Coefficient of level and lagged explanatory variable with different sign

[Marius van Oordt]

Hi,

Estimating by system GMM and treating the variable of interest as exogenous, without including the lagged of the variable of interest, the estimated coefficient of the variable of interest is about 0.025 (p<0.01). All variables are transformed to natural logarithms. Including the first lag of the variable of interest give a coefficient for the level of -0.925 (p<0.01) and a coefficient for the lag of 0.896 (p<0.01). The long run effect is therefore very similar to the level coefficient without including the lag (0.029 compared to 0.025).

Should I be concerned by the drastic difference between the level and the lag coefficient?

Thanks in advance.

[Sebastian Kripfganz]

Given that the coefficient of the contemporaneous and the lagged term are of similar magnitude, this indicates that it might be the first difference (i.e. the growth rate, given that your variables are in logs) rather than the level that matters in explaining your outcome variable. You could try including both x and D.x in your model (instead of x and L.x) and see whether the coefficient of x remains statistically significant. If both coefficients are statistically significant, then leaving one of them out might lead to an omitted variable bias.

[Marius van Oordt]

Given that the coefficient of the contemporaneous and the lagged term are of similar magnitude, this indicates that it might be the first difference (i.e. the growth rate, given that your variables are in logs) rather than the level that matters in explaining your outcome variable. You could try including both x and D.x in your model (instead of x and L.x) and see whether the coefficient of x remains statistically significant. If both coefficients are statistically significant, then leaving one of them out might lead to an omitted variable bias.

Thank you very much Sebastian. Including the first difference gives a coefficient for x of 0.029 and D.x of 0.896, both p<0.01. I therefore interpret the difference in the original coefficient as being due to both level and first difference influencing my outcome variable.

[Joro Kolev]

These two are different parametrisations of the same regression. That is, what Sebastian proposes (including x and d.x) is the same regression as including x and l.x in #1.

As to the original question, I do not see any problem with the fact that the contemporaneous and the lagged regressors have different signs.

The conclusion/interpretation Marius reaches in #3 is correct, but this is the same interpretation as "both x and l.x influencing the outcome variable".

On the claim that these are different parameterisations of the same regression, notice below in the two regressions that both the Residual Sum of Squares, and the Root Mean Square Error for the two parametrisations are numerically the same:

. tsset qtr
        time variable:  qtr, 1960q1 to 1982q4
                delta:  1 quarter

. reg cons inc l.inc

      Source |       SS           df       MS      Number of obs   =        91
-------------+----------------------------------   F(2, 88)        =  65113.33
       Model |  31184057.1         2  15592028.6   Prob > F        =    0.0000
    Residual |  21072.4675        88  239.459858   R-squared       =    0.9993
-------------+----------------------------------   Adj R-squared   =    0.9993
       Total |  31205129.6        90  346723.662   Root MSE        =    15.474

------------------------------------------------------------------------------
     consump |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         inc |
         --. |   .4917908    .107023     4.60   0.000     .2791051    .7044765
         L1. |   .3572938   .1081082     3.30   0.001     .1424515    .5721362
             |
       _cons |   24.55272   3.750001     6.55   0.000     17.10038    32.00506
------------------------------------------------------------------------------

. reg cons inc d.inc

      Source |       SS           df       MS      Number of obs   =        91
-------------+----------------------------------   F(2, 88)        =  65113.33
       Model |  31184057.1         2  15592028.6   Prob > F        =    0.0000
    Residual |  21072.4675        88  239.459858   R-squared       =    0.9993
-------------+----------------------------------   Adj R-squared   =    0.9993
       Total |  31205129.6        90  346723.662   Root MSE        =    15.474

------------------------------------------------------------------------------
     consump |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         inc |
         --. |   .8490847   .0025928   327.47   0.000     .8439319    .8542374
         D1. |  -.3572938   .1081082    -3.30   0.001    -.5721362   -.1424515
             |
       _cons |   24.55272   3.750001     6.55   0.000     17.10038    32.00506
------------------------------------------------------------------------------

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