Review of Classical Linear Regression
Model(CLRM)
1.1. Simple (Two-Variable) Liner Regression
1.2. Assumptions of CLRM
1.3. Hypothesis testing
1.4. Multiple liner regression-Matrix Form
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What is econometrics?
 It is a systematic study of the relationships among
variables on the basis of the observed data by using
statistical techniques(methods).
 It is the application of statistical and mathematical
methods to the analysis of economic data, with the
objective of giving empirical content to economic
theories either to verify or refute them.
 Let us elaborate this definition by taking an example.
We know that one of the simplest and well-known
example of economic theory in macroeconomics is
the Keynesian Theory of consumption which
states that consumption depends on disposable
income.
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What is econometrics?
 More specifically, the theory says that ‘As
disposable income increases, consumption will
also increase, but not as much as the increase
income’
 How could you write the mathematical and also the
econometric models for this theory?(Suppose that Y
denotes consumption and X income).
 One may write the relationship
consumption and income as:
between
yi  f ( xi )   0  1 xi
Where y is consumption and x is income
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Classical Linear Regression
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Example- Estimate the regression equation for the following data
and interpret the results and also plot the regression equation
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 The regression line represents the best fit to the
random sample of consumption-disposable
income observations in the sense that it minimizes
the sum of the squared (vertical) deviations from
the line.
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2.4. Multiple linear regression
 The simplest case of MLR- two explanatory
variable regression model is given as:
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ASSUMPTIONS
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The model is linear in parameters and correctly specified.
Y   1   2 X 2  ...   k X k  u
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There does not exist an exact linear relationship among the regressors
in the sample.
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The disturbance term has zero expectation
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The disturbance term is homoscedastic
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The values of the disturbance term have independent distributions
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The disturbance term has a normal distribution.
Moving from the simple to the multiple regression model, we start by restating the regression
model assumptions.
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OLS Estimators in the case of three variable regression
stated in the previous slide is given by:
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Illustrative Example- Given the following model and intermediate
results for a hypothetical data answer the questions that follow.
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QESTIONS!
1. Fit the regression model(estimate the parameters
and express the estimated equation).
2. Find the estimator of the population error variance.
3. Compute and interpret the coefficient of
determination .
4. Test the adequacy of the model.
5. Does food price significantly affect per capita food
consumption? Why?
6. Does per capita income significantly affect food
consumption? Why?
7. Interpret the results.
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Answer-For question No 1,use
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Q1. Fit the regression model
β̂1  86.083, β̂ 2  0.216, β̂ 3  0.378
Ŷi  86.083  0.216X 2i  0.378X 3i
-------------------------------Q2.Find the estimator of the population error vari ance
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ε̂
 i
8.56
σ̂ 

 0.714 - This is the variance of error term
n  3 15  3
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Q3. Compute and interpret the coefficien t of determinat ion .
Regression Sum of Squares RSS 91.36


 0.914
Total Sum of Squares
TSS 99.93
91 % of the variation in per capita
R2 
food consumptio n is explained by
food price and per capita income.
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Q4. Test the adequacy of the model.
 A model is said to be adequate when all non-
intercept coefficients are jointly significant.
 We use F-test statistics to check for model adequacy.
But first let us formulate the hypothesis;
H o : β 2  β 3  0 (slope coefficien ts are jointly equal to zero)
H A : β 2  β 3  0 (H 0is not true)
Calculate  F - Ratio as :
RSS/(k - 1)
91.36 /(3  1)
F

 63.98
ESS /( n  k ) 8.567 /(15  3)
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We compare this F - ratio with tabl e value
for a given degree of feedom and level of
significan ce, that is, F (k  1, n  k )  F (2,12)
For α  0.01, F (2,12)  6.93
For α  0.05, F (2,12)  3.89
Since the test statistics (computed F - value) is
greater th an both tabul ated values (at 1 and 5%),
we reject the null hyothesis and conclude that
the model is adequate because the coefficien ts
are jontly significan t.
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Q5. Does food price significantly affect per capita
food consumption? Why?
 The hypothesis to be tested is;
Null hypothesis is; H 0 :  2  0
Alternativ hypotheis is; H A :  2  0
The test statistics is calculated as :
ˆ2
 0.216
t

 4.155
0.052
se( ˆ )
2
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For significan t level α  0.01(1%) and degree of feedom
(n - 3)  12 the table value of t is t /2 (n  3)  t 0.005 (12)  3.055
Decision : Since t  3.055, we reject the null hypothesis and
conclude that food price significan tlly affects per capita food
consumptio n at 1 % level of significan ce.Why? because we can
see from the calculatio ns that the computed t - value is greater
than the tabulated value for given degree of freedom and level
of signifcanc e.
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Q6. Does per capita income significantly affect food
consumption? Why?
Null hypothesis is; H 0 :  3  0
Alternativ e hypotheis is; H A :  3  0
The test statistics is calculated as :
ˆ3
0.378
t

 11.18
ˆ
se(  ) 0.034
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Table value of t at   1% and n - 3  (15 - 3) degree of feedon is 3.055
Decision : Since t  3.055, we reject the null hypothesis and conclude that
per capita income significan tlly affects per capita consumptio n expenditur e
at 1% level of significan e.
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Q7. Interpret and conclude the result.
 Generally we have the following conclusions;
 Both food price and per capita income affect food
consumption.
 The estimated coefficient of food price is -0.22,
indicating that holding per capita income
constant , one dollar increase in food prices result
in 0.22 units decrease in per capita food
consumption .
 The estimated coefficient of per capita income is
0.38, implying that holding food price constant a
one dollar increase in per capita income results in
0.38 units increase in per capita food
consumption.
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Multiple Linear Regression-Matrix Notation
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