Online Econometrics Textbook - Regression Extensions - Assumption Violations of Linear Regression - Misspecification in Linear Regression

III.I.5 Misspecification in Linear Regression

In econometric theory it is almost always assumed that economic theory provides a "true" specification of the stochastic equation to be estimated. It is therefore obvious that if economic theory is incomplete, the estimated parameters will (almost surely) be biased. The bias induced by these "unobserved" variables is called the unobserved variables bias (UVB).

Formally the UVB for a linear statistical model

(III.I.5-1)

can be illustrated, assuming Z are the unobserved variables and c is the corresponding parameter vector, by rewriting the OLS parameter vector equation

(III.I.5-2)

where (X'X)^-1X'Zc are the so-called auxiliary regressions of the omitted variables on the exogenous variables of (III.I.5-1).

It is also interesting to see that the residual variance is biased since

(III.I.5-3)

or on using the true model y = Xb + Zc + e

(III.I.5-4)

and since

(III.I.5-5)

so that on combining (III.I.5-4) with (III.I.5-5) and the fact that E(e) = 0, it follows that

(III.I.5-6)

Now it is also obvious that

(III.I.5-7)

(Q.E.D.).

Detecting specification errors is sometimes possible by using Ramsey’s RESET specification test. This test is based on a test-equation where the original equation is expanded with powers of the interpolation fit series (powers 2, 3, etc...). The number of terms to be added is user-defined. An F-test is used to test if the added terms are significant or not.

Below you ‘ll find an example of how Ramsey’s RESET test can be applied to test for an inadequate specification (this test is applied to our example-equation):

Ramsey's RESET specification test by Least Squares:

Estimation with OLS:

Endogenous variable = y_name

Variable               Parameter           S.E.       t-stat

const(-0),1.,0,0       +1531.665134       685.7724822       +2.23       

employ(-0),1.,0,0      +1.778551911       16.31481316       +0.109      

expend(-0),1.,0,0      -12.90458938       7.248637814       -1.78      

forec(-0),2.,0,0       +2.841488934e-004  8.950629594e-005  +3.17       

forec(-0),3.,0,0       -1.30569336e-008   3.803412467e-009  -3.43      

2-tail-t at 95 percent = 2.042

1-tail-t at 95 percent = 1.697

R-squared of stationary series = 0.9450643085  Durbin-Watson = 1.995266891

Degrees of freedom           = 32

Variance       of regression = 788126.0046

Standard Error of regression = 887.764611

Sum of Squared Residuals     = 25220032.15



Correlation matrix of parameters:



  +1.00 -0.95 -0.83 +0.87 -0.81 

  -0.95 +1.00 +0.80 -0.89 +0.84 

  -0.83 +0.80 +1.00 -0.97 +0.96 

  +0.87 -0.89 -0.97 +1.00 -0.99 

  -0.81 +0.84 +0.96 -0.99 +1.00 

F-test = 8.00468

Critical F value (95%) = 2.61