Econometric Testing

for Laws of Demand

Assignment 2

ABSTRACT

The basic

economic theory states that for a normal good there exists a positive relation

between demand of good and income, ceteris paribus. We aim to assert and

examine such economic theory related to the consumer by using an annual time

series data of the US and have a critical examination with respect to two

commodities in the economy: Tobacco and Gas. The effects of price, income and

the sensitivity of the data with respect to these factors and the population is

taken into account. As we try to test various hypothesis related to consumer

behavior using OLS and assume that the demand equation takes a log linear

preference form. Also, a lag adjusted model is tested for both the commodities

and the trend values are checked for significance in the later parts of the

project.

Basic Demand Model

Hence, we

run OLS regression on the above equation and obtain the estimates of both the commodities.

It is interesting to note that the individual estimates actually represent the

elasticities.

Thus, the

income and own price elasticity of the commodities are as follows:

Income

elasticity: Gas:

0.82549

Tobacco: 0.50197

Own-price

elasticity: Gas: 0.00238

Tobacco: -0.5165

Thus, we

can infer that the demand for both the commodities has a strong positive

relationship with income changes while it is interesting to note that the

economic theory of reduction in demand with increase

in price is fact and is thus the elasticity is reflected for gas. However, as

tobacco is a substance with hardly any substitutes its highly inelastic and

thus the changes in price has a low or no effect on the demand for tobacco.

But, since gas has relatively number of substitutes it is seen to be more

elastic than tobacco.

Thus,

taking a 5% level of significance, we form a confidence interval for these

elasticities. Thus, computing the intervals with a value of 1.96 critical

value.

Income elasticity: 0.597209 ? Gas ? 1.05377

0.408690 ?

Tobacco ? 0.595246

Own-price

elasticity: -0.081110

? Gas ? 0.085870

-0.568832 ?

Tobacco ? -0.464168

Now we test the law of demand: the null hypothesis

?p = 0, against the alternative ?p<0. In this, we try to find out if our commodities are
perfectly inelastic in terms of their own price elasticity or not. From our computations, we check the t-ratio. This is a
one-tailed test with 5% significant level, from t-tables we get a critical
value of -1.68.
Hence, we
fail to reject our null hypothesis for tobacco as the absolute value of the
t-ratio is greater than the critical value:
|19.345| ? | 1.68 |
and thus,
confers to our initial assertions about tobacco being inelastic. However, we
reject the null hypothesis for gas as
0.029 | ? -1.68
Test
the homogeneity restriction
In the next
exercise we test the homogeneity restrictions for the existence of money illusion
for the demands of the specific commodities, i.e., we check H0:
against
the alternative that H1:
Thus, on
observation of the both the Wald test and the variable replacement significance
test (relative price is now regressed instead of price) on the quantity of each
commodity and we find that for both the commodities the t-ratio:
Gas: -3.273
and Tobacco: -6.077, is rejected for the critical value of 1.678. Hence, we
reject the null hypothesis implying that there exists no money illusion and the
tests are insignificant. Note, both the tests provide the same result as in the
Wald test we check the same equation's complimentary transposed version
.
As even in Wald test the p-value is less than 0.5 for both the commodities thus
we reject the null hypothesis in the latter as well.
FORECAST OF DEMAND FOR 2003
In the next section we create a model forecasting of demand using
the actual price and income data for that year - in other words, we fit the
model to 1959-2002 and check the Chow test to test that the sample has been
drawn from the same distribution as the residuals.
Figure forecast for Gas 2003:
Figure
forecast for Tobacco 2003:
The actual
forecast error computed by TSM for the commodities are:
Gas:
-0.14019 and Tobacco: 0.017695
Using
P-value approach we see that the Chow forecast test rejects the null hypothesis
for Gas which in this case is that the sample is
drawn from the same population as the residuals. As 0.023 is less than 0.05
SIGNIFICANCE OF
POPULATION ON THE DEMAND OF COMMODITIES
Now we
analyze how does the addition of another variable to the demand equation affect
the demand. The regression is carried out with the underlying assumption of the
null hypothesis being that the population doesn't affect the demand. However, for
both our commodities we observe that the p-values and also the t-ratios with
bounds of |1.96|.
It is
observed that for Gas the t-ratio is obtained as -3.045 which lies beyond the
acceptance region. Also, the p-value is 0.04 < 0.05, thus we reject the null
hypothesis and can comment that the population has a significance in terms of
affecting the demand for Gas.
Similarly,
it is observed that for Tobacco that the t-ratio obtained as -0.198 which lies within the acceptance
region. Also, the p-value is 0.844 > 0.05, thus we do not reject the null

hypothesis and can comment that the population is insignificance in terms of

affecting the demand for Tobacco.

Thus, in

line with economic theory, we see that there exists an inverse

relationship between elasticity and impact of population on the demand.

Greater elasticity reduces the impact of the population

due to the availability of alternate

substitutes.

Now, we

have also run another test for coefficients of the population to be equal to that of 1-

. The underlying theory being that whether

this is the best form of expression of the variables or not. For Tobacco we see

the p value is greater than 0.05, hence we see that the per capita form is

significant in this case. Theoretically it confers with economics of data as it

is easier to measure for tobacco as only a portion of the population has a

specific demand for Tobacco. However, the converse is true for Gas. As we have

already seen Gas acts as a necessary good and thus almost is demanded by the whole

population of an economy. Hence, as per our earlier results population affects

the demand for gas and hence per capita form can be asserted to be

insignificant and to be not represented by this form for expenditure and

income.

CHECKING FOR

STABILITY OF PARAMETERS:

The Chow test is

used to determine whether the independent variables have a structural break and

have different impacts on two subgroups of the population. Thus, we check the

stability of values over the whole period between 1981 to 2003 by dividing it

into two halves of 22 and 23 samples.

From TSM output of

Gas., we observe the stability test equals 32.0769 which is greater than the

critical value of F-test (4 and 37 degrees of freedom): 2.62605 and the p-value

of 0 is less than 0.05 at 5% significance

level.

32.0769 > 2.62605

0 < 0.05 Similarly, from TSM output of Tobacco, we observe the stability test equals 15.1298 which is greater than the critical value of F-test (4 and 37 degrees of freedom): 2.62605 and the p-value of 0 is less than 0.05 at 5% significance level. 15.1298 > 2.62605

0 < 0.05 This implies both Gas and Tobacco is unstable over the 10-year period of the sample. Note: Here our Ho is that our model is stable whereas H1 is that our model is unstable. Thus, we rejected Ho for Gas and Tobacco. TREND EFFECT Testing for an adjusted model that changes systematically with time. Thus, we observe that for both Gas and Tobacco the inclusion of the trend values gives us non-sensical parameter outputs as for example the effect of price and income is seen to be insignificant for tobacco (p-value < 0.05) and similarly for Gas as well. This might be due to the well-known factor of time series data to be suffering from multicollinearity. There may not be enough independent variation in the variables to measure the elasticities effectively, once the linear trend is taken into account. Thus, the income and price effect are not reflected in the demand equation when taken into account. Conclusion Regression analysis estimates the effect of each independent variable by seeing how much effect the independent variables has on the dependent variable when other independent variables are held constant. We recognise that not all of the econometric test statistics are adequately conclusive and entirely satisfactory. Alternative specifications or refinements may further improve on the present results. This may partly be due to multicollinearity between most price variables. But still, we obtain a significant empirical evidence of economic data of demand theory using real economy values, which give still a better proof than most other models.