When we interpret the linear regression model as describing the conditional expected value of yi given xi, that is, E{yi|xi} =xi𝛽, we are implicitly assuming that no other functions ofxiare relevant. This is restrictive, and it makes sense to test this restriction, or to compare the linear model against more general alternatives. In this section, we dis- cuss some tests on the functional form of the model, and introduce a class of nonlinear models that can be estimated using a nonlinear least squares approach. Subsection 3.3.3 presents a test for testing whether the model coefficients are constant across two (or more) subgroups in the sample, typically referred to as a test for a structural break.
3.3.1 Nonlinear Models
Nonlinearities can arise in two different ways. In the first case, the model is still linear in the parameters but nonlinear in its explanatory variables. This means that we include non- linear functions ofxias additional explanatory variables, for example, the variablesage2i andageimaleicould be included in an individual wage equation. The resulting model is still linear in the parameters and can still be estimated by ordinary least squares. In the second case, the model is nonlinear in its parameters and estimation is less easy. In gen- eral, this means that E{yi|xi} =g(xi, 𝛽), where g(.) is a regression function nonlinear in𝛽. For example, for a scalarxiwe could have
g(xi, 𝛽) =𝛽1+𝛽2x𝛽i3 (3.27) or for a two-dimensionalxi
g(xi, 𝛽) =𝛽1x𝛽i12x𝛽i23, (3.28) which corresponds to a Cobb–Douglas production function with two inputs. As the sec- ond function is linear in parameters after taking logarithms (assuming𝛽1>0), it is a common strategy in this case to model logyirather thanyi. This does not work for the first example.
Nonlinear models can also be estimated by a nonlinear version of the least squares method, by minimizing the objective function
S(̃𝛽) =
∑N i=1
(yi−g(xi, ̃𝛽))2 (3.29) with respect to ̃𝛽. This is called nonlinear least squares estimation. Unlike in the linear case, it is generally not possible analytically to solve for the value of ̃𝛽 that minimizesS(̃𝛽), and we need to use numerical procedures to obtain the nonlinear least
8It may be noted that with sufficiently general functional forms it is possible to obtain models foryiand logyi that are both correct in the sense that they representE{yi|xi}andE{logyi|xi}, respectively. It is not possible, however, that both specifications have a homoskedastic error term (see the example in Section 3.6).
k k squares estimator. A necessary condition for consistency is that there exists aunique
global minimum for S(̃𝛽), which means that the model is identified. An excellent treatment of such nonlinear models is given in Davidson and MacKinnon (1993), and we will not pursue it here.
It is possible to rule out functional form misspecifications completely, by saying that one is interested in the linearfunction of xi that approximatesyi as well as possible.
This goes back to the initial interpretation of ordinary least squares as determining the lin- ear combination ofxvariables that approximates a variableyas well as possible. We can do the same thing in a statistical setting by relaxing the assumption thatE{𝜀i|xi} =0 toE{𝜀ixi} =0. Recall thatE{𝜀i|xi} =0 implies thatE{𝜀ig(xi)} =0 for any functiong (see Appendix B.5), showing that imposingE{𝜀ixi} =0 is indeed weaker. In this case, we can interpret the linear regression model as describing the best linear approxima- tion of yi fromxi. In many cases, we would interpret the linear approximation as an estimate for its population equivalent rather than just an in-sample result. Note that the conditionE{𝜀ixi} =0 corresponds to condition (A7) from Chapter 2 and is necessary for consistency of the OLS estimator.
3.3.2 Testing the Functional Form A simple way to test the functional form of
E{yi|xi} =xi𝛽 (3.30)
would be to test whether additional nonlinear terms inxi are significant. This can be done using standardt-tests,F-tests, or, more generally, Wald tests. For example, to test whether individual wages depend linearly upon experience, one can test the significance of squared experience. Such an approach only works if one can be specific about the alternative. If the number of variables in xi is large, the number of possible tests is also large.
Ramsey (1969) has suggested a test based upon the idea that, under the null hypothesis, nonlinear functions ofŷi=xibshould not help in explainingyi. In particular, he tests whether powers ofŷihave nonzero coefficients in the auxiliary regression
yi=xi𝛽+𝛼2ŷ2i +𝛼3ŷ3i + ã ã ã +𝛼QŷQi +𝑣i. (3.31) Anauxiliary regression, and we shall see several below, is typically used to compute a test statistic only, and is not meant to represent a meaningful model. In this case we can use a standardF-test for theQ−1 restrictions inH0:𝛼2= ã ã ã =𝛼Q=0, or a more general Wald test (with an asymptotic𝜒2distribution withQ−1 degrees of freedom).
These tests are usually referred to asRESET tests(Regression Equation Specification Error Tests). Often, a test is performed forQ=2 only. It is not unlikely that a RESET test rejects because of the omission of relevant variables from the model (in the sense defined earlier) rather than just a functional form misspecification. That is, the inclusion of an additional variable may capture the nonlinearities indicated by the test.
3.3.3 Testing for a Structural Break
So far, we have assumed that the functional form of the model is the same for all obser- vations in the sample. As shown in Section 3.1, interacting dummy variables with other
k k
MISSPECIFYING THE FUNCTIONAL FORM 75
explanatory variables provides a useful tool to allow the marginal effects in the model to be different across subsamples. Sometimes it is interesting to consider an alternative specification in which all the coefficients are different across two or more subsamples.
In a cross-sectional context, we can think of subsamples containing males and females or married and unmarried workers. In a time series application, the subsamples are typically defined by time. For example, the coefficients in the model may be different before and after a major change in macro-economic policy. In such cases, the change in regression coefficients is referred to as astructural break.
Let us consider an alternative specification consisting of two groups, indicated bygi=0 andgi=1, respectively. A convenient way to express the general specification is given by yi=xi𝛽+gixi𝛾+𝜀i, (3.32) where the K-dimensional vectorgixi contains all explanatory variables (including the intercept), interacted with the indicator variablegi. This equation says that the coefficient vector for group 0 is𝛽, whereas for group 1 it is𝛽+𝛾. The null hypothesis is𝛾=0, in which case the model reduces to the restricted model.
A first way to test𝛾=0 is obtained by using theF-test from Subsection 2.5.4. Its test statistic is given by
F= (SR−SUR)∕K SUR∕(N−2K),
whereKis the number of regressors in the restricted model (including the intercept) and SUR andSR denote the residual sums of squares of the unrestricted and the restricted model, respectively. Alternatively, the general unrestricted model can be estimated by running a separate regression for each subsample. This leads to identical coefficient esti- mates as in (3.32), and consequently the unrestricted residual sum of squares can be obtained as SUR =S0+S1, where Sg denotes the residual sum of squares in subsam- ple g; see Section 3.6 for an illustration. The aboveF-test is typically referred to as the Chow testfor structural change (Chow, 1960).9 When using (3.32), it can easily be adjusted to check for a break in a subset of the coefficients by including only those interactions in the general model. Note that the degrees of freedom of the test should be adjusted accordingly.
Application of the Chow test is useful if one has some a priori idea that the regres- sion coefficients may be different across two well-defined subsamples. In a time series application, this requires a known break date, that is, a time period that indicates when the structural change occurred. Sometimes there are good economic reasons to identify the break dates, for example, the German unification in 1990, or the end of the Bretton Woods system of fixed exchange rates in 1973. If the date of a possible break is not known exactly, it is possible to adjust the Chow test by testing for all possible breaks in a given time interval. Although the test statistic is easily obtained as the maximum of all F-statistics, its distribution is nonstandard; see Stock and Watson (2007, Section 14.7) for additional discussion.
9The above version of the Chow test assumes homoskedastic error terms under the null hypothesis. That is, it assumes that the variance of𝜀iis constant and does not vary across subsamples or withxi. A version that allows for heteroskedasticity can be obtained by applying the Wald test to (3.32), combined with a heteroskedasticity-robust covariance matrix; see Subsections 4.3.2 and 4.3.4.
k k