for the level error term. The lag order for the dfuller command should be this lag order minus 1.

[John Riveros]

I've just run the command you gave me in order to establish the lag of dfuller test.

Selection-order criteria
   Sample:  1994 - 2017                         Number of obs      =        24
  +---------------------------------------------------------------------------+
  |lag |    LL      LR      df    p      FPE       AIC      HQIC      SBIC    |
  |----+----------------------------------------------------------------------|
  |  0 | -755.914                      1.4e+26*  63.0761*  63.0892*  63.1252* |
  |  1 |  -755.91  .00789    1  0.929  1.6e+26   63.1591   63.1852   63.2573  |
  |  2 | -755.901  .01772    1  0.894  1.7e+26   63.2417   63.2808    63.389  |
  |  3 | -755.898  .00475    1  0.945  1.9e+26   63.3249    63.377   63.5212  |
  |  4 | -755.833  .13018    1  0.718  2.0e+26   63.4028   63.4679   63.6482  |
  +---------------------------------------------------------------------------+

And dfuller test with lag order minus 1.

dfuller u, lags(3)

Augmented Dickey-Fuller test for unit root         Number of obs   =        24

                               ---------- Interpolated Dickey-Fuller ---------
                  Test         1% Critical       5% Critical      10% Critical
               Statistic           Value             Value             Value
------------------------------------------------------------------------------
 Z(t)             -2.510            -3.750            -3.000            -2.630
------------------------------------------------------------------------------
MacKinnon approximate p-value for Z(t) = 0.1131

So according to that, errors still have an unit roots. How do i proceed?

[Ariel Karlinsky]

Hi John,

Have you ever gotten a response to this?

[Sebastian Kripfganz]

John: Why did you choose 3 lags for the dfuller command? Your varsoc command indicates that 0 lags are optimal, which is already quite weird because that would not make much sense if there is serial correlation in the tested variable, and a unit root is an extreme form of serial correlation.

To answer the initial question: When the (augmented) Dickey-Fuller test indicates that the residuals do not have a unit root, you would conclude that you have estimated a long-run level relationship between the variables in your initial model. Otherwise, there is the risk of a spurious regression. But note that a high R² can also be the consequence of deterministically trending variables which is quite likely when you have macroeconomic variables that are not detrended. I would recommend to add a linear time trend to your model.

More generally, I recommend to estimate an error correction model as a reparameterization of an autoregressive distributed lag model. The dynamic nature circumvents potential spurious-regression problems that you would have in static models, and you can still infer the level relationship from the dynamic specification.

[Justin Niakamal]

I'm confused. Your sample-order criteria suggest no lags (lowest information criteria with *). And then you say a dfuller test with lag order minus 1, but your table shows lag order 3.

Also, which country has GDP that is I~(2)? I would suggest looking at real GDP, that should be I~(1). Here's a good rule of thumb: If your R-squared is greater than your Durbin Watson, you have a spurious regression. If your regression is spurious, remove the unit roots from the variables until they are stationary and run your regression on those series. Here's a helpful blog post related to this issue.