ECMT 5001 Year 2008 Semester 2_ECMT 5001 Solution Tutorial 6

Econometrics & Business Statistics

ECMT5001: Principles of Econometrics

Tutorial 6: Simple Regression Models

1.

Which of the following models are linear regression models?

a.

y

i

= B

1

+ B

2

x

i

2

+ u

i

d) y

i

= B

1

exp(B

2

< br>x

i

+ u

i

)

b.

y

i

= B

1

+ B

2

ln x

i

+ u

i

e) y

i

= B

1

+ B

2

3

x

i

+ u

i

c.

ln y

i

= B

1

+ B

2

x

i

+ u

i

f) y

i

= B

1

+ B

2

(1 / X

i

) + u

i

A linear regression model has to be linear in the parameters, i.e. in the B’s.

So, (a), (b), (c)

and (f) are linear regression models, but (d) and (e) are not. The formal definition is that

?

y

i

should NOT be a function of B

j

.

?

B

j

2.

(Gujarati, Ex. 6.9) The following table gives data on weekly family consumption

expenditure (Y) (in dollars) and weekly family income (X) (in dollars).

Weekly income (X)

Weekly consumption expenditure (Y)

80

55, 60, 65, 70, 75

100

65, 70, 74, 80, 85, 88

120

79, 84, 90, 94, 98

140

80, 93, 95, 103, 108, 113, 115

160

102, 107, 110, 116, 118, 125

180

110, 115, 120, 130, 135, 140

200

120, 136, 140, 144, 145

220

135, 137, 140, 152, 157, 160, 162

240

137, 145, 155, 165, 175, 189

260

150, 152, 175, 178, 180, 185, 191

(a)

For each income level, compute the mean consumption expenditure,

E

(

Y

X

i

)

, that is, the conditional expected value.

X

E(Y|X=x)

80

100

120

140

160

180

200

220

240

260

65

77

89

101

113

125

137

149

161

173

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(b)

Plot these data in a scattergram with income on the horizontal axis and consumption

expenditure on the vertical axis.

(c)

Plot the conditional means derived in part (a) on the same scattergram.

To get this in Excel

?

Make a list of all the data, X, Y and E(Y|X=x) (This is the average Y when

X=80 and the average when X = 100 etc.)

X

Y

E(Y | X=x)

80

55

65

80

65

65

80

65

65

80

70

65

80

75

65

100

65

77

100

70

77

100

74

77

100

80

77

100

85

77

You need to type out the whole list.

?

Go to Insert Chart: then choose XY scatter and make the graph.

(d)

What can you say about the relationship between

Y

and

X

and between mean

Y

and

X

.

Seems to be a positive relationship; as X increases, Y increases. Looks like a linear

relationship.

(e)

Write down the Population Regression Function (PRF) and the Sample Regression

Line (SRL) for this example.

PRF

:

y

i

?

B

?

B

1

x

i

?

u

i

?

i

?

b

0

?

b< /p>

1

x

i

?

17

?

0.6

x

i

y

(f)

Is the PRF linear or nonlinear?

Linear, because we defined it that way!

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3.

(Gujarati, Ex. 5.9) Suppose someone has presented the following regression results

for your consideration:

?

?

2.6911

?

0 .4795

X

Y

t

t

where

Y

= the coffee consumption in the United States (cups per person per day)

X

= the retail price of coffee ($$ per pound)

t

= the time period.

(a)

Is this a time series regression or a cross-sectional regression?

Time series

(b)

Sketch the regression line.

(c)

What is the interpretation of the intercept in this example? Does it make

economic sense?

If the retail price of coffee were $$0, then people would drink on average 2.7 cups per

day. Note that the intercept doesn’t have to make eco0

nomic sense. Realistically, we

wouldn’t have any data for when the price of coffee = $$0, so we can’t really say

anything about how people would behave.

(d)

How would you interpret the slope coefficient?

On average, holding everything else constant, we expect that, if the price of a cup of

coffee increases by $$1, then people will drink 0.48 fewer cups per day in the USA.

(e) Is it possible to tell what the true least squares line is? That is, can you find B

1

and B

2

?

NO; they are the true unknown values. The best we can do is to estimate them.

(f)

The

price elasticity

of demand is defined as the percentage change in the quantity

demanded for a percentage change in the price. Mathematically, it is expressed as

?

X

?

Ela sticity = slope *

?

?

?

Y

?

That is, elasticity is equal to the product of the slope and the ratio of

X

to

Y

, where

X

=

the price and

Y

= the quantity. From the regression results presented earlier, can you

tell what the price elasticity of demand is for coffee? If not, what additional

information would you need to compute the price elasticity?

From the definition given, we need to have values for X and Y before we can calculate

the elasticity. Often it is calculated at the sample mean values.

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