Functional Forms for Econometric Analysis
Suppose you have an economic model which predicts a relationship between a dependent variable Y and an independent variable X. In most cases the model will not tell you what functional form that relationship takes in the data, although it will usually give you some ideas about what shape it might take. The usual solution is to decide what sorts of shapes are most likely to describe the data, either from economic intuition or from examining the data, and then to try several different functional forms and see if they give similar results, and if not, see which gives the best results.
This page lists some of the most commonly used functional forms, shows what they look like and describes their properties, and gives you some ideas about how to choose between them.
Y = b0 + b1* X + e
and its graph looks like this:
The advantage of the linear functional form is its simplicity. Each time X goes up by 1 unit, Y goes up by b1 units, and this is true no matter what the values of X and Y are. The drawback of the linear functional form is also its simplicity; any time the effect of X should depend on the values of X or Y, the linear functional form can't be the right one. For example, if you have a cost curve of the form C = b0 + b1* Q + e, then the linear functional form implies that each time quantity Q goes up by one unit, costs C go up by b1 dollars. This can only be true if marginal costs are constant; it cannot be true if marginal costs are rising (or falling). If you think marginal costs are rising you would not want to use the linear functional form.
Y = b0 + b1* X + b2* X2 + e
and its graph looks like this:
The advantage of the quadratic form is that when X goes up by one unit, Y goes up by b1 + 2*b2*X units. (You can derive this yourself by calculating dY/dX from the equation if you know some calculus.) If b2>0, then as X gets bigger, the additional effect of X on Y gets bigger also; if b2<0, then as X gets bigger, the additional effect of X on Y gets smaller. If you had the cost curve C = b0 + b1* Q + b2* Q2 + e, then the marginal cost (the amount that C rises when Q goes up by one) would be MC = b1 + 2*b2*Q. If If b2>0, then you have increasing marginal costs; if b2<0, then you have falling marginal costs. Economic theory suggests that you should usually have increasing (or possibly constant) marginal costs; you could test whether b2>0 (or b2=0) to test this theory. The drawback is that this functional form will have (in different parts of the graph) both positive slope and negative slope; this may be a problem if your regression should always have the same slope (for instance, demand curves should always slope down) and if your data lies in parts of the estimated graph with both slopes.
log Y = b0 + b1* log X + e
and its graph looks like this:
There are two ways to think about this functional form. One way to think about it is that if X rises by 1%, then Y will rise by b1%; this is a special property of the logarithmic relationship. A second way is that b1 is the elasticity of Y with respect to X; this follows from the definition of elasticity. (You can also prove this with some calculus if you remember that d(logY)/d(logX) is equal to (dY/dX)*(X/Y) and rearrange terms.) This functional form is commonly used when you are interested in estimating an elasiticity of some kind. It is also commonly used any time a Cobb-Douglas function is likely; this is because the Cobb Douglas function is
Y = A * Xb1 [ *ee, if you want to include the error term]
and if you take logs on both sides, you get
log Y = log(A) + b1* log X + e
and you can let b0 = log(A). Therefore the logarithmic functional form is often used for cost functions, production functions, utility functions, and other functions that are often described with the Cobb-Douglas function.
log Y = b0 + b1* log X + b2* (log X)2 + e
and its graph can take a very wide variety of shapes.
This functional form has the same relationship to the logarithmic functional form that the quadratic has to the linear; it adds a squared term to the equation. In this functional form, the elasticity of Y with respect to X is b1 + 2*b2* log X, which means that the elasticity can change as X gets bigger or smaller, which is useful if you think the elasticity might not be constant with respect to X. It has the same problem of having both positively and negatively sloped regions that the quadratic functional form has. The translog functional form is a standard functional form for cost and production functions.
log Y = b0 + b1* X+ e
and its graph looks like this:
The semilog functional form has the property that if X rises by 1 unit, Y rises /100 by [b1*100] percent. This is not a commonly desired property but there are some applications where it is useful. For instance, the relationship between salaries and education is almost always expressed in this functional form as log Sal = b0 + b1* Ed + e, meaning that if a person's education goes up by 1 year, that person's salary rises by [b1*100] percent. For example, if b1=0.08, it means that one extra year of education increases your salary by 8%. As X gets large, the slope of the line will get quite large, because as X gets larger a percent increase in X also gets larger.
You can also put the log on the Y - that is, Y = b0 + b1* log X+ e - and this means when X rises by 1%, Y rises by [b1/100] units. In this case, as X gets larger, the slope gets flatter, because it takes a bigger increase in X to constitute a 1% increase. This functional form is not widely used; its graph is shown below.)
Y = b0 + b1* 1/X+ e
and its graph looks like this:
The reciprocal functional form is usually used when Y and X both have to be positive and when the relationship between them probably slopes down (that is, b0>0 and b1>0). In this case, the linear functional form isn't good because the line will eventually cross the axis and Y will become negative for high enough X values. The reciprocal functional form prevents this from happening. It's usually used for curves such as demand curves which need to have this property.
Here's a helpful chart which may help you decide which functional form best fits your data:
| Relationship looks like: | Shape is: | Mathematical Properties: | Functional form has these properties: |
 |
Rising straight line | dY/dX>0, d2Y/dX2=0 |
Linear: b1>0 |
| Quadratic: b1>0, b2=0 | |||
| Logarithmic: b1=1 | |||
| Translog: b1=1, b2=0 | |||
| Semilog: Cannot take this shape | |||
| Reciprocal: Cannot take this shape | |||
 |
Falling straight line | dY/dX<0, d2Y/dX2=0 |
Linear: b1<0 |
| Quadratic: b1<0, b2=0 | |||
| Logarithmic: Cannot take this shape | |||
| Translog: Cannot take this shape | |||
| Semilog: Cannot take this shape | |||
| Reciprocal: Cannot take this shape | |||
 |
Upward sloping curve, slope getting flatter (may become downward sloping in some cases). | dY/dX>0, d2Y/dX2<0 |
Linear: Cannot take this shape |
| Quadratic: b1>0, b2<0 | |||
| Logarithmic: 1>b1>0 | |||
| Translog: b1>0, b2<0, =0, or >0 and small | |||
| Semilog: b1>0 | |||
| Reciprocal: b1<0 | |||
 |
Upward sloping curve, slope getting steeper (may start downward sloping in some cases). | dY/dX>0, d2Y/dX2>0 |
Linear: Cannot take this shape |
| Quadratic: b1>0, b2>0 | |||
| Logarithmic: b1>1 | |||
| Translog: b1>1, b2>0, =0, or <0 and small | |||
| Semilog: Cannot take this shape | |||
| Reciprocal: Cannot take this shape | |||
 |
Downward sloping curve, slope getting flatter (may become upward sloping in some cases). | dY/dX<0, d2Y/dX2>0 |
Linear: Cannot take this shape |
| Quadratic: b1<0, b2>0 | |||
| Logarithmic: b1<0 | |||
| Translog: b1<0, b2>0 | |||
| Semilog: b1<0 | |||
| Reciprocal: b1>0 | |||
 |
Downward sloping curve, slope getting steepter (may start upward sloping in some cases). | dY/dX<0, d2Y/dX2<0 |
Linear: Cannot take this shape |
| Quadratic: b1<0, b2<0 | |||
| Logarithmic: Cannot take this shape | |||
| Translog: Various combinations | |||
| Semilog: Cannot take this shape | |||
| Reciprocal: Cannot take this shape |
If you have several different exogenous variables, then the shape of the relationship between Y and each exogenous variable might be the same or might be different. You can use the same functional form for each exogenous variable if you want to; for instance, if you think output depends on three different outputs, you might use the linear functional form for each:
Y = b0 + b1* X1 + b2* X2 + b3* X3 + e
Or the translog functional form for each one:
log Y = b0 + b1* log X1 + b2* (log X1)2+ b3* log X2 + b4* (log X2)2+ b5* log X3 + b6* (log X3)2 + e
You can also combine several different functional forms in one regression, for example:
Y = b0 + b1* log X + b2* X2 + b3* X22 + b4* 1/X3+ e
although you should usually have reasons to think that the shape of the relationship between Y and X1 is different than the shapes between Y and X2, and Y and X3, if you do.
