arises as to whether this type of Sherlock Holmes inference can be brought within the scope of statistical inference.

Leamers answer to this is that we have to view this new "data-instigated hypothesis" as something that was in the back of our mind all the time. We did not consider it explicitly because we thought it to be unimportant but now we change our mind after observing the data. Suppose that the model we want to consider h = + yz + u. Initially, we consider z to be uncorrelated with X or that 7 is negligibly small. So we start with the model = px - . If the resulting estimate of p has the wrong sign or the pattern of estimated residuals is peculiar, we change our mind and observe z. If the problem is viewed in this form of a decision problem. Leamer argues that we can attach the appropriate standard errors to the coefficients when the model = x + yz + is estimated.

In all cases of specification searches, there is the question of what the appropriate standard errors should be for the final model estimated. There is no easy answer to this question except that the standard errors are higher than those obtained from the estimation of the final model. The inference problem is more straightforward in the case of a simplification search. That is why David Hendry suggests starting with an extremely general model and then simplifying it progressively. We will, therefore, discuss this approach in greater detail.

Hendrys Approach to Model Selection

The approach to model building suggested by David Hendry," which is mainly applicable to dynamic time-series models, can be summarized as: intended ov-erparametrization witii data-based simplification. By contrast most of empirical econometric work can be characterized as excessive pre simplification with inadequate diagnostic testing. The latter method consists of the following steps:

1. Commence from theories which are drastic abstractions of reality.

2. Formulate highly parsimonious relationships to represent the theories.

3. Estimate the equations from the available data using techniques which are "optimal" only on the assumptions that the highly restricted model is correctly specified.

4. Test a few of the assumptions explicitly or implicitly (such as the conventional Durbin-Watson test for autocorrelation).

5. Revise the specification in the light of evidence acquired.

6. Reestimate accordingly.

According to Hendry, this approach to model building, which is a "specific to general" approach or a "bottoms-up" approach, has three main drawbacks:

"This approach is originally due to J. D. Sargan but has been popularized and expounded by David Hendry. Some representative papers are: D. F. Hendry, "Predictive Failure and Econometric Modelling in Macroeconomics: The Transactions Demand for Money," in Paul Ormerod (ed.). Economic Modelling (London: Heinemann, 1979), Chap. 9, pp. 217-242; and Mizon, "Model Selection Procedures."

1. Every test is conditional on arbitrary assumptions which are to be tested later, and if these are rejected, all earlier inferences are invalidated.

2. The significance levels of the unstructured sequence of tests actually conducted is unknown. For instance, suppose that we estimate an equation by OLS, find a significant DW statistic, and then reestimate the equation adjusting for the serial correlation. What are the significance levels for the estimated coefficients from this transformed equation? This is not very clear.

3. It is not always possible to end up with the best model by using this iterative method. We might get sidetracked by using the wrong (or inadequate) diagnostic tests. For instance, we might start with the equation = fix + and observing a significant DW test statistic, we reestimate the equation in a p-differenced form. If we now find that the coefficients of our equation are of the correct sign, we might rest satisfied. However, this procedure might lead us to the wrong model because as discussed in Section 6.9 the DW statistic could be significant not because of serial correlation in errors but because of misspecified dynamics. The model with serially correlated errors

y, = fix, + u, u,= p«,, + e,

implies the model

y, = Pr-i + Pr - Pp>:,-i + e, which is the same as

y, = Pi>,-i + ix, + -,-1 + e,

with the restriction fiifij = - - Now the true model could be the last one but with the restriction fiifin = - p, not satisfied. But there is no way of our arriving at this model by the modeling approach adopted. When we estimate the model y, ~ fix, + u, we would observe serial correlation in the residuals because y, and x, are "missing." But this does not mean that we have a model with serially correlated errors.

By contrast, the approach suggested by Hendry, which is a "top-down" or "general to specific" approach, starts with a very general dynamic model, which is "overparametrized," that is, has more lags than you would consider necessary. The model is then progressively simplified with a sequence of "simplification tests." The significance levels for the sequence of tests is known unlike the case of the sequence of tests we perform in the "specific-to-general" approach. The significance level for j\h test is*

1 - (1 - 7/)

*This result was derived by T. W. Anderson in the paper cited in Section 10.8. This result has been proved for dynamic econometric models by J. D. Sargan (in an unpublished London School of Economics discussion paper).

12.7 Selection of Regressors

In Section 12.6 we outlined some general approaches to the model selection problem. We now concentrate on a specific issue, the problem of selection of regressors.

In Chapter 4 we considered a multiple regression of an explained variable on a set of explanatory variables x,, Xj, . . . , It was assumed that the set of variables to be included in the equation is given. In practice this is rarely the case. There is typically a very large number of potential explanatory variables or regressors and one is faced with a problem of choosing a subset of these. This is the problem often referred to as the problem of selection of regressors.

Suppose that there is a potential set of regressors from which we have to select a smaller number. In the 1960s a number of stepwise regression methods were suggested." Some of these started with a regression, including all the variables and successively proceeded to eliminate variables with /-ratios less than a prespecified value (say, 1). This is called a backward selection procedure. Others started with a single variable which had the highest correlation with and then picked at each stage the variable with the highest partial correlation coefficient (this is called forward selection procedure). Some procedures combined the elements of the forward and backward selection procedures at each stage (i.e., deleted some variables and added some others).

This sort of mechanical picking of variables by the computer is no longer popular among econometricians. Fortunately, although economists do not have an exact idea of all the variables that have to be included in an equation, they do have an idea of what variables are likely to be very important and what variables are doubtful. In this case we can specify a small number of alternative models and then we need to choose one of these alternatives. Since theory cannot give us any guidance at this stage we have to make a choice on statistical grounds.

There are many criteria that have been suggested in the literature. Some of these are listed in Table 12.3. We discuss them briefly. The criteria that we have chosen in Table 12.3 are applicable to regression models only and we have chosen those criteria that depend on some summary measures commonly used like residual sum of squares.

Hendry, "Predictive Failure," p. 226.

"These methods are not popular at present. However, those interested in a discussion of stepwise procedures can refer to N. Draper and H. Smith, Applied Regression Analysis, 2nd ed. (New York: Wiley, 1981), Chap. 6.

where 7, is the significance level for the /th test in the sequence. The sequential testing procedures are used to select a "data coherent specialization."

Hendry argues that only after these steps should one test economic theories. "Until the model characterizes the data generation process, it seems rather pointless trying to test hypotheses of interest in economic theory."-