Department of Economics
W3412
Columbia University
Spring 2012
SOLUTIONS to Problem Set 1
Introduction to Econometrics
Profs: Seyhan E Arkonac and Christopher C Conlon
for all sections
1.
You are hired by the governor to study whether a tax on liquor has decreased average
liquor consumption in New York. You obtain from a random sample of n individuals in
New York, each person’
s liquor consumption both for the year before and for the year after
the introduction of the tax. From this data, you compute Y
i
="change in liquor
consumption" for individual i = 1
,….
n. Y
i
is measured in ounces so if, for example, Y
i
=
10, then individual i increased his liquor consumption by 10 ounces. Let the parameters
μ
y
and
σ
y
2
Y denote the population mean and variance of Y.
a)
You are interested in testing the hypothesis H
0
that there was no change in liquor
consumption due to the tax. State this formally in terms of the population
parameters.
b)
The alternative, H
1
, is that there was a decline in liquor consumption; state the
alternative in terms of the population parameters.
c)
Suppose that your sample size is n = 900 and you obtain estimates
μ
y
=

32.8 and
σ
y
2
= 466.4. Report the tstatistic for testing H
0
against H
1
. Obtain the pvalue for
the test [use Table 1 in Stock and Watson, p. 749750]. Do you reject at a 5%
level? At 1% level?
d)
Would you say that the estimated fall in consumption is large in magnitude?
Comment on the practical versus statistical significance of this estimate.
e)
In your analysis, what has been implicitly assumed about other determinants of
liquor consumption over the twoyear period in order to infer causality from the
tax change to liquor consumption?
Solution
a)
The formal statement of the null hypothesis is H
0
:
μ
y
= 0.
b)
The formal statement of the alternative hypothesis is H
1
:
μ
y
< 0.
c)
With s
y
= 466.4 and n = 900, we obtain SE (
Ỹ
) = s
y
/ √n =
466.4/√900 = 15.5467. Thus, the
observed value of the tstatistic is t
obs
= (32.80)/15.5467 = 2.1098.
The pvalue is then given the probability of observing something more extreme. That is,
p = P (t < t
obs
) =
Φ
(2:1098) = 0.0174
Thus, we reject the null hypothesis at a 5% but do not reject at a 1% level since 0.01 < p < 0.05.
d)
The estimated fall is quite large at 33 ounces of liquor. So people on average drink about one
bottle less per year due to the new tax. However, due to the relatively large standard error of
Ỹ
,
we cannot draw any strong conclusions from the data regarding the effect of the tax.
e)
We have implicitly assumed that all other determinants of liquor consumption have remained
unchanged over the twoyear period. If this is not the case, then we are comparing two different
populations in a sense. Imagine for example, that people’s preferences have changed and they
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(regardless of the new tax) decided to drink less liquor and more wine. Then we would also have
observed a decrease in the consumption of liquor, but that would not have been because of the
tax.
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 Spring '11
 Arkonac
 Econometrics, Normal Distribution, Standard Deviation, Variance, Probability theory, Statistical hypothesis testing

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