Specification

Model specification refers to the description of the process by which the dependent variable is generated by the independent variables. Thus, it encompasses the choice of independent (and dependent) variables, as well as the functional form connecting the independent variables to the dependent variable. Specification can also include any assumptions about the stochastic component of the model. Thus, specification occurs before estimation. A simple example of model specification with which most people are familiar is a model that assumes that some variable Y is a linear function of a random variable X and a stochastic disturbance term ε:

There are many alternative specifications that would relate Y to X, such as

Once a model is specified, an estimation technique is merely a tool for determining the values of the parameters in the model that best fits the observed data to the specification. However, without correct model specification, any estimation exercise is merely curve-fitting without any ability to make valid statistical inferences. To make inferences from the observed sample of data to the population, we are required to assume that all values of Y in the population are generated via the same process that generates the values in our observed sample. In other words, we must assume that the specification is correct. In the wellknown Gauss-Markov theorem, this is generally stated as “the model is correctly specified.”

Model specification should be based on some relevant theory. If the estimated parameters of the model are statistically significant, it may be regarded as confirmation of the theory. But in fact, it is technically a test of the model as specified versus the null model—a model that assumes there is no relationship between the dependent variable and the independent variables of the model. If we estimated any of the three models above and were able to reject the null hypothesis that β = 0 at the 95% level, we still could not say the specification we estimated was correct and the other two proposed specifications were incorrect.

Although infinitely many forms of specification error are possible, there are several common forms of specification error to beware of. Most ubiquitous is the problem of omitted variables. If the model specified omits an independent variable that is, in fact, part of the true model generating the data, then the model is misspecified. The consequences of this form of specification error a resevere. Even in the simple case of ordinary least squares and a linear model, the estimated parameters of other variables may be biased, and we can make no valid statistical inferences.

There is no test to determine that we absolutely have the correct specification. However, we can often compare our specification to alternative specifications with statistical tests. The simplest tests involve specifications that are nested within one another. One specification is nested in another if it has identical functional form, but the variables in it are a subset [Page 1056]of the variables in the other specification. So, for instance, the model Y = α0 + β1X1 + ε is nested within the model Y = α0 + β1X1 + β2X2 + ε, but the model Y = α0 + β3X3 + ε is not tested in that model. Nested models are typically tested against one another using F tests of linear restrictions (or Chow tests) for linear models, and log-likelihood ratio tests for maximum likelihood models. Such tests are described in almost all econometrics texts. In both cases, one estimates each of two proposed models and generates a test statistic based on the fit of each model. A classical hypothesis test can then be performed that may let one reject one model in favor of another. In the case of an F test to compare two linear specifications, one of which is nested within another, the test statistic is given

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