[Reply to OP and to the list.] I don't see the necessity for repeating the question four times.
On Sun, 22 Feb 2004, chokie wrote (edited to remove redundacies): > I have a qns, let say an original Regesssion to is be estimated as > follows: > > Y = B + b1x1 + b2x2 > > Then, if diagnostic test suggests that the relationship between Y and > x1 is nonlinear, add a variable X3 (which is square of X1). > > Y = B + b1x1 + b2x2 + b3(x1*x1) > > Qns: how to interpret X1 since it has both a linear and nonlinear > relationship ------------------------------------------------------------ For starters, it is not always necessary to model nonlinearity by adding squares (and cubes and fourth powers and...) of the original predictors. Some other form of nonlinearity might be much more pertinent. Once you start adding powers of any predictor, as a way of modelling some non-linear relationship between X1 and Y, you are nearly always immediately into unbridled empiricism. The only reasonable kind of "interpretation" (usually) is to display the function obtained, preferably in graphical form, and say "Look at it: this is how the relationship is currently being modeled." Or words to the same effect. And then try, if you can, to make sense of the SHAPE thus produced, not of the analytical algebraic function currently in hand. The slim volume on interpreting multiple regression by Aiken and West (about 1985 or 1990?) illustrates one way of interpreting shape information, although it doesn't make as effective use of that information as they might have done, and I disagree with a central facet of their approach. (It is of course possible that the relationship between the variables in question really is quadratic, and that this would be expected on some kind of theoretical basis. But in that case one would have set off at once with a quadratic function, and not justified including the square of X1 by a diagnostic test following the fitting of a linear function.) Also, ANY power function becomes dysfunctional when one gets into large enough values of X1: the predicted value heads for infinity (or negative infinity, depending on the sign of the coefficient for "X3") even more rapidly than does a linear function, and nearly always one KNOWS that cannot in fact be true, so that one's attention must be limited to a rather small region of values of X1. > Message 2 in thread > From: Don Libby ([EMAIL PROTECTED]) > Subject: Re: Weird instruction found in an econ text > Date: 2004-02-21 02:52:07 PST > < snip: copy of OP > > > Relationship between Y and X is quadradic: Well, no, not necessarily. The function one is using at this point is quadratic, right enough; but the relationship between X and Y is only being approximated by that function. Whether the true relationship is quadratic cannot be discovered at this point: one can only say (should it be true) that the functional fit is improved by adding a quadratic component to the linear function. > this is the first term of the Taylor Series appoximation to the > exponential function, ... True but probably irrelevant. If an exponential relationship made sense in the first place (or even in the second place or later), you would MODEL an exponential relationship, not futz around with crude quadratic approximations thereto. (Hint: you do this by taking logarithms. If Y is proportional to e^X: i.e., Y = a*e^(b*X) for real numbers "a" and "b", then log(Y) is linear in X: log(Y) = log(a) + b*X.) > which sometimes gives a not-too-bad approximation to the relationship > Y = e^X, which means Y accelerates (or decelerates) as X increases. Yes, well, if the real relationship was anything like "Y = e^X", you'd do much better modelling it directly. At least then you could talk learnedly, and perhaps even sensibly, about half-lives (or of doubling times, depending on the sign of "b") when X1 actually reports chronology. ------------------------------------------------------------ > Message 3 in thread > From: Robert Vienneau ([EMAIL PROTECTED]) > Subject: Re: Weird instruction found in an econ text > Date: 2004-02-21 13:30:10 PST < snip, copy of OP and previous reply > > The Taylor series for exp( x ) is > > 1 + x +x^2/2 + ... + x^n/n + ... > > It seems confused to describe the linear regression relationship > between Y and X above as the first term of the Taylor series > approximation to the exponential function, since, in most cases > the coefficients will not have desired ratios to one another. Another consequence of putting the cart before the horse. First thing is to try to apprehend, and then to describe, the SHAPE of the relationship you're trying to make sense of. And to fit it by a suitable function, once you discover that "linear" doesn't cut it. Hard to justify worrying about ratios of coefficients if one isn't yet sure one has a handle on an appropriate functional form. ------------------------------------------------------------ > Message 4 in thread > From: Robert Vienneau ([EMAIL PROTECTED]) > Subject: Re: Weird instruction found in an econ text > Date: 2004-02-21 16:08:07 PST < snip, copy of OP > > I don't find any problem of interpretation above. But perhaps the > following will help: < snip, some algebra > > That is, Y is linear in x1 and the square of x4. Hard to see how this is an improvement over "Y is linear in X1 and also quadratic in X1." It certainly isn't any simpler, and thus cannot be justified on grounds of parsimony (otherwise, by "Occam's razor"). ------------------------------------------------------------ > Message 5 in thread > From: chokie ([EMAIL PROTECTED]) > Subject: Re: Weird instruction found in an econ text > Date: 2004-02-22 08:01:46 PST < snip, copy of OP > > Maybe I should frame this in context. Suppose the dependent variable > is career prospects and X1 refers to the level of education. In such > an equation, am I saying that career prospects are both positively > (linear) associated with the level of education and also curvilinearly > associated with the level of education. > > In terms of econometric interpretation, I would not know how to > interpret X4. I do notice that x4 is additive, incuding the beta of > x2. Now, let say that X2 refers to a totally unrelated variable such > as race or location. How would you interpret the relationship? Sigh. You have a relationship in three dimensions: X1, X2, and Y. SHOW the relationship: draw pictures. Until you've done at least that much, if only for your own benefit if not for the benefit ofyour readers, all else is gas. ------------------------------------------------------------ Donald F. Burrill [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
