Asking for What you Don't Really Want

Sometimes, asking for something you don't really want can be an indirect way of getting something you actually do want or need. A million dollars? Not quite what I had in mind, actually. Nice thought, though!

Actually, what I had in mind was something a little more mundane. In particular, getting an econometric package to save you a lot of work by delivering up some information you need, when it's not at all apparent that the package is able to do so. It's a matter of asking it for something else, and getting what you really want as a by-product. A bonus, if you will!

The situation that I was thinking of is as follows.We've estimated a model, say by Maximum Likelihood estimation, and there is some nonlinear function of the parameters that we want an estimate for. Moreover, we want an interval estimate, so we need a standard error for the estimate of the nonlinear function.

More specifically, suppose we've estimated a CES production function, of the following form:

                 Q_i = γ [δ K_i^-ρ + (1 - δ)L_i^-ρ]^-ν/ρ exp(ε)   ;    i = 1, 2, ....., n ;

where is ε is a well-behaved, Normally distributed, error term, and γ, δ, ν, and ρ are unknown parameters, such that γ > 0; 0 < δ < 1; ν > 0; and ρ ≥ -1. The parameter γ is the “efficiency parameter”, δ is the “distribution parameter”, ν is the “returns to scale parameter”, and ρ is the “substitution parameter”.

It's easily shown that the elasticity of substitution between capital and labour in this model is given by the parameter, s = 1 / (1+r). This is a nonlinear function of ρ that I may want an interval estimate for.

Now, many econometrics packages make it easy construct such an interval estimate. Unfortunately, EViews doesn't appear to be one of these.

By the invariance of maximum likelihood, the MLE for σ is obtained immediately as 1 / (1 + r), where r is the MLE for ρ. But this is just a point estimate. To get an interval estimate what's needed, of course, is the complete estimated (asymptotic) covariance matrix for the estimated coefficients. Then, the Delta method can be used to get an asymptotic standard error for the estimate of the function of interest. The asymptotic normality of MLE's then enables us to construct an asymptotically valid confidence interval for σ.

Getting the asymptotic standard error by the Delta method would involve a bit of work in this example, but fear not! This is where we ask EViews for something we don't really want.

Once we've estimated the model, we pretend that we want to test some hypothesis involving the nonlinear function of interest. While we could easily, and legitimately test the hypothesis that ρ = 0 by using the asymptotic standard error for r, and a z-test, instead let's ask EViews to test the hypothesis that [1 / (1 + ρ)] = 1. As well as computing the associated Wald test statistic (which we're not really interested in), EViews will also report the MLE of the function [1 / ( 1 + ρ)], namely [1 / (1 + r)], as well as the asymptotic standard error of the latter, computed using the Delta method!

Let's illustrate this using some actual data. These data are available on the Data page that accompanies this blog, and the EViews file is available on the associated Code page.

I've taken the logarithm of both sides of  the CES production function in (1), and the parameters I'm going to estimate are log(γ), ν, ρ, δ, and the standard deviation of the Normally distributed errors (coefficients C(1) to C(5) in the EViews code). I've set up a "LOGL" object in my EViews workfile, and used the following starting values for the parameters (in the order above): 1.0, 0.8, 0.6, 0.4 and 1.0 respectively.

Here are the MLE results:

(You get the same results if you estimate the model by nonlinear least squares, of course, given the assumed normality of the errors.)

Now, if we select the "VIEW" tab, and then select "Wald coefficient tests...", we can formulate the restriction (nonlinear function) of interest to us:

And this is what we get:

Indirectly, this gives us the information that we wanted all along. The (asymptotic) standard error of the estimated normalized restriction is 0.116633. The fact that we've subtracted unity (or any other constant)value from the function we're interested in doesn't alter the standard error, so 0.116633 is the asymptotic standard error for [1 /(1 + r)] itself. Then, an asymptotically valid 95% confidence interval for the elasticity of substitution can be constructed as [1 /(1 + r)] +/- 1.96(0.11633), where [1 /(1 + r)] = 0.6623.

So the 95% confidence interval for the elasticity of substitution is [0.3943 ; 0.8503].

Notice that when we set up the restriction in order to "trick EViews", although we used [1 / (1 + ρ)] = 1, we could equally have used [1 / (1 + ρ)] = k, for any value of k. This would not have altered the standard error. (Try it!)

By the way, I know that when testing nonlinear restrictions, the value of the Wald test statistic not invariant to the way in which the restrictions are expressed. However, that doesn't affect anything here - we're not interested in the Wald test statistic per se - just in the asymptotic standard error associated with a particular nonlinear function.

So, sometimes asking for what we don't want will result in us getting something that we're actually hoping for. Of course, I wouldn't recommend doing this just before your next birthday!

© 2012, David E. Giles