Assignment 9 Econ 31 Fall 1999

Functional Form

Reading: Mirer p.111-130, 151-156, Beals Chtr. 12

Understanding Functional Form. ''An Equation consisting of two independentvariables can be linear in all variables, linear in some variables but not in others, or nonlinear in all variables'' (Johnson, Johnson, Buse 235).

Thus far, all of your regressions have involved equations which are linear
in all variables. Frequently, however, in applied economics, the
mathematical form of the equation to be estimated is not linear, and a ''an
estimating equation must be consistent with the underlying theory, in order
to get the desired estimates'' (234). The most common and basic non linear
functional forms involve reciprocal functions, quadratic functions, or
variables measured in Logarithms to the Base e (log-log, log-linear, or
linear-log). In each of these forms, we convert the data into a form which
is implicitly linear. (Table courtesy Johnson, Johnson, Buse, 11,1)

Name	Nonlinear Form	Linear and Additive Form	Marginal Effect (dy/dx)	Elasticity
Linear		Y=B0+B1X	B1	B1(X/Y)
Reciprocal (Hyperbola)		Y=B0+B1(1/X)	-B1(1/X^2)	-B1(1/YX)
Quadratic (polynomial degree 2)		Y=B0+B1X+B2X2	B1 +2B2X	(B1 +2B2X)(X/Y)
Log-Log	Y=B0XB1	ln Y=lnB0+B1ln X	B1(Y/X)	B1
Linear-Log (semi-log)	ey=eBoX1B1	Y=B0+B1lnX	B1(1/X)	B1(1/Y)
Log-Linear (expo)	Y=e(Bo+B1X)	ln Y=B0+B1X	B1e(Bo+B1X)	B1X

1 a.) Let Y be the output of wheat and X be the number of acres
cultivated. When acreage is low, the most fertile land will be cultivated
first. As acreage goes up, less fertile areas will be put to use; the
additional output from these areas will not be as high as the output from
the more fertile lands. This suggests a diminishing marginal product of
wheat acreage (Ramanathan).

i)What form would enable us to capture this relationship?

ii)Find the elasticity.

iii)Graph an equation for this line when B₀=0 and B₁<0.

2. A version of the so-called Phillips curve says that the rate of change
of money wages is a function of the reciprocal of the unemployment rate.
Specifically, let

w_t = money wage rate in year t
u_t = unemployment rate in year t

wpert_t = ((w_t -w_t-1 / w_t-1 ) * 100 = percentage rate of change in the wage rate

The Phillips curve is given by wpert_t = Y= B₀+B₁(1/u_t)+ e_t
where it is hypothesized that B₀ < 0 and B₁ >0

Use the country data given in stata file ''func3.dta'' to
a) Find the least squares estimates of B₀ and B₁
b) Test whether there is any relationship between wper and (1/u_t).
c) Find an estimate for the ''natural rate of unemployment'' (the natural
rate of unemployment is the rate for which wper = 0).

d) When does a change in the unemployment rate have the greatest impact on
the rate of change in wages? When does it have the smallest?

e) Find 95% interval estimates for B₀ and B₁ .

3. The log-linear model is frequently used in human capital literature;
the logarithm of earnings or wages is used as the dependent variable. One
general model would be of the form ln(WAGE)= B₀+B₁ED+B₂EX+B₃AGE+B₄ED^2+B₅EX^2+B₆AGE^2+e_t. Using the data from PS2 - File cps22an.

a) What signs do you expect on each of the variables?

b) Create variables ED^2,EX^2,AGE^2, and ln(HRWAGE).
(Command for creating LNWAGE . generate LNWAGE=ln(hourwg)
and replicate the above equation using cps22an data.

c) Test the significance of each of these variables individually and
interpret their meaning.

d) Test the hypothesis that B₄=B₅=B₆=0

e) Repeat a) replacing LNWAGE with HRWAGE.

f) Can you compare the R values from the above regression, and another
regression replacing LNWAGE with HOURWG? Which model makes more sense to
you? Can you explain the apparent insignificance or significance of certain
variables?

4. In the Keynesian theory of liquidity preference the transactions,
precautionary and speculative motives for holding money lead to a function
where the demand for money depends on income and the interest rate. Suppose
that we can write the demand for money as the linear function

M_t=B₀+B₁Y_t+B₂I_t+e_t where M_t represents money in the form currency and demand deposits,
Y_t is gross national product, and I_t is the interest rate on six-month U.S. treasury bills.

Observations on these variables in the U.S. economy for the period
1960-1983 are in a stata file entitled ''func1.dta''. Money, M, and Gross
National Product, Y, are in billions of dollars; the interest rate I is a
percentage.

b) Find least squares estimates of the coefficients B₀,B₁,and B₂ . Do these estimates have the expected
signs?

c) Give an economic interpretation for each of your estimated coefficients.

d) Predict money demanded for a gnp of 1,000 billion dollars and an interest rate of 12%,

5. A common specification of the Log-Log model, or double log model is the
Cobb-Douglas Production Function which has the following general form: Q_t=cK^a_tL^B_t
where c, a, b are unknown paramenters (Ramanathan). Taking the logarithms of both sides and
adding an error term, we get the following econometric forumulation (B₀=ln c). ln Q_t =B₀+ a ln K_t+
B ln L_t + u_t.

a) What is the elasticity of output with respect to capital?

b) What is the elasticity of output with respect to labor?

c) How could you test for constant returns to scale, increasing returns to
scale, and decreasing returns to scale? Write the null and alternative
hypothesis for each test (for further help see Micro textbook).

d) You have been approached by an agricultural firm; they want you to tell
them where they might best invest their money, whether in more labor, land,
buildings, machinery, other imputs, fertilizer, or pesticide in order to
improve output. Assuming the Cobb-Douglas function represents a valid model
for agricultural production, describe how you would go about finding the
marginal productivity of each of the inputs.

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