 # Model Misspecification

Model specification involves selecting independent variables to include in the regression and the functional form of the regression equation. We say that a model is misspecified when it violates the assumptions underlying linear regression, its functional form is incorrect, or it contains time series specification problems.

## Categories of Model Misspecification

#### Misspecified Functional Form

This type of model misspecification occurs when the regression formula is incorrect. Misspecification functional form can result from:

• The omission of important variables from the regression.
• Use of the wrong form of data in the regression. This may be due to failure to transform variables that are non-linear.
• Use of improperly pooled data.

#### Misspecified Time Series Data

This type of model misspecification occurs when there is a correlation between the independent variables and the error term. This is a violation of the multiple regression assumption that the error term has a mean of 0, conditioned on the independent variable results to biased and inconsistent estimated regression coefficients.

Time series model misspecification is created by:

i. Using lagged dependent variables as independent variables in regressions with serially correlated errors.

ii. Using a function of the dependent variable as an independent variable, for example, forecasting the past.

iii. Measuring independent variables with an error. An example is using forward rates instead of spot rates.

Nonstationarity is a type of time series misspecification that arises when a variable’s mean and variance vary with time.

## Effects of Model Misspecification

Misspecification of the main dependent variable and other covariates is very common. This has a great impact on tests of association between the dependent and the independent variables. More specifically, model misspecification leads to the following:

i. Biased and inconsistent regression coefficients

ii. Unreliable hypothesis test results

iii. Inaccurate predictions

## Avoiding Model Misspecification

i. Transform non-linear variables to a linear form—for example, the use of log-based transformations.

ii. Avoid independent variables that are mathematical functions of dependent variables.

iii. Omit spurious independent variables.

iv. Validate model estimations out-of-sample.

v. Use good samples when collecting data.

vi. Check for violations of linear regression assumptions using diagnostic tests.

## Question

The correlation of the independent variables with the error term may result in model misspecification. This type of time-series misspecification is most likely created by:

A. Using independent variables that are measured with an error.

B. The omission of some important variables from the regression.

C. Use of improperly pooled data.

### Solution

Measuring independent variables with an error creates time-series model misspecification. An example is using forward rates instead of spot rates.

Other common problems that create this type of time-series misspecification are:

i. Using lagged dependent variables as independent variables in regressions with serially correlated errors.

ii. Using a function of the dependent variable as an independent variable, for example, forecasting the past.

B and C are incorrect. They violate the assumption that a model has the correct functional form, when in fact, it does not, thus creating functional form model misspecification.

LOS 2 (m) Describe how model misspecification affects the results of regression analysis and describe how to avoid common forms of misspecification.

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