ienac19 / econometrics 2 / applied problem set 2

IENAC19 / Econometrics 2 / Applied Problem Set 2

Topic: Nonlinear restrictions, dummy variables, and nonlinear least squares

• This problem set addresses the problem of testing nonlinear hypotheses on parame-

ters estimated from a linear model, inclusion and interpretation of dummy variables

within a regression model, and nonlinear least squares estimation.

• We use data on annual U.S. aggregate income (Yt) and consumption (Ct) over the

period 1950 � 1985 (t = 1, 2, . . . , T = 36), available as income_consumption.txt on

the website. Import the data into a dated work�le as year, inc and cons.

• Refer to �gures 1 � 4, and perform the following:1

1. Perform a brief descriptive analysis of the dataset. Then estimate the following

simple consumption function (which is an example of a `distributed lag model'):

Ct = α + βYt + γCt−1 + ut.

Carefully interpret the regression output. Compute the estimated short-run

marginal propensity to consume (SRMPC), which is de�ned as:

SRMPC =∂Ct∂Yt

= β.

Set Ct = Ct−1 := C? and Yt = Yt−1 := Y ? (where C? and Y ? are `long-run'

values of consumption and income, respectively). Then, compute the estimated

1Note that not all of the necessary steps are shown in the �gures!

IENAC19 / Econometrics 2 / Applied Problem Set 2 2

long-run marginal propensity to consume (LRMPC) as:

LRMPC =∂C?

∂Y ?=

β

1− γ:= δ.

Test whether the LRMPC is equal to 1, at the 95% level of signi�cance, using

both a (linear hypothesis):

H0 : δ = 1 =⇒ H0 : β = 1− γ.

and a (nonlinear hypothesis):

H0 : δ = 1 =⇒ H0 :β

1− γ= 1,

What do you notice?!2

2. Was there a break in income growth between 1962 and 1963? (i.e., is the growth

rate from 1963 onwards di�erent to the growth rate up until 1962?) Hint:

estimate the following model:

Yt = (α + βDyeart≥1963) + (γ + λDyeart≥1963)yeart + ut,

whereDA is a dummy variable that takes value 1 when conditionA is satis�ed,

and 0 otherwise. Perform and interpret any relevant tests, and plot the �tted

regression. Can you suggest any possible limitations of using dummy variables

to test for structural breaks in this way?

2In this problem set, we do not assume any knowledge of the di�culties that can arise in modelling(these) time series! These issues will be dealt with in a later problem set, and in the Forecasting course.


3. Perform the nonlinear regression:

Ct = α + βY γt + ut, (1)

where starting values α† and β† are the estimated values from the restricted

regression:

Ct = α + βYt + ut.

Note that �nding initial values for a nonlinear procedure can be di�cult: there

is no good rule, but a suggestion is to use a consistent estimator to provide

starting values � here, a simple linear model is a special case, and provides `nat-

ural' starting values.

Test the null hypothesis H0 : γ = 1 in (1), at the 95% level of signi�cance.

Further, test whether the SRMPC is (a) constant (but not necessarily equal to

0), and (b) equal to 1, at the 95% level of signi�cance. Interpret your results.


Figure 1: Ordinary least squares regression of consumption on a constant, income, andconsumption lagged one period. Wald test of the null hypothesis H0 : β = 1− γ.


Figure 2: Plots of income and time dummy variable against observation number.


Figure 3: Ordinary least squares regression of income on a constant and year, and cross-products with a dummy variable Dyeart≥1963. Plot of �tted regression and residuals, andactual income data, against time.


Figure 4: Ordinary least squares regression of consumption on a constant and income. Usethe starting values α† = 11.37375 and β† = 0.898329 and γ† = 1 to perform the nonlinear

least squares regression of consumption on a constant and a coe�cient multiplied byincome raised to a power.


Figure 5: Statistical table for N(0, 1). These tables have been taken from:http://fsweb.berry.edu/academic/education/vbissonnette/tables/tables.html.


Figure 6: Statistical table for Student's t(r).


Figure 7: Statistical table for F (m,n) at the 5% level.


Figure 8: Statistical table for F (m,n) at the 1% level.


Figure 9: Statistical table for χ2(q).

ienac19 / econometrics 2 / applied problem set 2

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