ECON 3208

Econometric Methods

Tutorial 9 Week 10

Tutorial Question 1, Question 10.2 (Wooldridge, page 371)Let gGDPt denote the annual percentage change in gross domestic product

and let intt denote a short-term interest rate. Suppose that gGDPt is related

to interest rates by:

gGDPt = α0 + δ0intt + δ1intt-1 + ut

Where ut is uncorrelated with intt and intt-1 and all other past values of

interest rates. Suppose that the Federal Reserve follows the policy rule

intt = γ0 + γ1(gGDPt-1 – 3) + νt

Where γ1 > 0.

When last year’s GDP growth exceeds 3% the Federal Reserve increases

interest rates to prevent the economy “overheating”. If νt is uncorrelated

with all past values of intt and ut, argue that intt must be correlated with ut-1.

(Hint: Lag the first equation by one time period and substitute for gGDPt-1

in the second equation).

Which Gauss Markov assumption does this violate?

Argument: intt must be correlated with ut-1

Lag the equation for gGDP:

gGDPt-1 = α0 + δ0intt-1 + δ1intt-2 + ut-1

Now substitute this into the right-hand-side of the intt equation:

intt = γ0 + γ1(α0 + δ0intt-1 + δ1intt-2 + ut-1 – 3) + νt

Multiply out the bracket and rearrange

= (γ0 + γ1α0 – 3γ1) + γ1δ0intt-1 + γ1δ1intt-2 + γ1ut-1 + νt

By assumption, ut-1 has zero mean and is uncorrelated with all right-hand

side variables in the previous equation, except itself, so that:

Cov(int,ut-1) = E(intt⋅ut-1) = γ1E(u2t-1) > 0

This is because γ1 > 0.

Argument: intt must be correlated with ut-1 cont…

If σ2u=E(u2

t) for all t then it follows that Cov(int,ut-1) = γ1σ2

u.

Which Gauss Markov assumption does this violate?

This violates the strict exogeneity assumption, TS.3 (Wooldridge, 2013, p. 350).

Assumption TS 3

“For each, t, the expected value of the error ut given the explanatory variable for

all time periods, is zero. Mathematically:

E(ut|X) = 0 ∀ t)”

Here, while ut is uncorrelated with intt, intt-1, and so on, ut is correlated with

intt+1.

Tutorial Question 2, Question 10.6 (Wooldridge (2013, p. 376)

In example 10.4, p.354 (and replicated in computer exercise C10.4), we saw

that our estimates of the individual lag coefficients in a distributed lag model

were very imprecise. One way to alleviate the multicollinearity problem is to

assume that the parameters δj follow a relatively simple pattern. For

concreteness consider a model with four lags:

yt = α0 + δ0 zt + δ1 zt-1 + δ2 zt-2 + δ3 zt-3 + δ4 zt-4 + ut (1)

Now, let us assume that the δj follow a quadratic in the lag j:

δj = γ0 + γ1 j + γ2 j2

for parameters γ0, γ1, γ2.

This is an example of a polynomial distributed lag (PDL) model.

Substitute the formula for each δj into the distributed lag

model and write the model in terms of the parameters γh for h

= 0, 1, 2

Problem (i)

δ0 = γ0

δ1 = γ0 + γ1 + γ2

δ2 = γ0 + 2γ1 + 4γ2

δ3 = γ0 + 3γ1 + 9γ2

δ4 = γ0 + 4γ1 + 16γ2

Solution (i)First, recall the polynomial

δj = γ0 + γ1 j + γ2 j2, and

Define δ for for j = 0, 1, 2, 3, 4

Now substitute into the equation (1)

yt = α0 + γ0zt + (γ0 + γ1 + γ2)zt-1 + (γ0 + 2γ1 + 4γ2)zt-2

+ (γ0 + 3γ1 + 9γ2)zt-3 + (γ0 + 4γ1 + 16γ2)zt-4 + ut

Multiply out the brackets and collect terms:

yt = α0 + γ0(zt + zt-1 + zt-2 + zt-3 + zt-4 )

+ γ1(zt-1 + 2zt-2 + 3zt-3 + 4 zt-4 )

+ γ2(zt-1 + 4zt-2 + 9zt-3 + 16 zt-4 ) + ut

Solution (i) cont…..

Explain the regression you would run to estimate the

parameters γh.

Problem (ii)

Solution (ii)define three new variables, zt0, zt1,

zt2 in terms of t the variable zt to zt-4

zt0 = (zt + zt-1 + zt-2 + zt-3 + zt-4);

zt1 = (zt-1 + 2zt-2 + 3zt-3 + 4zt-4); and

zt2 = (zt-1 + 4zt-2 + 9zt-3 + 16zt-4).

Then, α0, γ0, γ1, and γ2 are obtained from the OLS regression of yt on

zt0, zt1, and zt2, t = 1, 2, … , n.

That is:

yt = α0 + γ0zt0 + γ1 zt1 + γ2 zt2 + ut (2)

Now you can retrieve the parameters of interest using:j

δ

2

0 1 2ˆ ˆ ˆ ˆ

j j jδ γ γ γ+ +=

The polynomial distributed lag of (2) is a is a restricted version

of the general model (1).

How many restrictions are imposed?

How would you test this or these restrictions?

Problem (iii)

The unrestricted model is the original equation (1), which has six

parameters (α0 and five δj, j = 0, 1, 2, 3 and 4).

The PDL model (the restricted model) has four parameters (α0 and

γ0, γ1, and γ2) .

Therefore, there are two restrictions imposed in moving from the

general model to the PDL model. (Note how we do not have to

actually write out what the restrictions are).

Solution (iii)How many restrictions are imposed?

Solution (iii) cont…

( )( )

( )( )

2 2 2 2

2 2

2

1 1 6

ur r ur r

ur ur ur

R R q R RF

R DoF R n

− −= =

− − −

which is F~F2,n-6 under the null of the restricted model.

The degrees of freedom in the unrestricted model are n – 6.

Therefore, we would obtain the unrestricted R-squared, R2ur from

the regression of yt on zt, zt-1, …, zt-4 and the restricted R-squared

from the regression in part (ii), R2r and the F statistic can be

computed as:

How would you test this or these restrictions?

Construct a simple F test on the basis of these two restrictions.

Tutorial Question 3, Question C10.1 (Wooldridge, p. 377)

In October 1979, the Federal Reserve changed its policy of targeting

the money supply and instead began to focus directly on short-term

interest rates. Using data in the intdef.dta file, define a dummy

variable, post79, equal to 1 for years after 1979. Include this dummy

in equation (10.15) to see if there is a shift in the interest rate equation

after 1979.

What do you conclude?

Equation (10.15), Example 10.2, p. 352

Where:

i3 is the three month T-bill rate;

inf is the annual inflation rate; and

def is the federal budget deficit, as a percentage of GDP.

Now augment this equation with:

Post79 a binary variable where Post79 = 1 if the

observation t occurs after the 1979 and zero else.

The Stata Output for the Two Models

What conclusions can you draw from the estimated coefficient and

its standard error?

The coefficient on post79 is statistically significant

(the t statistic is approximately 3.06) and economically large:

accounting for inflation and deficits, i3 was about

1.56 points higher on average in years after 1979.

The coefficient on def falls once post79 is included in

the regression.

Conclusion

Tutorial Question 4, Question C10.4 (Wooldridge, p. 377)

A distributed lag model, using fertl3.dta, based on the

distributed lag model of example 10.4, p. 358, equation 10.19.

The issue is that the estimated parameters on the distributed

lag are imprecise, due to multicollinearity. In order to estimate

and test the long run effect of the variable pe on the general

fertility rate, gfrt, the model is re parameterized so that the

long run propensity (LRP) can be estimated and tested.

Verify that the standard error for the LRP in equation (10.19

is about 0.030

gfrt = α0 + δ0 pet + δ1 pet-1 + δ2 pet-2 + β1 ww2t + β2 pillt + ut

(10.19)

where:

gfr is the general fertility rate (no of children born to 100

women;

pe is the real dollar personal tax exemption;

ww2 dummy variable =1 for years in world war two; and

pill dummy variable 1963 onwards when the birth control pill

available.

Equation (10.9)

The Estimate Equation (10.9), see p. 358

Recall equation (10.19) p. 355

gfrt = α0 + δ0 pet + δ1 pet-1 + δ2 pet-2 + β1 ww2t + β2 pillt + ut

Re parameterize the equation by defining:

θ0 = δ0+δ1+δ2 ;

as the long run propensity (LRP) of pe to affect gfr.

Solve this for δ0 : δ0 = θ0−δ1−δ2; and substitute into (10.19)

gfrt = α0 + (θ0−δ1−δ2) pet + δ1 pet-1 + δ2 pet-2 + ...

Solution to Problem

Recall

gfrt = α0 + (θ−δ1−δ2) pet + δ1 pet-1 + δ2 pet-2 + ...

Multiply out bracket and factor on common parameters:

gfrt = α0 + θ0 pet + δ1 (pet-1 − pet) + δ2(pet-2 − pet) + ...

The LRP θ0 can be estimated by regressing gfr on pet, (pet-1 – pet),

(pet-2 – pet ), ww2t and pillt

Solution to Problem cont…

The Estimate of the Re parameterized

Equation (10.9), see p. 358