ECMT 5001 Year 2008 Semester 2_ECMT 5001 Solution Tutorial 9

Econometrics & Business Statistics ECMT5001: Principles of Econometrics

Tutorial 9: Multiple Regression Applications

NOTE: Use EViews to solve the computational exercises. 1. The following fitted models are based on 11 annual observations:

Model A

2? 2.69110.47950.6628(.1216)(.1140)

t t

Y X r =-=

Model B

2?ln 0.77740.2530ln 0.7448

(.0152)(.0494)

t t

Y X r =-=

(Estimated standard errors are given in parentheses)

The variables are: Y - per capita number of cups of coffee consumed per day X - price of coffee A$ per 500gram packet. (a) Interpret the slope coefficients in the two estimated models

Model A: On average, all else held constant, a $1 increase in the price of coffee (per 500

gr packet) will decrease the number of cups consumed per day by 0.48.

Model B: in a model where both X and Y are logged, the slope coefficient is the elasticity. We say that, for a 1% increase in the price of coffee, there will be a 0.25% decrease in the number of cups consumed per day

(b)

Evaluate the price elasticities of demand for coffee in the two models. (For the

appropriate model, evaluate the elasticity at the point of the means (,)(1.11,2.43)X Y =.)

The elasticity at the means is defined by:

Y

X X Y δδε=

Model A:

1.11

0.47950.2192.43

y X x Y δεδ=

=-=-

Model B: ε = -0.2530

(c)

Test the hypotheses that the slope coefficients are equal to zero in each case.

Model A: H 0: βA = 0 H 1: βA ≠ 0

Level of Significance: α = 0.05

Distribution: ~()

A A n k A b

B t t SE b --=. A ssuming that the u i ~ iidN(0,σ2 ){ i.e. there is no

serial correlation, no heteroskedasticity and E(error|X i )=0} and assuming all other assumptions are met. (Recall that k is the number of regression coefficients estimated!) Critical Value: from the t-tables, the 5% critical value with 11 – 2 = 9 degrees of freedom is 2.262 for a two-tail test.

Rejection Rule: Reject H 0 if t* is greater than 2.262 or less than -2.262. Calculated Value: t* = -0.479/0.114 = -4.21.

Conclusion: Since t* is less than the critical value, we reject H 0 and we reject the null

hypothesis at the 5% level.

Conclude that, the evidence supports a negative linear relationship between the price of

coffee and the number of cups drunk per capita per day at the 5% level of significance.

Model B: H 0: βB = 0 H 1: βB ≠ 0

Level of Significance: α = 0.05

Distribution: ~()

B B n k B b B t t SE b --=. A ssuming that the u i ~ iid N(0,σ2) and assuming all

other assumptions are met.

Rejection Rule: Reject H 0 if t* is greater than 2.262 or less than -2.262. Calculated Value: t* = -0.2530/0.0494 = -5.121

Conclusion:. Since t* is less than the critical value, we reject H 0 and we reject the null

hypothesis at the 5% level.

Conclude that the evidence supports a negative logarithmic relationship between the

price of coffee and the number of cups drunk per capita per day at the 5% level of significance.

(d)

Evaluate the adjusted coefficient of determination in each case.

Model A:

()221110.63n R R n k -??=--= ?-??

Model B:

()2

21110.72n R R n k -??

=--= ?-??

(e)

Comment on the statement that since the r 2 for Model B is greater than that for Model A, the former is the preferred specification.

Can’t say this!!

We can only compare the R 2 values when the y variable is the same. So we can’t compare a regression on Y with a regression on ln(Y).

(f)

What would have been the estimates for Model A if price were measured in Australian cents instead of dollars?

Simple answer: if you multiply the scale in which Price is multiplied by 100. So the new Price variable would be 100 x Price = 100 x X

Therefore the new estimated coefficient will be b 2 / 100.

Proof: write the new model (with the different scaling) as:

12t t t Y G G X e =++

(100)12222

(100)

1

(100)(100)100/100

1000010000100000n

i

i

x y xy i n

x x

i i X Y n X Y

SS SS g b SS SS X n X ==-=

=

=

=-∑∑

2. Consider the following total cost function where Y i represents total cost for the ith firm (in A$) and X represents the quantity of outputs (in units). The data for 30 firms are in the file, Tut9Q2.wf1, on the Blackboard site under Data .

=++++231234i i i i i Y B B X B X B X u .

(a) What signs do you anticipate each of the β coefficients should have on the basis of economic theory? (Hint: what does economic theory tell you about the shapes of the average and marginal cost curves?)

Economic theory tells us that cost curves increase at a decreasing rate. One way to model something like this is to model it as an upside down parabola (sketch)

The equation for a parabola like this is y = -ax 2 + bx + c. We want the curve that we are modelling to have a shape like this, so we choose this functional form, and we expect that B 1 will be positive, B 2 will be positive and B 3 will be negative.

The point is: we choose the functional form to give us the shape that we want. The shape that we want is based on things like scatter graphs, or economic/financial theory.

(b)

Estimate the model on the basis of the data given.

(c) What is the value of R 2? See above – R 2 = 0.977

(d)

Test the hypotheses that each of the individual coefficients are significant (taking care to test the null against alternatives derived from your answers to part (a).

H 0: B 2=0 H A : B 2>0

Level of Significance: α = 0.05

Distribution: 222~()

n k b B t t SE b --=. A ssuming that the u i ~ iid N(0,σ2) and assuming all

other assumptions are met.

Critical Value: The one tailed critical value at the 5% level is 1.706 with 26 degrees of

freedom.

Rejection Rule: Reject H 0 if t* is greater than 1.706 Calculated Value: t* = 1.446

Conclusion: Since t* = 1.446 is less than 1.706, we do not reject the null hypothesis at

the 5% level.

Conclude that the evidence does not support a positive linear relationship between X and

Y at the 5% level of significance..

Repeat for B 3 and B 4

(e)

Test the joint significance of all of the slope coefficients.

H 0: B 2= B 3= B 4=0 H A : at least one B ≠ 0

Level of Significance: α = 0.05 Distribution: (),/~/()

R UR m n k UR RSS RSS m

F F RSS n k --=-

Assume that u i ~ iidN(0, σ2) and all other assumptions are met

Critical Value: Therefore the one tailed critical value from the F tables at the 5% level is

2.98 using 3 and 26 degrees of freedom.

Rejection Rule: Reject H 0 if F* is greater than 2.98. Calculated Value: F* = 363

Conclusion: As F* is greater than 2.98, so we reject the null hypothesis at the 5% level.

Conclude that at least one of the slope coefficients must be significant and the model is useful at the 5% level of significance.

(f)

Test whether the data suggest that a linear function will suffice.

This amounts to testing B 3 = B 4 = 0.

034:0H B B ==

H A : At least one B i ≠ 0 for i=3,4.

Level of Significance : α = 0.05

Distribution: (),/~/()

R UR m n k UR RSS RSS m

F F RSS n k --=

-

Assume that u i ~ iidN(0, σ2) and all other assumptions are met

Critical Value: Therefore the critical value from the F tables = 3.37 using 2 and 26 degrees of freedom.

Rejection rule: Reject H 0 if F*>F 2,26=3.37.

Calculated Value:

the relevant value is calculated as:

14563.235UR RSS =, 72538.64R RSS =

()()/*/()

72538.64-14563.235/2

14563.235/26

51.75

R UR UR RSS RSS m

F RSS n k -=-=

=

Conclusion: Therefore as F*> F 2,26=3.37, we reject the null hypothesis at the 5% level

Conclude that a linear model will not suffice at the 5% level of significance.

3. (Gujarati, Ex. 8.18) Consider the annual data for the period 1960-1982 on energy demand (D), real GDP (M) and real energy prices (P) in the OECD countries given in the file guj-table8-3.wf1 on the Blackboard site under Data .

(a) Fit a log-log model to the data (i.e. one where you have taken logarithms of all variables) (ld = log(demand), lgdp=log(gdp) and lp=log(price))

(b)

Fit a linear model to these data. 123t t t

t Y B B M B P u =+++

(c)

For each of your two estimated models, carry out statistical inference procedures to

assess the significance of each coefficient individually and all coefficients in an equation jointly.

Always state your hypotheses and distributions before you interpret the p-values.

By this stage, you should be able to do these just by looking at the t stats and the F stat. They all have low p values and high t stats, which always indicate that we would reject the null hypotheses.

(d)Now refit the two models to the period 1960-1981. Use the two new models to get a

prediction for the value in 1982 based on the first 22 observations.

Dependent Variable: LD

Method: Least Squares

Date: 05/24/07 Time: 09:48

Sample: 1960 1981

Included observations: 22

Variable Coefficient Std. Error t-Statistic Prob.

C 1.464640 0.096809 15.12916 0.0000

LGDP 0.995155 0.018111 54.94861 0.0000

LP -0.311661 0.025382 -12.27898 0.0000

Dependent Variable: DEMAND

Method: Least Squares

Date: 05/24/07 Time: 09:49

Sample: 1960 1981

Included observations: 22

Variable Coefficient Std. Error t-Statistic Prob.

C 27.31686 1.549455 17.62998 0.0000

GDP 0.978193 0.019172 51.02238 0.0000

PRICE -0.248012 0.016809 -14.75427 0.0000

Model A: ln(yhat) = 4.597, therefore, Yhat=exp(4.597)=98.202

Model B: yhat=97.941

(e)Explain with reasons whether you prefer the log-log specification to the linear one.

4.(Gujarati, Ex. 8.17) Based on the GNP/money supply data given in Table 8-5, the following

regression results were obtained (Y = GNP, X = M2)

a)For each model, interpret the slope coefficient.

log-log or double-log: slope coefficient measures constant elasticity:

For a 1% increase in money supply, we would expect a 0.9882% increase in GNP on average.

Log-lin: Slope coeff measures the relative change in Y for an absolute change in X:

For a unit increase in M2, we would expect a 0.00057*100=0.057% increase in GNP on average. In general – whichever variable has been logged, we talk about in terms of percentage changes. b)For each model, estimate the elasticity of the GNP with respect to money supply and interpret it.

c)Are all R2 values directly comparable? If not, which ones are?

The only comparable R2 are the ones with the same dependent variable. So the R2 of the linear and lin-log models are comparable. The R2 of the log-linear (log-log) and log-lin models are comparable.

d)Which model will you choose? What are the criteria that you consider in your choice?

e)According to the monetarists, there is a one-to-one relationship between the rates of change in

the money supply and the GNP. Do the preceding regressions support this view? How would you test this formally?

The rate of change would be the elasticity. I would test the slope coefficient is equal to one in the log-linear model.