Ordinary Least Squares (OLS) is a method used in econometrics to estimate the parameters of a linear regression model. There are 9 key assumptions of OLS: 1) The regression model is linear in parameters. 2) Regressor values are fixed in repeated samples. 3) The mean value of the disturbance term is zero. 4) The variance of the disturbance term is constant. 5) There is no autocorrelation between disturbance terms. 6) There is no covariance between regressors and disturbance terms. 7) The number of observations exceeds the number of parameters. 8) There is variability in regressor values. 9) The regression model is

1. Assumptions of Ordinary
Least Square

2. Intro
• Ordinary Least Squares (OLS) is a method used in
econometrics, which is a branch of economics that applies
statistical techniques to analyze economic data.
• OLS is a technique for estimating the parameters of a linear
regression model.
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3. Assumption 1: Linear regression model.
• The regression model is linear in the parameters, as shown in below
• Yi = β0 + β1Xi + ui
• that the regressand Y and the regressor X themselves may be nonlinear
• Yi = β0 + β1Xi
2+ ui
• Yi = β1 + 1
𝛽2 Xi + ui

4. Assumption 2: X values are fixed in repeated
sampling.
• Values taken by the regressor X are considered fixed in repeated
samples. More technically, X is assumed to be non-stochastic.
• It is very important to understand the concept of “fixed values in re-
peated sampling,”
• Conclusion
• conditional regression analysis, that is, conditional on the given values of the
regressor(s) X.

5. Y↓
X→
80 100 120 140 160 180 200 220 240 260
Weekly
family
consumption
expenditure
Y, $
55
60
65
70
75
–
–
65
70
74
80
85
88
–
79
84
90
94
98
–
–
80
93
95
103
108
113
115
102
107
110
116
118
125
–
110
115
120
130
135
140
–
120
136
140
144
145
–
–
135
137
140
152
157
160
162
137
145
155
165
175
189
–
150
152
175
178
180
185
191
Total 325 462 445 707 678 750 685 1043 966 1211
Conditional
means of Y,
E (Y | X )
65 77 89 101 113 125 137 149 161 173

6. Assumption 3: Zero mean value of disturbance ui.
• Given the value of X, the mean, or expected, value of the random
disturbance term ui is zero.
• Technically, the conditional mean value of ui is zero. Symbolically, we have
E(ui |Xi) = 0
• each Y population corresponding to a given X is distributed around its
mean value (shown by the circled points on the PRF) with some Y values
above the mean and some below it.
• It requires is that the average or mean value of these deviations
corresponding to any given X should be zero.9
• the positive ui values cancel out the negative ui values so that their average
or mean effect on Y is zero.10

7. Mean Y
PRF: Yi = b1 + b2 Xi
+ui
–ui
X1 X2 X3 X4

8. Assumption 4: Homoscedasticity or equal
variance of ui.
• states that the variance of ui for each Xi (i.e., the
conditional variance of ui) is some positive constant
number equal to σ 2.
• Technically, σ 2represents the assumption of
homoscedasticity, or equal (homo) spread (scedasticity)
or equal variance.

9. Continue…
The word comes from the Greek verb skedanime,
which means to disperse or scatter. Stated
differently. And the Y populations corresponding to
various X values have the same variance.
Put simply, the variation around the regression line
(which is the line of average relationship between Y
and X) is the same across the X values; it neither
increases or decreases as X varies.

10. Continue….

11. Y
X1
X2
X
i PRF: Yi = b1 + b 2Xi
Probability
density
of
ui
f (u)
X

12. Continue..
• In contrast, consider below the figure where the conditional variance of the Y
population varies with X.
• This situation is known appropriately as heteroscedasticity, or unequal spread, or
variance.
• Symbolically, can be written as
• Var (ui | Xi) = σ 2
• Notice the subscript on σ 2 which indicates that the variance of the Y population
is no longer constant.
• To understand the rationale behind this assumption, refer to Figure. As this figure
shows, var (u X1) < var (u X2), ... , < var (u Xi)

13. Continue…

14. Assumption 5: No autocorrelation between
the disturbances.
• Given any two X values,
• Xi and Xj (i ≠ j ), the correlation between any two ui and uj (i ≠ j ) is
zero.
• where i and j are two different observation
• the disturbances ui and uj are uncorrelated. Technically, this is the
assumption of no serial correlation, or no auto-correlation.

15. we see, that the u’s are positively correlated, a positive u
followed by a positive u or a negative u followed by a
negative u. Figure a

16. The u’s are negatively correlated, a positive u
followed by a negative u and vice versa. Figure b

17. Continue..
• If the disturbances (deviations) follow systematic
patterns, such as those shown in Figure a and b, there is
auto- or serial correlation,
• Assumption 5 requires is that such correlations be
absent.
• Figure c shows that, there is no systematic pattern to the
u’s, thus indicating zero correlation

18. Continue.. Figure c

19. Assumption 6: Zero covariance between ui and
Xi, or E(uiXi) = 0
• cov (ui, Xi) = E [ui − E(ui)][Xi − E(Xi)]
• = E [ui (Xi − E(Xi))]since E(ui) = 0
• = E (uiXi) − E(Xi)E(ui) since E(Xi) is nonstochastic
• = E(uiXi) since E(ui) = 0
• = 0 by assumption
• But if X and u are correlated, it is not possible to assess their individual
effects on Y.

20. Continue..
• Thus, if X and u are positively correlated, X increases when u increases and it
decreases when u decreases.
• Similarly, if X and u are negatively correlated, X increases when u decreases and it
decreases when u increases.
• In either case, it is difficult to isolate the influence of X and u on Y.
• Assumption 6 is automatically fulfilled if X variable is nonrandom or non-
stochastic and
• Assumption 3 holds, for in that case, cov (ui , Xi)= [Xi E(Xi)] E[ui E(ui)] = 0.
• But since we have assumed that our X variable not only is non-stochastic but also
assumes fixed values in repeated samples

21. Assumption 7: The number of observations n must be
greater than the number of parameters to be estimated.
• Alternatively, the number of observations n must be greater than the
number of explanatory variables
• From this single observation there is no way to estimate the two unknowns,
β1 and β2.
• We need at least two pairs of observations to estimate the two unknowns

22. Assumption 8: Variability in X values.
The X values in a given
sample must not all be the
same. Technically, var (X )
must be a finite positive
number.
=
𝛴𝑥𝑗𝑦𝑖
𝑥𝑖
2
If all the X values are
identical, the denominator
of that equation will be zero,
therefore, it’s not possible
to estimate parameters.
For example, if there is little
variation in family income,
we will not be able o explain
much of the variation in the
consumption expenditure.

23. Assumption 9: The regression model is correctly
specified.
• What is the functional form of the model?
• Whether your model has linear in the parameter or variables, or both?
• What are the probabilistic assumptions made about the Y and X, and the u
entering the model?
Yi = α0 + α1 Xi + ui ………………1
Xi
Y = 𝛽0 + 𝛽1
1
𝑥𝑖
+ 𝜇 … … … … … 2

24. Continue..
The regression model one(1) is linear both in the parameters and the variables,
whereas the model two(2) is linear in the parameters but nonlinear in the
variables X.
If the model two (2) is fitted to model one (1), it give us wrong predictions,
between A and B for any given X, i.e., the model one (1) going to overestimate the
true mean value of Y.
Whereas to the left of A is going to underestimate the true mean value of Y.

26. 10. There is no Perfect Multicollinearity.
You are correct. Perfect multicollinearity refers to a situation in which
two or more independent variables in a regression model have a
perfect linear relationship, meaning that one independent variable
can be expressed as a linear combination of the others.
In other words, there is a perfect correlation between these
variables.