2.3. Extending the Model

The model of equation (2.2) is somewhat restrictive because it assumes that the regression slopes are invariant across time. This assumption can be tested and relaxed. Consider the model

which is identical to equation (2.2) except that the β coefficient is allowed to differ for times 1 and 2. The difference score equation then becomes

which, with a little algebra, can also be written as

This equation says that both (x₂ – x₁) and x₁ should appear as independent variables in the difference equation. If the coefficient for x₁ is significantly different from 0, that's evidence that β₁ and β₂ are not equal.

Let's try it for the NLSY data. Here is the program:

PROC REG DATA=diff;
   MODEL antidiff=selfdiff povdiff self90 pov90;
RUN;

Results in Output 2.6 provide no evidence that the effects of poverty and self-esteem on antisocial behavior are different in 1990 and 1994. Both coefficients for the time 1 variables are far from statistically significant.

Table 2.6. Output 2.6 Difference Regression with Time 1 Variables

Variable	DF	Parameter Estimate	Standard Error	tValue	Pr>|t|
Intercept	1	0.64837	0.52113	1.24	0.2139
selfdiff	1	−0.06685	0.01960	−3.41	0.0007
povdiff	1	−0.01410	0.15110	−0.09	0.9257
self90	1	−0.02226	0.02511	−0.89	0.3757
pov90	1	0.04126	0.15626	0.26	0.7919

Another way to extend the model is to allow the coefficients for the time-invariant variables to change with time, as in

Here, the γ coefficient is allowed to differ at the two time points, leading to the difference equation

In this case z does not drop out of the equation and must be included in the regression model. This result teaches us that fixed effects regression only controls for those time-invariant variables whose effects on the dependent variable are also time invariant. If a variable does not have time-invariant effects, it must be explicitly included in the model.

The NLSY data set has several time-invariant variables that are worth examining as possible predictors:

BLACK

1 if child is black, otherwise 0

HISPANIC

1 if child is Hispanic, otherwise 0

CHILDAGE

child's age in 1990

MARRIED

1 if mother was currently married in 1990, otherwise 0

GENDER

1 if female, 0 if male

MOMAGE

mother's age at birth of child

MOMWORK

1 if mother was employed in 1990, otherwise 0

These variables are now included in the difference score regression:

PROC REG DATA=diff;
   MODEL antidiff=selfdiff povdiff black hispanic childage
         married gender momage momwork;
RUN;

Results in Output 2.7 show that only one of the time-invariant variables, CHILDAGE, has a coefficient that even approaches statistical significance. And the inclusion of the time-invariant variables has very little impact on the coefficient estimates for SELFDIFF and POVDIFF. We conclude that there is some evidence that the effect of CHILDAGE is different in 1990 and 1994, but there is little or no evidence for a change over time in the effects of the other time-invariant variables.

Table 2.7. Output 2.7 Difference Regression with Time Invariant Variables

Variable	DF	Parameter Estimate	Standard Error	tValue	Pr>|t|
Intercept	1	−0.82002	1.27227	−0.64	0.5195
selfdiff	1	−0.05275	0.01554	−3.39	0.0007
povdiff	1	−0.02670	0.12996	−0.21	0.8373
black	1	−0.06179	0.14583	−0.42	0.6720
hispanic	1	0.10518	0.16257	0.65	0.5179
childage	1	0.21953	0.10735	2.05	0.0413
married	1	−0.17487	0.14921	−1.17	0.2417
gender	1	0.11065	0.12584	0.88	0.3796
momage	1	−0.04415	0.02972	−1.49	0.1379
momwork	1	−0.12445	0.13270	−0.94	0.3487

Results using the difference score method can also be replicated on the person-year data set, using PROC GLM with the ABSORB statement. To test for changes in the coefficients for SELF and POV, we include interactions between these variables and TIME:

PROC GLM DATA=persyr2;
   ABSORB id;
   MODEL anti=self pov time self*time pov*time;
RUN;

Results in Output 2.8 are equivalent to those in Output 2.6.

Table 2.8. Output 2.8 GLM with Interactions between TIME and Time-Varying Covariates

Parameter	Estimate	Standard Error	tValue	Pr>|t|
self	−.0445833075	0.02002789	−2.23	0.0264
pov	−.0553562291	0.14951835	−0.37	0.7113
time	0.6483709742	0.52113018	1.24	0.2139
self*time	−.0222626258	0.02511098	−0.89	0.3757
pov*time	0.0412572182	0.15625947	0.26	0.7919

To replicate the results in Output 2.7, we can use PROC GLM with interactions between TIME and the time-invariant predictors:

PROC GLM DATA=persyr2;
   ABSORB id;
   MODEL anti=self pov time black*time hispanic*time
         childage*time married*time gender*time
         momage*time momwork*time;
RUN;

Results in Output 2.9 are, in fact, equivalent to those in Output 2.7. Note that although the model includes interactions between TIME and the time-invariant predictors, it does not include the "main effects" of those covariates. While this may seem contrary to conventional practice, it is not a problem for this kind of analysis. In fact, if you tried to include the main effects, GLM would report coefficients of 0 with 0 degrees of freedom. That's because the time-invariant variables are perfectly collinear with the α_i parameters that have been "conditioned" out of the regression equation.

Table 2.9. Output 2.9 GLM with Interactions between TIME and Time-Invariant Covariates

Parameter	Estimate	Standard Error	tValue	Pr>|t|
self	−.0527540181	0.01554026	−3.39	0.0007
pov	−.0267038294	0.12995869	−0.21	0.8373
time	−.8200195671	1.27227104	−0.64	0.5195
time*black	−.0617870014	0.14583481	−0.42	0.6720
time*hispanic	0.1051814604	0.16256679	0.65	0.5179
time*childage	0.2195306957	0.10734673	2.05	0.0413
time*married	−.1748734507	0.14921056	−1.17	0.2417
time*gender	0.1106505035	0.12583960	0.88	0.3796
time*momage	−.0441493754	0.02971813	−1.49	0.1379
time*momwork	−.1244505238	0.13269724	−0.94	0.3487

It's also worth noting that, like interactions in general, the interactions between time and the time-invariant covariates have a dual interpretation:

• The effect of the covariate varies with time.

• The effect of time varies with the level of the covariate.

Often the latter interpretation will be more compelling. For example, the coefficient of .219 for the CHILDAGE*TIME interaction says that with each one-year increase in age at time 1, the rate of change in antisocial behavior from time 1 to time 2 goes up by .219. That is, older children have a more rapid rate of increase.