The document discusses the assumptions of linear regression models. It outlines 10 key assumptions: 1) The regression model is linear with respect to parameters 2) X values are fixed in repeated sampling 3) The error term has a mean value of zero 4) The error term has constant variance 5) There is no autocorrelation between error terms 6) The error term is uncorrelated with the X values 7) The number of observations exceeds the number of parameters 8) The X values cannot all be the same 9) The regression model is correctly specified 10) There is no perfect multicollinearity between regressors

1. AMMARA AFTAB
ammara.aftab63@gmail.com
MSC (FINAL)
ECONOMETRICS

2. THE SHINNIG STAR FOR STATISTIC…
MY IDEALS…

3. REGRESSION seems like
MSC(FINAL) OF UOK DEPENDS ON QUALIFIED
ECONOMIST MR.ZOHAIB AZIZ
SIR
ZOHAIB
AZIZ
By Ammara Aftab

4. INTRODUCTION
By Ammara Aftab

5. HISTORICAL ORIGIN OF THE TERM
REGRESSION
By Ammara Aftab

6. Historical ORIGIN BY FRANCISGALTON
GALTON’S Law of universal regression was
confirmed by his friend CARL F .GAUSS
KARL PEARSONBy Ammara Aftab

7. Galton's universal regression law
Galton found that, although there was a tendency for tall parents to
have tall children and for short parents to have short children.
KARL PEARSON:
He is talking in average sense that average height (not single height of
children that may be high or low from tall fathers) of sons is less than
the fathers height means tendency to mid....
similarly, average height of sons (not single height of children that may
be high or low from short fathers) of short parents greater than from
them means tendency to mid.
Conclusion:
Tall parents have tall children but average height of their children will
be less from them similarly for short parents. That is why Karl
Pearson's endorses the Galton's theory.
By Ammara Aftab

8. MODERN
INTERPRETATION
OF REGRESSION
By Ammara Aftab

9. By Ammara Aftab

10. By Ammara Aftab

11. Reconsider Galton’s law of universal regression.
Galton was interested in finding out why there was a
stability in the distribution of heights
in a population.
But in the modern view our concern is not with this
explanation but rather with finding out how the
average height of sons changes, given the fathers’
height. In other words, our concern is with predicting
the average height of sons knowing the height of
their fathers
By Ammara Aftab

12. EXAMPLE
By Ammara Aftab

13. By Ammara Aftab

14. PUT THE REGRESSION DATA ON
EXCEL SHEET
By Ammara Aftab

15. The slope indicates the steepness of a line and the intercept indicates
the location where it intersects an axis.
By Ammara Aftab

16. By Ammara Aftab

17. By Ammara Aftab

18. By Ammara Aftab

19. By Ammara Aftab

20. REGRESSION
ASSUMPTIONS
By Ammara Aftab

21. WHY THE ASSUMPTION requirement
is needed??
By Ammara Aftab

22. LOOK AT PRF:
Yi=β1+(β2)Xi+µi
It shows that Yi depends on both Xi and ui . Therefore, unless
we are specific about how Xi and ui are created or generated,
there is no way we can make any statistical inference about
the Yi and also, as we shall see, about β1 and β2. Thus, the
assumptions made about the Xi variable(s) and the error term
are extremely critical to the valid interpretation of the
regression estimates.
By Ammara Aftab

23. the Gaussian or
classical linear regression ASSUMPTIONS(clrm)
By Ammara Aftab

24. By Ammara Aftab

25. LINEAR REGRESSION MODEL:
The regression model is linear with
respect to parameter.
1) Yi = β1 + β2X1 +µi
In this above equation the model is linear with
respect to parameter.
2)Yi=B1+(B2^2)X1+µi
In this above equation the model is non linear with
respect to parameter.
1
By Ammara Aftab

26. Linear with respect to
(Parameter)
• Yi=β1+(β2)Xi+µi
• Yi=β1+(β2)Xi^2+µi
Non-linear with respect to
(PARAMETER)
Yi=β1+(β2^2)Xi+µi
ß is the parameter,
If ß have power then it will be non linear with respect to
parameter, if it does not have then it is in Linear form.
By Ammara Aftab

27. Simple linear regression describes the
linear relationship between a predictor
variable, plotted on the x-axis, and a
response variable, plotted on the y-axis
Independent Variable (X)
dependentVariable(Y)
1
By Ammara Aftab

28. By Ammara Aftab

29. 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 no stochastic .
2
By Ammara Aftab

30. 2
By Ammara Aftab

31. X
Y
20$ 40$ 60$ 80$ 100$
JAPAN 18 23 47 78 89
CHINA 09 28 37 34 34
USA 16 35 48 45 67
RUSSIA 17 38 50 23 69
2
By Ammara Aftab

32. The X variable is measured without error X is
fixed and Y is dependent
X
Y
2
By Ammara Aftab

33. By Ammara Aftab

34. Zero mean value of disturbance ui. Giventhe 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
EXAMPLE:
If X= 3,6,9
Then the mean =6,after applying thisformula
(mean-X)
= (6-3)+(6-6)+(6-9)
= 0
so 1st moment is alwayszero.
3
By Ammara Aftab

35. 3
Mean =o
variance= constant
By Ammara Aftab

36. By Ammara Aftab

37. 4Homoscedasticity or equal variance of ui.
Given the value of X, the variance
of ui is the same for all observations. That is,
the conditional variances of ui are identical.
Symbolically, we have
var (ui |Xi) = E[ui − E(ui |Xi)]2
= E(ui2 | Xi ) because of Asp3
= σ2
By Ammara Aftab

38. 4
Homo= same
scedasticity=spreadness
By Ammara Aftab

39. By Ammara Aftab

40. 5No 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.
Symbolically,
cov (ui, uj |Xi, Xj) = E{[ui − E(ui)] | Xi }{[uj − E(uj)] | Xj }
= E(ui |Xi)(uj | Xj) (why?)
= 0
By Ammara Aftab

41. No Auto-Correlation
Positive
correlation
Negative
correlation
No
correlation
5
By Ammara Aftab

42. By Ammara Aftab

43. 6
Zero covariance between ui and
Xi,or E(uiXi) = 0.
We assumed that X and u have separate effect on Y,But if X
and u are corelated,it is not possible to assess their
individual effect on Y as well as they will be directly
proportional.
X  u
Formally,
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 ( NON RANDOM)
= E(uiXi) since E(ui) = 0
= 0
By Ammara Aftab

44. By Ammara Aftab

45. 7
The number of observations n must be greater
than the number of parameters to be
estimated.
Example:
Yi=β1+(β2)X1+(β3)X2+µi
As u can see that we have 3 parameters here so the number
of observation will be grater then 3.
Alternatively,
The number of observations n must be greater than the
number of regressors. 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.By Ammara Aftab

46. By Ammara Aftab

47. 8Variability in X values.
The X values in a given sample must not all be
the same.
If the X values are same then the Variance is
equal to zero and regression can’t be run
Technically,
var (X) must be a finite positive number.
By Ammara Aftab

48. 8If we have same values of X:
X = 2,2,2
Then, according to the variance formula
σ = [(x-µ)^2]/n
σ = [(2-2)^2]/3+[(2-2)^2]/3+[(2-2)^2]/3
σ = 0
As σ= o then regression could not be run
Variance = 0 ,regression can not run..
Positive Finite means:
variance could not be(-ve),because it has squaring σ^2
variance could not be 0 , regression could not run.
variance should be (+ve) , positive finite.By Ammara Aftab

49. By Ammara Aftab

50. 9
The regression model is correctly
specified.
Alternatively,
There is no specification bias or error in the
model used in empirical analysis.
Yi = α1 + α2Xi + ui
Yi = β1 + β2(1/Xi)+ ui
By Ammara Aftab

51. 9
Yi = β1 + β2(1/Xi)+ ui
By Ammara Aftab

52. By Ammara Aftab

53. 10
There is no perfect
multicollinearity.
Perfect multicollinearity=Strong Correlation b/w
regressors.
That is, there are no perfect linear
Relation b/w regressors.
Yi = β1 + β2X2i + β3X3i + ui
where Y is the dependent variable, X2 and X3 the
explanatory variables (or
regressors) or non random variables.
By Ammara Aftab