Solved – Interaction terms, dumthe variables, and fixed effects

categorical-encodingfixed-effects-modelinteraction

I am getting very strange results for an interaction term in a model I am fitting. I am interested in the impact of uncertainty on investment. The theory predicts that the effect of uncertainty on investment will be higher for manufacturing companies than for non-manufacturing ones. To tests this hypothesis I use the following fixed effects model in my longitudinal panel data set:

$$ Invest_{i,t} = \alpha_i + \beta_1 U_t + \beta_2 U_t \mathord{\cdot} M_{i} + \beta_3 X_{i,t} + \epsilon_{i,t} (1) $$

where $i=1,2,…,N$ indexes cross-section dimension and $t=1,2,…,T$ the time series dimension. $M_i$ is a dummy variable with values 0 and 1, none manufacturing firms and manufacturing firms respectively; $U_t$ is uncertainty which does not vary across firms but only through time, and $X_{i,t}$ is a set of controls. Given that $M_i$ is a time-invariant characteristic, my fixed effects variable $\alpha_i$ absorbs its effect in the interaction term: $ \beta_1 U_t + \beta_2 M_{i} + \beta_3 U_t \mathord{\cdot} M_{i} $. That is why $ \beta_2 = 0$. Now when running the regression of equation 1, I obtain the following results:

                  Estimate Std. Error t value  Pr(>|t|)    
U               -0.0102484  0.0031349 -3.2691  0.001088 *** 
UxM              0.0156437  0.0063180  2.4760  0.013326 **

which indicates that manufacturing companies react less to uncertainty since the interaction term $UxM$ is positive and significant. Nonetheless, when I do the regression separately using the two data sets (one for manufacturing companies and one for non-manufacturing) I get the opposite result (note that in this case, I do not need the interaction term so my equation is $ Invest_{i,t} = \alpha_i + \beta_1 U_t + \beta_2 X_{i,t} + \epsilon_{i,t} $:

Manufacturing firms:

         Estimate Std. Error t value  Pr(>|t|)    
U       -0.0092903  0.0048826 -1.9027 0.0573272 *

Non-Manufacturing firms:

          Estimate Std. Error t value  Pr(>|t|)    
U       -0.0046793  0.0032443 -1.4423  0.149320

Clearly, these results (separate regressions for each group) contradicts the results when I use the interaction term: these results suggest that Uncertainty correlates only with investment for manufacturing firms while not for non-manufacturing ones. Any ideas, suggestions of why this might be? might this be due to the fixed effects?

Best Answer

Interaction analyses and stratified (or separate) analyses have the same objective: predict and measure trend-line differences across levels of an interaction variable. Despite this shared objective, the two approaches rarely lead to the same estimates or inference.

Stratified/separate analyses are a much broader class of models and thus require a larger sample to obtain modestly powered inference. Interaction models, by contrast, are much more constrained and thus are more efficient when the modeling assumptions are true. If the analysis is decently powered, the stratified model is superior since there are fewer modeling assumptions, but this is rarely the case.

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